Group #19156263 — subgroups

Subgroups — C₂₃×#8637 #19156263

Order 5888.

336 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1566	C₂	1	2	yes	maximal
3	C₂₃×#1561	C₂	1	2	yes	maximal
4	C₂₃×#1309	C₂	1	2	yes	maximal
5	C₂₃×#533	C₂²	1	1	yes	characteristic
6	C₂₃×#446	—	2	2	no	—
7	C₈×Q₈×C₂₃	—	2	2	no	—
8	C₂₃×#424	C₄	1	2	yes	—
9	C₂₃×#423	C₂²	1	4	yes	—
10	C₂₃×#423	C₂²	1	1	yes	characteristic
11	C₂₃×#405	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#334	—	2	2	no	—
14	C₂₃×#334	—	2	2	no	—
15	C₂₃×#332	—	2	4	no	—
16	C₂×M₁₆×C₂₃	—	2	4	no	—
17	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
21	C₂₃×#125	Q₈	1	2	yes	—
22	C₂₃×C₂²⋊Q₈	—	2	2	no	—
23	C₄×Q₈×C₂₃	—	2	2	no	—
24	C₂₃×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	—	2	4	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	4	no	—
30	C₂₃×#97	—	2	2	no	—
31	C₄×C₈×C₂₃	—	2	2	no	—
32	C₄×C₈×C₂₃	D₄	1	2	yes	—
33	C₄×C₈×C₂₃	D₄	1	2	yes	—
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	#41	1	2	yes	—
37	M₁₆×C₂₃	—	4	2	no	—
38	M₁₆×C₂₃	—	8	4	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	4	1	no	—
42	C₂×C₈×C₂₃	—	4	4	no	—
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
49	C₂²⋊C₄×C₂₃	—	2	2	no	—
50	C₄²×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	C₂×D₄	1	2	yes	—
52	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	#8637	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#117	1	1	yes	characteristic
56	Q₈×C₂₃	—	4	4	no	—
57	C₂×C₄×C₂₃	—	2	1	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	4	no	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#121	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
65	C₈×C₂₃	—	2	1	no	—
66	C₈×C₂₃	—	2	2	no	—
67	C₈×C₂₃	—	4	2	no	—
68	C₈×C₂₃	—	8	4	no	—
69	C₈×C₂₃	#101	1	2	yes	—
70	#2960	C₂×C₂₃	1	1	yes	characteristic
71	#1566	C₂×C₂₃	1	2	yes	—
72	#1561	C₂×C₂₃	1	2	yes	—
73	#1309	C₂×C₂₃	1	2	yes	—
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	#380	1	1	yes	characteristic
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	—	2	2	no	—
79	C₄×C₂₃	—	4	4	no	—
80	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
81	C₄×C₂₃	#413	1	1	yes	characteristic
82	C₄×C₂₃	#471	1	4	yes	—
83	#533	C₂²×C₂₃	1	1	yes	characteristic
84	#446	—	2	2	no	—
85	C₈×Q₈	—	2	2	no	—
86	#424	C₄×C₂₃	1	2	yes	—
87	#423	C₂²×C₂₃	1	4	yes	—
88	#423	C₂²×C₂₃	1	1	yes	characteristic
89	#405	C₄×C₂₃	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
91	#334	—	2	2	no	—
92	#334	—	2	2	no	—
93	#332	—	2	4	no	—
94	C₂×C₂₃	—	4	1	no	—
95	C₂×C₂₃	#1801	1	1	yes	characteristic
96	C₂×C₂₃	#1815	1	2	yes	—
97	C₂×M₁₆	—	2	4	no	—
98	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
101	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
102	#125	Q₈×C₂₃	1	2	yes	—
103	C₂²⋊Q₈	—	2	2	no	—
104	C₄×Q₈	—	2	2	no	—
105	#117	C₂×C₄×C₂₃	1	2	yes	—
106	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	4	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	4	no	—
110	C₄⋊C₈	—	2	4	no	—
111	#97	—	2	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	D₄×C₂₃	1	2	yes	—
114	C₄×C₈	D₄×C₂₃	1	2	yes	—
115	C₂₃	#8637	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×Q₈	—	2	2	no	—
117	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
118	C₂²×C₄	#41×C₂₃	1	2	yes	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	8	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	4	no	—
125	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
127	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂×D₄×C₂₃	1	2	yes	—
134	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
135	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
136	C₂³	C₂₃×#117	1	1	yes	characteristic
137	Q₈	—	4	4	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₄	C₂₃×#121	1	2	yes	—
145	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
146	C₈	—	2	1	no	—
147	C₈	—	2	2	no	—
148	C₈	—	4	2	no	—
149	C₈	—	8	4	no	—
150	C₈	C₂₃×#101	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₂₃×#380	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	2	no	—
156	C₄	—	4	4	no	—
157	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
158	C₄	C₂₃×#413	1	1	yes	characteristic
159	C₄	C₂₃×#471	1	4	yes	—
160	C₂	—	4	1	no	—
161	C₂	C₂₃×#1801	1	1	yes	characteristic
162	C₂	C₂₃×#1815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2960

C₂

yes

maximal, characteristic

C₂₃×#1566

C₂

yes

maximal

C₂₃×#1561

C₂

yes

maximal

C₂₃×#1309

C₂

yes

maximal

C₂₃×#533

C₂²

yes

characteristic

C₂₃×#446

—

C₈×Q₈×C₂₃

—

C₂₃×#424

C₄

yes

—

C₂₃×#423

C₂²

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#405

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#334

—

C₂₃×#334

—

C₂₃×#332

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂₃×#125

Q₈

yes

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×Q₈

yes

characteristic

#8637

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#117

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#101

yes

—

#2960

C₂×C₂₃

yes

characteristic

#1566

C₂×C₂₃

yes

—

#1561

C₂×C₂₃

yes

—

#1309

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#380

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

#533

C₂²×C₂₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₄×C₂₃

yes

—

#423

C₂²×C₂₃

yes

—

#423

C₂²×C₂₃

yes

characteristic

#405

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

#1801

yes

characteristic

C₂×C₂₃

#1815

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

100

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

101

C₄⋊Q₈

D₄×C₂₃

yes

—

102

#125

Q₈×C₂₃

yes

—

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

C₂×C₄×C₂₃

yes

—

106

C₂×C₄²

C₂³×C₂₃

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

D₄×C₂₃

yes

—

114

C₄×C₈

D₄×C₂₃

yes

—

115

C₂₃

#8637

yes

characteristic, direct product factor, Sylow

116

C₂×Q₈

—

117

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

118

C₂²×C₄

#41×C₂₃

yes

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

126

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

127

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₄⋊C₄×C₂₃

yes

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

C₂×D₄×C₂₃

yes

—

134

C₄²

C₂×D₄×C₂₃

yes

characteristic

135

C₄²

C₂×Q₈×C₂₃

yes

characteristic

136

C₂³

C₂₃×#117

yes

characteristic

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

143

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

144

C₂×C₄

C₂₃×#121

yes

—

145

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

C₂₃×#101

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂₃×#380

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×C₂₃×#101

yes

characteristic

158

C₄

C₂₃×#413

yes

characteristic

159

C₄

C₂₃×#471

yes

—

160

C₂

—

161

C₂

C₂₃×#1801

yes

characteristic

162

C₂

C₂₃×#1815

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#380

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂³×C₂₃

#117

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
54. #8637	C₂₃	direct (N × H)
115. C₂₃	#8637	direct (N × H)

H ≅ G/N

type

54. #8637

C₂₃

direct (N × H)

115. C₂₃

#8637

direct (N × H)

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

subgroup	index
C₂₃×#2960	2
C₂₃×#1566	2
C₂₃×#1561	2
C₂₃×#1309	2
#8637	23

subgroup

index

C₂₃×#2960

C₂₃×#1566

C₂₃×#1561

C₂₃×#1309

#8637

Subgroups — C₂₃×#8637 #19156263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8637 #19156263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8637 #19156263