Group #19156257 — subgroups

Subgroups — C₂₃×#8631 #19156257

Order 5888.

320 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 130 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×#1568	C₂	1	2	yes	maximal
3	C₂₃×#1537	C₂	1	2	yes	maximal
4	C₂₃×#1307	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×#432	C₂²	1	4	yes	—
9	C₂₃×#424	C₄	1	2	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×#334	—	2	2	no	—
15	C₂₃×#330	—	2	4	no	—
16	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
20	C₂₃×#125	D₄	1	2	yes	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₂₃	—	2	2	no	—
23	C₂₃×#117	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₂₃×C₄⋊C₈	—	2	4	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	4	no	—
28	C₂₃×C₂²⋊C₈	—	2	4	no	—
29	C₂₃×#97	—	2	2	no	—
30	C₂₃×#97	—	2	4	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	C₂×C₈×C₂₃	—	2	2	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	4	1	no	—
41	C₂×C₈×C₂₃	—	4	4	no	—
42	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
48	C₂²⋊C₄×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	—	2	2	no	—
50	C₄²×C₂₃	C₂×D₄	1	2	yes	—
51	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	#8631	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	#117	1	1	yes	characteristic
55	Q₈×C₂₃	—	4	4	no	—
56	C₂×C₄×C₂₃	—	2	1	no	—
57	C₂×C₄×C₂₃	—	2	1	no	—
58	C₂×C₄×C₂₃	—	2	4	no	—
59	C₂×C₄×C₂₃	—	4	2	no	—
60	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#121	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	2	no	—
66	C₈×C₂₃	—	4	2	no	—
67	C₈×C₂₃	—	8	4	no	—
68	C₈×C₂₃	#101	1	2	yes	—
69	#2960	C₂×C₂₃	1	1	yes	characteristic
70	#1568	C₂×C₂₃	1	2	yes	—
71	#1537	C₂×C₂₃	1	2	yes	—
72	#1307	C₂×C₂₃	1	2	yes	—
73	C₂²×C₂₃	—	2	1	no	—
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	#380	1	1	yes	characteristic
76	C₄×C₂₃	—	2	1	no	—
77	C₄×C₂₃	—	2	2	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	#533	C₂²×C₂₃	1	1	yes	characteristic
83	#446	—	2	2	no	—
84	C₈×Q₈	—	2	2	no	—
85	#432	C₂²×C₂₃	1	4	yes	—
86	#424	C₄×C₂₃	1	2	yes	—
87	#423	C₂²×C₂₃	1	1	yes	characteristic
88	#405	C₄×C₂₃	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
90	#335	—	2	2	no	—
91	#334	—	2	2	no	—
92	#330	—	2	4	no	—
93	C₂×C₂₃	—	4	1	no	—
94	C₂×C₂₃	#1801	1	1	yes	characteristic
95	C₂×C₂₃	#1814	1	2	yes	—
96	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
97	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
99	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
100	#125	D₄×C₂₃	1	2	yes	—
101	C₂²⋊Q₈	—	2	2	no	—
102	C₄×Q₈	—	2	2	no	—
103	#117	C₂×C₄×C₂₃	1	2	yes	—
104	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
105	C₄⋊C₈	—	2	4	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	4	no	—
108	C₂²⋊C₈	—	2	4	no	—
109	#97	—	2	2	no	—
110	#97	—	2	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	D₄×C₂₃	1	2	yes	—
113	C₄×C₈	D₄×C₂₃	1	2	yes	—
114	C₂₃	#8631	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×Q₈	—	2	2	no	—
116	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₄	#41×C₂₃	1	2	yes	—
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	4	no	—
123	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
125	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	C₂×D₄×C₂₃	1	2	yes	—
132	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
133	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
134	C₂³	C₂₃×#117	1	1	yes	characteristic
135	Q₈	—	4	4	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₄	C₂₃×#121	1	2	yes	—
143	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	—	4	2	no	—
147	C₈	—	8	4	no	—
148	C₈	C₂₃×#101	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	C₂₃×#380	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
157	C₄	C₂₃×#413	1	1	yes	characteristic
158	C₂	—	4	1	no	—
159	C₂	C₂₃×#1801	1	1	yes	characteristic
160	C₂	C₂₃×#1814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2960

C₂

yes

maximal, characteristic

C₂₃×#1568

C₂

yes

maximal

C₂₃×#1537

C₂

yes

maximal

C₂₃×#1307

C₂

yes

maximal

C₂₃×#533

C₂²

yes

characteristic

C₂₃×#446

—

C₈×Q₈×C₂₃

—

C₂₃×#432

C₂²

yes

—

C₂₃×#424

C₄

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#405

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#335

—

C₂₃×#334

—

C₂₃×#330

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂₃×C₄⋊Q₈

Q₈

yes

—

C₂₃×#125

D₄

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×C₂²⋊C₈

—

C₂₃×#97

—

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

—

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

#8631

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

#1568

C₂×C₂₃

yes

—

#1537

C₂×C₂₃

yes

—

#1307

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#380

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂²×C₂₃

yes

characteristic

#446

—

C₈×Q₈

—

#432

C₂²×C₂₃

yes

—

#424

C₄×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

#1814

yes

—

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₄⋊Q₈

Q₈×C₂₃

yes

—

100

#125

D₄×C₂₃

yes

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

C₂×C₄×C₂₃

yes

—

104

C₂×C₄²

C₂³×C₂₃

yes

characteristic

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

#97

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

D₄×C₂₃

yes

—

113

C₄×C₈

D₄×C₂₃

yes

—

114

C₂₃

#8631

yes

characteristic, direct product factor, Sylow

115

C₂×Q₈

—

116

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

117

C₂²×C₄

#41×C₂₃

yes

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

124

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

125

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₄⋊C₄×C₂₃

yes

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

C₂×D₄×C₂₃

yes

—

132

C₄²

C₂×D₄×C₂₃

yes

characteristic

133

C₄²

C₂×Q₈×C₂₃

yes

characteristic

134

C₂³

C₂₃×#117

yes

characteristic

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

141

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

142

C₂×C₄

C₂₃×#121

yes

—

143

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₂₃×#101

yes

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂₃×#380

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×C₂₃×#101

yes

characteristic

157

C₄

C₂₃×#413

yes

characteristic

158

C₂

—

159

C₂

C₂₃×#1801

yes

characteristic

160

C₂

C₂₃×#1814

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
53. #8631	C₂₃	direct (N × H)
114. C₂₃	#8631	direct (N × H)

H ≅ G/N

type

53. #8631

C₂₃

direct (N × H)

114. C₂₃

#8631

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×#1568	2
C₂₃×#1537	2
C₂₃×#1307	2
#8631	23

subgroup

index

C₂₃×#2960

C₂₃×#1568

C₂₃×#1537

C₂₃×#1307

#8631

Subgroups — C₂₃×#8631 #19156257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8631 #19156257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8631 #19156257