Subgroups — C₂×#66634 #10621424

Order 672.

136 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	#66634	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1143	—	3	1	no	maximal, Hall
6	C₂²×D₂₁	—	2	1	no	—
7	C₃×C₇×Q₈	—	2	1	no	—
8	C₃×C₇×Q₈	C₂²	1	2	yes	complemented
9	C₂×C₃×C₄×C₇	—	2	1	no	—
10	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	D₈₄	—	2	1	no	—
12	D₈₄	C₂²	1	2	yes	—
13	C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
14	C₂×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
15	C₂×D₂₈	—	3	1	no	—
16	#1143	—	3	4	no	—
17	C₂×C₇⋊C₈	—	3	1	no	—
18	C₂×#215	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
20	C₂×D₂₁	—	2	2	no	—
21	C₂×D₂₁	—	4	2	no	—
22	C₃×C₄×C₇	—	2	2	no	—
23	C₃×C₄×C₇	D₄	1	1	yes	characteristic
24	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂²×D₇	—	6	1	no	—
26	C₇×Q₈	—	2	1	no	—
27	C₇×Q₈	C₂×D₃	1	2	yes	complemented
28	C₂×C₄×C₇	—	2	1	no	—
29	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
30	D₂₈	—	3	2	no	—
31	D₂₈	—	6	1	no	—
32	C₇⋊C₈	—	3	2	no	—
33	C₂×C₃×Q₈	D₇	1	1	yes	characteristic, complemented
34	C₂×D₁₂	—	7	1	no	—
35	#215	—	7	4	no	—
36	C₂×C₃⋊C₈	—	7	1	no	—
37	C₂×C₃×C₇	SD₁₆	1	2	yes	complemented
38	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
39	D₂₁	—	4	2	no	—
40	C₂×SD₁₆	—	21	1	no	Sylow
41	C₂²×C₇	#67	1	1	yes	characteristic
42	C₂×D₇	—	6	2	no	—
43	C₂×D₇	—	12	2	no	—
44	C₄×C₇	—	2	2	no	—
45	C₄×C₇	#67	1	1	yes	characteristic
46	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
47	C₂²×D₃	—	14	1	no	—
48	C₃×Q₈	—	2	1	no	—
49	C₃×Q₈	C₂×D₇	1	2	yes	complemented
50	C₂×C₃×C₄	—	2	1	no	—
51	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
52	D₁₂	—	7	2	no	—
53	D₁₂	—	14	1	no	—
54	C₃⋊C₈	—	7	2	no	—
55	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
56	C₂×Q₈	D₂₁	1	1	yes	characteristic, complemented
57	C₂×D₄	—	21	1	no	—
58	SD₁₆	—	21	4	no	—
59	C₂×C₈	—	21	1	no	—
60	C₂×C₇	#215	1	2	yes	complemented
61	C₂×C₇	C₂×#67	1	1	yes	characteristic
62	D₇	—	12	2	no	—
63	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
64	C₂×D₃	—	14	2	no	—
65	C₂×D₃	—	28	2	no	—
66	C₃×C₄	—	2	2	no	—
67	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
68	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
69	C₂³	—	42	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	C₂×D₂₁	1	2	yes	complemented
72	D₄	—	21	2	no	—
73	D₄	—	42	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
76	C₈	—	21	2	no	—
77	C₇	C₂×#215	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	#1143	1	2	yes	complemented
79	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
80	D₃	—	28	2	no	—
81	C₂²	—	42	2	no	—
82	C₂²	—	84	2	no	—
83	C₂²	#4268	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	#4268	1	1	yes	characteristic
86	C₄	C₂²×D₂₁	1	1	yes	characteristic
87	C₃	C₂×#1143	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	84	2	no	—
89	C₂	#66634	1	2	yes	direct product factor
90	C₂	C₂×#4268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4268
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₂×C₃×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	D₄
prime radical	C₂×D₈₄	C₂
hypercenter	C₂×Q₈	D₂₁
hypocenter	C₃×C₇	C₂×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ 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. #66634	C₂	direct (N × H)
4. C₂×C₃⋊(C₇⋊C₈)	C₂	semidirect
8. C₃×C₇×Q₈	C₂²	semidirect
13. C₃⋊(C₇⋊C₈)	C₂²	semidirect
14. C₂×C₇×Q₈	D₃	semidirect
27. C₇×Q₈	C₂×D₃	semidirect
33. C₂×C₃×Q₈	D₇	semidirect
37. C₂×C₃×C₇	SD₁₆	semidirect
49. C₃×Q₈	C₂×D₇	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
56. C₂×Q₈	D₂₁	semidirect
60. C₂×C₇	#215	semidirect
71. Q₈	C₂×D₂₁	semidirect
77. C₇	C₂×#215	semidirect
78. C₂×C₃	#1143	semidirect
87. C₃	C₂×#1143	semidirect
89. C₂	#66634	direct (N × H)

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
#66634	2
C₂×C₃⋊(C₇⋊C₈)	2
C₂×#1143	3
C₂×#215	7

Subgroups — C2×#66634 #10621424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66634 #10621424