Subgroups — C₁₀₉×#35064 #42153058

Order 27904.

2822 conjugacy classes of subgroups.

The subgroup lattice has 822 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#123×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2854	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2663	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#2661	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2602	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#2591	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#2583	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#2561	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#2539	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#2498	C₂	1	2	yes	maximal, complemented
12	C₁₀₉×#2430	C₂	1	2	yes	maximal, complemented
13	C₁₀₉×#2412	C₂	1	2	yes	maximal, complemented
14	C₁₀₉×#2410	C₂	1	2	yes	maximal, complemented
15	C₁₀₉×#2322	C₂	1	2	yes	maximal, complemented
16	C₁₀₉×#2321	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	#525×C₁₀₉	—	2	8	no	—
22	C₂×#123×C₁₀₉	—	2	8	no	—
23	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
24	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
25	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
27	C₂×#123×C₁₀₉	C₂²	1	2	yes	—
28	C₂×#123×C₁₀₉	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	C₂×#121×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂×#121×C₁₀₉	C₂²	1	4	yes	—
33	C₂×#121×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
34	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
35	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
38	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
39	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
42	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
43	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
45	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	complemented
46	#400×C₁₀₉	C₂²	1	2	yes	complemented
47	#399×C₁₀₉	C₂²	1	2	yes	complemented
48	#398×C₁₀₉	C₂²	1	2	yes	complemented
49	#397×C₁₀₉	C₂²	1	2	yes	complemented
50	#397×C₁₀₉	C₂²	1	4	yes	complemented
51	#395×C₁₀₉	C₂²	1	2	yes	complemented
52	#394×C₁₀₉	C₂²	1	4	yes	—
53	#394×C₁₀₉	C₂²	1	2	yes	complemented
54	#394×C₁₀₉	C₂²	1	4	yes	complemented
55	#393×C₁₀₉	C₂²	1	2	yes	complemented
56	#392×C₁₀₉	C₂²	1	4	yes	—
57	#392×C₁₀₉	C₂²	1	2	yes	complemented
58	#390×C₁₀₉	C₂²	1	4	yes	complemented
59	#390×C₁₀₉	C₂²	1	4	yes	complemented
60	#389×C₁₀₉	C₂²	1	4	yes	complemented
61	#389×C₁₀₉	C₂²	1	4	yes	complemented
62	#388×C₁₀₉	C₂²	1	4	yes	complemented
63	#388×C₁₀₉	C₂²	1	4	yes	complemented
64	#388×C₁₀₉	C₂²	1	4	yes	complemented
65	#388×C₁₀₉	C₂²	1	4	yes	complemented
66	#388×C₁₀₉	C₂²	1	4	yes	complemented
67	#387×C₁₀₉	C₂²	1	4	yes	complemented
68	#386×C₁₀₉	C₂²	1	4	yes	—
69	#386×C₁₀₉	C₂²	1	2	yes	—
70	#386×C₁₀₉	C₂²	1	2	yes	—
71	#386×C₁₀₉	C₂²	1	4	yes	complemented
72	#386×C₁₀₉	C₂²	1	4	yes	complemented
73	#385×C₁₀₉	C₂²	1	2	yes	—
74	#385×C₁₀₉	C₂²	1	2	yes	—
75	#385×C₁₀₉	C₂²	1	4	yes	complemented
76	#381×C₁₀₉	C₂²	1	2	yes	—
77	#380×C₁₀₉	C₂²	1	4	yes	complemented
78	#379×C₁₀₉	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
80	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
81	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
82	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
83	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
84	C₂×#95×C₁₀₉	C₂²	1	2	yes	complemented
85	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
87	C₂×#41×C₁₀₉	—	2	4	no	—
88	C₂×#41×C₁₀₉	—	4	4	no	—
89	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₁₀₉	—	2	4	no	—
91	C₂²×D₄×C₁₀₉	—	4	4	no	—
92	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
93	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
94	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
95	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
96	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
97	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
99	C₂³×C₄×C₁₀₉	—	4	4	no	—
100	C₂³×C₄×C₁₀₉	D₄	1	8	yes	complemented
101	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
103	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
108	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
112	#123×C₁₀₉	—	2	8	no	—
113	#123×C₁₀₉	—	2	8	no	—
114	#123×C₁₀₉	—	2	8	no	—
115	#123×C₁₀₉	—	2	8	no	—
116	#123×C₁₀₉	—	2	8	no	—
117	#123×C₁₀₉	—	2	8	no	—
118	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
119	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
120	#121×C₁₀₉	—	2	4	no	—
121	#121×C₁₀₉	—	2	4	no	—
122	#121×C₁₀₉	—	4	8	no	—
123	#120×C₁₀₉	—	2	8	no	—
124	#120×C₁₀₉	—	2	8	no	—
125	C₄×D₄×C₁₀₉	—	2	8	no	—
126	C₄×D₄×C₁₀₉	—	2	8	no	—
127	#117×C₁₀₉	—	2	4	no	—
128	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
129	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
130	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
133	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
134	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
135	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
136	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
137	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
138	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
139	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
140	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
141	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
142	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
143	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
144	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
145	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
146	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
147	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
148	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
149	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
150	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
151	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
152	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
153	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
154	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
155	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
156	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
157	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
158	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
159	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
160	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
161	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
162	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
163	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
164	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
165	C₂×C₄²×C₁₀₉	—	4	4	no	—
166	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
167	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
168	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
169	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
170	#95×C₁₀₉	C₂³	1	4	yes	—
171	#95×C₁₀₉	C₂³	1	2	yes	—
172	#95×C₁₀₉	C₂³	1	2	yes	—
173	#95×C₁₀₉	C₂³	1	4	yes	—
174	#95×C₁₀₉	C₂³	1	4	yes	—
175	#95×C₁₀₉	C₂³	1	4	yes	—
176	#95×C₁₀₉	C₂³	1	4	yes	—
177	#95×C₁₀₉	C₂³	1	4	yes	—
178	#95×C₁₀₉	C₂³	1	4	yes	—
179	C₂⁴×C₁₀₉	—	2	4	no	—
180	C₂⁴×C₁₀₉	—	2	4	no	—
181	C₂⁴×C₁₀₉	—	2	2	no	—
182	C₂⁴×C₁₀₉	—	2	4	no	—
183	C₂⁴×C₁₀₉	—	4	2	no	—
184	C₂⁴×C₁₀₉	—	4	2	no	—
185	C₂⁴×C₁₀₉	—	4	2	no	—
186	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
187	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
188	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
189	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
190	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
191	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
192	#41×C₁₀₉	—	4	16	no	—
193	C₂×Q₈×C₁₀₉	—	2	2	no	—
194	C₂×Q₈×C₁₀₉	—	4	2	no	—
195	C₂×D₄×C₁₀₉	—	2	4	no	—
196	C₂×D₄×C₁₀₉	—	2	4	no	—
197	C₂×D₄×C₁₀₉	—	2	4	no	—
198	C₂×D₄×C₁₀₉	—	2	4	no	—
199	C₂×D₄×C₁₀₉	—	2	4	no	—
200	C₂×D₄×C₁₀₉	—	2	4	no	—
201	C₂×D₄×C₁₀₉	—	2	4	no	—
202	C₂×D₄×C₁₀₉	—	2	8	no	—
203	C₂×D₄×C₁₀₉	—	2	8	no	—
204	C₂×D₄×C₁₀₉	—	2	2	no	—
205	C₂×D₄×C₁₀₉	—	2	8	no	—
206	C₂×D₄×C₁₀₉	—	4	4	no	—
207	C₂×D₄×C₁₀₉	—	4	4	no	—
208	C₂×D₄×C₁₀₉	—	4	2	no	—
209	C₂×D₄×C₁₀₉	—	4	4	no	—
210	C₂×D₄×C₁₀₉	—	4	4	no	—
211	C₂×D₄×C₁₀₉	—	4	16	no	—
212	C₂²×C₄×C₁₀₉	—	2	4	no	—
213	C₂²×C₄×C₁₀₉	—	2	4	no	—
214	C₂²×C₄×C₁₀₉	—	2	2	no	—
215	C₂²×C₄×C₁₀₉	—	2	2	no	—
216	C₂²×C₄×C₁₀₉	—	2	2	no	—
217	C₂²×C₄×C₁₀₉	—	2	4	no	—
218	C₂²×C₄×C₁₀₉	—	2	8	no	—
219	C₂²×C₄×C₁₀₉	—	2	4	no	—
220	C₂²×C₄×C₁₀₉	—	2	8	no	—
221	C₂²×C₄×C₁₀₉	—	4	16	no	—
222	C₂²×C₄×C₁₀₉	—	4	2	no	—
223	C₂²×C₄×C₁₀₉	—	4	2	no	—
224	C₂²×C₄×C₁₀₉	—	4	2	no	—
225	C₂²×C₄×C₁₀₉	—	4	2	no	—
226	C₂²×C₄×C₁₀₉	—	4	4	no	—
227	C₂²×C₄×C₁₀₉	—	4	2	no	—
228	C₂²×C₄×C₁₀₉	—	4	2	no	—
229	C₂²×C₄×C₁₀₉	—	4	4	no	—
230	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
231	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
232	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
233	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
234	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
235	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
236	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
237	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
238	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
239	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
240	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
241	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
242	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
243	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
244	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
245	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
246	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
247	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
248	C₄⋊C₄×C₁₀₉	—	2	4	no	—
249	C₄⋊C₄×C₁₀₉	—	2	4	no	—
250	C₄⋊C₄×C₁₀₉	—	2	4	no	—
251	C₄⋊C₄×C₁₀₉	—	2	4	no	—
252	C₄⋊C₄×C₁₀₉	—	2	2	no	—
253	C₄⋊C₄×C₁₀₉	—	2	2	no	—
254	C₄⋊C₄×C₁₀₉	—	2	4	no	—
255	C₄⋊C₄×C₁₀₉	—	2	4	no	—
256	C₄⋊C₄×C₁₀₉	—	2	8	no	—
257	C₄⋊C₄×C₁₀₉	—	4	4	no	—
258	C₄⋊C₄×C₁₀₉	—	4	4	no	—
259	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
260	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
261	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
262	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
263	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
264	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
265	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
266	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
267	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
268	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
269	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
270	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
271	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
272	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
273	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
274	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
275	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
276	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
277	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
278	C₂²⋊C₄×C₁₀₉	#41	1	8	yes	complemented
279	C₄²×C₁₀₉	—	2	4	no	—
280	C₄²×C₁₀₉	—	4	2	no	—
281	C₄²×C₁₀₉	—	4	4	no	—
282	C₄²×C₁₀₉	—	4	2	no	—
283	C₂³×C₁₀₉	—	2	2	no	—
284	C₂³×C₁₀₉	—	2	2	no	—
285	C₂³×C₁₀₉	—	2	2	no	—
286	C₂³×C₁₀₉	—	2	2	no	—
287	C₂³×C₁₀₉	—	2	8	no	—
288	C₂³×C₁₀₉	—	2	8	no	—
289	C₂³×C₁₀₉	—	2	2	no	—
290	C₂³×C₁₀₉	—	2	2	no	—
291	C₂³×C₁₀₉	—	2	2	no	—
292	C₂³×C₁₀₉	—	2	2	no	—
293	C₂³×C₁₀₉	—	2	2	no	—
294	C₂³×C₁₀₉	—	2	2	no	—
295	C₂³×C₁₀₉	—	2	4	no	—
296	C₂³×C₁₀₉	—	2	2	no	—
297	C₂³×C₁₀₉	—	2	4	no	—
298	C₂³×C₁₀₉	—	2	4	no	—
299	C₂³×C₁₀₉	—	2	2	no	—
300	C₂³×C₁₀₉	—	2	4	no	—
301	C₂³×C₁₀₉	—	2	2	no	—
302	C₂³×C₁₀₉	—	4	4	no	—
303	C₂³×C₁₀₉	—	4	16	no	—
304	C₂³×C₁₀₉	—	4	4	no	—
305	C₂³×C₁₀₉	—	4	4	no	—
306	C₂³×C₁₀₉	—	4	4	no	—
307	C₂³×C₁₀₉	—	8	8	no	—
308	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
309	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
310	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
311	C₂³×C₁₀₉	C₂×#41	1	4	yes	—
312	C₂³×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
313	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
314	Q₈×C₁₀₉	—	4	4	no	—
315	D₄×C₁₀₉	—	4	4	no	—
316	D₄×C₁₀₉	—	4	8	no	—
317	D₄×C₁₀₉	—	4	8	no	—
318	D₄×C₁₀₉	—	4	8	no	—
319	D₄×C₁₀₉	—	4	8	no	—
320	D₄×C₁₀₉	—	4	8	no	—
321	D₄×C₁₀₉	—	8	8	no	—
322	C₂×C₄×C₁₀₉	—	2	8	no	—
323	C₂×C₄×C₁₀₉	—	2	8	no	—
324	C₂×C₄×C₁₀₉	—	2	4	no	—
325	C₂×C₄×C₁₀₉	—	2	2	no	—
326	C₂×C₄×C₁₀₉	—	2	2	no	—
327	C₂×C₄×C₁₀₉	—	2	2	no	—
328	C₂×C₄×C₁₀₉	—	2	2	no	—
329	C₂×C₄×C₁₀₉	—	2	4	no	—
330	C₂×C₄×C₁₀₉	—	2	2	no	—
331	C₂×C₄×C₁₀₉	—	2	2	no	—
332	C₂×C₄×C₁₀₉	—	2	2	no	—
333	C₂×C₄×C₁₀₉	—	2	2	no	—
334	C₂×C₄×C₁₀₉	—	2	2	no	—
335	C₂×C₄×C₁₀₉	—	2	2	no	—
336	C₂×C₄×C₁₀₉	—	2	4	no	—
337	C₂×C₄×C₁₀₉	—	2	2	no	—
338	C₂×C₄×C₁₀₉	—	2	4	no	—
339	C₂×C₄×C₁₀₉	—	2	2	no	—
340	C₂×C₄×C₁₀₉	—	2	4	no	—
341	C₂×C₄×C₁₀₉	—	2	4	no	—
342	C₂×C₄×C₁₀₉	—	2	4	no	—
343	C₂×C₄×C₁₀₉	—	2	2	no	—
344	C₂×C₄×C₁₀₉	—	2	4	no	—
345	C₂×C₄×C₁₀₉	—	2	4	no	—
346	C₂×C₄×C₁₀₉	—	2	2	no	—
347	C₂×C₄×C₁₀₉	—	4	4	no	—
348	C₂×C₄×C₁₀₉	—	4	4	no	—
349	C₂×C₄×C₁₀₉	—	4	4	no	—
350	C₂×C₄×C₁₀₉	—	4	4	no	—
351	C₂×C₄×C₁₀₉	—	4	16	no	—
352	C₂×C₄×C₁₀₉	—	4	8	no	—
353	C₂×C₄×C₁₀₉	—	4	4	no	—
354	C₂×C₄×C₁₀₉	—	4	4	no	—
355	C₂×C₄×C₁₀₉	—	4	4	no	—
356	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
357	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
358	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
359	C₂×C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
360	C₂×C₄×C₁₀₉	C₂×#41	1	8	yes	—
361	C₂²×C₁₀₉	—	2	2	no	—
362	C₂²×C₁₀₉	—	2	2	no	—
363	C₂²×C₁₀₉	—	2	2	no	—
364	C₂²×C₁₀₉	—	2	2	no	—
365	C₂²×C₁₀₉	—	2	8	no	—
366	C₂²×C₁₀₉	—	2	8	no	—
367	C₂²×C₁₀₉	—	2	2	no	—
368	C₂²×C₁₀₉	—	2	2	no	—
369	C₂²×C₁₀₉	—	2	2	no	—
370	C₂²×C₁₀₉	—	4	16	no	—
371	C₂²×C₁₀₉	—	4	4	no	—
372	C₂²×C₁₀₉	—	4	2	no	—
373	C₂²×C₁₀₉	—	4	2	no	—
374	C₂²×C₁₀₉	—	4	2	no	—
375	C₂²×C₁₀₉	—	4	2	no	—
376	C₂²×C₁₀₉	—	4	2	no	—
377	C₂²×C₁₀₉	—	4	2	no	—
378	C₂²×C₁₀₉	—	4	4	no	—
379	C₂²×C₁₀₉	—	4	4	no	—
380	C₂²×C₁₀₉	—	4	4	no	—
381	C₂²×C₁₀₉	—	4	4	no	—
382	C₂²×C₁₀₉	—	8	8	no	—
383	C₂²×C₁₀₉	#532	1	4	yes	—
384	C₂²×C₁₀₉	#534	1	4	yes	—
385	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
386	C₂²×C₁₀₉	C₂²×#41	1	2	yes	—
387	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
388	C₂²×C₁₀₉	#585	1	2	yes	—
389	C₄×C₁₀₉	—	2	2	no	—
390	C₄×C₁₀₉	—	2	2	no	—
391	C₄×C₁₀₉	—	2	2	no	—
392	C₄×C₁₀₉	—	2	2	no	—
393	C₄×C₁₀₉	—	2	8	no	—
394	C₄×C₁₀₉	—	4	2	no	—
395	C₄×C₁₀₉	—	4	2	no	—
396	C₄×C₁₀₉	—	4	2	no	—
397	C₄×C₁₀₉	—	4	2	no	—
398	C₄×C₁₀₉	—	4	2	no	—
399	C₄×C₁₀₉	—	4	2	no	—
400	C₄×C₁₀₉	—	4	4	no	—
401	#35064	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
402	C₂×C₁₀₉	—	2	4	no	—
403	C₂×C₁₀₉	—	2	4	no	—
404	C₂×C₁₀₉	—	4	2	no	—
405	C₂×C₁₀₉	—	8	2	no	—
406	C₂×C₁₀₉	—	8	2	no	—
407	C₂×C₁₀₉	C₂×#532	1	1	yes	characteristic
408	C₂×C₁₀₉	C₂×#534	1	1	yes	characteristic
409	C₂×C₁₀₉	#3449	1	1	yes	characteristic
410	C₂×C₁₀₉	#3513	1	4	yes	—
411	C₂×#525	C₂×C₁₀₉	1	1	yes	characteristic, complemented
412	C₂²×#123	C₂×C₁₀₉	1	1	yes	characteristic, complemented
413	#2854	C₂×C₁₀₉	1	4	yes	complemented
414	#2663	C₂×C₁₀₉	1	2	yes	complemented
415	#2661	C₂×C₁₀₉	1	2	yes	complemented
416	#2602	C₂×C₁₀₉	1	4	yes	complemented
417	#2591	C₂×C₁₀₉	1	1	yes	characteristic, complemented
418	#2583	C₂×C₁₀₉	1	1	yes	characteristic, complemented
419	#2561	C₂×C₁₀₉	1	1	yes	characteristic, complemented
420	#2539	C₂×C₁₀₉	1	1	yes	characteristic, complemented
421	#2498	C₂×C₁₀₉	1	2	yes	complemented
422	#2430	C₂×C₁₀₉	1	2	yes	complemented
423	#2412	C₂×C₁₀₉	1	2	yes	complemented
424	#2410	C₂×C₁₀₉	1	2	yes	complemented
425	#2322	C₂×C₁₀₉	1	2	yes	complemented
426	#2321	C₂×C₁₀₉	1	2	yes	complemented
427	C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
428	C₁₀₉	#35064	1	1	yes	characteristic, direct product factor, Sylow
429	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
430	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
431	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
432	#525	—	2	8	no	—
433	C₂×#123	—	2	8	no	—
434	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
435	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
436	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
437	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
438	C₂×#123	C₂²×C₁₀₉	1	2	yes	—
439	C₂×#123	C₂²×C₁₀₉	1	2	yes	complemented
440	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
441	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
442	C₂×#121	C₂²×C₁₀₉	1	1	yes	characteristic
443	C₂×#121	C₂²×C₁₀₉	1	4	yes	—
444	C₂×#121	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
445	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
446	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
447	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
448	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	complemented
449	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
450	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
451	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
452	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
453	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
454	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
455	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
456	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	complemented
457	#400	C₂²×C₁₀₉	1	2	yes	complemented
458	#399	C₂²×C₁₀₉	1	2	yes	complemented
459	#398	C₂²×C₁₀₉	1	2	yes	complemented
460	#397	C₂²×C₁₀₉	1	2	yes	complemented
461	#397	C₂²×C₁₀₉	1	4	yes	complemented
462	#395	C₂²×C₁₀₉	1	2	yes	complemented
463	#394	C₂²×C₁₀₉	1	4	yes	—
464	#394	C₂²×C₁₀₉	1	2	yes	complemented
465	#394	C₂²×C₁₀₉	1	4	yes	complemented
466	#393	C₂²×C₁₀₉	1	2	yes	complemented
467	#392	C₂²×C₁₀₉	1	4	yes	—
468	#392	C₂²×C₁₀₉	1	2	yes	complemented
469	#390	C₂²×C₁₀₉	1	4	yes	complemented
470	#390	C₂²×C₁₀₉	1	4	yes	complemented
471	#389	C₂²×C₁₀₉	1	4	yes	complemented
472	#389	C₂²×C₁₀₉	1	4	yes	complemented
473	#388	C₂²×C₁₀₉	1	4	yes	complemented
474	#388	C₂²×C₁₀₉	1	4	yes	complemented
475	#388	C₂²×C₁₀₉	1	4	yes	complemented
476	#388	C₂²×C₁₀₉	1	4	yes	complemented
477	#388	C₂²×C₁₀₉	1	4	yes	complemented
478	#387	C₂²×C₁₀₉	1	4	yes	complemented
479	#386	C₂²×C₁₀₉	1	2	yes	—
480	#386	C₂²×C₁₀₉	1	2	yes	—
481	#386	C₂²×C₁₀₉	1	4	yes	—
482	#386	C₂²×C₁₀₉	1	4	yes	complemented
483	#386	C₂²×C₁₀₉	1	4	yes	complemented
484	#385	C₂²×C₁₀₉	1	2	yes	—
485	#385	C₂²×C₁₀₉	1	2	yes	—
486	#385	C₂²×C₁₀₉	1	4	yes	complemented
487	#381	C₂²×C₁₀₉	1	2	yes	—
488	#380	C₂²×C₁₀₉	1	4	yes	complemented
489	#379	C₂²×C₁₀₉	1	2	yes	—
490	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
491	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
492	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
493	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
494	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
495	C₂×#95	C₂²×C₁₀₉	1	2	yes	complemented
496	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
497	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
498	C₂×#41	—	2	4	no	—
499	C₂×#41	—	4	4	no	—
500	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
501	C₂²×D₄	—	2	4	no	—
502	C₂²×D₄	—	4	4	no	—
503	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
504	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
505	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
506	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
507	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
508	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
509	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
510	C₂³×C₄	—	4	4	no	—
511	C₂³×C₄	D₄×C₁₀₉	1	8	yes	complemented
512	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
513	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
514	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
515	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
516	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
517	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
518	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
519	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
520	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
521	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
522	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
523	#123	—	2	8	no	—
524	#123	—	2	8	no	—
525	#123	—	2	8	no	—
526	#123	—	2	8	no	—
527	#123	—	2	8	no	—
528	#123	—	2	8	no	—
529	C₂²⋊Q₈	—	2	4	no	—
530	C₂²⋊Q₈	—	2	4	no	—
531	#121	—	2	4	no	—
532	#121	—	2	4	no	—
533	#121	—	4	8	no	—
534	#120	—	2	8	no	—
535	#120	—	2	8	no	—
536	C₄×D₄	—	2	8	no	—
537	C₄×D₄	—	2	8	no	—
538	#117	—	2	4	no	—
539	C₂×C₄⋊C₄	—	2	4	no	—
540	C₂×C₄⋊C₄	—	2	4	no	—
541	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
542	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
543	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
544	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
545	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
546	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
547	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
548	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
549	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
550	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
551	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
552	C₂×C₂²⋊C₄	—	2	4	no	—
553	C₂×C₂²⋊C₄	—	2	4	no	—
554	C₂×C₂²⋊C₄	—	2	4	no	—
555	C₂×C₂²⋊C₄	—	2	8	no	—
556	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
557	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
558	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
559	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
560	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
561	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
562	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
563	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
564	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
565	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
566	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
567	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
568	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
569	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
570	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
571	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
572	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
573	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
574	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
575	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
576	C₂×C₄²	—	4	4	no	—
577	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
578	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
579	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
580	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
581	#95	C₂³×C₁₀₉	1	4	yes	—
582	#95	C₂³×C₁₀₉	1	2	yes	—
583	#95	C₂³×C₁₀₉	1	2	yes	—
584	#95	C₂³×C₁₀₉	1	4	yes	—
585	#95	C₂³×C₁₀₉	1	4	yes	—
586	#95	C₂³×C₁₀₉	1	4	yes	—
587	#95	C₂³×C₁₀₉	1	4	yes	—
588	#95	C₂³×C₁₀₉	1	4	yes	—
589	#95	C₂³×C₁₀₉	1	4	yes	—
590	C₂⁴	—	2	4	no	—
591	C₂⁴	—	2	4	no	—
592	C₂⁴	—	2	2	no	—
593	C₂⁴	—	2	4	no	—
594	C₂⁴	—	4	2	no	—
595	C₂⁴	—	4	2	no	—
596	C₂⁴	—	4	2	no	—
597	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
598	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
599	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
600	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
601	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
602	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
603	#41	—	4	16	no	—
604	C₂×Q₈	—	2	2	no	—
605	C₂×Q₈	—	4	2	no	—
606	C₂×D₄	—	2	4	no	—
607	C₂×D₄	—	2	4	no	—
608	C₂×D₄	—	2	4	no	—
609	C₂×D₄	—	2	4	no	—
610	C₂×D₄	—	2	4	no	—
611	C₂×D₄	—	2	4	no	—
612	C₂×D₄	—	2	4	no	—
613	C₂×D₄	—	2	8	no	—
614	C₂×D₄	—	2	8	no	—
615	C₂×D₄	—	2	2	no	—
616	C₂×D₄	—	2	8	no	—
617	C₂×D₄	—	4	4	no	—
618	C₂×D₄	—	4	4	no	—
619	C₂×D₄	—	4	2	no	—
620	C₂×D₄	—	4	4	no	—
621	C₂×D₄	—	4	4	no	—
622	C₂×D₄	—	4	16	no	—
623	C₂²×C₄	—	2	4	no	—
624	C₂²×C₄	—	2	4	no	—
625	C₂²×C₄	—	2	2	no	—
626	C₂²×C₄	—	2	2	no	—
627	C₂²×C₄	—	2	2	no	—
628	C₂²×C₄	—	2	4	no	—
629	C₂²×C₄	—	2	8	no	—
630	C₂²×C₄	—	2	4	no	—
631	C₂²×C₄	—	2	8	no	—
632	C₂²×C₄	—	4	16	no	—
633	C₂²×C₄	—	4	2	no	—
634	C₂²×C₄	—	4	2	no	—
635	C₂²×C₄	—	4	2	no	—
636	C₂²×C₄	—	4	2	no	—
637	C₂²×C₄	—	4	4	no	—
638	C₂²×C₄	—	4	2	no	—
639	C₂²×C₄	—	4	2	no	—
640	C₂²×C₄	—	4	4	no	—
641	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
642	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
643	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
644	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
645	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
646	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
647	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
648	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
649	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
650	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
651	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
652	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
653	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
654	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
655	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
656	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
657	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
658	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
659	C₄⋊C₄	—	2	4	no	—
660	C₄⋊C₄	—	2	4	no	—
661	C₄⋊C₄	—	2	4	no	—
662	C₄⋊C₄	—	2	4	no	—
663	C₄⋊C₄	—	2	2	no	—
664	C₄⋊C₄	—	2	2	no	—
665	C₄⋊C₄	—	2	4	no	—
666	C₄⋊C₄	—	2	4	no	—
667	C₄⋊C₄	—	2	8	no	—
668	C₄⋊C₄	—	4	4	no	—
669	C₄⋊C₄	—	4	4	no	—
670	C₂²⋊C₄	—	2	4	no	—
671	C₂²⋊C₄	—	2	4	no	—
672	C₂²⋊C₄	—	2	4	no	—
673	C₂²⋊C₄	—	2	4	no	—
674	C₂²⋊C₄	—	2	4	no	—
675	C₂²⋊C₄	—	2	4	no	—
676	C₂²⋊C₄	—	2	4	no	—
677	C₂²⋊C₄	—	2	4	no	—
678	C₂²⋊C₄	—	2	4	no	—
679	C₂²⋊C₄	—	2	4	no	—
680	C₂²⋊C₄	—	2	4	no	—
681	C₂²⋊C₄	—	2	4	no	—
682	C₂²⋊C₄	—	2	8	no	—
683	C₂²⋊C₄	—	2	8	no	—
684	C₂²⋊C₄	—	2	8	no	—
685	C₂²⋊C₄	—	4	4	no	—
686	C₂²⋊C₄	—	4	4	no	—
687	C₂²⋊C₄	—	4	4	no	—
688	C₂²⋊C₄	—	4	4	no	—
689	C₂²⋊C₄	#41×C₁₀₉	1	8	yes	complemented
690	C₄²	—	2	4	no	—
691	C₄²	—	4	2	no	—
692	C₄²	—	4	4	no	—
693	C₄²	—	4	2	no	—
694	C₂³	—	2	2	no	—
695	C₂³	—	2	2	no	—
696	C₂³	—	2	2	no	—
697	C₂³	—	2	2	no	—
698	C₂³	—	2	8	no	—
699	C₂³	—	2	8	no	—
700	C₂³	—	2	2	no	—
701	C₂³	—	2	2	no	—
702	C₂³	—	2	2	no	—
703	C₂³	—	2	2	no	—
704	C₂³	—	2	2	no	—
705	C₂³	—	2	2	no	—
706	C₂³	—	2	4	no	—
707	C₂³	—	2	2	no	—
708	C₂³	—	2	4	no	—
709	C₂³	—	2	4	no	—
710	C₂³	—	2	2	no	—
711	C₂³	—	2	4	no	—
712	C₂³	—	2	2	no	—
713	C₂³	—	4	4	no	—
714	C₂³	—	4	16	no	—
715	C₂³	—	4	4	no	—
716	C₂³	—	4	4	no	—
717	C₂³	—	4	4	no	—
718	C₂³	—	8	8	no	—
719	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
720	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
721	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
722	C₂³	C₂×#41×C₁₀₉	1	4	yes	—
723	C₂³	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
724	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
725	Q₈	—	4	4	no	—
726	D₄	—	4	4	no	—
727	D₄	—	4	8	no	—
728	D₄	—	4	8	no	—
729	D₄	—	4	8	no	—
730	D₄	—	4	8	no	—
731	D₄	—	4	8	no	—
732	D₄	—	8	8	no	—
733	C₂×C₄	—	2	8	no	—
734	C₂×C₄	—	2	8	no	—
735	C₂×C₄	—	2	4	no	—
736	C₂×C₄	—	2	2	no	—
737	C₂×C₄	—	2	2	no	—
738	C₂×C₄	—	2	2	no	—
739	C₂×C₄	—	2	2	no	—
740	C₂×C₄	—	2	4	no	—
741	C₂×C₄	—	2	2	no	—
742	C₂×C₄	—	2	2	no	—
743	C₂×C₄	—	2	2	no	—
744	C₂×C₄	—	2	2	no	—
745	C₂×C₄	—	2	2	no	—
746	C₂×C₄	—	2	2	no	—
747	C₂×C₄	—	2	4	no	—
748	C₂×C₄	—	2	2	no	—
749	C₂×C₄	—	2	4	no	—
750	C₂×C₄	—	2	2	no	—
751	C₂×C₄	—	2	4	no	—
752	C₂×C₄	—	2	4	no	—
753	C₂×C₄	—	2	4	no	—
754	C₂×C₄	—	2	4	no	—
755	C₂×C₄	—	2	2	no	—
756	C₂×C₄	—	2	4	no	—
757	C₂×C₄	—	2	2	no	—
758	C₂×C₄	—	4	4	no	—
759	C₂×C₄	—	4	4	no	—
760	C₂×C₄	—	4	4	no	—
761	C₂×C₄	—	4	4	no	—
762	C₂×C₄	—	4	16	no	—
763	C₂×C₄	—	4	8	no	—
764	C₂×C₄	—	4	4	no	—
765	C₂×C₄	—	4	4	no	—
766	C₂×C₄	—	4	4	no	—
767	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
768	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
769	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
770	C₂×C₄	C₂²×D₄×C₁₀₉	1	2	yes	—
771	C₂×C₄	C₂×#41×C₁₀₉	1	8	yes	—
772	C₂²	—	2	8	no	—
773	C₂²	—	2	8	no	—
774	C₂²	—	2	2	no	—
775	C₂²	—	2	2	no	—
776	C₂²	—	2	2	no	—
777	C₂²	—	2	2	no	—
778	C₂²	—	2	2	no	—
779	C₂²	—	2	2	no	—
780	C₂²	—	2	2	no	—
781	C₂²	—	4	16	no	—
782	C₂²	—	4	4	no	—
783	C₂²	—	4	4	no	—
784	C₂²	—	4	4	no	—
785	C₂²	—	4	4	no	—
786	C₂²	—	4	4	no	—
787	C₂²	—	4	2	no	—
788	C₂²	—	4	2	no	—
789	C₂²	—	4	2	no	—
790	C₂²	—	4	2	no	—
791	C₂²	—	4	2	no	—
792	C₂²	—	4	2	no	—
793	C₂²	—	8	8	no	—
794	C₂²	#532×C₁₀₉	1	4	yes	—
795	C₂²	#534×C₁₀₉	1	4	yes	—
796	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
797	C₂²	C₂²×#41×C₁₀₉	1	2	yes	—
798	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	2	yes	—
799	C₂²	#585×C₁₀₉	1	2	yes	—
800	C₄	—	2	8	no	—
801	C₄	—	2	2	no	—
802	C₄	—	2	2	no	—
803	C₄	—	2	2	no	—
804	C₄	—	2	2	no	—
805	C₄	—	4	4	no	—
806	C₄	—	4	2	no	—
807	C₄	—	4	2	no	—
808	C₄	—	4	2	no	—
809	C₄	—	4	2	no	—
810	C₄	—	4	2	no	—
811	C₄	—	4	2	no	—
812	C₂	—	2	4	no	—
813	C₂	—	2	4	no	—
814	C₂	—	4	2	no	—
815	C₂	—	8	2	no	—
816	C₂	—	8	2	no	—
817	C₂	C₂×#532×C₁₀₉	1	1	yes	characteristic
818	C₂	C₂×#534×C₁₀₉	1	1	yes	characteristic
819	C₂	C₁₀₉×#3449	1	1	yes	characteristic
820	C₂	C₁₀₉×#3513	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#525×C₁₀₉	C₂	semidirect
2. C₂²×#123×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2854	C₂	semidirect
4. C₁₀₉×#2663	C₂	semidirect
5. C₁₀₉×#2661	C₂	semidirect
6. C₁₀₉×#2602	C₂	semidirect
7. C₁₀₉×#2591	C₂	semidirect
8. C₁₀₉×#2583	C₂	semidirect
9. C₁₀₉×#2561	C₂	semidirect
10. C₁₀₉×#2539	C₂	semidirect
11. C₁₀₉×#2498	C₂	semidirect
12. C₁₀₉×#2430	C₂	semidirect
13. C₁₀₉×#2412	C₂	semidirect
14. C₁₀₉×#2410	C₂	semidirect
15. C₁₀₉×#2322	C₂	semidirect
16. C₁₀₉×#2321	C₂	semidirect
17. C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
18. C₂²×#41×C₁₀₉	C₂²	semidirect
28. C₂×#123×C₁₀₉	C₂²	semidirect
30. C₂×C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
33. C₂×#121×C₁₀₉	C₂²	semidirect
37. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
38. C₂×#117×C₁₀₉	C₂²	semidirect
45. C₂²×C₄²×C₁₀₉	C₂²	semidirect
46. #400×C₁₀₉	C₂²	semidirect
47. #399×C₁₀₉	C₂²	semidirect
48. #398×C₁₀₉	C₂²	semidirect
49. #397×C₁₀₉	C₂²	semidirect
50. #397×C₁₀₉	C₂²	semidirect
51. #395×C₁₀₉	C₂²	semidirect
53. #394×C₁₀₉	C₂²	semidirect
54. #394×C₁₀₉	C₂²	semidirect
55. #393×C₁₀₉	C₂²	semidirect
57. #392×C₁₀₉	C₂²	semidirect
58. #390×C₁₀₉	C₂²	semidirect
59. #390×C₁₀₉	C₂²	semidirect
60. #389×C₁₀₉	C₂²	semidirect
61. #389×C₁₀₉	C₂²	semidirect
62. #388×C₁₀₉	C₂²	semidirect
63. #388×C₁₀₉	C₂²	semidirect
64. #388×C₁₀₉	C₂²	semidirect
65. #388×C₁₀₉	C₂²	semidirect
66. #388×C₁₀₉	C₂²	semidirect
67. #387×C₁₀₉	C₂²	semidirect
71. #386×C₁₀₉	C₂²	semidirect
72. #386×C₁₀₉	C₂²	semidirect
75. #385×C₁₀₉	C₂²	semidirect
77. #380×C₁₀₉	C₂²	semidirect
82. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
83. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
84. C₂×#95×C₁₀₉	C₂²	semidirect
100. C₂³×C₄×C₁₀₉	D₄	semidirect
278. C₂²⋊C₄×C₁₀₉	#41	semidirect
401. #35064	C₁₀₉	direct (N × H)
411. C₂×#525	C₂×C₁₀₉	semidirect
412. C₂²×#123	C₂×C₁₀₉	semidirect
413. #2854	C₂×C₁₀₉	semidirect
414. #2663	C₂×C₁₀₉	semidirect
415. #2661	C₂×C₁₀₉	semidirect
416. #2602	C₂×C₁₀₉	semidirect
417. #2591	C₂×C₁₀₉	semidirect
418. #2583	C₂×C₁₀₉	semidirect
419. #2561	C₂×C₁₀₉	semidirect
420. #2539	C₂×C₁₀₉	semidirect
421. #2498	C₂×C₁₀₉	semidirect
422. #2430	C₂×C₁₀₉	semidirect
423. #2412	C₂×C₁₀₉	semidirect
424. #2410	C₂×C₁₀₉	semidirect
425. #2322	C₂×C₁₀₉	semidirect
426. #2321	C₂×C₁₀₉	semidirect
427. C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
428. C₁₀₉	#35064	direct (N × H)
429. C₂²×#41	C₂²×C₁₀₉	semidirect
439. C₂×#123	C₂²×C₁₀₉	semidirect
441. C₂×C₂²⋊Q₈	C₂²×C₁₀₉	semidirect
444. C₂×#121	C₂²×C₁₀₉	semidirect
448. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
449. C₂×#117	C₂²×C₁₀₉	semidirect
456. C₂²×C₄²	C₂²×C₁₀₉	semidirect
457. #400	C₂²×C₁₀₉	semidirect
458. #399	C₂²×C₁₀₉	semidirect
459. #398	C₂²×C₁₀₉	semidirect
460. #397	C₂²×C₁₀₉	semidirect
461. #397	C₂²×C₁₀₉	semidirect
462. #395	C₂²×C₁₀₉	semidirect
464. #394	C₂²×C₁₀₉	semidirect
465. #394	C₂²×C₁₀₉	semidirect
466. #393	C₂²×C₁₀₉	semidirect
468. #392	C₂²×C₁₀₉	semidirect
469. #390	C₂²×C₁₀₉	semidirect
470. #390	C₂²×C₁₀₉	semidirect
471. #389	C₂²×C₁₀₉	semidirect
472. #389	C₂²×C₁₀₉	semidirect
473. #388	C₂²×C₁₀₉	semidirect
474. #388	C₂²×C₁₀₉	semidirect
475. #388	C₂²×C₁₀₉	semidirect
476. #388	C₂²×C₁₀₉	semidirect
477. #388	C₂²×C₁₀₉	semidirect
478. #387	C₂²×C₁₀₉	semidirect
482. #386	C₂²×C₁₀₉	semidirect
483. #386	C₂²×C₁₀₉	semidirect
486. #385	C₂²×C₁₀₉	semidirect
488. #380	C₂²×C₁₀₉	semidirect
493. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
494. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
495. C₂×#95	C₂²×C₁₀₉	semidirect
511. C₂³×C₄	D₄×C₁₀₉	semidirect
689. C₂²⋊C₄	#41×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#525×C₁₀₉	2
C₂²×#123×C₁₀₉	2
C₁₀₉×#2854	2
C₁₀₉×#2663	2
C₁₀₉×#2661	2
C₁₀₉×#2602	2
C₁₀₉×#2591	2
C₁₀₉×#2583	2
C₁₀₉×#2561	2
C₁₀₉×#2539	2
C₁₀₉×#2498	2
C₁₀₉×#2430	2
C₁₀₉×#2412	2
C₁₀₉×#2410	2
C₁₀₉×#2322	2
C₁₀₉×#2321	2
C₂×C₄×C₂²⋊C₄×C₁₀₉	2
#35064	109

Subgroups — C109×#35064 #42153058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#35064 #42153058