Group #19156265 — subgroups

Subgroups — C₂₃×#8639 #19156265

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×#1567	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×#335	—	2	2	no	—
14	C₂₃×#335	—	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	#8639	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#117	1	1	yes	characteristic
56	Q₈×C₂₃	—	4	2	no	—
57	Q₈×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#121	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
67	C₈×C₂₃	—	2	1	no	—
68	C₈×C₂₃	—	2	2	no	—
69	C₈×C₂₃	—	4	2	no	—
70	C₈×C₂₃	—	8	4	no	—
71	C₈×C₂₃	#101	1	2	yes	—
72	#2960	C₂×C₂₃	1	1	yes	characteristic
73	#1567	C₂×C₂₃	1	2	yes	—
74	#1561	C₂×C₂₃	1	2	yes	—
75	#1309	C₂×C₂₃	1	2	yes	—
76	C₂²×C₂₃	—	2	1	no	—
77	C₂²×C₂₃	—	2	2	no	—
78	C₂²×C₂₃	#380	1	1	yes	characteristic
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	4	4	no	—
82	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
83	C₄×C₂₃	#413	1	1	yes	characteristic
84	C₄×C₂₃	#472	1	2	yes	—
85	C₄×C₂₃	#473	1	2	yes	—
86	#533	C₂²×C₂₃	1	1	yes	characteristic
87	#446	—	2	2	no	—
88	C₈×Q₈	—	2	2	no	—
89	#424	C₄×C₂₃	1	2	yes	—
90	#423	C₂²×C₂₃	1	4	yes	—
91	#423	C₂²×C₂₃	1	1	yes	characteristic
92	#405	C₄×C₂₃	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	#335	—	2	2	no	—
95	#335	—	2	2	no	—
96	#332	—	2	4	no	—
97	C₂×C₂₃	—	4	1	no	—
98	C₂×C₂₃	#1801	1	1	yes	characteristic
99	C₂×C₂₃	#1816	1	2	yes	—
100	C₂×M₁₆	—	2	4	no	—
101	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
102	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
105	#125	Q₈×C₂₃	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	#117	C₂×C₄×C₂₃	1	2	yes	—
109	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	4	no	—
114	#97	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	D₄×C₂₃	1	2	yes	—
117	C₄×C₈	D₄×C₂₃	1	2	yes	—
118	C₂₃	#8639	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×Q₈	—	2	2	no	—
120	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	#41×C₂₃	1	2	yes	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	8	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
130	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂×D₄×C₂₃	1	2	yes	—
137	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
138	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
139	C₂³	C₂₃×#117	1	1	yes	characteristic
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₄	C₂₃×#121	1	2	yes	—
150	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	—	8	4	no	—
155	C₈	C₂₃×#101	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	C₂₃×#380	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
163	C₄	C₂₃×#413	1	1	yes	characteristic
164	C₄	C₂₃×#472	1	2	yes	—
165	C₄	C₂₃×#473	1	2	yes	—
166	C₂	—	4	1	no	—
167	C₂	C₂₃×#1801	1	1	yes	characteristic
168	C₂	C₂₃×#1816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2960

C₂

yes

maximal, characteristic

C₂₃×#1567

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×#335

—

C₂₃×#335

—

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

#8639

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#117

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×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

#1567

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

—

C₄×C₂₃

#473

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

#1816

yes

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

102

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

103

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

104

C₄⋊Q₈

D₄×C₂₃

yes

—

105

#125

Q₈×C₂₃

yes

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

C₂×C₄×C₂₃

yes

—

109

C₂×C₄²

C₂³×C₂₃

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

#97

—

115

C₄×C₈

—

116

C₄×C₈

D₄×C₂₃

yes

—

117

C₄×C₈

D₄×C₂₃

yes

—

118

C₂₃

#8639

yes

characteristic, direct product factor, Sylow

119

C₂×Q₈

—

120

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

121

C₂²×C₄

#41×C₂₃

yes

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

129

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

130

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₄⋊C₄×C₂₃

yes

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×D₄×C₂₃

yes

—

137

C₄²

C₂×D₄×C₂₃

yes

characteristic

138

C₄²

C₂×Q₈×C₂₃

yes

characteristic

139

C₂³

C₂₃×#117

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

148

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

149

C₂×C₄

C₂₃×#121

yes

—

150

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

C₂₃×#101

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂₃×#380

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₂×C₂₃×#101

yes

characteristic

163

C₄

C₂₃×#413

yes

characteristic

164

C₄

C₂₃×#472

yes

—

165

C₄

C₂₃×#473

yes

—

166

C₂

—

167

C₂

C₂₃×#1801

yes

characteristic

168

C₂

C₂₃×#1816

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. #8639	C₂₃	direct (N × H)
118. C₂₃	#8639	direct (N × H)

H ≅ G/N

type

54. #8639

C₂₃

direct (N × H)

118. C₂₃

#8639

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×#1567	2
C₂₃×#1561	2
C₂₃×#1309	2
#8639	23

subgroup

index

C₂₃×#2960

C₂₃×#1567

C₂₃×#1561

C₂₃×#1309

#8639

Subgroups — C₂₃×#8639 #19156265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8639 #19156265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8639 #19156265