Group #15671718 — subgroups

Subgroups — #15671718

Order 2944.

312 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 113 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	#13092470	C₂	1	2	yes	maximal, complemented
2	#13092141	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₃	C₄	1	2	yes	—
6	C₄×D₄×C₂₃	—	2	2	no	—
7	C₄×D₄×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
13	C₄×C₂₃⋊C₈	—	2	2	no	—
14	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
15	C₂×D₄×C₂₃	—	2	2	no	—
16	C₂×D₄×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	C₈	1	4	yes	complemented
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	4	4	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂²⋊C₄×C₂₃	—	2	4	no	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	—	2	4	no	—
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	—	2	2	no	—
32	C₂³×C₂₃	—	4	2	no	—
33	C₂³×C₂₃	—	4	2	no	—
34	C₂³×C₂₃	C₂×C₈	1	4	yes	—
35	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	D₄×C₂₃	—	2	8	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	—	4	4	no	—
41	C₂×C₄×C₂₃	—	4	4	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	C₂₃⋊C₈	—	4	4	no	—
50	#1460	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	4	no	—
54	C₂²×C₂₃	—	4	4	no	—
55	C₂²×C₂₃	—	4	4	no	—
56	C₂²×C₂₃	—	4	2	no	—
57	C₂²×C₂₃	—	4	2	no	—
58	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	—	4	2	no	—
64	C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
65	C₄×C₂₃	#100	1	2	yes	—
66	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic
67	#431	—	23	2	no	—
68	#323	—	23	4	no	—
69	C₂×C₂₃	—	2	1	no	—
70	C₂×C₂₃	—	2	1	no	—
71	C₂×C₂₃	—	4	4	no	—
72	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₂₃	#410	1	1	yes	characteristic
74	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
75	C₂²×D₄	Dic₂₃	1	1	yes	characteristic
76	C₂³×C₄	Dic₂₃	1	2	yes	—
77	C₄×D₄	—	2	2	no	—
78	C₄×D₄	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
81	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
82	C₄⋊C₈	—	46	2	no	—
83	C₂²⋊C₈	—	23	4	no	—
84	C₄×C₈	—	46	2	no	—
85	C₂₃	#1460	1	1	yes	characteristic, complemented, Sylow
86	C₂⁴	C₂×Dic₂₃	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₂₃⋊C₈	1	4	yes	complemented
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	4	4	no	—
93	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
95	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	C₂×C₈	—	46	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	C₂²⋊D₂₃	1	2	yes	—
101	C₄²	C₂²⋊D₂₃	1	2	yes	—
102	C₂³	—	2	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂×C₂₃⋊C₈	1	4	yes	—
107	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
116	C₂×C₄	#67267	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
121	C₈	—	92	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
130	C₂²	C₂×#67267	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
136	C₄	#10625015	1	2	yes	—
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	4	4	no	—
140	C₂	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
141	C₂	#13092391	1	1	yes	characteristic
142	C₂	C₂×#10625015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092470

C₂

yes

maximal, complemented

#13092141

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₂₃

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₄

yes

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₄×C₂₃⋊C₈

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₈

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

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

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₈

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

#1460

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

#100

yes

—

C₂×C₄×D₄

D₂₃

yes

characteristic

#431

—

#323

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₂²⋊C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×D₄

Dic₂₃

yes

characteristic

C₂³×C₄

Dic₂₃

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

Dic₂₃

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₃

#1460

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂₃⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

C₂²⋊D₂₃

yes

—

101

C₄²

C₂²⋊D₂₃

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂×C₂₃⋊C₈

yes

—

107

C₂³

C₂²×Dic₂₃

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

116

C₂×C₄

#67267

yes

—

117

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₂₃

yes

—

119

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂²×C₂₃⋊C₈

yes

characteristic

130

C₂²

C₂×#67267

yes

characteristic

131

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

C₂²⋊(C₂₃⋊C₈)

yes

—

136

C₄

#10625015

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

141

C₂

#13092391

yes

characteristic

142

C₂

C₂×#10625015

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×D₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	#1460
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×D₄×C₂₃

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₂₃

C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×D₄

D₂₃

hypocenter

C₂₃

#1460

LCS

G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #13092470	C₂	semidirect
2. #13092141	C₂	semidirect
17. C₂×D₄×C₂₃	C₈	semidirect
85. C₂₃	#1460	semidirect
89. C₂×D₄	C₂₃⋊C₈	semidirect

H ≅ G/N

type

1. #13092470

C₂

semidirect

2. #13092141

C₂

semidirect

17. C₂×D₄×C₂₃

C₈

semidirect

85. C₂₃

#1460

semidirect

89. C₂×D₄

C₂₃⋊C₈

semidirect

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

subgroup	index
#13092470	2
#13092141	2
C₂×C₄×D₄×C₂₃	2
#1460	23

subgroup

index

#13092470

#13092141

C₂×C₄×D₄×C₂₃

#1460

Subgroups — #15671718

Conjugacy classes of subgroups

Complements

Maximal subgroups