Subgroups — #10621430

Order 672.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈₄	C₂	1	1	yes	maximal, characteristic
3	#66631	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#6510	—	3	1	no	maximal, Hall
6	C₂²×D₂₁	C₂²	1	2	yes	—
7	C₃×C₇×Q₈	—	2	2	no	—
8	C₂×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁	C₂²	1	2	yes	—
11	D₈₄	—	2	2	no	—
12	C₂×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
13	C₂×D₂₈	—	3	1	no	—
14	(C₂×D₇)⋊C₄	—	3	4	no	—
15	C₄×Dic₇	—	3	1	no	—
16	#947	—	7	1	no	maximal, Hall
17	C₂²×C₃×C₇	C₂³	1	1	yes	characteristic
18	C₂×D₂₁	—	2	2	no	—
19	C₂×D₂₁	—	2	4	no	—
20	C₃×C₄×C₇	—	2	2	no	—
21	C₃×C₄×C₇	D₄	1	2	yes	—
22	Dic₂₁	—	2	2	no	—
23	C₂²×D₇	—	3	2	no	—
24	C₇×Q₈	—	2	2	no	—
25	C₂×C₄×C₇	C₂×D₃	1	2	yes	—
26	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₇	—	3	2	no	—
28	D₂₈	—	6	2	no	—
29	C₂×C₃×Q₈	D₇	1	1	yes	characteristic, complemented
30	C₂×D₁₂	—	7	1	no	—
31	(C₂×D₃)⋊C₄	—	7	4	no	—
32	C₄×Dic₃	—	7	1	no	—
33	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₃×C₇	#41	1	2	yes	—
35	D₂₁	—	4	2	no	—
36	#124	—	21	1	no	Sylow
37	C₂²×C₇	C₂²×D₃	1	1	yes	characteristic
38	C₂×D₇	—	6	4	no	—
39	C₂×D₇	—	6	2	no	—
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	#67	1	2	yes	—
42	Dic₇	—	6	2	no	—
43	C₂²×D₃	—	7	2	no	—
44	C₃×Q₈	—	2	2	no	—
45	C₂×C₃×C₄	C₂×D₇	1	2	yes	—
46	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
47	C₂×Dic₃	—	7	2	no	—
48	D₁₂	—	14	2	no	—
49	C₃×C₇	#124	1	1	yes	characteristic, complemented, Hall
50	C₂×Q₈	D₂₁	1	1	yes	characteristic, complemented
51	C₂×D₄	—	21	1	no	—
52	C₂²⋊C₄	—	21	4	no	—
53	C₄²	—	21	1	no	—
54	C₂×C₇	#239	1	2	yes	—
55	C₂×C₇	C₂×#67	1	1	yes	characteristic
56	D₇	—	12	2	no	—
57	C₂²×C₃	C₂²×D₇	1	1	yes	characteristic
58	C₂×D₃	—	14	4	no	—
59	C₂×D₃	—	14	2	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	C₂²⋊D₇	1	2	yes	—
62	Dic₃	—	14	2	no	—
63	C₂³	—	21	2	no	—
64	Q₈	—	2	2	no	—
65	D₄	—	42	2	no	—
66	C₂×C₄	—	21	2	no	—
67	C₂×C₄	C₂×D₂₁	1	2	yes	—
68	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
69	C₇	#947	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	#1161	1	2	yes	—
71	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
72	D₃	—	28	2	no	—
73	C₂²	—	42	4	no	—
74	C₂²	—	42	2	no	—
75	C₂²	C₂²×D₂₁	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	42	2	no	—
78	C₄	#4268	1	2	yes	—
79	C₃	#6510	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	84	2	no	—
81	C₂	#66732	1	2	yes	—
82	C₂	C₂×#4268	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂³
Fitting	C₂×C₃×C₇×Q₈	C₂
prime radical	C₂×D₈₄	C₂
hypercenter	C₂×Q₈	D₂₁
hypocenter	C₃×C₇	#124
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₇×Q₈	C₂	semidirect
3. #66631	C₂	semidirect
4. C₄×Dic₂₁	C₂	semidirect
12. C₂×C₇×Q₈	D₃	semidirect
29. C₂×C₃×Q₈	D₇	semidirect
49. C₃×C₇	#124	semidirect
50. C₂×Q₈	D₂₁	semidirect
69. C₇	#947	semidirect
79. C₃	#6510	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×Q₈	2
C₂×D₈₄	2
#66631	2
C₄×Dic₂₁	2
#6510	3
#947	7