Subgroups — #10620521

Order 672.

72 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	C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	#6382	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₇×Dic₃	—	2	1	no	—
7	C₂×C₃×Dic₇	—	2	1	no	—
8	C₃⋊(C₇⋊C₈)	C₄	1	2	yes	complemented
9	C₇×C₄⋊C₄	—	3	1	no	—
10	C₄⋊Dic₇	D₃	1	1	yes	characteristic
11	C₂×C₇⋊C₈	—	3	1	no	—
12	#808	—	7	1	no	maximal, Hall
13	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
14	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄×C₇	Q₈	1	1	yes	characteristic
16	C₃×Dic₇	—	4	1	no	—
17	C₇×Dic₃	—	4	1	no	—
18	C₂×C₄×C₇	—	6	1	no	—
19	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂×Dic₇	—	2	1	no	—
21	C₇⋊C₈	—	3	2	no	—
22	C₃×C₄⋊C₄	—	7	1	no	—
23	C₄⋊Dic₃	D₇	1	1	yes	characteristic
24	C₂×C₃⋊C₈	—	7	1	no	—
25	C₂×C₃×C₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₃×C₇	SD₁₆	1	1	yes	characteristic
27	C₂×C₃×C₇	SD₁₆	1	1	yes	characteristic
28	#106	—	21	1	no	Sylow
29	C₂²×C₇	#67	1	1	yes	characteristic
30	C₄×C₇	—	12	1	no	—
31	C₄×C₇	Dic₆	1	1	yes	characteristic
32	C₄×C₇	C₄×D₃	1	1	yes	characteristic
33	Dic₇	—	4	1	no	—
34	C₂×C₃×C₄	—	14	1	no	—
35	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
36	C₂×Dic₃	—	2	1	no	—
37	C₃⋊C₈	—	7	2	no	—
38	C₃×C₇	#106	1	1	yes	characteristic, complemented, Hall
39	C₂×C₈	—	21	1	no	—
40	C₄⋊C₄	—	3	1	no	—
41	C₄⋊C₄	—	7	1	no	—
42	C₂×C₇	Dic₃⋊C₄	1	1	yes	characteristic
43	C₂×C₇	#214	1	1	yes	characteristic
44	C₂×C₇	#215	1	1	yes	characteristic
45	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
46	C₃×C₄	—	28	1	no	—
47	C₃×C₄	Dic₁₄	1	1	yes	characteristic
48	C₃×C₄	C₄×D₇	1	1	yes	characteristic
49	Dic₃	—	4	1	no	—
50	C₂×C₄	—	6	1	no	—
51	C₂×C₄	—	14	1	no	—
52	C₂×C₄	D₃×D₇	1	1	yes	characteristic
53	C₈	—	21	2	no	—
54	C₇	#808	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	Dic₇⋊C₄	1	1	yes	characteristic
56	C₂×C₃	#1142	1	1	yes	characteristic
57	C₂×C₃	#1143	1	1	yes	characteristic
58	C₂²	#4245	1	1	yes	characteristic
59	C₄	—	12	1	no	—
60	C₄	—	28	1	no	—
61	C₄	D₂₁⋊C₄	1	1	yes	characteristic
62	C₄	#4248	1	1	yes	characteristic
63	C₃	#6382	1	1	yes	characteristic, complemented, Sylow
64	C₂	#66563	1	1	yes	characteristic
65	C₂	#66564	1	1	yes	characteristic
66	C₂	#66577	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4245
commutator	C₃×C₄×C₇	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₇
Fitting, cosocle	C₂×C₃×C₄×C₇	C₂²
socle, prime radical	C₂²×C₃×C₇	D₄
hypocenter	C₃×C₇	#106
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. C₃⋊(C₇⋊C₈)	C₄	semidirect
38. C₃×C₇	#106	semidirect
54. C₇	#808	semidirect
63. C₃	#6382	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
C₇×C₄⋊Dic₃	2
C₃×C₄⋊Dic₇	2
#6382	3
#808	7