Group #19156384 — subgroups

Subgroups — C₂₃×#8758 #19156384

Order 5888.

394 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2313	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1463	C₂	1	2	yes	maximal
3	C₂₃×#1305	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#407	C₂²	1	4	yes	—
6	C₂₃×#387	—	2	4	no	—
7	C₂₃×#381	C₄	1	2	yes	—
8	C₂₃×#380	C₂²	1	2	yes	—
9	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
10	C₂₃×#324	—	2	4	no	—
11	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₂₃	—	2	2	no	—
20	C₂₃×C₂²⋊C₈	—	2	4	no	—
21	C₂₃×C₂²⋊C₈	—	2	4	no	—
22	C₂₃×#95	—	2	2	no	—
23	C₂₃×#95	—	2	4	no	—
24	C₂₃×#95	—	2	2	no	—
25	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	—	2	2	no	—
28	C₂²×C₄×C₂₃	—	2	4	no	—
29	C₂²×C₄×C₂₃	—	4	2	no	—
30	C₂²×C₄×C₂₃	—	8	1	no	—
31	C₂²×C₄×C₂₃	—	8	1	no	—
32	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₂₃	#41	1	4	yes	—
36	M₁₆×C₂₃	—	8	4	no	—
37	C₂×C₈×C₂₃	—	4	4	no	—
38	C₄⋊C₄×C₂₃	—	2	8	no	—
39	C₄⋊C₄×C₂₃	—	8	2	no	—
40	C₂²⋊C₄×C₂₃	—	4	4	no	—
41	C₄²×C₂₃	—	8	2	no	—
42	#8758	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	—	4	1	no	—
45	C₂³×C₂₃	—	4	1	no	—
46	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂³×C₂₃	#117	1	2	yes	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	—	4	4	no	—
53	C₂×C₄×C₂₃	—	4	4	no	—
54	C₂×C₄×C₂₃	—	4	2	no	—
55	C₂×C₄×C₂₃	—	4	4	no	—
56	C₂×C₄×C₂₃	—	4	4	no	—
57	C₂×C₄×C₂₃	—	4	4	no	—
58	C₂×C₄×C₂₃	—	8	2	no	—
59	C₂×C₄×C₂₃	#123	1	4	yes	—
60	C₈×C₂₃	—	8	4	no	—
61	#2313	C₂×C₂₃	1	1	yes	characteristic
62	#1463	C₂×C₂₃	1	2	yes	—
63	#1305	C₂×C₂₃	1	4	yes	—
64	C₂²×C₂₃	—	2	1	no	—
65	C₂²×C₂₃	—	2	1	no	—
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	—	2	2	no	—
68	C₂²×C₂₃	—	2	2	no	—
69	C₂²×C₂₃	—	4	4	no	—
70	C₂²×C₂₃	#381	1	1	yes	characteristic
71	C₂²×C₂₃	#413	1	2	yes	—
72	C₄×C₂₃	—	2	4	no	—
73	C₄×C₂₃	—	4	2	no	—
74	C₄×C₂₃	—	4	2	no	—
75	C₄×C₂₃	—	8	4	no	—
76	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
77	#407	C₂²×C₂₃	1	4	yes	—
78	#387	—	2	4	no	—
79	#381	C₄×C₂₃	1	2	yes	—
80	#380	C₂²×C₂₃	1	2	yes	—
81	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
82	#324	—	2	4	no	—
83	C₂×C₂₃	—	2	2	no	—
84	C₂×C₂₃	—	4	2	no	—
85	C₂×C₂₃	#1822	1	1	yes	characteristic
86	C₂×C₂₃	#1828	1	1	yes	characteristic
87	C₂×C₂₃	#1830	1	1	yes	characteristic
88	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
90	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
91	C₂×M₁₆	—	2	4	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
95	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂²⋊C₈	—	2	4	no	—
98	C₂²⋊C₈	—	2	4	no	—
99	#95	—	2	2	no	—
100	#95	—	2	4	no	—
101	#95	—	2	2	no	—
102	C₂₃	#8758	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
111	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
112	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
113	C₂²×C₄	#41×C₂₃	1	4	yes	—
114	M₁₆	—	8	4	no	—
115	C₂×C₈	—	4	4	no	—
116	C₄⋊C₄	—	2	8	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₄²	—	8	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	4	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
124	C₂³	C₂₃×#117	1	2	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	C₂₃×#123	1	4	yes	—
137	C₈	—	8	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	4	no	—
144	C₂²	C₂₃×#381	1	1	yes	characteristic
145	C₂²	C₂₃×#413	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	4	no	—
150	C₂	—	2	2	no	—
151	C₂	—	4	2	no	—
152	C₂	C₂₃×#1822	1	1	yes	characteristic
153	C₂	C₂₃×#1828	1	1	yes	characteristic
154	C₂	C₂₃×#1830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2313

C₂

yes

maximal, characteristic

C₂₃×#1463

C₂

yes

maximal

C₂₃×#1305

C₂

yes

maximal

C₂²×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂₃×#407

C₂²

yes

—

C₂₃×#387

—

C₂₃×#381

C₄

yes

—

C₂₃×#380

C₂²

yes

—

C₄×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂₃×#324

—

C₂³×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂³

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

—

C₂₃×C₂²⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#95

—

C₂₃×#95

—

C₂₃×#95

—

C₂⁴×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

#41

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

#8758

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₂₃

#117

yes

—

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

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#123

yes

—

C₈×C₂₃

—

#2313

C₂×C₂₃

yes

characteristic

#1463

C₂×C₂₃

yes

—

#1305

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#381

yes

characteristic

C₂²×C₂₃

#413

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂²×C₄⋊C₄

C₂²×C₂₃

yes

characteristic

#407

C₂²×C₂₃

yes

—

#387

—

#381

C₄×C₂₃

yes

—

#380

C₂²×C₂₃

yes

—

C₄×C₂²⋊C₄

C₄×C₂₃

yes

—

#324

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂³×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂³×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂³×C₄

C₂³×C₂₃

yes

characteristic

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂×C₄×C₂₃

yes

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#95

—

100

#95

—

101

#95

—

102

C₂₃

#8758

yes

characteristic, direct product factor, Sylow

103

C₂⁴

C₂²×C₄×C₂₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

111

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

112

C₂²×C₄

C₂×D₄×C₂₃

yes

—

113

C₂²×C₄

#41×C₂₃

yes

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

124

C₂³

C₂₃×#117

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂₃×#123

yes

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂₃×#381

yes

characteristic

145

C₂²

C₂₃×#413

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

C₂₃×#1822

yes

characteristic

153

C₂

C₂₃×#1828

yes

characteristic

154

C₂

C₂₃×#1830

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#381
commutator	C₂⁴	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#381

commutator

C₂⁴

C₂²×C₄×C₂₃

Frattini, cosocle

C₂³×C₄

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
42. #8758	C₂₃	direct (N × H)
102. C₂₃	#8758	direct (N × H)

H ≅ G/N

type

42. #8758

C₂₃

direct (N × H)

102. C₂₃

#8758

direct (N × H)

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

subgroup	index
C₂₃×#2313	2
C₂₃×#1463	2
C₂₃×#1305	2
#8758	23

subgroup

index

C₂₃×#2313

C₂₃×#1463

C₂₃×#1305

#8758

Subgroups — C₂₃×#8758 #19156384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8758 #19156384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8758 #19156384