Subgroups — C₇×#19632 #13196848

Order 1792.

750 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3230	C₂	1	2	yes	maximal
2	C₇×#3170	C₂	1	1	yes	maximal, characteristic
3	C₇×#3165	C₂	1	2	yes	maximal
4	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	4	yes	maximal, complemented
6	C₇×#1713	C₂	1	2	yes	maximal, complemented
7	C₇×#1707	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	—	2	2	no	—
10	C₇×#533	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₇×#501	—	2	4	no	—
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	4	yes	—
15	C₇×#500	C₂²	1	4	yes	—
16	C₇×#498	C₂²	1	4	yes	—
17	C₇×#494	—	2	2	no	—
18	C₇×#494	C₂²	1	2	yes	—
19	C₇×#494	C₂²	1	1	yes	characteristic
20	C₇×#494	C₂²	1	2	yes	—
21	C₇×#487	C₂²	1	4	yes	—
22	C₇×#487	C₂²	1	2	yes	—
23	C₇×#484	—	2	4	no	—
24	C₇×#477	—	2	4	no	—
25	C₇×#467	—	2	2	no	—
26	C₄×C₇×Q₁₆	—	2	2	no	—
27	C₇×#431	C₂²	1	1	yes	characteristic
28	C₇×#428	—	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	—
32	C₇×C₈²	C₂²	1	2	yes	—
33	#19632	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×Q₈	—	2	2	no	—
35	C₂×C₇×Q₁₆	—	2	2	no	—
36	C₂×C₇×Q₁₆	—	2	4	no	—
37	C₂×C₇×Q₁₆	—	2	4	no	—
38	C₂×C₇×Q₁₆	—	4	8	no	—
39	C₂×C₇×M₁₆	D₄	1	2	yes	—
40	C₂²×C₇×C₈	D₄	1	2	yes	—
41	C₇×C₄⋊Q₈	—	2	2	no	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×#125	—	2	2	no	—
48	C₇×C₂²⋊Q₈	—	2	2	no	—
49	C₇×C₂²⋊Q₈	—	2	2	no	—
50	C₄×C₇×Q₈	—	2	2	no	—
51	C₄×C₇×Q₈	—	2	2	no	—
52	C₇×#117	—	2	2	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×#107	—	2	2	no	—
57	C₇×#107	—	2	8	no	—
58	C₇×#107	—	2	8	no	—
59	C₇×#106	—	2	8	no	—
60	C₇×C₄⋊C₈	D₄	1	2	yes	—
61	C₇×#103	—	2	4	no	—
62	C₇×#103	—	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	#3230	C₂×C₇	1	2	yes	—
71	#3170	C₂×C₇	1	1	yes	characteristic
72	#3165	C₂×C₇	1	2	yes	—
73	C₂×#494	C₂×C₇	1	1	yes	characteristic
74	#1723	C₂×C₇	1	4	yes	complemented
75	#1713	C₂×C₇	1	2	yes	complemented
76	#1707	C₂×C₇	1	2	yes	complemented
77	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	—	2	2	no	—
80	C₂×C₇×Q₈	—	2	4	no	—
81	C₂×C₇×Q₈	—	4	8	no	—
82	C₂×C₇×Q₈	—	4	2	no	—
83	C₂²×C₄×C₇	—	4	2	no	—
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₇×Q₁₆	—	4	8	no	—
88	C₇×Q₁₆	—	4	4	no	—
89	C₇×Q₁₆	—	4	8	no	—
90	C₇×M₁₆	Q₁₆	1	8	yes	complemented
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
94	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
95	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	2	4	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	4	2	no	—
103	C₇×C₄⋊C₄	—	4	2	no	—
104	C₇×C₄⋊C₄	—	4	2	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₇	—	4	2	no	—
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇	C₂⁴	1	1	yes	characteristic
112	C₂²×Q₁₆	—	2	2	no	—
113	#533	C₂²×C₇	1	2	yes	—
114	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
115	#501	—	2	4	no	—
116	#500	C₂²×C₇	1	4	yes	—
117	#500	C₂²×C₇	1	1	yes	characteristic
118	#500	C₂²×C₇	1	4	yes	—
119	#498	C₂²×C₇	1	4	yes	—
120	#494	—	2	2	no	—
121	#494	C₂²×C₇	1	2	yes	—
122	#494	C₂²×C₇	1	2	yes	—
123	#494	C₂²×C₇	1	1	yes	characteristic
124	#487	C₂²×C₇	1	2	yes	—
125	#487	C₂²×C₇	1	4	yes	—
126	#484	—	2	4	no	—
127	#477	—	2	4	no	—
128	#467	—	2	2	no	—
129	C₄×Q₁₆	—	2	2	no	—
130	#431	C₂²×C₇	1	1	yes	characteristic
131	#428	—	2	4	no	—
132	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
134	#322	C₂²×C₇	1	2	yes	—
135	C₈²	C₂²×C₇	1	2	yes	—
136	C₂³×C₇	#127	1	1	yes	characteristic
137	C₇×Q₈	—	4	8	no	—
138	C₇×Q₈	—	4	4	no	—
139	C₇×Q₈	—	4	4	no	—
140	C₇×Q₈	—	8	4	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	4	2	no	—
144	C₂×C₄×C₇	—	4	2	no	—
145	C₂×C₄×C₇	—	4	4	no	—
146	C₂×C₄×C₇	—	8	4	no	—
147	C₂×C₄×C₇	#127	1	1	yes	characteristic
148	C₂×C₄×C₇	#127	1	1	yes	characteristic
149	C₂×C₄×C₇	#127	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
151	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
154	C₇×C₈	—	2	2	no	—
155	C₇×C₈	—	2	4	no	—
156	C₇×C₈	C₂×Q₁₆	1	8	yes	complemented
157	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	—	4	8	no	—
163	C₂×M₁₆	C₇×D₄	1	2	yes	—
164	C₂²×C₈	C₇×D₄	1	2	yes	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
169	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
170	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
171	#125	—	2	2	no	—
172	C₂²⋊Q₈	—	2	2	no	—
173	C₂²⋊Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	#117	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
180	#107	—	2	8	no	—
181	#107	—	2	2	no	—
182	#107	—	2	8	no	—
183	#106	—	2	8	no	—
184	C₄⋊C₈	C₇×D₄	1	2	yes	—
185	#103	—	2	4	no	—
186	#103	—	2	8	no	—
187	C₂²⋊C₈	C₇×D₄	1	2	yes	—
188	#97	C₇×D₄	1	2	yes	—
189	C₄×C₈	C₇×D₄	1	2	yes	—
190	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₇	1	2	yes	—
192	C₄×C₈	C₂³×C₇	1	2	yes	—
193	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
194	C₂²×C₇	—	2	1	no	—
195	C₂²×C₇	—	2	1	no	—
196	C₂²×C₇	#494	1	2	yes	—
197	C₂²×C₇	C₂×#127	1	1	yes	characteristic
198	C₄×C₇	—	2	1	no	—
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	8	4	no	—
201	C₄×C₇	—	8	2	no	—
202	C₄×C₇	—	8	2	no	—
203	C₄×C₇	#494	1	2	yes	—
204	C₄×C₇	C₂²×Q₁₆	1	2	yes	—
205	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
206	C₄×C₇	#578	1	2	yes	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	8	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
216	Q₁₆	—	4	8	no	—
217	Q₁₆	—	4	4	no	—
218	Q₁₆	—	4	8	no	—
219	M₁₆	C₇×Q₁₆	1	8	yes	complemented
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
222	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
223	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
224	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
238	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
239	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
240	C₄²	C₂⁴×C₇	1	1	yes	characteristic
241	C₂×C₇	—	2	2	no	—
242	C₂×C₇	C₂×#494	1	1	yes	characteristic
243	C₂×C₇	#3151	1	1	yes	characteristic
244	C₂×C₇	#3156	1	1	yes	characteristic
245	C₂³	C₇×#127	1	1	yes	characteristic
246	Q₈	—	4	8	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	8	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	C₇×#127	1	1	yes	characteristic
257	C₂×C₄	C₇×#127	1	1	yes	characteristic
258	C₂×C₄	C₇×#127	1	1	yes	characteristic
259	C₂×C₄	C₂×C₇×Q₁₆	1	4	yes	—
260	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	C₂×C₇×Q₁₆	1	8	yes	complemented
266	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
267	C₇	#19632	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₇×#494	1	2	yes	—
271	C₂²	C₂×C₇×#127	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	2	no	—
275	C₄	—	8	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₇×#494	1	2	yes	—
278	C₄	C₂²×C₇×Q₁₆	1	2	yes	—
279	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
280	C₄	C₇×#578	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	C₂×C₇×#494	1	1	yes	characteristic
283	C₂	C₇×#3151	1	1	yes	characteristic
284	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂³×C₇	#127
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
5. C₇×#1723	C₂	semidirect
6. C₇×#1713	C₂	semidirect
7. C₇×#1707	C₂	semidirect
33. #19632	C₇	direct (N × H)
74. #1723	C₂×C₇	semidirect
75. #1713	C₂×C₇	semidirect
76. #1707	C₂×C₇	semidirect
90. C₇×M₁₆	Q₁₆	semidirect
156. C₇×C₈	C₂×Q₁₆	semidirect
219. M₁₆	C₇×Q₁₆	semidirect
265. C₈	C₂×C₇×Q₁₆	semidirect
267. C₇	#19632	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×#3230	2
C₇×#3170	2
C₇×#3165	2
C₂×C₇×#494	2
C₇×#1723	2
C₇×#1713	2
C₇×#1707	2
C₇×C₈×M₁₆	2
#19632	7

Subgroups — C7×#19632 #13196848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19632 #13196848