Subgroups — #10620488

Order 672.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#6377	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×Dic₃	C₄	1	2	yes	—
7	C₃⋊(C₇⋊C₈)	C₄	1	2	yes	complemented
8	C₃×C₇⋊C₈	C₄	1	2	yes	complemented
9	C₄²×C₇	—	3	1	no	—
10	C₂×C₇⋊C₈	—	3	1	no	—
11	C₂×C₇⋊C₈	D₃	1	1	yes	characteristic
12	#815	—	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	C₇×Dic₃	—	2	2	no	—
17	C₂×C₄×C₇	—	3	2	no	—
18	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
19	C₇⋊C₈	—	3	2	no	—
20	C₇⋊C₈	Dic₃	1	2	yes	complemented
21	C₂×C₃×C₈	—	7	1	no	—
22	C₄×Dic₃	D₇	1	1	yes	characteristic
23	C₂×C₃⋊C₈	—	7	1	no	—
24	C₂×C₃×C₇	C₄²	1	1	yes	characteristic
25	C₂×C₃×C₇	M₁₆	1	2	yes	—
26	#97	—	21	1	no	Sylow
27	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
28	C₄×C₇	—	6	2	no	—
29	C₄×C₇	C₄×D₃	1	1	yes	characteristic
30	C₄×C₇	C₂×Dic₃	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
32	C₂×Dic₃	Dic₇	1	2	yes	—
33	C₃×C₈	—	7	2	no	—
34	C₃⋊C₈	—	7	2	no	—
35	C₃×C₇	#97	1	1	yes	characteristic, complemented, Hall
36	C₂×C₈	—	7	1	no	—
37	C₂×C₈	—	21	1	no	—
38	C₄²	—	3	1	no	—
39	C₂×C₇	ΓL(1,𝔽₂₅)	1	2	yes	—
40	C₂×C₇	C₄×Dic₃	1	1	yes	characteristic
41	C₂²×C₃	C₂×Dic₇	1	1	yes	characteristic
42	C₃×C₄	C₄×D₇	1	1	yes	characteristic
43	C₃×C₄	C₄×D₇	1	1	yes	characteristic
44	Dic₃	—	2	2	no	—
45	C₂×C₄	—	3	2	no	—
46	C₂×C₄	D₃×D₇	1	1	yes	characteristic
47	C₈	—	7	2	no	—
48	C₈	—	21	2	no	—
49	C₇	#815	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	#1136	1	2	yes	—
51	C₂×C₃	C₄×Dic₇	1	1	yes	characteristic
52	C₂²	D₃×Dic₇	1	1	yes	characteristic
53	C₄	—	6	2	no	—
54	C₄	Dic₃×D₇	1	1	yes	characteristic
55	C₄	D₂₁⋊C₄	1	1	yes	characteristic
56	C₃	#6377	1	1	yes	characteristic, complemented, Sylow
57	C₂	#66558	1	2	yes	—
58	C₂	Dic₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃×D₇
commutator	C₂×C₃×C₇	C₄²
Fitting, cosocle	C₂×C₃×C₄×C₇	C₂²
socle, prime radical	C₂²×C₃×C₇	C₂×C₄
hypocenter	C₃×C₇	#97
LCS	G ▷ C₂×C₃×C₇ ▷ 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. C₃×C₇⋊C₈	C₄	semidirect
20. C₇⋊C₈	Dic₃	semidirect
35. C₃×C₇	#97	semidirect
49. C₇	#815	semidirect
56. C₃	#6377	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊(C₇⋊C₈)	2
C₄×C₇×Dic₃	2
C₂×C₃×C₇⋊C₈	2
#6377	3
#815	7