Subgroups — C₇×#18661 #13195877

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3515	C₂	1	1	yes	maximal, characteristic
2	C₇×#3233	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1648	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1647	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1644	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1643	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1636	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1635	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1554	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1494	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1476	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#557	C₂²	1	1	yes	characteristic
17	C₇×#554	C₂²	1	1	yes	characteristic
18	C₇×#554	C₂²	1	1	yes	characteristic
19	C₇×#553	—	2	1	no	—
20	C₇×#552	—	2	1	no	—
21	C₇×#552	—	2	2	no	—
22	C₇×#552	—	2	1	no	—
23	C₇×#547	—	2	1	no	—
24	C₇×#547	C₂²	1	1	yes	characteristic
25	C₇×#541	—	2	1	no	—
26	C₇×#539	—	2	2	no	—
27	C₇×#533	C₂²	1	1	yes	characteristic
28	C₇×#533	C₂²	1	1	yes	characteristic
29	C₇×#529	—	2	1	no	—
30	C₇×#529	C₂²	1	1	yes	characteristic
31	C₂×C₇×#125	—	2	1	no	—
32	C₇×#520	—	2	1	no	—
33	C₄×C₇×#41	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	1	yes	characteristic
35	C₇×#499	—	2	1	no	—
36	C₇×#494	C₂²	1	1	yes	characteristic
37	C₇×#488	—	2	1	no	—
38	C₇×#487	C₂²	1	1	yes	characteristic
39	C₇×#487	C₂²	1	1	yes	characteristic
40	C₇×#485	—	2	1	no	—
41	C₇×#485	—	2	1	no	—
42	C₇×#482	—	2	1	no	—
43	C₇×#477	C₂²	1	1	yes	characteristic
44	C₇×#477	C₂²	1	1	yes	characteristic
45	C₇×#476	C₂²	1	1	yes	characteristic
46	C₇×#476	C₂²	1	1	yes	characteristic
47	C₇×#475	C₂²	1	1	yes	characteristic
48	C₇×#474	C₂²	1	1	yes	characteristic
49	C₇×#467	—	2	2	no	—
50	C₇×#462	C₂²	1	1	yes	characteristic
51	C₇×#461	C₂²	1	1	yes	characteristic
52	C₇×#452	—	2	1	no	—
53	C₇×#441	—	2	1	no	—
54	C₇×#440	—	2	1	no	—
55	C₄×C₇×Q₁₆	—	2	2	no	—
56	C₇×#432	C₂²	1	1	yes	characteristic
57	C₇×#431	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₇×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	#18661	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	D₄	1	1	yes	characteristic
74	C₂×C₇×Q₁₆	—	2	2	no	—
75	C₂×C₇×Q₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	—	2	1	no	—
77	C₂²×C₇×C₈	—	2	1	no	—
78	C₇×C₄⋊Q₈	—	2	1	no	—
79	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₇×#126	—	2	2	no	—
84	C₇×#126	—	2	2	no	—
85	C₇×#126	—	2	2	no	—
86	C₇×#126	—	2	1	no	—
87	C₇×#125	—	2	2	no	—
88	C₇×#125	—	2	2	no	—
89	C₇×#125	—	2	2	no	—
90	C₇×#125	—	2	1	no	—
91	C₇×#125	—	2	1	no	—
92	C₇×#125	—	2	2	no	—
93	C₇×#125	—	2	1	no	—
94	C₇×#125	—	2	1	no	—
95	C₇×#125	—	2	1	no	—
96	C₇×#125	D₄	1	1	yes	characteristic
97	C₇×#124	D₄	1	1	yes	characteristic
98	C₇×#123	—	2	2	no	—
99	C₇×#123	—	2	2	no	—
100	C₇×#123	—	2	2	no	—
101	C₇×#123	—	2	1	no	—
102	C₇×#123	—	2	2	no	—
103	C₇×C₂²⋊Q₈	—	2	1	no	—
104	C₇×C₂²⋊Q₈	—	2	2	no	—
105	C₇×C₂²⋊Q₈	—	2	2	no	—
106	C₇×C₂²⋊Q₈	—	2	1	no	—
107	C₇×C₂²⋊Q₈	—	2	1	no	—
108	C₇×C₂²⋊Q₈	—	2	1	no	—
109	C₇×C₂²⋊Q₈	—	2	1	no	—
110	C₇×C₂²⋊Q₈	—	2	1	no	—
111	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
112	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
113	C₇×#121	—	2	2	no	—
114	C₇×#121	—	2	1	no	—
115	C₇×#121	D₄	1	1	yes	characteristic
116	C₄×C₇×Q₈	—	2	1	no	—
117	C₄×C₇×Q₈	—	2	1	no	—
118	C₄×C₇×Q₈	—	2	1	no	—
119	C₄×C₇×Q₈	—	2	1	no	—
120	C₄×C₇×Q₈	—	2	1	no	—
121	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
122	C₄×C₇×Q₈	D₄	1	2	yes	—
123	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
124	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
125	C₄×C₇×D₄	—	2	2	no	—
126	C₄×C₇×D₄	—	2	2	no	—
127	C₄×C₇×D₄	—	2	1	no	—
128	C₄×C₇×D₄	—	2	1	no	—
129	C₄×C₇×D₄	—	2	1	no	—
130	C₄×C₇×D₄	—	2	1	no	—
131	C₄×C₇×D₄	—	2	1	no	—
132	C₄×C₇×D₄	D₄	1	1	yes	characteristic
133	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
134	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
135	C₇×#117	—	2	2	no	—
136	C₇×#117	—	2	2	no	—
137	C₇×#117	—	2	1	no	—
138	C₇×#117	—	2	1	no	—
139	C₇×#117	—	2	1	no	—
140	C₇×#117	—	2	1	no	—
141	C₇×#117	D₄	1	1	yes	characteristic
142	C₂×C₇×C₄⋊C₄	—	2	2	no	—
143	C₂×C₇×C₄⋊C₄	—	2	2	no	—
144	C₂×C₇×C₄⋊C₄	—	2	1	no	—
145	C₂×C₇×C₄⋊C₄	—	2	1	no	—
146	C₂×C₇×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄²×C₇	—	2	1	no	—
148	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
149	C₇×#107	—	2	2	no	—
150	C₇×#107	—	2	1	no	—
151	C₇×#107	—	2	1	no	—
152	C₇×#107	—	2	1	no	—
153	C₇×#106	—	2	1	no	—
154	C₇×#106	—	2	1	no	—
155	C₇×#106	—	2	1	no	—
156	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₇×#103	—	2	2	no	—
161	C₇×#103	—	2	1	no	—
162	C₇×#103	—	2	1	no	—
163	C₇×#103	—	2	1	no	—
164	C₇×#103	—	2	2	no	—
165	C₇×#103	—	2	1	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#102	—	2	1	no	—
168	C₇×#102	—	2	1	no	—
169	C₇×#102	—	2	1	no	—
170	C₇×C₂²⋊C₈	—	2	1	no	—
171	C₇×C₂²⋊C₈	—	2	1	no	—
172	C₇×#97	—	2	1	no	—
173	C₄×C₇×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	#3515	C₂×C₇	1	1	yes	characteristic
178	#3233	C₂×C₇	1	1	yes	characteristic, complemented
179	#3170	C₂×C₇	1	1	yes	characteristic, complemented
180	#3116	C₂×C₇	1	1	yes	characteristic, complemented
181	#1648	C₂×C₇	1	1	yes	characteristic, complemented
182	#1647	C₂×C₇	1	1	yes	characteristic, complemented
183	#1644	C₂×C₇	1	1	yes	characteristic, complemented
184	#1643	C₂×C₇	1	1	yes	characteristic, complemented
185	#1636	C₂×C₇	1	1	yes	characteristic, complemented
186	#1635	C₂×C₇	1	1	yes	characteristic, complemented
187	#1632	C₂×C₇	1	1	yes	characteristic, complemented
188	#1632	C₂×C₇	1	1	yes	characteristic, complemented
189	#1554	C₂×C₇	1	1	yes	characteristic, complemented
190	#1494	C₂×C₇	1	1	yes	characteristic, complemented
191	#1476	C₂×C₇	1	1	yes	characteristic, complemented
192	C₇×#41	—	2	2	no	—
193	C₇×#41	—	4	1	no	—
194	C₇×#41	—	4	2	no	—
195	C₇×#41	—	4	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	C₂×D₄	1	1	yes	characteristic
200	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
202	C₂×C₇×D₄	—	2	2	no	—
203	C₂×C₇×D₄	—	2	1	no	—
204	C₂×C₇×D₄	—	2	1	no	—
205	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
206	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₇	—	2	2	no	—
208	C₂²×C₄×C₇	—	2	2	no	—
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	4	no	—
218	C₇×Q₁₆	—	4	2	no	—
219	C₇×SD₁₆	—	4	2	no	—
220	C₂×C₇×C₈	—	2	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	—	4	2	no	—
227	C₇×C₄⋊C₄	—	2	2	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	2	no	—
233	C₇×C₄⋊C₄	—	2	2	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	2	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	—	2	1	no	—
248	C₇×C₄⋊C₄	—	2	1	no	—
249	C₇×C₄⋊C₄	—	2	1	no	—
250	C₇×C₄⋊C₄	—	4	1	no	—
251	C₇×C₄⋊C₄	—	4	1	no	—
252	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
253	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
254	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
255	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
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	2	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	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	4	1	no	—
270	C₄²×C₇	—	2	2	no	—
271	C₄²×C₇	—	2	1	no	—
272	C₄²×C₇	—	2	1	no	—
273	C₄²×C₇	—	2	2	no	—
274	C₄²×C₇	—	2	1	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	4	1	no	—
277	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
278	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₇	C₂⁴	1	1	yes	characteristic
281	#557	C₂²×C₇	1	1	yes	characteristic
282	#554	C₂²×C₇	1	1	yes	characteristic
283	#554	C₂²×C₇	1	1	yes	characteristic
284	#553	—	2	1	no	—
285	#552	—	2	1	no	—
286	#552	—	2	2	no	—
287	#552	—	2	1	no	—
288	#547	—	2	1	no	—
289	#547	C₂²×C₇	1	1	yes	characteristic
290	#541	—	2	1	no	—
291	#539	—	2	2	no	—
292	#533	C₂²×C₇	1	1	yes	characteristic
293	#533	C₂²×C₇	1	1	yes	characteristic
294	#529	—	2	1	no	—
295	#529	C₂²×C₇	1	1	yes	characteristic
296	C₂×#125	—	2	1	no	—
297	#520	—	2	1	no	—
298	C₄×#41	C₂²×C₇	1	1	yes	characteristic
299	#500	C₂²×C₇	1	1	yes	characteristic
300	#499	—	2	1	no	—
301	#494	C₂²×C₇	1	1	yes	characteristic
302	#488	—	2	1	no	—
303	#487	C₂²×C₇	1	1	yes	characteristic
304	#487	C₂²×C₇	1	1	yes	characteristic
305	#485	—	2	1	no	—
306	#485	—	2	1	no	—
307	#482	—	2	1	no	—
308	#477	C₂²×C₇	1	1	yes	characteristic
309	#477	C₂²×C₇	1	1	yes	characteristic
310	#476	C₂²×C₇	1	1	yes	characteristic
311	#476	C₂²×C₇	1	1	yes	characteristic
312	#475	C₂²×C₇	1	1	yes	characteristic
313	#474	C₂²×C₇	1	1	yes	characteristic
314	#467	—	2	2	no	—
315	#462	C₂²×C₇	1	1	yes	characteristic
316	#461	C₂²×C₇	1	1	yes	characteristic
317	#452	—	2	1	no	—
318	#441	—	2	1	no	—
319	#440	—	2	1	no	—
320	C₄×Q₁₆	—	2	2	no	—
321	#432	C₂²×C₇	1	1	yes	characteristic
322	#431	C₂²×C₇	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
324	#333	C₂²×C₇	1	1	yes	characteristic
325	#333	C₂²×C₇	1	1	yes	characteristic
326	#332	C₂²×C₇	1	1	yes	characteristic
327	#332	C₂²×C₇	1	1	yes	characteristic
328	Q₈⋊C₈	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	D₄⋊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	C₂³×C₇	—	2	1	no	—
337	C₂³×C₇	—	2	1	no	—
338	C₂³×C₇	#120	1	1	yes	characteristic
339	C₇×Q₈	—	2	1	no	—
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	2	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×D₄	—	2	1	no	—
350	C₇×D₄	—	2	1	no	—
351	C₇×D₄	—	2	1	no	—
352	C₇×D₄	—	4	2	no	—
353	C₇×D₄	—	4	4	no	—
354	C₇×D₄	—	8	1	no	—
355	C₇×D₄	#135	1	2	yes	—
356	C₂×C₄×C₇	—	2	1	no	—
357	C₂×C₄×C₇	—	2	1	no	—
358	C₂×C₄×C₇	—	2	1	no	—
359	C₂×C₄×C₇	—	2	1	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	2	no	—
365	C₂×C₄×C₇	—	2	2	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	4	1	no	—
369	C₂×C₄×C₇	—	4	2	no	—
370	C₂×C₄×C₇	—	4	4	no	—
371	C₂×C₄×C₇	—	4	1	no	—
372	C₂×C₄×C₇	—	4	1	no	—
373	C₂×C₄×C₇	—	4	1	no	—
374	C₂×C₄×C₇	—	4	1	no	—
375	C₂×C₄×C₇	—	4	1	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	8	1	no	—
378	C₂×C₄×C₇	—	8	2	no	—
379	C₂×C₄×C₇	—	8	2	no	—
380	C₂×C₄×C₇	#120	1	1	yes	characteristic
381	C₂×C₄×C₇	#120	1	1	yes	characteristic
382	C₂×C₄×C₇	#120	1	1	yes	characteristic
383	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
384	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₇×C₈	—	2	2	no	—
387	C₇×C₈	—	4	1	no	—
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	C₂×#41	—	2	1	no	—
393	C₂×#41	C₇×D₄	1	1	yes	characteristic
394	C₂×Q₁₆	—	2	2	no	—
395	C₂×Q₁₆	—	2	1	no	—
396	C₂×SD₁₆	—	2	1	no	—
397	C₂²×C₈	—	2	1	no	—
398	C₄⋊Q₈	—	2	1	no	—
399	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
400	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
401	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
402	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
403	#126	—	2	2	no	—
404	#126	—	2	2	no	—
405	#126	—	2	2	no	—
406	#126	—	2	1	no	—
407	#125	—	2	2	no	—
408	#125	—	2	2	no	—
409	#125	—	2	2	no	—
410	#125	—	2	1	no	—
411	#125	—	2	1	no	—
412	#125	—	2	2	no	—
413	#125	—	2	1	no	—
414	#125	—	2	1	no	—
415	#125	—	2	1	no	—
416	#125	C₇×D₄	1	1	yes	characteristic
417	#124	C₇×D₄	1	1	yes	characteristic
418	#123	—	2	2	no	—
419	#123	—	2	2	no	—
420	#123	—	2	2	no	—
421	#123	—	2	1	no	—
422	#123	—	2	2	no	—
423	C₂²⋊Q₈	—	2	1	no	—
424	C₂²⋊Q₈	—	2	2	no	—
425	C₂²⋊Q₈	—	2	2	no	—
426	C₂²⋊Q₈	—	2	1	no	—
427	C₂²⋊Q₈	—	2	1	no	—
428	C₂²⋊Q₈	—	2	1	no	—
429	C₂²⋊Q₈	—	2	1	no	—
430	C₂²⋊Q₈	—	2	1	no	—
431	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
432	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
433	#121	—	2	2	no	—
434	#121	—	2	1	no	—
435	#121	C₇×D₄	1	1	yes	characteristic
436	C₄×Q₈	—	2	1	no	—
437	C₄×Q₈	—	2	1	no	—
438	C₄×Q₈	—	2	1	no	—
439	C₄×Q₈	—	2	1	no	—
440	C₄×Q₈	—	2	1	no	—
441	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
442	C₄×Q₈	C₇×D₄	1	2	yes	—
443	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
444	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	1	no	—
448	C₄×D₄	—	2	1	no	—
449	C₄×D₄	—	2	1	no	—
450	C₄×D₄	—	2	1	no	—
451	C₄×D₄	—	2	1	no	—
452	C₄×D₄	C₇×D₄	1	1	yes	characteristic
453	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
454	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
455	#117	—	2	2	no	—
456	#117	—	2	2	no	—
457	#117	—	2	1	no	—
458	#117	—	2	1	no	—
459	#117	—	2	1	no	—
460	#117	—	2	1	no	—
461	#117	C₇×D₄	1	1	yes	characteristic
462	C₂×C₄⋊C₄	—	2	2	no	—
463	C₂×C₄⋊C₄	—	2	2	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
469	#107	—	2	2	no	—
470	#107	—	2	1	no	—
471	#107	—	2	1	no	—
472	#107	—	2	1	no	—
473	#106	—	2	1	no	—
474	#106	—	2	1	no	—
475	#106	—	2	1	no	—
476	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
477	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
478	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
479	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
480	#103	—	2	2	no	—
481	#103	—	2	1	no	—
482	#103	—	2	1	no	—
483	#103	—	2	1	no	—
484	#103	—	2	2	no	—
485	#103	—	2	1	no	—
486	#103	—	2	1	no	—
487	#102	—	2	1	no	—
488	#102	—	2	1	no	—
489	#102	—	2	1	no	—
490	C₂²⋊C₈	—	2	1	no	—
491	C₂²⋊C₈	—	2	1	no	—
492	#97	—	2	1	no	—
493	C₄×C₈	—	2	1	no	—
494	C₄×C₈	—	2	1	no	—
495	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
496	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
497	C₂²×C₇	—	2	1	no	—
498	C₂²×C₇	—	2	2	no	—
499	C₂²×C₇	—	2	1	no	—
500	C₂²×C₇	—	2	1	no	—
501	C₂²×C₇	—	4	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	C₂×#120	1	1	yes	characteristic
507	C₄×C₇	—	2	1	no	—
508	C₄×C₇	—	2	1	no	—
509	C₄×C₇	—	2	1	no	—
510	C₄×C₇	—	2	1	no	—
511	C₄×C₇	—	4	1	no	—
512	C₄×C₇	—	4	1	no	—
513	C₄×C₇	—	4	2	no	—
514	C₄×C₇	—	4	1	no	—
515	C₄×C₇	—	4	1	no	—
516	C₄×C₇	—	4	1	no	—
517	C₄×C₇	—	4	1	no	—
518	C₄×C₇	—	4	2	no	—
519	C₄×C₇	—	8	1	no	—
520	C₄×C₇	—	8	1	no	—
521	C₄×C₇	C₂×#135	1	1	yes	characteristic
522	C₄×C₇	#575	1	1	yes	characteristic
523	C₄×C₇	#578	1	1	yes	characteristic
524	C₄×C₇	#578	1	1	yes	characteristic
525	#41	—	2	2	no	—
526	#41	—	4	1	no	—
527	#41	—	4	2	no	—
528	#41	—	4	2	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
533	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
534	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
535	C₂×D₄	—	2	2	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	2	1	no	—
538	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
539	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	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	Q₁₆	—	4	4	no	—
551	Q₁₆	—	4	2	no	—
552	SD₁₆	—	4	2	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	4	2	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	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	2	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	1	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	—	2	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	—	4	1	no	—
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	2	yes	—
588	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
589	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
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	1	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	—	4	1	no	—
603	C₄²	—	2	2	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	1	no	—
606	C₄²	—	2	2	no	—
607	C₄²	—	2	1	no	—
608	C₄²	—	2	1	no	—
609	C₄²	—	4	1	no	—
610	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
611	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
612	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
613	C₄²	C₂⁴×C₇	1	1	yes	characteristic
614	C₂×C₇	—	4	1	no	—
615	C₂×C₇	—	4	2	no	—
616	C₂×C₇	—	8	1	no	—
617	C₂×C₇	#3002	1	1	yes	characteristic
618	C₂×C₇	#3011	1	1	yes	characteristic
619	C₂×C₇	#3018	1	1	yes	characteristic
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	C₇×#120	1	1	yes	characteristic
623	Q₈	—	2	1	no	—
624	Q₈	—	2	1	no	—
625	Q₈	—	2	1	no	—
626	Q₈	—	2	1	no	—
627	Q₈	—	4	2	no	—
628	Q₈	—	4	2	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	D₄	—	2	1	no	—
634	D₄	—	2	1	no	—
635	D₄	—	2	1	no	—
636	D₄	—	4	2	no	—
637	D₄	—	4	4	no	—
638	D₄	—	8	1	no	—
639	D₄	C₇×#135	1	2	yes	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	2	no	—
654	C₂×C₄	—	4	4	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	8	1	no	—
662	C₂×C₄	—	8	2	no	—
663	C₂×C₄	—	8	2	no	—
664	C₂×C₄	C₇×#120	1	1	yes	characteristic
665	C₂×C₄	C₇×#120	1	1	yes	characteristic
666	C₂×C₄	C₇×#120	1	1	yes	characteristic
667	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
668	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
669	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
670	C₈	—	2	2	no	—
671	C₈	—	4	1	no	—
672	C₈	—	4	1	no	—
673	C₈	—	4	1	no	—
674	C₈	—	4	1	no	—
675	C₈	—	4	1	no	—
676	C₇	#18661	1	1	yes	characteristic, direct product factor, Sylow
677	C₂²	—	2	1	no	—
678	C₂²	—	2	2	no	—
679	C₂²	—	2	1	no	—
680	C₂²	—	2	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	1	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	C₂×C₇×#120	1	1	yes	characteristic
687	C₄	—	2	1	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	1	no	—
690	C₄	—	2	1	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	2	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	C₂×C₇×#135	1	1	yes	characteristic
702	C₄	C₇×#575	1	1	yes	characteristic
703	C₄	C₇×#578	1	1	yes	characteristic
704	C₄	C₇×#578	1	1	yes	characteristic
705	C₂	—	4	1	no	—
706	C₂	—	4	2	no	—
707	C₂	—	8	1	no	—
708	C₂	C₇×#3002	1	1	yes	characteristic
709	C₂	C₇×#3011	1	1	yes	characteristic
710	C₂	C₇×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3515	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×#3233	C₂	semidirect
3. C₇×#3170	C₂	semidirect
4. C₇×#3116	C₂	semidirect
5. C₇×#1648	C₂	semidirect
6. C₇×#1647	C₂	semidirect
7. C₇×#1644	C₂	semidirect
8. C₇×#1643	C₂	semidirect
9. C₇×#1636	C₂	semidirect
10. C₇×#1635	C₂	semidirect
11. C₇×#1632	C₂	semidirect
12. C₇×#1632	C₂	semidirect
13. C₇×#1554	C₂	semidirect
14. C₇×#1494	C₂	semidirect
15. C₇×#1476	C₂	semidirect
71. #18661	C₇	direct (N × H)
178. #3233	C₂×C₇	semidirect
179. #3170	C₂×C₇	semidirect
180. #3116	C₂×C₇	semidirect
181. #1648	C₂×C₇	semidirect
182. #1647	C₂×C₇	semidirect
183. #1644	C₂×C₇	semidirect
184. #1643	C₂×C₇	semidirect
185. #1636	C₂×C₇	semidirect
186. #1635	C₂×C₇	semidirect
187. #1632	C₂×C₇	semidirect
188. #1632	C₂×C₇	semidirect
189. #1554	C₂×C₇	semidirect
190. #1494	C₂×C₇	semidirect
191. #1476	C₂×C₇	semidirect
676. C₇	#18661	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×#3515	2
C₇×#3233	2
C₇×#3170	2
C₇×#3116	2
C₇×#1648	2
C₇×#1647	2
C₇×#1644	2
C₇×#1643	2
C₇×#1636	2
C₇×#1635	2
C₇×#1632	2
C₇×#1554	2
C₇×#1494	2
C₇×#1476	2
#18661	7

Subgroups — C7×#18661 #13195877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18661 #13195877