Subgroups — #89313

Order 448.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6467	C₂	1	1	yes	maximal, characteristic
2	C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#6391	C₂	1	1	yes	maximal, characteristic, complemented
4	#1157	—	2	1	no	—
5	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₇	—	2	1	no	—
7	Dic₂₈	C₄	1	1	yes	characteristic, complemented
8	D₅₆	C₄	1	1	yes	characteristic, complemented
9	#1132	—	2	1	no	—
10	C₇×C₁₆	—	2	1	no	—
11	#360	—	7	1	no	maximal, Sylow
12	C₂×C₄×C₇	D₄	1	1	yes	characteristic
13	C₂²⋊D₇	—	4	1	no	—
14	C₂×Dic₇	—	4	1	no	—
15	D₂₈	—	2	1	no	—
16	C₄×D₇	—	4	1	no	—
17	Dic₁₄	—	2	1	no	—
18	C₇×C₈	C₂×C₄	1	1	yes	characteristic
19	C₇×C₈	D₄	1	1	yes	characteristic
20	#135	—	7	1	no	—
21	M₃₂	D₇	1	1	yes	characteristic, complemented
22	#106	—	7	1	no	—
23	C₂²×C₇	D₈	1	1	yes	characteristic
24	C₂×D₇	—	4	1	no	—
25	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₇	SD₁₆	1	1	yes	characteristic
27	Dic₇	—	4	1	no	—
28	Dic₇	—	8	1	no	—
29	#41	—	14	1	no	—
30	Q₁₆	—	7	1	no	—
31	SD₁₆	—	14	1	no	—
32	D₈	—	7	1	no	—
33	C₂×C₈	C₂×D₇	1	1	yes	characteristic
34	C₄⋊C₄	—	14	1	no	—
35	C₁₆	—	2	1	no	—
36	C₂×C₇	—	2	1	no	—
37	C₂×C₇	#102	1	1	yes	characteristic
38	D₇	—	8	1	no	—
39	Q₈	—	14	1	no	—
40	D₄	—	14	1	no	—
41	D₄	—	28	1	no	—
42	C₂×C₄	—	28	1	no	—
43	C₂×C₄	—	28	1	no	—
44	C₂×C₄	D₂₈	1	1	yes	characteristic
45	C₈	C₄×D₇	1	1	yes	characteristic
46	C₈	C₂²⋊D₇	1	1	yes	characteristic
47	C₇	#360	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	28	1	no	—
49	C₂²	D₅₆	1	1	yes	characteristic
50	C₄	—	28	1	no	—
51	C₄	—	56	1	no	—
52	C₄	#1132	1	1	yes	characteristic
53	C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	56	1	no	—
56	C₂	#6395	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6395
commutator	C₇×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₇
Fitting	C₇×M₃₂	C₂
socle	C₂×C₇	#102
prime radical	#6467	C₂
cosocle	C₂×C₇×C₈	C₂²
hypercenter	M₃₂	D₇
hypocenter	C₇	#360
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
2. C₇×M₃₂	C₂	semidirect
3. #6391	C₂	semidirect
7. Dic₂₈	C₄	semidirect
8. D₅₆	C₄	semidirect
21. M₃₂	D₇	semidirect
47. C₇	#360	semidirect

Maximal subgroups

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

subgroup	index
#6467	2
C₇×M₃₂	2
#6391	2
#360	7