Group #15674230 — subgroups

Subgroups — #15674230

Order 2944.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625019	C₂	1	4	yes	maximal, complemented
3	#13092263	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₂₃	C₂²	1	2	yes	—
6	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₂₃×#127	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#124	C₄	1	2	yes	complemented
9	C₂₃×#121	—	2	2	no	—
10	C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
13	C₂×#67267	C₂²	1	2	yes	—
14	#10625019	—	2	8	no	—
15	C₂₃⋊#95	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
17	#41×C₂₃	—	2	4	no	—
18	#41×C₂₃	—	2	4	no	—
19	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₂₃	—	2	2	no	—
21	C₂×D₄×C₂₃	—	2	2	no	—
22	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂²⋊C₄×C₂₃	—	2	4	no	—
29	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₃	D₄	1	2	yes	—
32	C₄×Dic₂₃	—	2	2	no	—
33	C₄×Dic₂₃	D₄	1	4	yes	complemented
34	#67267	—	2	4	no	—
35	C₂×C₂₃⋊C₈	—	2	2	no	—
36	C₂³×C₂₃	—	2	2	no	—
37	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₂₃	—	2	4	no	—
39	D₄×C₂₃	—	2	4	no	—
40	D₄×C₂₃	—	4	4	no	—
41	D₄×C₂₃	—	4	4	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	4	2	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	2	yes	—
52	C₂×Dic₂₃	—	2	4	no	—
53	C₂×Dic₂₃	—	4	4	no	—
54	C₂₃⋊C₈	—	4	2	no	—
55	#1963	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	—	4	2	no	—
58	C₂²×C₂₃	—	4	4	no	—
59	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	#121	1	2	yes	—
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	4	2	no	—
64	C₄×C₂₃	C₄×D₄	1	2	yes	—
65	C₄×C₂₃	#124	1	1	yes	characteristic
66	C₄×C₂₃	#127	1	1	yes	characteristic
67	Dic₂₃	—	4	4	no	—
68	#532	D₂₃	1	1	yes	characteristic
69	#423	—	23	1	no	—
70	C₂×#104	—	23	4	no	—
71	#376	—	23	1	no	—
72	C₂×C₂₃	—	2	2	no	—
73	C₂×C₂₃	—	8	2	no	—
74	C₂×C₂₃	#390	1	1	yes	characteristic
75	C₂×C₂₃	#421	1	2	yes	—
76	C₂×#41	C₂×D₂₃	1	2	yes	—
77	C₂×M₁₆	—	23	2	no	—
78	C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic, complemented
79	#127	Dic₂₃	1	1	yes	characteristic, complemented
80	#124	Dic₂₃	1	2	yes	complemented
81	#121	—	2	2	no	—
82	C₄×D₄	—	2	2	no	—
83	C₂×C₄²	—	23	2	no	—
84	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
85	C₄⋊C₈	—	46	2	no	—
86	#104	—	46	8	no	—
87	#95	—	46	2	no	—
88	C₂₃	#1963	1	1	yes	characteristic, complemented, Sylow
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₂₃	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	46	2	no	—
97	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	M₁₆	—	46	4	no	—
100	C₂×C₈	—	46	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	23	4	no	—
104	C₄²	—	46	2	no	—
105	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
106	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
107	C₄²	C₂²⋊D₂₃	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	46	4	no	—
119	C₂×C₄	—	92	4	no	—
120	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
122	C₂×C₄	D₄×D₂₃	1	2	yes	—
123	C₂×C₄	#67290	1	2	yes	—
124	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
126	C₈	—	92	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	4	2	no	—
130	C₂²	#10625109	1	2	yes	—
131	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	92	4	no	—
136	C₄	D₄×Dic₂₃	1	2	yes	—
137	C₄	#10625107	1	1	yes	characteristic
138	C₄	#10625111	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#13092320	1	1	yes	characteristic
142	C₂	#13092434	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic, complemented

C₂×#10625019

C₂

yes

maximal, complemented

#13092263

C₂

yes

maximal, characteristic, complemented

C₂₃×#532

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₄

yes

characteristic, complemented

C₂₃×#127

C₄

yes

characteristic, complemented

C₂₃×#124

C₄

yes

complemented

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

#10625019

—

C₂₃⋊#95

—

C₄⋊(C₂₃⋊C₈)

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

complemented

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×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

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1963

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

#532

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#421

yes

—

C₂×#41

C₂×D₂₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₂₃

yes

characteristic, complemented

#127

Dic₂₃

yes

characteristic, complemented

#124

Dic₂₃

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₂₃

#1963

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×Dic₂₃

yes

characteristic

106

C₄²

C₂×Dic₂₃

yes

characteristic

107

C₄²

C₂²⋊D₂₃

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²⋊Dic₂₃

yes

—

121

C₂×C₄

C₂²⋊Dic₂₃

yes

—

122

C₂×C₄

D₄×D₂₃

yes

—

123

C₂×C₄

#67290

yes

—

124

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#10625109

yes

—

131

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×Dic₂₃

yes

—

137

C₄

#10625107

yes

characteristic

138

C₄

#10625111

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#13092320

yes

characteristic

142

C₂

#13092434

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting, prime radical	C₂₃×#532	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#532	D₂₃
hypocenter	C₂₃	#1963
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting, prime radical

C₂₃×#532

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #13092441	C₂	semidirect
2. C₂×#10625019	C₂	semidirect
3. #13092263	C₂	semidirect
6. C₂₃×C₄⋊Q₈	C₄	semidirect
7. C₂₃×#127	C₄	semidirect
8. C₂₃×#124	C₄	semidirect
33. C₄×Dic₂₃	D₄	semidirect
78. C₄⋊Q₈	Dic₂₃	semidirect
79. #127	Dic₂₃	semidirect
80. #124	Dic₂₃	semidirect
88. C₂₃	#1963	semidirect

H ≅ G/N

type

1. #13092441

C₂

semidirect

2. C₂×#10625019

C₂

semidirect

3. #13092263

C₂

semidirect

6. C₂₃×C₄⋊Q₈

C₄

semidirect

7. C₂₃×#127

C₄

semidirect

8. C₂₃×#124

C₄

semidirect

33. C₄×Dic₂₃

D₄

semidirect

78. C₄⋊Q₈

Dic₂₃

semidirect

79. #127

Dic₂₃

semidirect

80. #124

Dic₂₃

semidirect

88. C₂₃

#1963

semidirect

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

subgroup	index
#13092441	2
C₂×#10625019	2
#13092263	2
C₂₃×#532	2
#1963	23

subgroup

index

C₂₃×#532

Subgroups — #15674230

Conjugacy classes of subgroups

Complements

Maximal subgroups