Group #21462713 — subgroups

Subgroups — #21462713

Order 7808.

396 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840811	C₂	1	6	yes	maximal, complemented
2	Q₈×#92420	C₂	1	2	yes	maximal, complemented
3	#16840781	C₂	1	3	yes	maximal
4	#16840780	C₂	1	3	yes	maximal
5	C₄×Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519844	C₂²	1	6	yes	—
7	C₄⋊#92420	C₂²	1	6	yes	—
8	#14519834	C₂²	1	1	yes	characteristic
9	#14519834	C₂²	1	3	yes	—
10	C₄×#92420	C₂²	1	6	yes	—
11	C₄×#92420	C₂²	1	6	yes	—
12	C₂×Q₈×D₆₁	C₄	1	1	yes	characteristic
13	Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₂₂	C₄	1	3	yes	—
15	#14519742	C₂²	1	3	yes	—
16	C₄⋊C₄×D₆₁	C₄	1	3	yes	—
17	C₄²×D₆₁	C₂²	1	3	yes	—
18	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	Q₈×D₆₁	C₈	1	8	yes	complemented
21	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₆₁	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₂₂	C₂×C₄	1	3	yes	—
25	C₂×#92420	C₂³	1	6	yes	—
26	C₂×#92420	C₂³	1	2	yes	—
27	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
28	C₄²×C₆₁	C₂×C₄	1	3	yes	—
29	C₄⋊Dic₆₁	C₂×C₄	1	3	yes	—
30	#11758196	C₂×C₄	1	6	yes	—
31	C₄×Dic₆₁	C₂×C₄	1	3	yes	—
32	C₄×Dic₆₁	C₂³	1	3	yes	—
33	C₄×Dic₆₁	C₂³	1	3	yes	—
34	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₆₁	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
39	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
40	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
41	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
42	C₄×D₆₁	—	2	2	no	—
43	C₄×D₆₁	—	4	3	no	—
44	C₄×D₆₁	C₂×C₈	1	12	yes	—
45	Dic₁₂₂	C₂×C₈	1	12	yes	—
46	#92420	—	2	6	no	—
47	#92420	—	2	2	no	—
48	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₆₁	—	2	1	no	—
50	C₂×D₆₁	—	2	1	no	—
51	C₂×D₆₁	C₂²×C₈	1	2	yes	—
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	2	3	no	—
54	C₄×C₆₁	C₂²×C₈	1	6	yes	—
55	Dic₆₁	—	2	3	no	—
56	Dic₆₁	C₂²×C₈	1	6	yes	—
57	Dic₆₁	2₋¹⁺⁴	1	2	yes	—
58	#2982	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	#519	1	1	yes	characteristic
60	C₂×C₆₁	C₂³×C₈	1	1	yes	characteristic
61	C₂×C₆₁	#568	1	1	yes	characteristic
62	D₆₁	—	2	2	no	—
63	C₂×C₄×Q₈	—	61	1	no	—
64	C₈×Q₈	—	61	2	no	—
65	C₈×Q₈	—	61	6	no	—
66	#431	—	61	3	no	—
67	#431	—	61	3	no	—
68	C₆₁	#2982	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	61	1	no	—
70	C₄×Q₈	—	61	1	no	—
71	C₄×Q₈	—	61	3	no	—
72	C₄×Q₈	—	61	3	no	—
73	C₄×Q₈	#6923	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	61	3	no	—
75	C₂×C₄²	—	61	3	no	—
76	C₄⋊C₈	—	61	6	no	—
77	C₄⋊C₈	—	61	6	no	—
78	C₂²⋊C₈	—	61	1	no	—
79	C₂²⋊C₈	—	61	3	no	—
80	C₄×C₈	—	61	6	no	—
81	C₄×C₈	—	61	6	no	—
82	C₂×Q₈	—	61	8	no	—
83	C₂×Q₈	—	61	3	no	—
84	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
85	C₂²×C₄	—	61	3	no	—
86	C₂²×C₄	—	61	1	no	—
87	C₂²×C₄	—	61	3	no	—
88	C₂×C₈	—	61	6	no	—
89	C₂×C₈	—	61	2	no	—
90	C₄⋊C₄	—	61	6	no	—
91	C₄⋊C₄	—	61	3	no	—
92	C₄⋊C₄	C₂×#6923	1	3	yes	—
93	C₄²	—	61	3	no	—
94	C₄²	—	61	3	no	—
95	C₄²	—	61	3	no	—
96	C₄²	C₂×#6923	1	3	yes	—
97	C₂³	—	61	1	no	—
98	Q₈	—	61	12	no	—
99	Q₈	D₆₁⋊C₈	1	4	yes	—
100	C₂×C₄	—	61	3	no	—
101	C₂×C₄	—	61	1	no	—
102	C₂×C₄	—	61	12	no	—
103	C₂×C₄	—	61	3	no	—
104	C₂×C₄	—	122	2	no	—
105	C₂×C₄	—	244	3	no	—
106	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
107	C₂×C₄	C₂²×#6923	1	3	yes	—
108	C₂×C₄	C₂²×#6923	1	3	yes	—
109	C₈	—	122	6	no	—
110	C₈	—	122	2	no	—
111	C₂²	—	61	2	no	—
112	C₂²	—	122	1	no	—
113	C₂²	—	122	1	no	—
114	C₂²	C₂³×#6923	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	61	2	no	—
118	C₄	—	61	6	no	—
119	C₄	—	122	3	no	—
120	C₄	C₂×D₆₁⋊C₈	1	6	yes	—
121	C₂	—	122	2	no	—
122	C₂	#16840830	1	1	yes	characteristic
123	C₂	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
124	C₂	#16840841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840811

C₂

yes

maximal, complemented

Q₈×#92420

C₂

yes

maximal, complemented

#16840781

C₂

yes

maximal

#16840780

C₂

yes

maximal

C₄×Q₈×D₆₁

C₂

yes

maximal, characteristic

#14519844

C₂²

yes

—

C₄⋊#92420

C₂²

yes

—

#14519834

C₂²

yes

characteristic

#14519834

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₄×#92420

C₂²

yes

—

C₂×Q₈×D₆₁

C₄

yes

characteristic

Q₈×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₁₂₂

C₄

yes

—

#14519742

C₂²

yes

—

C₄⋊C₄×D₆₁

C₄

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

Q₈×D₆₁

C₈

yes

complemented

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂²×D₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×C₈

yes

—

Dic₁₂₂

C₂×C₈

yes

—

#92420

—

#92420

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂²×C₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²×C₈

yes

—

Dic₆₁

—

Dic₆₁

C₂²×C₈

yes

—

Dic₆₁

2₋¹⁺⁴

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂³×C₈

yes

characteristic

C₂×C₆₁

#568

yes

characteristic

D₆₁

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

#431

—

#431

—

C₆₁

#2982

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#6923

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6923

yes

—

C₂³

—

Q₈

—

Q₈

D₆₁⋊C₈

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

244

—

106

C₂×C₄

C₂²×#6923

yes

characteristic

107

C₂×C₄

C₂²×#6923

yes

—

108

C₂×C₄

C₂²×#6923

yes

—

109

C₈

—

122

—

110

C₈

—

122

—

111

C₂²

—

112

C₂²

—

122

—

113

C₂²

—

122

—

114

C₂²

C₂³×#6923

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

122

—

120

C₄

C₂×D₆₁⋊C₈

yes

—

121

C₂

—

122

—

122

C₂

#16840830

yes

characteristic

123

C₂

C₂²×D₆₁⋊C₈

yes

characteristic

124

C₂

#16840841

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6923
commutator	C₂×C₆₁	C₂³×C₈
Fitting	C₄×Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂³×C₄
hypercenter	C₄×Q₈	#6923
hypocenter	C₆₁	#2982
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6923

commutator

C₂×C₆₁

C₂³×C₈

Fitting

C₄×Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂³×C₄

C₄×Q₈

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #16840811	C₂	semidirect
2. Q₈×#92420	C₂	semidirect
20. Q₈×D₆₁	C₈	semidirect
68. C₆₁	#2982	semidirect

H ≅ G/N

type

1. #16840811

C₂

semidirect

2. Q₈×#92420

C₂

semidirect

20. Q₈×D₆₁

C₈

semidirect

68. C₆₁

#2982

semidirect

Subgroups — #21462713

Conjugacy classes of subgroups

Complements