Subgroups — C₁₀₉×#34198 #42152192

Order 27904.

4982 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#120×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2358	C₂	1	8	yes	maximal, complemented
5	C₁₀₉×#2342	C₂	1	4	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₁₀₉×#2321	C₂	1	4	yes	maximal, complemented
8	C₁₀₉×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#2308	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#120×C₁₀₉	C₂	1	4	yes	maximal, complemented
11	C₂×#379×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₀₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁶×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂×#120×C₁₀₉	C₄	1	16	yes	complemented
18	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
19	C₂×#120×C₁₀₉	C₂²	1	2	yes	—
20	C₂×#120×C₁₀₉	C₂²	1	4	yes	—
21	#518×C₁₀₉	—	2	8	no	—
22	C₂×C₄×D₄×C₁₀₉	C₂²	1	4	yes	complemented
23	C₂×C₄×D₄×C₁₀₉	C₂²	1	4	yes	complemented
24	C₂×C₄×D₄×C₁₀₉	C₂²	1	8	yes	complemented
25	C₂×#117×C₁₀₉	C₂²	1	2	yes	complemented
26	C₂²×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
27	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
31	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
35	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
36	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
37	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
38	#390×C₁₀₉	C₂²	1	8	yes	complemented
39	#388×C₁₀₉	C₂²	1	8	yes	complemented
40	#388×C₁₀₉	C₂²	1	8	yes	complemented
41	#386×C₁₀₉	C₂²	1	8	yes	complemented
42	#386×C₁₀₉	C₂²	1	4	yes	complemented
43	#386×C₁₀₉	C₂²	1	8	yes	complemented
44	#386×C₁₀₉	C₂²	1	4	yes	complemented
45	#385×C₁₀₉	C₂²	1	4	yes	complemented
46	#385×C₁₀₉	C₂²	1	8	yes	complemented
47	#385×C₁₀₉	C₂²	1	4	yes	complemented
48	#381×C₁₀₉	C₂²	1	2	yes	complemented
49	#380×C₁₀₉	C₂²	1	4	yes	complemented
50	#379×C₁₀₉	C₂²	1	4	yes	—
51	#379×C₁₀₉	C₂²	1	2	yes	complemented
52	#379×C₁₀₉	C₂²	1	2	yes	complemented
53	#379×C₁₀₉	C₂²	1	2	yes	complemented
54	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
55	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
56	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	complemented
57	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
58	C₂⁵×C₁₀₉	—	2	4	no	—
59	C₂⁵×C₁₀₉	—	2	4	no	—
60	C₂⁵×C₁₀₉	—	4	8	no	—
61	C₂⁵×C₁₀₉	C₂×C₄	1	8	yes	complemented
62	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
63	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
64	C₂⁵×C₁₀₉	C₂³	1	2	yes	—
65	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
66	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₁₀₉	—	2	4	no	—
69	C₂²×D₄×C₁₀₉	—	2	4	no	—
70	C₂²×D₄×C₁₀₉	—	4	8	no	—
71	C₂²×D₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
72	C₂²×D₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
73	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
74	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
75	C₂²×D₄×C₁₀₉	C₂³	1	2	yes	—
76	C₂²×D₄×C₁₀₉	C₂³	1	4	yes	—
77	C₂³×C₄×C₁₀₉	—	2	8	no	—
78	C₂³×C₄×C₁₀₉	—	4	8	no	—
79	C₂³×C₄×C₁₀₉	C₂³	1	8	yes	—
80	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
82	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
83	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
84	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
85	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
86	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	complemented
87	#120×C₁₀₉	C₂×C₄	1	16	yes	complemented
88	#120×C₁₀₉	C₂×C₄	1	32	yes	complemented
89	#120×C₁₀₉	C₂×C₄	1	16	yes	complemented
90	C₄×D₄×C₁₀₉	—	2	16	no	—
91	C₄×D₄×C₁₀₉	—	2	16	no	—
92	C₄×D₄×C₁₀₉	—	4	16	no	—
93	#117×C₁₀₉	—	2	8	no	—
94	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
95	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	8	yes	complemented
96	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
100	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
101	C₂×C₂²⋊C₄×C₁₀₉	—	2	8	no	—
102	C₂×C₂²⋊C₄×C₁₀₉	—	2	16	no	—
103	C₂×C₂²⋊C₄×C₁₀₉	—	4	8	no	—
104	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
105	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
106	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	8	yes	complemented
107	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	—
116	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
123	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	8	yes	complemented
124	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	complemented
125	C₂×C₄²×C₁₀₉	C₂³	1	8	yes	complemented
126	#95×C₁₀₉	C₂³	1	8	yes	complemented
127	#95×C₁₀₉	C₂³	1	8	yes	complemented
128	#95×C₁₀₉	C₂³	1	8	yes	complemented
129	C₂⁴×C₁₀₉	—	2	2	no	—
130	C₂⁴×C₁₀₉	—	2	2	no	—
131	C₂⁴×C₁₀₉	—	2	8	no	—
132	C₂⁴×C₁₀₉	—	2	32	no	—
133	C₂⁴×C₁₀₉	—	2	4	no	—
134	C₂⁴×C₁₀₉	—	2	4	no	—
135	C₂⁴×C₁₀₉	—	2	2	no	—
136	C₂⁴×C₁₀₉	—	2	4	no	—
137	C₂⁴×C₁₀₉	—	2	4	no	—
138	C₂⁴×C₁₀₉	—	2	2	no	—
139	C₂⁴×C₁₀₉	—	2	8	no	—
140	C₂⁴×C₁₀₉	—	2	8	no	—
141	C₂⁴×C₁₀₉	—	2	2	no	—
142	C₂⁴×C₁₀₉	—	2	2	no	—
143	C₂⁴×C₁₀₉	—	4	32	no	—
144	C₂⁴×C₁₀₉	—	4	4	no	—
145	C₂⁴×C₁₀₉	—	4	4	no	—
146	C₂⁴×C₁₀₉	—	4	4	no	—
147	C₂⁴×C₁₀₉	—	4	4	no	—
148	C₂⁴×C₁₀₉	—	4	4	no	—
149	C₂⁴×C₁₀₉	—	4	32	no	—
150	C₂⁴×C₁₀₉	—	4	32	no	—
151	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
152	C₂⁴×C₁₀₉	C₂²×C₄	1	8	yes	—
153	C₂⁴×C₁₀₉	C₂²×C₄	1	8	yes	—
154	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
155	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
156	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	complemented
157	C₂⁴×C₁₀₉	C₂×D₄	1	4	yes	—
158	C₂⁴×C₁₀₉	#41	1	4	yes	—
159	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
160	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
161	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
162	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
163	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
164	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
165	C₂×D₄×C₁₀₉	—	2	32	no	—
166	C₂×D₄×C₁₀₉	—	2	4	no	—
167	C₂×D₄×C₁₀₉	—	2	4	no	—
168	C₂×D₄×C₁₀₉	—	2	4	no	—
169	C₂×D₄×C₁₀₉	—	2	4	no	—
170	C₂×D₄×C₁₀₉	—	2	16	no	—
171	C₂×D₄×C₁₀₉	—	4	8	no	—
172	C₂×D₄×C₁₀₉	—	4	32	no	—
173	C₂×D₄×C₁₀₉	C₂²×C₄	1	8	yes	—
174	C₂×D₄×C₁₀₉	C₂²×C₄	1	16	yes	—
175	C₂×D₄×C₁₀₉	C₂²×C₄	1	8	yes	complemented
176	C₂×D₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
177	C₂²×C₄×C₁₀₉	—	2	2	no	—
178	C₂²×C₄×C₁₀₉	—	2	2	no	—
179	C₂²×C₄×C₁₀₉	—	2	4	no	—
180	C₂²×C₄×C₁₀₉	—	2	16	no	—
181	C₂²×C₄×C₁₀₉	—	2	4	no	—
182	C₂²×C₄×C₁₀₉	—	2	4	no	—
183	C₂²×C₄×C₁₀₉	—	2	16	no	—
184	C₂²×C₄×C₁₀₉	—	2	8	no	—
185	C₂²×C₄×C₁₀₉	—	4	4	no	—
186	C₂²×C₄×C₁₀₉	—	4	8	no	—
187	C₂²×C₄×C₁₀₉	—	4	4	no	—
188	C₂²×C₄×C₁₀₉	—	4	4	no	—
189	C₂²×C₄×C₁₀₉	—	4	4	no	—
190	C₂²×C₄×C₁₀₉	—	4	4	no	—
191	C₂²×C₄×C₁₀₉	—	4	8	no	—
192	C₂²×C₄×C₁₀₉	—	4	32	no	—
193	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	complemented
194	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	complemented
195	C₂²×C₄×C₁₀₉	C₂×D₄	1	8	yes	complemented
196	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
197	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₁₀₉	C₂⁴	1	4	yes	—
202	C₄⋊C₄×C₁₀₉	—	2	8	no	—
203	C₄⋊C₄×C₁₀₉	—	4	8	no	—
204	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
205	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
206	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
207	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
208	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
209	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
210	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
211	C₂²⋊C₄×C₁₀₉	—	2	16	no	—
212	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
213	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
214	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
215	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
216	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
217	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
218	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	8	yes	complemented
219	C₂²⋊C₄×C₁₀₉	C₂×D₄	1	8	yes	—
220	C₂²⋊C₄×C₁₀₉	C₂×D₄	1	8	yes	complemented
221	C₄²×C₁₀₉	—	2	8	no	—
222	C₄²×C₁₀₉	—	4	8	no	—
223	C₂³×C₁₀₉	—	2	16	no	—
224	C₂³×C₁₀₉	—	2	16	no	—
225	C₂³×C₁₀₉	—	2	8	no	—
226	C₂³×C₁₀₉	—	2	16	no	—
227	C₂³×C₁₀₉	—	2	2	no	—
228	C₂³×C₁₀₉	—	2	2	no	—
229	C₂³×C₁₀₉	—	2	4	no	—
230	C₂³×C₁₀₉	—	2	4	no	—
231	C₂³×C₁₀₉	—	2	4	no	—
232	C₂³×C₁₀₉	—	2	4	no	—
233	C₂³×C₁₀₉	—	2	16	no	—
234	C₂³×C₁₀₉	—	2	16	no	—
235	C₂³×C₁₀₉	—	2	8	no	—
236	C₂³×C₁₀₉	—	2	16	no	—
237	C₂³×C₁₀₉	—	2	16	no	—
238	C₂³×C₁₀₉	—	2	2	no	—
239	C₂³×C₁₀₉	—	2	2	no	—
240	C₂³×C₁₀₉	—	2	2	no	—
241	C₂³×C₁₀₉	—	2	2	no	—
242	C₂³×C₁₀₉	—	2	4	no	—
243	C₂³×C₁₀₉	—	2	8	no	—
244	C₂³×C₁₀₉	—	2	4	no	—
245	C₂³×C₁₀₉	—	2	8	no	—
246	C₂³×C₁₀₉	—	4	8	no	—
247	C₂³×C₁₀₉	—	4	8	no	—
248	C₂³×C₁₀₉	—	4	8	no	—
249	C₂³×C₁₀₉	—	4	8	no	—
250	C₂³×C₁₀₉	—	4	8	no	—
251	C₂³×C₁₀₉	—	4	8	no	—
252	C₂³×C₁₀₉	—	4	8	no	—
253	C₂³×C₁₀₉	—	4	16	no	—
254	C₂³×C₁₀₉	—	4	16	no	—
255	C₂³×C₁₀₉	—	4	16	no	—
256	C₂³×C₁₀₉	—	4	16	no	—
257	C₂³×C₁₀₉	—	4	128	no	—
258	C₂³×C₁₀₉	—	4	8	no	—
259	C₂³×C₁₀₉	—	4	8	no	—
260	C₂³×C₁₀₉	—	8	4	no	—
261	C₂³×C₁₀₉	—	8	4	no	—
262	C₂³×C₁₀₉	C₄×D₄	1	16	yes	complemented
263	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
264	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
265	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
266	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
267	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
268	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
269	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
270	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
271	C₂³×C₁₀₉	C₂²×D₄	1	2	yes	—
272	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
273	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
274	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
275	C₂³×C₁₀₉	H(5,𝔽₂)	1	8	yes	—
276	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
277	D₄×C₁₀₉	—	2	16	no	—
278	D₄×C₁₀₉	—	2	16	no	—
279	D₄×C₁₀₉	—	2	32	no	—
280	D₄×C₁₀₉	—	4	16	no	—
281	C₂×C₄×C₁₀₉	—	2	8	no	—
282	C₂×C₄×C₁₀₉	—	2	2	no	—
283	C₂×C₄×C₁₀₉	—	2	2	no	—
284	C₂×C₄×C₁₀₉	—	2	8	no	—
285	C₂×C₄×C₁₀₉	—	2	8	no	—
286	C₂×C₄×C₁₀₉	—	2	4	no	—
287	C₂×C₄×C₁₀₉	—	2	4	no	—
288	C₂×C₄×C₁₀₉	—	2	4	no	—
289	C₂×C₄×C₁₀₉	—	2	8	no	—
290	C₂×C₄×C₁₀₉	—	2	8	no	—
291	C₂×C₄×C₁₀₉	—	2	8	no	—
292	C₂×C₄×C₁₀₉	—	4	8	no	—
293	C₂×C₄×C₁₀₉	—	4	8	no	—
294	C₂×C₄×C₁₀₉	—	4	32	no	—
295	C₂×C₄×C₁₀₉	—	4	8	no	—
296	C₂×C₄×C₁₀₉	—	8	8	no	—
297	C₂×C₄×C₁₀₉	—	8	4	no	—
298	C₂×C₄×C₁₀₉	—	8	4	no	—
299	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
300	C₂×C₄×C₁₀₉	C₂³×C₄	1	8	yes	—
301	C₂×C₄×C₁₀₉	C₂²×D₄	1	8	yes	—
302	C₂²×C₁₀₉	—	2	32	no	—
303	C₂²×C₁₀₉	—	2	4	no	—
304	C₂²×C₁₀₉	—	2	4	no	—
305	C₂²×C₁₀₉	—	2	4	no	—
306	C₂²×C₁₀₉	—	2	4	no	—
307	C₂²×C₁₀₉	—	2	4	no	—
308	C₂²×C₁₀₉	—	2	4	no	—
309	C₂²×C₁₀₉	—	2	8	no	—
310	C₂²×C₁₀₉	—	2	4	no	—
311	C₂²×C₁₀₉	—	2	4	no	—
312	C₂²×C₁₀₉	—	2	4	no	—
313	C₂²×C₁₀₉	—	2	4	no	—
314	C₂²×C₁₀₉	—	2	8	no	—
315	C₂²×C₁₀₉	—	2	4	no	—
316	C₂²×C₁₀₉	—	2	4	no	—
317	C₂²×C₁₀₉	—	2	4	no	—
318	C₂²×C₁₀₉	—	2	8	no	—
319	C₂²×C₁₀₉	—	2	4	no	—
320	C₂²×C₁₀₉	—	2	4	no	—
321	C₂²×C₁₀₉	—	4	16	no	—
322	C₂²×C₁₀₉	—	4	32	no	—
323	C₂²×C₁₀₉	—	4	32	no	—
324	C₂²×C₁₀₉	—	4	16	no	—
325	C₂²×C₁₀₉	—	4	4	no	—
326	C₂²×C₁₀₉	—	4	4	no	—
327	C₂²×C₁₀₉	—	4	4	no	—
328	C₂²×C₁₀₉	—	4	4	no	—
329	C₂²×C₁₀₉	—	8	8	no	—
330	C₂²×C₁₀₉	C₂×C₄×D₄	1	4	yes	—
331	C₂²×C₁₀₉	C₂×C₄×D₄	1	4	yes	—
332	C₂²×C₁₀₉	C₂×C₄×D₄	1	4	yes	—
333	C₂²×C₁₀₉	#518	1	8	yes	complemented
334	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
335	C₂²×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
336	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
337	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	4	yes	—
338	C₄×C₁₀₉	—	2	4	no	—
339	C₄×C₁₀₉	—	2	8	no	—
340	C₄×C₁₀₉	—	2	8	no	—
341	C₄×C₁₀₉	—	4	4	no	—
342	C₄×C₁₀₉	—	4	4	no	—
343	C₄×C₁₀₉	—	4	4	no	—
344	#34198	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
345	C₂×C₁₀₉	—	2	4	no	—
346	C₂×C₁₀₉	—	2	8	no	—
347	C₂×C₁₀₉	—	2	4	no	—
348	C₂×C₁₀₉	—	2	4	no	—
349	C₂×C₁₀₉	—	4	4	no	—
350	C₂×C₁₀₉	—	4	4	no	—
351	C₂×C₁₀₉	C₂²×C₄×D₄	1	1	yes	characteristic
352	C₂×C₁₀₉	C₂×#518	1	2	yes	—
353	C₂×C₁₀₉	#3484	1	2	yes	—
354	C₂×C₁₀₉	#3490	1	2	yes	—
355	C₂²×#120	C₂×C₁₀₉	1	1	yes	characteristic
356	C₂×#518	C₂×C₁₀₉	1	1	yes	characteristic, complemented
357	C₂³×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
358	#2358	C₂×C₁₀₉	1	8	yes	complemented
359	#2342	C₂×C₁₀₉	1	4	yes	complemented
360	D₄×C₂²⋊C₄	C₂×C₁₀₉	1	4	yes	complemented
361	#2321	C₂×C₁₀₉	1	4	yes	complemented
362	#2315	C₂×C₁₀₉	1	1	yes	characteristic, complemented
363	#2308	C₂×C₁₀₉	1	1	yes	characteristic, complemented
364	C₄×#120	C₂×C₁₀₉	1	4	yes	complemented
365	C₂×#379	C₂×C₁₀₉	1	1	yes	characteristic, complemented
366	#2273	C₂×C₁₀₉	1	1	yes	characteristic, complemented
367	C₁₀₉	#34198	1	1	yes	characteristic, direct product factor, Sylow
368	C₂⁶	C₂²×C₁₀₉	1	1	yes	characteristic
369	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
370	C₂³×D₄	C₂²×C₁₀₉	1	2	yes	—
371	C₂⁴×C₄	C₂²×C₁₀₉	1	2	yes	complemented
372	C₂×#120	C₄×C₁₀₉	1	16	yes	complemented
373	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
374	C₂×#120	C₂²×C₁₀₉	1	4	yes	—
375	C₂×#120	C₂²×C₁₀₉	1	2	yes	—
376	#518	—	2	8	no	—
377	C₂×C₄×D₄	C₂²×C₁₀₉	1	4	yes	complemented
378	C₂×C₄×D₄	C₂²×C₁₀₉	1	4	yes	complemented
379	C₂×C₄×D₄	C₂²×C₁₀₉	1	8	yes	complemented
380	C₂×#117	C₂²×C₁₀₉	1	2	yes	complemented
381	C₂²×C₂²⋊C₄	—	2	8	no	—
382	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
383	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
384	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
385	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
386	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
387	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	—
388	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
389	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
390	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
391	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	2	yes	complemented
392	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
393	#390	C₂²×C₁₀₉	1	8	yes	complemented
394	#388	C₂²×C₁₀₉	1	8	yes	complemented
395	#388	C₂²×C₁₀₉	1	8	yes	complemented
396	#386	C₂²×C₁₀₉	1	8	yes	complemented
397	#386	C₂²×C₁₀₉	1	4	yes	complemented
398	#386	C₂²×C₁₀₉	1	4	yes	complemented
399	#386	C₂²×C₁₀₉	1	8	yes	complemented
400	#385	C₂²×C₁₀₉	1	4	yes	complemented
401	#385	C₂²×C₁₀₉	1	4	yes	complemented
402	#385	C₂²×C₁₀₉	1	8	yes	complemented
403	#381	C₂²×C₁₀₉	1	2	yes	complemented
404	#380	C₂²×C₁₀₉	1	4	yes	complemented
405	#379	C₂²×C₁₀₉	1	4	yes	—
406	#379	C₂²×C₁₀₉	1	2	yes	complemented
407	#379	C₂²×C₁₀₉	1	2	yes	complemented
408	#379	C₂²×C₁₀₉	1	2	yes	complemented
409	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
410	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
411	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
412	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	complemented
413	C₂⁵	—	2	4	no	—
414	C₂⁵	—	2	4	no	—
415	C₂⁵	—	4	8	no	—
416	C₂⁵	C₂×C₄×C₁₀₉	1	8	yes	complemented
417	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
418	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
419	C₂⁵	C₂³×C₁₀₉	1	2	yes	—
420	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
421	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
422	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
423	C₂²×D₄	—	2	4	no	—
424	C₂²×D₄	—	2	4	no	—
425	C₂²×D₄	—	4	8	no	—
426	C₂²×D₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
427	C₂²×D₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
428	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
429	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
430	C₂²×D₄	C₂³×C₁₀₉	1	2	yes	—
431	C₂²×D₄	C₂³×C₁₀₉	1	4	yes	—
432	C₂³×C₄	—	2	8	no	—
433	C₂³×C₄	—	4	8	no	—
434	C₂³×C₄	C₂³×C₁₀₉	1	8	yes	—
435	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
436	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
437	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
438	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
439	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
440	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
441	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	complemented
442	#120	C₂×C₄×C₁₀₉	1	16	yes	complemented
443	#120	C₂×C₄×C₁₀₉	1	32	yes	complemented
444	#120	C₂×C₄×C₁₀₉	1	16	yes	complemented
445	C₄×D₄	—	2	16	no	—
446	C₄×D₄	—	2	16	no	—
447	C₄×D₄	—	4	16	no	—
448	#117	—	2	8	no	—
449	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	complemented
450	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	8	yes	complemented
451	C₂×C₂²⋊C₄	—	2	4	no	—
452	C₂×C₂²⋊C₄	—	2	4	no	—
453	C₂×C₂²⋊C₄	—	2	4	no	—
454	C₂×C₂²⋊C₄	—	2	8	no	—
455	C₂×C₂²⋊C₄	—	2	16	no	—
456	C₂×C₂²⋊C₄	—	2	8	no	—
457	C₂×C₂²⋊C₄	—	2	16	no	—
458	C₂×C₂²⋊C₄	—	4	8	no	—
459	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	16	yes	complemented
460	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
461	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	8	yes	complemented
462	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
469	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
470	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	—
471	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
472	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
474	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
475	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
476	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	2	yes	—
477	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	complemented
478	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	8	yes	complemented
479	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	complemented
480	C₂×C₄²	C₂³×C₁₀₉	1	8	yes	complemented
481	#95	C₂³×C₁₀₉	1	8	yes	complemented
482	#95	C₂³×C₁₀₉	1	8	yes	complemented
483	#95	C₂³×C₁₀₉	1	8	yes	complemented
484	C₂⁴	—	2	2	no	—
485	C₂⁴	—	2	2	no	—
486	C₂⁴	—	2	8	no	—
487	C₂⁴	—	2	32	no	—
488	C₂⁴	—	2	4	no	—
489	C₂⁴	—	2	4	no	—
490	C₂⁴	—	2	2	no	—
491	C₂⁴	—	2	4	no	—
492	C₂⁴	—	2	4	no	—
493	C₂⁴	—	2	2	no	—
494	C₂⁴	—	2	8	no	—
495	C₂⁴	—	2	8	no	—
496	C₂⁴	—	2	2	no	—
497	C₂⁴	—	2	2	no	—
498	C₂⁴	—	4	32	no	—
499	C₂⁴	—	4	4	no	—
500	C₂⁴	—	4	4	no	—
501	C₂⁴	—	4	4	no	—
502	C₂⁴	—	4	4	no	—
503	C₂⁴	—	4	4	no	—
504	C₂⁴	—	4	32	no	—
505	C₂⁴	—	4	32	no	—
506	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
507	C₂⁴	C₂²×C₄×C₁₀₉	1	8	yes	—
508	C₂⁴	C₂²×C₄×C₁₀₉	1	8	yes	—
509	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
510	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
511	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	complemented
512	C₂⁴	C₂×D₄×C₁₀₉	1	4	yes	—
513	C₂⁴	#41×C₁₀₉	1	4	yes	—
514	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
515	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
516	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
517	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
518	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
519	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
520	C₂×D₄	—	2	32	no	—
521	C₂×D₄	—	2	4	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	4	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	16	no	—
526	C₂×D₄	—	4	8	no	—
527	C₂×D₄	—	4	32	no	—
528	C₂×D₄	C₂²×C₄×C₁₀₉	1	8	yes	—
529	C₂×D₄	C₂²×C₄×C₁₀₉	1	16	yes	—
530	C₂×D₄	C₂²×C₄×C₁₀₉	1	8	yes	complemented
531	C₂×D₄	C₂²×C₄×C₁₀₉	1	16	yes	complemented
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	4	no	—
535	C₂²×C₄	—	2	16	no	—
536	C₂²×C₄	—	2	4	no	—
537	C₂²×C₄	—	2	4	no	—
538	C₂²×C₄	—	2	16	no	—
539	C₂²×C₄	—	2	8	no	—
540	C₂²×C₄	—	4	4	no	—
541	C₂²×C₄	—	4	8	no	—
542	C₂²×C₄	—	4	4	no	—
543	C₂²×C₄	—	4	4	no	—
544	C₂²×C₄	—	4	4	no	—
545	C₂²×C₄	—	4	4	no	—
546	C₂²×C₄	—	4	8	no	—
547	C₂²×C₄	—	4	32	no	—
548	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	complemented
549	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	complemented
550	C₂²×C₄	C₂×D₄×C₁₀₉	1	8	yes	complemented
551	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
552	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
553	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
554	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
555	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
556	C₂²×C₄	C₂⁴×C₁₀₉	1	4	yes	—
557	C₄⋊C₄	—	2	8	no	—
558	C₄⋊C₄	—	4	8	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	16	no	—
566	C₂²⋊C₄	—	2	16	no	—
567	C₂²⋊C₄	—	2	8	no	—
568	C₂²⋊C₄	—	4	8	no	—
569	C₂²⋊C₄	—	4	8	no	—
570	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
571	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
572	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	complemented
573	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	8	yes	complemented
574	C₂²⋊C₄	C₂×D₄×C₁₀₉	1	8	yes	—
575	C₂²⋊C₄	C₂×D₄×C₁₀₉	1	8	yes	complemented
576	C₄²	—	2	8	no	—
577	C₄²	—	4	8	no	—
578	C₂³	—	2	16	no	—
579	C₂³	—	2	16	no	—
580	C₂³	—	2	8	no	—
581	C₂³	—	2	16	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	4	no	—
586	C₂³	—	2	4	no	—
587	C₂³	—	2	4	no	—
588	C₂³	—	2	16	no	—
589	C₂³	—	2	16	no	—
590	C₂³	—	2	8	no	—
591	C₂³	—	2	16	no	—
592	C₂³	—	2	16	no	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	2	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	4	no	—
598	C₂³	—	2	8	no	—
599	C₂³	—	2	4	no	—
600	C₂³	—	2	8	no	—
601	C₂³	—	4	8	no	—
602	C₂³	—	4	8	no	—
603	C₂³	—	4	8	no	—
604	C₂³	—	4	8	no	—
605	C₂³	—	4	8	no	—
606	C₂³	—	4	8	no	—
607	C₂³	—	4	8	no	—
608	C₂³	—	4	16	no	—
609	C₂³	—	4	16	no	—
610	C₂³	—	4	16	no	—
611	C₂³	—	4	16	no	—
612	C₂³	—	4	128	no	—
613	C₂³	—	4	8	no	—
614	C₂³	—	4	8	no	—
615	C₂³	—	8	4	no	—
616	C₂³	—	8	4	no	—
617	C₂³	C₄×D₄×C₁₀₉	1	16	yes	complemented
618	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
619	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
620	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
621	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
622	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
623	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
624	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
625	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
626	C₂³	C₂²×D₄×C₁₀₉	1	2	yes	—
627	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
628	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
629	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
630	C₂³	H(5,𝔽₂)×C₁₀₉	1	8	yes	—
631	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
632	D₄	—	2	16	no	—
633	D₄	—	2	16	no	—
634	D₄	—	2	32	no	—
635	D₄	—	4	16	no	—
636	C₂×C₄	—	2	8	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	8	no	—
640	C₂×C₄	—	2	8	no	—
641	C₂×C₄	—	2	4	no	—
642	C₂×C₄	—	2	4	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	2	8	no	—
645	C₂×C₄	—	2	8	no	—
646	C₂×C₄	—	2	8	no	—
647	C₂×C₄	—	4	8	no	—
648	C₂×C₄	—	4	8	no	—
649	C₂×C₄	—	4	32	no	—
650	C₂×C₄	—	4	8	no	—
651	C₂×C₄	—	8	8	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
655	C₂×C₄	C₂³×C₄×C₁₀₉	1	8	yes	—
656	C₂×C₄	C₂²×D₄×C₁₀₉	1	8	yes	—
657	C₂²	—	2	32	no	—
658	C₂²	—	2	4	no	—
659	C₂²	—	2	8	no	—
660	C₂²	—	2	4	no	—
661	C₂²	—	2	4	no	—
662	C₂²	—	2	4	no	—
663	C₂²	—	2	4	no	—
664	C₂²	—	2	8	no	—
665	C₂²	—	2	4	no	—
666	C₂²	—	2	4	no	—
667	C₂²	—	2	4	no	—
668	C₂²	—	2	4	no	—
669	C₂²	—	2	4	no	—
670	C₂²	—	2	4	no	—
671	C₂²	—	2	4	no	—
672	C₂²	—	2	4	no	—
673	C₂²	—	2	8	no	—
674	C₂²	—	2	4	no	—
675	C₂²	—	2	4	no	—
676	C₂²	—	4	16	no	—
677	C₂²	—	4	32	no	—
678	C₂²	—	4	32	no	—
679	C₂²	—	4	16	no	—
680	C₂²	—	4	4	no	—
681	C₂²	—	4	4	no	—
682	C₂²	—	4	4	no	—
683	C₂²	—	4	4	no	—
684	C₂²	—	8	8	no	—
685	C₂²	C₂×C₄×D₄×C₁₀₉	1	4	yes	—
686	C₂²	C₂×C₄×D₄×C₁₀₉	1	4	yes	—
687	C₂²	C₂×C₄×D₄×C₁₀₉	1	4	yes	—
688	C₂²	#518×C₁₀₉	1	8	yes	complemented
689	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
690	C₂²	C₂³×D₄×C₁₀₉	1	1	yes	characteristic
691	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
692	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	4	yes	—
693	C₄	—	2	8	no	—
694	C₄	—	2	4	no	—
695	C₄	—	2	8	no	—
696	C₄	—	4	4	no	—
697	C₄	—	4	4	no	—
698	C₄	—	4	4	no	—
699	C₂	—	2	4	no	—
700	C₂	—	2	8	no	—
701	C₂	—	2	4	no	—
702	C₂	—	2	4	no	—
703	C₂	—	4	4	no	—
704	C₂	—	4	4	no	—
705	C₂	C₂²×C₄×D₄×C₁₀₉	1	1	yes	characteristic
706	C₂	C₂×#518×C₁₀₉	1	2	yes	—
707	C₂	C₁₀₉×#3484	1	2	yes	—
708	C₂	C₁₀₉×#3490	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₀₉
Frattini, cosocle	C₂³	C₂⁵×C₁₀₉
prime radical	C₂²×#120×C₁₀₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#518×C₁₀₉	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₁₀₉	C₂	semidirect
4. C₁₀₉×#2358	C₂	semidirect
5. C₁₀₉×#2342	C₂	semidirect
6. D₄×C₂²⋊C₄×C₁₀₉	C₂	semidirect
7. C₁₀₉×#2321	C₂	semidirect
8. C₁₀₉×#2315	C₂	semidirect
9. C₁₀₉×#2308	C₂	semidirect
10. C₄×#120×C₁₀₉	C₂	semidirect
11. C₂×#379×C₁₀₉	C₂	semidirect
12. C₁₀₉×#2273	C₂	semidirect
16. C₂⁴×C₄×C₁₀₉	C₂²	semidirect
17. C₂×#120×C₁₀₉	C₄	semidirect
22. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
23. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
24. C₂×C₄×D₄×C₁₀₉	C₂²	semidirect
25. C₂×#117×C₁₀₉	C₂²	semidirect
34. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
35. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
36. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
37. C₂²×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
38. #390×C₁₀₉	C₂²	semidirect
39. #388×C₁₀₉	C₂²	semidirect
40. #388×C₁₀₉	C₂²	semidirect
41. #386×C₁₀₉	C₂²	semidirect
42. #386×C₁₀₉	C₂²	semidirect
43. #386×C₁₀₉	C₂²	semidirect
44. #386×C₁₀₉	C₂²	semidirect
45. #385×C₁₀₉	C₂²	semidirect
46. #385×C₁₀₉	C₂²	semidirect
47. #385×C₁₀₉	C₂²	semidirect
48. #381×C₁₀₉	C₂²	semidirect
49. #380×C₁₀₉	C₂²	semidirect
51. #379×C₁₀₉	C₂²	semidirect
52. #379×C₁₀₉	C₂²	semidirect
53. #379×C₁₀₉	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
55. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
57. C₄×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
61. C₂⁵×C₁₀₉	C₂×C₄	semidirect
71. C₂²×D₄×C₁₀₉	C₂×C₄	semidirect
72. C₂²×D₄×C₁₀₉	C₂×C₄	semidirect
86. C₂³×C₄×C₁₀₉	C₂³	semidirect
87. #120×C₁₀₉	C₂×C₄	semidirect
88. #120×C₁₀₉	C₂×C₄	semidirect
89. #120×C₁₀₉	C₂×C₄	semidirect
94. C₂×C₄⋊C₄×C₁₀₉	C₂³	semidirect
95. C₂×C₄⋊C₄×C₁₀₉	C₂³	semidirect
104. C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
105. C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	semidirect
106. C₂×C₂²⋊C₄×C₁₀₉	D₄	semidirect
122. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
123. C₂×C₂²⋊C₄×C₁₀₉	C₂³	semidirect
124. C₂×C₄²×C₁₀₉	C₂³	semidirect
125. C₂×C₄²×C₁₀₉	C₂³	semidirect
126. #95×C₁₀₉	C₂³	semidirect
127. #95×C₁₀₉	C₂³	semidirect
128. #95×C₁₀₉	C₂³	semidirect
156. C₂⁴×C₁₀₉	C₂²×C₄	semidirect
175. C₂×D₄×C₁₀₉	C₂²×C₄	semidirect
176. C₂×D₄×C₁₀₉	C₂²×C₄	semidirect
193. C₂²×C₄×C₁₀₉	C₂²×C₄	semidirect
194. C₂²×C₄×C₁₀₉	C₂²×C₄	semidirect
195. C₂²×C₄×C₁₀₉	C₂×D₄	semidirect
217. C₂²⋊C₄×C₁₀₉	C₂²×C₄	semidirect
218. C₂²⋊C₄×C₁₀₉	C₂²×C₄	semidirect
220. C₂²⋊C₄×C₁₀₉	C₂×D₄	semidirect
262. C₂³×C₁₀₉	C₄×D₄	semidirect
333. C₂²×C₁₀₉	#518	semidirect
344. #34198	C₁₀₉	direct (N × H)
356. C₂×#518	C₂×C₁₀₉	semidirect
357. C₂³×C₂²⋊C₄	C₂×C₁₀₉	semidirect
358. #2358	C₂×C₁₀₉	semidirect
359. #2342	C₂×C₁₀₉	semidirect
360. D₄×C₂²⋊C₄	C₂×C₁₀₉	semidirect
361. #2321	C₂×C₁₀₉	semidirect
362. #2315	C₂×C₁₀₉	semidirect
363. #2308	C₂×C₁₀₉	semidirect
364. C₄×#120	C₂×C₁₀₉	semidirect
365. C₂×#379	C₂×C₁₀₉	semidirect
366. #2273	C₂×C₁₀₉	semidirect
367. C₁₀₉	#34198	direct (N × H)
371. C₂⁴×C₄	C₂²×C₁₀₉	semidirect
372. C₂×#120	C₄×C₁₀₉	semidirect
377. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
378. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
379. C₂×C₄×D₄	C₂²×C₁₀₉	semidirect
380. C₂×#117	C₂²×C₁₀₉	semidirect
389. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
390. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
391. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
392. C₂²×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
393. #390	C₂²×C₁₀₉	semidirect
394. #388	C₂²×C₁₀₉	semidirect
395. #388	C₂²×C₁₀₉	semidirect
396. #386	C₂²×C₁₀₉	semidirect
397. #386	C₂²×C₁₀₉	semidirect
398. #386	C₂²×C₁₀₉	semidirect
399. #386	C₂²×C₁₀₉	semidirect
400. #385	C₂²×C₁₀₉	semidirect
401. #385	C₂²×C₁₀₉	semidirect
402. #385	C₂²×C₁₀₉	semidirect
403. #381	C₂²×C₁₀₉	semidirect
404. #380	C₂²×C₁₀₉	semidirect
406. #379	C₂²×C₁₀₉	semidirect
407. #379	C₂²×C₁₀₉	semidirect
408. #379	C₂²×C₁₀₉	semidirect
409. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
410. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
411. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
412. C₄×C₂²⋊C₄	C₂²×C₁₀₉	semidirect
416. C₂⁵	C₂×C₄×C₁₀₉	semidirect
426. C₂²×D₄	C₂×C₄×C₁₀₉	semidirect
427. C₂²×D₄	C₂×C₄×C₁₀₉	semidirect
441. C₂³×C₄	C₂³×C₁₀₉	semidirect
442. #120	C₂×C₄×C₁₀₉	semidirect
443. #120	C₂×C₄×C₁₀₉	semidirect
444. #120	C₂×C₄×C₁₀₉	semidirect
449. C₂×C₄⋊C₄	C₂³×C₁₀₉	semidirect
450. C₂×C₄⋊C₄	C₂³×C₁₀₉	semidirect
459. C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect
460. C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	semidirect
461. C₂×C₂²⋊C₄	D₄×C₁₀₉	semidirect
477. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
478. C₂×C₂²⋊C₄	C₂³×C₁₀₉	semidirect
479. C₂×C₄²	C₂³×C₁₀₉	semidirect
480. C₂×C₄²	C₂³×C₁₀₉	semidirect
481. #95	C₂³×C₁₀₉	semidirect
482. #95	C₂³×C₁₀₉	semidirect
483. #95	C₂³×C₁₀₉	semidirect
511. C₂⁴	C₂²×C₄×C₁₀₉	semidirect
530. C₂×D₄	C₂²×C₄×C₁₀₉	semidirect
531. C₂×D₄	C₂²×C₄×C₁₀₉	semidirect
548. C₂²×C₄	C₂²×C₄×C₁₀₉	semidirect
549. C₂²×C₄	C₂²×C₄×C₁₀₉	semidirect
550. C₂²×C₄	C₂×D₄×C₁₀₉	semidirect
572. C₂²⋊C₄	C₂²×C₄×C₁₀₉	semidirect
573. C₂²⋊C₄	C₂²×C₄×C₁₀₉	semidirect
575. C₂²⋊C₄	C₂×D₄×C₁₀₉	semidirect
617. C₂³	C₄×D₄×C₁₀₉	semidirect
688. C₂²	#518×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#120×C₁₀₉	2
C₂×#518×C₁₀₉	2
C₂³×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#2358	2
C₁₀₉×#2342	2
D₄×C₂²⋊C₄×C₁₀₉	2
C₁₀₉×#2321	2
C₁₀₉×#2315	2
C₁₀₉×#2308	2
C₄×#120×C₁₀₉	2
C₂×#379×C₁₀₉	2
C₁₀₉×#2273	2
#34198	109

Subgroups — C109×#34198 #42152192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#34198 #42152192