Group #15673465 — subgroups

Subgroups — #15673465

Order 2944.

380 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
2	#13092172	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₃	—	2	4	no	—
8	C₂×C₄²×C₂₃	C₄	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
11	C₂²⋊(C₂₃⋊C₈)	—	2	4	no	—
12	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	4	no	—
17	C₂²×C₄×C₂₃	—	2	4	no	—
18	C₂²×C₄×C₂₃	C₈	1	4	yes	complemented
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	D₄	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₄²×C₂₃	—	2	4	no	—
25	C₄²×C₂₃	—	2	4	no	—
26	C₂×C₂₃⋊C₈	—	2	4	no	—
27	C₂×C₂₃⋊C₈	—	2	8	no	—
28	C₂³×C₂₃	—	2	2	no	—
29	C₂³×C₂₃	—	2	2	no	—
30	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
31	C₂³×C₂₃	C₂×C₈	1	2	yes	—
32	C₂³×C₂₃	M₁₆	1	2	yes	—
33	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	8	no	—
35	C₂×C₄×C₂₃	—	2	8	no	—
36	C₂×C₄×C₂₃	—	2	2	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	8	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
44	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	4	yes	—
47	C₂₃⋊C₈	—	4	4	no	—
48	#1817	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	2	4	no	—
52	C₂²×C₂₃	—	2	4	no	—
53	C₂²×C₂₃	C₂²⋊C₈	1	4	yes	—
54	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₃	#117	1	2	yes	—
56	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	2	4	no	—
61	C₄×C₂₃	—	2	4	no	—
62	C₄×C₂₃	#123	1	4	yes	—
63	C₂²×C₄²	D₂₃	1	1	yes	characteristic
64	C₂×C₂²⋊C₈	—	23	2	no	—
65	#336	—	23	4	no	—
66	C₂×C₂₃	—	2	4	no	—
67	C₂×C₂₃	#381	1	1	yes	characteristic
68	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₂₃	#407	1	1	yes	characteristic
70	C₂×C₂₃	#431	1	2	yes	—
71	C₂×C₂₃	#432	1	2	yes	—
72	C₂³×C₄	Dic₂₃	1	1	yes	characteristic
73	C₂³×C₄	Dic₂₃	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
75	C₂²×C₈	—	23	4	no	—
76	C₂×C₄²	—	2	4	no	—
77	C₂×C₄²	Dic₂₃	1	2	yes	—
78	C₂×C₄²	C₂×D₂₃	1	2	yes	—
79	C₂²⋊C₈	—	46	4	no	—
80	C₂₃	#1817	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	C₂₃⋊C₈	1	4	yes	complemented
88	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
89	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
90	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
91	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
92	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
93	C₂×C₈	—	46	8	no	—
94	C₂×C₈	—	46	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	C₂×C₂₃⋊C₈	1	2	yes	—
100	C₂³	#67267	1	2	yes	—
101	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
102	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
112	C₂×C₄	#67267	1	4	yes	—
113	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
114	C₂×C₄	#67288	1	4	yes	—
115	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
116	C₈	—	92	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	C₂²⋊(C₂₃⋊C₈)	1	4	yes	—
122	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
123	C₂²	C₂×#67267	1	1	yes	characteristic
124	C₂²	C₂×#67267	1	2	yes	—
125	C₂²	#10625097	1	2	yes	—
126	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	#10625100	1	4	yes	—
131	C₂	—	2	4	no	—
132	C₂	#13092280	1	1	yes	characteristic
133	C₂	C₂×C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
134	C₂	#13092381	1	1	yes	characteristic
135	C₂	#13092470	1	2	yes	—
136	C₂	#13092474	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal

#13092172

C₂

yes

maximal, complemented

C₂²×C₄²×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₈

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×C₈

yes

—

C₂³×C₂₃

M₁₆

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×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

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

#1817

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊C₈

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₂²×C₄²

D₂₃

yes

characteristic

C₂×C₂²⋊C₈

—

#336

—

C₂×C₂₃

—

C₂×C₂₃

#381

yes

characteristic

C₂×C₂₃

C₂×C₂²⋊C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#432

yes

—

C₂³×C₄

Dic₂₃

yes

characteristic

C₂³×C₄

Dic₂₃

yes

characteristic

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

C₂²⋊C₈

—

C₂₃

#1817

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂₃⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×C₂₃⋊C₈

yes

—

100

C₂³

#67267

yes

—

101

C₂³

C₂²⋊Dic₂₃

yes

—

102

C₂³

C₂²×Dic₂₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

112

C₂×C₄

#67267

yes

—

113

C₂×C₄

C₂²⋊Dic₂₃

yes

—

114

C₂×C₄

#67288

yes

—

115

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²⋊(C₂₃⋊C₈)

yes

—

122

C₂²

C₂²×C₂₃⋊C₈

yes

characteristic

123

C₂²

C₂×#67267

yes

characteristic

124

C₂²

C₂×#67267

yes

—

125

C₂²

#10625097

yes

—

126

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10625100

yes

—

131

C₂

—

132

C₂

#13092280

yes

characteristic

133

C₂

C₂×C₂²⋊(C₂₃⋊C₈)

yes

characteristic

134

C₂

#13092381

yes

characteristic

135

C₂

#13092470

yes

—

136

C₂

#13092474

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₂²×C₄²×C₂₃	C₂
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	#1817
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Fitting

C₂²×C₄²×C₂₃

C₂

socle

C₂³×C₂₃

C₂²×C₄

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×C₄²

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #13092172	C₂	semidirect
18. C₂²×C₄×C₂₃	C₈	semidirect
80. C₂₃	#1817	semidirect
87. C₂²×C₄	C₂₃⋊C₈	semidirect

H ≅ G/N

type

2. #13092172

C₂

semidirect

18. C₂²×C₄×C₂₃

C₈

semidirect

80. C₂₃

#1817

semidirect

87. C₂²×C₄

C₂₃⋊C₈

semidirect

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

subgroup	index
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092172	2
C₂²×C₄²×C₂₃	2
#1817	23

subgroup

index

C₂×C₂²⋊(C₂₃⋊C₈)

#13092172

C₂²×C₄²×C₂₃

#1817

Subgroups — #15673465

Conjugacy classes of subgroups

Complements

Maximal subgroups