Group #15674334 — subgroups

Subgroups — #15674334

Order 2944.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092172	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₂₃	—	2	2	no	—
7	C₄×Q₈×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₄×C₂₃⋊C₈	—	2	2	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	C₈	1	4	yes	complemented
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	—	2	2	no	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	4	no	—
34	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₂₃	—	2	8	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₂₃	#41	1	2	yes	—
51	C₂₃⋊C₈	—	2	4	no	—
52	C₂₃⋊C₈	—	4	2	no	—
53	#1984	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
56	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₂₃	#131	1	2	yes	—
59	C₄×C₂₃	—	2	4	no	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	—	4	2	no	—
62	C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
63	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
64	C₄×C₂₃	#124	1	1	yes	characteristic
65	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	23	1	no	—
68	C₂×C₄×C₈	—	23	1	no	—
69	#336	—	23	4	no	—
70	C₂×C₂₃	#391	1	1	yes	characteristic
71	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₂₃	#408	1	1	yes	characteristic
73	C₂×C₂₃	C₈×Q₈	1	2	yes	—
74	C₂×C₂₃	#446	1	2	yes	—
75	C₂²×Q₈	Dic₂₃	1	1	yes	characteristic
76	C₂²×C₈	—	23	2	no	—
77	C₂²×C₈	—	23	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
81	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
82	C₂×C₄²	C₂×D₂₃	1	2	yes	—
83	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
84	C₄⋊C₈	—	46	2	no	—
85	C₄×C₈	—	46	2	no	—
86	C₂₃	#1984	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₂₃⋊C₈	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
91	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
92	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
95	C₂×C₈	—	23	4	no	—
96	C₂×C₈	—	46	2	no	—
97	C₂×C₈	—	46	2	no	—
98	C₂×C₈	—	46	4	no	—
99	C₂×C₈	—	46	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₂₃	1	2	yes	—
105	C₄²	C₂²⋊D₂₃	1	2	yes	—
106	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
114	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
115	C₂×C₄	#67267	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
118	C₂×C₄	Q₈×D₂₃	1	2	yes	—
119	C₂×C₄	#67292	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
122	C₈	—	46	4	no	—
123	C₈	—	92	2	no	—
124	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
125	C₂²	C₂×#67267	1	1	yes	characteristic
126	C₂²	Q₈×Dic₂₃	1	2	yes	—
127	C₂²	#10625119	1	2	yes	—
128	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
133	C₄	#10625115	1	1	yes	characteristic
134	C₄	#10625117	1	2	yes	—
135	C₄	#10625118	1	1	yes	characteristic
136	C₂	#13092324	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
138	C₂	#13092384	1	1	yes	characteristic
139	C₂	Q₈×C₂₃⋊C₈	1	2	yes	—
140	C₂	#13092548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal

C₂×C₄×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₄

yes

characteristic

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄×C₂₃⋊C₈

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₈

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1984

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×Q₈

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#336

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₃

#408

yes

characteristic

C₂×C₂₃

C₈×Q₈

yes

—

C₂×C₂₃

#446

yes

—

C₂²×Q₈

Dic₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₃

yes

characteristic

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂₃

#1984

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂₃⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₂₃

yes

—

105

C₄²

C₂²⋊D₂₃

yes

—

106

C₂³

C₂²×Dic₂₃

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

114

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

115

C₂×C₄

#67267

yes

—

116

C₂×C₄

C₂²⋊Dic₂₃

yes

—

117

C₂×C₄

C₂²⋊Dic₂₃

yes

—

118

C₂×C₄

Q₈×D₂₃

yes

—

119

C₂×C₄

#67292

yes

—

120

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

122

C₈

—

123

C₈

—

124

C₂²

C₂²×C₂₃⋊C₈

yes

characteristic

125

C₂²

C₂×#67267

yes

characteristic

126

C₂²

Q₈×Dic₂₃

yes

—

127

C₂²

#10625119

yes

—

128

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊(C₂₃⋊C₈)

yes

—

133

C₄

#10625115

yes

characteristic

134

C₄

#10625117

yes

—

135

C₄

#10625118

yes

characteristic

136

C₂

#13092324

yes

characteristic

137

C₂

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

138

C₂

#13092384

yes

characteristic

139

C₂

Q₈×C₂₃⋊C₈

yes

—

140

C₂

#13092548

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	#1984
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
18. C₂×Q₈×C₂₃	C₈	semidirect
34. C₂×C₂₃⋊C₈	Q₈	semidirect
86. C₂₃	#1984	semidirect
89. C₂×Q₈	C₂₃⋊C₈	semidirect

H ≅ G/N

type

18. C₂×Q₈×C₂₃

C₈

semidirect

34. C₂×C₂₃⋊C₈

Q₈

semidirect

86. C₂₃

#1984

semidirect

89. C₂×Q₈

C₂₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₄×C₂₃⋊C₈	2
#13092172	2
C₂×C₄×Q₈×C₂₃	2
#1984	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₄×C₂₃⋊C₈

#13092172

C₂×C₄×Q₈×C₂₃

#1984

Subgroups — #15674334

Conjugacy classes of subgroups

Complements

Maximal subgroups