Group #15673579 — subgroups

Subgroups — #15673579

Order 2944.

324 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092171	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
11	C₂×#67267	—	2	2	no	—
12	C₂×#67267	—	2	4	no	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂₃⋊#95	C₄	1	1	yes	characteristic
16	C₂₃⋊#95	C₄	1	1	yes	characteristic
17	Dic₂₃⋊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	1	yes	characteristic
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
26	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂²⋊Dic₂₃	—	2	4	no	—
32	C₄×Dic₂₃	—	2	2	no	—
33	C₄×Dic₂₃	—	2	2	no	—
34	#67267	—	2	8	no	—
35	C₂×C₂₃⋊C₈	—	2	2	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	4	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	2	yes	—
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	—	2	2	no	—
58	C₂×Dic₂₃	M₁₆	1	4	yes	complemented
59	C₈×C₂₃	—	4	1	no	—
60	C₈×C₂₃	—	4	1	no	—
61	C₂₃⋊C₈	—	2	4	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	#120	1	1	yes	characteristic
75	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
76	C₄×C₂₃	#123	1	1	yes	characteristic
77	Dic₂₃	—	2	4	no	—
78	Dic₂₃	—	4	2	no	—
79	C₂²×M₁₆	—	23	1	no	—
80	C₂×C₄⋊C₈	—	23	2	no	—
81	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic
82	C₄×C₂²⋊C₄	—	23	1	no	—
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	C₂×D₂₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	—	23	2	no	—
97	C₂×C₄²	—	23	2	no	—
98	C₄⋊C₈	—	46	4	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	#95	—	23	1	no	—
101	#95	—	23	1	no	—
102	C₂₃	#1851	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	23	2	no	—
107	C₂²×C₄	—	23	2	no	—
108	C₂²×C₄	Dic₄₆	1	2	yes	—
109	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	M₁₆	—	46	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	23	4	no	—
118	C₂×C₈	—	46	2	no	—
119	C₂×C₈	—	46	2	no	—
120	C₂²⋊C₄	—	46	4	no	—
121	C₄²	—	46	2	no	—
122	C₄²	—	46	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	1	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	2	no	—
127	C₂³	Dic₂₃⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	23	4	no	—
134	C₂×C₄	—	46	2	no	—
135	C₂×C₄	—	46	2	no	—
136	C₂×C₄	—	46	2	no	—
137	C₂×C₄	—	46	2	no	—
138	C₂×C₄	—	46	2	no	—
139	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
141	C₂×C₄	D₄×D₂₃	1	2	yes	—
142	C₂×C₄	#67290	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	46	4	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	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
153	C₂²	#10625072	1	1	yes	characteristic
154	C₂²	#10625072	1	1	yes	characteristic
155	C₂²	M₁₆×D₂₃	1	2	yes	—
156	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	46	4	no	—
159	C₄	—	92	2	no	—
160	C₄	#10625049	1	2	yes	—
161	C₄	#10625106	1	1	yes	characteristic
162	C₄	#10625108	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#13092289	1	1	yes	characteristic
165	C₂	#13092440	1	1	yes	characteristic
166	C₂	#13092443	1	1	yes	characteristic
167	C₂	#13092491	1	2	yes	—
168	C₂	#13092492	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#67267

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, complemented

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×#67267

—

C₂×#67267

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂₃⋊#95

C₄

yes

characteristic

C₂₃⋊#95

C₄

yes

characteristic

Dic₂₃⋊C₈

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

M₁₆

yes

complemented

C₈×C₂₃

—

C₈×C₂₃

—

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

#120

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#123

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

D₂₃

yes

characteristic

C₄×C₂²⋊C₄

—

#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

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

#95

—

101

#95

—

102

C₂₃

#1851

yes

characteristic, complemented, Sylow

103

C₂⁴

C₄×D₂₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

Dic₄₆

yes

—

109

C₂²×C₄

C₄×D₂₃

yes

characteristic

110

C₂²×C₄

C₂²⋊D₂₃

yes

—

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

Dic₂₃⋊C₄

yes

—

128

C₂³

C₂×C₄×D₂₃

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

—

139

C₂×C₄

Dic₂₃⋊C₄

yes

—

140

C₂×C₄

C₂×Dic₄₆

yes

characteristic

141

C₂×C₄

D₄×D₂₃

yes

—

142

C₂×C₄

#67290

yes

—

143

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

(C₂²⋊D₂₃)⋊C₄

yes

—

153

C₂²

#10625072

yes

characteristic

154

C₂²

#10625072

yes

characteristic

155

C₂²

M₁₆×D₂₃

yes

—

156

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#10625049

yes

—

161

C₄

#10625106

yes

characteristic

162

C₄

#10625108

yes

characteristic

163

C₂

—

164

C₂

#13092289

yes

characteristic

165

C₂

#13092440

yes

characteristic

166

C₂

#13092443

yes

characteristic

167

C₂

#13092491

yes

—

168

C₂

#13092492

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#1851
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×C₂²⋊C₈

C₂

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₂₃⋊C₈	C₂	semidirect
4. #13092171	C₂	semidirect
58. C₂×Dic₂₃	M₁₆	semidirect
102. C₂₃	#1851	semidirect

H ≅ G/N

type

2. C₂×Dic₂₃⋊C₈

C₂

semidirect

4. #13092171

C₂

semidirect

58. C₂×Dic₂₃

M₁₆

semidirect

102. C₂₃

#1851

semidirect

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

subgroup	index
C₂²×#67267	2
C₂×Dic₂₃⋊C₈	2
C₄×C₂²⋊Dic₂₃	2
#13092171	2
C₂×C₂₃×C₂²⋊C₈	2
#1851	23

subgroup

index

C₂²×#67267

C₂×Dic₂₃⋊C₈

C₄×C₂²⋊Dic₂₃

#13092171

C₂×C₂₃×C₂²⋊C₈

#1851

Subgroups — #15673579

Conjugacy classes of subgroups

Complements

Maximal subgroups