Subgroups — C₇×#60627 #13237843

Order 1792.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3499	C₂	1	1	yes	maximal, characteristic
3	C₇×#3142	C₂	1	3	yes	maximal, complemented
4	C₇×#3139	C₂	1	3	yes	maximal, complemented
5	C₇×#3137	C₂	1	12	yes	maximal, complemented
6	C₇×#3135	C₂	1	3	yes	maximal, complemented
7	C₂×C₇×#491	C₂	1	3	yes	maximal, complemented
8	C₇×#2898	C₂	1	4	yes	maximal, complemented
9	C₇×#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×2₋¹⁺⁴	C₂²	1	2	yes	—
11	C₂×C₇×#131	C₂²	1	2	yes	—
12	C₇×#557	—	2	2	no	—
13	C₇×#554	—	2	6	no	—
14	C₇×D₄×Q₈	—	2	2	no	—
15	C₇×#547	—	2	6	no	—
16	C₇×#544	C₂²	1	6	yes	—
17	C₇×#536	C₂²	1	6	yes	—
18	C₇×#533	C₂²	1	3	yes	—
19	C₇×#532	C₂²	1	3	yes	—
20	C₂×C₇×C₄⋊Q₈	C₂²	1	3	yes	—
21	C₂×C₇×#125	C₂²	1	3	yes	—
22	C₇×#520	C₂²	1	2	yes	—
23	C₄×C₇×#41	C₂²	1	1	yes	characteristic
24	C₇×#491	C₂²	1	3	yes	—
25	C₇×#491	C₂²	1	12	yes	—
26	C₇×#491	C₂²	1	6	yes	—
27	C₇×#491	C₂²	1	3	yes	—
28	C₇×#490	C₂²	1	1	yes	characteristic
29	C₇×#490	C₂²	1	3	yes	—
30	C₇×#489	C₂²	1	12	yes	—
31	C₇×#488	C₂²	1	12	yes	—
32	C₇×#487	C₂²	1	12	yes	—
33	C₇×#419	C₂²	1	12	yes	—
34	C₇×#417	C₂²	1	12	yes	—
35	C₇×#416	C₂²	1	12	yes	—
36	C₂×C₇×#103	C₂²	1	12	yes	—
37	C₇×#405	C₂²	1	6	yes	—
38	C₄×C₇×M₁₆	C₂²	1	3	yes	—
39	C₂×C₇×#97	C₂²	1	3	yes	—
40	#60627	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₇×2₋¹⁺⁴	—	2	16	no	—
42	C₂×C₇×#41	—	2	6	no	—
43	C₂×C₇×#41	C₂³	1	6	yes	—
44	C₂×C₇×#41	C₂³	1	1	yes	characteristic
45	C₂²×C₇×Q₈	—	2	2	no	—
46	C₂²×C₇×Q₈	C₂³	1	6	yes	—
47	C₇×#131	—	2	8	no	—
48	C₂×C₇×M₁₆	C₂³	1	6	yes	—
49	C₂²×C₇×C₈	C₂³	1	6	yes	—
50	C₇×C₄⋊Q₈	—	2	6	no	—
51	C₇×C₄⋊Q₈	—	2	6	no	—
52	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
53	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
54	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
55	C₇×#127	C₂³	1	1	yes	characteristic
56	C₇×#125	—	2	6	no	—
57	C₇×#125	—	2	6	no	—
58	C₇×#125	C₂³	1	6	yes	—
59	C₇×#125	C₂³	1	3	yes	—
60	C₇×#125	C₂³	1	1	yes	characteristic
61	C₇×#124	C₂³	1	6	yes	—
62	C₇×#123	—	2	12	no	—
63	C₇×C₂²⋊Q₈	—	2	6	no	—
64	C₇×C₂²⋊Q₈	—	2	12	no	—
65	C₇×C₂²⋊Q₈	—	2	12	no	—
66	C₇×#121	—	2	6	no	—
67	C₄×C₇×Q₈	—	2	2	no	—
68	C₄×C₇×Q₈	—	2	6	no	—
69	C₄×C₇×Q₈	D₄	1	2	yes	—
70	C₄×C₇×D₄	—	2	6	no	—
71	C₄×C₇×D₄	—	2	6	no	—
72	C₄×C₇×D₄	D₄	1	6	yes	—
73	C₇×#117	C₂³	1	6	yes	—
74	C₇×#117	C₂³	1	3	yes	—
75	C₂×C₇×C₄⋊C₄	—	2	6	no	—
76	C₂×C₇×C₄⋊C₄	—	2	6	no	—
77	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
78	C₂×C₄²×C₇	C₂³	1	3	yes	—
79	C₇×#103	C₂³	1	12	yes	—
80	C₇×#103	C₂³	1	24	yes	—
81	C₇×#103	C₂³	1	12	yes	—
82	C₇×#102	C₂³	1	12	yes	—
83	C₇×#102	C₂³	1	4	yes	—
84	C₇×#97	C₂³	1	6	yes	—
85	C₇×#97	C₂³	1	1	yes	characteristic
86	C₇×#97	C₂³	1	3	yes	—
87	C₄×C₇×C₈	C₂³	1	6	yes	—
88	#3504	C₂×C₇	1	1	yes	characteristic, complemented
89	#3499	C₂×C₇	1	1	yes	characteristic
90	#3142	C₂×C₇	1	3	yes	complemented
91	#3139	C₂×C₇	1	3	yes	complemented
92	#3137	C₂×C₇	1	12	yes	complemented
93	#3135	C₂×C₇	1	3	yes	complemented
94	C₂×#491	C₂×C₇	1	3	yes	complemented
95	#2898	C₂×C₇	1	4	yes	complemented
96	#2875	C₂×C₇	1	1	yes	characteristic, complemented
97	C₇×#41	—	2	24	no	—
98	C₇×#41	—	4	24	no	—
99	C₇×#41	#41	1	8	yes	—
100	C₂×C₇×Q₈	—	2	6	no	—
101	C₂×C₇×Q₈	—	2	6	no	—
102	C₂×C₇×Q₈	—	2	24	no	—
103	C₂×C₇×Q₈	—	4	8	no	—
104	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₇×Q₈	C₂⁴	1	6	yes	—
106	C₂×C₇×D₄	—	2	6	no	—
107	C₂×C₇×D₄	—	2	2	no	—
108	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
109	C₂×C₇×D₄	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₇	—	2	6	no	—
111	C₂²×C₄×C₇	—	2	6	no	—
112	C₂²×C₄×C₇	—	2	6	no	—
113	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
114	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
115	C₇×M₁₆	—	2	12	no	—
116	C₂×C₇×C₈	—	2	12	no	—
117	C₂×C₇×C₈	C₂⁴	1	6	yes	—
118	C₂×C₇×C₈	C₂⁴	1	2	yes	—
119	C₇×C₄⋊C₄	—	2	6	no	—
120	C₇×C₄⋊C₄	—	2	6	no	—
121	C₇×C₄⋊C₄	—	2	12	no	—
122	C₇×C₄⋊C₄	—	2	12	no	—
123	C₇×C₄⋊C₄	—	2	6	no	—
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	—	2	6	no	—
126	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
127	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
128	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
129	C₇×C₂²⋊C₄	—	2	6	no	—
130	C₇×C₂²⋊C₄	—	2	12	no	—
131	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
132	C₄²×C₇	—	2	6	no	—
133	C₄²×C₇	C₂×D₄	1	6	yes	—
134	C₄²×C₇	C₂⁴	1	3	yes	—
135	C₄²×C₇	C₂⁴	1	1	yes	characteristic
136	C₂×2₋¹⁺⁴	C₂²×C₇	1	2	yes	—
137	C₂×#131	C₂²×C₇	1	2	yes	—
138	#557	—	2	2	no	—
139	#554	—	2	6	no	—
140	D₄×Q₈	—	2	2	no	—
141	#547	—	2	6	no	—
142	#544	C₂²×C₇	1	6	yes	—
143	#536	C₂²×C₇	1	6	yes	—
144	#533	C₂²×C₇	1	3	yes	—
145	#532	C₂²×C₇	1	3	yes	—
146	C₂×C₄⋊Q₈	C₂²×C₇	1	3	yes	—
147	C₂×#125	C₂²×C₇	1	3	yes	—
148	#520	C₂²×C₇	1	2	yes	—
149	C₄×#41	C₂²×C₇	1	1	yes	characteristic
150	#491	C₂²×C₇	1	3	yes	—
151	#491	C₂²×C₇	1	12	yes	—
152	#491	C₂²×C₇	1	6	yes	—
153	#491	C₂²×C₇	1	3	yes	—
154	#490	C₂²×C₇	1	1	yes	characteristic
155	#490	C₂²×C₇	1	3	yes	—
156	#489	C₂²×C₇	1	12	yes	—
157	#488	C₂²×C₇	1	12	yes	—
158	#487	C₂²×C₇	1	12	yes	—
159	#419	C₂²×C₇	1	12	yes	—
160	#417	C₂²×C₇	1	12	yes	—
161	#416	C₂²×C₇	1	12	yes	—
162	C₂×#103	C₂²×C₇	1	12	yes	—
163	#405	C₂²×C₇	1	6	yes	—
164	C₄×M₁₆	C₂²×C₇	1	3	yes	—
165	C₂×#97	C₂²×C₇	1	3	yes	—
166	C₂³×C₇	—	2	2	no	—
167	C₂³×C₇	C₂²×D₄	1	3	yes	—
168	C₇×Q₈	—	2	12	no	—
169	C₇×Q₈	—	4	12	no	—
170	C₇×Q₈	—	8	6	no	—
171	C₇×Q₈	#124	1	4	yes	—
172	C₇×D₄	—	2	4	no	—
173	C₇×D₄	—	4	4	no	—
174	C₇×D₄	—	4	12	no	—
175	C₇×D₄	#124	1	12	yes	—
176	C₂×C₄×C₇	—	2	2	no	—
177	C₂×C₄×C₇	—	2	6	no	—
178	C₂×C₄×C₇	—	2	6	no	—
179	C₂×C₄×C₇	—	4	6	no	—
180	C₂×C₄×C₇	—	4	12	no	—
181	C₂×C₄×C₇	—	4	12	no	—
182	C₂×C₄×C₇	—	8	6	no	—
183	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
184	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
185	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
186	C₂×C₄×C₇	C₂×#41	1	12	yes	—
187	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
188	C₇×C₈	—	2	2	no	—
189	C₇×C₈	—	2	6	no	—
190	2₋¹⁺⁴	—	2	16	no	—
191	C₂×#41	—	2	6	no	—
192	C₂×#41	C₂³×C₇	1	6	yes	—
193	C₂×#41	C₂³×C₇	1	1	yes	characteristic
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	C₂³×C₇	1	6	yes	—
196	#131	—	2	8	no	—
197	C₂×M₁₆	C₂³×C₇	1	6	yes	—
198	C₂²×C₈	C₂³×C₇	1	6	yes	—
199	C₄⋊Q₈	—	2	6	no	—
200	C₄⋊Q₈	—	2	6	no	—
201	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
202	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
203	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
204	#127	C₂³×C₇	1	1	yes	characteristic
205	#125	—	2	6	no	—
206	#125	—	2	6	no	—
207	#125	C₂³×C₇	1	3	yes	—
208	#125	C₂³×C₇	1	6	yes	—
209	#125	C₂³×C₇	1	1	yes	characteristic
210	#124	C₂³×C₇	1	6	yes	—
211	#123	—	2	12	no	—
212	C₂²⋊Q₈	—	2	6	no	—
213	C₂²⋊Q₈	—	2	12	no	—
214	C₂²⋊Q₈	—	2	12	no	—
215	#121	—	2	6	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	6	no	—
218	C₄×Q₈	C₇×D₄	1	2	yes	—
219	C₄×D₄	—	2	6	no	—
220	C₄×D₄	—	2	6	no	—
221	C₄×D₄	C₇×D₄	1	6	yes	—
222	#117	C₂³×C₇	1	6	yes	—
223	#117	C₂³×C₇	1	3	yes	—
224	C₂×C₄⋊C₄	—	2	6	no	—
225	C₂×C₄⋊C₄	—	2	6	no	—
226	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
227	C₂×C₄²	C₂³×C₇	1	3	yes	—
228	#103	C₂³×C₇	1	12	yes	—
229	#103	C₂³×C₇	1	24	yes	—
230	#103	C₂³×C₇	1	12	yes	—
231	#102	C₂³×C₇	1	4	yes	—
232	#102	C₂³×C₇	1	12	yes	—
233	#97	C₂³×C₇	1	1	yes	characteristic
234	#97	C₂³×C₇	1	3	yes	—
235	#97	C₂³×C₇	1	6	yes	—
236	C₄×C₈	C₂³×C₇	1	6	yes	—
237	C₂²×C₇	—	2	6	no	—
238	C₂²×C₇	—	4	4	no	—
239	C₂²×C₇	—	4	2	no	—
240	C₂²×C₇	C₂×#124	1	6	yes	—
241	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₇	—	2	2	no	—
243	C₄×C₇	—	2	6	no	—
244	C₄×C₇	—	4	6	no	—
245	C₄×C₇	—	4	2	no	—
246	C₄×C₇	—	4	6	no	—
247	C₄×C₇	C₂×#124	1	6	yes	—
248	C₄×C₇	C₂²×#41	1	2	yes	—
249	#41	—	2	24	no	—
250	#41	—	4	24	no	—
251	#41	C₇×#41	1	8	yes	—
252	C₂×Q₈	—	2	6	no	—
253	C₂×Q₈	—	2	6	no	—
254	C₂×Q₈	—	2	24	no	—
255	C₂×Q₈	—	4	8	no	—
256	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂×Q₈	C₂⁴×C₇	1	6	yes	—
258	C₂×D₄	—	2	6	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
261	C₂×D₄	C₂⁴×C₇	1	2	yes	—
262	C₂²×C₄	—	2	6	no	—
263	C₂²×C₄	—	2	6	no	—
264	C₂²×C₄	—	2	6	no	—
265	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
266	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
267	M₁₆	—	2	12	no	—
268	C₂×C₈	—	2	12	no	—
269	C₂×C₈	C₂⁴×C₇	1	6	yes	—
270	C₂×C₈	C₂⁴×C₇	1	2	yes	—
271	C₄⋊C₄	—	2	6	no	—
272	C₄⋊C₄	—	2	6	no	—
273	C₄⋊C₄	—	2	12	no	—
274	C₄⋊C₄	—	2	12	no	—
275	C₄⋊C₄	—	2	6	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	6	no	—
278	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
279	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
280	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
281	C₂²⋊C₄	—	2	6	no	—
282	C₂²⋊C₄	—	2	12	no	—
283	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
284	C₄²	—	2	6	no	—
285	C₄²	C₂×C₇×D₄	1	6	yes	—
286	C₄²	C₂⁴×C₇	1	3	yes	—
287	C₄²	C₂⁴×C₇	1	1	yes	characteristic
288	C₂×C₇	—	2	6	no	—
289	C₂×C₇	—	8	2	no	—
290	C₂×C₇	C₂²×#124	1	1	yes	characteristic
291	C₂×C₇	#3586	1	2	yes	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂²×C₇×D₄	1	3	yes	—
294	Q₈	—	2	12	no	—
295	Q₈	—	4	12	no	—
296	Q₈	—	8	6	no	—
297	Q₈	C₇×#124	1	4	yes	—
298	D₄	—	2	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	12	no	—
301	D₄	C₇×#124	1	12	yes	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	6	no	—
304	C₂×C₄	—	2	6	no	—
305	C₂×C₄	—	4	6	no	—
306	C₂×C₄	—	4	12	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	8	6	no	—
309	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
310	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
311	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
312	C₂×C₄	C₂×C₇×#41	1	12	yes	—
313	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	6	no	—
316	C₇	#60627	1	1	yes	characteristic, direct product factor, Sylow
317	C₂²	—	2	6	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₂×C₇×#124	1	6	yes	—
321	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	6	no	—
324	C₄	—	4	6	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂×C₇×#124	1	6	yes	—
328	C₄	C₂²×C₇×#41	1	2	yes	—
329	C₂	—	2	6	no	—
330	C₂	—	8	2	no	—
331	C₂	C₂²×C₇×#124	1	1	yes	characteristic
332	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3499	C₂
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
1. C₇×#3504	C₂	semidirect
3. C₇×#3142	C₂	semidirect
4. C₇×#3139	C₂	semidirect
5. C₇×#3137	C₂	semidirect
6. C₇×#3135	C₂	semidirect
7. C₂×C₇×#491	C₂	semidirect
8. C₇×#2898	C₂	semidirect
9. C₇×#2875	C₂	semidirect
40. #60627	C₇	direct (N × H)
88. #3504	C₂×C₇	semidirect
90. #3142	C₂×C₇	semidirect
91. #3139	C₂×C₇	semidirect
92. #3137	C₂×C₇	semidirect
93. #3135	C₂×C₇	semidirect
94. C₂×#491	C₂×C₇	semidirect
95. #2898	C₂×C₇	semidirect
96. #2875	C₂×C₇	semidirect
316. C₇	#60627	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×#3504	2
C₇×#3499	2
C₇×#3142	2
C₇×#3139	2
C₇×#3137	2
C₇×#3135	2
C₂×C₇×#491	2
C₇×#2898	2
C₇×#2875	2
#60627	7

Subgroups — C7×#60627 #13237843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60627 #13237843