Group #21454908 — subgroups

Subgroups — #21454908

Order 7808.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#16839893	C₂	1	6	yes	maximal, complemented
2	Q₈×#92418	C₂	1	2	yes	maximal, complemented
3	#16839819	C₂	1	3	yes	maximal
4	#16839815	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#92418	C₂²	1	6	yes	—
7	C₄⋊#92418	C₂²	1	6	yes	—
8	C₂²⋊#92418	C₂²	1	1	yes	characteristic
9	C₂²⋊#92418	C₂²	1	3	yes	—
10	#14519688	C₂²	1	6	yes	—
11	C₄×#92418	C₂²	1	6	yes	—
12	C₂²×Q₈×C₆₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₆₁	C₄	1	3	yes	—
14	C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₆₁	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₆₁	C₄	1	3	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	3	yes	—
19	C₂×Q₈×C₆₁	—	2	4	no	—
20	C₂×Q₈×C₆₁	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₁	C₂×C₄	1	6	yes	—
26	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
27	C₄⋊C₄×C₆₁	C₂×C₄	1	3	yes	—
28	C₄²×C₆₁	C₂×C₄	1	3	yes	—
29	C₄²×C₆₁	C₂×C₄	1	3	yes	—
30	C₄²×C₆₁	C₂³	1	3	yes	—
31	C₄²×C₆₁	C₂³	1	3	yes	—
32	C₂×#92418	C₂³	1	2	yes	—
33	C₂×#92418	C₂³	1	6	yes	—
34	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₆₁	—	2	6	no	—
36	Q₈×C₆₁	M₁₆	1	4	yes	—
37	C₂×C₄×C₆₁	—	2	6	no	—
38	C₂×C₄×C₆₁	—	2	6	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	#92418	—	2	2	no	—
47	#92418	—	2	6	no	—
48	C₂²×C₆₁	—	2	1	no	—
49	C₂²×C₆₁	—	2	1	no	—
50	C₂²×C₆₁	#131	1	2	yes	—
51	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	2	3	no	—
54	C₄×C₆₁	—	2	3	no	—
55	C₄×C₆₁	—	2	3	no	—
56	C₄×C₆₁	C₂×M₁₆	1	6	yes	—
57	C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
58	#2984	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	—	2	2	no	—
60	C₂×C₆₁	#519	1	1	yes	characteristic
61	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
62	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
63	C₂×C₄×Q₈	D₆₁	1	1	yes	characteristic
64	#446	—	61	6	no	—
65	C₈×Q₈	—	61	2	no	—
66	#432	—	61	3	no	—
67	#431	—	61	3	no	—
68	C₆₁	#2984	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	Dic₆₁	1	1	yes	characteristic
70	C₄×Q₈	Dic₆₁	1	3	yes	—
71	C₄×Q₈	Dic₆₁	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₆₁	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₆₁	1	3	yes	—
75	C₂×C₄²	C₂×D₆₁	1	3	yes	—
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	#97	—	61	6	no	—
81	C₄×C₈	—	61	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₆₁	1	3	yes	—
84	C₂×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
86	C₂²×C₄	C₂×Dic₆₁	1	3	yes	—
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	C₂×C₈	—	61	2	no	—
89	C₂×C₈	—	61	6	no	—
90	C₄⋊C₄	C₂×Dic₆₁	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₆₁	1	3	yes	—
93	C₄²	C₂×Dic₆₁	1	3	yes	—
94	C₄²	C₂×Dic₆₁	1	3	yes	—
95	C₄²	C₂²×D₆₁	1	3	yes	—
96	C₄²	C₂²×D₆₁	1	3	yes	—
97	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#11758194	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
105	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
106	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
107	C₂×C₄	C₂²×Dic₆₁	1	3	yes	—
108	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
109	C₈	—	122	2	no	—
110	C₈	—	122	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#14519784	1	2	yes	—
114	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#11758194	1	6	yes	—
120	C₄	#14519825	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#16840240	1	1	yes	characteristic
123	C₂	C₂²×#11758194	1	1	yes	characteristic
124	C₂	C₂×#14519784	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839893

C₂

yes

maximal, complemented

Q₈×#92418

C₂

yes

maximal, complemented

#16839819

C₂

yes

maximal

#16839815

C₂

yes

maximal

C₂×C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₄⋊#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

characteristic

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₂²×Q₈×C₆₁

C₄

yes

characteristic

C₄×Q₈×C₆₁

C₄

yes

—

C₄×Q₈×C₆₁

C₄

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#131

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

2₋¹⁺⁴

yes

—

#2984

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#519

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

C₂×C₄×Q₈

D₆₁

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₆₁

#2984

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₆₁

yes

characteristic

C₄×Q₈

Dic₆₁

yes

—

C₄×Q₈

Dic₆₁

yes

characteristic

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂×Q₈

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₂³

C₂²×Dic₆₁

yes

characteristic

Q₈

—

Q₈

#11758194

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

104

C₂×C₄

C₂²×Dic₆₁

yes

—

105

C₂×C₄

C₂²×Dic₆₁

yes

—

106

C₂×C₄

C₂²×Dic₆₁

yes

—

107

C₂×C₄

C₂²×Dic₆₁

yes

—

108

C₂×C₄

C₂³×D₆₁

yes

characteristic

109

C₈

—

122

—

110

C₈

—

122

—

111

C₂²

—

112

C₂²

—

113

C₂²

#14519784

yes

—

114

C₂²

C₂³×Dic₆₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#11758194

yes

—

120

C₄

#14519825

yes

—

121

C₂

—

122

C₂

#16840240

yes

characteristic

123

C₂

C₂²×#11758194

yes

characteristic

124

C₂

C₂×#14519784

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₆₁	C₂
hypercenter	C₂×C₄×Q₈	D₆₁
hypocenter	C₆₁	#2984
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₆₁

C₂

hypercenter

C₂×C₄×Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16839893	C₂	semidirect
2. Q₈×#92418	C₂	semidirect
68. C₆₁	#2984	semidirect

H ≅ G/N

type

1. #16839893

C₂

semidirect

2. Q₈×#92418

C₂

semidirect

68. C₆₁

#2984

semidirect

Subgroups — #21454908

Conjugacy classes of subgroups

Complements