Subgroups — C₇×#19635 #13196851

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3208	C₂	1	2	yes	maximal, complemented
2	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
4	C₇×#3138	C₂	1	2	yes	maximal, complemented
5	C₇×#1717	C₂	1	4	yes	maximal, complemented
6	C₇×#1709	C₂	1	2	yes	maximal, complemented
7	C₇×#1706	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	—	2	2	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₂×C₇×#127	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	1	yes	characteristic, complemented
13	C₇×#494	C₂²	1	1	yes	characteristic, complemented
14	C₇×#493	—	2	2	no	—
15	C₇×#493	C₂²	1	2	yes	—
16	C₇×#493	C₂²	1	2	yes	—
17	C₇×#492	C₂²	1	4	yes	complemented
18	C₇×#490	—	2	4	no	—
19	C₇×#487	C₂²	1	2	yes	complemented
20	C₇×#486	C₂²	1	4	yes	complemented
21	C₇×#486	C₂²	1	4	yes	complemented
22	C₇×#486	C₂²	1	4	yes	complemented
23	C₇×#467	—	2	2	no	—
24	C₇×#460	—	2	4	no	—
25	C₇×#450	—	2	4	no	—
26	C₄×C₇×Q₁₆	—	2	2	no	—
27	C₇×#431	C₂²	1	1	yes	characteristic
28	C₇×#418	—	2	4	no	—
29	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
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	#19635	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×D₄	—	2	2	no	—
35	C₂²×C₇×D₄	—	2	2	no	—
36	C₂²×C₇×D₄	—	2	2	no	—
37	C₂×C₇×Q₁₆	—	2	2	no	—
38	C₂×C₇×SD₁₆	—	2	8	no	—
39	C₂×C₇×D₈	—	2	4	no	—
40	C₂×C₇×D₈	—	2	4	no	—
41	C₂×C₇×D₈	—	4	8	no	—
42	C₂×C₇×M₁₆	D₄	1	2	yes	—
43	C₂²×C₇×C₈	D₄	1	2	yes	—
44	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
46	C₇×#127	—	2	2	no	—
47	C₇×#127	—	2	2	no	—
48	C₇×#127	—	2	2	no	—
49	C₇×#127	C₂³	1	4	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×C₄²×C₇	C₂³	1	1	yes	characteristic
57	C₇×#107	—	2	2	no	—
58	C₇×C₄⋊C₈	D₄	1	2	yes	—
59	C₇×#103	—	2	4	no	—
60	C₇×#102	—	2	8	no	—
61	C₇×#102	—	2	8	no	—
62	C₇×#102	—	2	8	no	—
63	C₇×C₂²⋊C₈	D₄	1	2	yes	—
64	C₇×#97	D₄	1	2	yes	—
65	C₄×C₇×C₈	D₄	1	2	yes	—
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂³	1	2	yes	—
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	#3208	C₂×C₇	1	2	yes	complemented
71	#3170	C₂×C₇	1	1	yes	characteristic, complemented
72	C₂×#493	C₂×C₇	1	1	yes	characteristic
73	#3138	C₂×C₇	1	2	yes	complemented
74	#1717	C₂×C₇	1	4	yes	complemented
75	#1709	C₂×C₇	1	2	yes	complemented
76	#1706	C₂×C₇	1	2	yes	complemented
77	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂⁴×C₇	—	4	2	no	—
79	C₂×C₇×Q₈	—	2	2	no	—
80	C₂×C₇×Q₈	—	2	2	no	—
81	C₂×C₇×D₄	—	2	4	no	—
82	C₂×C₇×D₄	—	2	4	no	—
83	C₂×C₇×D₄	—	2	4	no	—
84	C₂×C₇×D₄	—	4	8	no	—
85	C₂×C₇×D₄	—	4	2	no	—
86	C₂×C₇×D₄	—	4	2	no	—
87	C₂×C₇×D₄	—	4	2	no	—
88	C₂×C₇×D₄	—	4	8	no	—
89	C₂×C₇×D₄	—	8	4	no	—
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₇×Q₁₆	—	4	4	no	—
94	C₇×SD₁₆	—	8	8	no	—
95	C₇×D₈	—	4	8	no	—
96	C₇×D₈	—	4	8	no	—
97	C₇×M₁₆	—	2	4	no	—
98	C₂×C₇×C₈	—	2	4	no	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
101	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
102	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₄⋊C₄	—	4	2	no	—
108	C₇×C₄⋊C₄	—	4	2	no	—
109	C₇×C₂²⋊C₄	—	4	2	no	—
110	C₄²×C₇	—	4	2	no	—
111	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	C₂⁴	1	1	yes	characteristic
115	C₂²×D₈	—	2	2	no	—
116	#533	C₂²×C₇	1	2	yes	—
117	C₂×#127	C₂²×C₇	1	2	yes	—
118	#500	C₂²×C₇	1	1	yes	characteristic, complemented
119	#494	C₂²×C₇	1	1	yes	characteristic, complemented
120	#493	—	2	2	no	—
121	#493	C₂²×C₇	1	2	yes	—
122	#493	C₂²×C₇	1	2	yes	—
123	#492	C₂²×C₇	1	4	yes	complemented
124	#490	—	2	4	no	—
125	#487	C₂²×C₇	1	2	yes	complemented
126	#486	C₂²×C₇	1	4	yes	complemented
127	#486	C₂²×C₇	1	4	yes	complemented
128	#486	C₂²×C₇	1	4	yes	complemented
129	#467	—	2	2	no	—
130	#460	—	2	4	no	—
131	#450	—	2	4	no	—
132	C₄×Q₁₆	—	2	2	no	—
133	#431	C₂²×C₇	1	1	yes	characteristic
134	#418	—	2	4	no	—
135	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
137	#322	C₂²×C₇	1	2	yes	complemented
138	C₈²	C₂²×C₇	1	2	yes	complemented
139	C₂³×C₇	—	4	4	no	—
140	C₂³×C₇	—	8	4	no	—
141	C₂³×C₇	—	8	4	no	—
142	C₂³×C₇	—	8	4	no	—
143	C₂³×C₇	#127	1	1	yes	characteristic
144	C₇×Q₈	—	4	4	no	—
145	C₇×Q₈	—	4	4	no	—
146	C₇×D₄	—	4	8	no	—
147	C₇×D₄	—	4	8	no	—
148	C₇×D₄	—	4	8	no	—
149	C₇×D₄	—	8	4	no	—
150	C₇×D₄	—	8	4	no	—
151	C₇×D₄	—	8	4	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	4	2	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	#127	1	1	yes	characteristic
157	C₂×C₄×C₇	#127	1	1	yes	characteristic
158	C₂×C₄×C₇	#127	1	1	yes	characteristic
159	C₂×C₄×C₇	#135	1	4	yes	—
160	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₇×C₈	—	2	4	no	—
164	C₇×C₈	—	2	2	no	—
165	C₇×C₈	—	2	4	no	—
166	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂×Q₁₆	—	2	2	no	—
171	C₂×SD₁₆	—	2	8	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	4	8	no	—
175	C₂×M₁₆	C₇×D₄	1	2	yes	—
176	C₂²×C₈	C₇×D₄	1	2	yes	—
177	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
178	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
179	#127	—	2	2	no	—
180	#127	—	2	2	no	—
181	#127	—	2	2	no	—
182	#127	C₂³×C₇	1	4	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	#117	—	2	2	no	—
189	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
190	#107	—	2	2	no	—
191	C₄⋊C₈	C₇×D₄	1	2	yes	—
192	#103	—	2	4	no	—
193	#102	—	2	8	no	—
194	#102	—	2	8	no	—
195	#102	—	2	8	no	—
196	C₂²⋊C₈	C₇×D₄	1	2	yes	—
197	#97	C₇×D₄	1	2	yes	—
198	C₄×C₈	C₇×D₄	1	2	yes	—
199	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₇	1	2	yes	—
201	C₄×C₈	C₂³×C₇	1	2	yes	—
202	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
203	C₂²×C₇	—	2	1	no	—
204	C₂²×C₇	—	2	1	no	—
205	C₂²×C₇	—	8	4	no	—
206	C₂²×C₇	—	8	4	no	—
207	C₂²×C₇	—	8	4	no	—
208	C₂²×C₇	—	16	8	no	—
209	C₂²×C₇	#495	1	2	yes	—
210	C₂²×C₇	C₂×#127	1	1	yes	characteristic
211	C₄×C₇	—	2	1	no	—
212	C₄×C₇	—	2	1	no	—
213	C₄×C₇	—	8	2	no	—
214	C₄×C₇	—	8	2	no	—
215	C₄×C₇	#495	1	2	yes	—
216	C₄×C₇	C₂×#135	1	2	yes	—
217	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
218	C₄×C₇	#576	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	8	no	—
230	C₂×D₄	—	8	4	no	—
231	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
234	Q₁₆	—	4	4	no	—
235	SD₁₆	—	8	8	no	—
236	D₈	—	4	8	no	—
237	D₈	—	4	8	no	—
238	M₁₆	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
241	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
242	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
243	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	4	2	no	—
252	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
253	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
254	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
255	C₄²	C₂⁴×C₇	1	1	yes	characteristic
256	C₂×C₇	—	2	2	no	—
257	C₂×C₇	—	16	4	no	—
258	C₂×C₇	C₂×#495	1	1	yes	characteristic
259	C₂×C₇	#3151	1	1	yes	characteristic
260	C₂×C₇	#3155	1	1	yes	characteristic
261	C₂³	—	4	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	8	4	no	—
265	C₂³	C₇×#127	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	D₄	—	4	8	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	8	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	C₇×#127	1	1	yes	characteristic
279	C₂×C₄	C₇×#127	1	1	yes	characteristic
280	C₂×C₄	C₇×#127	1	1	yes	characteristic
281	C₂×C₄	C₇×#135	1	4	yes	—
282	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	2	2	no	—
287	C₈	—	2	4	no	—
288	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
289	C₇	#19635	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	16	8	no	—
296	C₂²	C₇×#495	1	2	yes	—
297	C₂²	C₂×C₇×#127	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₇×#495	1	2	yes	—
303	C₄	C₂×C₇×#135	1	2	yes	—
304	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
305	C₄	C₇×#576	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	C₂×C₇×#495	1	1	yes	characteristic
309	C₂	C₇×#3151	1	1	yes	characteristic
310	C₂	C₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#493	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
1. C₇×#3208	C₂	semidirect
2. C₇×#3170	C₂	semidirect
4. C₇×#3138	C₂	semidirect
5. C₇×#1717	C₂	semidirect
6. C₇×#1709	C₂	semidirect
7. C₇×#1706	C₂	semidirect
8. C₇×C₈×M₁₆	C₂	semidirect
12. C₇×#500	C₂²	semidirect
13. C₇×#494	C₂²	semidirect
17. C₇×#492	C₂²	semidirect
19. C₇×#487	C₂²	semidirect
20. C₇×#486	C₂²	semidirect
21. C₇×#486	C₂²	semidirect
22. C₇×#486	C₂²	semidirect
31. C₇×#322	C₂²	semidirect
32. C₇×C₈²	C₂²	semidirect
33. #19635	C₇	direct (N × H)
70. #3208	C₂×C₇	semidirect
71. #3170	C₂×C₇	semidirect
73. #3138	C₂×C₇	semidirect
74. #1717	C₂×C₇	semidirect
75. #1709	C₂×C₇	semidirect
76. #1706	C₂×C₇	semidirect
77. C₈×M₁₆	C₂×C₇	semidirect
118. #500	C₂²×C₇	semidirect
119. #494	C₂²×C₇	semidirect
123. #492	C₂²×C₇	semidirect
125. #487	C₂²×C₇	semidirect
126. #486	C₂²×C₇	semidirect
127. #486	C₂²×C₇	semidirect
128. #486	C₂²×C₇	semidirect
137. #322	C₂²×C₇	semidirect
138. C₈²	C₂²×C₇	semidirect
289. C₇	#19635	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×#3208	2
C₇×#3170	2
C₂×C₇×#493	2
C₇×#3138	2
C₇×#1717	2
C₇×#1709	2
C₇×#1706	2
C₇×C₈×M₁₆	2
#19635	7

Subgroups — C7×#19635 #13196851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19635 #13196851