Subgroups — C₇×#19651 #13196867

Order 1792.

806 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3229	C₂	1	2	yes	maximal, complemented
2	C₇×#3208	C₂	1	2	yes	maximal
3	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic
5	C₇×#1714	C₂	1	4	yes	maximal, complemented
6	C₇×#1708	C₂	1	2	yes	maximal, complemented
7	C₇×#1705	C₂	1	2	yes	maximal, complemented
8	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×SD₁₆	—	2	2	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×#127	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	2	yes	complemented
14	C₇×#499	—	2	2	no	—
15	C₇×#498	C₂²	1	1	yes	characteristic, complemented
16	C₇×#498	C₂²	1	1	yes	characteristic, complemented
17	C₇×#498	C₂²	1	4	yes	complemented
18	C₇×#492	—	2	2	no	—
19	C₇×#492	C₂²	1	2	yes	—
20	C₇×#492	C₂²	1	2	yes	—
21	C₇×#492	C₂²	1	4	yes	—
22	C₇×#487	C₂²	1	4	yes	complemented
23	C₇×#486	C₂²	1	4	yes	—
24	C₇×#483	—	2	4	no	—
25	C₇×#475	—	2	4	no	—
26	C₇×#460	—	2	4	no	—
27	C₇×#450	—	2	4	no	—
28	C₇×#431	C₂²	1	2	yes	—
29	C₇×#427	—	2	2	no	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	complemented
32	C₇×C₈²	C₂²	1	2	yes	complemented
33	#19651	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×Q₈	—	2	1	no	—
35	C₂²×C₇×D₄	—	2	1	no	—
36	C₂²×C₇×D₄	—	2	2	no	—
37	C₂×C₇×SD₁₆	—	2	4	no	—
38	C₂×C₇×SD₁₆	—	2	4	no	—
39	C₂×C₇×SD₁₆	—	2	8	no	—
40	C₂×C₇×SD₁₆	—	4	8	no	—
41	C₂²×C₇×C₈	D₄	1	2	yes	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	1	no	—
44	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×#127	—	2	1	no	—
48	C₇×#127	—	2	2	no	—
49	C₇×#127	C₂³	1	2	yes	—
50	C₇×#125	—	2	2	no	—
51	C₇×C₂²⋊Q₈	—	2	2	no	—
52	C₇×C₂²⋊Q₈	—	2	2	no	—
53	C₄×C₇×Q₈	—	2	2	no	—
54	C₄×C₇×Q₈	—	2	2	no	—
55	C₇×#117	—	2	2	no	—
56	C₂×C₇×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₇×#107	—	2	4	no	—
59	C₇×#106	—	2	8	no	—
60	C₇×#106	—	2	2	no	—
61	C₇×#106	—	2	2	no	—
62	C₇×C₄⋊C₈	D₄	1	4	yes	—
63	C₇×#103	—	2	8	no	—
64	C₇×#102	—	2	8	no	—
65	C₇×C₂²⋊C₈	D₄	1	4	yes	—
66	C₄×C₇×C₈	D₄	1	2	yes	—
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	4	yes	—
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	#3229	C₂×C₇	1	2	yes	complemented
71	#3208	C₂×C₇	1	2	yes	—
72	#3160	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂×#492	C₂×C₇	1	1	yes	characteristic
74	#1714	C₂×C₇	1	4	yes	complemented
75	#1708	C₂×C₇	1	2	yes	complemented
76	#1705	C₂×C₇	1	2	yes	complemented
77	#1450	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂⁴×C₇	—	4	1	no	—
79	C₂×C₇×Q₈	—	2	2	no	—
80	C₂×C₇×Q₈	—	2	2	no	—
81	C₂×C₇×Q₈	—	2	2	no	—
82	C₂×C₇×Q₈	—	4	1	no	—
83	C₂×C₇×Q₈	—	4	4	no	—
84	C₂×C₇×D₄	—	2	4	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	4	2	no	—
87	C₂×C₇×D₄	—	4	1	no	—
88	C₂×C₇×D₄	—	4	4	no	—
89	C₂×C₇×D₄	—	8	4	no	—
90	C₂²×C₄×C₇	—	4	1	no	—
91	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₇×SD₁₆	—	4	8	no	—
94	C₇×SD₁₆	—	4	8	no	—
95	C₇×SD₁₆	—	8	8	no	—
96	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
97	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
98	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₄⋊C₄	—	4	2	no	—
107	C₇×C₄⋊C₄	—	4	2	no	—
108	C₇×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₇	—	4	2	no	—
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇	C₂×D₄	1	2	yes	—
112	C₄²×C₇	C₂⁴	1	1	yes	characteristic
113	C₂²×SD₁₆	—	2	2	no	—
114	#533	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
116	C₂×#127	C₂²×C₇	1	1	yes	characteristic
117	#500	C₂²×C₇	1	2	yes	complemented
118	#499	—	2	2	no	—
119	#498	C₂²×C₇	1	1	yes	characteristic, complemented
120	#498	C₂²×C₇	1	1	yes	characteristic, complemented
121	#498	C₂²×C₇	1	4	yes	complemented
122	#492	—	2	2	no	—
123	#492	C₂²×C₇	1	2	yes	—
124	#492	C₂²×C₇	1	2	yes	—
125	#492	C₂²×C₇	1	4	yes	—
126	#487	C₂²×C₇	1	4	yes	complemented
127	#486	C₂²×C₇	1	4	yes	—
128	#483	—	2	4	no	—
129	#475	—	2	4	no	—
130	#460	—	2	4	no	—
131	#450	—	2	4	no	—
132	#431	C₂²×C₇	1	2	yes	—
133	#427	—	2	2	no	—
134	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
135	#322	C₂²×C₇	1	2	yes	complemented
136	C₈²	C₂²×C₇	1	2	yes	complemented
137	C₂³×C₇	—	4	2	no	—
138	C₂³×C₇	—	8	4	no	—
139	C₂³×C₇	—	8	2	no	—
140	C₂³×C₇	#127	1	1	yes	characteristic
141	C₇×Q₈	—	4	4	no	—
142	C₇×Q₈	—	4	4	no	—
143	C₇×Q₈	—	4	4	no	—
144	C₇×Q₈	—	8	2	no	—
145	C₇×D₄	—	4	4	no	—
146	C₇×D₄	—	4	8	no	—
147	C₇×D₄	—	8	4	no	—
148	C₇×D₄	—	8	2	no	—
149	C₂×C₄×C₇	—	2	4	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	—	4	2	no	—
153	C₂×C₄×C₇	—	8	2	no	—
154	C₂×C₄×C₇	#127	1	2	yes	—
155	C₂×C₄×C₇	#127	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
157	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
159	C₇×C₈	—	2	8	no	—
160	C₇×C₈	C₂×SD₁₆	1	4	yes	—
161	C₇×C₈	C₂×SD₁₆	1	4	yes	—
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	2	2	no	—
165	C₂×SD₁₆	—	2	4	no	—
166	C₂×SD₁₆	—	2	4	no	—
167	C₂×SD₁₆	—	2	8	no	—
168	C₂×SD₁₆	—	4	8	no	—
169	C₂²×C₈	C₇×D₄	1	2	yes	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	1	no	—
172	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
173	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
174	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
175	#127	—	2	2	no	—
176	#127	—	2	1	no	—
177	#127	C₂³×C₇	1	2	yes	—
178	#125	—	2	2	no	—
179	C₂²⋊Q₈	—	2	2	no	—
180	C₂²⋊Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×Q₈	—	2	2	no	—
183	#117	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
186	#107	—	2	4	no	—
187	#106	—	2	2	no	—
188	#106	—	2	2	no	—
189	#106	—	2	8	no	—
190	C₄⋊C₈	C₇×D₄	1	4	yes	—
191	#103	—	2	8	no	—
192	#102	—	2	8	no	—
193	C₂²⋊C₈	C₇×D₄	1	4	yes	—
194	C₄×C₈	C₇×D₄	1	2	yes	—
195	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₇	1	4	yes	—
197	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
198	C₂²×C₇	—	2	2	no	—
199	C₂²×C₇	—	8	2	no	—
200	C₂²×C₇	—	8	4	no	—
201	C₂²×C₇	—	16	4	no	—
202	C₂²×C₇	#492	1	2	yes	—
203	C₂²×C₇	C₂×#127	1	1	yes	characteristic
204	C₄×C₇	—	2	2	no	—
205	C₄×C₇	—	8	2	no	—
206	C₄×C₇	—	8	2	no	—
207	C₄×C₇	—	8	2	no	—
208	C₄×C₇	#492	1	2	yes	—
209	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
210	C₄×C₇	#577	1	4	yes	—
211	C₂⁴	—	4	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	8	4	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
225	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
226	SD₁₆	—	4	8	no	—
227	SD₁₆	—	4	8	no	—
228	SD₁₆	—	8	8	no	—
229	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
230	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
231	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
232	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
244	C₄²	C₂×C₇×D₄	1	2	yes	—
245	C₄²	C₂⁴×C₇	1	1	yes	characteristic
246	C₂×C₇	—	2	2	no	—
247	C₂×C₇	—	16	2	no	—
248	C₂×C₇	C₂×#492	1	1	yes	characteristic
249	C₂×C₇	#3154	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	—	8	4	no	—
252	C₂³	—	8	2	no	—
253	C₂³	C₇×#127	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	2	no	—
258	D₄	—	4	4	no	—
259	D₄	—	4	8	no	—
260	D₄	—	8	4	no	—
261	D₄	—	8	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	C₇×#127	1	2	yes	—
268	C₂×C₄	C₇×#127	1	1	yes	characteristic
269	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
270	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
272	C₈	—	2	8	no	—
273	C₈	C₂×C₇×SD₁₆	1	4	yes	—
274	C₈	C₂×C₇×SD₁₆	1	4	yes	—
275	C₇	#19651	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	16	4	no	—
280	C₂²	C₇×#492	1	2	yes	—
281	C₂²	C₂×C₇×#127	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₇×#492	1	2	yes	—
287	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
288	C₄	C₇×#577	1	4	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₇×#492	1	1	yes	characteristic
292	C₂	C₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂²
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
1. C₇×#3229	C₂	semidirect
3. C₇×#3160	C₂	semidirect
5. C₇×#1714	C₂	semidirect
6. C₇×#1708	C₂	semidirect
7. C₇×#1705	C₂	semidirect
8. C₇×#1450	C₂	semidirect
13. C₇×#500	C₂²	semidirect
15. C₇×#498	C₂²	semidirect
16. C₇×#498	C₂²	semidirect
17. C₇×#498	C₂²	semidirect
22. C₇×#487	C₂²	semidirect
31. C₇×#322	C₂²	semidirect
32. C₇×C₈²	C₂²	semidirect
33. #19651	C₇	direct (N × H)
70. #3229	C₂×C₇	semidirect
72. #3160	C₂×C₇	semidirect
74. #1714	C₂×C₇	semidirect
75. #1708	C₂×C₇	semidirect
76. #1705	C₂×C₇	semidirect
77. #1450	C₂×C₇	semidirect
96. C₂×C₇×C₈	SD₁₆	semidirect
117. #500	C₂²×C₇	semidirect
119. #498	C₂²×C₇	semidirect
120. #498	C₂²×C₇	semidirect
121. #498	C₂²×C₇	semidirect
126. #487	C₂²×C₇	semidirect
135. #322	C₂²×C₇	semidirect
136. C₈²	C₂²×C₇	semidirect
229. C₂×C₈	C₇×SD₁₆	semidirect
275. C₇	#19651	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×#3229	2
C₇×#3208	2
C₇×#3160	2
C₂×C₇×#492	2
C₇×#1714	2
C₇×#1708	2
C₇×#1705	2
C₇×#1450	2
#19651	7

Subgroups — C7×#19651 #13196867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19651 #13196867