Subgroups — #89304

Order 448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#6392	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₈	—	2	1	no	—
5	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₇	—	2	1	no	—
7	D₅₆	—	2	1	no	—
8	D₅₆	C₄	1	2	yes	complemented
9	C₇×C₁₆	—	2	1	no	—
10	#357	—	7	1	no	maximal, Sylow
11	C₂²×D₇	—	4	1	no	—
12	C₂×C₄×C₇	D₄	1	1	yes	characteristic
13	C₂×Dic₇	—	4	1	no	—
14	D₂₈	—	2	2	no	—
15	D₂₈	—	4	1	no	—
16	C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×D₈	—	7	1	no	—
19	C₂×C₁₆	D₇	1	1	yes	characteristic, complemented
20	#107	—	7	1	no	—
21	C₂²×C₇	D₈	1	1	yes	characteristic
22	C₂×D₇	—	4	2	no	—
23	C₂×D₇	—	8	1	no	—
24	C₂×D₇	—	8	1	no	—
25	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₇	SD₁₆	1	1	yes	characteristic
27	Dic₇	—	8	1	no	—
28	C₂×D₄	—	14	1	no	—
29	D₈	—	7	2	no	—
30	D₈	—	14	1	no	—
31	C₂×C₈	C₂×D₇	1	1	yes	characteristic
32	C₄⋊C₄	—	14	1	no	—
33	C₁₆	—	2	1	no	—
34	C₂×C₇	#102	1	1	yes	characteristic
35	C₂×C₇	D₁₆	1	1	yes	characteristic
36	C₂×C₇	SD₃₂	1	1	yes	characteristic
37	D₇	—	8	2	no	—
38	C₂³	—	28	1	no	—
39	D₄	—	14	2	no	—
40	D₄	—	28	1	no	—
41	C₂×C₄	—	28	1	no	—
42	C₂×C₄	D₂₈	1	1	yes	characteristic
43	C₈	C₄×D₇	1	1	yes	characteristic
44	C₈	C₂²⋊D₇	1	1	yes	characteristic
45	C₇	#357	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	28	2	no	—
47	C₂²	—	56	1	no	—
48	C₂²	—	56	1	no	—
49	C₂²	D₅₆	1	1	yes	characteristic
50	C₄	—	56	1	no	—
51	C₄	#1132	1	1	yes	characteristic
52	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
53	C₂	—	56	2	no	—
54	C₂	D₁₁₂	1	1	yes	characteristic
55	C₂	#6374	1	1	yes	characteristic
56	C₂	#6395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅₆
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₂×C₇×C₁₆	C₂
socle	C₂²×C₇	D₈
prime radical	C₂×D₅₆	C₂
cosocle	C₂×C₇×C₈	C₂²
hypercenter	C₂×C₁₆	D₇
hypocenter	C₇	#357
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₁₆	C₂	semidirect
3. #6392	C₂	semidirect
8. D₅₆	C₄	semidirect
19. C₂×C₁₆	D₇	semidirect
45. C₇	#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₆	2
C₂×C₇×C₁₆	2
#6392	2
#357	7