Group #15673488 — subgroups

Subgroups — #15673488

Order 2944.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092277	C₂	1	1	yes	maximal, characteristic
3	#13092180	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₃	—	2	1	no	—
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
9	C₂×C₄⋊Dic₂₃	—	2	1	no	—
10	C₂×C₄⋊Dic₂₃	—	2	2	no	—
11	C₂×C₄⋊Dic₂₃	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂₃⋊#95	—	2	1	no	—
16	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	—	2	4	no	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	—	2	4	no	—
23	C₂²×Dic₂₃	—	2	2	no	—
24	C₂²×Dic₂₃	—	2	2	no	—
25	C₂²×Dic₂₃	—	4	4	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	4	no	—
28	C₂²⋊Dic₂₃	—	4	2	no	—
29	C₄⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	—	2	4	no	—
32	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	—	2	2	no	—
35	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₂₃	SD₁₆	1	2	yes	—
37	C₂³×C₂₃	Q₁₆	1	2	yes	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	4	yes	—
46	C₂×Dic₂₃	—	2	4	no	—
47	C₂×Dic₂₃	—	4	8	no	—
48	C₂×Dic₂₃	—	4	4	no	—
49	C₂×Dic₂₃	—	4	2	no	—
50	C₂×Dic₂₃	—	4	2	no	—
51	C₂×Dic₂₃	—	4	2	no	—
52	C₈×C₂₃	—	4	2	no	—
53	#1824	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	—	2	4	no	—
57	C₂²×C₂₃	—	2	4	no	—
58	C₂²×C₂₃	#103	1	4	yes	—
59	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	#123	1	2	yes	—
61	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₂₃	—	2	2	no	—
65	C₄×C₂₃	#117	1	2	yes	—
66	C₄×C₂₃	#123	1	2	yes	—
67	Dic₂₃	—	4	4	no	—
68	Dic₂₃	—	8	2	no	—
69	C₂²×C₄⋊C₄	—	23	1	no	—
70	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic
71	#380	—	23	1	no	—
72	#340	—	23	4	no	—
73	C₂×C₂₃	—	2	4	no	—
74	C₂×C₂₃	#381	1	1	yes	characteristic
75	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₂₃	#418	1	1	yes	characteristic
77	C₂×C₂₃	#481	1	2	yes	—
78	C₂×C₂₃	#484	1	2	yes	—
79	C₂³×C₄	—	46	1	no	—
80	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₂₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	23	4	no	—
83	C₂×C₄⋊C₄	—	23	2	no	—
84	C₂×C₄⋊C₄	—	23	2	no	—
85	C₂×C₄⋊C₄	—	46	1	no	—
86	C₂×C₄⋊C₄	—	46	2	no	—
87	C₂×C₂²⋊C₄	—	46	1	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	#95	—	46	1	no	—
90	C₂₃	#1824	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	D₉₂	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	46	4	no	—
94	C₂²×C₄	—	46	2	no	—
95	C₂²×C₄	—	46	2	no	—
96	C₂²×C₄	—	92	4	no	—
97	C₂²×C₄	C₄×D₂₃	1	2	yes	—
98	C₂²×C₄	D₉₂	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	23	4	no	—
104	C₄⋊C₄	—	46	4	no	—
105	C₄⋊C₄	—	46	2	no	—
106	C₄⋊C₄	—	46	4	no	—
107	C₂²⋊C₄	—	92	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#67263	1	2	yes	—
111	C₂³	Dic₉₂	1	2	yes	—
112	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
113	C₂³	C₂×D₉₂	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	46	4	no	—
118	C₂×C₄	—	92	8	no	—
119	C₂×C₄	—	92	4	no	—
120	C₂×C₄	—	92	2	no	—
121	C₂×C₄	—	92	2	no	—
122	C₂×C₄	—	92	2	no	—
123	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	#67290	1	4	yes	—
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#10624998	1	4	yes	—
133	C₂²	#10625057	1	2	yes	—
134	C₂²	C₂×#67263	1	1	yes	characteristic
135	C₂²	C₂×Dic₉₂	1	1	yes	characteristic
136	C₂²	#10625079	1	2	yes	—
137	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	92	4	no	—
140	C₄	—	184	2	no	—
141	C₄	#10625048	1	2	yes	—
142	C₄	#10625106	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#13092278	1	1	yes	characteristic
145	C₂	C₂×#10624998	1	1	yes	characteristic
146	C₂	#13092423	1	1	yes	characteristic
147	C₂	#13092736	1	2	yes	—
148	C₂	#13092753	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092277

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal, complemented

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂³×Dic₂₃

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₄

yes

complemented

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂₃⋊#95

—

C₂⁴×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

SD₁₆

yes

—

C₂³×C₂₃

Q₁₆

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

#1824

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#103

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

C₂²×C₄⋊C₄

—

C₂×C₂²⋊C₈

D₂₃

yes

characteristic

#380

—

#340

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#484

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂₃

#1824

yes

characteristic, complemented, Sylow

C₂⁴

D₉₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

100

C₂²×C₄

C₂²×D₂₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

#67263

yes

—

111

C₂³

Dic₉₂

yes

—

112

C₂³

(C₂×D₂₃)⋊C₄

yes

—

113

C₂³

C₂×D₉₂

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

124

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

125

C₂×C₄

#67290

yes

—

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#10624998

yes

—

133

C₂²

#10625057

yes

—

134

C₂²

C₂×#67263

yes

characteristic

135

C₂²

C₂×Dic₉₂

yes

characteristic

136

C₂²

#10625079

yes

—

137

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

184

—

141

C₄

#10625048

yes

—

142

C₄

#10625106

yes

—

143

C₂

—

144

C₂

#13092278

yes

characteristic

145

C₂

C₂×#10624998

yes

characteristic

146

C₂

#13092423

yes

characteristic

147

C₂

#13092736

yes

—

148

C₂

#13092753

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#1824
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×C₂²⋊C₈

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂⁴×C₂₃

D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₃

hypocenter

C₂₃

#1824

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #13092180	C₂	semidirect
11. C₂×C₄⋊Dic₂₃	C₄	semidirect
90. C₂₃	#1824	semidirect

H ≅ G/N

type

3. #13092180

C₂

semidirect

11. C₂×C₄⋊Dic₂₃

C₄

semidirect

90. C₂₃

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₃	2
#13092277	2
#13092180	2
C₂×C₂₃×C₂²⋊C₈	2
#1824	23

subgroup

index

C₂²×C₄⋊Dic₂₃

#13092277

#13092180

C₂×C₂₃×C₂²⋊C₈

#1824

Subgroups — #15673488

Conjugacy classes of subgroups

Complements

Maximal subgroups