Group #15673796 — subgroups

Subgroups — #15673796

Order 2944.

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

normal subgroups only

This group has 444 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	#13092892	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092422	C₂	1	2	yes	maximal, complemented
3	C₂×#10625015	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092177	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₂₃	—	2	2	no	—
7	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#127	—	2	2	no	—
10	C₂₃×#127	C₄	1	4	yes	complemented
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Dic₂₃	—	2	2	no	—
13	D₄×Dic₂₃	—	2	4	no	—
14	#10625097	C₂²	1	2	yes	—
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625015	—	2	2	no	—
17	#10625014	—	2	4	no	—
18	C₂⁴×C₂₃	—	2	1	no	—
19	C₂⁴×C₂₃	D₄	1	2	yes	—
20	C₂×D₄×C₂₃	—	2	8	no	—
21	C₂×D₄×C₂₃	—	2	2	no	—
22	C₂×D₄×C₂₃	—	2	2	no	—
23	C₂×D₄×C₂₃	—	2	1	no	—
24	C₂×D₄×C₂₃	—	2	2	no	—
25	C₂×D₄×C₂₃	—	2	2	no	—
26	C₂×D₄×C₂₃	—	4	8	no	—
27	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	complemented
29	C₂×D₄×C₂₃	D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	—	2	1	no	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₂₃	—	2	4	no	—
33	C₄²×C₂₃	—	2	1	no	—
34	C₄²×C₂₃	C₂×C₄	1	2	yes	complemented
35	C₂²⋊Dic₂₃	—	2	4	no	—
36	C₂²⋊Dic₂₃	—	2	2	no	—
37	C₄⋊Dic₂₃	D₄	1	4	yes	complemented
38	C₄×Dic₂₃	—	2	2	no	—
39	#67267	—	2	2	no	—
40	#67267	—	2	2	no	—
41	C₂×C₂₃⋊C₈	—	2	2	no	—
42	C₂³×C₂₃	—	2	2	no	—
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	—	2	2	no	—
45	C₂³×C₂₃	—	4	4	no	—
46	C₂³×C₂₃	—	4	4	no	—
47	C₂³×C₂₃	—	4	1	no	—
48	C₂³×C₂₃	—	4	1	no	—
49	C₂³×C₂₃	C₂²⋊C₄	1	4	yes	—
50	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
51	D₄×C₂₃	—	2	4	no	—
52	D₄×C₂₃	—	2	4	no	—
53	D₄×C₂₃	—	2	4	no	—
54	D₄×C₂₃	—	4	16	no	—
55	D₄×C₂₃	—	4	2	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#41	1	2	yes	—
62	C₂×Dic₂₃	—	2	4	no	—
63	C₂×Dic₂₃	—	8	4	no	—
64	C₂₃⋊C₈	—	4	2	no	—
65	#1887	—	23	1	no	maximal, Sylow
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	—	2	4	no	—
68	C₂²×C₂₃	—	4	4	no	—
69	C₂²×C₂₃	—	4	4	no	—
70	C₂²×C₂₃	—	4	4	no	—
71	C₂²×C₂₃	—	4	2	no	—
72	C₂²×C₂₃	—	8	2	no	—
73	C₂²×C₂₃	—	8	2	no	—
74	C₂²×C₂₃	—	8	2	no	—
75	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₃	#120	1	1	yes	characteristic
77	C₂²×C₂₃	#123	1	1	yes	characteristic
78	C₄×C₂₃	—	2	4	no	—
79	C₄×C₂₃	C₄×D₄	1	2	yes	—
80	C₄×C₂₃	#121	1	2	yes	—
81	Dic₂₃	—	4	4	no	—
82	C₂×#127	D₂₃	1	1	yes	characteristic
83	#518	—	23	1	no	—
84	#418	—	23	2	no	—
85	C₂×#100	—	23	1	no	—
86	#339	—	23	2	no	—
87	C₂×C₂₃	—	2	1	no	—
88	C₂×C₂₃	—	2	1	no	—
89	C₂×C₂₃	—	4	4	no	—
90	C₂×C₂₃	—	8	2	no	—
91	C₂×C₂₃	#386	1	1	yes	characteristic
92	C₂×C₂₃	#453	1	2	yes	—
93	C₂²×D₄	—	2	2	no	—
94	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
95	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
96	C₂×M₁₆	—	23	2	no	—
97	#127	—	2	2	no	—
98	#127	Dic₂₃	1	4	yes	complemented
99	C₄×D₄	—	46	4	no	—
100	#117	—	23	2	no	—
101	C₂×C₂²⋊C₄	—	46	2	no	—
102	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
103	#102	—	46	4	no	—
104	#100	—	46	2	no	—
105	C₂₃	#1887	1	1	yes	characteristic, complemented, Sylow
106	C₂⁴	—	2	1	no	—
107	C₂⁴	C₂²⋊D₂₃	1	2	yes	—
108	C₂×D₄	—	2	8	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	4	8	no	—
115	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
116	C₂×D₄	C₂×Dic₂₃	1	2	yes	complemented
117	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	46	4	no	—
120	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	M₁₆	—	46	2	no	—
122	M₁₆	—	46	2	no	—
123	C₂×C₈	—	46	2	no	—
124	C₄⋊C₄	—	23	4	no	—
125	C₂²⋊C₄	—	46	4	no	—
126	C₂²⋊C₄	—	46	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	46	2	no	—
129	C₄²	C₂×Dic₂₃	1	2	yes	complemented
130	C₂³	—	2	2	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	4	4	no	—
134	C₂³	—	4	4	no	—
135	C₂³	—	4	1	no	—
136	C₂³	—	4	1	no	—
137	C₂³	C₂²⋊Dic₂₃	1	4	yes	—
138	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
139	D₄	—	2	4	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	46	4	no	—
147	C₂×C₄	—	184	4	no	—
148	C₂×C₄	D₄×D₂₃	1	2	yes	—
149	C₂×C₄	#67290	1	2	yes	—
150	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
152	C₈	—	92	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	#10625106	1	1	yes	characteristic
163	C₂²	#10625108	1	1	yes	characteristic
164	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	92	4	no	—
167	C₄	D₄×Dic₂₃	1	2	yes	—
168	C₄	#10625110	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	4	4	no	—
172	C₂	—	8	2	no	—
173	C₂	#13092297	1	1	yes	characteristic
174	C₂	#13092584	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092892

C₂

yes

maximal, characteristic, complemented

#13092422

C₂

yes

maximal, complemented

C₂×#10625015

C₂

yes

maximal, characteristic, complemented

#13092177

C₂

yes

maximal, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂₃×#127

—

C₂₃×#127

C₄

yes

complemented

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

—

D₄×Dic₂₃

—

#10625097

C₂²

yes

—

C₂×#67267

C₂²

yes

—

#10625015

—

#10625014

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

complemented

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

complemented

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

complemented

C₄×Dic₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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

—

#1887

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

C₂×#127

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

C₂×C₂₃

#453

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂²×D₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

#127

—

#127

Dic₂₃

yes

complemented

C₄×D₄

—

100

#117

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₄²

C₂×D₂₃

yes

characteristic

103

#102

—

104

#100

—

105

C₂₃

#1887

yes

characteristic, complemented, Sylow

106

C₂⁴

—

107

C₂⁴

C₂²⋊D₂₃

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₂×Dic₂₃

yes

—

116

C₂×D₄

C₂×Dic₂₃

yes

complemented

117

C₂×D₄

C₂²⋊D₂₃

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²×D₂₃

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂×Dic₂₃

yes

complemented

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂²⋊Dic₂₃

yes

—

138

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

184

—

148

C₂×C₄

D₄×D₂₃

yes

—

149

C₂×C₄

#67290

yes

—

150

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

152

C₈

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#10625106

yes

characteristic

163

C₂²

#10625108

yes

characteristic

164

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

D₄×Dic₂₃

yes

—

168

C₄

#10625110

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

—

173

C₂

#13092297

yes

characteristic

174

C₂

#13092584

yes

—

	subgroup	G/H
center	C₂²	#10625108
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle	C₂²×C₂₃	#120
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	#1887
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#127

subgroup

G/H

center

C₂²

#10625108

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle

C₂²×C₂₃

#120

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#127

N	H ≅ G/N	type
1. #13092892	C₂	semidirect
2. #13092422	C₂	semidirect
3. C₂×#10625015	C₂	semidirect
4. #13092177	C₂	semidirect
10. C₂₃×#127	C₄	semidirect
28. C₂×D₄×C₂₃	C₂×C₄	semidirect
34. C₄²×C₂₃	C₂×C₄	semidirect
37. C₄⋊Dic₂₃	D₄	semidirect
98. #127	Dic₂₃	semidirect
105. C₂₃	#1887	semidirect
116. C₂×D₄	C₂×Dic₂₃	semidirect
129. C₄²	C₂×Dic₂₃	semidirect

H ≅ G/N

type

1. #13092892

C₂

semidirect

2. #13092422

C₂

semidirect

3. C₂×#10625015

C₂

semidirect

4. #13092177

C₂

semidirect

10. C₂₃×#127

C₄

semidirect

28. C₂×D₄×C₂₃

C₂×C₄

semidirect

34. C₄²×C₂₃

C₂×C₄

semidirect

37. C₄⋊Dic₂₃

D₄

semidirect

98. #127

Dic₂₃

semidirect

105. C₂₃

#1887

semidirect

116. C₂×D₄

C₂×Dic₂₃

semidirect

129. C₄²

C₂×Dic₂₃

semidirect

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

subgroup	index
#13092892	2
#13092422	2
C₂×#10625015	2
#13092177	2
C₂×C₂₃×#127	2
#1887	23

subgroup

index

C₂×C₂₃×#127

Subgroups — #15673796

Conjugacy classes of subgroups

Complements

Maximal subgroups