Subgroups — #89680

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6467	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#6374	C₂	1	2	yes	maximal, complemented
5	D₁₁₂	C₂	1	2	yes	maximal, complemented
6	#1157	—	2	1	no	—
7	C₂×D₂₈	—	2	1	no	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	Dic₂₈	C₂²	1	1	yes	characteristic, complemented
10	D₅₆	—	2	1	no	—
11	D₅₆	C₂²	1	2	yes	—
12	D₅₆	C₂²	1	1	yes	characteristic, complemented
13	#1132	—	2	1	no	—
14	C₇×C₁₆	C₂²	1	2	yes	complemented
15	#509	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	4	1	no	—
17	C₂×C₄×C₇	D₄	1	1	yes	characteristic
18	C₂²⋊D₇	—	4	1	no	—
19	D₂₈	—	2	1	no	—
20	D₂₈	—	2	2	no	—
21	D₂₈	—	4	1	no	—
22	C₄×D₇	—	4	1	no	—
23	Dic₁₄	—	2	1	no	—
24	C₇×C₈	D₄	1	1	yes	characteristic
25	C₇×C₈	C₂³	1	1	yes	characteristic
26	#135	—	7	1	no	—
27	C₂×D₈	—	7	1	no	—
28	SD₃₂	—	7	2	no	—
29	D₁₆	—	7	2	no	—
30	M₃₂	D₇	1	1	yes	characteristic, complemented
31	C₂²×C₇	D₈	1	1	yes	characteristic
32	C₂×D₇	—	4	1	no	—
33	C₂×D₇	—	4	2	no	—
34	C₂×D₇	—	8	2	no	—
35	C₄×C₇	D₈	1	1	yes	characteristic
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	Dic₇	—	4	1	no	—
38	#41	—	14	1	no	—
39	C₂×D₄	—	14	1	no	—
40	Q₁₆	—	7	1	no	—
41	SD₁₆	—	14	1	no	—
42	D₈	—	7	2	no	—
43	D₈	—	7	1	no	—
44	D₈	—	14	1	no	—
45	C₂×C₈	C₂×D₇	1	1	yes	characteristic
46	C₁₆	C₂×D₇	1	2	yes	complemented
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	C₂×D₈	1	1	yes	characteristic
49	D₇	—	8	1	no	—
50	D₇	—	8	2	no	—
51	C₂³	—	28	1	no	—
52	Q₈	—	14	1	no	—
53	D₄	—	14	2	no	—
54	D₄	—	14	1	no	—
55	D₄	—	28	1	no	—
56	D₄	—	28	1	no	—
57	C₂×C₄	—	28	1	no	—
58	C₂×C₄	D₂₈	1	1	yes	characteristic
59	C₈	D₂₈	1	1	yes	characteristic
60	C₈	C₂²×D₇	1	1	yes	characteristic
61	C₇	#509	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	28	1	no	—
63	C₂²	—	28	2	no	—
64	C₂²	—	56	2	no	—
65	C₂²	D₅₆	1	1	yes	characteristic
66	C₄	—	28	1	no	—
67	C₄	D₅₆	1	1	yes	characteristic
68	C₄	C₂×D₂₈	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	56	2	no	—
71	C₂	—	56	1	no	—
72	C₂	C₂×D₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅₆
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×M₃₂	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	M₃₂	D₇
hypocenter	C₇	#509
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #6467	C₂	semidirect
2. C₂×D₅₆	C₂	semidirect
3. C₇×M₃₂	C₂	semidirect
4. #6374	C₂	semidirect
5. D₁₁₂	C₂	semidirect
9. Dic₂₈	C₂²	semidirect
12. D₅₆	C₂²	semidirect
14. C₇×C₁₆	C₂²	semidirect
30. M₃₂	D₇	semidirect
46. C₁₆	C₂×D₇	semidirect
61. C₇	#509	semidirect

Maximal subgroups

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

subgroup	index
#6467	2
C₂×D₅₆	2
C₇×M₃₂	2
#6374	2
D₁₁₂	2
#509	7