Group #15682332 — subgroups

Subgroups — #15682332

Order 2944.

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	#13092548	C₂	1	4	yes	maximal, complemented
2	Q₈×C₂₃⋊C₈	C₂	1	4	yes	maximal, complemented
3	#13092444	C₂	1	2	yes	maximal
4	#13092441	C₂	1	1	yes	maximal, characteristic
5	#13092373	C₂	1	2	yes	maximal
6	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
8	C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
9	C₂₃×#125	C₄	1	2	yes	—
10	C₂₃×C₂²⋊Q₈	C₄	1	4	yes	—
11	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
12	C₂₃×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67267	C₂²	1	2	yes	—
15	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
18	C₄⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
19	#10624985	C₂²	1	4	yes	—
20	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
21	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
22	C₄×C₂₃⋊C₈	C₂²	1	4	yes	—
23	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₂₃	C₂³	1	2	yes	—
33	C₄²×C₂₃	C₂³	1	4	yes	—
34	#67267	—	2	4	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
37	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
38	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
39	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₂₃	—	2	8	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₃	#41	1	2	yes	—
50	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂₃⋊C₈	—	2	2	no	—
52	C₂₃⋊C₈	—	2	4	no	—
53	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
54	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
55	#2960	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
58	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	2	8	no	—
61	C₄×C₂₃	C₄×Q₈	1	2	yes	—
62	C₄×C₂₃	#131	1	4	yes	—
63	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
65	#533	D₂₃	1	1	yes	characteristic
66	#446	—	23	4	no	—
67	C₈×Q₈	—	23	4	no	—
68	#424	—	23	2	no	—
69	#423	—	23	1	no	—
70	#405	—	23	2	no	—
71	C₂×C₄×C₈	—	23	1	no	—
72	C₂×C₂₃	—	2	2	no	—
73	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
74	C₂×C₂₃	C₂×#131	1	2	yes	—
75	C₂×M₁₆	—	23	2	no	—
76	C₂²×C₈	—	23	1	no	—
77	C₂²×C₈	—	23	1	no	—
78	C₄⋊Q₈	Dic₂₃	1	2	yes	—
79	#125	Dic₂₃	1	2	yes	—
80	C₂²⋊Q₈	Dic₂₃	1	4	yes	—
81	C₄×Q₈	C₂×D₂₃	1	4	yes	—
82	#117	C₂×D₂₃	1	2	yes	—
83	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
84	C₄⋊C₈	—	23	4	no	—
85	C₄⋊C₈	—	23	8	no	—
86	#97	—	23	4	no	—
87	C₄×C₈	—	23	2	no	—
88	C₄×C₈	—	23	2	no	—
89	C₄×C₈	—	23	4	no	—
90	C₂₃	#2960	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
92	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	M₁₆	—	46	4	no	—
95	C₂×C₈	—	23	4	no	—
96	C₂×C₈	—	23	1	no	—
97	C₂×C₈	—	23	1	no	—
98	C₂×C₈	—	23	2	no	—
99	C₂×C₈	—	23	4	no	—
100	C₂×C₈	—	46	2	no	—
101	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₂₃	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
105	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
106	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
107	C₄²	C₂²×D₂₃	1	2	yes	—
108	C₄²	C₂²×D₂₃	1	4	yes	—
109	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	Q₈×D₂₃	1	2	yes	—
113	C₂×C₄	#67292	1	2	yes	—
114	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
115	C₂×C₄	C₂²×Dic₂₃	1	8	yes	—
116	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
118	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
119	C₈	—	23	2	no	—
120	C₈	—	23	2	no	—
121	C₈	—	46	2	no	—
122	C₈	—	46	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₂₃	1	2	yes	—
125	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₂₃	1	2	yes	—
129	C₄	#10625119	1	4	yes	—
130	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
131	C₄	C₂×#67292	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₂₃	1	1	yes	characteristic
134	C₂	C₂×#10625119	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092548

C₂

yes

maximal, complemented

Q₈×C₂₃⋊C₈

C₂

yes

maximal, complemented

#13092444

C₂

yes

maximal

#13092441

C₂

yes

maximal, characteristic

#13092373

C₂

yes

maximal

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₄

yes

—

C₂₃×#125

C₄

yes

—

C₂₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

C₂³

yes

characteristic

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#2960

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#131

yes

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

#533

D₂₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₃

C₂×#131

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₂₃

yes

—

#125

Dic₂₃

yes

—

C₂²⋊Q₈

Dic₂₃

yes

—

C₄×Q₈

C₂×D₂₃

yes

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#2960

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

C₂×Dic₂₃

yes

—

102

C₄⋊C₄

C₂×Dic₂₃

yes

—

103

C₄⋊C₄

C₂×Dic₂₃

yes

—

104

C₂²⋊C₄

C₂×Dic₂₃

yes

—

105

C₄²

C₂×Dic₂₃

yes

characteristic

106

C₄²

C₂×Dic₂₃

yes

characteristic

107

C₄²

C₂²×D₂₃

yes

—

108

C₄²

C₂²×D₂₃

yes

—

109

C₂³

C₂²×Dic₂₃

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₂₃

yes

—

113

C₂×C₄

#67292

yes

—

114

C₂×C₄

C₂²×Dic₂₃

yes

—

115

C₂×C₄

C₂²×Dic₂₃

yes

—

116

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

117

C₂×C₄

C₂²×Dic₂₃

yes

—

118

C₂×C₄

C₂³×D₂₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

Q₈×Dic₂₃

yes

—

125

C₂²

C₂³×Dic₂₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₂₃

yes

—

129

C₄

#10625119

yes

—

130

C₄

C₂×Q₈×D₂₃

yes

characteristic

131

C₄

C₂×#67292

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₂₃

yes

characteristic

134

C₂

C₂×#10625119

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#533	C₂
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#533	D₂₃
hypocenter	C₂₃	#2960
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Fitting

C₂₃×#533

C₂

prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂×C₄×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #13092548	C₂	semidirect
2. Q₈×C₂₃⋊C₈	C₂	semidirect
36. C₂×C₂₃⋊C₈	Q₈	semidirect
90. C₂₃	#2960	semidirect

H ≅ G/N

type

1. #13092548

C₂

semidirect

2. Q₈×C₂₃⋊C₈

C₂

semidirect

36. C₂×C₂₃⋊C₈

Q₈

semidirect

90. C₂₃

#2960

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092548	2
Q₈×C₂₃⋊C₈	2
#13092444	2
#13092441	2
#13092373	2
C₂×C₄×C₂₃⋊C₈	2
C₂₃×#533	2
#2960	23

subgroup

index

Q₈×C₂₃⋊C₈

C₂×C₄×C₂₃⋊C₈

C₂₃×#533

#2960

Subgroups — #15682332

Conjugacy classes of subgroups

Complements

Maximal subgroups