Subgroups — C₂×#66623 #10621385

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#66623	C₂	1	4	yes	maximal, direct product factor
5	C₂×#1132	—	3	1	no	maximal, Hall
6	C₂²×D₂₁	—	2	1	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₁	—	2	1	no	—
9	D₈₄	—	2	1	no	—
10	D₈₄	C₂²	1	2	yes	—
11	Dic₄₂	—	2	1	no	—
12	Dic₄₂	C₂²	1	2	yes	complemented
13	C₃×C₇×C₈	C₂²	1	2	yes	complemented
14	C₂×D₂₈	—	3	1	no	—
15	C₂×Dic₁₄	—	3	1	no	—
16	C₂×C₇×C₈	D₃	1	1	yes	characteristic, complemented
17	#1132	—	3	4	no	—
18	C₂×#204	—	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	D₄	1	1	yes	characteristic
23	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	Dic₂₁	—	2	2	no	—
25	C₂²×D₇	—	6	1	no	—
26	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₇	—	6	1	no	—
28	D₂₈	—	3	2	no	—
29	D₂₈	—	6	1	no	—
30	Dic₁₄	—	3	2	no	—
31	Dic₁₄	—	6	1	no	—
32	C₇×C₈	C₂×D₃	1	2	yes	complemented
33	C₂×D₁₂	—	7	1	no	—
34	C₂×Dic₆	—	7	1	no	—
35	C₂×C₃×C₈	D₇	1	1	yes	characteristic, complemented
36	#204	—	7	4	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	D₁₂	1	1	yes	characteristic
42	C₂×D₇	—	6	2	no	—
43	C₂×D₇	—	12	2	no	—
44	C₄×C₇	D₁₂	1	1	yes	characteristic
45	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
46	Dic₇	—	6	2	no	—
47	C₂²×D₃	—	14	1	no	—
48	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
49	C₂×Dic₃	—	14	1	no	—
50	D₁₂	—	7	2	no	—
51	D₁₂	—	14	1	no	—
52	Dic₆	—	7	2	no	—
53	Dic₆	—	14	1	no	—
54	C₃×C₈	C₂×D₇	1	2	yes	complemented
55	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
56	C₂×Q₈	—	21	1	no	—
57	C₂×D₄	—	21	1	no	—
58	SD₁₆	—	21	4	no	—
59	C₂×C₈	D₂₁	1	1	yes	characteristic, complemented
60	C₂×C₇	#204	1	2	yes	complemented
61	C₂×C₇	C₂×D₁₂	1	1	yes	characteristic
62	D₇	—	12	2	no	—
63	C₂²×C₃	D₂₈	1	1	yes	characteristic
64	C₂×D₃	—	14	2	no	—
65	C₂×D₃	—	28	2	no	—
66	C₃×C₄	D₂₈	1	1	yes	characteristic
67	C₃×C₄	C₂²×D₇	1	1	yes	characteristic
68	Dic₃	—	14	2	no	—
69	C₂³	—	42	1	no	—
70	Q₈	—	21	2	no	—
71	Q₈	—	42	1	no	—
72	D₄	—	21	2	no	—
73	D₄	—	42	1	no	—
74	C₂×C₄	—	42	1	no	—
75	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
76	C₈	C₂×D₂₁	1	2	yes	complemented
77	C₇	C₂×#204	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃	#1132	1	2	yes	complemented
79	C₂×C₃	C₂×D₂₈	1	1	yes	characteristic
80	D₃	—	28	2	no	—
81	C₂²	—	42	2	no	—
82	C₂²	—	84	2	no	—
83	C₂²	D₈₄	1	1	yes	characteristic
84	C₄	—	42	2	no	—
85	C₄	D₈₄	1	1	yes	characteristic
86	C₄	C₂²×D₂₁	1	1	yes	characteristic
87	C₃	C₂×#1132	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	84	2	no	—
89	C₂	#66623	1	2	yes	direct product factor
90	C₂	C₂×D₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈₄
commutator, cosocle	C₃×C₄×C₇	C₂³
Frattini	C₄	C₂²×D₂₁
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	D₄
prime radical	C₂×D₈₄	C₂
hypercenter	C₂×C₈	D₂₁
hypocenter	C₃×C₇	C₂×SD₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂	C₂	semidirect
3. C₂×C₃×C₇×C₈	C₂	semidirect
4. #66623	C₂	direct (N × H)
12. Dic₄₂	C₂²	semidirect
13. C₃×C₇×C₈	C₂²	semidirect
16. C₂×C₇×C₈	D₃	semidirect
32. C₇×C₈	C₂×D₃	semidirect
35. C₂×C₃×C₈	D₇	semidirect
37. C₂×C₃×C₇	SD₁₆	semidirect
54. C₃×C₈	C₂×D₇	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
59. C₂×C₈	D₂₁	semidirect
60. C₂×C₇	#204	semidirect
76. C₈	C₂×D₂₁	semidirect
77. C₇	C₂×#204	semidirect
78. C₂×C₃	#1132	semidirect
87. C₃	C₂×#1132	semidirect
89. C₂	#66623	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×D₈₄	2
C₂×Dic₄₂	2
C₂×C₃×C₇×C₈	2
#66623	2
C₂×#1132	3
C₂×#204	7

Subgroups — C2×#66623 #10621385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66623 #10621385