Subgroups — #10620474

Order 672.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6400	—	3	1	no	maximal, Hall
5	D₈₄	—	2	1	no	—
6	C₇×D₁₂	—	2	1	no	—
7	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×D₈	—	3	1	no	—
9	D₅₆	—	3	1	no	—
10	C₇⋊C₁₆	D₃	1	1	yes	characteristic, complemented
11	D₄₈	—	7	1	no	maximal, Hall
12	C₂×D₂₁	—	4	1	no	—
13	C₂×D₃×C₇	—	4	1	no	—
14	C₃×C₄×C₇	D₄	1	1	yes	characteristic
15	C₇×D₄	—	6	1	no	—
16	D₂₈	—	6	1	no	—
17	C₇×C₈	C₂×D₃	1	1	yes	characteristic
18	D₂₄	—	7	1	no	—
19	D₂₄	D₇	1	1	yes	characteristic, complemented
20	C₃×C₁₆	—	7	1	no	—
21	C₂×C₃×C₇	D₈	1	1	yes	characteristic
22	D₂₁	—	8	1	no	—
23	D₃×C₇	—	8	1	no	—
24	D₁₆	—	21	1	no	Sylow
25	C₂²×C₇	—	12	1	no	—
26	C₂×D₇	—	12	1	no	—
27	C₄×C₇	D₁₂	1	1	yes	characteristic
28	D₁₂	—	2	1	no	—
29	D₁₂	—	14	1	no	—
30	C₃×C₈	C₂×D₇	1	1	yes	characteristic
31	C₃×C₇	D₁₆	1	1	yes	characteristic, complemented, Hall
32	D₈	—	3	1	no	—
33	D₈	—	21	1	no	—
34	C₁₆	—	7	1	no	—
35	C₂×C₇	—	24	1	no	—
36	C₂×C₇	D₂₄	1	1	yes	characteristic
37	D₇	—	24	1	no	—
38	C₂×D₃	—	4	1	no	—
39	C₂×D₃	—	28	1	no	—
40	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
41	D₄	—	6	1	no	—
42	D₄	—	42	1	no	—
43	C₈	D₃×D₇	1	1	yes	characteristic
44	C₇	D₄₈	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃	#1141	1	1	yes	characteristic
46	D₃	—	8	1	no	—
47	D₃	—	56	1	no	—
48	C₂²	—	12	1	no	—
49	C₂²	—	84	1	no	—
50	C₄	#4247	1	1	yes	characteristic
51	C₃	#6400	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	24	1	no	—
53	C₂	—	168	1	no	—
54	C₂	#66562	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#66562
commutator, Fitting, cosocle	C₃×C₇×C₈	C₂²
Frattini, hypercenter	C₈	D₃×D₇
socle	C₂×C₃×C₇	D₈
hypocenter	C₃×C₇	D₁₆
LCS	G ▷ C₃×C₇×C₈ ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₁₆₈	C₂	semidirect
2. C₇×D₂₄	C₂	semidirect
3. C₃×C₇⋊C₁₆	C₂	semidirect
10. C₇⋊C₁₆	D₃	semidirect
19. D₂₄	D₇	semidirect
31. C₃×C₇	D₁₆	semidirect
44. C₇	D₄₈	semidirect
51. C₃	#6400	semidirect

Maximal subgroups

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

subgroup	index
D₁₆₈	2
C₇×D₂₄	2
C₃×C₇⋊C₁₆	2
#6400	3
D₄₈	7