Subgroups — C₇×#60715 #13237931

Order 1792.

1830 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3509	C₂	1	2	yes	maximal
2	C₇×#3185	C₂	1	4	yes	maximal, complemented
3	C₇×#3177	C₂	1	8	yes	maximal, complemented
4	C₇×#3170	C₂	1	4	yes	maximal, complemented
5	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3139	C₂	1	2	yes	maximal, complemented
8	C₇×#3059	C₂	1	4	yes	maximal
9	C₇×#2950	C₂	1	4	yes	maximal, complemented
10	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
11	C₂²×C₇×Q₁₆	C₂²	1	4	yes	—
12	C₂×C₇×#131	C₂²	1	2	yes	—
13	C₇×#557	—	2	4	no	—
14	C₇×#556	—	2	4	no	—
15	C₇×#552	—	2	8	no	—
16	C₇×#541	C₂²	1	8	yes	—
17	C₇×#536	C₂²	1	4	yes	—
18	C₇×#533	C₂²	1	8	yes	—
19	C₇×#533	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
21	C₇×#519	C₂²	1	4	yes	—
22	C₄×C₇×#41	C₂²	1	1	yes	characteristic
23	C₇×#500	C₂²	1	2	yes	—
24	C₇×#499	C₂²	1	4	yes	—
25	C₇×#498	C₂²	1	2	yes	—
26	C₇×#494	C₂²	1	2	yes	—
27	C₇×#494	C₂²	1	4	yes	—
28	C₇×#494	C₂²	1	2	yes	—
29	C₇×#491	C₂²	1	8	yes	—
30	C₇×#487	C₂²	1	8	yes	—
31	C₇×#470	C₂²	1	16	yes	—
32	C₇×#467	C₂²	1	16	yes	—
33	C₇×#441	C₂²	1	16	yes	—
34	C₄×C₇×Q₁₆	C₂²	1	16	yes	—
35	C₇×#427	C₂²	1	4	yes	—
36	C₇×#419	C₂²	1	8	yes	—
37	C₇×#405	C₂²	1	4	yes	—
38	C₇×#405	C₂²	1	2	yes	—
39	C₄×C₇×M₁₆	C₂²	1	2	yes	—
40	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
43	#60715	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	C₂³	1	1	yes	characteristic
45	C₂²×C₇×Q₈	C₂³	1	4	yes	—
46	C₂×C₇×Q₁₆	—	2	16	no	—
47	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
48	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
49	C₇×#131	—	2	8	no	—
50	C₂×C₇×M₁₆	C₂³	1	4	yes	—
51	C₂×C₇×M₁₆	C₂³	1	2	yes	—
52	C₂²×C₇×C₈	C₂³	1	4	yes	—
53	C₂²×C₇×C₈	C₂³	1	2	yes	—
54	C₇×C₄⋊Q₈	—	2	4	no	—
55	C₇×C₄⋊Q₈	—	2	4	no	—
56	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
57	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
58	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
59	C₇×#126	—	2	16	no	—
60	C₇×#125	—	2	16	no	—
61	C₇×#125	—	2	8	no	—
62	C₇×#125	C₂³	1	4	yes	—
63	C₇×#124	—	2	4	no	—
64	C₇×#123	—	2	8	no	—
65	C₇×C₂²⋊Q₈	—	2	4	no	—
66	C₇×C₂²⋊Q₈	—	2	8	no	—
67	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
68	C₄×C₇×Q₈	—	2	8	no	—
69	C₄×C₇×Q₈	—	2	8	no	—
70	C₄×C₇×Q₈	—	2	4	no	—
71	C₄×C₇×Q₈	D₄	1	2	yes	—
72	C₄×C₇×Q₈	C₂³	1	16	yes	—
73	C₄×C₇×D₄	D₄	1	4	yes	—
74	C₄×C₇×D₄	D₄	1	2	yes	—
75	C₇×#117	—	2	8	no	—
76	C₇×#117	—	2	8	no	—
77	C₇×#117	C₂³	1	4	yes	—
78	C₇×#117	C₂³	1	1	yes	characteristic
79	C₇×#117	C₂³	1	2	yes	—
80	C₂×C₇×C₄⋊C₄	—	2	4	no	—
81	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₇	C₂³	1	2	yes	—
83	C₇×#107	C₂³	1	8	yes	—
84	C₇×#106	C₂³	1	8	yes	—
85	C₇×#103	C₂³	1	32	yes	—
86	C₇×#97	C₂³	1	2	yes	—
87	C₇×#97	C₂³	1	4	yes	—
88	C₄×C₇×C₈	C₂³	1	2	yes	—
89	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₇×C₈	C₂³	1	2	yes	—
91	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₇×C₈	C₂³	1	4	yes	—
93	#3509	C₂×C₇	1	2	yes	—
94	#3185	C₂×C₇	1	4	yes	complemented
95	#3177	C₂×C₇	1	8	yes	complemented
96	#3170	C₂×C₇	1	4	yes	complemented
97	#3160	C₂×C₇	1	1	yes	characteristic, complemented
98	C₂×#494	C₂×C₇	1	1	yes	characteristic, complemented
99	#3139	C₂×C₇	1	2	yes	complemented
100	#3059	C₂×C₇	1	4	yes	—
101	#2950	C₂×C₇	1	4	yes	complemented
102	C₄×#131	C₂×C₇	1	1	yes	characteristic
103	C₇×#41	—	2	4	no	—
104	C₂×C₇×Q₈	—	2	4	no	—
105	C₂×C₇×Q₈	—	2	4	no	—
106	C₂×C₇×Q₈	—	2	16	no	—
107	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇×Q₈	C₂⁴	1	8	yes	—
109	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
110	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	—	2	4	no	—
112	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
115	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
116	C₇×Q₁₆	—	2	16	no	—
117	C₇×Q₁₆	—	2	16	no	—
118	C₇×M₁₆	—	2	4	no	—
119	C₇×M₁₆	—	2	8	no	—
120	C₂×C₇×C₈	—	2	8	no	—
121	C₂×C₇×C₈	#41	1	8	yes	—
122	C₂×C₇×C₈	C₂⁴	1	4	yes	—
123	C₂×C₇×C₈	C₂⁴	1	2	yes	—
124	C₂×C₇×C₈	C₂⁴	1	2	yes	—
125	C₇×C₄⋊C₄	—	2	8	no	—
126	C₇×C₄⋊C₄	—	2	8	no	—
127	C₇×C₄⋊C₄	—	2	4	no	—
128	C₇×C₄⋊C₄	—	2	16	no	—
129	C₇×C₄⋊C₄	—	2	8	no	—
130	C₇×C₄⋊C₄	—	2	16	no	—
131	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
132	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
133	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
134	C₇×C₂²⋊C₄	—	2	4	no	—
135	C₇×C₂²⋊C₄	—	2	16	no	—
136	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
137	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
138	C₄²×C₇	—	2	16	no	—
139	C₄²×C₇	—	2	4	no	—
140	C₄²×C₇	C₂×D₄	1	4	yes	—
141	C₄²×C₇	C₂×D₄	1	2	yes	—
142	C₄²×C₇	C₂⁴	1	1	yes	characteristic
143	C₄²×C₇	C₂⁴	1	2	yes	—
144	C₄²×C₇	C₂⁴	1	1	yes	characteristic
145	C₂²×Q₁₆	C₂²×C₇	1	4	yes	—
146	C₂×#131	C₂²×C₇	1	2	yes	—
147	#557	—	2	4	no	—
148	#556	—	2	4	no	—
149	#552	—	2	8	no	—
150	#541	C₂²×C₇	1	8	yes	—
151	#536	C₂²×C₇	1	4	yes	—
152	#533	C₂²×C₇	1	8	yes	—
153	#533	C₂²×C₇	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
155	#519	C₂²×C₇	1	4	yes	—
156	C₄×#41	C₂²×C₇	1	1	yes	characteristic
157	#500	C₂²×C₇	1	2	yes	—
158	#499	C₂²×C₇	1	4	yes	—
159	#498	C₂²×C₇	1	2	yes	—
160	#494	C₂²×C₇	1	2	yes	—
161	#494	C₂²×C₇	1	4	yes	—
162	#494	C₂²×C₇	1	2	yes	—
163	#491	C₂²×C₇	1	8	yes	—
164	#487	C₂²×C₇	1	8	yes	—
165	#470	C₂²×C₇	1	16	yes	—
166	#467	C₂²×C₇	1	16	yes	—
167	#441	C₂²×C₇	1	16	yes	—
168	C₄×Q₁₆	C₂²×C₇	1	16	yes	—
169	#427	C₂²×C₇	1	4	yes	—
170	#419	C₂²×C₇	1	8	yes	—
171	#405	C₂²×C₇	1	4	yes	—
172	#405	C₂²×C₇	1	2	yes	—
173	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
174	C₄×M₁₆	C₂²×C₇	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
177	C₂³×C₇	C₂²×D₄	1	2	yes	—
178	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
179	C₇×Q₈	—	2	2	no	—
180	C₇×Q₈	—	2	16	no	—
181	C₇×Q₈	—	4	8	no	—
182	C₇×Q₈	—	4	8	no	—
183	C₇×D₄	—	2	4	no	—
184	C₇×D₄	—	2	2	no	—
185	C₂×C₄×C₇	—	2	4	no	—
186	C₂×C₄×C₇	—	2	8	no	—
187	C₂×C₄×C₇	—	2	4	no	—
188	C₂×C₄×C₇	—	2	8	no	—
189	C₂×C₄×C₇	—	2	8	no	—
190	C₂×C₄×C₇	—	4	8	no	—
191	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
193	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₇	C₂×#41	1	4	yes	—
197	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
198	C₇×C₈	—	2	4	no	—
199	C₇×C₈	C₂×#41	1	4	yes	—
200	C₇×C₈	C₂×#41	1	4	yes	—
201	C₂×#41	C₂³×C₇	1	1	yes	characteristic
202	C₂²×Q₈	C₂³×C₇	1	4	yes	—
203	C₂×Q₁₆	—	2	16	no	—
204	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
205	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
206	#131	—	2	8	no	—
207	C₂×M₁₆	C₂³×C₇	1	2	yes	—
208	C₂×M₁₆	C₂³×C₇	1	4	yes	—
209	C₂²×C₈	C₂³×C₇	1	4	yes	—
210	C₂²×C₈	C₂³×C₇	1	2	yes	—
211	C₄⋊Q₈	—	2	4	no	—
212	C₄⋊Q₈	—	2	4	no	—
213	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
214	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
215	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
216	#126	—	2	16	no	—
217	#125	—	2	16	no	—
218	#125	—	2	8	no	—
219	#125	C₂³×C₇	1	4	yes	—
220	#124	—	2	4	no	—
221	#123	—	2	8	no	—
222	C₂²⋊Q₈	—	2	4	no	—
223	C₂²⋊Q₈	—	2	8	no	—
224	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
225	C₄×Q₈	—	2	8	no	—
226	C₄×Q₈	—	2	8	no	—
227	C₄×Q₈	—	2	4	no	—
228	C₄×Q₈	C₇×D₄	1	2	yes	—
229	C₄×Q₈	C₂³×C₇	1	16	yes	—
230	C₄×D₄	C₇×D₄	1	4	yes	—
231	C₄×D₄	C₇×D₄	1	2	yes	—
232	#117	—	2	8	no	—
233	#117	—	2	8	no	—
234	#117	C₂³×C₇	1	4	yes	—
235	#117	C₂³×C₇	1	1	yes	characteristic
236	#117	C₂³×C₇	1	2	yes	—
237	C₂×C₄⋊C₄	—	2	4	no	—
238	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₇	1	2	yes	—
240	#107	C₂³×C₇	1	8	yes	—
241	#106	C₂³×C₇	1	8	yes	—
242	#103	C₂³×C₇	1	32	yes	—
243	#97	C₂³×C₇	1	4	yes	—
244	#97	C₂³×C₇	1	2	yes	—
245	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₇	1	2	yes	—
247	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₇	1	4	yes	—
249	C₄×C₈	C₂³×C₇	1	2	yes	—
250	C₂²×C₇	—	2	2	no	—
251	C₂²×C₇	—	2	2	no	—
252	C₂²×C₇	—	2	4	no	—
253	C₂²×C₇	#532	1	2	yes	—
254	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
255	C₄×C₇	—	2	2	no	—
256	C₄×C₇	—	2	4	no	—
257	C₄×C₇	—	2	2	no	—
258	C₄×C₇	—	4	8	no	—
259	C₄×C₇	—	4	8	no	—
260	C₄×C₇	#532	1	2	yes	—
261	C₄×C₇	#578	1	4	yes	—
262	C₄×C₇	C₂²×#41	1	2	yes	—
263	#41	—	2	4	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	4	no	—
266	C₂×Q₈	—	2	16	no	—
267	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
268	C₂×Q₈	C₂⁴×C₇	1	8	yes	—
269	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
270	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
273	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
274	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
276	Q₁₆	—	2	16	no	—
277	Q₁₆	—	2	16	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	8	no	—
280	C₂×C₈	—	2	8	no	—
281	C₂×C₈	C₇×#41	1	8	yes	—
282	C₂×C₈	C₂⁴×C₇	1	4	yes	—
283	C₂×C₈	C₂⁴×C₇	1	2	yes	—
284	C₂×C₈	C₂⁴×C₇	1	2	yes	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	16	no	—
289	C₄⋊C₄	—	2	8	no	—
290	C₄⋊C₄	—	2	16	no	—
291	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
292	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
293	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	2	16	no	—
296	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
297	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
298	C₄²	—	2	16	no	—
299	C₄²	—	2	4	no	—
300	C₄²	C₂×C₇×D₄	1	4	yes	—
301	C₄²	C₂×C₇×D₄	1	2	yes	—
302	C₄²	C₂⁴×C₇	1	1	yes	characteristic
303	C₄²	C₂⁴×C₇	1	2	yes	—
304	C₄²	C₂⁴×C₇	1	1	yes	characteristic
305	C₂×C₇	—	2	2	no	—
306	C₂×C₇	—	4	2	no	—
307	C₂×C₇	C₂×#532	1	1	yes	characteristic
308	C₂×C₇	C₂×#578	1	2	yes	—
309	C₂³	C₂²×C₇×D₄	1	2	yes	—
310	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
311	Q₈	—	2	2	no	—
312	Q₈	—	2	16	no	—
313	Q₈	—	4	8	no	—
314	Q₈	—	4	8	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	2	8	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
324	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
325	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
326	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
328	C₂×C₄	C₂×C₇×#41	1	4	yes	—
329	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	C₂×C₇×#41	1	4	yes	—
332	C₈	C₂×C₇×#41	1	4	yes	—
333	C₇	#60715	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	2	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	2	2	no	—
337	C₂²	C₇×#532	1	2	yes	—
338	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
339	C₄	—	2	4	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	8	no	—
343	C₄	—	4	8	no	—
344	C₄	C₇×#532	1	2	yes	—
345	C₄	C₇×#578	1	4	yes	—
346	C₄	C₂²×C₇×#41	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	4	2	no	—
349	C₂	C₂×C₇×#532	1	1	yes	characteristic
350	C₂	C₂×C₇×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3185	C₂	semidirect
3. C₇×#3177	C₂	semidirect
4. C₇×#3170	C₂	semidirect
5. C₇×#3160	C₂	semidirect
6. C₂×C₇×#494	C₂	semidirect
7. C₇×#3139	C₂	semidirect
9. C₇×#2950	C₂	semidirect
43. #60715	C₇	direct (N × H)
94. #3185	C₂×C₇	semidirect
95. #3177	C₂×C₇	semidirect
96. #3170	C₂×C₇	semidirect
97. #3160	C₂×C₇	semidirect
98. C₂×#494	C₂×C₇	semidirect
99. #3139	C₂×C₇	semidirect
101. #2950	C₂×C₇	semidirect
333. C₇	#60715	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×#3509	2
C₇×#3185	2
C₇×#3177	2
C₇×#3170	2
C₇×#3160	2
C₂×C₇×#494	2
C₇×#3139	2
C₇×#3059	2
C₇×#2950	2
C₄×C₇×#131	2
#60715	7

Subgroups — C7×#60715 #13237931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60715 #13237931