Subgroups — C₇×#60939 #13238155

Order 1792.

1758 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3495	C₂	1	2	yes	maximal, complemented
2	C₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₇×#3261	C₂	1	2	yes	maximal, complemented
4	C₇×#3258	C₂	1	4	yes	maximal, complemented
5	C₇×#3256	C₂	1	2	yes	maximal, complemented
6	C₇×#3255	C₂	1	2	yes	maximal, complemented
7	C₇×#3252	C₂	1	4	yes	maximal, complemented
8	C₇×#3084	C₂	1	2	yes	maximal, complemented
9	C₇×#3082	C₂	1	2	yes	maximal, complemented
10	C₇×#3056	C₂	1	2	yes	maximal, complemented
11	C₇×#3054	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#3046	C₂	1	2	yes	maximal, complemented
14	C₇×#3044	C₂	1	2	yes	maximal, complemented
15	C₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#137	C₂²	1	2	yes	—
17	C₂×C₇×#135	C₂²	1	1	yes	characteristic
18	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
21	C₇×#556	—	2	2	no	—
22	C₇×#554	—	2	2	no	—
23	C₇×#552	—	2	4	no	—
24	C₇×#552	—	2	2	no	—
25	C₇×#552	—	2	2	no	—
26	C₇×#551	—	2	4	no	—
27	C₇×#544	C₂²	1	2	yes	—
28	C₇×#542	C₂²	1	4	yes	—
29	C₇×#541	C₂²	1	4	yes	—
30	C₇×#536	C₂²	1	2	yes	—
31	C₇×#533	C₂²	1	4	yes	—
32	C₇×#529	C₂²	1	4	yes	—
33	C₇×#525	C₂²	1	2	yes	—
34	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₇×#520	C₂²	1	2	yes	—
36	C₇×#519	C₂²	1	2	yes	—
37	C₂×C₇×#117	C₂²	1	1	yes	characteristic
38	C₇×#479	C₂²	1	4	yes	complemented
39	C₇×#479	C₂²	1	4	yes	complemented
40	C₇×#479	C₂²	1	4	yes	complemented
41	C₇×#478	C₂²	1	4	yes	complemented
42	C₇×#477	C₂²	1	4	yes	complemented
43	C₇×#476	C₂²	1	4	yes	complemented
44	C₇×#475	C₂²	1	4	yes	complemented
45	C₇×#475	C₂²	1	4	yes	complemented
46	C₇×#470	C₂²	1	2	yes	—
47	C₇×#470	C₂²	1	4	yes	—
48	C₇×#470	C₂²	1	1	yes	characteristic
49	C₇×#470	C₂²	1	2	yes	—
50	C₇×#470	C₂²	1	2	yes	—
51	C₇×#470	C₂²	1	1	yes	characteristic
52	C₇×#470	C₂²	1	2	yes	—
53	C₇×#469	C₂²	1	1	yes	characteristic
54	C₇×#469	C₂²	1	1	yes	characteristic
55	C₇×#468	C₂²	1	2	yes	—
56	C₇×#468	C₂²	1	4	yes	—
57	C₇×#468	C₂²	1	2	yes	—
58	C₇×#467	C₂²	1	4	yes	—
59	C₇×#465	C₂²	1	4	yes	—
60	C₇×#464	C₂²	1	4	yes	complemented
61	C₇×#464	C₂²	1	4	yes	complemented
62	C₇×#464	C₂²	1	4	yes	complemented
63	C₇×#463	C₂²	1	4	yes	complemented
64	C₇×#462	C₂²	1	4	yes	complemented
65	C₇×#462	C₂²	1	4	yes	complemented
66	C₇×#462	C₂²	1	4	yes	complemented
67	C₇×#461	C₂²	1	4	yes	complemented
68	C₇×#428	C₂²	1	2	yes	—
69	C₇×#427	C₂²	1	1	yes	characteristic
70	C₂×C₇×#106	C₂²	1	1	yes	characteristic
71	C₇×#424	C₂²	1	2	yes	—
72	C₇×#424	C₂²	1	2	yes	—
73	C₇×#424	C₂²	1	2	yes	—
74	C₇×#424	C₂²	1	2	yes	—
75	C₇×#423	C₂²	1	2	yes	—
76	C₇×#423	C₂²	1	2	yes	—
77	C₇×#419	C₂²	1	2	yes	—
78	C₇×#417	C₂²	1	2	yes	—
79	C₇×#416	C₂²	1	2	yes	—
80	C₂×C₇×#103	C₂²	1	2	yes	—
81	#60939	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₇×#41	C₂³	1	2	yes	—
83	C₂²×C₇×Q₈	C₂³	1	2	yes	—
84	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
85	C₇×#137	—	4	4	no	—
86	C₇×#135	—	4	2	no	—
87	C₂×C₇×Q₁₆	—	4	2	no	—
88	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
89	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
90	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
91	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
92	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
93	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
94	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
95	C₂×C₇×M₁₆	—	4	2	no	—
96	C₂×C₇×M₁₆	—	4	2	no	—
97	C₂×C₇×M₁₆	C₂³	1	2	yes	—
98	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
99	C₂×C₇×M₁₆	C₂³	1	2	yes	—
100	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
101	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
102	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
103	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
104	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
105	C₂²×C₇×C₈	C₂³	1	2	yes	—
106	C₇×C₄⋊Q₈	—	2	2	no	—
107	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
108	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
109	C₇×#126	—	2	4	no	—
110	C₇×#126	—	2	4	no	—
111	C₇×#126	—	2	4	no	—
112	C₇×#126	—	2	4	no	—
113	C₇×#125	—	2	4	no	—
114	C₇×#125	—	2	4	no	—
115	C₇×#125	—	2	2	no	—
116	C₇×#125	—	2	4	no	—
117	C₇×#125	C₂³	1	4	yes	—
118	C₇×#124	—	2	4	no	—
119	C₇×#124	C₂³	1	4	yes	—
120	C₇×#123	—	2	2	no	—
121	C₇×#123	—	2	4	no	—
122	C₇×#123	—	2	4	no	—
123	C₇×#123	—	2	4	no	—
124	C₇×C₂²⋊Q₈	—	2	2	no	—
125	C₇×C₂²⋊Q₈	—	2	2	no	—
126	C₇×C₂²⋊Q₈	—	2	4	no	—
127	C₇×C₂²⋊Q₈	—	2	2	no	—
128	C₇×C₂²⋊Q₈	—	2	2	no	—
129	C₇×C₂²⋊Q₈	—	2	4	no	—
130	C₇×C₂²⋊Q₈	—	2	4	no	—
131	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
132	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
133	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
134	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
135	C₇×#121	C₂³	1	2	yes	—
136	C₇×#121	C₂³	1	2	yes	—
137	C₇×#120	—	2	2	no	—
138	C₄×C₇×Q₈	—	2	2	no	—
139	C₄×C₇×Q₈	—	2	2	no	—
140	C₄×C₇×Q₈	—	2	4	no	—
141	C₄×C₇×Q₈	C₂³	1	4	yes	—
142	C₄×C₇×Q₈	C₂³	1	4	yes	—
143	C₄×C₇×Q₈	C₂³	1	4	yes	—
144	C₄×C₇×D₄	—	2	2	no	—
145	C₄×C₇×D₄	—	2	2	no	—
146	C₄×C₇×D₄	—	2	4	no	—
147	C₄×C₇×D₄	—	2	4	no	—
148	C₄×C₇×D₄	C₂³	1	4	yes	—
149	C₇×#117	—	2	4	no	—
150	C₇×#117	—	2	2	no	—
151	C₇×#117	—	2	4	no	—
152	C₇×#117	—	2	4	no	—
153	C₇×#117	D₄	1	1	yes	characteristic
154	C₇×#117	D₄	1	2	yes	—
155	C₇×#117	D₄	1	1	yes	characteristic
156	C₇×#117	C₂³	1	2	yes	—
157	C₇×#117	C₂³	1	2	yes	—
158	C₇×#117	C₂³	1	2	yes	—
159	C₂×C₇×C₄⋊C₄	—	2	2	no	—
160	C₂×C₇×C₄⋊C₄	—	2	2	no	—
161	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
162	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
163	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
165	C₂×C₄²×C₇	C₂³	1	2	yes	—
166	C₇×#107	C₂³	1	4	yes	—
167	C₇×#107	C₂³	1	2	yes	—
168	C₇×#106	C₂³	1	1	yes	characteristic
169	C₇×#106	C₂³	1	1	yes	characteristic
170	C₇×#106	C₂³	1	2	yes	—
171	C₇×#106	C₂³	1	2	yes	—
172	C₇×#106	C₂³	1	4	yes	—
173	C₇×C₄⋊C₈	C₂³	1	4	yes	—
174	C₇×C₄⋊C₈	C₂³	1	4	yes	—
175	C₇×C₄⋊C₈	C₂³	1	4	yes	—
176	C₇×C₄⋊C₈	C₂³	1	4	yes	—
177	C₇×#103	C₂³	1	4	yes	—
178	C₇×#103	C₂³	1	4	yes	—
179	C₇×#103	C₂³	1	2	yes	—
180	C₇×#103	C₂³	1	2	yes	—
181	C₇×#103	C₂³	1	4	yes	—
182	C₇×#103	C₂³	1	4	yes	—
183	C₇×#103	C₂³	1	4	yes	—
184	C₇×#102	C₂³	1	2	yes	—
185	C₇×#102	C₂³	1	2	yes	—
186	C₇×#102	C₂³	1	4	yes	—
187	#3495	C₂×C₇	1	2	yes	complemented
188	#3263	C₂×C₇	1	2	yes	complemented
189	#3261	C₂×C₇	1	2	yes	complemented
190	#3258	C₂×C₇	1	4	yes	complemented
191	#3256	C₂×C₇	1	2	yes	complemented
192	#3255	C₂×C₇	1	2	yes	complemented
193	#3252	C₂×C₇	1	4	yes	complemented
194	#3084	C₂×C₇	1	2	yes	complemented
195	#3082	C₂×C₇	1	2	yes	complemented
196	#3056	C₂×C₇	1	2	yes	complemented
197	#3054	C₂×C₇	1	1	yes	characteristic
198	C₂×#470	C₂×C₇	1	1	yes	characteristic, complemented
199	#3046	C₂×C₇	1	2	yes	complemented
200	#3044	C₂×C₇	1	2	yes	complemented
201	#2905	C₂×C₇	1	1	yes	characteristic, complemented
202	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
203	C₇×#41	—	2	4	no	—
204	C₇×#41	—	2	4	no	—
205	C₂×C₇×Q₈	—	2	2	no	—
206	C₂×C₇×Q₈	—	2	4	no	—
207	C₂×C₇×Q₈	—	2	4	no	—
208	C₂×C₇×Q₈	—	2	4	no	—
209	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
210	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
211	C₂×C₇×D₄	—	2	2	no	—
212	C₂×C₇×D₄	—	2	2	no	—
213	C₂×C₇×D₄	—	2	2	no	—
214	C₂×C₇×D₄	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₇	—	2	2	no	—
216	C₂²×C₄×C₇	—	2	2	no	—
217	C₂²×C₄×C₇	—	2	2	no	—
218	C₂²×C₄×C₇	—	2	2	no	—
219	C₂²×C₄×C₇	—	2	2	no	—
220	C₂²×C₄×C₇	—	2	4	no	—
221	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
222	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
223	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
225	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
226	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
227	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
228	C₇×Q₁₆	—	4	2	no	—
229	C₇×Q₁₆	—	4	2	no	—
230	C₇×Q₁₆	—	4	1	no	—
231	C₇×Q₁₆	—	4	4	no	—
232	C₇×SD₁₆	—	4	2	no	—
233	C₇×SD₁₆	—	4	4	no	—
234	C₇×D₈	—	4	1	no	—
235	C₇×M₁₆	—	2	4	no	—
236	C₇×M₁₆	—	4	1	no	—
237	C₇×M₁₆	—	4	1	no	—
238	C₇×M₁₆	—	4	1	no	—
239	C₇×M₁₆	—	4	1	no	—
240	C₇×M₁₆	—	4	2	no	—
241	C₂×C₇×C₈	—	2	2	no	—
242	C₂×C₇×C₈	—	2	2	no	—
243	C₂×C₇×C₈	—	4	2	no	—
244	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
245	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
246	C₂×C₇×C₈	C₂⁴	1	4	yes	—
247	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
248	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
249	C₇×C₄⋊C₄	—	2	2	no	—
250	C₇×C₄⋊C₄	—	2	4	no	—
251	C₇×C₄⋊C₄	—	2	4	no	—
252	C₇×C₄⋊C₄	—	2	4	no	—
253	C₇×C₄⋊C₄	—	2	4	no	—
254	C₇×C₄⋊C₄	—	2	4	no	—
255	C₇×C₄⋊C₄	—	2	4	no	—
256	C₇×C₄⋊C₄	—	2	4	no	—
257	C₇×C₄⋊C₄	—	2	4	no	—
258	C₇×C₄⋊C₄	—	2	2	no	—
259	C₇×C₄⋊C₄	—	2	4	no	—
260	C₇×C₄⋊C₄	—	2	2	no	—
261	C₇×C₄⋊C₄	—	2	4	no	—
262	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
263	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
264	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
265	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
266	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
267	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
268	C₇×C₂²⋊C₄	—	2	4	no	—
269	C₇×C₂²⋊C₄	—	2	4	no	—
270	C₇×C₂²⋊C₄	—	2	2	no	—
271	C₇×C₂²⋊C₄	—	2	2	no	—
272	C₇×C₂²⋊C₄	—	2	2	no	—
273	C₇×C₂²⋊C₄	—	2	2	no	—
274	C₇×C₂²⋊C₄	—	2	4	no	—
275	C₇×C₂²⋊C₄	—	2	4	no	—
276	C₇×C₂²⋊C₄	—	2	2	no	—
277	C₇×C₂²⋊C₄	—	2	4	no	—
278	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
279	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
280	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
281	C₄²×C₇	—	2	4	no	—
282	C₄²×C₇	—	2	4	no	—
283	C₄²×C₇	—	2	4	no	—
284	C₄²×C₇	—	2	2	no	—
285	C₄²×C₇	C₂×D₄	1	2	yes	—
286	C₄²×C₇	C₂×D₄	1	2	yes	—
287	C₄²×C₇	C₂⁴	1	4	yes	—
288	C₂×#137	C₂²×C₇	1	2	yes	—
289	C₂×#135	C₂²×C₇	1	1	yes	characteristic
290	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
291	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
292	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
293	#556	—	2	2	no	—
294	#554	—	2	2	no	—
295	#552	—	2	4	no	—
296	#552	—	2	2	no	—
297	#552	—	2	2	no	—
298	#551	—	2	4	no	—
299	#544	C₂²×C₇	1	2	yes	—
300	#542	C₂²×C₇	1	4	yes	—
301	#541	C₂²×C₇	1	4	yes	—
302	#536	C₂²×C₇	1	2	yes	—
303	#533	C₂²×C₇	1	4	yes	—
304	#529	C₂²×C₇	1	4	yes	—
305	#525	C₂²×C₇	1	2	yes	—
306	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
307	#520	C₂²×C₇	1	2	yes	—
308	#519	C₂²×C₇	1	2	yes	—
309	C₂×#117	C₂²×C₇	1	1	yes	characteristic
310	#479	C₂²×C₇	1	4	yes	complemented
311	#479	C₂²×C₇	1	4	yes	complemented
312	#479	C₂²×C₇	1	4	yes	complemented
313	#478	C₂²×C₇	1	4	yes	complemented
314	#477	C₂²×C₇	1	4	yes	complemented
315	#476	C₂²×C₇	1	4	yes	complemented
316	#475	C₂²×C₇	1	4	yes	complemented
317	#475	C₂²×C₇	1	4	yes	complemented
318	#470	C₂²×C₇	1	2	yes	—
319	#470	C₂²×C₇	1	4	yes	—
320	#470	C₂²×C₇	1	1	yes	characteristic
321	#470	C₂²×C₇	1	2	yes	—
322	#470	C₂²×C₇	1	2	yes	—
323	#470	C₂²×C₇	1	1	yes	characteristic
324	#470	C₂²×C₇	1	2	yes	—
325	#469	C₂²×C₇	1	1	yes	characteristic
326	#469	C₂²×C₇	1	1	yes	characteristic
327	#468	C₂²×C₇	1	2	yes	—
328	#468	C₂²×C₇	1	4	yes	—
329	#468	C₂²×C₇	1	2	yes	—
330	#467	C₂²×C₇	1	4	yes	—
331	#465	C₂²×C₇	1	4	yes	—
332	#464	C₂²×C₇	1	4	yes	complemented
333	#464	C₂²×C₇	1	4	yes	complemented
334	#464	C₂²×C₇	1	4	yes	complemented
335	#463	C₂²×C₇	1	4	yes	complemented
336	#462	C₂²×C₇	1	4	yes	complemented
337	#462	C₂²×C₇	1	4	yes	complemented
338	#462	C₂²×C₇	1	4	yes	complemented
339	#461	C₂²×C₇	1	4	yes	complemented
340	#428	C₂²×C₇	1	2	yes	—
341	#427	C₂²×C₇	1	1	yes	characteristic
342	C₂×#106	C₂²×C₇	1	1	yes	characteristic
343	#424	C₂²×C₇	1	2	yes	—
344	#424	C₂²×C₇	1	2	yes	—
345	#424	C₂²×C₇	1	2	yes	—
346	#424	C₂²×C₇	1	2	yes	—
347	#423	C₂²×C₇	1	2	yes	—
348	#423	C₂²×C₇	1	2	yes	—
349	#419	C₂²×C₇	1	2	yes	—
350	#417	C₂²×C₇	1	2	yes	—
351	#416	C₂²×C₇	1	2	yes	—
352	C₂×#103	C₂²×C₇	1	2	yes	—
353	C₂³×C₇	—	2	2	no	—
354	C₂³×C₇	—	2	2	no	—
355	C₂³×C₇	—	4	2	no	—
356	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
357	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
358	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
359	C₇×Q₈	—	2	4	no	—
360	C₇×Q₈	—	2	2	no	—
361	C₇×Q₈	—	2	4	no	—
362	C₇×Q₈	—	2	2	no	—
363	C₇×Q₈	—	4	4	no	—
364	C₇×Q₈	—	8	4	no	—
365	C₇×D₄	—	2	2	no	—
366	C₇×D₄	—	2	2	no	—
367	C₇×D₄	—	4	4	no	—
368	C₇×D₄	—	4	4	no	—
369	C₇×D₄	—	8	2	no	—
370	C₂×C₄×C₇	—	2	2	no	—
371	C₂×C₄×C₇	—	2	2	no	—
372	C₂×C₄×C₇	—	2	2	no	—
373	C₂×C₄×C₇	—	2	1	no	—
374	C₂×C₄×C₇	—	2	1	no	—
375	C₂×C₄×C₇	—	2	1	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	2	2	no	—
378	C₂×C₄×C₇	—	2	2	no	—
379	C₂×C₄×C₇	—	2	4	no	—
380	C₂×C₄×C₇	—	2	2	no	—
381	C₂×C₄×C₇	—	2	4	no	—
382	C₂×C₄×C₇	—	4	4	no	—
383	C₂×C₄×C₇	—	4	4	no	—
384	C₂×C₄×C₇	—	4	4	no	—
385	C₂×C₄×C₇	—	8	4	no	—
386	C₂×C₄×C₇	—	8	2	no	—
387	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
388	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
389	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
390	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
391	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
392	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
393	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
394	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
395	C₇×C₈	—	2	1	no	—
396	C₇×C₈	—	2	1	no	—
397	C₇×C₈	—	2	1	no	—
398	C₇×C₈	—	2	1	no	—
399	C₇×C₈	—	2	4	no	—
400	C₂×#41	C₂³×C₇	1	2	yes	—
401	C₂²×Q₈	C₂³×C₇	1	2	yes	—
402	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
403	#137	—	4	4	no	—
404	#135	—	4	2	no	—
405	C₂×Q₁₆	—	4	2	no	—
406	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
407	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
408	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
409	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
410	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
411	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
412	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
413	C₂×M₁₆	—	4	2	no	—
414	C₂×M₁₆	—	4	2	no	—
415	C₂×M₁₆	C₂³×C₇	1	2	yes	—
416	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
417	C₂×M₁₆	C₂³×C₇	1	2	yes	—
418	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
419	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
420	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₇	1	2	yes	—
424	C₄⋊Q₈	—	2	2	no	—
425	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
426	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
427	#126	—	2	4	no	—
428	#126	—	2	4	no	—
429	#126	—	2	4	no	—
430	#126	—	2	4	no	—
431	#125	—	2	4	no	—
432	#125	—	2	4	no	—
433	#125	—	2	2	no	—
434	#125	—	2	4	no	—
435	#125	C₂³×C₇	1	4	yes	—
436	#124	—	2	4	no	—
437	#124	C₂³×C₇	1	4	yes	—
438	#123	—	2	2	no	—
439	#123	—	2	4	no	—
440	#123	—	2	4	no	—
441	#123	—	2	4	no	—
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	4	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	4	no	—
448	C₂²⋊Q₈	—	2	4	no	—
449	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
450	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
451	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
452	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
453	#121	C₂³×C₇	1	2	yes	—
454	#121	C₂³×C₇	1	2	yes	—
455	#120	—	2	2	no	—
456	C₄×Q₈	—	2	2	no	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×Q₈	—	2	4	no	—
459	C₄×Q₈	C₂³×C₇	1	4	yes	—
460	C₄×Q₈	C₂³×C₇	1	4	yes	—
461	C₄×Q₈	C₂³×C₇	1	4	yes	—
462	C₄×D₄	—	2	2	no	—
463	C₄×D₄	—	2	2	no	—
464	C₄×D₄	—	2	4	no	—
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	C₂³×C₇	1	4	yes	—
467	#117	—	2	4	no	—
468	#117	—	2	2	no	—
469	#117	—	2	4	no	—
470	#117	—	2	4	no	—
471	#117	C₇×D₄	1	1	yes	characteristic
472	#117	C₇×D₄	1	2	yes	—
473	#117	C₇×D₄	1	1	yes	characteristic
474	#117	C₂³×C₇	1	2	yes	—
475	#117	C₂³×C₇	1	2	yes	—
476	#117	C₂³×C₇	1	2	yes	—
477	C₂×C₄⋊C₄	—	2	2	no	—
478	C₂×C₄⋊C₄	—	2	2	no	—
479	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
480	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
481	C₂×C₂²⋊C₄	—	2	2	no	—
482	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
483	C₂×C₄²	C₂³×C₇	1	2	yes	—
484	#107	C₂³×C₇	1	2	yes	—
485	#107	C₂³×C₇	1	4	yes	—
486	#106	C₂³×C₇	1	2	yes	—
487	#106	C₂³×C₇	1	2	yes	—
488	#106	C₂³×C₇	1	1	yes	characteristic
489	#106	C₂³×C₇	1	1	yes	characteristic
490	#106	C₂³×C₇	1	4	yes	—
491	C₄⋊C₈	C₂³×C₇	1	4	yes	—
492	C₄⋊C₈	C₂³×C₇	1	4	yes	—
493	C₄⋊C₈	C₂³×C₇	1	4	yes	—
494	C₄⋊C₈	C₂³×C₇	1	4	yes	—
495	#103	C₂³×C₇	1	4	yes	—
496	#103	C₂³×C₇	1	4	yes	—
497	#103	C₂³×C₇	1	2	yes	—
498	#103	C₂³×C₇	1	2	yes	—
499	#103	C₂³×C₇	1	4	yes	—
500	#103	C₂³×C₇	1	4	yes	—
501	#103	C₂³×C₇	1	4	yes	—
502	#102	C₂³×C₇	1	2	yes	—
503	#102	C₂³×C₇	1	2	yes	—
504	#102	C₂³×C₇	1	4	yes	—
505	C₂²×C₇	—	2	2	no	—
506	C₂²×C₇	—	2	2	no	—
507	C₂²×C₇	—	2	2	no	—
508	C₂²×C₇	—	2	1	no	—
509	C₂²×C₇	—	2	1	no	—
510	C₂²×C₇	—	4	2	no	—
511	C₂²×C₇	—	4	2	no	—
512	C₂²×C₇	—	4	2	no	—
513	C₂²×C₇	—	8	2	no	—
514	C₂²×C₇	#537	1	2	yes	—
515	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
516	C₄×C₇	—	2	4	no	—
517	C₄×C₇	—	2	4	no	—
518	C₄×C₇	—	2	1	no	—
519	C₄×C₇	—	2	1	no	—
520	C₄×C₇	—	4	4	no	—
521	C₄×C₇	—	4	2	no	—
522	C₄×C₇	—	4	2	no	—
523	C₄×C₇	—	4	2	no	—
524	C₄×C₇	—	4	4	no	—
525	C₄×C₇	#537	1	2	yes	—
526	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
527	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
528	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
529	#41	—	2	4	no	—
530	#41	—	2	4	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	2	4	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	—	2	4	no	—
535	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
536	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	C₂⁴×C₇	1	2	yes	—
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	2	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
548	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
549	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
550	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
551	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
552	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
553	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
554	Q₁₆	—	4	2	no	—
555	Q₁₆	—	4	2	no	—
556	Q₁₆	—	4	1	no	—
557	Q₁₆	—	4	4	no	—
558	SD₁₆	—	4	2	no	—
559	SD₁₆	—	4	4	no	—
560	D₈	—	4	1	no	—
561	M₁₆	—	2	4	no	—
562	M₁₆	—	4	1	no	—
563	M₁₆	—	4	1	no	—
564	M₁₆	—	4	1	no	—
565	M₁₆	—	4	1	no	—
566	M₁₆	—	4	2	no	—
567	C₂×C₈	—	2	2	no	—
568	C₂×C₈	—	2	2	no	—
569	C₂×C₈	—	4	2	no	—
570	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
571	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
572	C₂×C₈	C₂⁴×C₇	1	4	yes	—
573	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
574	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	4	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	4	no	—
588	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
589	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
590	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
591	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
592	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
593	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	2	no	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	—	2	2	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	4	no	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
605	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
606	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	4	no	—
610	C₄²	—	2	2	no	—
611	C₄²	C₂×C₇×D₄	1	2	yes	—
612	C₄²	C₂×C₇×D₄	1	2	yes	—
613	C₄²	C₂⁴×C₇	1	4	yes	—
614	C₂×C₇	—	2	2	no	—
615	C₂×C₇	—	4	1	no	—
616	C₂×C₇	—	4	1	no	—
617	C₂×C₇	—	8	2	no	—
618	C₂×C₇	C₂×#537	1	1	yes	characteristic
619	C₂×C₇	#3586	1	2	yes	—
620	C₂³	—	2	2	no	—
621	C₂³	—	2	2	no	—
622	C₂³	—	4	2	no	—
623	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
624	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
625	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
626	Q₈	—	2	4	no	—
627	Q₈	—	2	2	no	—
628	Q₈	—	2	4	no	—
629	Q₈	—	2	2	no	—
630	Q₈	—	4	4	no	—
631	Q₈	—	8	4	no	—
632	D₄	—	2	2	no	—
633	D₄	—	2	2	no	—
634	D₄	—	4	4	no	—
635	D₄	—	4	4	no	—
636	D₄	—	8	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	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	4	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	4	no	—
649	C₂×C₄	—	4	4	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	4	no	—
652	C₂×C₄	—	8	4	no	—
653	C₂×C₄	—	8	2	no	—
654	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
655	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
656	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
657	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
658	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
659	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
660	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
661	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
662	C₈	—	2	1	no	—
663	C₈	—	2	1	no	—
664	C₈	—	2	1	no	—
665	C₈	—	2	1	no	—
666	C₈	—	2	4	no	—
667	C₇	#60939	1	1	yes	characteristic, direct product factor, Sylow
668	C₂²	—	2	2	no	—
669	C₂²	—	2	2	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	1	no	—
672	C₂²	—	2	1	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	8	2	no	—
677	C₂²	C₇×#537	1	2	yes	—
678	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
679	C₄	—	2	4	no	—
680	C₄	—	2	4	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	1	no	—
683	C₄	—	4	4	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	4	2	no	—
687	C₄	—	4	4	no	—
688	C₄	C₇×#537	1	2	yes	—
689	C₄	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic
690	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
691	C₂	—	2	2	no	—
692	C₂	—	4	1	no	—
693	C₂	—	4	1	no	—
694	C₂	—	8	2	no	—
695	C₂	C₂×C₇×#537	1	1	yes	characteristic
696	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3495	C₂	semidirect
2. C₇×#3263	C₂	semidirect
3. C₇×#3261	C₂	semidirect
4. C₇×#3258	C₂	semidirect
5. C₇×#3256	C₂	semidirect
6. C₇×#3255	C₂	semidirect
7. C₇×#3252	C₂	semidirect
8. C₇×#3084	C₂	semidirect
9. C₇×#3082	C₂	semidirect
10. C₇×#3056	C₂	semidirect
12. C₂×C₇×#470	C₂	semidirect
13. C₇×#3046	C₂	semidirect
14. C₇×#3044	C₂	semidirect
15. C₇×#2905	C₂	semidirect
38. C₇×#479	C₂²	semidirect
39. C₇×#479	C₂²	semidirect
40. C₇×#479	C₂²	semidirect
41. C₇×#478	C₂²	semidirect
42. C₇×#477	C₂²	semidirect
43. C₇×#476	C₂²	semidirect
44. C₇×#475	C₂²	semidirect
45. C₇×#475	C₂²	semidirect
60. C₇×#464	C₂²	semidirect
61. C₇×#464	C₂²	semidirect
62. C₇×#464	C₂²	semidirect
63. C₇×#463	C₂²	semidirect
64. C₇×#462	C₂²	semidirect
65. C₇×#462	C₂²	semidirect
66. C₇×#462	C₂²	semidirect
67. C₇×#461	C₂²	semidirect
81. #60939	C₇	direct (N × H)
187. #3495	C₂×C₇	semidirect
188. #3263	C₂×C₇	semidirect
189. #3261	C₂×C₇	semidirect
190. #3258	C₂×C₇	semidirect
191. #3256	C₂×C₇	semidirect
192. #3255	C₂×C₇	semidirect
193. #3252	C₂×C₇	semidirect
194. #3084	C₂×C₇	semidirect
195. #3082	C₂×C₇	semidirect
196. #3056	C₂×C₇	semidirect
198. C₂×#470	C₂×C₇	semidirect
199. #3046	C₂×C₇	semidirect
200. #3044	C₂×C₇	semidirect
201. #2905	C₂×C₇	semidirect
310. #479	C₂²×C₇	semidirect
311. #479	C₂²×C₇	semidirect
312. #479	C₂²×C₇	semidirect
313. #478	C₂²×C₇	semidirect
314. #477	C₂²×C₇	semidirect
315. #476	C₂²×C₇	semidirect
316. #475	C₂²×C₇	semidirect
317. #475	C₂²×C₇	semidirect
332. #464	C₂²×C₇	semidirect
333. #464	C₂²×C₇	semidirect
334. #464	C₂²×C₇	semidirect
335. #463	C₂²×C₇	semidirect
336. #462	C₂²×C₇	semidirect
337. #462	C₂²×C₇	semidirect
338. #462	C₂²×C₇	semidirect
339. #461	C₂²×C₇	semidirect
667. C₇	#60939	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×#3495	2
C₇×#3263	2
C₇×#3261	2
C₇×#3258	2
C₇×#3256	2
C₇×#3255	2
C₇×#3252	2
C₇×#3084	2
C₇×#3082	2
C₇×#3056	2
C₇×#3054	2
C₂×C₇×#470	2
C₇×#3046	2
C₇×#3044	2
C₇×#2905	2
#60939	7

Subgroups — C7×#60939 #13238155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60939 #13238155