Subgroups — #10620486

Order 672.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₇	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₂₁	C₄	1	2	yes	—
7	C₃×C₇⋊C₈	C₄	1	2	yes	complemented
8	C₇×C₃⋊C₈	C₄	1	2	yes	complemented
9	C₂×C₇×C₈	—	3	1	no	—
10	C₄×Dic₇	—	3	1	no	—
11	C₂×C₇⋊C₈	D₃	1	1	yes	characteristic
12	C₈×Dic₃	—	7	1	no	maximal, Hall
13	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
14	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	Dic₂₁	C₈	1	4	yes	complemented
17	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
18	C₂×Dic₇	—	3	2	no	—
19	C₇×C₈	—	3	2	no	—
20	C₇⋊C₈	Dic₃	1	2	yes	complemented
21	C₂×C₃×C₈	—	7	1	no	—
22	C₄×Dic₃	—	7	1	no	—
23	C₂×C₃⋊C₈	D₇	1	1	yes	characteristic
24	C₂×C₃×C₇	C₄²	1	1	yes	characteristic
25	C₂×C₃×C₇	C₂×C₈	1	2	yes	—
26	C₄×C₈	—	21	1	no	Sylow
27	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
28	C₄×C₇	C₄×D₃	1	1	yes	characteristic
29	C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
30	Dic₇	—	3	4	no	—
31	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
32	C₂×Dic₃	—	7	2	no	—
33	C₃×C₈	—	7	2	no	—
34	C₃⋊C₈	Dic₇	1	2	yes	complemented
35	C₃×C₇	C₄×C₈	1	1	yes	characteristic, complemented, Hall
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	—	7	1	no	—
38	C₄²	—	21	1	no	—
39	C₂×C₇	D₃×C₈	1	2	yes	—
40	C₂×C₇	C₄×Dic₃	1	1	yes	characteristic
41	C₂²×C₃	C₄×D₇	1	1	yes	characteristic
42	C₃×C₄	C₄×D₇	1	1	yes	characteristic
43	C₃×C₄	C₂×Dic₇	1	1	yes	characteristic
44	Dic₃	—	7	4	no	—
45	C₂×C₄	—	21	2	no	—
46	C₂×C₄	D₃×D₇	1	1	yes	characteristic
47	C₈	—	3	2	no	—
48	C₈	—	7	2	no	—
49	C₇	C₈×Dic₃	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	C₈×D₇	1	2	yes	—
51	C₂×C₃	C₄×Dic₇	1	1	yes	characteristic
52	C₂²	D₂₁⋊C₄	1	1	yes	characteristic
53	C₄	—	21	4	no	—
54	C₄	Dic₃×D₇	1	1	yes	characteristic
55	C₄	D₃×Dic₇	1	1	yes	characteristic
56	C₃	C₈×Dic₇	1	1	yes	characteristic, complemented, Sylow
57	C₂	D₂₁⋊C₈	1	2	yes	—
58	C₂	Dic₃×Dic₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₃×C₇⋊C₈	C₄	semidirect
8. C₇×C₃⋊C₈	C₄	semidirect
16. Dic₂₁	C₈	semidirect
20. C₇⋊C₈	Dic₃	semidirect
34. C₃⋊C₈	Dic₇	semidirect
35. C₃×C₇	C₄×C₈	semidirect
49. C₇	C₈×Dic₃	semidirect
56. C₃	C₈×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₂₁	2
C₂×C₇×C₃⋊C₈	2
C₂×C₃×C₇⋊C₈	2
C₈×Dic₇	3
C₈×Dic₃	7