Subgroups — C₇×#61881 #13239097

Order 1792.

2038 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic
3	C₇×#3392	C₂	1	8	yes	maximal, complemented
4	C₇×#3362	C₂	1	8	yes	maximal, complemented
5	C₇×#3208	C₂	1	6	yes	maximal, complemented
6	C₇×#3101	C₂	1	6	yes	maximal, complemented
7	C₇×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×#41	—	2	1	no	—
9	C₇×Q₈²	C₂²	1	4	yes	complemented
10	C₇×#557	—	2	4	no	—
11	C₇×#557	C₂²	1	4	yes	complemented
12	C₇×#550	—	2	4	no	—
13	C₇×#550	C₂²	1	8	yes	—
14	C₇×#544	—	2	3	no	—
15	C₇×#533	C₂²	1	6	yes	—
16	C₂×C₇×#127	C₂²	1	3	yes	—
17	C₂×C₇×#121	—	2	3	no	—
18	C₇×#519	—	2	1	no	—
19	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
21	C₇×#492	C₂²	1	12	yes	complemented
22	C₇×#486	C₂²	1	12	yes	complemented
23	C₇×#483	C₂²	1	4	yes	—
24	C₇×#476	C₂²	1	12	yes	complemented
25	C₇×#461	C₂²	1	12	yes	complemented
26	C₇×#460	C₂²	1	24	yes	complemented
27	C₇×#450	C₂²	1	12	yes	—
28	C₇×C₈×Q₈	C₂²	1	8	yes	complemented
29	C₄×C₇×SD₁₆	C₂²	1	24	yes	complemented
30	C₇×#431	C₂²	1	6	yes	—
31	#61881	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×#41	—	2	8	no	—
33	C₂×C₇×#41	—	4	8	no	—
34	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×D₄	—	2	3	no	—
36	C₂²×C₇×D₄	—	2	3	no	—
37	C₂²×C₇×D₄	C₂³	1	3	yes	—
38	C₂³×C₄×C₇	—	2	3	no	—
39	C₂×C₇×SD₁₆	C₂³	1	24	yes	—
40	C₇×C₄⋊Q₈	—	2	12	no	—
41	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
42	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
43	C₇×#127	—	2	6	no	—
44	C₇×#127	C₂³	1	12	yes	—
45	C₇×#125	—	2	12	no	—
46	C₇×#125	—	2	6	no	—
47	C₇×C₂²⋊Q₈	—	2	6	no	—
48	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
49	C₇×#121	—	2	24	no	—
50	C₄×C₇×Q₈	—	2	4	no	—
51	C₄×C₇×Q₈	—	2	4	no	—
52	C₄×C₇×Q₈	—	2	6	no	—
53	C₄×C₇×Q₈	D₄	1	4	yes	—
54	C₄×C₇×Q₈	C₂³	1	12	yes	—
55	C₄×C₇×Q₈	C₂³	1	4	yes	—
56	C₄×C₇×D₄	—	2	6	no	—
57	C₄×C₇×D₄	C₂³	1	12	yes	—
58	C₇×#117	—	2	6	no	—
59	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
60	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
61	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
62	C₂×C₄²×C₇	C₂³	1	3	yes	—
63	C₇×#106	C₂³	1	8	yes	—
64	C₇×C₄⋊C₈	C₂³	1	12	yes	—
65	C₇×#103	C₂³	1	8	yes	—
66	C₇×#102	C₂³	1	24	yes	—
67	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
68	C₄×C₇×C₈	C₂³	1	12	yes	—
69	#3502	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂×#550	C₂×C₇	1	1	yes	characteristic
71	#3392	C₂×C₇	1	8	yes	complemented
72	#3362	C₂×C₇	1	8	yes	complemented
73	#3208	C₂×C₇	1	6	yes	complemented
74	#3101	C₂×C₇	1	6	yes	complemented
75	#2982	C₂×C₇	1	1	yes	characteristic, complemented
76	C₂⁴×C₇	—	2	3	no	—
77	C₇×#41	—	4	32	no	—
78	C₂×C₇×Q₈	—	2	6	no	—
79	C₂×C₇×Q₈	SD₁₆	1	8	yes	complemented
80	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
81	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
82	C₂×C₇×D₄	—	2	3	no	—
83	C₂×C₇×D₄	—	2	12	no	—
84	C₂×C₇×D₄	—	2	12	no	—
85	C₂×C₇×D₄	—	4	6	no	—
86	C₂×C₇×D₄	—	4	12	no	—
87	C₂×C₇×D₄	—	8	6	no	—
88	C₂×C₇×D₄	C₂⁴	1	6	yes	—
89	C₂²×C₄×C₇	—	2	1	no	—
90	C₂²×C₄×C₇	—	2	12	no	—
91	C₂²×C₄×C₇	—	4	12	no	—
92	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₇×SD₁₆	—	4	24	no	—
96	C₂×C₇×C₈	C₂⁴	1	8	yes	—
97	C₇×C₄⋊C₄	—	2	1	no	—
98	C₇×C₄⋊C₄	—	2	6	no	—
99	C₇×C₄⋊C₄	—	2	6	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	6	no	—
102	C₇×C₄⋊C₄	—	2	12	no	—
103	C₇×C₄⋊C₄	C₂×D₄	1	12	yes	—
104	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
105	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
106	C₇×C₂²⋊C₄	—	2	4	no	—
107	C₇×C₂²⋊C₄	—	2	12	no	—
108	C₄²×C₇	—	2	12	no	—
109	C₄²×C₇	C₂×D₄	1	6	yes	—
110	C₄²×C₇	C₂⁴	1	6	yes	—
111	C₂²×#41	—	2	1	no	—
112	Q₈²	C₂²×C₇	1	4	yes	complemented
113	#557	—	2	4	no	—
114	#557	C₂²×C₇	1	4	yes	complemented
115	#550	—	2	4	no	—
116	#550	C₂²×C₇	1	8	yes	—
117	#544	—	2	3	no	—
118	#533	C₂²×C₇	1	6	yes	—
119	C₂×#127	C₂²×C₇	1	3	yes	—
120	C₂×#121	—	2	3	no	—
121	#519	—	2	1	no	—
122	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
124	#492	C₂²×C₇	1	12	yes	complemented
125	#486	C₂²×C₇	1	12	yes	complemented
126	#483	C₂²×C₇	1	4	yes	—
127	#476	C₂²×C₇	1	12	yes	complemented
128	#461	C₂²×C₇	1	12	yes	complemented
129	#460	C₂²×C₇	1	24	yes	complemented
130	#450	C₂²×C₇	1	12	yes	—
131	C₈×Q₈	C₂²×C₇	1	8	yes	complemented
132	C₄×SD₁₆	C₂²×C₇	1	24	yes	complemented
133	#431	C₂²×C₇	1	6	yes	—
134	C₂³×C₇	—	2	6	no	—
135	C₂³×C₇	—	4	6	no	—
136	C₂³×C₇	—	4	6	no	—
137	C₂³×C₇	—	4	6	no	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₇×Q₈	—	2	8	no	—
140	C₇×Q₈	—	8	6	no	—
141	C₇×Q₈	C₂×SD₁₆	1	16	yes	—
142	C₇×D₄	—	2	12	no	—
143	C₇×D₄	—	4	6	no	—
144	C₇×D₄	—	4	24	no	—
145	C₇×D₄	—	8	12	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	2	4	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	4	3	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	24	no	—
153	C₂×C₄×C₇	C₂×SD₁₆	1	12	yes	—
154	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
156	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
157	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
158	C₇×C₈	—	2	8	no	—
159	C₂×#41	—	2	8	no	—
160	C₂×#41	—	4	8	no	—
161	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
162	C₂²×D₄	—	2	3	no	—
163	C₂²×D₄	—	2	3	no	—
164	C₂²×D₄	C₂³×C₇	1	3	yes	—
165	C₂³×C₄	—	2	3	no	—
166	C₂×SD₁₆	C₂³×C₇	1	24	yes	—
167	C₄⋊Q₈	—	2	12	no	—
168	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
169	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
170	#127	—	2	6	no	—
171	#127	C₂³×C₇	1	12	yes	—
172	#125	—	2	12	no	—
173	#125	—	2	6	no	—
174	C₂²⋊Q₈	—	2	6	no	—
175	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
176	#121	—	2	24	no	—
177	C₄×Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	6	no	—
180	C₄×Q₈	C₇×D₄	1	4	yes	—
181	C₄×Q₈	C₂³×C₇	1	12	yes	—
182	C₄×Q₈	C₂³×C₇	1	4	yes	—
183	C₄×D₄	—	2	6	no	—
184	C₄×D₄	C₂³×C₇	1	12	yes	—
185	#117	—	2	6	no	—
186	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
187	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	—	2	3	no	—
189	C₂×C₄²	C₂³×C₇	1	3	yes	—
190	#106	C₂³×C₇	1	8	yes	—
191	C₄⋊C₈	C₂³×C₇	1	12	yes	—
192	#103	C₂³×C₇	1	8	yes	—
193	#102	C₂³×C₇	1	24	yes	—
194	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
195	C₄×C₈	C₂³×C₇	1	12	yes	—
196	C₂²×C₇	—	2	1	no	—
197	C₂²×C₇	—	2	1	no	—
198	C₂²×C₇	—	4	6	no	—
199	C₂²×C₇	—	4	6	no	—
200	C₂²×C₇	—	4	6	no	—
201	C₂²×C₇	—	8	12	no	—
202	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
203	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₇	—	2	6	no	—
205	C₄×C₇	—	2	1	no	—
206	C₄×C₇	—	4	4	no	—
207	C₄×C₇	—	4	4	no	—
208	C₄×C₇	—	4	2	no	—
209	C₄×C₇	C₂²×SD₁₆	1	12	yes	—
210	C₄×C₇	#585	1	2	yes	—
211	C₂⁴	—	2	3	no	—
212	#41	—	4	32	no	—
213	C₂×Q₈	—	2	6	no	—
214	C₂×Q₈	C₇×SD₁₆	1	8	yes	complemented
215	C₂×Q₈	C₂×C₇×D₄	1	4	yes	—
216	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
217	C₂×D₄	—	2	3	no	—
218	C₂×D₄	—	2	12	no	—
219	C₂×D₄	—	2	12	no	—
220	C₂×D₄	—	4	6	no	—
221	C₂×D₄	—	4	12	no	—
222	C₂×D₄	—	8	6	no	—
223	C₂×D₄	C₂⁴×C₇	1	6	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	12	no	—
226	C₂²×C₄	—	4	12	no	—
227	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
229	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
230	SD₁₆	—	4	24	no	—
231	C₂×C₈	C₂⁴×C₇	1	8	yes	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	6	no	—
234	C₄⋊C₄	—	2	6	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	6	no	—
237	C₄⋊C₄	—	2	12	no	—
238	C₄⋊C₄	C₂×C₇×D₄	1	12	yes	—
239	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
240	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	12	no	—
243	C₄²	—	2	12	no	—
244	C₄²	C₂×C₇×D₄	1	6	yes	—
245	C₄²	C₂⁴×C₇	1	6	yes	—
246	C₂×C₇	—	2	2	no	—
247	C₂×C₇	—	8	6	no	—
248	C₂×C₇	#3487	1	1	yes	characteristic
249	C₂×C₇	C₂³×SD₁₆	1	1	yes	characteristic
250	C₂×C₇	#3585	1	1	yes	characteristic
251	C₂³	—	2	6	no	—
252	C₂³	—	4	6	no	—
253	C₂³	—	4	6	no	—
254	C₂³	—	4	6	no	—
255	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
256	Q₈	—	2	8	no	—
257	Q₈	—	8	6	no	—
258	Q₈	C₂×C₇×SD₁₆	1	16	yes	—
259	D₄	—	2	12	no	—
260	D₄	—	4	6	no	—
261	D₄	—	4	24	no	—
262	D₄	—	8	12	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	3	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	24	no	—
270	C₂×C₄	C₂×C₇×SD₁₆	1	12	yes	—
271	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
273	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
274	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₇	#61881	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	6	no	—
280	C₂²	—	4	6	no	—
281	C₂²	—	4	6	no	—
282	C₂²	—	8	12	no	—
283	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
284	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
285	C₄	—	2	6	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂²×C₇×SD₁₆	1	12	yes	—
291	C₄	C₇×#585	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	8	6	no	—
294	C₂	C₇×#3487	1	1	yes	characteristic
295	C₂	C₂³×C₇×SD₁₆	1	1	yes	characteristic
296	C₂	C₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3502	C₂	semidirect
3. C₇×#3392	C₂	semidirect
4. C₇×#3362	C₂	semidirect
5. C₇×#3208	C₂	semidirect
6. C₇×#3101	C₂	semidirect
7. C₇×#2982	C₂	semidirect
9. C₇×Q₈²	C₂²	semidirect
11. C₇×#557	C₂²	semidirect
21. C₇×#492	C₂²	semidirect
22. C₇×#486	C₂²	semidirect
24. C₇×#476	C₂²	semidirect
25. C₇×#461	C₂²	semidirect
26. C₇×#460	C₂²	semidirect
28. C₇×C₈×Q₈	C₂²	semidirect
29. C₄×C₇×SD₁₆	C₂²	semidirect
31. #61881	C₇	direct (N × H)
69. #3502	C₂×C₇	semidirect
71. #3392	C₂×C₇	semidirect
72. #3362	C₂×C₇	semidirect
73. #3208	C₂×C₇	semidirect
74. #3101	C₂×C₇	semidirect
75. #2982	C₂×C₇	semidirect
79. C₂×C₇×Q₈	SD₁₆	semidirect
112. Q₈²	C₂²×C₇	semidirect
114. #557	C₂²×C₇	semidirect
124. #492	C₂²×C₇	semidirect
125. #486	C₂²×C₇	semidirect
127. #476	C₂²×C₇	semidirect
128. #461	C₂²×C₇	semidirect
129. #460	C₂²×C₇	semidirect
131. C₈×Q₈	C₂²×C₇	semidirect
132. C₄×SD₁₆	C₂²×C₇	semidirect
214. C₂×Q₈	C₇×SD₁₆	semidirect
276. C₇	#61881	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×#3502	2
C₂×C₇×#550	2
C₇×#3392	2
C₇×#3362	2
C₇×#3208	2
C₇×#3101	2
C₇×#2982	2
#61881	7

Subgroups — C7×#61881 #13239097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61881 #13239097