Subgroups — #10621395

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66730	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#66633	C₂	1	2	yes	maximal, complemented
4	Dic₈₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#6475	—	3	1	no	maximal, Hall
7	C₃×C₇×D₄	C₂²	1	2	yes	—
8	#4268	—	2	2	no	—
9	C₂×Dic₂₁	—	2	2	no	—
10	C₄×D₂₁	C₂²	1	1	yes	characteristic
11	Dic₄₂	C₂²	1	2	yes	—
12	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#1159	—	3	2	no	—
15	C₇×D₈	D₃	1	1	yes	characteristic, complemented
16	#1142	—	3	2	no	—
17	Dic₂₈	—	3	1	no	—
18	C₈×D₇	—	3	1	no	—
19	#912	—	7	1	no	maximal, Hall
20	C₂²×C₃×C₇	—	2	2	no	—
21	C₂×D₂₁	D₄	1	1	yes	characteristic
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	Dic₂₁	—	2	2	no	—
24	Dic₂₁	D₄	1	1	yes	characteristic
25	C₇×D₄	C₂×D₃	1	2	yes	—
26	C₂²⋊D₇	—	6	2	no	—
27	C₂×Dic₇	—	6	2	no	—
28	C₄×D₇	—	3	1	no	—
29	Dic₁₄	—	3	2	no	—
30	C₇×C₈	C₂×D₃	1	1	yes	characteristic
31	C₇⋊C₈	—	3	1	no	—
32	#237	—	7	2	no	—
33	C₃×D₈	D₇	1	1	yes	characteristic, complemented
34	#214	—	7	2	no	—
35	Dic₁₂	—	7	1	no	—
36	D₃×C₈	—	7	1	no	—
37	C₂×C₃×C₇	—	4	2	no	—
38	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
39	D₂₁	—	2	1	no	—
40	#135	—	21	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂×D₇	—	3	1	no	—
43	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
44	Dic₇	—	3	1	no	—
45	Dic₇	—	6	2	no	—
46	C₃×D₄	C₂×D₇	1	2	yes	—
47	#67	—	14	2	no	—
48	C₂×Dic₃	—	14	2	no	—
49	C₄×D₃	—	7	1	no	—
50	Dic₆	—	7	2	no	—
51	C₃×C₈	C₂×D₇	1	1	yes	characteristic
52	C₃⋊C₈	—	7	1	no	—
53	C₃×C₇	#135	1	1	yes	characteristic, complemented, Hall
54	#41	—	21	2	no	—
55	Q₁₆	—	21	1	no	—
56	SD₁₆	—	21	2	no	—
57	D₈	D₂₁	1	1	yes	characteristic, complemented
58	C₂×C₈	—	21	1	no	—
59	C₂×C₇	—	4	2	no	—
60	C₂×C₇	D₃×D₄	1	1	yes	characteristic
61	D₇	—	6	1	no	—
62	C₂²×C₃	—	2	2	no	—
63	C₂×D₃	—	7	1	no	—
64	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
65	Dic₃	—	7	1	no	—
66	Dic₃	—	14	2	no	—
67	Q₈	—	21	2	no	—
68	D₄	—	42	2	no	—
69	D₄	C₂×D₂₁	1	2	yes	—
70	C₂×C₄	—	21	1	no	—
71	C₂×C₄	—	42	2	no	—
72	C₈	—	21	1	no	—
73	C₈	C₂×D₂₁	1	1	yes	characteristic
74	C₇	#912	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	4	2	no	—
76	C₂×C₃	D₄×D₇	1	1	yes	characteristic
77	D₃	—	14	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	21	1	no	—
80	C₄	—	21	1	no	—
81	C₄	—	42	2	no	—
82	C₄	C₂²×D₂₁	1	1	yes	characteristic
83	C₃	#6475	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	4	2	no	—
85	C₂	—	42	1	no	—
86	C₂	D₄×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₁
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₃×C₇×D₈	C₂
socle	C₂×C₃×C₇	C₂×D₄
hypercenter	D₈	D₂₁
hypocenter	C₃×C₇	#135
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #66730	C₂	semidirect
2. C₃×C₇×D₈	C₂	semidirect
3. #66633	C₂	semidirect
4. Dic₈₄	C₂	semidirect
5. C₈×D₂₁	C₂	semidirect
15. C₇×D₈	D₃	semidirect
33. C₃×D₈	D₇	semidirect
53. C₃×C₇	#135	semidirect
57. D₈	D₂₁	semidirect
74. C₇	#912	semidirect
83. C₃	#6475	semidirect

Maximal subgroups

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

subgroup	index
#66730	2
C₃×C₇×D₈	2
#66633	2
Dic₈₄	2
C₈×D₂₁	2
#6475	3
#912	7