Subgroups — C₇×#18921 #13196137

Order 1792.

864 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3518	C₂	1	1	yes	maximal, characteristic
2	C₇×#3248	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3231	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1652	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1651	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1646	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1646	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1645	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1645	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1482	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×Q₈²	C₂²	1	1	yes	characteristic
17	C₇×#556	—	2	1	no	—
18	C₇×#556	—	2	2	no	—
19	C₇×#556	—	2	1	no	—
20	C₇×#556	C₂²	1	1	yes	characteristic
21	C₇×#555	—	2	1	no	—
22	C₇×#555	C₂²	1	1	yes	characteristic
23	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
24	C₇×#548	—	2	1	no	—
25	C₇×#542	—	2	2	no	—
26	C₇×#542	—	2	1	no	—
27	C₇×#533	—	2	1	no	—
28	C₇×#533	C₂²	1	1	yes	characteristic
29	C₇×#533	C₂²	1	1	yes	characteristic
30	C₇×#529	—	2	1	no	—
31	C₇×#529	C₂²	1	1	yes	characteristic
32	C₇×#519	—	2	1	no	—
33	C₄×C₇×#41	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	1	yes	characteristic
35	C₇×#498	C₂²	1	1	yes	characteristic
36	C₇×#487	C₂²	1	1	yes	characteristic
37	C₇×#487	C₂²	1	1	yes	characteristic
38	C₇×#485	—	2	2	no	—
39	C₇×#485	—	2	1	no	—
40	C₇×#479	—	2	1	no	—
41	C₇×#479	—	2	2	no	—
42	C₇×#475	C₂²	1	1	yes	characteristic
43	C₇×#475	C₂²	1	1	yes	characteristic
44	C₇×#474	C₂²	1	1	yes	characteristic
45	C₇×#474	C₂²	1	1	yes	characteristic
46	C₇×#467	—	2	1	no	—
47	C₇×#465	—	2	1	no	—
48	C₇×#464	—	2	1	no	—
49	C₇×#462	C₂²	1	1	yes	characteristic
50	C₇×#462	C₂²	1	1	yes	characteristic
51	C₇×#461	C₂²	1	1	yes	characteristic
52	C₇×#461	C₂²	1	1	yes	characteristic
53	C₇×#452	—	2	1	no	—
54	C₄×C₇×Q₁₆	—	2	1	no	—
55	C₄×C₇×SD₁₆	—	2	1	no	—
56	C₇×#431	C₂²	1	1	yes	characteristic
57	C₇×#431	C₂²	1	1	yes	characteristic
58	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₇×#333	C₂²	1	1	yes	characteristic
60	C₇×#333	C₂²	1	1	yes	characteristic
61	C₇×#332	C₂²	1	1	yes	characteristic
62	C₇×#332	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
68	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
69	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	#18921	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	D₄	1	1	yes	characteristic
73	C₂²×C₇×Q₈	—	2	1	no	—
74	C₂×C₇×Q₁₆	—	2	1	no	—
75	C₂×C₇×Q₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂²×C₇×C₈	—	2	1	no	—
79	C₇×C₄⋊Q₈	—	2	2	no	—
80	C₇×C₄⋊Q₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	—	2	1	no	—
82	C₇×C₄⋊Q₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₇×#126	—	2	2	no	—
88	C₇×#126	—	2	2	no	—
89	C₇×#126	—	2	2	no	—
90	C₇×#126	—	2	2	no	—
91	C₇×#126	—	2	1	no	—
92	C₇×#125	—	2	1	no	—
93	C₇×#125	—	2	1	no	—
94	C₇×#125	D₄	1	1	yes	characteristic
95	C₇×#124	—	2	2	no	—
96	C₇×#124	—	2	2	no	—
97	C₇×#124	—	2	2	no	—
98	C₇×#124	—	2	2	no	—
99	C₇×#124	—	2	1	no	—
100	C₇×#124	D₄	1	1	yes	characteristic
101	C₇×#123	—	2	2	no	—
102	C₇×#123	—	2	1	no	—
103	C₇×C₂²⋊Q₈	—	2	2	no	—
104	C₇×C₂²⋊Q₈	—	2	2	no	—
105	C₇×C₂²⋊Q₈	—	2	1	no	—
106	C₇×C₂²⋊Q₈	—	2	1	no	—
107	C₇×C₂²⋊Q₈	—	2	1	no	—
108	C₇×C₂²⋊Q₈	—	2	2	no	—
109	C₇×C₂²⋊Q₈	—	2	1	no	—
110	C₇×C₂²⋊Q₈	—	2	1	no	—
111	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
112	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
113	C₇×#121	—	2	2	no	—
114	C₇×#121	—	2	1	no	—
115	C₇×#121	—	2	1	no	—
116	C₇×#121	D₄	1	1	yes	characteristic
117	C₄×C₇×Q₈	—	2	2	no	—
118	C₄×C₇×Q₈	—	2	2	no	—
119	C₄×C₇×Q₈	—	2	2	no	—
120	C₄×C₇×Q₈	—	2	1	no	—
121	C₄×C₇×Q₈	—	2	1	no	—
122	C₄×C₇×Q₈	—	2	1	no	—
123	C₄×C₇×Q₈	—	2	1	no	—
124	C₄×C₇×Q₈	—	2	1	no	—
125	C₄×C₇×Q₈	—	2	1	no	—
126	C₄×C₇×Q₈	D₄	1	2	yes	—
127	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
128	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
129	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
130	C₄×C₇×D₄	—	2	2	no	—
131	C₄×C₇×D₄	—	2	1	no	—
132	C₄×C₇×D₄	—	2	1	no	—
133	C₄×C₇×D₄	—	2	1	no	—
134	C₄×C₇×D₄	—	2	1	no	—
135	C₄×C₇×D₄	D₄	1	1	yes	characteristic
136	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
137	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
138	C₇×#117	—	2	2	no	—
139	C₇×#117	—	2	2	no	—
140	C₇×#117	—	2	1	no	—
141	C₇×#117	—	2	1	no	—
142	C₇×#117	—	2	1	no	—
143	C₇×#117	—	2	1	no	—
144	C₇×#117	—	2	1	no	—
145	C₇×#117	—	2	1	no	—
146	C₇×#117	—	2	1	no	—
147	C₇×#117	D₄	1	1	yes	characteristic
148	C₂×C₄²×C₇	—	2	1	no	—
149	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
150	C₇×#107	—	2	1	no	—
151	C₇×#107	—	2	1	no	—
152	C₇×#107	—	2	2	no	—
153	C₇×#106	—	2	1	no	—
154	C₇×#106	—	2	1	no	—
155	C₇×#106	—	2	2	no	—
156	C₇×C₄⋊C₈	—	2	1	no	—
157	C₇×C₄⋊C₈	—	2	1	no	—
158	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₇×#103	—	2	1	no	—
163	C₇×#103	—	2	1	no	—
164	C₇×#103	—	2	2	no	—
165	C₇×#103	—	2	2	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#103	—	2	1	no	—
168	C₇×#103	—	2	1	no	—
169	C₇×#103	—	2	1	no	—
170	C₇×#102	—	2	1	no	—
171	C₇×#102	—	2	1	no	—
172	C₇×C₂²⋊C₈	—	2	1	no	—
173	C₇×C₂²⋊C₈	—	2	1	no	—
174	C₄×C₇×C₈	—	2	1	no	—
175	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
176	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
177	#3518	C₂×C₇	1	1	yes	characteristic
178	#3248	C₂×C₇	1	1	yes	characteristic, complemented
179	#3231	C₂×C₇	1	1	yes	characteristic, complemented
180	#3107	C₂×C₇	1	1	yes	characteristic, complemented
181	#1652	C₂×C₇	1	1	yes	characteristic, complemented
182	#1651	C₂×C₇	1	1	yes	characteristic, complemented
183	#1650	C₂×C₇	1	1	yes	characteristic, complemented
184	#1649	C₂×C₇	1	1	yes	characteristic, complemented
185	#1646	C₂×C₇	1	1	yes	characteristic, complemented
186	#1646	C₂×C₇	1	1	yes	characteristic, complemented
187	#1645	C₂×C₇	1	1	yes	characteristic, complemented
188	#1645	C₂×C₇	1	1	yes	characteristic, complemented
189	#1542	C₂×C₇	1	1	yes	characteristic, complemented
190	#1482	C₂×C₇	1	1	yes	characteristic, complemented
191	#1477	C₂×C₇	1	1	yes	characteristic, complemented
192	C₇×#41	—	2	2	no	—
193	C₇×#41	—	4	1	no	—
194	C₂×C₇×Q₈	—	2	2	no	—
195	C₂×C₇×Q₈	—	2	2	no	—
196	C₂×C₇×Q₈	—	2	1	no	—
197	C₂×C₇×Q₈	—	2	1	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	4	2	no	—
200	C₂×C₇×Q₈	—	4	2	no	—
201	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
202	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
204	C₂×C₇×D₄	—	2	1	no	—
205	C₂×C₇×D₄	—	2	2	no	—
206	C₂×C₇×D₄	—	2	1	no	—
207	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
208	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₇	—	2	2	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	1	no	—
213	C₂²×C₄×C₇	—	2	1	no	—
214	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
217	C₇×Q₁₆	—	4	2	no	—
218	C₇×Q₁₆	—	8	1	no	—
219	C₇×SD₁₆	—	4	2	no	—
220	C₇×SD₁₆	—	8	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	—	2	1	no	—
224	C₂×C₇×C₈	—	2	1	no	—
225	C₂×C₇×C₈	—	2	1	no	—
226	C₂×C₇×C₈	—	2	1	no	—
227	C₂×C₇×C₈	—	8	1	no	—
228	C₇×C₄⋊C₄	—	2	2	no	—
229	C₇×C₄⋊C₄	—	2	2	no	—
230	C₇×C₄⋊C₄	—	2	2	no	—
231	C₇×C₄⋊C₄	—	2	2	no	—
232	C₇×C₄⋊C₄	—	2	1	no	—
233	C₇×C₄⋊C₄	—	2	1	no	—
234	C₇×C₄⋊C₄	—	2	2	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	1	no	—
237	C₇×C₄⋊C₄	—	2	2	no	—
238	C₇×C₄⋊C₄	—	2	1	no	—
239	C₇×C₄⋊C₄	—	2	1	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	1	no	—
242	C₇×C₄⋊C₄	—	2	1	no	—
243	C₇×C₄⋊C₄	—	2	1	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	1	no	—
246	C₇×C₄⋊C₄	—	2	1	no	—
247	C₇×C₄⋊C₄	—	4	1	no	—
248	C₇×C₄⋊C₄	—	4	1	no	—
249	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
250	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
251	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
252	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
253	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
254	C₇×C₂²⋊C₄	—	2	2	no	—
255	C₇×C₂²⋊C₄	—	2	2	no	—
256	C₇×C₂²⋊C₄	—	2	2	no	—
257	C₇×C₂²⋊C₄	—	2	2	no	—
258	C₇×C₂²⋊C₄	—	2	2	no	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	1	no	—
261	C₇×C₂²⋊C₄	—	2	1	no	—
262	C₇×C₂²⋊C₄	—	2	1	no	—
263	C₇×C₂²⋊C₄	—	2	1	no	—
264	C₇×C₂²⋊C₄	—	2	1	no	—
265	C₇×C₂²⋊C₄	—	2	1	no	—
266	C₇×C₂²⋊C₄	—	2	1	no	—
267	C₇×C₂²⋊C₄	—	2	1	no	—
268	C₇×C₂²⋊C₄	—	4	1	no	—
269	C₄²×C₇	—	2	2	no	—
270	C₄²×C₇	—	2	2	no	—
271	C₄²×C₇	—	2	2	no	—
272	C₄²×C₇	—	2	1	no	—
273	C₄²×C₇	—	2	1	no	—
274	C₄²×C₇	—	2	2	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	2	1	no	—
277	C₄²×C₇	—	4	1	no	—
278	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₇	C₂⁴	1	1	yes	characteristic
282	Q₈²	C₂²×C₇	1	1	yes	characteristic
283	#556	—	2	1	no	—
284	#556	—	2	2	no	—
285	#556	—	2	1	no	—
286	#556	C₂²×C₇	1	1	yes	characteristic
287	#555	—	2	1	no	—
288	#555	C₂²×C₇	1	1	yes	characteristic
289	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
290	#548	—	2	1	no	—
291	#542	—	2	2	no	—
292	#542	—	2	1	no	—
293	#533	—	2	1	no	—
294	#533	C₂²×C₇	1	1	yes	characteristic
295	#533	C₂²×C₇	1	1	yes	characteristic
296	#529	—	2	1	no	—
297	#529	C₂²×C₇	1	1	yes	characteristic
298	#519	—	2	1	no	—
299	C₄×#41	C₂²×C₇	1	1	yes	characteristic
300	#500	C₂²×C₇	1	1	yes	characteristic
301	#498	C₂²×C₇	1	1	yes	characteristic
302	#487	C₂²×C₇	1	1	yes	characteristic
303	#487	C₂²×C₇	1	1	yes	characteristic
304	#485	—	2	2	no	—
305	#485	—	2	1	no	—
306	#479	—	2	1	no	—
307	#479	—	2	2	no	—
308	#475	C₂²×C₇	1	1	yes	characteristic
309	#475	C₂²×C₇	1	1	yes	characteristic
310	#474	C₂²×C₇	1	1	yes	characteristic
311	#474	C₂²×C₇	1	1	yes	characteristic
312	#467	—	2	1	no	—
313	#465	—	2	1	no	—
314	#464	—	2	1	no	—
315	#462	C₂²×C₇	1	1	yes	characteristic
316	#462	C₂²×C₇	1	1	yes	characteristic
317	#461	C₂²×C₇	1	1	yes	characteristic
318	#461	C₂²×C₇	1	1	yes	characteristic
319	#452	—	2	1	no	—
320	C₄×Q₁₆	—	2	1	no	—
321	C₄×SD₁₆	—	2	1	no	—
322	#431	C₂²×C₇	1	1	yes	characteristic
323	#431	C₂²×C₇	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
325	#333	C₂²×C₇	1	1	yes	characteristic
326	#333	C₂²×C₇	1	1	yes	characteristic
327	#332	C₂²×C₇	1	1	yes	characteristic
328	#332	C₂²×C₇	1	1	yes	characteristic
329	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
331	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
333	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
334	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
335	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
336	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
337	C₂³×C₇	—	2	1	no	—
338	C₂³×C₇	—	2	1	no	—
339	C₂³×C₇	#120	1	1	yes	characteristic
340	C₇×Q₈	—	2	1	no	—
341	C₇×Q₈	—	2	1	no	—
342	C₇×Q₈	—	2	1	no	—
343	C₇×Q₈	—	4	2	no	—
344	C₇×Q₈	—	4	4	no	—
345	C₇×Q₈	—	4	1	no	—
346	C₇×Q₈	—	4	1	no	—
347	C₇×Q₈	—	4	1	no	—
348	C₇×Q₈	—	4	1	no	—
349	C₇×Q₈	—	4	1	no	—
350	C₇×Q₈	—	4	1	no	—
351	C₇×Q₈	—	8	2	no	—
352	C₇×Q₈	#135	1	2	yes	—
353	C₇×D₄	—	2	1	no	—
354	C₇×D₄	—	2	1	no	—
355	C₇×D₄	—	2	1	no	—
356	C₇×D₄	—	2	1	no	—
357	C₇×D₄	—	4	2	no	—
358	C₇×D₄	—	8	1	no	—
359	C₇×D₄	—	8	2	no	—
360	C₂×C₄×C₇	—	2	1	no	—
361	C₂×C₄×C₇	—	2	1	no	—
362	C₂×C₄×C₇	—	2	1	no	—
363	C₂×C₄×C₇	—	2	1	no	—
364	C₂×C₄×C₇	—	2	1	no	—
365	C₂×C₄×C₇	—	2	1	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	2	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	2	2	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	2	no	—
374	C₂×C₄×C₇	—	4	4	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	1	no	—
379	C₂×C₄×C₇	—	4	1	no	—
380	C₂×C₄×C₇	—	4	1	no	—
381	C₂×C₄×C₇	—	8	1	no	—
382	C₂×C₄×C₇	#120	1	1	yes	characteristic
383	C₂×C₄×C₇	#120	1	1	yes	characteristic
384	C₂×C₄×C₇	#120	1	1	yes	characteristic
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
387	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
388	C₇×C₈	—	4	1	no	—
389	C₇×C₈	—	4	1	no	—
390	C₇×C₈	—	4	1	no	—
391	C₇×C₈	—	4	1	no	—
392	C₇×C₈	—	4	1	no	—
393	C₇×C₈	—	4	1	no	—
394	C₂×#41	C₇×D₄	1	1	yes	characteristic
395	C₂²×Q₈	—	2	1	no	—
396	C₂×Q₁₆	—	2	1	no	—
397	C₂×Q₁₆	—	2	1	no	—
398	C₂×SD₁₆	—	2	1	no	—
399	C₂×SD₁₆	—	2	1	no	—
400	C₂²×C₈	—	2	1	no	—
401	C₄⋊Q₈	—	2	2	no	—
402	C₄⋊Q₈	—	2	1	no	—
403	C₄⋊Q₈	—	2	1	no	—
404	C₄⋊Q₈	—	2	1	no	—
405	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
406	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
407	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
408	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
409	#126	—	2	2	no	—
410	#126	—	2	2	no	—
411	#126	—	2	2	no	—
412	#126	—	2	2	no	—
413	#126	—	2	1	no	—
414	#125	—	2	1	no	—
415	#125	—	2	1	no	—
416	#125	C₇×D₄	1	1	yes	characteristic
417	#124	—	2	2	no	—
418	#124	—	2	2	no	—
419	#124	—	2	2	no	—
420	#124	—	2	2	no	—
421	#124	—	2	1	no	—
422	#124	C₇×D₄	1	1	yes	characteristic
423	#123	—	2	2	no	—
424	#123	—	2	1	no	—
425	C₂²⋊Q₈	—	2	2	no	—
426	C₂²⋊Q₈	—	2	2	no	—
427	C₂²⋊Q₈	—	2	1	no	—
428	C₂²⋊Q₈	—	2	1	no	—
429	C₂²⋊Q₈	—	2	1	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	1	no	—
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
434	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
435	#121	—	2	2	no	—
436	#121	—	2	1	no	—
437	#121	—	2	1	no	—
438	#121	C₇×D₄	1	1	yes	characteristic
439	C₄×Q₈	—	2	2	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	2	2	no	—
442	C₄×Q₈	—	2	1	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
449	C₄×Q₈	C₇×D₄	1	2	yes	—
450	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
451	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
452	C₄×D₄	—	2	2	no	—
453	C₄×D₄	—	2	1	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	C₇×D₄	1	1	yes	characteristic
458	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
459	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
460	#117	—	2	2	no	—
461	#117	—	2	2	no	—
462	#117	—	2	1	no	—
463	#117	—	2	1	no	—
464	#117	—	2	1	no	—
465	#117	—	2	1	no	—
466	#117	—	2	1	no	—
467	#117	—	2	1	no	—
468	#117	—	2	1	no	—
469	#117	C₇×D₄	1	1	yes	characteristic
470	C₂×C₄²	—	2	1	no	—
471	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
472	#107	—	2	1	no	—
473	#107	—	2	1	no	—
474	#107	—	2	2	no	—
475	#106	—	2	1	no	—
476	#106	—	2	1	no	—
477	#106	—	2	2	no	—
478	C₄⋊C₈	—	2	1	no	—
479	C₄⋊C₈	—	2	1	no	—
480	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
481	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
482	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
483	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
484	#103	—	2	1	no	—
485	#103	—	2	1	no	—
486	#103	—	2	2	no	—
487	#103	—	2	2	no	—
488	#103	—	2	1	no	—
489	#103	—	2	1	no	—
490	#103	—	2	1	no	—
491	#103	—	2	1	no	—
492	#102	—	2	1	no	—
493	#102	—	2	1	no	—
494	C₂²⋊C₈	—	2	1	no	—
495	C₂²⋊C₈	—	2	1	no	—
496	C₄×C₈	—	2	1	no	—
497	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
498	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
499	C₂²×C₇	—	2	1	no	—
500	C₂²×C₇	—	2	1	no	—
501	C₂²×C₇	—	2	1	no	—
502	C₂²×C₇	—	4	1	no	—
503	C₂²×C₇	—	4	1	no	—
504	C₂²×C₇	—	4	1	no	—
505	C₂²×C₇	—	4	1	no	—
506	C₂²×C₇	—	4	1	no	—
507	C₂²×C₇	—	4	1	no	—
508	C₂²×C₇	C₂×#120	1	1	yes	characteristic
509	C₄×C₇	—	2	1	no	—
510	C₄×C₇	—	2	2	no	—
511	C₄×C₇	—	2	1	no	—
512	C₄×C₇	—	2	1	no	—
513	C₄×C₇	—	2	1	no	—
514	C₄×C₇	—	4	2	no	—
515	C₄×C₇	—	4	1	no	—
516	C₄×C₇	—	4	1	no	—
517	C₄×C₇	—	4	1	no	—
518	C₄×C₇	—	4	1	no	—
519	C₄×C₇	—	4	2	no	—
520	C₄×C₇	—	4	1	no	—
521	C₄×C₇	—	8	1	no	—
522	C₄×C₇	—	8	1	no	—
523	C₄×C₇	C₂×#135	1	1	yes	characteristic
524	C₄×C₇	#575	1	1	yes	characteristic
525	C₄×C₇	#577	1	1	yes	characteristic
526	C₄×C₇	#578	1	1	yes	characteristic
527	#41	—	2	2	no	—
528	#41	—	4	1	no	—
529	C₂×Q₈	—	2	2	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	4	2	no	—
535	C₂×Q₈	—	4	2	no	—
536	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
537	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
538	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
539	C₂×D₄	—	2	1	no	—
540	C₂×D₄	—	2	2	no	—
541	C₂×D₄	—	2	1	no	—
542	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
543	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
550	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
551	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
552	Q₁₆	—	4	2	no	—
553	Q₁₆	—	8	1	no	—
554	SD₁₆	—	4	2	no	—
555	SD₁₆	—	8	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	8	1	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	4	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
585	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
586	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
587	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
588	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	2	2	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	1	no	—
596	C₂²⋊C₄	—	2	1	no	—
597	C₂²⋊C₄	—	2	1	no	—
598	C₂²⋊C₄	—	2	1	no	—
599	C₂²⋊C₄	—	2	1	no	—
600	C₂²⋊C₄	—	2	1	no	—
601	C₂²⋊C₄	—	2	1	no	—
602	C₂²⋊C₄	—	2	1	no	—
603	C₂²⋊C₄	—	4	1	no	—
604	C₄²	—	2	2	no	—
605	C₄²	—	2	2	no	—
606	C₄²	—	2	2	no	—
607	C₄²	—	2	1	no	—
608	C₄²	—	2	1	no	—
609	C₄²	—	2	2	no	—
610	C₄²	—	2	1	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	4	1	no	—
613	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
614	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
615	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
616	C₄²	C₂⁴×C₇	1	1	yes	characteristic
617	C₂×C₇	—	4	1	no	—
618	C₂×C₇	—	8	1	no	—
619	C₂×C₇	—	8	1	no	—
620	C₂×C₇	#3002	1	1	yes	characteristic
621	C₂×C₇	#3013	1	1	yes	characteristic
622	C₂×C₇	#3022	1	1	yes	characteristic
623	C₂³	—	2	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	C₇×#120	1	1	yes	characteristic
626	Q₈	—	2	1	no	—
627	Q₈	—	2	1	no	—
628	Q₈	—	2	1	no	—
629	Q₈	—	4	2	no	—
630	Q₈	—	4	4	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	8	2	no	—
638	Q₈	C₇×#135	1	2	yes	—
639	D₄	—	2	1	no	—
640	D₄	—	2	1	no	—
641	D₄	—	2	1	no	—
642	D₄	—	2	1	no	—
643	D₄	—	4	2	no	—
644	D₄	—	8	1	no	—
645	D₄	—	8	2	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	2	no	—
660	C₂×C₄	—	4	4	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	8	1	no	—
668	C₂×C₄	C₇×#120	1	1	yes	characteristic
669	C₂×C₄	C₇×#120	1	1	yes	characteristic
670	C₂×C₄	C₇×#120	1	1	yes	characteristic
671	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
672	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
673	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
674	C₈	—	4	1	no	—
675	C₈	—	4	1	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₈	—	4	1	no	—
680	C₇	#18921	1	1	yes	characteristic, direct product factor, Sylow
681	C₂²	—	2	1	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	4	1	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	1	no	—
690	C₂²	C₂×C₇×#120	1	1	yes	characteristic
691	C₄	—	2	1	no	—
692	C₄	—	2	2	no	—
693	C₄	—	2	1	no	—
694	C₄	—	2	1	no	—
695	C₄	—	2	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	1	no	—
700	C₄	—	4	1	no	—
701	C₄	—	4	2	no	—
702	C₄	—	4	1	no	—
703	C₄	—	8	1	no	—
704	C₄	—	8	1	no	—
705	C₄	C₂×C₇×#135	1	1	yes	characteristic
706	C₄	C₇×#575	1	1	yes	characteristic
707	C₄	C₇×#577	1	1	yes	characteristic
708	C₄	C₇×#578	1	1	yes	characteristic
709	C₂	—	4	1	no	—
710	C₂	—	8	1	no	—
711	C₂	—	8	1	no	—
712	C₂	C₇×#3002	1	1	yes	characteristic
713	C₂	C₇×#3013	1	1	yes	characteristic
714	C₂	C₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3518	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3248	C₂	semidirect
3. C₇×#3231	C₂	semidirect
4. C₇×#3107	C₂	semidirect
5. C₇×#1652	C₂	semidirect
6. C₇×#1651	C₂	semidirect
7. C₇×#1650	C₂	semidirect
8. C₇×#1649	C₂	semidirect
9. C₇×#1646	C₂	semidirect
10. C₇×#1646	C₂	semidirect
11. C₇×#1645	C₂	semidirect
12. C₇×#1645	C₂	semidirect
13. C₇×#1542	C₂	semidirect
14. C₇×#1482	C₂	semidirect
15. C₇×#1477	C₂	semidirect
71. #18921	C₇	direct (N × H)
178. #3248	C₂×C₇	semidirect
179. #3231	C₂×C₇	semidirect
180. #3107	C₂×C₇	semidirect
181. #1652	C₂×C₇	semidirect
182. #1651	C₂×C₇	semidirect
183. #1650	C₂×C₇	semidirect
184. #1649	C₂×C₇	semidirect
185. #1646	C₂×C₇	semidirect
186. #1646	C₂×C₇	semidirect
187. #1645	C₂×C₇	semidirect
188. #1645	C₂×C₇	semidirect
189. #1542	C₂×C₇	semidirect
190. #1482	C₂×C₇	semidirect
191. #1477	C₂×C₇	semidirect
680. C₇	#18921	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×#3518	2
C₇×#3248	2
C₇×#3231	2
C₇×#3107	2
C₇×#1652	2
C₇×#1651	2
C₇×#1650	2
C₇×#1649	2
C₇×#1646	2
C₇×#1645	2
C₇×#1542	2
C₇×#1482	2
C₇×#1477	2
#18921	7

Subgroups — C7×#18921 #13196137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18921 #13196137