Subgroups — #11729067

Order 896.

196 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, complemented
2	C₂×#6402	C₂	1	4	yes	maximal, complemented
3	#89467	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×Q₁₆	—	2	1	no	—
6	C₂×C₇×Q₁₆	C₂²	1	2	yes	—
7	C₇×C₄⋊Q₈	—	2	1	no	—
8	C₂×D₅₆	—	2	1	no	—
9	C₂×D₅₆	C₂²	1	2	yes	—
10	#6437	—	2	1	no	—
11	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#6402	—	2	8	no	—
13	C₂×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#2248	—	7	1	no	maximal, Sylow
15	C₂×C₇×Q₈	—	2	2	no	—
16	C₂×D₂₈	—	2	2	no	—
17	C₂×D₂₈	—	4	2	no	—
18	C₇×Q₁₆	—	2	4	no	—
19	C₇×Q₁₆	—	4	2	no	—
20	C₂×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇×C₄⋊C₄	—	4	2	no	—
23	C₄²×C₇	D₄	1	1	yes	characteristic
24	D₅₆	—	2	4	no	—
25	D₅₆	—	4	2	no	—
26	C₇⋊C₁₆	D₄	1	4	yes	complemented
27	C₂×SD₃₂	—	7	4	no	—
28	#494	D₇	1	1	yes	characteristic, complemented
29	#493	—	7	1	no	—
30	C₄×C₁₆	—	7	1	no	—
31	C₂²×D₇	—	4	2	no	—
32	C₇×Q₈	—	4	2	no	—
33	C₇×Q₈	—	4	2	no	—
34	C₂×C₄×C₇	—	4	2	no	—
35	C₂×C₄×C₇	D₈	1	2	yes	—
36	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	D₂₈	—	4	2	no	—
38	D₂₈	—	4	2	no	—
39	D₂₈	—	8	4	no	—
40	C₇×C₈	D₈	1	2	yes	—
41	C₇×C₈	C₂×D₄	1	2	yes	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	C₂×D₇	1	2	yes	—
44	C₂×D₈	—	7	2	no	—
45	C₂×D₈	—	14	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	#127	—	14	1	no	—
48	SD₃₂	—	14	8	no	—
49	C₂×C₁₆	—	7	2	no	—
50	C₄×C₈	C₂×D₇	1	1	yes	characteristic
51	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
52	C₂×D₇	—	8	2	no	—
53	C₂×D₇	—	8	4	no	—
54	C₄×C₇	—	8	2	no	—
55	C₄×C₇	SD₃₂	1	4	yes	complemented
56	C₄×C₇	#127	1	1	yes	characteristic
57	C₄×C₇	C₂×D₈	1	1	yes	characteristic
58	C₂×Q₈	—	2	2	no	—
59	C₂×D₄	—	14	2	no	—
60	C₂×D₄	—	28	2	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	D₈	—	14	4	no	—
64	D₈	—	28	2	no	—
65	C₂×C₈	C₂²⋊D₇	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊D₇	1	1	yes	characteristic
69	C₁₆	—	7	4	no	—
70	C₂×C₇	#493	1	1	yes	characteristic
71	C₂×C₇	C₂×SD₃₂	1	2	yes	—
72	D₇	—	16	2	no	—
73	C₂³	—	28	2	no	—
74	Q₈	—	4	2	no	—
75	Q₈	—	4	2	no	—
76	D₄	—	28	2	no	—
77	D₄	—	28	2	no	—
78	D₄	—	56	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#1141	1	2	yes	—
81	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
82	C₈	#1141	1	2	yes	—
83	C₈	D₄×D₇	1	2	yes	—
84	C₇	#2248	1	1	yes	characteristic, complemented, Sylow
85	C₂²	—	56	4	no	—
86	C₂²	—	56	2	no	—
87	C₂²	C₂×#1141	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#6402	1	4	yes	complemented
90	C₄	C₂×#1141	1	1	yes	characteristic
91	C₄	#6503	1	1	yes	characteristic
92	C₂	—	112	2	no	—
93	C₂	#89845	1	1	yes	characteristic
94	C₂	C₂×#6402	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#494	C₂
socle	C₂²×C₇	C₂×D₈
prime radical	#89467	C₂
hypercenter	#494	D₇
hypocenter	C₇	#2248
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₇×#494	C₂	semidirect
2. C₂×#6402	C₂	semidirect
4. C₄×C₇⋊C₁₆	C₂	semidirect
26. C₇⋊C₁₆	D₄	semidirect
28. #494	D₇	semidirect
55. C₄×C₇	SD₃₂	semidirect
84. C₇	#2248	semidirect
89. C₄	#6402	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
C₂×#6402	2
#89467	2
C₄×C₇⋊C₁₆	2
#2248	7