Group #21454754 — subgroups

Subgroups — #21454754

Order 7808.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	4	yes	maximal, complemented
2	Q₈×#92418	C₂	1	4	yes	maximal, complemented
3	#16839789	C₂	1	2	yes	maximal
4	#16839786	C₂	1	1	yes	maximal, characteristic
5	#16839718	C₂	1	2	yes	maximal
6	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#533	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758194	C₂²	1	2	yes	—
9	C₂²×#92418	C₂²	1	1	yes	characteristic
10	C₂²×#92418	C₂²	1	1	yes	characteristic
11	C₄⋊#92418	C₂²	1	4	yes	—
12	C₄⋊#92418	C₂²	1	8	yes	—
13	#14519688	C₂²	1	4	yes	—
14	C₄×#92418	C₂²	1	2	yes	—
15	C₄×#92418	C₂²	1	2	yes	—
16	C₄×#92418	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₆₁	C₄	1	2	yes	—
18	#125×C₆₁	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₆₁	C₄	1	4	yes	—
20	C₄×Q₈×C₆₁	C₂²	1	4	yes	—
21	#117×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
27	C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
29	C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
30	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂³	1	2	yes	—
33	C₄²×C₆₁	C₂³	1	4	yes	—
34	#11758194	—	2	4	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	Q₈	1	4	yes	complemented
37	C₂×#92418	C₂³	1	2	yes	—
38	C₂×#92418	C₂³	1	4	yes	—
39	C₂×#92418	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂³	1	1	yes	characteristic
41	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₆₁	—	2	8	no	—
43	C₂×C₄×C₆₁	—	2	4	no	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₆₁	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₆₁	#41	1	2	yes	—
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	#92418	—	2	2	no	—
52	#92418	—	2	4	no	—
53	#92418	C₂×Q₈	1	2	yes	—
54	#92418	C₂×Q₈	1	2	yes	—
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
57	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	—	2	8	no	—
60	C₄×C₆₁	C₄×Q₈	1	2	yes	—
61	C₄×C₆₁	#131	1	4	yes	—
62	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
64	#2960	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	—	2	2	no	—
66	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₂×C₆₁	C₂×#131	1	2	yes	—
68	#533	D₆₁	1	1	yes	characteristic
69	#446	—	61	4	no	—
70	C₈×Q₈	—	61	4	no	—
71	#424	—	61	2	no	—
72	#423	—	61	1	no	—
73	#405	—	61	2	no	—
74	C₂×C₄×C₈	—	61	1	no	—
75	C₆₁	#2960	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	61	2	no	—
77	C₂²×C₈	—	61	1	no	—
78	C₂²×C₈	—	61	1	no	—
79	C₄⋊Q₈	Dic₆₁	1	2	yes	—
80	#125	Dic₆₁	1	2	yes	—
81	C₂²⋊Q₈	Dic₆₁	1	4	yes	—
82	C₄×Q₈	C₂×D₆₁	1	4	yes	—
83	#117	C₂×D₆₁	1	2	yes	—
84	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
85	C₄⋊C₈	—	61	4	no	—
86	C₄⋊C₈	—	61	8	no	—
87	#97	—	61	4	no	—
88	C₄×C₈	—	61	2	no	—
89	C₄×C₈	—	61	2	no	—
90	C₄×C₈	—	61	4	no	—
91	C₂×Q₈	C₂×Dic₆₁	1	4	yes	—
92	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
94	M₁₆	—	122	4	no	—
95	C₂×C₈	—	61	4	no	—
96	C₂×C₈	—	61	1	no	—
97	C₂×C₈	—	61	1	no	—
98	C₂×C₈	—	61	2	no	—
99	C₂×C₈	—	61	4	no	—
100	C₂×C₈	—	122	2	no	—
101	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₆₁	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₆₁	1	4	yes	—
105	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
106	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
107	C₄²	C₂²×D₆₁	1	2	yes	—
108	C₄²	C₂²×D₆₁	1	4	yes	—
109	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	Q₈×D₆₁	1	2	yes	—
113	C₂×C₄	#11758226	1	2	yes	—
114	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
115	C₂×C₄	C₂²×Dic₆₁	1	8	yes	—
116	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
118	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
119	C₈	—	61	2	no	—
120	C₈	—	61	2	no	—
121	C₈	—	122	2	no	—
122	C₈	—	122	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₆₁	1	2	yes	—
125	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₆₁	1	2	yes	—
129	C₄	#14519784	1	4	yes	—
130	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
131	C₄	C₂×#11758226	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₆₁	1	1	yes	characteristic
134	C₂	C₂×#14519784	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839893

C₂

yes

maximal, complemented

Q₈×#92418

C₂

yes

maximal, complemented

#16839789

C₂

yes

maximal

#16839786

C₂

yes

maximal, characteristic

#16839718

C₂

yes

maximal

C₂×C₄×#92418

C₂

yes

maximal, characteristic

C₆₁×#533

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂²×#92418

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₄

yes

—

#125×C₆₁

C₄

yes

—

C₂²⋊Q₈×C₆₁

C₄

yes

—

C₄×Q₈×C₆₁

C₂²

yes

—

#117×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×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

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

#2960

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂×#131

yes

—

#533

D₆₁

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₆₁

#2960

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₆₁

yes

—

#125

Dic₆₁

yes

—

C₂²⋊Q₈

Dic₆₁

yes

—

C₄×Q₈

C₂×D₆₁

yes

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

122

—

101

C₄⋊C₄

C₂×Dic₆₁

yes

—

102

C₄⋊C₄

C₂×Dic₆₁

yes

—

103

C₄⋊C₄

C₂×Dic₆₁

yes

—

104

C₂²⋊C₄

C₂×Dic₆₁

yes

—

105

C₄²

C₂×Dic₆₁

yes

characteristic

106

C₄²

C₂×Dic₆₁

yes

characteristic

107

C₄²

C₂²×D₆₁

yes

—

108

C₄²

C₂²×D₆₁

yes

—

109

C₂³

C₂²×Dic₆₁

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₆₁

yes

—

113

C₂×C₄

#11758226

yes

—

114

C₂×C₄

C₂²×Dic₆₁

yes

—

115

C₂×C₄

C₂²×Dic₆₁

yes

—

116

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

117

C₂×C₄

C₂²×Dic₆₁

yes

—

118

C₂×C₄

C₂³×D₆₁

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

—

122

C₈

—

122

—

123

C₂²

—

124

C₂²

Q₈×Dic₆₁

yes

—

125

C₂²

C₂³×Dic₆₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₆₁

yes

—

129

C₄

#14519784

yes

—

130

C₄

C₂×Q₈×D₆₁

yes

characteristic

131

C₄

C₂×#11758226

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₆₁

yes

characteristic

134

C₂

C₂×#14519784

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#533	C₂
hypercenter	#533	D₆₁
hypocenter	C₆₁	#2960
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#533

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #16839893	C₂	semidirect
2. Q₈×#92418	C₂	semidirect
36. C₂×#92418	Q₈	semidirect
75. C₆₁	#2960	semidirect

H ≅ G/N

type

1. #16839893

C₂

semidirect

2. Q₈×#92418

C₂

semidirect

36. C₂×#92418

Q₈

semidirect

75. C₆₁

#2960

semidirect

Subgroups — #21454754

Conjugacy classes of subgroups

Complements