Subgroups — #10620498

Order 672.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic
2	C₇×#208	C₂	1	1	yes	maximal, characteristic
3	C₃×#1136	C₂	1	1	yes	maximal, characteristic
4	#6398	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×Dic₂₁	C₄	1	2	yes	—
7	Dic₄₂	—	2	1	no	—
8	Dic₄₂	—	2	1	no	—
9	C₃×C₇⋊C₈	—	2	1	no	—
10	C₇×C₃⋊C₈	—	2	1	no	—
11	C₂×Dic₁₄	—	3	1	no	—
12	C₇×M₁₆	—	3	1	no	—
13	#1136	D₃	1	1	yes	characteristic
14	#824	—	7	1	no	maximal, Hall
15	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄×C₇	D₄	1	1	yes	characteristic
17	C₃×C₄×C₇	D₄	1	1	yes	characteristic
18	Dic₂₁	—	2	2	no	—
19	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×Dic₇	—	3	2	no	—
21	Dic₁₄	—	6	1	no	—
22	Dic₁₄	—	6	1	no	—
23	C₇×C₈	—	6	1	no	—
24	C₇⋊C₈	—	2	1	no	—
25	C₂×Dic₆	—	7	1	no	—
26	C₃×M₁₆	—	7	1	no	—
27	#208	D₇	1	1	yes	characteristic
28	C₂×C₃×C₇	—	2	1	no	—
29	C₂×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
30	#101	—	21	1	no	Sylow
31	C₂²×C₇	C₄×D₃	1	1	yes	characteristic
32	C₄×C₇	D₁₂	1	1	yes	characteristic
33	C₄×C₇	#67	1	1	yes	characteristic
34	Dic₇	—	6	2	no	—
35	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
36	C₂×Dic₃	—	7	2	no	—
37	Dic₆	—	14	1	no	—
38	Dic₆	—	14	1	no	—
39	C₃×C₈	—	14	1	no	—
40	C₃⋊C₈	—	2	1	no	—
41	C₃×C₇	#101	1	1	yes	characteristic, complemented, Hall
42	C₂×Q₈	—	21	1	no	—
43	M₁₆	—	3	1	no	—
44	M₁₆	—	7	1	no	—
45	C₂×C₇	—	2	1	no	—
46	C₂×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
47	C₂²×C₃	C₄×D₇	1	1	yes	characteristic
48	C₃×C₄	D₂₈	1	1	yes	characteristic
49	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
50	Dic₃	—	14	2	no	—
51	Q₈	—	42	1	no	—
52	Q₈	—	42	1	no	—
53	C₂×C₄	—	21	2	no	—
54	C₂×C₄	D₃×D₇	1	1	yes	characteristic
55	C₈	—	6	1	no	—
56	C₈	—	14	1	no	—
57	C₇	#824	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	(C₂×D₇)⋊C₄	1	1	yes	characteristic
60	C₂²	D₂₁⋊C₄	1	1	yes	characteristic
61	C₄	—	42	2	no	—
62	C₄	#4246	1	1	yes	characteristic
63	C₄	#4247	1	1	yes	characteristic
64	C₃	#6398	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	2	1	no	—
66	C₂	#66575	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#66575
commutator, prime radical	C₂²×C₃×C₇	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₇
Fitting, cosocle	C₂×C₃×C₄×C₇	C₂²
socle	C₂×C₃×C₇	C₂²⋊C₄
hypocenter	C₃×C₇	#101
LCS	G ▷ C₂²×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
41. C₃×C₇	#101	semidirect
57. C₇	#824	semidirect
64. C₃	#6398	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₄₂	2
C₇×#208	2
C₃×#1136	2
#6398	3
#824	7