Subgroups — C₇×#35086 #13212302

Order 1792.

1974 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2848	C₂	1	2	yes	maximal
4	C₇×#2652	C₂	1	4	yes	maximal, complemented
5	C₇×#2505	C₂	1	4	yes	maximal, complemented
6	C₇×#2496	C₂	1	2	yes	maximal, complemented
7	C₇×#2489	C₂	1	2	yes	maximal, complemented
8	C₇×#2480	C₂	1	4	yes	maximal
9	C₇×#2429	C₂	1	2	yes	maximal, complemented
10	C₇×#2421	C₂	1	2	yes	maximal, complemented
11	C₇×#2334	C₂	1	4	yes	maximal, complemented
12	C₄×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
14	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₇×#126	C₂²	1	4	yes	—
17	C₂×C₇×#125	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	complemented
22	C₂²×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₇×#400	C₂²	1	4	yes	complemented
26	C₇×#400	C₂²	1	4	yes	complemented
27	C₇×#400	C₂²	1	4	yes	complemented
28	C₇×#398	C₂²	1	4	yes	complemented
29	C₇×#397	C₂²	1	4	yes	—
30	C₇×#397	C₂²	1	4	yes	—
31	C₇×#397	C₂²	1	4	yes	—
32	C₇×#396	C₂²	1	4	yes	—
33	C₇×#391	C₂²	1	2	yes	complemented
34	C₇×#391	C₂²	1	2	yes	complemented
35	C₇×#389	C₂²	1	4	yes	complemented
36	C₇×#389	C₂²	1	2	yes	complemented
37	C₇×#389	C₂²	1	2	yes	complemented
38	C₇×#389	C₂²	1	4	yes	complemented
39	C₇×#388	C₂²	1	4	yes	—
40	C₇×#388	C₂²	1	4	yes	—
41	C₇×#388	C₂²	1	4	yes	—
42	C₇×#387	C₂²	1	4	yes	complemented
43	C₇×#387	C₂²	1	4	yes	complemented
44	C₇×#387	C₂²	1	4	yes	complemented
45	C₇×#387	C₂²	1	4	yes	complemented
46	C₇×#385	C₂²	1	4	yes	—
47	C₇×#385	C₂²	1	4	yes	—
48	C₇×#385	C₂²	1	4	yes	—
49	C₇×#385	C₂²	1	4	yes	—
50	C₇×#385	C₂²	1	4	yes	—
51	C₇×#384	C₂²	1	1	yes	characteristic, complemented
52	C₇×#382	C₂²	1	1	yes	characteristic, complemented
53	C₇×#382	C₂²	1	4	yes	complemented
54	C₇×#380	C₂²	1	2	yes	—
55	C₇×#380	C₂²	1	4	yes	—
56	C₇×#380	C₂²	1	2	yes	—
57	C₇×#380	C₂²	1	2	yes	—
58	C₇×#379	C₂²	1	4	yes	—
59	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
60	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	complemented
61	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
62	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
63	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
64	C₄×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
65	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
66	C₂×C₇×#95	C₂²	1	2	yes	—
67	#35086	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁵×C₇	C₂³	1	1	yes	characteristic
69	C₂²×C₇×Q₈	C₂³	1	2	yes	—
70	C₂³×C₄×C₇	—	2	4	no	—
71	C₂³×C₄×C₇	—	4	2	no	—
72	C₂³×C₄×C₇	—	4	2	no	—
73	C₂³×C₄×C₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₇	C₂³	1	2	yes	—
79	C₇×C₄⋊Q₈	—	4	2	no	—
80	C₇×#126	—	2	16	no	—
81	C₇×#125	—	4	2	no	—
82	C₇×C₂²⋊Q₈	—	2	8	no	—
83	C₇×C₂²⋊Q₈	—	4	4	no	—
84	C₇×C₂²⋊Q₈	—	4	4	no	—
85	C₄×C₇×Q₈	—	4	4	no	—
86	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
87	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
88	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
89	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
90	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
91	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
92	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
93	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
94	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
95	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
96	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
97	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
98	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
99	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
100	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
101	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
102	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
103	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
104	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
105	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
106	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
107	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
108	C₂×C₇×C₂²⋊C₄	C₂³	1	2	yes	—
109	C₂×C₄²×C₇	—	4	2	no	—
110	C₂×C₄²×C₇	—	4	2	no	—
111	C₂×C₄²×C₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₇	C₂³	1	4	yes	—
115	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₇	C₂³	1	4	yes	—
119	C₇×#95	C₂³	1	2	yes	—
120	C₇×#95	C₂³	1	2	yes	—
121	C₇×#95	C₂³	1	2	yes	—
122	C₇×#95	C₂³	1	2	yes	—
123	C₇×#95	C₂³	1	4	yes	—
124	C₇×#95	C₂³	1	4	yes	—
125	C₇×#95	C₂³	1	4	yes	—
126	C₇×#95	C₂³	1	4	yes	—
127	C₇×#95	C₂³	1	4	yes	—
128	C₇×#95	C₂³	1	4	yes	—
129	C₇×#95	C₂³	1	4	yes	—
130	C₇×#95	C₂³	1	4	yes	—
131	C₇×#95	C₂³	1	4	yes	—
132	C₇×#95	C₂³	1	4	yes	—
133	C₇×#95	C₂³	1	4	yes	—
134	#2857	C₂×C₇	1	1	yes	characteristic, complemented
135	#2851	C₂×C₇	1	1	yes	characteristic, complemented
136	#2848	C₂×C₇	1	2	yes	—
137	#2652	C₂×C₇	1	4	yes	complemented
138	#2505	C₂×C₇	1	4	yes	complemented
139	#2496	C₂×C₇	1	2	yes	complemented
140	#2489	C₂×C₇	1	2	yes	complemented
141	#2480	C₂×C₇	1	4	yes	—
142	#2429	C₂×C₇	1	2	yes	complemented
143	#2421	C₂×C₇	1	2	yes	complemented
144	#2334	C₂×C₇	1	4	yes	complemented
145	C₄×C₂²⋊Q₈	C₂×C₇	1	2	yes	complemented
146	C₂×C₄×C₂²⋊C₄	C₂×C₇	1	1	yes	characteristic
147	C₂⁴×C₇	—	2	4	no	—
148	C₂⁴×C₇	—	2	2	no	—
149	C₂⁴×C₇	—	2	4	no	—
150	C₂⁴×C₇	—	4	1	no	—
151	C₂⁴×C₇	—	4	1	no	—
152	C₂⁴×C₇	C₂⁴	1	2	yes	—
153	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
154	C₂×C₇×Q₈	—	2	4	no	—
155	C₂×C₇×Q₈	—	2	4	no	—
156	C₂×C₇×Q₈	—	2	4	no	—
157	C₂²×C₄×C₇	—	2	2	no	—
158	C₂²×C₄×C₇	—	2	2	no	—
159	C₂²×C₄×C₇	—	2	2	no	—
160	C₂²×C₄×C₇	—	2	4	no	—
161	C₂²×C₄×C₇	—	2	4	no	—
162	C₂²×C₄×C₇	—	2	4	no	—
163	C₂²×C₄×C₇	—	2	4	no	—
164	C₂²×C₄×C₇	—	2	4	no	—
165	C₂²×C₄×C₇	—	2	4	no	—
166	C₂²×C₄×C₇	—	2	4	no	—
167	C₂²×C₄×C₇	—	2	4	no	—
168	C₂²×C₄×C₇	—	4	2	no	—
169	C₂²×C₄×C₇	—	4	2	no	—
170	C₂²×C₄×C₇	—	4	2	no	—
171	C₂²×C₄×C₇	—	4	2	no	—
172	C₂²×C₄×C₇	—	4	1	no	—
173	C₂²×C₄×C₇	—	4	1	no	—
174	C₂²×C₄×C₇	—	4	16	no	—
175	C₂²×C₄×C₇	#41	1	4	yes	—
176	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
187	C₇×C₄⋊C₄	—	2	4	no	—
188	C₇×C₄⋊C₄	—	2	8	no	—
189	C₇×C₄⋊C₄	—	2	8	no	—
190	C₇×C₄⋊C₄	—	2	8	no	—
191	C₇×C₄⋊C₄	—	2	8	no	—
192	C₇×C₄⋊C₄	—	2	4	no	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	4	4	no	—
195	C₇×C₄⋊C₄	—	4	4	no	—
196	C₇×C₄⋊C₄	—	4	4	no	—
197	C₇×C₄⋊C₄	—	4	4	no	—
198	C₇×C₄⋊C₄	—	4	4	no	—
199	C₇×C₂²⋊C₄	—	2	4	no	—
200	C₇×C₂²⋊C₄	—	2	8	no	—
201	C₇×C₂²⋊C₄	—	2	8	no	—
202	C₇×C₂²⋊C₄	—	2	8	no	—
203	C₇×C₂²⋊C₄	—	2	8	no	—
204	C₇×C₂²⋊C₄	—	2	4	no	—
205	C₇×C₂²⋊C₄	—	2	4	no	—
206	C₇×C₂²⋊C₄	—	4	4	no	—
207	C₇×C₂²⋊C₄	—	4	4	no	—
208	C₇×C₂²⋊C₄	#41	1	8	yes	—
209	C₄²×C₇	—	2	8	no	—
210	C₄²×C₇	—	2	4	no	—
211	C₄²×C₇	—	2	2	no	—
212	C₄²×C₇	—	2	2	no	—
213	C₄²×C₇	—	4	1	no	—
214	C₄²×C₇	—	4	2	no	—
215	C₄²×C₇	—	4	1	no	—
216	C₄²×C₇	—	4	4	no	—
217	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
218	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
219	C₂×#126	C₂²×C₇	1	4	yes	—
220	C₂×#125	C₂²×C₇	1	1	yes	characteristic, complemented
221	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
222	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
223	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
224	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	complemented
225	C₂²×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
226	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
227	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
228	#400	C₂²×C₇	1	4	yes	complemented
229	#400	C₂²×C₇	1	4	yes	complemented
230	#400	C₂²×C₇	1	4	yes	complemented
231	#398	C₂²×C₇	1	4	yes	complemented
232	#397	C₂²×C₇	1	4	yes	—
233	#397	C₂²×C₇	1	4	yes	—
234	#397	C₂²×C₇	1	4	yes	—
235	#396	C₂²×C₇	1	4	yes	—
236	#391	C₂²×C₇	1	2	yes	complemented
237	#391	C₂²×C₇	1	2	yes	complemented
238	#389	C₂²×C₇	1	4	yes	complemented
239	#389	C₂²×C₇	1	4	yes	complemented
240	#389	C₂²×C₇	1	2	yes	complemented
241	#389	C₂²×C₇	1	2	yes	complemented
242	#388	C₂²×C₇	1	4	yes	—
243	#388	C₂²×C₇	1	4	yes	—
244	#388	C₂²×C₇	1	4	yes	—
245	#387	C₂²×C₇	1	4	yes	complemented
246	#387	C₂²×C₇	1	4	yes	complemented
247	#387	C₂²×C₇	1	4	yes	complemented
248	#387	C₂²×C₇	1	4	yes	complemented
249	#385	C₂²×C₇	1	4	yes	—
250	#385	C₂²×C₇	1	4	yes	—
251	#385	C₂²×C₇	1	4	yes	—
252	#385	C₂²×C₇	1	4	yes	—
253	#385	C₂²×C₇	1	4	yes	—
254	#384	C₂²×C₇	1	1	yes	characteristic, complemented
255	#382	C₂²×C₇	1	4	yes	complemented
256	#382	C₂²×C₇	1	1	yes	characteristic, complemented
257	#380	C₂²×C₇	1	2	yes	—
258	#380	C₂²×C₇	1	2	yes	—
259	#380	C₂²×C₇	1	2	yes	—
260	#380	C₂²×C₇	1	4	yes	—
261	#379	C₂²×C₇	1	4	yes	—
262	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	complemented
263	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
264	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
265	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
266	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
267	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
268	C₄×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
269	C₂×#95	C₂²×C₇	1	2	yes	—
270	C₂³×C₇	—	2	2	no	—
271	C₂³×C₇	—	2	1	no	—
272	C₂³×C₇	—	2	1	no	—
273	C₂³×C₇	—	2	8	no	—
274	C₂³×C₇	—	2	2	no	—
275	C₂³×C₇	—	2	2	no	—
276	C₂³×C₇	—	2	2	no	—
277	C₂³×C₇	—	2	2	no	—
278	C₂³×C₇	—	2	2	no	—
279	C₂³×C₇	—	2	2	no	—
280	C₂³×C₇	—	4	8	no	—
281	C₂³×C₇	—	4	8	no	—
282	C₂³×C₇	—	4	4	no	—
283	C₂³×C₇	—	4	4	no	—
284	C₂³×C₇	C₂×#41	1	4	yes	—
285	C₂³×C₇	C₂×#41	1	2	yes	—
286	C₂³×C₇	2₋¹⁺⁴	1	4	yes	—
287	C₂³×C₇	C₂⁵	1	1	yes	characteristic
288	C₇×Q₈	—	4	8	no	—
289	C₂×C₄×C₇	—	2	2	no	—
290	C₂×C₄×C₇	—	2	2	no	—
291	C₂×C₄×C₇	—	2	1	no	—
292	C₂×C₄×C₇	—	2	1	no	—
293	C₂×C₄×C₇	—	2	2	no	—
294	C₂×C₄×C₇	—	2	2	no	—
295	C₂×C₄×C₇	—	2	8	no	—
296	C₂×C₄×C₇	—	2	4	no	—
297	C₂×C₄×C₇	—	2	4	no	—
298	C₂×C₄×C₇	—	2	2	no	—
299	C₂×C₄×C₇	—	2	2	no	—
300	C₂×C₄×C₇	—	2	4	no	—
301	C₂×C₄×C₇	—	2	4	no	—
302	C₂×C₄×C₇	—	2	4	no	—
303	C₂×C₄×C₇	—	2	4	no	—
304	C₂×C₄×C₇	—	2	2	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	2	no	—
307	C₂×C₄×C₇	—	2	4	no	—
308	C₂×C₄×C₇	—	2	4	no	—
309	C₂×C₄×C₇	—	2	4	no	—
310	C₂×C₄×C₇	—	2	4	no	—
311	C₂×C₄×C₇	—	2	2	no	—
312	C₂×C₄×C₇	—	2	2	no	—
313	C₂×C₄×C₇	—	2	4	no	—
314	C₂×C₄×C₇	—	2	4	no	—
315	C₂×C₄×C₇	—	2	2	no	—
316	C₂×C₄×C₇	—	2	2	no	—
317	C₂×C₄×C₇	—	4	8	no	—
318	C₂×C₄×C₇	—	4	8	no	—
319	C₂×C₄×C₇	—	4	8	no	—
320	C₂×C₄×C₇	—	4	4	no	—
321	C₂×C₄×C₇	—	4	8	no	—
322	C₂×C₄×C₇	C₂×#41	1	4	yes	—
323	C₂×C₄×C₇	C₂×#41	1	4	yes	—
324	C₂×C₄×C₇	C₂×#41	1	2	yes	—
325	C₂×C₄×C₇	C₂×#41	1	2	yes	—
326	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
327	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
328	C₂⁵	C₂³×C₇	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₇	1	2	yes	—
330	C₂³×C₄	—	2	4	no	—
331	C₂³×C₄	—	4	2	no	—
332	C₂³×C₄	—	4	2	no	—
333	C₂³×C₄	C₂³×C₇	1	2	yes	—
334	C₂³×C₄	C₂³×C₇	1	2	yes	—
335	C₂³×C₄	C₂³×C₇	1	2	yes	—
336	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₇	1	2	yes	—
339	C₄⋊Q₈	—	4	2	no	—
340	#126	—	2	16	no	—
341	#125	—	4	2	no	—
342	C₂²⋊Q₈	—	2	8	no	—
343	C₂²⋊Q₈	—	4	4	no	—
344	C₂²⋊Q₈	—	4	4	no	—
345	C₄×Q₈	—	4	4	no	—
346	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
348	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
349	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
350	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
357	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
358	C₂×C₂²⋊C₄	—	2	8	no	—
359	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₇	1	2	yes	—
369	C₂×C₄²	—	4	2	no	—
370	C₂×C₄²	—	4	2	no	—
371	C₂×C₄²	C₂³×C₇	1	2	yes	—
372	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
373	C₂×C₄²	C₂³×C₇	1	4	yes	—
374	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₇	1	4	yes	—
376	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₇	1	2	yes	—
379	#95	C₂³×C₇	1	2	yes	—
380	#95	C₂³×C₇	1	2	yes	—
381	#95	C₂³×C₇	1	2	yes	—
382	#95	C₂³×C₇	1	2	yes	—
383	#95	C₂³×C₇	1	4	yes	—
384	#95	C₂³×C₇	1	4	yes	—
385	#95	C₂³×C₇	1	4	yes	—
386	#95	C₂³×C₇	1	4	yes	—
387	#95	C₂³×C₇	1	4	yes	—
388	#95	C₂³×C₇	1	4	yes	—
389	#95	C₂³×C₇	1	4	yes	—
390	#95	C₂³×C₇	1	4	yes	—
391	#95	C₂³×C₇	1	4	yes	—
392	#95	C₂³×C₇	1	4	yes	—
393	#95	C₂³×C₇	1	4	yes	—
394	C₂²×C₇	—	2	2	no	—
395	C₂²×C₇	—	2	2	no	—
396	C₂²×C₇	—	2	4	no	—
397	C₂²×C₇	—	2	4	no	—
398	C₂²×C₇	—	2	4	no	—
399	C₂²×C₇	—	2	4	no	—
400	C₂²×C₇	—	2	4	no	—
401	C₂²×C₇	—	4	16	no	—
402	C₂²×C₇	—	4	2	no	—
403	C₂²×C₇	—	4	2	no	—
404	C₂²×C₇	—	4	2	no	—
405	C₂²×C₇	—	4	2	no	—
406	C₂²×C₇	#541	1	4	yes	—
407	C₂²×C₇	C₂²×#41	1	1	yes	characteristic
408	C₂²×C₇	C₂²×#41	1	2	yes	—
409	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
410	C₂²×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
411	C₂²×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
412	C₄×C₇	—	2	2	no	—
413	C₄×C₇	—	2	4	no	—
414	C₄×C₇	—	2	4	no	—
415	C₄×C₇	—	4	4	no	—
416	C₄×C₇	—	4	2	no	—
417	C₄×C₇	—	4	4	no	—
418	C₄×C₇	—	4	4	no	—
419	C₄×C₇	—	4	4	no	—
420	C₄×C₇	—	4	2	no	—
421	C₄×C₇	—	4	2	no	—
422	C₄×C₇	#538	1	4	yes	—
423	C₂⁴	—	2	4	no	—
424	C₂⁴	—	2	2	no	—
425	C₂⁴	—	2	4	no	—
426	C₂⁴	—	4	1	no	—
427	C₂⁴	—	4	1	no	—
428	C₂⁴	C₂⁴×C₇	1	2	yes	—
429	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	16	no	—
451	C₂²×C₄	C₇×#41	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
455	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
461	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	4	4	no	—
471	C₄⋊C₄	—	4	4	no	—
472	C₄⋊C₄	—	4	4	no	—
473	C₄⋊C₄	—	4	4	no	—
474	C₄⋊C₄	—	4	4	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	4	4	no	—
483	C₂²⋊C₄	—	4	4	no	—
484	C₂²⋊C₄	C₇×#41	1	8	yes	—
485	C₄²	—	2	8	no	—
486	C₄²	—	2	4	no	—
487	C₄²	—	2	2	no	—
488	C₄²	—	2	2	no	—
489	C₄²	—	4	1	no	—
490	C₄²	—	4	2	no	—
491	C₄²	—	4	1	no	—
492	C₄²	—	4	4	no	—
493	C₂×C₇	—	2	4	no	—
494	C₂×C₇	—	4	4	no	—
495	C₂×C₇	#3444	1	1	yes	characteristic
496	C₂×C₇	C₂×#538	1	1	yes	characteristic
497	C₂×C₇	C₂×#541	1	1	yes	characteristic
498	C₂×C₇	#3520	1	2	yes	—
499	C₂×C₇	#3520	1	2	yes	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	8	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	4	8	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	C₂×C₇×#41	1	4	yes	—
515	C₂³	C₂×C₇×#41	1	2	yes	—
516	C₂³	C₇×2₋¹⁺⁴	1	4	yes	—
517	C₂³	C₂⁵×C₇	1	1	yes	characteristic
518	Q₈	—	4	8	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	8	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	4	8	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	C₂×C₇×#41	1	4	yes	—
553	C₂×C₄	C₂×C₇×#41	1	4	yes	—
554	C₂×C₄	C₂×C₇×#41	1	2	yes	—
555	C₂×C₄	C₂×C₇×#41	1	2	yes	—
556	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
557	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
558	C₇	#35086	1	1	yes	characteristic, direct product factor, Sylow
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	16	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	2	no	—
571	C₂²	C₇×#541	1	4	yes	—
572	C₂²	C₂²×C₇×#41	1	1	yes	characteristic
573	C₂²	C₂²×C₇×#41	1	2	yes	—
574	C₂²	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
575	C₂²	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
576	C₂²	C₂×C₇×2₋¹⁺⁴	1	2	yes	—
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	C₇×#538	1	4	yes	—
588	C₂	—	2	4	no	—
589	C₂	—	4	4	no	—
590	C₂	C₇×#3444	1	1	yes	characteristic
591	C₂	C₂×C₇×#538	1	1	yes	characteristic
592	C₂	C₂×C₇×#541	1	1	yes	characteristic
593	C₂	C₇×#3520	1	2	yes	—
594	C₂	C₇×#3520	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#2857	C₂	semidirect
2. C₇×#2851	C₂	semidirect
4. C₇×#2652	C₂	semidirect
5. C₇×#2505	C₂	semidirect
6. C₇×#2496	C₂	semidirect
7. C₇×#2489	C₂	semidirect
9. C₇×#2429	C₂	semidirect
10. C₇×#2421	C₂	semidirect
11. C₇×#2334	C₂	semidirect
12. C₄×C₇×C₂²⋊Q₈	C₂	semidirect
15. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₇×#125	C₂²	semidirect
21. C₂×C₄×C₇×Q₈	C₂²	semidirect
25. C₇×#400	C₂²	semidirect
26. C₇×#400	C₂²	semidirect
27. C₇×#400	C₂²	semidirect
28. C₇×#398	C₂²	semidirect
33. C₇×#391	C₂²	semidirect
34. C₇×#391	C₂²	semidirect
35. C₇×#389	C₂²	semidirect
36. C₇×#389	C₂²	semidirect
37. C₇×#389	C₂²	semidirect
38. C₇×#389	C₂²	semidirect
42. C₇×#387	C₂²	semidirect
43. C₇×#387	C₂²	semidirect
44. C₇×#387	C₂²	semidirect
45. C₇×#387	C₂²	semidirect
51. C₇×#384	C₂²	semidirect
52. C₇×#382	C₂²	semidirect
53. C₇×#382	C₂²	semidirect
59. C₄×C₇×C₄⋊C₄	C₂²	semidirect
60. C₄×C₇×C₄⋊C₄	C₂²	semidirect
67. #35086	C₇	direct (N × H)
134. #2857	C₂×C₇	semidirect
135. #2851	C₂×C₇	semidirect
137. #2652	C₂×C₇	semidirect
138. #2505	C₂×C₇	semidirect
139. #2496	C₂×C₇	semidirect
140. #2489	C₂×C₇	semidirect
142. #2429	C₂×C₇	semidirect
143. #2421	C₂×C₇	semidirect
144. #2334	C₂×C₇	semidirect
145. C₄×C₂²⋊Q₈	C₂×C₇	semidirect
218. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
220. C₂×#125	C₂²×C₇	semidirect
224. C₂×C₄×Q₈	C₂²×C₇	semidirect
228. #400	C₂²×C₇	semidirect
229. #400	C₂²×C₇	semidirect
230. #400	C₂²×C₇	semidirect
231. #398	C₂²×C₇	semidirect
236. #391	C₂²×C₇	semidirect
237. #391	C₂²×C₇	semidirect
238. #389	C₂²×C₇	semidirect
239. #389	C₂²×C₇	semidirect
240. #389	C₂²×C₇	semidirect
241. #389	C₂²×C₇	semidirect
245. #387	C₂²×C₇	semidirect
246. #387	C₂²×C₇	semidirect
247. #387	C₂²×C₇	semidirect
248. #387	C₂²×C₇	semidirect
254. #384	C₂²×C₇	semidirect
255. #382	C₂²×C₇	semidirect
256. #382	C₂²×C₇	semidirect
262. C₄×C₄⋊C₄	C₂²×C₇	semidirect
263. C₄×C₄⋊C₄	C₂²×C₇	semidirect
558. C₇	#35086	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2857	2
C₇×#2851	2
C₇×#2848	2
C₇×#2652	2
C₇×#2505	2
C₇×#2496	2
C₇×#2489	2
C₇×#2480	2
C₇×#2429	2
C₇×#2421	2
C₇×#2334	2
C₄×C₇×C₂²⋊Q₈	2
C₂×C₄×C₇×C₂²⋊C₄	2
#35086	7

Subgroups — C7×#35086 #13212302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#35086 #13212302