Subgroups — C₇×#18807 #13196023

Order 1792.

922 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3475	C₂	1	1	yes	maximal, characteristic
2	C₇×#3231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1654	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1652	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1647	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×#1644	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1643	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1554	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#556	C₂²	1	1	yes	characteristic
17	C₇×#555	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	1	no	—
22	C₇×#551	—	2	2	no	—
23	C₇×#547	C₂²	1	1	yes	characteristic
24	C₇×#546	—	2	1	no	—
25	C₇×#533	C₂²	1	1	yes	characteristic
26	C₇×#533	C₂²	1	1	yes	characteristic
27	C₇×#529	—	2	2	no	—
28	C₇×#529	—	2	1	no	—
29	C₇×#525	—	2	1	no	—
30	C₄×C₇×#41	—	2	1	no	—
31	C₄×C₇×#41	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
33	C₂×C₇×#117	—	2	1	no	—
34	C₇×#500	C₂²	1	1	yes	characteristic, complemented
35	C₇×#498	C₂²	1	1	yes	characteristic, complemented
36	C₇×#487	C₂²	1	1	yes	characteristic, complemented
37	C₇×#487	C₂²	1	1	yes	characteristic, complemented
38	C₇×#485	—	2	2	no	—
39	C₇×#485	—	2	1	no	—
40	C₇×#484	—	2	1	no	—
41	C₇×#479	—	2	2	no	—
42	C₇×#475	C₂²	1	1	yes	characteristic, complemented
43	C₇×#474	C₂²	1	1	yes	characteristic
44	C₇×#474	C₂²	1	1	yes	characteristic
45	C₇×#474	C₂²	1	1	yes	characteristic, complemented
46	C₇×#462	C₂²	1	1	yes	characteristic, complemented
47	C₇×#461	C₂²	1	1	yes	characteristic
48	C₇×#461	C₂²	1	1	yes	characteristic
49	C₇×#461	C₂²	1	1	yes	characteristic, complemented
50	C₇×#452	—	2	1	no	—
51	C₇×#450	—	2	1	no	—
52	C₇×#440	—	2	1	no	—
53	C₇×#440	—	2	1	no	—
54	C₄×C₇×Q₁₆	—	2	1	no	—
55	C₄×C₇×SD₁₆	—	2	1	no	—
56	C₇×#432	C₂²	1	1	yes	characteristic
57	C₇×#431	C₂²	1	1	yes	characteristic
58	C₇×#431	C₂²	1	1	yes	characteristic
59	C₇×#333	C₂²	1	1	yes	characteristic, complemented
60	C₇×#333	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	#18807	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	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
75	C₂³×C₄×C₇	—	2	1	no	—
76	C₂×C₇×Q₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×SD₁₆	—	2	1	no	—
79	C₂×C₇×SD₁₆	—	2	1	no	—
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×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×#126	—	2	2	no	—
85	C₇×#126	—	2	2	no	—
86	C₇×#126	—	2	2	no	—
87	C₇×#125	—	2	2	no	—
88	C₇×#125	—	2	1	no	—
89	C₇×#125	—	2	1	no	—
90	C₇×#125	—	2	1	no	—
91	C₇×#124	—	2	2	no	—
92	C₇×#124	—	2	1	no	—
93	C₇×#123	—	2	2	no	—
94	C₇×#123	—	2	2	no	—
95	C₇×#123	—	2	2	no	—
96	C₇×#123	—	2	2	no	—
97	C₇×C₂²⋊Q₈	—	2	1	no	—
98	C₇×C₂²⋊Q₈	—	2	1	no	—
99	C₇×C₂²⋊Q₈	—	2	2	no	—
100	C₇×C₂²⋊Q₈	—	2	1	no	—
101	C₇×C₂²⋊Q₈	—	2	1	no	—
102	C₇×C₂²⋊Q₈	—	2	1	no	—
103	C₇×C₂²⋊Q₈	—	2	1	no	—
104	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
105	C₇×#121	—	2	2	no	—
106	C₇×#121	—	2	1	no	—
107	C₇×#121	—	2	1	no	—
108	C₇×#121	—	2	1	no	—
109	C₇×#120	—	2	2	no	—
110	C₄×C₇×Q₈	—	2	1	no	—
111	C₄×C₇×Q₈	—	2	1	no	—
112	C₄×C₇×Q₈	—	2	1	no	—
113	C₄×C₇×Q₈	—	2	1	no	—
114	C₄×C₇×Q₈	D₄	1	2	yes	—
115	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
116	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
117	C₄×C₇×D₄	—	2	2	no	—
118	C₄×C₇×D₄	—	2	2	no	—
119	C₄×C₇×D₄	—	2	2	no	—
120	C₄×C₇×D₄	—	2	1	no	—
121	C₄×C₇×D₄	—	2	1	no	—
122	C₄×C₇×D₄	—	2	1	no	—
123	C₄×C₇×D₄	—	2	1	no	—
124	C₄×C₇×D₄	—	2	1	no	—
125	C₄×C₇×D₄	D₄	1	1	yes	characteristic
126	C₄×C₇×D₄	D₄	1	1	yes	characteristic
127	C₄×C₇×D₄	D₄	1	1	yes	characteristic
128	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
129	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
130	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
131	C₇×#117	—	2	2	no	—
132	C₇×#117	—	2	2	no	—
133	C₇×#117	—	2	2	no	—
134	C₇×#117	—	2	1	no	—
135	C₇×#117	—	2	1	no	—
136	C₇×#117	—	2	1	no	—
137	C₇×#117	—	2	1	no	—
138	C₇×#117	—	2	1	no	—
139	C₇×#117	—	2	1	no	—
140	C₇×#117	D₄	1	1	yes	characteristic
141	C₂×C₇×C₄⋊C₄	—	2	1	no	—
142	C₂×C₇×C₄⋊C₄	—	2	1	no	—
143	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
144	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
145	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²×C₇	—	2	2	no	—
147	C₂×C₄²×C₇	—	2	1	no	—
148	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
149	C₇×#107	—	2	1	no	—
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	2	no	—
155	C₇×C₄⋊C₈	—	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	1	no	—
161	C₇×#103	—	2	2	no	—
162	C₇×#103	—	2	2	no	—
163	C₇×#103	—	2	1	no	—
164	C₇×#103	—	2	1	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×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	#3475	C₂×C₇	1	1	yes	characteristic
178	#3231	C₂×C₇	1	1	yes	characteristic, complemented
179	#3110	C₂×C₇	1	1	yes	characteristic, complemented
180	#3107	C₂×C₇	1	1	yes	characteristic, complemented
181	#1654	C₂×C₇	1	1	yes	characteristic, complemented
182	#1652	C₂×C₇	1	1	yes	characteristic, complemented
183	#1650	C₂×C₇	1	1	yes	characteristic, complemented
184	#1647	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	#1644	C₂×C₇	1	1	yes	characteristic, complemented
188	#1643	C₂×C₇	1	1	yes	characteristic, complemented
189	#1554	C₂×C₇	1	1	yes	characteristic, complemented
190	#1488	C₂×C₇	1	1	yes	characteristic, complemented
191	#1478	C₂×C₇	1	1	yes	characteristic, complemented
192	C₂⁴×C₇	—	2	1	no	—
193	C₇×#41	—	4	1	no	—
194	C₇×#41	—	4	1	no	—
195	C₇×#41	—	4	2	no	—
196	C₇×#41	—	4	2	no	—
197	C₂×C₇×Q₈	—	2	1	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	2	1	no	—
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	2	no	—
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	—	4	1	no	—
208	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
209	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
210	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₇	—	2	2	no	—
212	C₂²×C₄×C₇	—	2	1	no	—
213	C₂²×C₄×C₇	—	2	2	no	—
214	C₂²×C₄×C₇	—	2	1	no	—
215	C₂²×C₄×C₇	—	2	1	no	—
216	C₂²×C₄×C₇	—	2	1	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	—	4	2	no	—
222	C₂²×C₄×C₇	—	4	2	no	—
223	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
226	C₇×Q₁₆	—	4	2	no	—
227	C₇×SD₁₆	—	4	2	no	—
228	C₇×SD₁₆	—	4	2	no	—
229	C₇×SD₁₆	—	4	2	no	—
230	C₂×C₇×C₈	—	2	1	no	—
231	C₂×C₇×C₈	—	2	1	no	—
232	C₂×C₇×C₈	—	2	1	no	—
233	C₂×C₇×C₈	—	2	1	no	—
234	C₂×C₇×C₈	—	2	1	no	—
235	C₂×C₇×C₈	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	2	no	—
237	C₇×C₄⋊C₄	—	2	2	no	—
238	C₇×C₄⋊C₄	—	2	2	no	—
239	C₇×C₄⋊C₄	—	2	2	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	2	no	—
242	C₇×C₄⋊C₄	—	2	1	no	—
243	C₇×C₄⋊C₄	—	2	2	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	—	2	1	no	—
251	C₇×C₄⋊C₄	—	2	1	no	—
252	C₇×C₄⋊C₄	—	4	1	no	—
253	C₇×C₄⋊C₄	—	4	1	no	—
254	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
255	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
258	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	2	no	—
261	C₇×C₂²⋊C₄	—	2	2	no	—
262	C₇×C₂²⋊C₄	—	2	2	no	—
263	C₇×C₂²⋊C₄	—	2	2	no	—
264	C₇×C₂²⋊C₄	—	2	2	no	—
265	C₇×C₂²⋊C₄	—	2	1	no	—
266	C₇×C₂²⋊C₄	—	2	2	no	—
267	C₇×C₂²⋊C₄	—	2	1	no	—
268	C₇×C₂²⋊C₄	—	2	1	no	—
269	C₇×C₂²⋊C₄	—	2	1	no	—
270	C₇×C₂²⋊C₄	—	2	1	no	—
271	C₇×C₂²⋊C₄	—	2	1	no	—
272	C₇×C₂²⋊C₄	—	2	1	no	—
273	C₇×C₂²⋊C₄	—	4	1	no	—
274	C₄²×C₇	—	2	2	no	—
275	C₄²×C₇	—	2	2	no	—
276	C₄²×C₇	—	2	1	no	—
277	C₄²×C₇	—	2	2	no	—
278	C₄²×C₇	—	2	1	no	—
279	C₄²×C₇	—	2	1	no	—
280	C₄²×C₇	—	4	1	no	—
281	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₇	C₂⁴	1	1	yes	characteristic
285	#556	C₂²×C₇	1	1	yes	characteristic
286	#555	C₂²×C₇	1	1	yes	characteristic
287	#554	C₂²×C₇	1	1	yes	characteristic
288	#553	—	2	1	no	—
289	#552	—	2	1	no	—
290	#552	—	2	1	no	—
291	#551	—	2	2	no	—
292	#547	C₂²×C₇	1	1	yes	characteristic
293	#546	—	2	1	no	—
294	#533	C₂²×C₇	1	1	yes	characteristic
295	#533	C₂²×C₇	1	1	yes	characteristic
296	#529	—	2	2	no	—
297	#529	—	2	1	no	—
298	#525	—	2	1	no	—
299	C₄×#41	—	2	1	no	—
300	C₄×#41	C₂²×C₇	1	1	yes	characteristic
301	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
302	C₂×#117	—	2	1	no	—
303	#500	C₂²×C₇	1	1	yes	characteristic, complemented
304	#498	C₂²×C₇	1	1	yes	characteristic, complemented
305	#487	C₂²×C₇	1	1	yes	characteristic, complemented
306	#487	C₂²×C₇	1	1	yes	characteristic, complemented
307	#485	—	2	2	no	—
308	#485	—	2	1	no	—
309	#484	—	2	1	no	—
310	#479	—	2	2	no	—
311	#475	C₂²×C₇	1	1	yes	characteristic, complemented
312	#474	C₂²×C₇	1	1	yes	characteristic
313	#474	C₂²×C₇	1	1	yes	characteristic
314	#474	C₂²×C₇	1	1	yes	characteristic, complemented
315	#462	C₂²×C₇	1	1	yes	characteristic, complemented
316	#461	C₂²×C₇	1	1	yes	characteristic
317	#461	C₂²×C₇	1	1	yes	characteristic
318	#461	C₂²×C₇	1	1	yes	characteristic, complemented
319	#452	—	2	1	no	—
320	#450	—	2	1	no	—
321	#440	—	2	1	no	—
322	#440	—	2	1	no	—
323	C₄×Q₁₆	—	2	1	no	—
324	C₄×SD₁₆	—	2	1	no	—
325	#432	C₂²×C₇	1	1	yes	characteristic
326	#431	C₂²×C₇	1	1	yes	characteristic
327	#431	C₂²×C₇	1	1	yes	characteristic
328	#333	C₂²×C₇	1	1	yes	characteristic, complemented
329	#333	C₂²×C₇	1	1	yes	characteristic, complemented
330	#332	C₂²×C₇	1	1	yes	characteristic, complemented
331	#332	C₂²×C₇	1	1	yes	characteristic, complemented
332	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
333	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
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	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
338	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
339	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
340	C₂³×C₇	—	2	1	no	—
341	C₂³×C₇	—	2	1	no	—
342	C₂³×C₇	—	2	1	no	—
343	C₂³×C₇	—	4	2	no	—
344	C₂³×C₇	—	4	2	no	—
345	C₂³×C₇	—	8	1	no	—
346	C₂³×C₇	#120	1	1	yes	characteristic
347	C₇×Q₈	—	2	1	no	—
348	C₇×Q₈	—	2	1	no	—
349	C₇×Q₈	—	4	2	no	—
350	C₇×Q₈	—	4	1	no	—
351	C₇×Q₈	—	4	1	no	—
352	C₇×Q₈	—	4	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×Q₈	—	4	1	no	—
355	C₇×Q₈	—	4	1	no	—
356	C₇×D₄	—	2	1	no	—
357	C₇×D₄	—	2	1	no	—
358	C₇×D₄	—	2	1	no	—
359	C₇×D₄	—	2	1	no	—
360	C₇×D₄	—	2	1	no	—
361	C₇×D₄	—	4	2	no	—
362	C₇×D₄	—	4	4	no	—
363	C₇×D₄	—	4	2	no	—
364	C₇×D₄	—	8	2	no	—
365	C₇×D₄	#135	1	2	yes	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	1	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	2	1	no	—
373	C₂×C₄×C₇	—	2	1	no	—
374	C₂×C₄×C₇	—	2	2	no	—
375	C₂×C₄×C₇	—	2	2	no	—
376	C₂×C₄×C₇	—	2	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	2	no	—
379	C₂×C₄×C₇	—	4	2	no	—
380	C₂×C₄×C₇	—	4	2	no	—
381	C₂×C₄×C₇	—	4	2	no	—
382	C₂×C₄×C₇	—	4	4	no	—
383	C₂×C₄×C₇	—	4	1	no	—
384	C₂×C₄×C₇	—	4	1	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	—	8	1	no	—
392	C₂×C₄×C₇	—	8	2	no	—
393	C₂×C₄×C₇	#120	1	1	yes	characteristic
394	C₂×C₄×C₇	#120	1	1	yes	characteristic
395	C₂×C₄×C₇	#120	1	1	yes	characteristic
396	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
397	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
398	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
399	C₇×C₈	—	4	1	no	—
400	C₇×C₈	—	4	1	no	—
401	C₇×C₈	—	4	1	no	—
402	C₇×C₈	—	4	1	no	—
403	C₇×C₈	—	4	1	no	—
404	C₇×C₈	—	4	1	no	—
405	C₂×#41	—	2	1	no	—
406	C₂×#41	C₇×D₄	1	1	yes	characteristic
407	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
408	C₂³×C₄	—	2	1	no	—
409	C₂×Q₁₆	—	2	1	no	—
410	C₂×SD₁₆	—	2	1	no	—
411	C₂×SD₁₆	—	2	1	no	—
412	C₂×SD₁₆	—	2	1	no	—
413	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
414	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
415	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
416	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
417	#126	—	2	2	no	—
418	#126	—	2	2	no	—
419	#126	—	2	2	no	—
420	#125	—	2	2	no	—
421	#125	—	2	1	no	—
422	#125	—	2	1	no	—
423	#125	—	2	1	no	—
424	#124	—	2	2	no	—
425	#124	—	2	1	no	—
426	#123	—	2	2	no	—
427	#123	—	2	2	no	—
428	#123	—	2	2	no	—
429	#123	—	2	2	no	—
430	C₂²⋊Q₈	—	2	1	no	—
431	C₂²⋊Q₈	—	2	1	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	1	no	—
434	C₂²⋊Q₈	—	2	1	no	—
435	C₂²⋊Q₈	—	2	1	no	—
436	C₂²⋊Q₈	—	2	1	no	—
437	C₂²⋊Q₈	C₇×D₄	1	2	yes	—
438	#121	—	2	2	no	—
439	#121	—	2	1	no	—
440	#121	—	2	1	no	—
441	#121	—	2	1	no	—
442	#120	—	2	2	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	C₇×D₄	1	1	yes	characteristic
448	C₄×Q₈	C₇×D₄	1	2	yes	—
449	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
450	C₄×D₄	—	2	2	no	—
451	C₄×D₄	—	2	2	no	—
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	—	2	1	no	—
458	C₄×D₄	C₇×D₄	1	1	yes	characteristic
459	C₄×D₄	C₇×D₄	1	1	yes	characteristic
460	C₄×D₄	C₇×D₄	1	1	yes	characteristic
461	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
462	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
463	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
464	#117	—	2	2	no	—
465	#117	—	2	2	no	—
466	#117	—	2	2	no	—
467	#117	—	2	1	no	—
468	#117	—	2	1	no	—
469	#117	—	2	1	no	—
470	#117	—	2	1	no	—
471	#117	—	2	1	no	—
472	#117	—	2	1	no	—
473	#117	C₇×D₄	1	1	yes	characteristic
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	—	2	1	no	—
476	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
477	C₂×C₂²⋊C₄	—	2	2	no	—
478	C₂×C₂²⋊C₄	—	2	1	no	—
479	C₂×C₄²	—	2	2	no	—
480	C₂×C₄²	—	2	1	no	—
481	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
482	#107	—	2	1	no	—
483	#107	—	2	1	no	—
484	#107	—	2	1	no	—
485	#107	—	2	2	no	—
486	#106	—	2	1	no	—
487	#106	—	2	2	no	—
488	C₄⋊C₈	—	2	1	no	—
489	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
490	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
491	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
492	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
493	#103	—	2	1	no	—
494	#103	—	2	2	no	—
495	#103	—	2	2	no	—
496	#103	—	2	1	no	—
497	#103	—	2	1	no	—
498	#103	—	2	1	no	—
499	#103	—	2	1	no	—
500	#102	—	2	1	no	—
501	#102	—	2	1	no	—
502	#102	—	2	1	no	—
503	C₂²⋊C₈	—	2	1	no	—
504	C₂²⋊C₈	—	2	1	no	—
505	C₂²⋊C₈	—	2	1	no	—
506	#97	—	2	1	no	—
507	C₄×C₈	—	2	1	no	—
508	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
509	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
510	C₂²×C₇	—	2	1	no	—
511	C₂²×C₇	—	2	2	no	—
512	C₂²×C₇	—	2	1	no	—
513	C₂²×C₇	—	2	1	no	—
514	C₂²×C₇	—	4	1	no	—
515	C₂²×C₇	—	4	1	no	—
516	C₂²×C₇	—	4	1	no	—
517	C₂²×C₇	—	4	1	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	—	8	4	no	—
523	C₂²×C₇	C₂×#120	1	1	yes	characteristic
524	C₄×C₇	—	2	1	no	—
525	C₄×C₇	—	2	1	no	—
526	C₄×C₇	—	2	1	no	—
527	C₄×C₇	—	2	1	no	—
528	C₄×C₇	—	4	1	no	—
529	C₄×C₇	—	4	2	no	—
530	C₄×C₇	—	4	1	no	—
531	C₄×C₇	—	4	1	no	—
532	C₄×C₇	—	4	1	no	—
533	C₄×C₇	—	4	1	no	—
534	C₄×C₇	—	4	2	no	—
535	C₄×C₇	—	8	1	no	—
536	C₄×C₇	—	8	1	no	—
537	C₄×C₇	C₂×#135	1	1	yes	characteristic
538	C₄×C₇	#575	1	1	yes	characteristic
539	C₄×C₇	#575	1	1	yes	characteristic
540	C₄×C₇	#575	1	1	yes	characteristic
541	C₂⁴	—	2	1	no	—
542	#41	—	4	1	no	—
543	#41	—	4	1	no	—
544	#41	—	4	2	no	—
545	#41	—	4	2	no	—
546	C₂×Q₈	—	2	1	no	—
547	C₂×Q₈	—	2	1	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
550	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	2	no	—
553	C₂×D₄	—	2	1	no	—
554	C₂×D₄	—	2	2	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	4	1	no	—
557	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
558	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
559	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
560	C₂²×C₄	—	2	2	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	2	no	—
563	C₂²×C₄	—	2	1	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	1	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	4	2	no	—
571	C₂²×C₄	—	4	2	no	—
572	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
573	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
574	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
575	Q₁₆	—	4	2	no	—
576	SD₁₆	—	4	2	no	—
577	SD₁₆	—	4	2	no	—
578	SD₁₆	—	4	2	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	2	1	no	—
581	C₂×C₈	—	2	1	no	—
582	C₂×C₈	—	2	1	no	—
583	C₂×C₈	—	2	1	no	—
584	C₂×C₈	—	2	1	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	1	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	1	no	—
601	C₄⋊C₄	—	4	1	no	—
602	C₄⋊C₄	—	4	1	no	—
603	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
604	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
605	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
606	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
607	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
608	C₂²⋊C₄	—	2	2	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	2	2	no	—
611	C₂²⋊C₄	—	2	2	no	—
612	C₂²⋊C₄	—	2	2	no	—
613	C₂²⋊C₄	—	2	2	no	—
614	C₂²⋊C₄	—	2	1	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	1	no	—
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	2	1	no	—
621	C₂²⋊C₄	—	2	1	no	—
622	C₂²⋊C₄	—	4	1	no	—
623	C₄²	—	2	2	no	—
624	C₄²	—	2	2	no	—
625	C₄²	—	2	1	no	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	1	no	—
628	C₄²	—	2	1	no	—
629	C₄²	—	4	1	no	—
630	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
631	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
632	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
633	C₄²	C₂⁴×C₇	1	1	yes	characteristic
634	C₂×C₇	—	4	1	no	—
635	C₂×C₇	—	4	2	no	—
636	C₂×C₇	—	8	1	no	—
637	C₂×C₇	—	8	1	no	—
638	C₂×C₇	#3002	1	1	yes	characteristic
639	C₂×C₇	#3005	1	1	yes	characteristic
640	C₂×C₇	#3022	1	1	yes	characteristic
641	C₂³	—	2	1	no	—
642	C₂³	—	2	1	no	—
643	C₂³	—	2	1	no	—
644	C₂³	—	4	2	no	—
645	C₂³	—	4	2	no	—
646	C₂³	—	8	1	no	—
647	C₂³	C₇×#120	1	1	yes	characteristic
648	Q₈	—	2	1	no	—
649	Q₈	—	2	1	no	—
650	Q₈	—	4	2	no	—
651	Q₈	—	4	1	no	—
652	Q₈	—	4	1	no	—
653	Q₈	—	4	1	no	—
654	Q₈	—	4	1	no	—
655	Q₈	—	4	1	no	—
656	Q₈	—	4	1	no	—
657	D₄	—	2	1	no	—
658	D₄	—	2	1	no	—
659	D₄	—	2	1	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	1	no	—
662	D₄	—	4	2	no	—
663	D₄	—	4	4	no	—
664	D₄	—	4	2	no	—
665	D₄	—	8	2	no	—
666	D₄	C₇×#135	1	2	yes	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	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	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	2	no	—
676	C₂×C₄	—	2	2	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	2	no	—
680	C₂×C₄	—	4	2	no	—
681	C₂×C₄	—	4	2	no	—
682	C₂×C₄	—	4	2	no	—
683	C₂×C₄	—	4	4	no	—
684	C₂×C₄	—	4	1	no	—
685	C₂×C₄	—	4	1	no	—
686	C₂×C₄	—	4	1	no	—
687	C₂×C₄	—	4	1	no	—
688	C₂×C₄	—	4	1	no	—
689	C₂×C₄	—	4	1	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	8	1	no	—
693	C₂×C₄	—	8	2	no	—
694	C₂×C₄	C₇×#120	1	1	yes	characteristic
695	C₂×C₄	C₇×#120	1	1	yes	characteristic
696	C₂×C₄	C₇×#120	1	1	yes	characteristic
697	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
698	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
699	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
700	C₈	—	4	1	no	—
701	C₈	—	4	1	no	—
702	C₈	—	4	1	no	—
703	C₈	—	4	1	no	—
704	C₈	—	4	1	no	—
705	C₈	—	4	1	no	—
706	C₇	#18807	1	1	yes	characteristic, direct product factor, Sylow
707	C₂²	—	2	1	no	—
708	C₂²	—	2	2	no	—
709	C₂²	—	2	1	no	—
710	C₂²	—	2	1	no	—
711	C₂²	—	4	1	no	—
712	C₂²	—	4	1	no	—
713	C₂²	—	4	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	4	1	no	—
716	C₂²	—	4	1	no	—
717	C₂²	—	4	1	no	—
718	C₂²	—	4	1	no	—
719	C₂²	—	8	4	no	—
720	C₂²	C₂×C₇×#120	1	1	yes	characteristic
721	C₄	—	2	1	no	—
722	C₄	—	2	1	no	—
723	C₄	—	2	1	no	—
724	C₄	—	2	1	no	—
725	C₄	—	4	1	no	—
726	C₄	—	4	2	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	1	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	2	no	—
732	C₄	—	8	1	no	—
733	C₄	—	8	1	no	—
734	C₄	C₂×C₇×#135	1	1	yes	characteristic
735	C₄	C₇×#575	1	1	yes	characteristic
736	C₄	C₇×#575	1	1	yes	characteristic
737	C₄	C₇×#575	1	1	yes	characteristic
738	C₂	—	4	1	no	—
739	C₂	—	4	2	no	—
740	C₂	—	8	1	no	—
741	C₂	—	8	1	no	—
742	C₂	C₇×#3002	1	1	yes	characteristic
743	C₂	C₇×#3005	1	1	yes	characteristic
744	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×#3475	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×#3231	C₂	semidirect
3. C₇×#3110	C₂	semidirect
4. C₇×#3107	C₂	semidirect
5. C₇×#1654	C₂	semidirect
6. C₇×#1652	C₂	semidirect
7. C₇×#1650	C₂	semidirect
8. C₇×#1647	C₂	semidirect
9. C₇×#1646	C₂	semidirect
10. C₇×#1646	C₂	semidirect
11. C₇×#1644	C₂	semidirect
12. C₇×#1643	C₂	semidirect
13. C₇×#1554	C₂	semidirect
14. C₇×#1488	C₂	semidirect
15. C₇×#1478	C₂	semidirect
34. C₇×#500	C₂²	semidirect
35. C₇×#498	C₂²	semidirect
36. C₇×#487	C₂²	semidirect
37. C₇×#487	C₂²	semidirect
42. C₇×#475	C₂²	semidirect
45. C₇×#474	C₂²	semidirect
46. C₇×#462	C₂²	semidirect
49. C₇×#461	C₂²	semidirect
59. C₇×#333	C₂²	semidirect
60. C₇×#333	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. #18807	C₇	direct (N × H)
178. #3231	C₂×C₇	semidirect
179. #3110	C₂×C₇	semidirect
180. #3107	C₂×C₇	semidirect
181. #1654	C₂×C₇	semidirect
182. #1652	C₂×C₇	semidirect
183. #1650	C₂×C₇	semidirect
184. #1647	C₂×C₇	semidirect
185. #1646	C₂×C₇	semidirect
186. #1646	C₂×C₇	semidirect
187. #1644	C₂×C₇	semidirect
188. #1643	C₂×C₇	semidirect
189. #1554	C₂×C₇	semidirect
190. #1488	C₂×C₇	semidirect
191. #1478	C₂×C₇	semidirect
303. #500	C₂²×C₇	semidirect
304. #498	C₂²×C₇	semidirect
305. #487	C₂²×C₇	semidirect
306. #487	C₂²×C₇	semidirect
311. #475	C₂²×C₇	semidirect
314. #474	C₂²×C₇	semidirect
315. #462	C₂²×C₇	semidirect
318. #461	C₂²×C₇	semidirect
328. #333	C₂²×C₇	semidirect
329. #333	C₂²×C₇	semidirect
330. #332	C₂²×C₇	semidirect
331. #332	C₂²×C₇	semidirect
332. Q₈⋊C₈	C₂²×C₇	semidirect
333. Q₈⋊C₈	C₂²×C₇	semidirect
338. D₄⋊C₈	C₂²×C₇	semidirect
339. D₄⋊C₈	C₂²×C₇	semidirect
706. C₇	#18807	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×#3475	2
C₇×#3231	2
C₇×#3110	2
C₇×#3107	2
C₇×#1654	2
C₇×#1652	2
C₇×#1650	2
C₇×#1647	2
C₇×#1646	2
C₇×#1644	2
C₇×#1643	2
C₇×#1554	2
C₇×#1488	2
C₇×#1478	2
#18807	7

Subgroups — C7×#18807 #13196023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18807 #13196023