Subgroups — C₂×C₁₀₉×#2373 #42152142

Order 27904.

2726 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2373	C₂	1	16	yes	maximal, direct product factor
3	C₂×#389×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#388×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#387×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#385×C₁₀₉	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
15	C₂×C₄×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂⁴×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#126×C₁₀₉	C₄	1	16	yes	complemented
19	C₂×#126×C₁₀₉	C₂²	1	8	yes	—
20	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊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	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
42	#389×C₁₀₉	C₂²	1	8	yes	complemented
43	#388×C₁₀₉	C₂²	1	8	yes	—
44	#388×C₁₀₉	C₂²	1	8	yes	—
45	#388×C₁₀₉	C₂²	1	8	yes	—
46	#387×C₁₀₉	C₂²	1	8	yes	complemented
47	#387×C₁₀₉	C₂²	1	8	yes	complemented
48	#387×C₁₀₉	C₂²	1	8	yes	complemented
49	#385×C₁₀₉	C₂²	1	8	yes	—
50	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
51	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
52	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	complemented
53	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
54	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	8	yes	—
55	#376×C₁₀₉	C₂²	1	8	yes	complemented
56	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
60	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
61	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
62	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
63	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
64	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
65	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
66	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
67	C₂×#95×C₁₀₉	C₂²	1	1	yes	characteristic
68	C₂⁵×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₁₀₉	—	2	4	no	—
70	C₂³×C₄×C₁₀₉	—	2	4	no	—
71	C₂³×C₄×C₁₀₉	—	4	2	no	—
72	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₁₀₉	C₂³	1	4	yes	—
77	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
87	#126×C₁₀₉	C₂×C₄	1	64	yes	complemented
88	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
89	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
90	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
91	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
92	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
94	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
95	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
96	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
106	C₂×C₂²⋊C₄×C₁₀₉	—	4	2	no	—
107	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
108	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
109	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
110	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
116	C₂×C₄²×C₁₀₉	—	2	4	no	—
117	C₂×C₄²×C₁₀₉	—	2	4	no	—
118	C₂×C₄²×C₁₀₉	—	2	4	no	—
119	C₂×C₄²×C₁₀₉	—	2	4	no	—
120	C₂×C₄²×C₁₀₉	—	4	2	no	—
121	C₂×C₄²×C₁₀₉	—	4	2	no	—
122	C₂×C₄²×C₁₀₉	—	4	2	no	—
123	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	complemented
124	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
125	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
126	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
127	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
128	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
129	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
130	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
131	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
132	#95×C₁₀₉	C₂³	1	4	yes	—
133	#95×C₁₀₉	C₂³	1	4	yes	—
134	#95×C₁₀₉	C₂³	1	4	yes	—
135	#95×C₁₀₉	C₂³	1	4	yes	—
136	#95×C₁₀₉	C₂³	1	4	yes	—
137	#95×C₁₀₉	C₂³	1	4	yes	—
138	#95×C₁₀₉	C₂³	1	4	yes	—
139	#95×C₁₀₉	C₂³	1	4	yes	—
140	#95×C₁₀₉	C₂³	1	4	yes	—
141	#95×C₁₀₉	C₂³	1	4	yes	—
142	#95×C₁₀₉	C₂³	1	4	yes	—
143	#95×C₁₀₉	C₂³	1	4	yes	—
144	C₂⁴×C₁₀₉	—	2	2	no	—
145	C₂⁴×C₁₀₉	—	2	2	no	—
146	C₂⁴×C₁₀₉	—	2	2	no	—
147	C₂⁴×C₁₀₉	—	2	2	no	—
148	C₂⁴×C₁₀₉	—	2	2	no	—
149	C₂⁴×C₁₀₉	—	2	2	no	—
150	C₂⁴×C₁₀₉	C₂²×C₄	1	4	yes	—
151	C₂⁴×C₁₀₉	C₂⁴	1	2	yes	—
152	C₂⁴×C₁₀₉	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₁₀₉	—	2	2	no	—
154	C₂²×C₄×C₁₀₉	—	2	2	no	—
155	C₂²×C₄×C₁₀₉	—	2	2	no	—
156	C₂²×C₄×C₁₀₉	—	2	2	no	—
157	C₂²×C₄×C₁₀₉	—	2	2	no	—
158	C₂²×C₄×C₁₀₉	—	2	2	no	—
159	C₂²×C₄×C₁₀₉	—	2	2	no	—
160	C₂²×C₄×C₁₀₉	—	2	2	no	—
161	C₂²×C₄×C₁₀₉	—	2	2	no	—
162	C₂²×C₄×C₁₀₉	—	2	2	no	—
163	C₂²×C₄×C₁₀₉	—	2	2	no	—
164	C₂²×C₄×C₁₀₉	—	2	2	no	—
165	C₂²×C₄×C₁₀₉	—	2	2	no	—
166	C₂²×C₄×C₁₀₉	—	2	2	no	—
167	C₂²×C₄×C₁₀₉	—	2	2	no	—
168	C₂²×C₄×C₁₀₉	—	2	2	no	—
169	C₂²×C₄×C₁₀₉	—	2	2	no	—
170	C₂²×C₄×C₁₀₉	—	2	2	no	—
171	C₂²×C₄×C₁₀₉	—	2	2	no	—
172	C₂²×C₄×C₁₀₉	—	2	2	no	—
173	C₂²×C₄×C₁₀₉	—	2	2	no	—
174	C₂²×C₄×C₁₀₉	—	2	2	no	—
175	C₂²×C₄×C₁₀₉	—	2	2	no	—
176	C₂²×C₄×C₁₀₉	—	2	2	no	—
177	C₂²×C₄×C₁₀₉	—	2	2	no	—
178	C₂²×C₄×C₁₀₉	—	2	2	no	—
179	C₂²×C₄×C₁₀₉	—	2	2	no	—
180	C₂²×C₄×C₁₀₉	—	2	2	no	—
181	C₂²×C₄×C₁₀₉	—	2	2	no	—
182	C₂²×C₄×C₁₀₉	—	2	2	no	—
183	C₂²×C₄×C₁₀₉	—	2	2	no	—
184	C₂²×C₄×C₁₀₉	—	2	2	no	—
185	C₂²×C₄×C₁₀₉	—	2	8	no	—
186	C₂²×C₄×C₁₀₉	—	2	8	no	—
187	C₂²×C₄×C₁₀₉	—	4	4	no	—
188	C₂²×C₄×C₁₀₉	—	4	16	no	—
189	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
190	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
191	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
192	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
193	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
194	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
195	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
196	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
197	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
198	C₂²×C₄×C₁₀₉	#41	1	4	yes	—
199	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₁₀₉	C₂⁴	1	2	yes	—
213	C₄⋊C₄×C₁₀₉	—	2	8	no	—
214	C₄⋊C₄×C₁₀₉	—	2	8	no	—
215	C₄⋊C₄×C₁₀₉	—	2	8	no	—
216	C₄⋊C₄×C₁₀₉	—	2	8	no	—
217	C₄⋊C₄×C₁₀₉	—	4	4	no	—
218	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
219	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
220	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	complemented
221	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
222	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
223	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
224	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
225	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
226	C₂²⋊C₄×C₁₀₉	C₂²×C₄	1	16	yes	—
227	C₄²×C₁₀₉	—	2	8	no	—
228	C₄²×C₁₀₉	—	2	8	no	—
229	C₄²×C₁₀₉	—	2	8	no	—
230	C₄²×C₁₀₉	—	2	8	no	—
231	C₄²×C₁₀₉	—	4	4	no	—
232	C₄²×C₁₀₉	—	4	4	no	—
233	C₄²×C₁₀₉	—	4	4	no	—
234	C₄²×C₁₀₉	C₂²×C₄	1	16	yes	complemented
235	C₂³×C₁₀₉	—	2	8	no	—
236	C₂³×C₁₀₉	—	2	8	no	—
237	C₂³×C₁₀₉	—	2	8	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	2	no	—
243	C₂³×C₁₀₉	—	2	2	no	—
244	C₂³×C₁₀₉	—	4	4	no	—
245	C₂³×C₁₀₉	—	4	4	no	—
246	C₂³×C₁₀₉	—	4	4	no	—
247	C₂³×C₁₀₉	—	4	4	no	—
248	C₂³×C₁₀₉	C₂³×C₄	1	4	yes	—
249	C₂³×C₁₀₉	C₂³×C₄	1	2	yes	—
250	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
251	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
252	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
253	C₂³×C₁₀₉	C₂×#41	1	2	yes	—
254	C₂³×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
255	C₂³×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
256	C₂³×C₁₀₉	C₂⁵	1	1	yes	characteristic
257	C₂×C₄×C₁₀₉	—	2	8	no	—
258	C₂×C₄×C₁₀₉	—	2	8	no	—
259	C₂×C₄×C₁₀₉	—	2	8	no	—
260	C₂×C₄×C₁₀₉	—	2	8	no	—
261	C₂×C₄×C₁₀₉	—	2	8	no	—
262	C₂×C₄×C₁₀₉	—	2	8	no	—
263	C₂×C₄×C₁₀₉	—	2	8	no	—
264	C₂×C₄×C₁₀₉	—	2	8	no	—
265	C₂×C₄×C₁₀₉	—	2	8	no	—
266	C₂×C₄×C₁₀₉	—	2	8	no	—
267	C₂×C₄×C₁₀₉	—	2	2	no	—
268	C₂×C₄×C₁₀₉	—	2	2	no	—
269	C₂×C₄×C₁₀₉	—	2	2	no	—
270	C₂×C₄×C₁₀₉	—	2	2	no	—
271	C₂×C₄×C₁₀₉	—	2	2	no	—
272	C₂×C₄×C₁₀₉	—	2	2	no	—
273	C₂×C₄×C₁₀₉	—	2	2	no	—
274	C₂×C₄×C₁₀₉	—	2	2	no	—
275	C₂×C₄×C₁₀₉	—	2	2	no	—
276	C₂×C₄×C₁₀₉	—	2	2	no	—
277	C₂×C₄×C₁₀₉	—	2	2	no	—
278	C₂×C₄×C₁₀₉	—	2	2	no	—
279	C₂×C₄×C₁₀₉	—	2	2	no	—
280	C₂×C₄×C₁₀₉	—	2	2	no	—
281	C₂×C₄×C₁₀₉	—	2	2	no	—
282	C₂×C₄×C₁₀₉	—	2	2	no	—
283	C₂×C₄×C₁₀₉	—	2	2	no	—
284	C₂×C₄×C₁₀₉	—	2	2	no	—
285	C₂×C₄×C₁₀₉	—	2	2	no	—
286	C₂×C₄×C₁₀₉	—	2	2	no	—
287	C₂×C₄×C₁₀₉	—	2	2	no	—
288	C₂×C₄×C₁₀₉	—	2	2	no	—
289	C₂×C₄×C₁₀₉	—	2	2	no	—
290	C₂×C₄×C₁₀₉	—	2	2	no	—
291	C₂×C₄×C₁₀₉	—	2	2	no	—
292	C₂×C₄×C₁₀₉	—	2	2	no	—
293	C₂×C₄×C₁₀₉	—	2	2	no	—
294	C₂×C₄×C₁₀₉	—	2	2	no	—
295	C₂×C₄×C₁₀₉	—	2	2	no	—
296	C₂×C₄×C₁₀₉	—	2	2	no	—
297	C₂×C₄×C₁₀₉	—	2	2	no	—
298	C₂×C₄×C₁₀₉	—	2	2	no	—
299	C₂×C₄×C₁₀₉	—	4	16	no	—
300	C₂×C₄×C₁₀₉	—	4	4	no	—
301	C₂×C₄×C₁₀₉	—	4	4	no	—
302	C₂×C₄×C₁₀₉	—	4	4	no	—
303	C₂×C₄×C₁₀₉	—	4	4	no	—
304	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
305	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
306	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
307	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
308	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
309	C₂×C₄×C₁₀₉	C₂³×C₄	1	4	yes	—
310	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
311	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
312	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
313	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
314	C₂²×C₁₀₉	—	2	4	no	—
315	C₂²×C₁₀₉	—	2	4	no	—
316	C₂²×C₁₀₉	—	2	4	no	—
317	C₂²×C₁₀₉	—	4	16	no	—
318	C₂²×C₁₀₉	—	4	2	no	—
319	C₂²×C₁₀₉	—	4	2	no	—
320	C₂²×C₁₀₉	—	4	2	no	—
321	C₂²×C₁₀₉	—	4	2	no	—
322	C₂²×C₁₀₉	C₄×#41	1	4	yes	—
323	C₂²×C₁₀₉	C₄×#41	1	4	yes	—
324	C₂²×C₁₀₉	#520	1	4	yes	—
325	C₂²×C₁₀₉	#551	1	4	yes	—
326	C₂²×C₁₀₉	#552	1	4	yes	—
327	C₂²×C₁₀₉	#553	1	4	yes	—
328	C₂²×C₁₀₉	#556	1	4	yes	—
329	C₂²×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
330	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
331	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
332	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
333	C₂²×C₁₀₉	C₂²×#41	1	1	yes	characteristic
334	C₂²×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
335	C₂²×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
336	C₄×C₁₀₉	—	2	4	no	—
337	C₄×C₁₀₉	—	2	4	no	—
338	C₄×C₁₀₉	—	2	4	no	—
339	C₄×C₁₀₉	—	2	4	no	—
340	C₄×C₁₀₉	—	2	4	no	—
341	C₄×C₁₀₉	—	2	4	no	—
342	C₄×C₁₀₉	—	2	4	no	—
343	C₄×C₁₀₉	—	2	4	no	—
344	C₄×C₁₀₉	—	2	4	no	—
345	C₄×C₁₀₉	—	2	4	no	—
346	C₄×C₁₀₉	—	4	2	no	—
347	C₄×C₁₀₉	—	4	2	no	—
348	C₄×C₁₀₉	—	4	2	no	—
349	C₄×C₁₀₉	—	4	2	no	—
350	C₂×#2373	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
351	C₂×C₁₀₉	—	4	4	no	—
352	C₂×C₁₀₉	#2373	1	8	yes	direct product factor
353	C₂×C₁₀₉	C₂×C₄×#41	1	1	yes	characteristic
354	C₂×C₁₀₉	C₂×C₄×#41	1	1	yes	characteristic
355	C₂×C₁₀₉	C₂×#520	1	1	yes	characteristic
356	C₂×C₁₀₉	C₂×#551	1	1	yes	characteristic
357	C₂×C₁₀₉	C₂×#552	1	1	yes	characteristic
358	C₂×C₁₀₉	C₂×#553	1	1	yes	characteristic
359	C₂×C₁₀₉	C₂×#556	1	1	yes	characteristic
360	C₂²×#126	C₂×C₁₀₉	1	1	yes	characteristic
361	#2373	C₂×C₁₀₉	1	16	yes	direct product factor
362	C₂×#389	C₂×C₁₀₉	1	1	yes	characteristic, complemented
363	C₂×#388	C₂×C₁₀₉	1	1	yes	characteristic
364	C₂×#388	C₂×C₁₀₉	1	1	yes	characteristic
365	C₂×#388	C₂×C₁₀₉	1	1	yes	characteristic
366	C₂×#387	C₂×C₁₀₉	1	1	yes	characteristic, complemented
367	C₂×#387	C₂×C₁₀₉	1	1	yes	characteristic, complemented
368	C₂×#387	C₂×C₁₀₉	1	1	yes	characteristic, complemented
369	C₂×#385	C₂×C₁₀₉	1	1	yes	characteristic
370	C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
371	C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
372	C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
373	C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
374	C₂×C₄×C₂²⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
375	C₂×#376	C₂×C₁₀₉	1	1	yes	characteristic, complemented
376	C₁₀₉	C₂×#2373	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
378	C₂×#126	C₄×C₁₀₉	1	16	yes	complemented
379	C₂×#126	C₂²×C₁₀₉	1	8	yes	—
380	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
381	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
382	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
383	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
384	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
385	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
386	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
387	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
388	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
389	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
390	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
391	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
392	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
393	C₂²×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
394	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
395	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
396	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
397	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
398	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
399	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
400	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
401	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
402	#389	C₂²×C₁₀₉	1	8	yes	complemented
403	#388	C₂²×C₁₀₉	1	8	yes	—
404	#388	C₂²×C₁₀₉	1	8	yes	—
405	#388	C₂²×C₁₀₉	1	8	yes	—
406	#387	C₂²×C₁₀₉	1	8	yes	complemented
407	#387	C₂²×C₁₀₉	1	8	yes	complemented
408	#387	C₂²×C₁₀₉	1	8	yes	complemented
409	#385	C₂²×C₁₀₉	1	8	yes	—
410	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
411	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
412	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	8	yes	complemented
413	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
414	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	8	yes	—
415	#376	C₂²×C₁₀₉	1	8	yes	complemented
416	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
417	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
418	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
419	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
420	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
421	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
422	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
423	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
424	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
425	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
426	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
427	C₂×#95	C₂²×C₁₀₉	1	1	yes	characteristic
428	C₂⁵	C₂³×C₁₀₉	1	1	yes	characteristic
429	C₂³×C₄	—	2	4	no	—
430	C₂³×C₄	—	2	4	no	—
431	C₂³×C₄	—	4	2	no	—
432	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
433	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
434	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
435	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
436	C₂³×C₄	C₂³×C₁₀₉	1	4	yes	—
437	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
438	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
439	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
440	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
441	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
442	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
443	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
444	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
445	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
446	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
447	#126	C₂×C₄×C₁₀₉	1	64	yes	complemented
448	C₂×C₄⋊C₄	—	2	4	no	—
449	C₂×C₄⋊C₄	—	2	4	no	—
450	C₂×C₄⋊C₄	—	2	4	no	—
451	C₂×C₄⋊C₄	—	2	4	no	—
452	C₂×C₄⋊C₄	—	4	2	no	—
453	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
454	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
455	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
456	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
458	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
459	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
460	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
461	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
462	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
463	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
464	C₂×C₂²⋊C₄	—	2	4	no	—
465	C₂×C₂²⋊C₄	—	2	4	no	—
466	C₂×C₂²⋊C₄	—	4	2	no	—
467	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
468	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
469	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	—
470	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
471	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
472	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
474	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
475	C₂×C₂²⋊C₄	C₂³×C₁₀₉	1	4	yes	—
476	C₂×C₄²	—	2	4	no	—
477	C₂×C₄²	—	2	4	no	—
478	C₂×C₄²	—	2	4	no	—
479	C₂×C₄²	—	2	4	no	—
480	C₂×C₄²	—	4	2	no	—
481	C₂×C₄²	—	4	2	no	—
482	C₂×C₄²	—	4	2	no	—
483	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	complemented
484	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
485	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
486	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
487	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
488	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
489	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
490	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
491	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
492	#95	C₂³×C₁₀₉	1	4	yes	—
493	#95	C₂³×C₁₀₉	1	4	yes	—
494	#95	C₂³×C₁₀₉	1	4	yes	—
495	#95	C₂³×C₁₀₉	1	4	yes	—
496	#95	C₂³×C₁₀₉	1	4	yes	—
497	#95	C₂³×C₁₀₉	1	4	yes	—
498	#95	C₂³×C₁₀₉	1	4	yes	—
499	#95	C₂³×C₁₀₉	1	4	yes	—
500	#95	C₂³×C₁₀₉	1	4	yes	—
501	#95	C₂³×C₁₀₉	1	4	yes	—
502	#95	C₂³×C₁₀₉	1	4	yes	—
503	#95	C₂³×C₁₀₉	1	4	yes	—
504	C₂⁴	—	2	2	no	—
505	C₂⁴	—	2	2	no	—
506	C₂⁴	—	2	2	no	—
507	C₂⁴	—	2	2	no	—
508	C₂⁴	—	2	2	no	—
509	C₂⁴	—	2	2	no	—
510	C₂⁴	C₂²×C₄×C₁₀₉	1	4	yes	—
511	C₂⁴	C₂⁴×C₁₀₉	1	2	yes	—
512	C₂⁴	C₂⁴×C₁₀₉	1	1	yes	characteristic
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	2	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	8	no	—
546	C₂²×C₄	—	2	8	no	—
547	C₂²×C₄	—	4	4	no	—
548	C₂²×C₄	—	4	16	no	—
549	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
550	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
551	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
552	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
553	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
554	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
555	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
556	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
557	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
558	C₂²×C₄	#41×C₁₀₉	1	4	yes	—
559	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
560	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
561	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
562	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
563	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
564	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
565	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
566	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
567	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
568	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
569	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
570	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
571	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
572	C₂²×C₄	C₂⁴×C₁₀₉	1	2	yes	—
573	C₄⋊C₄	—	2	8	no	—
574	C₄⋊C₄	—	2	8	no	—
575	C₄⋊C₄	—	2	8	no	—
576	C₄⋊C₄	—	2	8	no	—
577	C₄⋊C₄	—	4	4	no	—
578	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	complemented
579	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	complemented
580	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	complemented
581	C₂²⋊C₄	—	2	8	no	—
582	C₂²⋊C₄	—	2	8	no	—
583	C₂²⋊C₄	—	4	4	no	—
584	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
585	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
586	C₂²⋊C₄	C₂²×C₄×C₁₀₉	1	16	yes	—
587	C₄²	—	2	8	no	—
588	C₄²	—	2	8	no	—
589	C₄²	—	2	8	no	—
590	C₄²	—	2	8	no	—
591	C₄²	—	4	4	no	—
592	C₄²	—	4	4	no	—
593	C₄²	—	4	4	no	—
594	C₄²	C₂²×C₄×C₁₀₉	1	16	yes	complemented
595	C₂³	—	2	8	no	—
596	C₂³	—	2	8	no	—
597	C₂³	—	2	8	no	—
598	C₂³	—	2	2	no	—
599	C₂³	—	2	2	no	—
600	C₂³	—	2	2	no	—
601	C₂³	—	2	2	no	—
602	C₂³	—	2	2	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	4	4	no	—
605	C₂³	—	4	4	no	—
606	C₂³	—	4	4	no	—
607	C₂³	—	4	4	no	—
608	C₂³	C₂³×C₄×C₁₀₉	1	4	yes	—
609	C₂³	C₂³×C₄×C₁₀₉	1	2	yes	—
610	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
611	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
612	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
613	C₂³	C₂×#41×C₁₀₉	1	2	yes	—
614	C₂³	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
615	C₂³	2₋¹⁺⁴×C₁₀₉	1	2	yes	—
616	C₂³	C₂⁵×C₁₀₉	1	1	yes	characteristic
617	C₂×C₄	—	2	8	no	—
618	C₂×C₄	—	2	8	no	—
619	C₂×C₄	—	2	8	no	—
620	C₂×C₄	—	2	8	no	—
621	C₂×C₄	—	2	8	no	—
622	C₂×C₄	—	2	8	no	—
623	C₂×C₄	—	2	8	no	—
624	C₂×C₄	—	2	8	no	—
625	C₂×C₄	—	2	8	no	—
626	C₂×C₄	—	2	8	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	4	16	no	—
660	C₂×C₄	—	4	4	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	4	no	—
663	C₂×C₄	—	4	4	no	—
664	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
665	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
666	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
667	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
668	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
669	C₂×C₄	C₂³×C₄×C₁₀₉	1	4	yes	—
670	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
671	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
672	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
673	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
674	C₂²	—	2	4	no	—
675	C₂²	—	2	4	no	—
676	C₂²	—	2	4	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	4	16	no	—
682	C₂²	C₄×#41×C₁₀₉	1	4	yes	—
683	C₂²	C₄×#41×C₁₀₉	1	4	yes	—
684	C₂²	#520×C₁₀₉	1	4	yes	—
685	C₂²	#551×C₁₀₉	1	4	yes	—
686	C₂²	#552×C₁₀₉	1	4	yes	—
687	C₂²	#553×C₁₀₉	1	4	yes	—
688	C₂²	#556×C₁₀₉	1	4	yes	—
689	C₂²	C₂⁴×C₄×C₁₀₉	1	1	yes	characteristic
690	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
691	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
692	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
693	C₂²	C₂²×#41×C₁₀₉	1	1	yes	characteristic
694	C₂²	C₂×H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
695	C₂²	C₂×2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
696	C₄	—	2	4	no	—
697	C₄	—	2	4	no	—
698	C₄	—	2	4	no	—
699	C₄	—	2	4	no	—
700	C₄	—	2	4	no	—
701	C₄	—	2	4	no	—
702	C₄	—	2	4	no	—
703	C₄	—	2	4	no	—
704	C₄	—	2	4	no	—
705	C₄	—	2	4	no	—
706	C₄	—	4	2	no	—
707	C₄	—	4	2	no	—
708	C₄	—	4	2	no	—
709	C₄	—	4	2	no	—
710	C₂	—	4	4	no	—
711	C₂	C₁₀₉×#2373	1	8	yes	direct product factor
712	C₂	C₂×C₄×#41×C₁₀₉	1	1	yes	characteristic
713	C₂	C₂×C₄×#41×C₁₀₉	1	1	yes	characteristic
714	C₂	C₂×#520×C₁₀₉	1	1	yes	characteristic
715	C₂	C₂×#551×C₁₀₉	1	1	yes	characteristic
716	C₂	C₂×#552×C₁₀₉	1	1	yes	characteristic
717	C₂	C₂×#553×C₁₀₉	1	1	yes	characteristic
718	C₂	C₂×#556×C₁₀₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₁₀₉×#2373	C₂	direct (N × H)
3. C₂×#389×C₁₀₉	C₂	semidirect
7. C₂×#387×C₁₀₉	C₂	semidirect
8. C₂×#387×C₁₀₉	C₂	semidirect
9. C₂×#387×C₁₀₉	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
12. C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
16. C₂×#376×C₁₀₉	C₂	semidirect
18. C₂×#126×C₁₀₉	C₄	semidirect
42. #389×C₁₀₉	C₂²	semidirect
46. #387×C₁₀₉	C₂²	semidirect
47. #387×C₁₀₉	C₂²	semidirect
48. #387×C₁₀₉	C₂²	semidirect
50. C₄×C₄⋊C₄×C₁₀₉	C₂²	semidirect
51. C₄×C₄⋊C₄×C₁₀₉	C₂²	semidirect
52. C₄×C₄⋊C₄×C₁₀₉	C₂²	semidirect
55. #376×C₁₀₉	C₂²	semidirect
87. #126×C₁₀₉	C₂×C₄	semidirect
93. C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
94. C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
95. C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
123. C₂×C₄²×C₁₀₉	C₂×C₄	semidirect
218. C₄⋊C₄×C₁₀₉	C₂²×C₄	semidirect
219. C₄⋊C₄×C₁₀₉	C₂²×C₄	semidirect
220. C₄⋊C₄×C₁₀₉	C₂²×C₄	semidirect
234. C₄²×C₁₀₉	C₂²×C₄	semidirect
350. C₂×#2373	C₁₀₉	direct (N × H)
352. C₂×C₁₀₉	#2373	direct (N × H)
361. #2373	C₂×C₁₀₉	direct (N × H)
362. C₂×#389	C₂×C₁₀₉	semidirect
366. C₂×#387	C₂×C₁₀₉	semidirect
367. C₂×#387	C₂×C₁₀₉	semidirect
368. C₂×#387	C₂×C₁₀₉	semidirect
370. C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
371. C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
372. C₂×C₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
375. C₂×#376	C₂×C₁₀₉	semidirect
376. C₁₀₉	C₂×#2373	direct (N × H)
378. C₂×#126	C₄×C₁₀₉	semidirect
402. #389	C₂²×C₁₀₉	semidirect
406. #387	C₂²×C₁₀₉	semidirect
407. #387	C₂²×C₁₀₉	semidirect
408. #387	C₂²×C₁₀₉	semidirect
410. C₄×C₄⋊C₄	C₂²×C₁₀₉	semidirect
411. C₄×C₄⋊C₄	C₂²×C₁₀₉	semidirect
412. C₄×C₄⋊C₄	C₂²×C₁₀₉	semidirect
415. #376	C₂²×C₁₀₉	semidirect
447. #126	C₂×C₄×C₁₀₉	semidirect
453. C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
454. C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
455. C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
483. C₂×C₄²	C₂×C₄×C₁₀₉	semidirect
578. C₄⋊C₄	C₂²×C₄×C₁₀₉	semidirect
579. C₄⋊C₄	C₂²×C₄×C₁₀₉	semidirect
580. C₄⋊C₄	C₂²×C₄×C₁₀₉	semidirect
594. C₄²	C₂²×C₄×C₁₀₉	semidirect
711. C₂	C₁₀₉×#2373	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#126×C₁₀₉	2
C₁₀₉×#2373	2
C₂×#389×C₁₀₉	2
C₂×#388×C₁₀₉	2
C₂×#387×C₁₀₉	2
C₂×#385×C₁₀₉	2
C₂×C₄×C₄⋊C₄×C₁₀₉	2
C₂×C₄×C₂²⋊C₄×C₁₀₉	2
C₂×#376×C₁₀₉	2
C₂×#2373	109

Subgroups — C2×C109×#2373 #42152142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×#2373 #42152142