Subgroups — C₂×#66635 #10621426

Order 672.

120 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
2	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic
3	#66635	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×#1144	—	3	1	no	maximal, Hall
6	C₃×C₇×Q₈	—	2	1	no	—
7	C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₇	—	2	1	no	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₁	—	2	1	no	—
11	Dic₄₂	—	2	1	no	—
12	Dic₄₂	C₂²	1	2	yes	—
13	C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₇×Q₈	D₃	1	1	yes	characteristic
15	C₂×Dic₁₄	—	3	1	no	—
16	#1144	—	3	4	no	—
17	C₂×C₇⋊C₈	—	3	1	no	—
18	C₂×#216	—	7	1	no	maximal, Hall
19	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
20	C₃×C₄×C₇	—	2	2	no	—
21	C₃×C₄×C₇	D₄	1	1	yes	characteristic
22	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
23	Dic₂₁	—	2	2	no	—
24	C₇×Q₈	—	2	1	no	—
25	C₇×Q₈	C₂×D₃	1	2	yes	—
26	C₂×C₄×C₇	—	2	1	no	—
27	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
28	C₂×Dic₇	—	6	1	no	—
29	Dic₁₄	—	3	2	no	—
30	Dic₁₄	—	6	1	no	—
31	C₇⋊C₈	—	3	2	no	—
32	C₂×C₃×Q₈	D₇	1	1	yes	characteristic
33	C₂×Dic₆	—	7	1	no	—
34	#216	—	7	4	no	—
35	C₂×C₃⋊C₈	—	7	1	no	—
36	C₂×C₃×C₇	Q₁₆	1	2	yes	complemented
37	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Q₁₆	—	21	1	no	Sylow
39	C₂²×C₇	#67	1	1	yes	characteristic
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	#67	1	1	yes	characteristic
42	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
43	Dic₇	—	6	2	no	—
44	C₃×Q₈	—	2	1	no	—
45	C₃×Q₈	C₂×D₇	1	2	yes	—
46	C₂×C₃×C₄	—	2	1	no	—
47	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
48	C₂×Dic₃	—	14	1	no	—
49	Dic₆	—	7	2	no	—
50	Dic₆	—	14	1	no	—
51	C₃⋊C₈	—	7	2	no	—
52	C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
53	C₂×Q₈	—	21	1	no	—
54	C₂×Q₈	D₂₁	1	1	yes	characteristic
55	Q₁₆	—	21	4	no	—
56	C₂×C₈	—	21	1	no	—
57	C₂×C₇	#216	1	2	yes	complemented
58	C₂×C₇	C₂×#67	1	1	yes	characteristic
59	C₂²×C₃	C₂²⋊D₇	1	1	yes	characteristic
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	C₂²⋊D₇	1	1	yes	characteristic
62	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
63	Dic₃	—	14	2	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	21	2	no	—
66	Q₈	—	42	1	no	—
67	Q₈	C₂×D₂₁	1	2	yes	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	42	1	no	—
70	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
71	C₈	—	21	2	no	—
72	C₇	C₂×#216	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	#1144	1	2	yes	complemented
74	C₂×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
75	C₂²	#4268	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	42	2	no	—
78	C₄	#4268	1	1	yes	characteristic
79	C₄	C₂²×D₂₁	1	1	yes	characteristic
80	C₃	C₂×#1144	1	1	yes	characteristic, complemented, Sylow
81	C₂	#66635	1	2	yes	direct product factor
82	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, prime radical	C₂²×C₃×C₇	D₄
hypercenter	C₂×Q₈	D₂₁
hypocenter	C₃×C₇	C₂×Q₁₆
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
3. #66635	C₂	direct (N × H)
36. C₂×C₃×C₇	Q₁₆	semidirect
52. C₃×C₇	C₂×Q₁₆	semidirect
57. C₂×C₇	#216	semidirect
72. C₇	C₂×#216	semidirect
73. C₂×C₃	#1144	semidirect
80. C₃	C₂×#1144	semidirect
81. C₂	#66635	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×Dic₄₂	2
#66635	2
C₂×C₃⋊(C₇⋊C₈)	2
C₂×#1144	3
C₂×#216	7

Subgroups — C2×#66635 #10621426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66635 #10621426