Subgroups — #10620649

Order 672.

84 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	C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#6384	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₇	—	2	1	no	—
6	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₁	—	2	1	no	—
8	Dic₄₂	—	2	1	no	—
9	Dic₄₂	C₄	1	2	yes	complemented
10	C₃⋊(C₇⋊C₈)	—	2	1	no	—
11	C₂×Dic₁₄	—	3	1	no	—
12	C₇×C₄⋊C₄	D₃	1	1	yes	characteristic
13	C₂×C₇⋊C₈	—	3	1	no	—
14	#810	—	7	1	no	maximal, Hall
15	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
16	C₃×C₄×C₇	—	4	1	no	—
17	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₇	D₄	1	1	yes	characteristic
19	Dic₂₁	—	2	2	no	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
22	C₂×Dic₇	—	6	1	no	—
23	Dic₁₄	—	3	2	no	—
24	Dic₁₄	—	6	1	no	—
25	C₇⋊C₈	—	6	1	no	—
26	C₂×Dic₆	—	7	1	no	—
27	C₃×C₄⋊C₄	D₇	1	1	yes	characteristic
28	C₂×C₃⋊C₈	—	7	1	no	—
29	C₂×C₃×C₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₃×C₇	SD₁₆	1	1	yes	characteristic
31	C₂×C₃×C₇	Q₁₆	1	1	yes	characteristic
32	#103	—	21	1	no	Sylow
33	C₂²×C₇	#67	1	1	yes	characteristic
34	C₄×C₇	—	4	1	no	—
35	C₄×C₇	C₄×D₃	1	1	yes	characteristic
36	C₄×C₇	D₁₂	1	1	yes	characteristic
37	Dic₇	—	6	2	no	—
38	C₂×C₃×C₄	—	2	1	no	—
39	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
40	C₂×Dic₃	—	14	1	no	—
41	Dic₆	—	7	2	no	—
42	Dic₆	—	14	1	no	—
43	C₃⋊C₈	—	14	1	no	—
44	C₃×C₇	#103	1	1	yes	characteristic, complemented, Hall
45	C₂×Q₈	—	21	1	no	—
46	C₂×C₈	—	21	1	no	—
47	C₄⋊C₄	D₂₁	1	1	yes	characteristic
48	C₂×C₇	(C₂×D₃)⋊C₄	1	1	yes	characteristic
49	C₂×C₇	#214	1	1	yes	characteristic
50	C₂×C₇	#216	1	1	yes	characteristic
51	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
52	C₃×C₄	—	4	1	no	—
53	C₃×C₄	C₄×D₇	1	1	yes	characteristic
54	C₃×C₄	D₂₈	1	1	yes	characteristic
55	Dic₃	—	14	2	no	—
56	Q₈	—	21	2	no	—
57	Q₈	—	42	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	42	1	no	—
60	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
61	C₈	—	42	1	no	—
62	C₇	#810	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₃	(C₂×D₇)⋊C₄	1	1	yes	characteristic
64	C₂×C₃	#1142	1	1	yes	characteristic
65	C₂×C₃	#1144	1	1	yes	characteristic
66	C₂²	#4268	1	1	yes	characteristic
67	C₄	—	4	1	no	—
68	C₄	—	42	2	no	—
69	C₄	C₄×D₂₁	1	1	yes	characteristic
70	C₄	D₈₄	1	1	yes	characteristic
71	C₃	#6384	1	1	yes	characteristic, complemented, Sylow
72	C₂	#66631	1	1	yes	characteristic
73	C₂	#66633	1	1	yes	characteristic
74	C₂	#66635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4268
commutator	C₃×C₄×C₇	C₂×C₄
Frattini	C₂×C₄	C₂×D₂₁
Fitting	C₃×C₇×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃×C₇	D₄
cosocle	C₂×C₃×C₄×C₇	C₂²
hypercenter	C₄⋊C₄	D₂₁
hypocenter	C₃×C₇	#103
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Dic₄₂	C₄	semidirect
44. C₃×C₇	#103	semidirect
62. C₇	#810	semidirect
71. C₃	#6384	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
C₃×C₇×C₄⋊C₄	2
C₂×C₃⋊(C₇⋊C₈)	2
#6384	3
#810	7