Subgroups — #89676

Order 448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6467	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6374	C₂	1	2	yes	maximal, complemented
5	D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#1157	—	2	2	no	—
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	Dic₂₈	C₂²	1	2	yes	—
9	D₅₆	C₂²	1	2	yes	—
10	#1132	—	2	2	no	—
11	C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₇×C₁₆	C₂²	1	1	yes	characteristic
13	#508	—	7	1	no	maximal, Sylow
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂²⋊D₇	—	4	2	no	—
16	D₂₈	—	2	2	no	—
17	C₄×D₇	—	4	2	no	—
18	Dic₁₄	—	2	2	no	—
19	C₇×C₈	D₄	1	1	yes	characteristic
20	C₇×C₈	C₂³	1	1	yes	characteristic
21	#135	—	7	2	no	—
22	Q₃₂	—	7	1	no	—
23	SD₃₂	—	7	2	no	—
24	D₁₆	—	7	1	no	—
25	C₂×C₁₆	D₇	1	1	yes	characteristic, complemented
26	C₂²×C₇	D₈	1	1	yes	characteristic
27	C₂×D₇	—	4	2	no	—
28	C₄×C₇	D₈	1	1	yes	characteristic
29	C₄×C₇	C₂×D₄	1	1	yes	characteristic
30	Dic₇	—	4	2	no	—
31	#41	—	14	2	no	—
32	Q₁₆	—	7	2	no	—
33	SD₁₆	—	14	2	no	—
34	D₈	—	7	2	no	—
35	C₂×C₈	C₂×D₇	1	1	yes	characteristic
36	C₁₆	C₂×D₇	1	1	yes	characteristic
37	C₁₆	C₂×D₇	1	1	yes	characteristic
38	C₂×C₇	—	2	1	no	—
39	C₂×C₇	C₂×D₈	1	1	yes	characteristic
40	D₇	—	8	2	no	—
41	Q₈	—	14	2	no	—
42	D₄	—	14	2	no	—
43	D₄	—	28	2	no	—
44	C₂×C₄	—	28	2	no	—
45	C₂×C₄	D₂₈	1	1	yes	characteristic
46	C₈	D₂₈	1	1	yes	characteristic
47	C₈	C₂²×D₇	1	1	yes	characteristic
48	C₇	#508	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	28	2	no	—
50	C₂²	D₅₆	1	1	yes	characteristic
51	C₄	—	28	2	no	—
52	C₄	D₅₆	1	1	yes	characteristic
53	C₄	C₂×D₂₈	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	56	2	no	—
56	C₂	C₂×D₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅₆
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#508
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #6467	C₂	semidirect
2. C₂×C₇×C₁₆	C₂	semidirect
3. Dic₅₆	C₂	semidirect
4. #6374	C₂	semidirect
5. D₁₁₂	C₂	semidirect
25. C₂×C₁₆	D₇	semidirect
48. C₇	#508	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×C₇×C₁₆	2
Dic₅₆	2
#6374	2
D₁₁₂	2
#508	7