Subgroups — #10621431

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	#66627	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₇⋊C₈)	C₂	1	3	yes	maximal, complemented
4	#6511	—	3	1	no	maximal, Hall
5	C₃×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₇×D₄	C₄	1	3	yes	—
7	C₂×C₃×C₄×C₇	C₂²	1	3	yes	—
8	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₃⋊(C₇⋊C₈)	C₂²	1	3	yes	—
10	C₇×#41	D₃	1	1	yes	characteristic
11	#1136	—	3	3	no	—
12	C₂×C₇⋊C₈	—	3	3	no	—
13	#948	—	7	1	no	maximal, Hall
14	C₂²×C₃×C₇	C₂×C₄	1	3	yes	—
15	C₃×C₄×C₇	C₂×C₄	1	3	yes	—
16	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₇×Q₈	Dic₃	1	1	yes	characteristic
18	C₇×D₄	Dic₃	1	3	yes	—
19	C₂×C₄×C₇	C₂×D₃	1	3	yes	—
20	C₇⋊C₈	—	3	1	no	—
21	C₇⋊C₈	—	3	3	no	—
22	C₃×#41	D₇	1	1	yes	characteristic
23	#208	—	7	3	no	—
24	C₂×C₃⋊C₈	—	7	3	no	—
25	C₂×C₃×C₇	—	2	3	no	—
26	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
27	#131	—	21	1	no	Sylow
28	C₂²×C₇	C₂×Dic₃	1	3	yes	—
29	C₄×C₇	C₂×Dic₃	1	3	yes	—
30	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
31	C₃×Q₈	Dic₇	1	1	yes	characteristic
32	C₃×D₄	Dic₇	1	3	yes	—
33	C₂×C₃×C₄	C₂×D₇	1	3	yes	—
34	C₃⋊C₈	—	7	1	no	—
35	C₃⋊C₈	—	7	3	no	—
36	C₃×C₇	#131	1	1	yes	characteristic, complemented, Hall
37	#41	D₂₁	1	1	yes	characteristic
38	M₁₆	—	21	3	no	—
39	C₂×C₈	—	21	3	no	—
40	C₂×C₇	—	2	3	no	—
41	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
42	C₂²×C₃	C₂×Dic₇	1	3	yes	—
43	C₃×C₄	C₂×Dic₇	1	3	yes	—
44	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
45	Q₈	Dic₂₁	1	1	yes	characteristic
46	D₄	Dic₂₁	1	3	yes	—
47	C₂×C₄	C₂×D₂₁	1	3	yes	—
48	C₈	—	21	1	no	—
49	C₈	—	21	3	no	—
50	C₇	#948	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	2	3	no	—
52	C₂×C₃	C₂²×Dic₇	1	1	yes	characteristic
53	C₂²	C₂×Dic₂₁	1	3	yes	—
54	C₄	C₂×Dic₂₁	1	3	yes	—
55	C₄	C₂²×D₂₁	1	1	yes	characteristic
56	C₃	#6511	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	2	3	no	—
58	C₂	C₂²×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₂₁
commutator, socle	C₂×C₃×C₇	C₂²×C₄
Fitting, prime radical	C₃×C₇×#41	C₂
cosocle	C₃×C₄×C₇	C₂³
hypercenter	#41	D₂₁
hypocenter	C₃×C₇	#131
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #66627	C₂	semidirect
3. C₂×C₃⋊(C₇⋊C₈)	C₂	semidirect
36. C₃×C₇	#131	semidirect
50. C₇	#948	semidirect
56. C₃	#6511	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#41	2
#66627	2
C₂×C₃⋊(C₇⋊C₈)	2
#6511	3
#948	7