Subgroups — C₇×#60711 #13237927

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 730 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×#3185	C₂	1	2	yes	maximal, complemented
3	C₇×#3183	C₂	1	2	yes	maximal, complemented
4	C₇×#3178	C₂	1	4	yes	maximal, complemented
5	C₇×#3177	C₂	1	2	yes	maximal, complemented
6	C₇×#3173	C₂	1	2	yes	maximal, complemented
7	C₇×#3172	C₂	1	4	yes	maximal, complemented
8	C₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3137	C₂	1	2	yes	maximal, complemented
11	C₇×#3056	C₂	1	2	yes	maximal, complemented
12	C₇×#3054	C₂	1	1	yes	maximal, characteristic
13	C₂×C₇×#470	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2952	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#2947	C₂	1	2	yes	maximal, complemented
17	C₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₇×#137	C₂²	1	2	yes	—
19	C₂×C₇×#135	C₂²	1	1	yes	characteristic
20	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
23	C₇×#556	—	2	2	no	—
24	C₇×#554	—	2	2	no	—
25	C₇×#552	—	2	4	no	—
26	C₇×#552	—	2	2	no	—
27	C₇×#552	—	2	2	no	—
28	C₇×#551	—	2	4	no	—
29	C₇×#544	C₂²	1	2	yes	—
30	C₇×#542	C₂²	1	4	yes	—
31	C₇×#541	C₂²	1	4	yes	—
32	C₇×#536	C₂²	1	2	yes	—
33	C₇×#533	C₂²	1	4	yes	—
34	C₇×#529	C₂²	1	4	yes	—
35	C₇×#525	C₂²	1	2	yes	—
36	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
37	C₇×#520	C₂²	1	2	yes	—
38	C₇×#519	C₂²	1	2	yes	—
39	C₂×C₇×#117	C₂²	1	1	yes	characteristic
40	C₇×#501	C₂²	1	4	yes	complemented
41	C₇×#499	C₂²	1	2	yes	complemented
42	C₇×#498	C₂²	1	2	yes	complemented
43	C₇×#497	C₂²	1	4	yes	complemented
44	C₇×#495	C₂²	1	2	yes	complemented
45	C₇×#494	C₂²	1	2	yes	complemented
46	C₇×#491	C₂²	1	4	yes	complemented
47	C₇×#489	C₂²	1	4	yes	complemented
48	C₇×#488	C₂²	1	4	yes	complemented
49	C₇×#487	C₂²	1	4	yes	complemented
50	C₇×#470	C₂²	1	2	yes	—
51	C₇×#470	C₂²	1	4	yes	—
52	C₇×#470	C₂²	1	2	yes	—
53	C₇×#470	C₂²	1	1	yes	characteristic
54	C₇×#470	C₂²	1	2	yes	—
55	C₇×#470	C₂²	1	2	yes	—
56	C₇×#470	C₂²	1	1	yes	characteristic
57	C₇×#469	C₂²	1	1	yes	characteristic
58	C₇×#469	C₂²	1	1	yes	characteristic
59	C₇×#468	C₂²	1	2	yes	—
60	C₇×#468	C₂²	1	4	yes	—
61	C₇×#468	C₂²	1	2	yes	—
62	C₇×#467	C₂²	1	4	yes	—
63	C₇×#465	C₂²	1	4	yes	—
64	C₇×#442	C₂²	1	2	yes	complemented
65	C₇×#441	C₂²	1	2	yes	complemented
66	C₇×#441	C₂²	1	2	yes	complemented
67	C₇×#441	C₂²	1	2	yes	complemented
68	C₇×#441	C₂²	1	4	yes	complemented
69	C₇×#440	C₂²	1	4	yes	complemented
70	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
71	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
72	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
73	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
74	C₇×#428	C₂²	1	2	yes	—
75	C₇×#427	C₂²	1	1	yes	characteristic
76	C₂×C₇×#106	C₂²	1	1	yes	characteristic
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	C₇×#405	C₂²	1	1	yes	characteristic
82	C₇×#405	C₂²	1	2	yes	—
83	C₇×#405	C₂²	1	2	yes	—
84	C₇×#405	C₂²	1	1	yes	characteristic
85	C₇×#405	C₂²	1	1	yes	characteristic
86	C₇×#405	C₂²	1	1	yes	characteristic
87	C₄×C₇×M₁₆	C₂²	1	2	yes	—
88	C₄×C₇×M₁₆	C₂²	1	2	yes	—
89	#60711	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₇×#41	C₂³	1	2	yes	—
91	C₂²×C₇×Q₈	C₂³	1	2	yes	—
92	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
93	C₇×#137	—	2	8	no	—
94	C₇×#135	—	2	4	no	—
95	C₂×C₇×Q₁₆	—	2	4	no	—
96	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
97	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
98	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
99	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
100	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
101	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
102	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
103	C₂×C₇×M₁₆	—	2	4	no	—
104	C₂×C₇×M₁₆	—	2	4	no	—
105	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
106	C₂×C₇×M₁₆	C₂³	1	2	yes	—
107	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
108	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
109	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
111	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
112	C₂²×C₇×C₈	C₂³	1	2	yes	—
113	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
114	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
115	C₇×C₄⋊Q₈	—	2	2	no	—
116	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
117	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
118	C₇×#126	—	2	4	no	—
119	C₇×#126	—	2	4	no	—
120	C₇×#126	—	2	4	no	—
121	C₇×#126	—	2	4	no	—
122	C₇×#125	—	2	4	no	—
123	C₇×#125	—	2	4	no	—
124	C₇×#125	—	2	2	no	—
125	C₇×#125	—	2	4	no	—
126	C₇×#125	C₂³	1	4	yes	—
127	C₇×#124	—	2	4	no	—
128	C₇×#124	C₂³	1	4	yes	—
129	C₇×#123	—	2	2	no	—
130	C₇×#123	—	2	4	no	—
131	C₇×#123	—	2	4	no	—
132	C₇×#123	—	2	4	no	—
133	C₇×C₂²⋊Q₈	—	2	2	no	—
134	C₇×C₂²⋊Q₈	—	2	2	no	—
135	C₇×C₂²⋊Q₈	—	2	4	no	—
136	C₇×C₂²⋊Q₈	—	2	2	no	—
137	C₇×C₂²⋊Q₈	—	2	2	no	—
138	C₇×C₂²⋊Q₈	—	2	4	no	—
139	C₇×C₂²⋊Q₈	—	2	4	no	—
140	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
141	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
142	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
143	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
144	C₇×#121	C₂³	1	2	yes	—
145	C₇×#121	C₂³	1	2	yes	—
146	C₇×#120	—	2	2	no	—
147	C₄×C₇×Q₈	—	2	2	no	—
148	C₄×C₇×Q₈	—	2	2	no	—
149	C₄×C₇×Q₈	—	2	4	no	—
150	C₄×C₇×Q₈	C₂³	1	4	yes	—
151	C₄×C₇×Q₈	C₂³	1	4	yes	—
152	C₄×C₇×Q₈	C₂³	1	4	yes	—
153	C₄×C₇×D₄	—	2	2	no	—
154	C₄×C₇×D₄	—	2	2	no	—
155	C₄×C₇×D₄	—	2	4	no	—
156	C₄×C₇×D₄	—	2	4	no	—
157	C₄×C₇×D₄	C₂³	1	4	yes	—
158	C₇×#117	—	2	4	no	—
159	C₇×#117	—	2	2	no	—
160	C₇×#117	—	2	4	no	—
161	C₇×#117	—	2	4	no	—
162	C₇×#117	D₄	1	2	yes	—
163	C₇×#117	D₄	1	2	yes	—
164	C₇×#117	C₂³	1	2	yes	—
165	C₇×#117	C₂³	1	1	yes	characteristic
166	C₇×#117	C₂³	1	1	yes	characteristic
167	C₇×#117	C₂³	1	2	yes	—
168	C₂×C₇×C₄⋊C₄	—	2	2	no	—
169	C₂×C₇×C₄⋊C₄	—	2	2	no	—
170	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
171	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
172	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
174	C₂×C₄²×C₇	C₂³	1	2	yes	—
175	C₇×#107	C₂³	1	4	yes	—
176	C₇×#107	C₂³	1	2	yes	—
177	C₇×#106	C₂³	1	1	yes	characteristic
178	C₇×#106	C₂³	1	1	yes	characteristic
179	C₇×#106	C₂³	1	2	yes	—
180	C₇×#106	C₂³	1	2	yes	—
181	C₇×#106	C₂³	1	4	yes	—
182	C₇×#103	C₂³	1	4	yes	—
183	C₇×#103	C₂³	1	4	yes	—
184	C₇×#103	C₂³	1	2	yes	—
185	C₇×#103	C₂³	1	2	yes	—
186	C₇×#103	C₂³	1	4	yes	—
187	C₇×#103	C₂³	1	4	yes	—
188	C₇×#103	C₂³	1	4	yes	—
189	C₇×#102	C₂³	1	2	yes	—
190	C₇×#102	C₂³	1	2	yes	—
191	C₇×#102	C₂³	1	4	yes	—
192	C₇×#97	C₂³	1	2	yes	—
193	C₇×#97	C₂³	1	2	yes	—
194	C₇×#97	C₂³	1	2	yes	—
195	C₇×#97	C₂³	1	2	yes	—
196	C₄×C₇×C₈	C₂³	1	2	yes	—
197	C₄×C₇×C₈	C₂³	1	2	yes	—
198	C₄×C₇×C₈	C₂³	1	2	yes	—
199	C₄×C₇×C₈	C₂³	1	2	yes	—
200	#3495	C₂×C₇	1	2	yes	complemented
201	#3185	C₂×C₇	1	2	yes	complemented
202	#3183	C₂×C₇	1	2	yes	complemented
203	#3178	C₂×C₇	1	4	yes	complemented
204	#3177	C₂×C₇	1	2	yes	complemented
205	#3173	C₂×C₇	1	2	yes	complemented
206	#3172	C₂×C₇	1	4	yes	complemented
207	#3163	C₂×C₇	1	1	yes	characteristic, complemented
208	#3156	C₂×C₇	1	1	yes	characteristic, complemented
209	#3137	C₂×C₇	1	2	yes	complemented
210	#3056	C₂×C₇	1	2	yes	complemented
211	#3054	C₂×C₇	1	1	yes	characteristic
212	C₂×#470	C₂×C₇	1	1	yes	characteristic, complemented
213	#2952	C₂×C₇	1	1	yes	characteristic, complemented
214	#2950	C₂×C₇	1	1	yes	characteristic, complemented
215	#2947	C₂×C₇	1	2	yes	complemented
216	#2873	C₂×C₇	1	1	yes	characteristic, complemented
217	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
218	C₇×#41	—	2	4	no	—
219	C₇×#41	—	2	4	no	—
220	C₂×C₇×Q₈	—	2	2	no	—
221	C₂×C₇×Q₈	—	2	4	no	—
222	C₂×C₇×Q₈	—	2	4	no	—
223	C₂×C₇×Q₈	—	2	4	no	—
224	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
225	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
226	C₂×C₇×D₄	—	2	2	no	—
227	C₂×C₇×D₄	—	2	2	no	—
228	C₂×C₇×D₄	—	2	2	no	—
229	C₂×C₇×D₄	C₂⁴	1	2	yes	—
230	C₂²×C₄×C₇	—	2	2	no	—
231	C₂²×C₄×C₇	—	2	2	no	—
232	C₂²×C₄×C₇	—	2	2	no	—
233	C₂²×C₄×C₇	—	2	2	no	—
234	C₂²×C₄×C₇	—	2	2	no	—
235	C₂²×C₄×C₇	—	2	4	no	—
236	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
237	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
238	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
239	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
240	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
241	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
242	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
243	C₇×Q₁₆	—	2	4	no	—
244	C₇×Q₁₆	—	2	4	no	—
245	C₇×Q₁₆	—	2	2	no	—
246	C₇×Q₁₆	—	2	8	no	—
247	C₇×SD₁₆	—	2	4	no	—
248	C₇×SD₁₆	—	2	8	no	—
249	C₇×D₈	—	2	2	no	—
250	C₇×M₁₆	—	2	2	no	—
251	C₇×M₁₆	—	2	4	no	—
252	C₇×M₁₆	—	2	2	no	—
253	C₇×M₁₆	—	2	2	no	—
254	C₇×M₁₆	—	2	2	no	—
255	C₇×M₁₆	—	2	2	no	—
256	C₇×M₁₆	#41	1	4	yes	—
257	C₂×C₇×C₈	—	2	2	no	—
258	C₂×C₇×C₈	—	2	4	no	—
259	C₂×C₇×C₈	#41	1	4	yes	—
260	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
261	C₂×C₇×C₈	C₂⁴	1	2	yes	—
262	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
263	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
264	C₂×C₇×C₈	C₂⁴	1	2	yes	—
265	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
266	C₇×C₄⋊C₄	—	2	2	no	—
267	C₇×C₄⋊C₄	—	2	4	no	—
268	C₇×C₄⋊C₄	—	2	4	no	—
269	C₇×C₄⋊C₄	—	2	4	no	—
270	C₇×C₄⋊C₄	—	2	4	no	—
271	C₇×C₄⋊C₄	—	2	4	no	—
272	C₇×C₄⋊C₄	—	2	4	no	—
273	C₇×C₄⋊C₄	—	2	4	no	—
274	C₇×C₄⋊C₄	—	2	4	no	—
275	C₇×C₄⋊C₄	—	2	2	no	—
276	C₇×C₄⋊C₄	—	2	4	no	—
277	C₇×C₄⋊C₄	—	2	2	no	—
278	C₇×C₄⋊C₄	—	2	4	no	—
279	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
280	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
281	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
282	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
283	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
284	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
285	C₇×C₂²⋊C₄	—	2	4	no	—
286	C₇×C₂²⋊C₄	—	2	4	no	—
287	C₇×C₂²⋊C₄	—	2	2	no	—
288	C₇×C₂²⋊C₄	—	2	2	no	—
289	C₇×C₂²⋊C₄	—	2	2	no	—
290	C₇×C₂²⋊C₄	—	2	2	no	—
291	C₇×C₂²⋊C₄	—	2	4	no	—
292	C₇×C₂²⋊C₄	—	2	4	no	—
293	C₇×C₂²⋊C₄	—	2	2	no	—
294	C₇×C₂²⋊C₄	—	2	4	no	—
295	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
296	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
297	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
298	C₄²×C₇	—	2	4	no	—
299	C₄²×C₇	—	2	4	no	—
300	C₄²×C₇	—	2	4	no	—
301	C₄²×C₇	—	2	2	no	—
302	C₄²×C₇	C₂×D₄	1	4	yes	—
303	C₄²×C₇	C₂⁴	1	2	yes	—
304	C₄²×C₇	C₂⁴	1	2	yes	—
305	C₂×#137	C₂²×C₇	1	2	yes	—
306	C₂×#135	C₂²×C₇	1	1	yes	characteristic
307	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
308	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
309	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
310	#556	—	2	2	no	—
311	#554	—	2	2	no	—
312	#552	—	2	4	no	—
313	#552	—	2	2	no	—
314	#552	—	2	2	no	—
315	#551	—	2	4	no	—
316	#544	C₂²×C₇	1	2	yes	—
317	#542	C₂²×C₇	1	4	yes	—
318	#541	C₂²×C₇	1	4	yes	—
319	#536	C₂²×C₇	1	2	yes	—
320	#533	C₂²×C₇	1	4	yes	—
321	#529	C₂²×C₇	1	4	yes	—
322	#525	C₂²×C₇	1	2	yes	—
323	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
324	#520	C₂²×C₇	1	2	yes	—
325	#519	C₂²×C₇	1	2	yes	—
326	C₂×#117	C₂²×C₇	1	1	yes	characteristic
327	#501	C₂²×C₇	1	4	yes	complemented
328	#499	C₂²×C₇	1	2	yes	complemented
329	#498	C₂²×C₇	1	2	yes	complemented
330	#497	C₂²×C₇	1	4	yes	complemented
331	#495	C₂²×C₇	1	2	yes	complemented
332	#494	C₂²×C₇	1	2	yes	complemented
333	#491	C₂²×C₇	1	4	yes	complemented
334	#489	C₂²×C₇	1	4	yes	complemented
335	#488	C₂²×C₇	1	4	yes	complemented
336	#487	C₂²×C₇	1	4	yes	complemented
337	#470	C₂²×C₇	1	2	yes	—
338	#470	C₂²×C₇	1	4	yes	—
339	#470	C₂²×C₇	1	2	yes	—
340	#470	C₂²×C₇	1	1	yes	characteristic
341	#470	C₂²×C₇	1	2	yes	—
342	#470	C₂²×C₇	1	2	yes	—
343	#470	C₂²×C₇	1	1	yes	characteristic
344	#469	C₂²×C₇	1	1	yes	characteristic
345	#469	C₂²×C₇	1	1	yes	characteristic
346	#468	C₂²×C₇	1	2	yes	—
347	#468	C₂²×C₇	1	4	yes	—
348	#468	C₂²×C₇	1	2	yes	—
349	#467	C₂²×C₇	1	4	yes	—
350	#465	C₂²×C₇	1	4	yes	—
351	#442	C₂²×C₇	1	2	yes	complemented
352	#441	C₂²×C₇	1	2	yes	complemented
353	#441	C₂²×C₇	1	2	yes	complemented
354	#441	C₂²×C₇	1	2	yes	complemented
355	#441	C₂²×C₇	1	4	yes	complemented
356	#440	C₂²×C₇	1	4	yes	complemented
357	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
358	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
359	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
360	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
361	#428	C₂²×C₇	1	2	yes	—
362	#427	C₂²×C₇	1	1	yes	characteristic
363	C₂×#106	C₂²×C₇	1	1	yes	characteristic
364	#419	C₂²×C₇	1	2	yes	—
365	#417	C₂²×C₇	1	2	yes	—
366	#416	C₂²×C₇	1	2	yes	—
367	C₂×#103	C₂²×C₇	1	2	yes	—
368	#405	C₂²×C₇	1	1	yes	characteristic
369	#405	C₂²×C₇	1	1	yes	characteristic
370	#405	C₂²×C₇	1	2	yes	—
371	#405	C₂²×C₇	1	2	yes	—
372	#405	C₂²×C₇	1	1	yes	characteristic
373	#405	C₂²×C₇	1	1	yes	characteristic
374	C₄×M₁₆	C₂²×C₇	1	2	yes	—
375	C₄×M₁₆	C₂²×C₇	1	2	yes	—
376	C₂³×C₇	—	2	2	no	—
377	C₂³×C₇	—	2	2	no	—
378	C₂³×C₇	—	4	2	no	—
379	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
380	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
381	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
382	C₇×Q₈	—	2	4	no	—
383	C₇×Q₈	—	2	2	no	—
384	C₇×Q₈	—	2	4	no	—
385	C₇×Q₈	—	2	2	no	—
386	C₇×Q₈	—	4	4	no	—
387	C₇×Q₈	—	8	4	no	—
388	C₇×D₄	—	2	2	no	—
389	C₇×D₄	—	2	2	no	—
390	C₇×D₄	—	4	4	no	—
391	C₇×D₄	—	4	4	no	—
392	C₇×D₄	—	8	2	no	—
393	C₂×C₄×C₇	—	2	2	no	—
394	C₂×C₄×C₇	—	2	2	no	—
395	C₂×C₄×C₇	—	2	2	no	—
396	C₂×C₄×C₇	—	2	1	no	—
397	C₂×C₄×C₇	—	2	1	no	—
398	C₂×C₄×C₇	—	2	1	no	—
399	C₂×C₄×C₇	—	2	1	no	—
400	C₂×C₄×C₇	—	2	2	no	—
401	C₂×C₄×C₇	—	2	2	no	—
402	C₂×C₄×C₇	—	2	4	no	—
403	C₂×C₄×C₇	—	2	2	no	—
404	C₂×C₄×C₇	—	2	4	no	—
405	C₂×C₄×C₇	—	4	4	no	—
406	C₂×C₄×C₇	—	4	4	no	—
407	C₂×C₄×C₇	—	4	4	no	—
408	C₂×C₄×C₇	—	8	4	no	—
409	C₂×C₄×C₇	—	8	2	no	—
410	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
411	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
413	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
415	C₂×C₄×C₇	C₂×#41	1	2	yes	—
416	C₂×C₄×C₇	C₂×#41	1	2	yes	—
417	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
418	C₇×C₈	—	2	1	no	—
419	C₇×C₈	—	2	1	no	—
420	C₇×C₈	—	2	2	no	—
421	C₇×C₈	C₂×#41	1	4	yes	—
422	C₇×C₈	C₂×#41	1	2	yes	—
423	C₇×C₈	C₂×#41	1	2	yes	—
424	C₂×#41	C₂³×C₇	1	2	yes	—
425	C₂²×Q₈	C₂³×C₇	1	2	yes	—
426	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
427	#137	—	2	8	no	—
428	#135	—	2	4	no	—
429	C₂×Q₁₆	—	2	4	no	—
430	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
431	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
432	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
433	C₂×Q₁₆	C₂³×C₇	1	4	yes	—
434	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
435	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
436	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
437	C₂×M₁₆	—	2	4	no	—
438	C₂×M₁₆	—	2	4	no	—
439	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₇	1	2	yes	—
441	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
442	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
445	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
446	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₇	1	2	yes	—
448	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
449	C₄⋊Q₈	—	2	2	no	—
450	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
451	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
452	#126	—	2	4	no	—
453	#126	—	2	4	no	—
454	#126	—	2	4	no	—
455	#126	—	2	4	no	—
456	#125	—	2	4	no	—
457	#125	—	2	4	no	—
458	#125	—	2	2	no	—
459	#125	—	2	4	no	—
460	#125	C₂³×C₇	1	4	yes	—
461	#124	—	2	4	no	—
462	#124	C₂³×C₇	1	4	yes	—
463	#123	—	2	2	no	—
464	#123	—	2	4	no	—
465	#123	—	2	4	no	—
466	#123	—	2	4	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	2	no	—
469	C₂²⋊Q₈	—	2	4	no	—
470	C₂²⋊Q₈	—	2	2	no	—
471	C₂²⋊Q₈	—	2	2	no	—
472	C₂²⋊Q₈	—	2	4	no	—
473	C₂²⋊Q₈	—	2	4	no	—
474	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
475	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
476	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
477	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
478	#121	C₂³×C₇	1	2	yes	—
479	#121	C₂³×C₇	1	2	yes	—
480	#120	—	2	2	no	—
481	C₄×Q₈	—	2	2	no	—
482	C₄×Q₈	—	2	2	no	—
483	C₄×Q₈	—	2	4	no	—
484	C₄×Q₈	C₂³×C₇	1	4	yes	—
485	C₄×Q₈	C₂³×C₇	1	4	yes	—
486	C₄×Q₈	C₂³×C₇	1	4	yes	—
487	C₄×D₄	—	2	2	no	—
488	C₄×D₄	—	2	2	no	—
489	C₄×D₄	—	2	4	no	—
490	C₄×D₄	—	2	4	no	—
491	C₄×D₄	C₂³×C₇	1	4	yes	—
492	#117	—	2	4	no	—
493	#117	—	2	2	no	—
494	#117	—	2	4	no	—
495	#117	—	2	4	no	—
496	#117	C₇×D₄	1	2	yes	—
497	#117	C₇×D₄	1	2	yes	—
498	#117	C₂³×C₇	1	2	yes	—
499	#117	C₂³×C₇	1	1	yes	characteristic
500	#117	C₂³×C₇	1	1	yes	characteristic
501	#117	C₂³×C₇	1	2	yes	—
502	C₂×C₄⋊C₄	—	2	2	no	—
503	C₂×C₄⋊C₄	—	2	2	no	—
504	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
505	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
506	C₂×C₂²⋊C₄	—	2	2	no	—
507	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
508	C₂×C₄²	C₂³×C₇	1	2	yes	—
509	#107	C₂³×C₇	1	2	yes	—
510	#107	C₂³×C₇	1	4	yes	—
511	#106	C₂³×C₇	1	2	yes	—
512	#106	C₂³×C₇	1	2	yes	—
513	#106	C₂³×C₇	1	1	yes	characteristic
514	#106	C₂³×C₇	1	1	yes	characteristic
515	#106	C₂³×C₇	1	4	yes	—
516	#103	C₂³×C₇	1	4	yes	—
517	#103	C₂³×C₇	1	4	yes	—
518	#103	C₂³×C₇	1	2	yes	—
519	#103	C₂³×C₇	1	2	yes	—
520	#103	C₂³×C₇	1	4	yes	—
521	#103	C₂³×C₇	1	4	yes	—
522	#103	C₂³×C₇	1	4	yes	—
523	#102	C₂³×C₇	1	2	yes	—
524	#102	C₂³×C₇	1	2	yes	—
525	#102	C₂³×C₇	1	4	yes	—
526	#97	C₂³×C₇	1	2	yes	—
527	#97	C₂³×C₇	1	2	yes	—
528	#97	C₂³×C₇	1	2	yes	—
529	#97	C₂³×C₇	1	2	yes	—
530	C₄×C₈	C₂³×C₇	1	2	yes	—
531	C₄×C₈	C₂³×C₇	1	2	yes	—
532	C₄×C₈	C₂³×C₇	1	2	yes	—
533	C₄×C₈	C₂³×C₇	1	2	yes	—
534	C₂²×C₇	—	2	2	no	—
535	C₂²×C₇	—	2	2	no	—
536	C₂²×C₇	—	2	2	no	—
537	C₂²×C₇	—	2	1	no	—
538	C₂²×C₇	—	2	1	no	—
539	C₂²×C₇	—	4	2	no	—
540	C₂²×C₇	—	4	2	no	—
541	C₂²×C₇	—	4	2	no	—
542	C₂²×C₇	—	8	2	no	—
543	C₂²×C₇	#532	1	2	yes	—
544	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
545	C₄×C₇	—	2	4	no	—
546	C₄×C₇	—	2	4	no	—
547	C₄×C₇	—	2	1	no	—
548	C₄×C₇	—	2	1	no	—
549	C₄×C₇	—	4	4	no	—
550	C₄×C₇	—	4	2	no	—
551	C₄×C₇	—	4	2	no	—
552	C₄×C₇	—	4	2	no	—
553	C₄×C₇	—	4	4	no	—
554	C₄×C₇	#532	1	2	yes	—
555	C₄×C₇	C₂²×#41	1	1	yes	characteristic
556	C₄×C₇	C₂²×#41	1	1	yes	characteristic
557	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
558	#41	—	2	4	no	—
559	#41	—	2	4	no	—
560	C₂×Q₈	—	2	2	no	—
561	C₂×Q₈	—	2	4	no	—
562	C₂×Q₈	—	2	4	no	—
563	C₂×Q₈	—	2	4	no	—
564	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
565	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
566	C₂×D₄	—	2	2	no	—
567	C₂×D₄	—	2	2	no	—
568	C₂×D₄	—	2	2	no	—
569	C₂×D₄	C₂⁴×C₇	1	2	yes	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	—	2	2	no	—
573	C₂²×C₄	—	2	2	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	4	no	—
576	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
577	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
578	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
579	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
580	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
581	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
582	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
583	Q₁₆	—	2	4	no	—
584	Q₁₆	—	2	4	no	—
585	Q₁₆	—	2	2	no	—
586	Q₁₆	—	2	8	no	—
587	SD₁₆	—	2	4	no	—
588	SD₁₆	—	2	8	no	—
589	D₈	—	2	2	no	—
590	M₁₆	—	2	2	no	—
591	M₁₆	—	2	4	no	—
592	M₁₆	—	2	2	no	—
593	M₁₆	—	2	2	no	—
594	M₁₆	—	2	2	no	—
595	M₁₆	—	2	2	no	—
596	M₁₆	C₇×#41	1	4	yes	—
597	C₂×C₈	—	2	2	no	—
598	C₂×C₈	—	2	4	no	—
599	C₂×C₈	C₇×#41	1	4	yes	—
600	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
601	C₂×C₈	C₂⁴×C₇	1	2	yes	—
602	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
603	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
604	C₂×C₈	C₂⁴×C₇	1	2	yes	—
605	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
606	C₄⋊C₄	—	2	2	no	—
607	C₄⋊C₄	—	2	4	no	—
608	C₄⋊C₄	—	2	4	no	—
609	C₄⋊C₄	—	2	4	no	—
610	C₄⋊C₄	—	2	4	no	—
611	C₄⋊C₄	—	2	4	no	—
612	C₄⋊C₄	—	2	4	no	—
613	C₄⋊C₄	—	2	4	no	—
614	C₄⋊C₄	—	2	4	no	—
615	C₄⋊C₄	—	2	2	no	—
616	C₄⋊C₄	—	2	4	no	—
617	C₄⋊C₄	—	2	2	no	—
618	C₄⋊C₄	—	2	4	no	—
619	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
620	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
621	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
622	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
623	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
624	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
625	C₂²⋊C₄	—	2	4	no	—
626	C₂²⋊C₄	—	2	4	no	—
627	C₂²⋊C₄	—	2	2	no	—
628	C₂²⋊C₄	—	2	2	no	—
629	C₂²⋊C₄	—	2	2	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	2	4	no	—
632	C₂²⋊C₄	—	2	4	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₂²⋊C₄	—	2	4	no	—
635	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
636	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
637	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
638	C₄²	—	2	4	no	—
639	C₄²	—	2	4	no	—
640	C₄²	—	2	4	no	—
641	C₄²	—	2	2	no	—
642	C₄²	C₂×C₇×D₄	1	4	yes	—
643	C₄²	C₂⁴×C₇	1	2	yes	—
644	C₄²	C₂⁴×C₇	1	2	yes	—
645	C₂×C₇	—	2	2	no	—
646	C₂×C₇	—	4	1	no	—
647	C₂×C₇	—	4	1	no	—
648	C₂×C₇	—	8	2	no	—
649	C₂×C₇	C₂×#532	1	1	yes	characteristic
650	C₂×C₇	#3586	1	2	yes	—
651	C₂³	—	2	2	no	—
652	C₂³	—	2	2	no	—
653	C₂³	—	4	2	no	—
654	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
655	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
656	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
657	Q₈	—	2	4	no	—
658	Q₈	—	2	2	no	—
659	Q₈	—	2	4	no	—
660	Q₈	—	2	2	no	—
661	Q₈	—	4	4	no	—
662	Q₈	—	8	4	no	—
663	D₄	—	2	2	no	—
664	D₄	—	2	2	no	—
665	D₄	—	4	4	no	—
666	D₄	—	4	4	no	—
667	D₄	—	8	2	no	—
668	C₂×C₄	—	2	2	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	1	no	—
671	C₂×C₄	—	2	1	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	2	2	no	—
674	C₂×C₄	—	2	2	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	4	no	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	2	4	no	—
680	C₂×C₄	—	4	4	no	—
681	C₂×C₄	—	4	4	no	—
682	C₂×C₄	—	4	4	no	—
683	C₂×C₄	—	8	4	no	—
684	C₂×C₄	—	8	2	no	—
685	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
686	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
687	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
688	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
689	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
690	C₂×C₄	C₂×C₇×#41	1	2	yes	—
691	C₂×C₄	C₂×C₇×#41	1	2	yes	—
692	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
693	C₈	—	2	1	no	—
694	C₈	—	2	1	no	—
695	C₈	—	2	2	no	—
696	C₈	C₂×C₇×#41	1	4	yes	—
697	C₈	C₂×C₇×#41	1	2	yes	—
698	C₈	C₂×C₇×#41	1	2	yes	—
699	C₇	#60711	1	1	yes	characteristic, direct product factor, Sylow
700	C₂²	—	2	2	no	—
701	C₂²	—	2	2	no	—
702	C₂²	—	2	2	no	—
703	C₂²	—	2	1	no	—
704	C₂²	—	2	1	no	—
705	C₂²	—	4	2	no	—
706	C₂²	—	4	2	no	—
707	C₂²	—	4	2	no	—
708	C₂²	—	8	2	no	—
709	C₂²	C₇×#532	1	2	yes	—
710	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
711	C₄	—	2	4	no	—
712	C₄	—	2	4	no	—
713	C₄	—	2	1	no	—
714	C₄	—	2	1	no	—
715	C₄	—	4	4	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	2	no	—
719	C₄	—	4	4	no	—
720	C₄	C₇×#532	1	2	yes	—
721	C₄	C₂²×C₇×#41	1	1	yes	characteristic
722	C₄	C₂²×C₇×#41	1	1	yes	characteristic
723	C₂	—	2	2	no	—
724	C₂	—	4	1	no	—
725	C₂	—	4	1	no	—
726	C₂	—	8	2	no	—
727	C₂	C₂×C₇×#532	1	1	yes	characteristic
728	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×#3185	C₂	semidirect
3. C₇×#3183	C₂	semidirect
4. C₇×#3178	C₂	semidirect
5. C₇×#3177	C₂	semidirect
6. C₇×#3173	C₂	semidirect
7. C₇×#3172	C₂	semidirect
8. C₇×#3163	C₂	semidirect
9. C₇×#3156	C₂	semidirect
10. C₇×#3137	C₂	semidirect
11. C₇×#3056	C₂	semidirect
13. C₂×C₇×#470	C₂	semidirect
14. C₇×#2952	C₂	semidirect
15. C₇×#2950	C₂	semidirect
16. C₇×#2947	C₂	semidirect
17. C₇×#2873	C₂	semidirect
40. C₇×#501	C₂²	semidirect
41. C₇×#499	C₂²	semidirect
42. C₇×#498	C₂²	semidirect
43. C₇×#497	C₂²	semidirect
44. C₇×#495	C₂²	semidirect
45. C₇×#494	C₂²	semidirect
46. C₇×#491	C₂²	semidirect
47. C₇×#489	C₂²	semidirect
48. C₇×#488	C₂²	semidirect
49. C₇×#487	C₂²	semidirect
64. C₇×#442	C₂²	semidirect
65. C₇×#441	C₂²	semidirect
66. C₇×#441	C₂²	semidirect
67. C₇×#441	C₂²	semidirect
68. C₇×#441	C₂²	semidirect
69. C₇×#440	C₂²	semidirect
70. C₄×C₇×Q₁₆	C₂²	semidirect
71. C₄×C₇×Q₁₆	C₂²	semidirect
72. C₄×C₇×SD₁₆	C₂²	semidirect
73. C₄×C₇×SD₁₆	C₂²	semidirect
89. #60711	C₇	direct (N × H)
200. #3495	C₂×C₇	semidirect
201. #3185	C₂×C₇	semidirect
202. #3183	C₂×C₇	semidirect
203. #3178	C₂×C₇	semidirect
204. #3177	C₂×C₇	semidirect
205. #3173	C₂×C₇	semidirect
206. #3172	C₂×C₇	semidirect
207. #3163	C₂×C₇	semidirect
208. #3156	C₂×C₇	semidirect
209. #3137	C₂×C₇	semidirect
210. #3056	C₂×C₇	semidirect
212. C₂×#470	C₂×C₇	semidirect
213. #2952	C₂×C₇	semidirect
214. #2950	C₂×C₇	semidirect
215. #2947	C₂×C₇	semidirect
216. #2873	C₂×C₇	semidirect
327. #501	C₂²×C₇	semidirect
328. #499	C₂²×C₇	semidirect
329. #498	C₂²×C₇	semidirect
330. #497	C₂²×C₇	semidirect
331. #495	C₂²×C₇	semidirect
332. #494	C₂²×C₇	semidirect
333. #491	C₂²×C₇	semidirect
334. #489	C₂²×C₇	semidirect
335. #488	C₂²×C₇	semidirect
336. #487	C₂²×C₇	semidirect
351. #442	C₂²×C₇	semidirect
352. #441	C₂²×C₇	semidirect
353. #441	C₂²×C₇	semidirect
354. #441	C₂²×C₇	semidirect
355. #441	C₂²×C₇	semidirect
356. #440	C₂²×C₇	semidirect
357. C₄×Q₁₆	C₂²×C₇	semidirect
358. C₄×Q₁₆	C₂²×C₇	semidirect
359. C₄×SD₁₆	C₂²×C₇	semidirect
360. C₄×SD₁₆	C₂²×C₇	semidirect
699. C₇	#60711	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×#3185	2
C₇×#3183	2
C₇×#3178	2
C₇×#3177	2
C₇×#3173	2
C₇×#3172	2
C₇×#3163	2
C₇×#3156	2
C₇×#3137	2
C₇×#3056	2
C₇×#3054	2
C₂×C₇×#470	2
C₇×#2952	2
C₇×#2950	2
C₇×#2947	2
C₇×#2873	2
#60711	7

Subgroups — C7×#60711 #13237927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60711 #13237927