Subgroups — C₇×#18648 #13195864

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3520	C₂	1	1	yes	maximal, characteristic
2	C₇×#3248	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1654	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1652	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1651	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1647	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1647	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1552	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1494	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₇×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#556	C₂²	1	1	yes	characteristic
17	C₇×#554	—	2	1	no	—
18	C₇×#552	—	2	1	no	—
19	C₇×#551	—	2	2	no	—
20	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
21	C₇×#548	—	2	1	no	—
22	C₇×#547	C₂²	1	1	yes	characteristic
23	C₇×#547	C₂²	1	1	yes	characteristic
24	C₇×#546	—	2	1	no	—
25	C₇×#542	—	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	C₂²	1	1	yes	characteristic
30	C₇×#525	—	2	1	no	—
31	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
32	C₇×#520	—	2	1	no	—
33	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	1	yes	characteristic, complemented
35	C₇×#500	C₂²	1	1	yes	characteristic, complemented
36	C₇×#499	—	2	2	no	—
37	C₇×#498	C₂²	1	1	yes	characteristic, complemented
38	C₇×#498	C₂²	1	1	yes	characteristic, complemented
39	C₇×#484	—	2	1	no	—
40	C₇×#479	—	2	1	no	—
41	C₇×#475	C₂²	1	1	yes	characteristic, complemented
42	C₇×#474	C₂²	1	1	yes	characteristic
43	C₇×#474	C₂²	1	1	yes	characteristic
44	C₇×#474	C₂²	1	1	yes	characteristic, complemented
45	C₇×#467	—	2	1	no	—
46	C₇×#465	—	2	1	no	—
47	C₇×#464	—	2	1	no	—
48	C₇×#462	C₂²	1	1	yes	characteristic, complemented
49	C₇×#461	C₂²	1	1	yes	characteristic
50	C₇×#461	C₂²	1	1	yes	characteristic
51	C₇×#461	C₂²	1	1	yes	characteristic, complemented
52	C₇×#450	—	2	1	no	—
53	C₇×#440	—	2	1	no	—
54	C₇×#440	—	2	1	no	—
55	C₇×#432	C₂²	1	1	yes	characteristic
56	C₇×#427	—	2	2	no	—
57	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic, complemented
60	C₇×#332	C₂²	1	1	yes	characteristic, complemented
61	C₇×#332	C₂²	1	1	yes	characteristic, complemented
62	C₇×#332	C₂²	1	1	yes	characteristic, complemented
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×D₄⋊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, complemented
70	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
71	#18648	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	—	2	1	no	—
73	C₂×C₇×#41	—	2	1	no	—
74	C₂²×C₇×Q₈	—	2	1	no	—
75	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₇	—	2	1	no	—
77	C₂×C₇×Q₁₆	—	2	1	no	—
78	C₂×C₇×SD₁₆	—	2	1	no	—
79	C₂×C₇×SD₁₆	—	2	1	no	—
80	C₂×C₇×SD₁₆	—	2	1	no	—
81	C₂²×C₇×C₈	—	2	1	no	—
82	C₂²×C₇×C₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	—	2	1	no	—
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×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₇×#126	—	2	2	no	—
89	C₇×#126	—	2	2	no	—
90	C₇×#126	—	2	1	no	—
91	C₇×#125	—	2	2	no	—
92	C₇×#125	—	2	1	no	—
93	C₇×#125	D₄	1	1	yes	characteristic
94	C₇×#125	D₄	1	2	yes	—
95	C₇×#124	—	2	2	no	—
96	C₇×#124	—	2	1	no	—
97	C₇×#124	D₄	1	1	yes	characteristic
98	C₇×#123	—	2	2	no	—
99	C₇×#123	—	2	2	no	—
100	C₇×#123	—	2	1	no	—
101	C₇×#123	—	2	2	no	—
102	C₇×#123	—	2	2	no	—
103	C₇×C₂²⋊Q₈	—	2	2	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	—	2	2	no	—
112	C₇×C₂²⋊Q₈	—	2	1	no	—
113	C₇×C₂²⋊Q₈	—	2	1	no	—
114	C₇×C₂²⋊Q₈	—	2	1	no	—
115	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
116	C₇×#121	—	2	2	no	—
117	C₇×#121	—	2	1	no	—
118	C₇×#121	—	2	1	no	—
119	C₇×#121	D₄	1	1	yes	characteristic
120	C₇×#120	—	2	2	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	C₂³	1	1	yes	characteristic
127	C₄×C₇×D₄	—	2	2	no	—
128	C₄×C₇×D₄	—	2	2	no	—
129	C₄×C₇×D₄	—	2	2	no	—
130	C₄×C₇×D₄	—	2	1	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	D₄	1	1	yes	characteristic
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	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
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	D₄	1	2	yes	—
144	C₂×C₇×C₄⋊C₄	—	2	2	no	—
145	C₂×C₇×C₄⋊C₄	—	2	1	no	—
146	C₂×C₇×C₄⋊C₄	—	2	1	no	—
147	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
148	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
149	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
151	C₇×#107	—	2	1	no	—
152	C₇×#107	—	2	1	no	—
153	C₇×#107	—	2	1	no	—
154	C₇×#107	—	2	2	no	—
155	C₇×#107	—	2	2	no	—
156	C₇×#106	—	2	1	no	—
157	C₇×#106	—	2	2	no	—
158	C₇×#106	—	2	2	no	—
159	C₇×C₄⋊C₈	—	2	1	no	—
160	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₇×#103	—	2	1	no	—
165	C₇×#103	—	2	1	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#103	—	2	1	no	—
168	C₇×#103	—	2	1	no	—
169	C₇×#102	—	2	1	no	—
170	C₇×#102	—	2	1	no	—
171	C₇×#102	—	2	1	no	—
172	C₇×C₂²⋊C₈	—	2	1	no	—
173	C₇×#97	—	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	#3520	C₂×C₇	1	1	yes	characteristic
178	#3248	C₂×C₇	1	1	yes	characteristic, complemented
179	#3160	C₂×C₇	1	1	yes	characteristic, complemented
180	#3110	C₂×C₇	1	1	yes	characteristic, complemented
181	#1654	C₂×C₇	1	1	yes	characteristic, complemented
182	#1654	C₂×C₇	1	1	yes	characteristic, complemented
183	#1652	C₂×C₇	1	1	yes	characteristic, complemented
184	#1651	C₂×C₇	1	1	yes	characteristic, complemented
185	#1650	C₂×C₇	1	1	yes	characteristic, complemented
186	#1649	C₂×C₇	1	1	yes	characteristic, complemented
187	#1647	C₂×C₇	1	1	yes	characteristic, complemented
188	#1647	C₂×C₇	1	1	yes	characteristic, complemented
189	#1552	C₂×C₇	1	1	yes	characteristic, complemented
190	#1494	C₂×C₇	1	1	yes	characteristic, complemented
191	C₂×D₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
192	C₂⁴×C₇	—	2	1	no	—
193	C₇×#41	—	4	2	no	—
194	C₇×#41	—	4	2	no	—
195	C₇×#41	—	8	1	no	—
196	C₇×#41	—	8	1	no	—
197	C₂×C₇×Q₈	—	2	2	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	2	1	no	—
200	C₂×C₇×Q₈	—	2	1	no	—
201	C₂×C₇×Q₈	—	2	1	no	—
202	C₂×C₇×Q₈	—	8	1	no	—
203	C₂×C₇×Q₈	—	8	1	no	—
204	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×D₄	—	2	2	no	—
206	C₂×C₇×D₄	—	2	2	no	—
207	C₂×C₇×D₄	—	2	1	no	—
208	C₂×C₇×D₄	—	2	2	no	—
209	C₂×C₇×D₄	—	2	1	no	—
210	C₂×C₇×D₄	—	4	1	no	—
211	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
212	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₇	—	2	2	no	—
215	C₂²×C₄×C₇	—	2	2	no	—
216	C₂²×C₄×C₇	—	2	2	no	—
217	C₂²×C₄×C₇	—	2	1	no	—
218	C₂²×C₄×C₇	—	2	1	no	—
219	C₂²×C₄×C₇	—	2	1	no	—
220	C₂²×C₄×C₇	—	2	1	no	—
221	C₂²×C₄×C₇	—	2	1	no	—
222	C₂²×C₄×C₇	—	2	1	no	—
223	C₂²×C₄×C₇	—	8	1	no	—
224	C₂²×C₄×C₇	—	8	1	no	—
225	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
228	C₇×Q₁₆	—	8	1	no	—
229	C₇×SD₁₆	—	4	2	no	—
230	C₇×SD₁₆	—	4	2	no	—
231	C₇×SD₁₆	—	8	1	no	—
232	C₂×C₇×C₈	—	2	4	no	—
233	C₂×C₇×C₈	—	2	1	no	—
234	C₂×C₇×C₈	—	2	1	no	—
235	C₂×C₇×C₈	—	2	1	no	—
236	C₂×C₇×C₈	—	2	1	no	—
237	C₂×C₇×C₈	—	2	1	no	—
238	C₂×C₇×C₈	—	2	1	no	—
239	C₂×C₇×C₈	—	8	1	no	—
240	C₇×C₄⋊C₄	—	2	2	no	—
241	C₇×C₄⋊C₄	—	2	2	no	—
242	C₇×C₄⋊C₄	—	2	2	no	—
243	C₇×C₄⋊C₄	—	2	2	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	2	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	2	no	—
250	C₇×C₄⋊C₄	—	2	1	no	—
251	C₇×C₄⋊C₄	—	2	1	no	—
252	C₇×C₄⋊C₄	—	2	1	no	—
253	C₇×C₄⋊C₄	—	2	1	no	—
254	C₇×C₄⋊C₄	—	2	1	no	—
255	C₇×C₄⋊C₄	—	2	1	no	—
256	C₇×C₄⋊C₄	—	2	1	no	—
257	C₇×C₄⋊C₄	—	4	1	no	—
258	C₇×C₄⋊C₄	—	4	1	no	—
259	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
260	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
261	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
262	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
263	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
264	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
265	C₇×C₂²⋊C₄	—	2	2	no	—
266	C₇×C₂²⋊C₄	—	2	2	no	—
267	C₇×C₂²⋊C₄	—	2	2	no	—
268	C₇×C₂²⋊C₄	—	2	2	no	—
269	C₇×C₂²⋊C₄	—	2	2	no	—
270	C₇×C₂²⋊C₄	—	2	1	no	—
271	C₇×C₂²⋊C₄	—	2	2	no	—
272	C₇×C₂²⋊C₄	—	2	1	no	—
273	C₇×C₂²⋊C₄	—	2	1	no	—
274	C₇×C₂²⋊C₄	—	2	1	no	—
275	C₇×C₂²⋊C₄	—	2	1	no	—
276	C₇×C₂²⋊C₄	—	2	1	no	—
277	C₇×C₂²⋊C₄	—	2	1	no	—
278	C₇×C₂²⋊C₄	—	2	1	no	—
279	C₇×C₂²⋊C₄	—	2	1	no	—
280	C₇×C₂²⋊C₄	—	4	1	no	—
281	C₄²×C₇	—	2	2	no	—
282	C₄²×C₇	—	2	2	no	—
283	C₄²×C₇	—	2	1	no	—
284	C₄²×C₇	—	4	1	no	—
285	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
286	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
287	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
288	C₄²×C₇	C₂⁴	1	1	yes	characteristic
289	#556	C₂²×C₇	1	1	yes	characteristic
290	#554	—	2	1	no	—
291	#552	—	2	1	no	—
292	#551	—	2	2	no	—
293	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
294	#548	—	2	1	no	—
295	#547	C₂²×C₇	1	1	yes	characteristic
296	#547	C₂²×C₇	1	1	yes	characteristic
297	#546	—	2	1	no	—
298	#542	—	2	1	no	—
299	#539	—	2	2	no	—
300	#533	C₂²×C₇	1	1	yes	characteristic
301	#533	C₂²×C₇	1	1	yes	characteristic
302	#529	C₂²×C₇	1	1	yes	characteristic
303	#525	—	2	1	no	—
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	#520	—	2	1	no	—
306	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
307	#500	C₂²×C₇	1	1	yes	characteristic, complemented
308	#500	C₂²×C₇	1	1	yes	characteristic, complemented
309	#499	—	2	2	no	—
310	#498	C₂²×C₇	1	1	yes	characteristic, complemented
311	#498	C₂²×C₇	1	1	yes	characteristic, complemented
312	#484	—	2	1	no	—
313	#479	—	2	1	no	—
314	#475	C₂²×C₇	1	1	yes	characteristic, complemented
315	#474	C₂²×C₇	1	1	yes	characteristic
316	#474	C₂²×C₇	1	1	yes	characteristic
317	#474	C₂²×C₇	1	1	yes	characteristic, complemented
318	#467	—	2	1	no	—
319	#465	—	2	1	no	—
320	#464	—	2	1	no	—
321	#462	C₂²×C₇	1	1	yes	characteristic, complemented
322	#461	C₂²×C₇	1	1	yes	characteristic
323	#461	C₂²×C₇	1	1	yes	characteristic
324	#461	C₂²×C₇	1	1	yes	characteristic, complemented
325	#450	—	2	1	no	—
326	#440	—	2	1	no	—
327	#440	—	2	1	no	—
328	#432	C₂²×C₇	1	1	yes	characteristic
329	#427	—	2	2	no	—
330	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
331	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
332	#332	C₂²×C₇	1	1	yes	characteristic, complemented
333	#332	C₂²×C₇	1	1	yes	characteristic, complemented
334	#332	C₂²×C₇	1	1	yes	characteristic, complemented
335	#332	C₂²×C₇	1	1	yes	characteristic, complemented
336	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
337	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
338	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
339	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
340	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
341	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
343	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
344	C₂³×C₇	—	2	1	no	—
345	C₂³×C₇	—	2	1	no	—
346	C₂³×C₇	—	2	1	no	—
347	C₂³×C₇	—	4	2	no	—
348	C₂³×C₇	—	4	2	no	—
349	C₂³×C₇	—	8	1	no	—
350	C₂³×C₇	#120	1	1	yes	characteristic
351	C₇×Q₈	—	2	1	no	—
352	C₇×Q₈	—	2	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×Q₈	—	4	1	no	—
355	C₇×Q₈	—	4	1	no	—
356	C₇×Q₈	—	4	1	no	—
357	C₇×Q₈	—	4	1	no	—
358	C₇×Q₈	—	4	1	no	—
359	C₇×Q₈	—	4	1	no	—
360	C₇×Q₈	—	4	1	no	—
361	C₇×Q₈	—	8	2	no	—
362	C₇×D₄	—	2	1	no	—
363	C₇×D₄	—	2	1	no	—
364	C₇×D₄	—	2	1	no	—
365	C₇×D₄	—	2	1	no	—
366	C₇×D₄	—	2	1	no	—
367	C₇×D₄	—	4	2	no	—
368	C₇×D₄	—	4	4	no	—
369	C₇×D₄	—	4	2	no	—
370	C₇×D₄	—	8	2	no	—
371	C₇×D₄	#135	1	2	yes	—
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	1	no	—
378	C₂×C₄×C₇	—	2	2	no	—
379	C₂×C₄×C₇	—	2	1	no	—
380	C₂×C₄×C₇	—	2	1	no	—
381	C₂×C₄×C₇	—	2	2	no	—
382	C₂×C₄×C₇	—	2	1	no	—
383	C₂×C₄×C₇	—	2	1	no	—
384	C₂×C₄×C₇	—	4	4	no	—
385	C₂×C₄×C₇	—	4	1	no	—
386	C₂×C₄×C₇	—	4	1	no	—
387	C₂×C₄×C₇	—	4	1	no	—
388	C₂×C₄×C₇	—	4	1	no	—
389	C₂×C₄×C₇	—	4	1	no	—
390	C₂×C₄×C₇	—	4	1	no	—
391	C₂×C₄×C₇	—	4	1	no	—
392	C₂×C₄×C₇	—	4	1	no	—
393	C₂×C₄×C₇	—	8	1	no	—
394	C₂×C₄×C₇	—	8	1	no	—
395	C₂×C₄×C₇	—	8	1	no	—
396	C₂×C₄×C₇	—	8	2	no	—
397	C₂×C₄×C₇	—	8	2	no	—
398	C₂×C₄×C₇	#120	1	1	yes	characteristic
399	C₂×C₄×C₇	#120	1	1	yes	characteristic
400	C₂×C₄×C₇	#120	1	1	yes	characteristic
401	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
402	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
403	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
404	C₇×C₈	—	2	2	no	—
405	C₇×C₈	—	2	2	no	—
406	C₇×C₈	—	4	1	no	—
407	C₇×C₈	—	4	1	no	—
408	C₇×C₈	—	4	1	no	—
409	C₇×C₈	—	4	1	no	—
410	C₂×#41	—	2	1	no	—
411	C₂×#41	—	2	1	no	—
412	C₂²×Q₈	—	2	1	no	—
413	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
414	C₂³×C₄	—	2	1	no	—
415	C₂×Q₁₆	—	2	1	no	—
416	C₂×SD₁₆	—	2	1	no	—
417	C₂×SD₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂²×C₈	—	2	1	no	—
420	C₂²×C₈	—	2	1	no	—
421	C₄⋊Q₈	—	2	1	no	—
422	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
423	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
424	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
425	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
426	#126	—	2	2	no	—
427	#126	—	2	2	no	—
428	#126	—	2	1	no	—
429	#125	—	2	2	no	—
430	#125	—	2	1	no	—
431	#125	C₇×D₄	1	1	yes	characteristic
432	#125	C₇×D₄	1	2	yes	—
433	#124	—	2	2	no	—
434	#124	—	2	1	no	—
435	#124	C₇×D₄	1	1	yes	characteristic
436	#123	—	2	2	no	—
437	#123	—	2	2	no	—
438	#123	—	2	1	no	—
439	#123	—	2	2	no	—
440	#123	—	2	2	no	—
441	C₂²⋊Q₈	—	2	2	no	—
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	1	no	—
445	C₂²⋊Q₈	—	2	1	no	—
446	C₂²⋊Q₈	—	2	1	no	—
447	C₂²⋊Q₈	—	2	1	no	—
448	C₂²⋊Q₈	—	2	1	no	—
449	C₂²⋊Q₈	—	2	2	no	—
450	C₂²⋊Q₈	—	2	1	no	—
451	C₂²⋊Q₈	—	2	1	no	—
452	C₂²⋊Q₈	—	2	1	no	—
453	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
454	#121	—	2	2	no	—
455	#121	—	2	1	no	—
456	#121	—	2	1	no	—
457	#121	C₇×D₄	1	1	yes	characteristic
458	#120	—	2	2	no	—
459	C₄×Q₈	—	2	1	no	—
460	C₄×Q₈	—	2	1	no	—
461	C₄×Q₈	—	2	1	no	—
462	C₄×Q₈	—	2	1	no	—
463	C₄×Q₈	—	2	1	no	—
464	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	—	2	1	no	—
469	C₄×D₄	—	2	1	no	—
470	C₄×D₄	—	2	1	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	C₇×D₄	1	1	yes	characteristic
473	C₄×D₄	C₇×D₄	1	1	yes	characteristic
474	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
475	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
476	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
477	#117	—	2	2	no	—
478	#117	—	2	1	no	—
479	#117	—	2	1	no	—
480	#117	—	2	1	no	—
481	#117	C₇×D₄	1	2	yes	—
482	C₂×C₄⋊C₄	—	2	2	no	—
483	C₂×C₄⋊C₄	—	2	1	no	—
484	C₂×C₄⋊C₄	—	2	1	no	—
485	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
486	C₂×C₂²⋊C₄	—	2	2	no	—
487	C₂×C₂²⋊C₄	—	2	1	no	—
488	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
489	#107	—	2	2	no	—
490	#107	—	2	2	no	—
491	#107	—	2	1	no	—
492	#107	—	2	1	no	—
493	#107	—	2	1	no	—
494	#106	—	2	2	no	—
495	#106	—	2	2	no	—
496	#106	—	2	1	no	—
497	C₄⋊C₈	—	2	1	no	—
498	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
499	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
500	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
501	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
502	#103	—	2	1	no	—
503	#103	—	2	1	no	—
504	#103	—	2	1	no	—
505	#103	—	2	1	no	—
506	#103	—	2	1	no	—
507	#102	—	2	1	no	—
508	#102	—	2	1	no	—
509	#102	—	2	1	no	—
510	C₂²⋊C₈	—	2	1	no	—
511	#97	—	2	1	no	—
512	C₄×C₈	—	2	1	no	—
513	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
514	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
515	C₂²×C₇	—	2	1	no	—
516	C₂²×C₇	—	2	1	no	—
517	C₂²×C₇	—	2	2	no	—
518	C₂²×C₇	—	4	1	no	—
519	C₂²×C₇	—	4	1	no	—
520	C₂²×C₇	—	4	1	no	—
521	C₂²×C₇	—	4	1	no	—
522	C₂²×C₇	—	4	1	no	—
523	C₂²×C₇	—	4	1	no	—
524	C₂²×C₇	—	4	1	no	—
525	C₂²×C₇	—	4	1	no	—
526	C₂²×C₇	—	8	4	no	—
527	C₂²×C₇	#456	1	2	yes	—
528	C₂²×C₇	C₂×#120	1	1	yes	characteristic
529	C₄×C₇	—	2	1	no	—
530	C₄×C₇	—	2	1	no	—
531	C₄×C₇	—	2	1	no	—
532	C₄×C₇	—	2	1	no	—
533	C₄×C₇	—	4	1	no	—
534	C₄×C₇	—	4	2	no	—
535	C₄×C₇	—	4	1	no	—
536	C₄×C₇	—	4	1	no	—
537	C₄×C₇	—	4	2	no	—
538	C₄×C₇	—	4	1	no	—
539	C₄×C₇	—	4	1	no	—
540	C₄×C₇	—	8	1	no	—
541	C₄×C₇	—	8	1	no	—
542	C₄×C₇	C₂×#135	1	1	yes	characteristic
543	C₄×C₇	#575	1	1	yes	characteristic
544	C₄×C₇	#576	1	1	yes	characteristic
545	C₄×C₇	#577	1	1	yes	characteristic
546	C₂⁴	—	2	1	no	—
547	#41	—	4	2	no	—
548	#41	—	4	2	no	—
549	#41	—	8	1	no	—
550	#41	—	8	1	no	—
551	C₂×Q₈	—	2	2	no	—
552	C₂×Q₈	—	2	1	no	—
553	C₂×Q₈	—	2	1	no	—
554	C₂×Q₈	—	2	1	no	—
555	C₂×Q₈	—	2	1	no	—
556	C₂×Q₈	—	8	1	no	—
557	C₂×Q₈	—	8	1	no	—
558	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
559	C₂×D₄	—	2	2	no	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	4	1	no	—
565	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
566	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
567	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	2	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	2	1	no	—
573	C₂²×C₄	—	2	1	no	—
574	C₂²×C₄	—	2	1	no	—
575	C₂²×C₄	—	2	1	no	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	8	1	no	—
578	C₂²×C₄	—	8	1	no	—
579	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
580	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
581	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
582	Q₁₆	—	8	1	no	—
583	SD₁₆	—	4	2	no	—
584	SD₁₆	—	4	2	no	—
585	SD₁₆	—	8	1	no	—
586	C₂×C₈	—	2	4	no	—
587	C₂×C₈	—	2	1	no	—
588	C₂×C₈	—	2	1	no	—
589	C₂×C₈	—	2	1	no	—
590	C₂×C₈	—	2	1	no	—
591	C₂×C₈	—	2	1	no	—
592	C₂×C₈	—	2	1	no	—
593	C₂×C₈	—	8	1	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	2	no	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	2	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	4	1	no	—
612	C₄⋊C₄	—	4	1	no	—
613	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
614	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
615	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
616	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
617	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
618	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	2	no	—
622	C₂²⋊C₄	—	2	2	no	—
623	C₂²⋊C₄	—	2	2	no	—
624	C₂²⋊C₄	—	2	1	no	—
625	C₂²⋊C₄	—	2	2	no	—
626	C₂²⋊C₄	—	2	1	no	—
627	C₂²⋊C₄	—	2	1	no	—
628	C₂²⋊C₄	—	2	1	no	—
629	C₂²⋊C₄	—	2	1	no	—
630	C₂²⋊C₄	—	2	1	no	—
631	C₂²⋊C₄	—	2	1	no	—
632	C₂²⋊C₄	—	2	1	no	—
633	C₂²⋊C₄	—	2	1	no	—
634	C₂²⋊C₄	—	4	1	no	—
635	C₄²	—	2	2	no	—
636	C₄²	—	2	2	no	—
637	C₄²	—	2	1	no	—
638	C₄²	—	4	1	no	—
639	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
640	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
641	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
642	C₄²	C₂⁴×C₇	1	1	yes	characteristic
643	C₂×C₇	—	2	2	no	—
644	C₂×C₇	—	4	2	no	—
645	C₂×C₇	—	8	1	no	—
646	C₂×C₇	—	8	1	no	—
647	C₂×C₇	#3002	1	1	yes	characteristic
648	C₂×C₇	#3012	1	1	yes	characteristic
649	C₂×C₇	C₂×#456	1	1	yes	characteristic
650	C₂³	—	2	1	no	—
651	C₂³	—	2	1	no	—
652	C₂³	—	2	1	no	—
653	C₂³	—	4	2	no	—
654	C₂³	—	4	2	no	—
655	C₂³	—	8	1	no	—
656	C₂³	C₇×#120	1	1	yes	characteristic
657	Q₈	—	2	1	no	—
658	Q₈	—	2	1	no	—
659	Q₈	—	4	1	no	—
660	Q₈	—	4	1	no	—
661	Q₈	—	4	1	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	1	no	—
664	Q₈	—	4	1	no	—
665	Q₈	—	4	1	no	—
666	Q₈	—	4	1	no	—
667	Q₈	—	8	2	no	—
668	D₄	—	2	1	no	—
669	D₄	—	2	1	no	—
670	D₄	—	2	1	no	—
671	D₄	—	2	1	no	—
672	D₄	—	2	1	no	—
673	D₄	—	4	2	no	—
674	D₄	—	4	4	no	—
675	D₄	—	4	2	no	—
676	D₄	—	8	2	no	—
677	D₄	C₇×#135	1	2	yes	—
678	C₂×C₄	—	2	2	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	1	no	—
682	C₂×C₄	—	2	1	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	2	1	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	1	no	—
689	C₂×C₄	—	2	1	no	—
690	C₂×C₄	—	4	4	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	1	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	—	4	1	no	—
696	C₂×C₄	—	4	1	no	—
697	C₂×C₄	—	4	1	no	—
698	C₂×C₄	—	4	1	no	—
699	C₂×C₄	—	8	1	no	—
700	C₂×C₄	—	8	1	no	—
701	C₂×C₄	—	8	1	no	—
702	C₂×C₄	—	8	2	no	—
703	C₂×C₄	—	8	2	no	—
704	C₂×C₄	C₇×#120	1	1	yes	characteristic
705	C₂×C₄	C₇×#120	1	1	yes	characteristic
706	C₂×C₄	C₇×#120	1	1	yes	characteristic
707	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
708	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
709	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
710	C₈	—	2	2	no	—
711	C₈	—	2	2	no	—
712	C₈	—	4	1	no	—
713	C₈	—	4	1	no	—
714	C₈	—	4	1	no	—
715	C₈	—	4	1	no	—
716	C₇	#18648	1	1	yes	characteristic, direct product factor, Sylow
717	C₂²	—	2	1	no	—
718	C₂²	—	2	1	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	4	1	no	—
721	C₂²	—	4	1	no	—
722	C₂²	—	4	1	no	—
723	C₂²	—	4	1	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	1	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	4	1	no	—
728	C₂²	—	8	4	no	—
729	C₂²	C₇×#456	1	2	yes	—
730	C₂²	C₂×C₇×#120	1	1	yes	characteristic
731	C₄	—	2	1	no	—
732	C₄	—	2	1	no	—
733	C₄	—	2	1	no	—
734	C₄	—	2	1	no	—
735	C₄	—	4	1	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	1	no	—
738	C₄	—	4	1	no	—
739	C₄	—	4	2	no	—
740	C₄	—	4	1	no	—
741	C₄	—	4	1	no	—
742	C₄	—	8	1	no	—
743	C₄	—	8	1	no	—
744	C₄	C₂×C₇×#135	1	1	yes	characteristic
745	C₄	C₇×#575	1	1	yes	characteristic
746	C₄	C₇×#576	1	1	yes	characteristic
747	C₄	C₇×#577	1	1	yes	characteristic
748	C₂	—	2	2	no	—
749	C₂	—	4	2	no	—
750	C₂	—	8	1	no	—
751	C₂	—	8	1	no	—
752	C₂	C₇×#3002	1	1	yes	characteristic
753	C₂	C₇×#3012	1	1	yes	characteristic
754	C₂	C₂×C₇×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3520	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×#3160	C₂	semidirect
4. C₇×#3110	C₂	semidirect
5. C₇×#1654	C₂	semidirect
6. C₇×#1654	C₂	semidirect
7. C₇×#1652	C₂	semidirect
8. C₇×#1651	C₂	semidirect
9. C₇×#1650	C₂	semidirect
10. C₇×#1649	C₂	semidirect
11. C₇×#1647	C₂	semidirect
12. C₇×#1647	C₂	semidirect
13. C₇×#1552	C₂	semidirect
14. C₇×#1494	C₂	semidirect
15. C₂×C₇×D₄⋊C₈	C₂	semidirect
34. C₇×#500	C₂²	semidirect
35. C₇×#500	C₂²	semidirect
37. C₇×#498	C₂²	semidirect
38. C₇×#498	C₂²	semidirect
41. C₇×#475	C₂²	semidirect
44. C₇×#474	C₂²	semidirect
48. C₇×#462	C₂²	semidirect
51. C₇×#461	C₂²	semidirect
59. C₇×#332	C₂²	semidirect
60. C₇×#332	C₂²	semidirect
61. C₇×#332	C₂²	semidirect
62. C₇×#332	C₂²	semidirect
63. C₇×Q₈⋊C₈	C₂²	semidirect
64. C₇×Q₈⋊C₈	C₂²	semidirect
69. C₇×D₄⋊C₈	C₂²	semidirect
70. C₇×D₄⋊C₈	C₂²	semidirect
71. #18648	C₇	direct (N × H)
178. #3248	C₂×C₇	semidirect
179. #3160	C₂×C₇	semidirect
180. #3110	C₂×C₇	semidirect
181. #1654	C₂×C₇	semidirect
182. #1654	C₂×C₇	semidirect
183. #1652	C₂×C₇	semidirect
184. #1651	C₂×C₇	semidirect
185. #1650	C₂×C₇	semidirect
186. #1649	C₂×C₇	semidirect
187. #1647	C₂×C₇	semidirect
188. #1647	C₂×C₇	semidirect
189. #1552	C₂×C₇	semidirect
190. #1494	C₂×C₇	semidirect
191. C₂×D₄⋊C₈	C₂×C₇	semidirect
307. #500	C₂²×C₇	semidirect
308. #500	C₂²×C₇	semidirect
310. #498	C₂²×C₇	semidirect
311. #498	C₂²×C₇	semidirect
314. #475	C₂²×C₇	semidirect
317. #474	C₂²×C₇	semidirect
321. #462	C₂²×C₇	semidirect
324. #461	C₂²×C₇	semidirect
332. #332	C₂²×C₇	semidirect
333. #332	C₂²×C₇	semidirect
334. #332	C₂²×C₇	semidirect
335. #332	C₂²×C₇	semidirect
336. Q₈⋊C₈	C₂²×C₇	semidirect
337. Q₈⋊C₈	C₂²×C₇	semidirect
342. D₄⋊C₈	C₂²×C₇	semidirect
343. D₄⋊C₈	C₂²×C₇	semidirect
716. C₇	#18648	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×#3520	2
C₇×#3248	2
C₇×#3160	2
C₇×#3110	2
C₇×#1654	2
C₇×#1652	2
C₇×#1651	2
C₇×#1650	2
C₇×#1649	2
C₇×#1647	2
C₇×#1552	2
C₇×#1494	2
C₂×C₇×D₄⋊C₈	2
#18648	7

Subgroups — C7×#18648 #13195864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18648 #13195864