Group #15673582 — subgroups

Subgroups — #15673582

Order 2944.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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×#67267	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
9	C₂×#67267	—	2	2	no	—
10	C₂×#67267	—	2	4	no	—
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃×#95	C₄	1	1	yes	characteristic
15	C₂₃×#95	C₄	1	1	yes	characteristic
16	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
17	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
18	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₃	—	2	4	no	—
28	C₄²×C₂₃	—	2	2	no	—
29	C₄²×C₂₃	—	2	2	no	—
30	#67267	—	2	8	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂×C₂₃⋊C₈	—	2	1	no	—
36	C₂×C₂₃⋊C₈	—	2	2	no	—
37	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
38	C₂³×C₂₃	—	2	2	no	—
39	C₂³×C₂₃	—	2	2	no	—
40	C₂³×C₂₃	—	2	1	no	—
41	C₂³×C₂₃	—	2	1	no	—
42	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₃	M₁₆	1	4	yes	complemented
55	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#41	1	2	yes	—
59	C₂₃⋊C₈	—	2	4	no	—
60	C₂₃⋊C₈	—	4	1	no	—
61	C₂₃⋊C₈	—	4	1	no	—
62	#1851	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	2	2	no	—
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	4	2	no	—
67	C₂²×C₂₃	—	4	2	no	—
68	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₃	C₄×D₄	1	2	yes	—
70	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
71	C₂²×C₂₃	#131	1	1	yes	characteristic
72	C₂²×C₂₃	#131	1	1	yes	characteristic
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	—	2	4	no	—
75	C₄×C₂₃	—	4	2	no	—
76	C₄×C₂₃	#120	1	1	yes	characteristic
77	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
78	C₄×C₂₃	#123	1	1	yes	characteristic
79	C₂²×M₁₆	—	23	1	no	—
80	C₂×C₄⋊C₈	—	23	2	no	—
81	C₂×C₂²⋊C₈	—	23	1	no	—
82	C₄×C₂²⋊C₄	D₂₃	1	1	yes	characteristic
83	#336	—	23	2	no	—
84	C₂×C₂₃	—	4	2	no	—
85	C₂×C₂₃	#385	1	1	yes	characteristic
86	C₂×C₂₃	#423	1	1	yes	characteristic
87	C₂×C₂₃	#424	1	1	yes	characteristic
88	C₂×C₂₃	#435	1	2	yes	—
89	C₂×C₂₃	#436	1	2	yes	—
90	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
91	C₂×M₁₆	—	46	4	no	—
92	C₂×M₁₆	—	46	2	no	—
93	C₂²×C₈	—	23	2	no	—
94	C₂²×C₈	—	23	1	no	—
95	C₂²×C₈	—	23	1	no	—
96	C₂×C₂²⋊C₄	Dic₂₃	1	2	yes	—
97	C₂×C₄²	C₂×D₂₃	1	2	yes	—
98	C₄⋊C₈	—	46	4	no	—
99	C₂²⋊C₈	—	46	2	no	—
100	#95	Dic₂₃	1	1	yes	characteristic
101	#95	Dic₂₃	1	1	yes	characteristic
102	C₂₃	#1851	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	Dic₄₆	1	2	yes	—
107	C₂²×C₄	D₉₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
109	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
110	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	8	no	—
113	C₂×C₈	—	23	4	no	—
114	C₂×C₈	—	46	2	no	—
115	C₂×C₈	—	46	2	no	—
116	C₂×C₈	—	46	1	no	—
117	C₂×C₈	—	46	1	no	—
118	C₂×C₈	—	46	2	no	—
119	C₂×C₈	—	46	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	1	no	—
127	C₂³	C₄⋊Dic₂₃	1	2	yes	—
128	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	#67267	1	4	yes	complemented
139	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
140	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
141	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
142	C₂×C₄	D₄×D₂₃	1	2	yes	—
143	C₂×C₄	#67290	1	2	yes	—
144	C₈	—	46	4	no	—
145	C₈	—	92	1	no	—
146	C₈	—	92	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×#67267	1	2	yes	—
153	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
154	C₂²	D₄×Dic₂₃	1	2	yes	—
155	C₂²	#10625119	1	1	yes	characteristic
156	C₂²	#10625119	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	#10625049	1	2	yes	—
161	C₄	#10625053	1	1	yes	characteristic
162	C₄	#10625057	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#13092292	1	1	yes	characteristic
165	C₂	#13092441	1	1	yes	characteristic
166	C₂	#13092444	1	1	yes	characteristic
167	C₂	#13092489	1	2	yes	—
168	C₂	#13092495	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#67267

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×#67267

—

C₂×#67267

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×#95

C₄

yes

characteristic

C₂₃×#95

C₄

yes

characteristic

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

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

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

M₁₆

yes

complemented

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1851

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#123

yes

characteristic

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#436

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₂²⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

#95

Dic₂₃

yes

characteristic

101

#95

Dic₂₃

yes

characteristic

102

C₂₃

#1851

yes

characteristic, complemented, Sylow

103

C₂⁴

C₂×Dic₂₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

Dic₄₆

yes

—

107

C₂²×C₄

D₉₂

yes

—

108

C₂²×C₄

C₂×Dic₂₃

yes

—

109

C₂²×C₄

C₂×Dic₂₃

yes

—

110

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

111

C₂²×C₄

C₂²×D₂₃

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₂₃

yes

—

128

C₂³

C₂²×Dic₂₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#67267

yes

complemented

139

C₂×C₄

C₄⋊Dic₂₃

yes

—

140

C₂×C₄

C₂×Dic₄₆

yes

characteristic

141

C₂×C₄

C₂×D₉₂

yes

characteristic

142

C₂×C₄

D₄×D₂₃

yes

—

143

C₂×C₄

#67290

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×#67267

yes

—

153

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

154

C₂²

D₄×Dic₂₃

yes

—

155

C₂²

#10625119

yes

characteristic

156

C₂²

#10625119

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#10625049

yes

—

161

C₄

#10625053

yes

characteristic

162

C₄

#10625057

yes

characteristic

163

C₂

—

164

C₂

#13092292

yes

characteristic

165

C₂

#13092441

yes

characteristic

166

C₂

#13092444

yes

characteristic

167

C₂

#13092489

yes

—

168

C₂

#13092495

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₄×C₂²⋊C₄×C₂₃	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄×C₂²⋊C₄	D₂₃
hypocenter	C₂₃	#1851
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₄×C₂²⋊C₄×C₂₃

C₂

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄×C₂²⋊C₄

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
4. #13092172	C₂	semidirect
54. C₂×C₄×C₂₃	M₁₆	semidirect
102. C₂₃	#1851	semidirect
138. C₂×C₄	#67267	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₂₃⋊C₈)

C₂

semidirect

4. #13092172

C₂

semidirect

54. C₂×C₄×C₂₃

M₁₆

semidirect

102. C₂₃

#1851

semidirect

138. C₂×C₄

#67267

semidirect

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

subgroup	index
C₂²×#67267	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092172	2
C₄×C₂²⋊C₄×C₂₃	2
#1851	23

subgroup

index

C₂²×#67267

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₂²⋊(C₂₃⋊C₈)

#13092172

C₄×C₂²⋊C₄×C₂₃

#1851

Subgroups — #15673582

Conjugacy classes of subgroups

Complements

Maximal subgroups