Group #19157402 — subgroups

Subgroups — C₂₃×#9776 #19157402

Order 5888.

390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1549	C₂	1	2	yes	maximal
4	C₂₃×#1548	C₂	1	2	yes	maximal
5	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#498	—	2	2	no	—
8	C₂₃×#498	C₄	1	4	yes	—
9	C₂₃×#431	C₂²	1	2	yes	—
10	C₂×C₂₃×#106	—	2	2	no	—
11	C₂₃×#423	C₂²	1	1	yes	characteristic
12	C₂₃×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#341	—	2	2	no	—
15	C₂₃×#333	—	2	4	no	—
16	C₂₃×#332	—	2	4	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂²×Q₈×C₂₃	—	2	2	no	—
19	C₂×M₁₆×C₂₃	—	2	1	no	—
20	C₂×M₁₆×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	D₄	1	2	yes	—
22	C₂₃×C₄⋊Q₈	—	2	2	no	—
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#106	—	2	8	no	—
30	C₂₃×C₄⋊C₈	—	2	2	no	—
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
33	C₂₃×C₂²⋊C₈	—	2	2	no	—
34	C₄×C₈×C₂₃	—	2	2	no	—
35	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₂₃	D₄	1	2	yes	—
37	C₂×Q₈×C₂₃	—	2	4	no	—
38	C₂×Q₈×C₂₃	—	4	2	no	—
39	C₂×Q₈×C₂₃	—	8	4	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	#41	1	2	yes	—
43	M₁₆×C₂₃	—	4	2	no	—
44	M₁₆×C₂₃	—	4	2	no	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	4	no	—
48	C₂×C₈×C₂₃	—	4	1	no	—
49	C₂×C₈×C₂₃	—	4	1	no	—
50	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₂₃	C₂×D₄	1	2	yes	—
58	#9776	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	#124	1	1	yes	characteristic
60	Q₈×C₂₃	—	8	4	no	—
61	Q₈×C₂₃	—	8	4	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	4	4	no	—
64	C₂×C₄×C₂₃	—	8	4	no	—
65	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₂₃	#121	1	2	yes	—
68	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#137	1	2	yes	—
70	C₂×C₄×C₂₃	#137	1	2	yes	—
71	C₈×C₂₃	—	2	4	no	—
72	C₈×C₂₃	—	4	4	no	—
73	C₈×C₂₃	—	8	1	no	—
74	C₈×C₂₃	—	8	1	no	—
75	C₂×#498	C₂×C₂₃	1	1	yes	characteristic
76	#1837	C₂×C₂₃	1	1	yes	characteristic
77	#1549	C₂×C₂₃	1	2	yes	—
78	#1548	C₂×C₂₃	1	2	yes	—
79	#1452	C₂×C₂₃	1	1	yes	characteristic
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	#390	1	1	yes	characteristic
82	C₂²×C₂₃	#491	1	2	yes	—
83	C₄×C₂₃	—	2	2	no	—
84	C₄×C₂₃	—	8	4	no	—
85	C₄×C₂₃	#419	1	2	yes	—
86	C₄×C₂₃	#471	1	4	yes	—
87	C₄×C₂₃	#497	1	1	yes	characteristic
88	C₄×C₂₃	#497	1	1	yes	characteristic
89	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
90	#498	—	2	2	no	—
91	#498	C₄×C₂₃	1	4	yes	—
92	#431	C₂²×C₂₃	1	2	yes	—
93	C₂×#106	—	2	2	no	—
94	#423	C₂²×C₂₃	1	1	yes	characteristic
95	#423	C₂²×C₂₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
97	#341	—	2	2	no	—
98	#333	—	2	4	no	—
99	#332	—	2	4	no	—
100	#322	—	2	2	no	—
101	C₂×C₂₃	—	2	2	no	—
102	C₂×C₂₃	#1960	1	1	yes	characteristic
103	C₂×C₂₃	#1977	1	2	yes	—
104	C₂²×Q₈	—	2	2	no	—
105	C₂×M₁₆	—	2	1	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	D₄×C₂₃	1	2	yes	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
115	#106	—	2	8	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
122	C₄×C₈	D₄×C₂₃	1	2	yes	—
123	C₂₃	#9776	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	8	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₄	#41×C₂₃	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
143	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
144	C₄²	C₂×D₄×C₂₃	1	2	yes	—
145	C₂³	C₂₃×#124	1	1	yes	characteristic
146	Q₈	—	8	4	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
153	C₂×C₄	C₂₃×#121	1	2	yes	—
154	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#137	1	2	yes	—
156	C₂×C₄	C₂₃×#137	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	C₂₃×#390	1	1	yes	characteristic
163	C₂²	C₂₃×#491	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	8	4	no	—
166	C₄	C₂₃×#419	1	2	yes	—
167	C₄	C₂₃×#471	1	4	yes	—
168	C₄	C₂₃×#497	1	1	yes	characteristic
169	C₄	C₂₃×#497	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	C₂₃×#1960	1	1	yes	characteristic
172	C₂	C₂₃×#1977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#498

C₂

yes

maximal, characteristic

C₂₃×#1837

C₂

yes

maximal, characteristic

C₂₃×#1549

C₂

yes

maximal

C₂₃×#1548

C₂

yes

maximal

C₂₃×#1452

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#498

—

C₂₃×#498

C₄

yes

—

C₂₃×#431

C₂²

yes

—

C₂×C₂₃×#106

—

C₂₃×#423

C₂²

yes

characteristic

C₂₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#341

—

C₂₃×#333

—

C₂₃×#332

—

C₂₃×#322

—

C₂²×Q₈×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

D₄

yes

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

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

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

—

#9776

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#124

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄×D₄

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

#137

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂×#498

C₂×C₂₃

yes

characteristic

#1837

C₂×C₂₃

yes

characteristic

#1549

C₂×C₂₃

yes

—

#1548

C₂×C₂₃

yes

—

#1452

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#390

yes

characteristic

C₂²×C₂₃

#491

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#419

yes

—

C₄×C₂₃

#471

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×C₄⋊Q₈

C₂²×C₂₃

yes

—

#498

—

#498

C₄×C₂₃

yes

—

#431

C₂²×C₂₃

yes

—

C₂×#106

—

#423

C₂²×C₂₃

yes

characteristic

#423

C₂²×C₂₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₂₃

yes

—

—

—

100

#322

—

101

C₂×C₂₃

—

102

C₂×C₂₃

#1960

yes

characteristic

103

C₂×C₂₃

#1977

yes

—

104

C₂²×Q₈

—

105

C₂×M₁₆

—

106

C₂×M₁₆

—

107

C₂²×C₈

D₄×C₂₃

yes

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

C₂×C₄×C₂₃

yes

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄²

C₂³×C₂₃

yes

characteristic

115

#106

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

D₄×C₂₃

yes

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₂×C₄×C₂₃

yes

—

122

C₄×C₈

D₄×C₂₃

yes

—

123

C₂₃

#9776

yes

characteristic, direct product factor, Sylow

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

129

C₂²×C₄

#41×C₂₃

yes

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

C₂²×C₄×C₂₃

yes

characteristic

143

C₄²

C₂×D₄×C₂₃

yes

characteristic

144

C₄²

C₂×D₄×C₂₃

yes

—

145

C₂³

C₂₃×#124

yes

characteristic

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

152

C₂×C₄

C₄×D₄×C₂₃

yes

—

153

C₂×C₄

C₂₃×#121

yes

—

154

C₂×C₄

C₂₃×#127

yes

characteristic

155

C₂×C₄

C₂₃×#137

yes

—

156

C₂×C₄

C₂₃×#137

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

C₂₃×#390

yes

characteristic

163

C₂²

C₂₃×#491

yes

—

164

C₄

—

165

C₄

—

166

C₄

C₂₃×#419

yes

—

167

C₄

C₂₃×#471

yes

—

168

C₄

C₂₃×#497

yes

characteristic

169

C₄

C₂₃×#497

yes

characteristic

170

C₂

—

171

C₂

C₂₃×#1960

yes

characteristic

172

C₂

C₂₃×#1977

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#390
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#390

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

C₂³×C₂₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
58. #9776	C₂₃	direct (N × H)
123. C₂₃	#9776	direct (N × H)

H ≅ G/N

type

58. #9776

C₂₃

direct (N × H)

123. C₂₃

#9776

direct (N × H)

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

subgroup	index
C₂×C₂₃×#498	2
C₂₃×#1837	2
C₂₃×#1549	2
C₂₃×#1548	2
C₂₃×#1452	2
#9776	23

subgroup

index

C₂×C₂₃×#498

C₂₃×#1837

C₂₃×#1549

C₂₃×#1548

C₂₃×#1452

#9776

Subgroups — C₂₃×#9776 #19157402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#9776 #19157402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9776 #19157402