Group #15673147 — subgroups

Subgroups — #15673147

Order 2944.

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

normal subgroups only

This group has 240 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	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092263	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	#13092171	C₂	1	1	yes	maximal, characteristic
6	#13092171	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃⋊#95	C₄	1	2	yes	—
16	C₂₃⋊#95	C₄	1	2	yes	—
17	#10624997	C₄	1	4	yes	complemented
18	C₈×Dic₂₃	C₄	1	4	yes	complemented
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	—	2	1	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₄²×C₂₃	—	2	1	no	—
31	C₄²×C₂₃	—	2	1	no	—
32	C₄×Dic₂₃	—	2	1	no	—
33	C₄×Dic₂₃	—	2	1	no	—
34	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₂₃⋊C₈	—	2	1	no	—
37	C₂×C₂₃⋊C₈	—	2	1	no	—
38	C₂×C₂₃⋊C₈	—	2	1	no	—
39	C₂×C₂₃⋊C₈	—	2	1	no	—
40	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×Dic₂₃	—	2	2	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	C₄²	1	2	yes	—
58	C₂×Dic₂₃	C₄²	1	2	yes	—
59	C₈×C₂₃	—	2	2	no	—
60	C₈×C₂₃	—	2	2	no	—
61	C₂₃⋊C₈	—	2	2	no	—
62	C₂₃⋊C₈	—	2	2	no	—
63	#1746	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
65	C₂²×C₂₃	#117	1	1	yes	characteristic
66	C₂²×C₂₃	#117	1	1	yes	characteristic
67	C₂²×C₂₃	#131	1	2	yes	—
68	C₂²×C₂₃	#131	1	2	yes	—
69	C₄×C₂₃	—	4	2	no	—
70	C₄×C₂₃	#117	1	1	yes	characteristic
71	C₄×C₂₃	#117	1	1	yes	characteristic
72	C₄×C₂₃	#117	1	1	yes	characteristic
73	C₄×C₂₃	#117	1	1	yes	characteristic
74	Dic₂₃	—	2	2	no	—
75	Dic₂₃	—	2	2	no	—
76	Dic₂₃	—	4	2	no	—
77	C₂×#97	—	23	1	no	—
78	C₂×C₄×C₈	—	23	1	no	—
79	#376	—	23	1	no	—
80	#336	—	23	1	no	—
81	#336	—	23	1	no	—
82	#336	—	23	1	no	—
83	#336	D₂₃	1	1	yes	characteristic
84	C₂×C₂₃	#376	1	1	yes	characteristic
85	C₂×C₂₃	#405	1	1	yes	characteristic
86	C₂×C₂₃	#405	1	1	yes	characteristic
87	C₂×C₂₃	#433	1	2	yes	—
88	C₂×C₂₃	#433	1	2	yes	—
89	C₂²×C₈	—	23	1	no	—
90	C₂²×C₈	—	23	1	no	—
91	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
93	C₂×C₄²	—	23	1	no	—
94	C₂×C₄²	—	23	1	no	—
95	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
96	#97	—	23	4	no	—
97	C₄×C₈	—	23	4	no	—
98	#95	—	23	2	no	—
99	#95	—	23	2	no	—
100	C₂₃	#1746	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	23	1	no	—
102	C₂²×C₄	—	23	1	no	—
103	C₂²×C₄	—	23	2	no	—
104	C₂²×C₄	C₄×D₂₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	23	2	no	—
111	C₂×C₈	—	23	2	no	—
112	C₂×C₈	—	46	1	no	—
113	C₂×C₈	—	46	1	no	—
114	C₂×C₈	—	46	1	no	—
115	C₂×C₈	—	46	1	no	—
116	C₂×C₈	C₄×D₂₃	1	2	yes	—
117	C₂×C₈	C₄×D₂₃	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	23	2	no	—
121	C₄²	—	23	2	no	—
122	C₄²	—	46	1	no	—
123	C₄²	—	46	1	no	—
124	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	23	2	no	—
129	C₂×C₄	—	23	2	no	—
130	C₂×C₄	—	46	2	no	—
131	C₂×C₄	—	46	2	no	—
132	C₂×C₄	—	46	2	no	—
133	C₂×C₄	—	46	2	no	—
134	C₂×C₄	—	46	2	no	—
135	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
137	C₂×C₄	#67290	1	1	yes	characteristic
138	C₂×C₄	#67290	1	1	yes	characteristic
139	C₂×C₄	#67290	1	1	yes	characteristic
140	C₂×C₄	#67292	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	46	2	no	—
144	C₈	—	46	2	no	—
145	C₂²	C₄²×D₂₃	1	1	yes	characteristic
146	C₂²	#10625048	1	1	yes	characteristic
147	C₂²	#10625063	1	1	yes	characteristic
148	C₂²	#10625072	1	2	yes	—
149	C₂²	#10625078	1	2	yes	—
150	C₄	—	4	2	no	—
151	C₄	—	46	2	no	—
152	C₄	—	46	2	no	—
153	C₄	—	92	2	no	—
154	C₄	#10625048	1	1	yes	characteristic
155	C₄	#10625048	1	1	yes	characteristic
156	C₄	#10625063	1	1	yes	characteristic
157	C₄	#10625063	1	1	yes	characteristic
158	C₂	#13092262	1	1	yes	characteristic
159	C₂	#13092371	1	1	yes	characteristic
160	C₂	#13092374	1	1	yes	characteristic
161	C₂	#13092475	1	2	yes	—
162	C₂	#13092477	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624997

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

#13092263

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, characteristic

C₂₃×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃⋊#95

C₄

yes

—

C₂₃⋊#95

C₄

yes

—

#10624997

C₄

yes

complemented

C₈×Dic₂₃

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

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

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄²

yes

—

C₂×Dic₂₃

C₄²

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1746

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#433

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#97

—

C₄×C₈

—

#95

—

#95

—

100

C₂₃

#1746

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₂₃

yes

—

105

C₂²×C₄

C₂²×D₂₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×D₂₃

yes

—

117

C₂×C₈

C₄×D₂₃

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₂³

C₂×C₄×D₂₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

136

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

137

C₂×C₄

#67290

yes

characteristic

138

C₂×C₄

#67290

yes

characteristic

139

C₂×C₄

#67290

yes

characteristic

140

C₂×C₄

#67292

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₂²

C₄²×D₂₃

yes

characteristic

146

C₂²

#10625048

yes

characteristic

147

C₂²

#10625063

yes

characteristic

148

C₂²

#10625072

yes

—

149

C₂²

#10625078

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#10625048

yes

characteristic

155

C₄

#10625048

yes

characteristic

156

C₄

#10625063

yes

characteristic

157

C₄

#10625063

yes

characteristic

158

C₂

#13092262

yes

characteristic

159

C₂

#13092371

yes

characteristic

160

C₂

#13092374

yes

characteristic

161

C₂

#13092475

yes

—

162

C₂

#13092477

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂₃×#336	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#336	D₂₃
hypocenter	C₂₃	#1746
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂×C₄²

Fitting

C₂₃×#336

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
17. #10624997	C₄	semidirect
18. C₈×Dic₂₃	C₄	semidirect
100. C₂₃	#1746	semidirect

H ≅ G/N

type

17. #10624997

C₄

semidirect

18. C₈×Dic₂₃

C₄

semidirect

100. C₂₃

#1746

semidirect

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

subgroup	index
C₂×#10624997	2
C₂×C₈×Dic₂₃	2
#13092263	2
#13092172	2
#13092171	2
C₂₃×#336	2
#1746	23

subgroup

index

C₂×C₈×Dic₂₃

C₂₃×#336

Subgroups — #15673147

Conjugacy classes of subgroups

Complements

Maximal subgroups