Subgroups — #11729053

Order 896.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#494	C₂	1	1	yes	maximal, characteristic
2	#89865	C₂	1	2	yes	maximal
3	#89851	C₂	1	2	yes	maximal
4	#89452	C₂	1	1	yes	maximal, characteristic
5	#89250	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₁₆	—	2	2	no	—
7	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#6432	C₂²	1	2	yes	—
10	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	#6391	—	2	2	no	—
12	#6384	—	2	4	no	—
13	#6382	—	2	4	no	—
14	C₄×C₇⋊C₈	C₂²	1	2	yes	—
15	#1716	—	7	1	no	maximal, Sylow
16	C₂×C₇×Q₈	—	2	1	no	—
17	C₂×C₇×Q₈	—	2	1	no	—
18	C₂×Dic₁₄	—	2	2	no	—
19	C₇×Q₁₆	—	4	4	no	—
20	C₂×C₇×C₈	D₄	1	2	yes	—
21	C₇×C₄⋊C₄	—	2	2	no	—
22	C₇×C₄⋊C₄	—	2	2	no	—
23	C₄²×C₇	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₇	—	2	2	no	—
25	C₄⋊Dic₇	—	2	2	no	—
26	C₂×C₇⋊C₈	D₄	1	4	yes	—
27	#498	—	7	2	no	—
28	#498	—	7	1	no	—
29	#494	D₇	1	1	yes	characteristic
30	#487	—	7	2	no	—
31	#322	—	7	1	no	—
32	C₇×Q₈	—	4	2	no	—
33	C₇×Q₈	—	4	2	no	—
34	C₂×C₄×C₇	—	4	1	no	—
35	C₂×C₄×C₇	—	4	1	no	—
36	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₇	—	4	2	no	—
39	Dic₁₄	—	4	4	no	—
40	C₇×C₈	SD₁₆	1	4	yes	—
41	C₇⋊C₈	—	2	4	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₄⋊Q₈	—	7	2	no	—
44	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
45	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
46	#106	—	14	4	no	—
47	#106	—	14	2	no	—
48	#103	—	14	4	no	—
49	C₄×C₈	—	7	2	no	—
50	C₄×C₈	C₂×D₇	1	1	yes	characteristic
51	C₂²×C₇	#127	1	1	yes	characteristic
52	C₄×C₇	—	8	1	no	—
53	C₄×C₇	—	8	1	no	—
54	C₄×C₇	C₂×SD₁₆	1	2	yes	—
55	C₄×C₇	#137	1	4	yes	—
56	Dic₇	—	8	2	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	2	1	no	—
59	C₂×Q₈	—	14	2	no	—
60	Q₁₆	—	4	4	no	—
61	C₂×C₈	—	7	4	no	—
62	C₂×C₈	C₂²⋊D₇	1	2	yes	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	14	2	no	—
66	C₄⋊C₄	—	14	2	no	—
67	C₄²	C₂²×D₇	1	1	yes	characteristic
68	C₂×C₇	#492	1	1	yes	characteristic
69	C₂×C₇	#497	1	2	yes	—
70	Q₈	—	4	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	28	4	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	28	2	no	—
76	C₂×C₄	D₄×D₇	1	2	yes	—
77	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
78	C₈	—	14	4	no	—
79	C₈	#1142	1	4	yes	—
80	C₇	#1716	1	1	yes	characteristic, complemented, Sylow
81	C₂²	#6503	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	56	2	no	—
85	C₄	#6481	1	4	yes	—
86	C₄	C₂×#1142	1	2	yes	—
87	C₂	#89848	1	1	yes	characteristic
88	C₂	#89962	1	2	yes	—

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#494	C₂
socle, prime radical	C₂²×C₇	#127
hypercenter	#494	D₇
hypocenter	C₇	#1716
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
80. C₇	#1716	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#494	2
#89865	2
#89851	2
#89452	2
#89250	2
#1716	7