Subgroups — C₇×#60915 #13238131

Order 1792.

1790 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	2	yes	maximal, complemented
2	C₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₇×#3262	C₂	1	2	yes	maximal, complemented
4	C₇×#3261	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#3260	C₂	1	2	yes	maximal, complemented
6	C₇×#3259	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3258	C₂	1	4	yes	maximal, complemented
8	C₇×#3257	C₂	1	4	yes	maximal, complemented
9	C₇×#3076	C₂	1	4	yes	maximal, complemented
10	C₇×#3054	C₂	1	2	yes	maximal, complemented
11	C₇×#3050	C₂	1	2	yes	maximal, complemented
12	C₇×#3035	C₂	1	4	yes	maximal, complemented
13	C₂×C₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₇×#135	C₂²	1	2	yes	—
15	C₂×C₇×#135	C₂²	1	2	yes	—
16	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#542	C₂²	1	4	yes	—
19	C₇×#542	C₂²	1	4	yes	—
20	C₇×#541	C₂²	1	2	yes	—
21	C₇×#541	C₂²	1	2	yes	—
22	C₇×#540	C₂²	1	2	yes	—
23	C₇×#540	C₂²	1	2	yes	—
24	C₇×#539	—	2	4	no	—
25	C₇×#538	—	2	4	no	—
26	C₇×#533	C₂²	1	2	yes	—
27	C₇×#532	C₂²	1	2	yes	—
28	C₇×#529	—	2	4	no	—
29	C₇×#529	—	2	4	no	—
30	C₇×#525	C₂²	1	4	yes	—
31	C₄×C₇×#41	C₂²	1	4	yes	—
32	C₂×C₇×#117	C₂²	1	1	yes	characteristic
33	C₇×#479	C₂²	1	4	yes	complemented
34	C₇×#479	C₂²	1	4	yes	complemented
35	C₇×#478	C₂²	1	4	yes	complemented
36	C₇×#478	C₂²	1	4	yes	complemented
37	C₇×#477	C₂²	1	4	yes	complemented
38	C₇×#476	C₂²	1	4	yes	complemented
39	C₇×#475	C₂²	1	4	yes	complemented
40	C₇×#474	C₂²	1	4	yes	complemented
41	C₇×#470	C₂²	1	2	yes	—
42	C₇×#470	C₂²	1	2	yes	—
43	C₇×#469	C₂²	1	2	yes	—
44	C₇×#469	C₂²	1	2	yes	—
45	C₇×#468	C₂²	1	4	yes	—
46	C₇×#468	C₂²	1	4	yes	—
47	C₇×#467	C₂²	1	2	yes	—
48	C₇×#467	C₂²	1	2	yes	—
49	C₇×#466	C₂²	1	2	yes	—
50	C₇×#466	C₂²	1	2	yes	—
51	C₇×#465	C₂²	1	4	yes	—
52	C₇×#465	C₂²	1	4	yes	—
53	C₇×#464	C₂²	1	4	yes	complemented
54	C₇×#464	C₂²	1	4	yes	complemented
55	C₇×#463	C₂²	1	4	yes	complemented
56	C₇×#463	C₂²	1	4	yes	complemented
57	C₇×#462	C₂²	1	4	yes	complemented
58	C₇×#461	C₂²	1	4	yes	complemented
59	C₇×#460	C₂²	1	4	yes	complemented
60	C₇×#459	C₂²	1	4	yes	complemented
61	C₇×#427	C₂²	1	2	yes	—
62	C₇×#427	C₂²	1	2	yes	—
63	C₇×#424	C₂²	1	4	yes	—
64	C₇×#424	C₂²	1	1	yes	characteristic
65	C₇×#424	C₂²	1	1	yes	characteristic
66	C₇×#424	C₂²	1	1	yes	characteristic
67	C₇×#424	C₂²	1	1	yes	characteristic
68	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
69	C₇×#416	C₂²	1	4	yes	—
70	C₇×#416	C₂²	1	4	yes	—
71	#60915	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	C₂³	1	4	yes	—
73	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
74	C₇×#135	—	4	4	no	—
75	C₇×#135	—	4	4	no	—
76	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
77	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
78	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
79	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
80	C₂×C₇×D₈	C₂³	1	2	yes	—
81	C₂×C₇×D₈	C₂³	1	2	yes	—
82	C₂×C₇×M₁₆	—	4	2	no	—
83	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₇×M₁₆	C₂³	1	2	yes	—
85	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
86	C₂²×C₇×C₈	—	4	2	no	—
87	C₂²×C₇×C₈	C₂³	1	2	yes	—
88	C₂²×C₇×C₈	C₂³	1	2	yes	—
89	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
92	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
93	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
94	C₇×#127	C₂³	1	2	yes	—
95	C₇×#126	—	2	4	no	—
96	C₇×#126	—	2	4	no	—
97	C₇×#126	—	2	4	no	—
98	C₇×#126	—	2	4	no	—
99	C₇×#125	—	2	4	no	—
100	C₇×#125	—	2	4	no	—
101	C₇×#125	C₂³	1	2	yes	—
102	C₇×#125	C₂³	1	2	yes	—
103	C₇×#124	—	2	4	no	—
104	C₇×#124	—	2	4	no	—
105	C₇×#124	C₂³	1	4	yes	—
106	C₇×#123	—	2	4	no	—
107	C₇×#123	—	2	4	no	—
108	C₇×#123	—	2	4	no	—
109	C₇×#123	—	2	4	no	—
110	C₇×#123	—	2	4	no	—
111	C₇×C₂²⋊Q₈	—	2	4	no	—
112	C₇×C₂²⋊Q₈	—	2	4	no	—
113	C₇×C₂²⋊Q₈	—	2	4	no	—
114	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
115	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
116	C₇×#121	—	2	4	no	—
117	C₇×#121	—	2	4	no	—
118	C₇×#121	—	2	4	no	—
119	C₇×#121	C₂³	1	4	yes	—
120	C₇×#121	C₂³	1	4	yes	—
121	C₇×#120	—	2	4	no	—
122	C₄×C₇×Q₈	—	2	4	no	—
123	C₄×C₇×Q₈	C₂³	1	4	yes	—
124	C₄×C₇×Q₈	C₂³	1	4	yes	—
125	C₄×C₇×D₄	—	2	4	no	—
126	C₄×C₇×D₄	—	2	4	no	—
127	C₄×C₇×D₄	—	2	4	no	—
128	C₄×C₇×D₄	—	2	4	no	—
129	C₄×C₇×D₄	—	2	4	no	—
130	C₄×C₇×D₄	C₂³	1	4	yes	—
131	C₄×C₇×D₄	C₂³	1	4	yes	—
132	C₇×#117	—	2	4	no	—
133	C₇×#117	D₄	1	2	yes	—
134	C₇×#117	D₄	1	2	yes	—
135	C₇×#117	C₂³	1	4	yes	—
136	C₇×#117	C₂³	1	1	yes	characteristic
137	C₇×#117	C₂³	1	2	yes	—
138	C₇×#117	C₂³	1	1	yes	characteristic
139	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
140	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
141	C₂×C₄²×C₇	—	2	4	no	—
142	C₂×C₄²×C₇	C₂³	1	2	yes	—
143	C₇×#107	C₂³	1	2	yes	—
144	C₇×#107	C₂³	1	2	yes	—
145	C₇×#107	C₂³	1	4	yes	—
146	C₇×#106	C₂³	1	2	yes	—
147	C₇×#106	C₂³	1	2	yes	—
148	C₇×#106	C₂³	1	4	yes	—
149	C₇×C₄⋊C₈	C₂³	1	4	yes	—
150	C₇×C₄⋊C₈	C₂³	1	4	yes	—
151	C₇×C₄⋊C₈	C₂³	1	4	yes	—
152	C₇×C₄⋊C₈	C₂³	1	4	yes	—
153	C₇×#103	C₂³	1	4	yes	—
154	C₇×#103	C₂³	1	4	yes	—
155	C₇×#103	C₂³	1	4	yes	—
156	C₇×#103	C₂³	1	4	yes	—
157	C₇×#102	C₂³	1	4	yes	—
158	C₇×#102	C₂³	1	4	yes	—
159	C₇×#102	C₂³	1	4	yes	—
160	C₇×#102	C₂³	1	4	yes	—
161	#3455	C₂×C₇	1	2	yes	complemented
162	#3263	C₂×C₇	1	2	yes	complemented
163	#3262	C₂×C₇	1	2	yes	complemented
164	#3261	C₂×C₇	1	1	yes	characteristic, complemented
165	#3260	C₂×C₇	1	2	yes	complemented
166	#3259	C₂×C₇	1	1	yes	characteristic, complemented
167	#3258	C₂×C₇	1	4	yes	complemented
168	#3257	C₂×C₇	1	4	yes	complemented
169	#3076	C₂×C₇	1	4	yes	complemented
170	#3054	C₂×C₇	1	2	yes	complemented
171	#3050	C₂×C₇	1	2	yes	complemented
172	#3035	C₂×C₇	1	4	yes	complemented
173	C₂×#424	C₂×C₇	1	1	yes	characteristic, complemented
174	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
175	C₇×#41	—	2	8	no	—
176	C₇×#41	—	2	8	no	—
177	C₂×C₇×Q₈	—	2	4	no	—
178	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
179	C₂×C₇×D₄	—	2	4	no	—
180	C₂×C₇×D₄	—	2	4	no	—
181	C₂×C₇×D₄	—	2	4	no	—
182	C₂×C₇×D₄	—	2	4	no	—
183	C₂×C₇×D₄	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₇	—	2	2	no	—
185	C₂²×C₄×C₇	—	2	2	no	—
186	C₂²×C₄×C₇	—	2	4	no	—
187	C₂²×C₄×C₇	—	2	4	no	—
188	C₂²×C₄×C₇	—	2	4	no	—
189	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
193	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
196	C₇×Q₁₆	—	4	2	no	—
197	C₇×Q₁₆	—	4	2	no	—
198	C₇×SD₁₆	—	4	4	no	—
199	C₇×SD₁₆	—	4	4	no	—
200	C₇×D₈	—	4	2	no	—
201	C₇×D₈	—	4	2	no	—
202	C₇×M₁₆	—	4	1	no	—
203	C₇×M₁₆	—	4	2	no	—
204	C₇×M₁₆	—	4	1	no	—
205	C₂×C₇×C₈	—	2	4	no	—
206	C₂×C₇×C₈	—	2	4	no	—
207	C₂×C₇×C₈	—	4	1	no	—
208	C₂×C₇×C₈	—	4	2	no	—
209	C₂×C₇×C₈	—	4	1	no	—
210	C₂×C₇×C₈	C₂⁴	1	2	yes	—
211	C₂×C₇×C₈	C₂⁴	1	2	yes	—
212	C₂×C₇×C₈	C₂⁴	1	2	yes	—
213	C₂×C₇×C₈	C₂⁴	1	2	yes	—
214	C₇×C₄⋊C₄	—	2	4	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	4	no	—
217	C₇×C₄⋊C₄	—	2	4	no	—
218	C₇×C₄⋊C₄	—	2	4	no	—
219	C₇×C₄⋊C₄	—	2	4	no	—
220	C₇×C₄⋊C₄	—	2	4	no	—
221	C₇×C₄⋊C₄	—	2	4	no	—
222	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
223	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
224	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
225	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
226	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
227	C₇×C₂²⋊C₄	—	2	4	no	—
228	C₇×C₂²⋊C₄	—	2	4	no	—
229	C₇×C₂²⋊C₄	—	2	4	no	—
230	C₇×C₂²⋊C₄	—	2	4	no	—
231	C₇×C₂²⋊C₄	—	2	4	no	—
232	C₇×C₂²⋊C₄	—	2	4	no	—
233	C₇×C₂²⋊C₄	—	2	4	no	—
234	C₇×C₂²⋊C₄	—	2	4	no	—
235	C₇×C₂²⋊C₄	—	2	4	no	—
236	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
237	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
238	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
239	C₄²×C₇	—	2	4	no	—
240	C₄²×C₇	—	2	4	no	—
241	C₄²×C₇	—	2	4	no	—
242	C₄²×C₇	C₂×D₄	1	4	yes	—
243	C₄²×C₇	C₂⁴	1	2	yes	—
244	C₄²×C₇	C₂⁴	1	2	yes	—
245	C₂×#135	C₂²×C₇	1	2	yes	—
246	C₂×#135	C₂²×C₇	1	2	yes	—
247	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
248	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
249	#542	C₂²×C₇	1	4	yes	—
250	#542	C₂²×C₇	1	4	yes	—
251	#541	C₂²×C₇	1	2	yes	—
252	#541	C₂²×C₇	1	2	yes	—
253	#540	C₂²×C₇	1	2	yes	—
254	#540	C₂²×C₇	1	2	yes	—
255	#539	—	2	4	no	—
256	#538	—	2	4	no	—
257	#533	C₂²×C₇	1	2	yes	—
258	#532	C₂²×C₇	1	2	yes	—
259	#529	—	2	4	no	—
260	#529	—	2	4	no	—
261	#525	C₂²×C₇	1	4	yes	—
262	C₄×#41	C₂²×C₇	1	4	yes	—
263	C₂×#117	C₂²×C₇	1	1	yes	characteristic
264	#479	C₂²×C₇	1	4	yes	complemented
265	#479	C₂²×C₇	1	4	yes	complemented
266	#478	C₂²×C₇	1	4	yes	complemented
267	#478	C₂²×C₇	1	4	yes	complemented
268	#477	C₂²×C₇	1	4	yes	complemented
269	#476	C₂²×C₇	1	4	yes	complemented
270	#475	C₂²×C₇	1	4	yes	complemented
271	#474	C₂²×C₇	1	4	yes	complemented
272	#470	C₂²×C₇	1	2	yes	—
273	#470	C₂²×C₇	1	2	yes	—
274	#469	C₂²×C₇	1	2	yes	—
275	#469	C₂²×C₇	1	2	yes	—
276	#468	C₂²×C₇	1	4	yes	—
277	#468	C₂²×C₇	1	4	yes	—
278	#467	C₂²×C₇	1	2	yes	—
279	#467	C₂²×C₇	1	2	yes	—
280	#466	C₂²×C₇	1	2	yes	—
281	#466	C₂²×C₇	1	2	yes	—
282	#465	C₂²×C₇	1	4	yes	—
283	#465	C₂²×C₇	1	4	yes	—
284	#464	C₂²×C₇	1	4	yes	complemented
285	#464	C₂²×C₇	1	4	yes	complemented
286	#463	C₂²×C₇	1	4	yes	complemented
287	#463	C₂²×C₇	1	4	yes	complemented
288	#462	C₂²×C₇	1	4	yes	complemented
289	#461	C₂²×C₇	1	4	yes	complemented
290	#460	C₂²×C₇	1	4	yes	complemented
291	#459	C₂²×C₇	1	4	yes	complemented
292	#427	C₂²×C₇	1	2	yes	—
293	#427	C₂²×C₇	1	2	yes	—
294	#424	C₂²×C₇	1	4	yes	—
295	#424	C₂²×C₇	1	1	yes	characteristic
296	#424	C₂²×C₇	1	1	yes	characteristic
297	#424	C₂²×C₇	1	1	yes	characteristic
298	#424	C₂²×C₇	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
300	#416	C₂²×C₇	1	4	yes	—
301	#416	C₂²×C₇	1	4	yes	—
302	C₂³×C₇	—	2	2	no	—
303	C₂³×C₇	—	2	4	no	—
304	C₂³×C₇	—	4	2	no	—
305	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
306	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
308	C₇×Q₈	—	2	4	no	—
309	C₇×Q₈	—	2	4	no	—
310	C₇×Q₈	—	8	4	no	—
311	C₇×D₄	—	2	4	no	—
312	C₇×D₄	—	2	4	no	—
313	C₇×D₄	—	4	8	no	—
314	C₇×D₄	—	8	4	no	—
315	C₇×D₄	—	8	4	no	—
316	C₇×D₄	—	8	4	no	—
317	C₂×C₄×C₇	—	2	2	no	—
318	C₂×C₄×C₇	—	2	2	no	—
319	C₂×C₄×C₇	—	2	2	no	—
320	C₂×C₄×C₇	—	2	1	no	—
321	C₂×C₄×C₇	—	2	1	no	—
322	C₂×C₄×C₇	—	2	1	no	—
323	C₂×C₄×C₇	—	2	1	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	4	no	—
326	C₂×C₄×C₇	—	2	4	no	—
327	C₂×C₄×C₇	—	4	4	no	—
328	C₂×C₄×C₇	—	4	8	no	—
329	C₂×C₄×C₇	—	8	4	no	—
330	C₂×C₄×C₇	—	8	4	no	—
331	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
332	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
333	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
337	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
338	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
339	C₇×C₈	—	2	2	no	—
340	C₇×C₈	—	2	2	no	—
341	C₇×C₈	—	2	2	no	—
342	C₇×C₈	—	2	2	no	—
343	C₂×#41	C₂³×C₇	1	4	yes	—
344	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
345	#135	—	4	4	no	—
346	#135	—	4	4	no	—
347	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
348	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
349	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
350	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
351	C₂×D₈	C₂³×C₇	1	2	yes	—
352	C₂×D₈	C₂³×C₇	1	2	yes	—
353	C₂×M₁₆	—	4	2	no	—
354	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
355	C₂×M₁₆	C₂³×C₇	1	2	yes	—
356	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
357	C₂²×C₈	—	4	2	no	—
358	C₂²×C₈	C₂³×C₇	1	2	yes	—
359	C₂²×C₈	C₂³×C₇	1	2	yes	—
360	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₇	1	2	yes	—
362	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
363	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
364	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
365	#127	C₂³×C₇	1	2	yes	—
366	#126	—	2	4	no	—
367	#126	—	2	4	no	—
368	#126	—	2	4	no	—
369	#126	—	2	4	no	—
370	#125	—	2	4	no	—
371	#125	—	2	4	no	—
372	#125	C₂³×C₇	1	2	yes	—
373	#125	C₂³×C₇	1	2	yes	—
374	#124	—	2	4	no	—
375	#124	—	2	4	no	—
376	#124	C₂³×C₇	1	4	yes	—
377	#123	—	2	4	no	—
378	#123	—	2	4	no	—
379	#123	—	2	4	no	—
380	#123	—	2	4	no	—
381	#123	—	2	4	no	—
382	C₂²⋊Q₈	—	2	4	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
386	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	4	no	—
390	#121	C₂³×C₇	1	4	yes	—
391	#121	C₂³×C₇	1	4	yes	—
392	#120	—	2	4	no	—
393	C₄×Q₈	—	2	4	no	—
394	C₄×Q₈	C₂³×C₇	1	4	yes	—
395	C₄×Q₈	C₂³×C₇	1	4	yes	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	C₂³×C₇	1	4	yes	—
402	C₄×D₄	C₂³×C₇	1	4	yes	—
403	#117	—	2	4	no	—
404	#117	C₇×D₄	1	2	yes	—
405	#117	C₇×D₄	1	2	yes	—
406	#117	C₂³×C₇	1	4	yes	—
407	#117	C₂³×C₇	1	1	yes	characteristic
408	#117	C₂³×C₇	1	2	yes	—
409	#117	C₂³×C₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
411	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
412	C₂×C₄²	—	2	4	no	—
413	C₂×C₄²	C₂³×C₇	1	2	yes	—
414	#107	C₂³×C₇	1	2	yes	—
415	#107	C₂³×C₇	1	2	yes	—
416	#107	C₂³×C₇	1	4	yes	—
417	#106	C₂³×C₇	1	4	yes	—
418	#106	C₂³×C₇	1	2	yes	—
419	#106	C₂³×C₇	1	2	yes	—
420	C₄⋊C₈	C₂³×C₇	1	4	yes	—
421	C₄⋊C₈	C₂³×C₇	1	4	yes	—
422	C₄⋊C₈	C₂³×C₇	1	4	yes	—
423	C₄⋊C₈	C₂³×C₇	1	4	yes	—
424	#103	C₂³×C₇	1	4	yes	—
425	#103	C₂³×C₇	1	4	yes	—
426	#103	C₂³×C₇	1	4	yes	—
427	#103	C₂³×C₇	1	4	yes	—
428	#102	C₂³×C₇	1	4	yes	—
429	#102	C₂³×C₇	1	4	yes	—
430	#102	C₂³×C₇	1	4	yes	—
431	#102	C₂³×C₇	1	4	yes	—
432	C₂²×C₇	—	2	2	no	—
433	C₂²×C₇	—	2	2	no	—
434	C₂²×C₇	—	2	2	no	—
435	C₂²×C₇	—	2	1	no	—
436	C₂²×C₇	—	2	1	no	—
437	C₂²×C₇	—	4	4	no	—
438	C₂²×C₇	—	4	4	no	—
439	C₂²×C₇	—	4	4	no	—
440	C₂²×C₇	—	4	4	no	—
441	C₂²×C₇	#537	1	2	yes	—
442	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
443	C₄×C₇	—	2	4	no	—
444	C₄×C₇	—	2	4	no	—
445	C₄×C₇	—	2	1	no	—
446	C₄×C₇	—	2	1	no	—
447	C₄×C₇	—	4	4	no	—
448	C₄×C₇	—	4	4	no	—
449	C₄×C₇	—	4	4	no	—
450	C₄×C₇	#537	1	2	yes	—
451	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
452	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
453	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
454	#41	—	2	8	no	—
455	#41	—	2	8	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	C₂⁴×C₇	1	4	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
469	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
470	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
471	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
475	Q₁₆	—	4	2	no	—
476	Q₁₆	—	4	2	no	—
477	SD₁₆	—	4	4	no	—
478	SD₁₆	—	4	4	no	—
479	D₈	—	4	2	no	—
480	D₈	—	4	2	no	—
481	M₁₆	—	4	1	no	—
482	M₁₆	—	4	2	no	—
483	M₁₆	—	4	1	no	—
484	C₂×C₈	—	2	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	—	4	1	no	—
487	C₂×C₈	—	4	2	no	—
488	C₂×C₈	—	4	1	no	—
489	C₂×C₈	C₂⁴×C₇	1	2	yes	—
490	C₂×C₈	C₂⁴×C₇	1	2	yes	—
491	C₂×C₈	C₂⁴×C₇	1	2	yes	—
492	C₂×C₈	C₂⁴×C₇	1	2	yes	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
502	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
503	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
504	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
505	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
516	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
517	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
518	C₄²	—	2	4	no	—
519	C₄²	—	2	4	no	—
520	C₄²	—	2	4	no	—
521	C₄²	C₂×C₇×D₄	1	4	yes	—
522	C₄²	C₂⁴×C₇	1	2	yes	—
523	C₄²	C₂⁴×C₇	1	2	yes	—
524	C₂×C₇	—	2	2	no	—
525	C₂×C₇	—	4	1	no	—
526	C₂×C₇	—	4	1	no	—
527	C₂×C₇	—	8	4	no	—
528	C₂×C₇	C₂×#537	1	1	yes	characteristic
529	C₂×C₇	#3584	1	2	yes	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	4	2	no	—
533	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
534	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
535	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	8	4	no	—
539	D₄	—	2	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	4	8	no	—
542	D₄	—	8	4	no	—
543	D₄	—	8	4	no	—
544	D₄	—	8	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	—	8	4	no	—
559	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
560	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
561	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
563	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
564	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
565	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
566	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
567	C₈	—	2	2	no	—
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	2	2	no	—
571	C₇	#60915	1	1	yes	characteristic, direct product factor, Sylow
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	C₇×#537	1	2	yes	—
582	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
583	C₄	—	2	4	no	—
584	C₄	—	2	4	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	4	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	4	no	—
590	C₄	C₇×#537	1	2	yes	—
591	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
592	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
593	C₂	—	2	2	no	—
594	C₂	—	4	1	no	—
595	C₂	—	4	1	no	—
596	C₂	—	8	4	no	—
597	C₂	C₂×C₇×#537	1	1	yes	characteristic
598	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3455	C₂	semidirect
2. C₇×#3263	C₂	semidirect
3. C₇×#3262	C₂	semidirect
4. C₇×#3261	C₂	semidirect
5. C₇×#3260	C₂	semidirect
6. C₇×#3259	C₂	semidirect
7. C₇×#3258	C₂	semidirect
8. C₇×#3257	C₂	semidirect
9. C₇×#3076	C₂	semidirect
10. C₇×#3054	C₂	semidirect
11. C₇×#3050	C₂	semidirect
12. C₇×#3035	C₂	semidirect
13. C₂×C₇×#424	C₂	semidirect
33. C₇×#479	C₂²	semidirect
34. C₇×#479	C₂²	semidirect
35. C₇×#478	C₂²	semidirect
36. C₇×#478	C₂²	semidirect
37. C₇×#477	C₂²	semidirect
38. C₇×#476	C₂²	semidirect
39. C₇×#475	C₂²	semidirect
40. C₇×#474	C₂²	semidirect
53. C₇×#464	C₂²	semidirect
54. C₇×#464	C₂²	semidirect
55. C₇×#463	C₂²	semidirect
56. C₇×#463	C₂²	semidirect
57. C₇×#462	C₂²	semidirect
58. C₇×#461	C₂²	semidirect
59. C₇×#460	C₂²	semidirect
60. C₇×#459	C₂²	semidirect
71. #60915	C₇	direct (N × H)
161. #3455	C₂×C₇	semidirect
162. #3263	C₂×C₇	semidirect
163. #3262	C₂×C₇	semidirect
164. #3261	C₂×C₇	semidirect
165. #3260	C₂×C₇	semidirect
166. #3259	C₂×C₇	semidirect
167. #3258	C₂×C₇	semidirect
168. #3257	C₂×C₇	semidirect
169. #3076	C₂×C₇	semidirect
170. #3054	C₂×C₇	semidirect
171. #3050	C₂×C₇	semidirect
172. #3035	C₂×C₇	semidirect
173. C₂×#424	C₂×C₇	semidirect
264. #479	C₂²×C₇	semidirect
265. #479	C₂²×C₇	semidirect
266. #478	C₂²×C₇	semidirect
267. #478	C₂²×C₇	semidirect
268. #477	C₂²×C₇	semidirect
269. #476	C₂²×C₇	semidirect
270. #475	C₂²×C₇	semidirect
271. #474	C₂²×C₇	semidirect
284. #464	C₂²×C₇	semidirect
285. #464	C₂²×C₇	semidirect
286. #463	C₂²×C₇	semidirect
287. #463	C₂²×C₇	semidirect
288. #462	C₂²×C₇	semidirect
289. #461	C₂²×C₇	semidirect
290. #460	C₂²×C₇	semidirect
291. #459	C₂²×C₇	semidirect
571. C₇	#60915	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×#3455	2
C₇×#3263	2
C₇×#3262	2
C₇×#3261	2
C₇×#3260	2
C₇×#3259	2
C₇×#3258	2
C₇×#3257	2
C₇×#3076	2
C₇×#3054	2
C₇×#3050	2
C₇×#3035	2
C₂×C₇×#424	2
#60915	7

Subgroups — C7×#60915 #13238131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60915 #13238131