Group #19158261 — subgroups

Subgroups — C₂₃×#10635 #19158261

Order 5888.

294 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1838	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1546	C₂	1	2	yes	maximal
4	C₂₃×#1286	C₂	1	2	yes	maximal
5	C₂₃×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₂²	1	2	yes	—
8	C₂₃×#423	C₂²	1	2	yes	—
9	C₂×C₂₃×#103	—	2	2	no	—
10	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
11	C₂₃×#382	C₂²	1	1	yes	characteristic
12	C₂₃×#341	—	2	2	no	—
13	C₂₃×#332	—	2	4	no	—
14	C₂²×Q₈×C₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	D₄	1	2	yes	—
17	C₂₃×C₄⋊Q₈	—	2	2	no	—
18	C₂₃×C₄⋊Q₈	—	2	1	no	—
19	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
22	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×C₄⋊C₈	—	2	2	no	—
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×#103	—	4	4	no	—
27	C₂₃×C₂²⋊C₈	—	2	2	no	—
28	C₂₃×#97	—	2	2	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₂₃×#95	—	2	2	no	—
31	C₂×Q₈×C₂₃	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	4	1	no	—
33	C₂×Q₈×C₂₃	—	4	2	no	—
34	C₂×Q₈×C₂₃	—	4	2	no	—
35	C₂²×C₄×C₂₃	—	4	1	no	—
36	C₂²×C₄×C₂₃	—	4	1	no	—
37	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	#41	1	2	yes	—
39	M₁₆×C₂₃	—	4	2	no	—
40	M₁₆×C₂₃	—	4	2	no	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	4	2	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₂₃	#41	1	1	yes	characteristic
51	C₄²×C₂₃	#41	1	2	yes	—
52	#10635	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	#124	1	1	yes	characteristic
54	Q₈×C₂₃	—	4	4	no	—
55	Q₈×C₂₃	—	8	2	no	—
56	C₂×C₄×C₂₃	—	4	2	no	—
57	C₂×C₄×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	—	8	1	no	—
59	C₂×C₄×C₂₃	—	8	1	no	—
60	C₂×C₄×C₂₃	—	8	2	no	—
61	C₂×C₄×C₂₃	—	8	1	no	—
62	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#123	1	2	yes	—
64	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#126	1	2	yes	—
66	C₂×C₄×C₂₃	#135	1	2	yes	—
67	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₈×C₂₃	—	4	2	no	—
69	C₈×C₂₃	—	8	2	no	—
70	C₈×C₂₃	—	8	2	no	—
71	#1960	C₂×C₂₃	1	1	yes	characteristic
72	#1838	C₂×C₂₃	1	1	yes	characteristic
73	#1546	C₂×C₂₃	1	2	yes	—
74	#1286	C₂×C₂₃	1	2	yes	—
75	#1281	C₂×C₂₃	1	1	yes	characteristic
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	#397	1	1	yes	characteristic
78	C₂²×C₂₃	#488	1	1	yes	characteristic
79	C₂²×C₂₃	#490	1	1	yes	characteristic
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	2	2	no	—
82	C₄×C₂₃	—	8	2	no	—
83	C₄×C₂₃	—	16	1	no	—
84	C₄×C₂₃	#463	1	2	yes	—
85	C₄×C₂₃	#482	1	2	yes	—
86	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
87	#431	C₂²×C₂₃	1	2	yes	—
88	#423	C₂²×C₂₃	1	2	yes	—
89	C₂×#103	—	2	2	no	—
90	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
91	#382	C₂²×C₂₃	1	1	yes	characteristic
92	#341	—	2	2	no	—
93	#332	—	2	4	no	—
94	C₂×C₂₃	—	2	1	no	—
95	C₂×C₂₃	—	2	1	no	—
96	C₂×C₂₃	#2073	1	1	yes	characteristic
97	C₂×C₂₃	#2082	1	2	yes	—
98	C₂²×Q₈	—	2	1	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂²×C₈	D₄×C₂₃	1	2	yes	—
101	C₄⋊Q₈	—	2	2	no	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	#103	—	4	4	no	—
111	C₂²⋊C₈	—	2	2	no	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	#95	—	2	2	no	—
115	C₂₃	#10635	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
123	C₂²×C₄	#41×C₂₃	1	2	yes	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
135	C₄²	#41×C₂₃	1	1	yes	characteristic
136	C₄²	#41×C₂₃	1	2	yes	—
137	C₂³	C₂₃×#124	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
147	C₂×C₄	C₂₃×#123	1	2	yes	—
148	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
149	C₂×C₄	C₂₃×#126	1	2	yes	—
150	C₂×C₄	C₂₃×#135	1	2	yes	—
151	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂₃×#397	1	1	yes	characteristic
157	C₂²	C₂₃×#488	1	1	yes	characteristic
158	C₂²	C₂₃×#490	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	C₂₃×#463	1	2	yes	—
164	C₄	C₂₃×#482	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	C₂₃×#2073	1	1	yes	characteristic
168	C₂	C₂₃×#2082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1960

C₂

yes

maximal, characteristic

C₂₃×#1838

C₂

yes

maximal, characteristic

C₂₃×#1546

C₂

yes

maximal

C₂₃×#1286

C₂

yes

maximal

C₂₃×#1281

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#431

C₂²

yes

—

C₂₃×#423

C₂²

yes

—

C₂×C₂₃×#103

—

C₂×C₂₃×#97

C₂²

yes

characteristic

C₂₃×#382

C₂²

yes

characteristic

C₂₃×#341

—

C₂₃×#332

—

C₂²×Q₈×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×#103

—

C₂₃×C₂²⋊C₈

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×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

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

C₄²×C₂₃

yes

C₄²×C₂₃

yes

—

#10635

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×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#121

yes

characteristic

C₂×C₄×C₂₃

#123

yes

—

C₂×C₄×C₂₃

#123

yes

characteristic

C₂×C₄×C₂₃

#126

yes

—

C₂×C₄×C₂₃

#135

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1960

C₂×C₂₃

yes

characteristic

#1838

C₂×C₂₃

yes

characteristic

#1546

C₂×C₂₃

yes

—

#1286

C₂×C₂₃

yes

—

#1281

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#463

yes

—

C₄×C₂₃

#482

yes

—

C₂×C₄⋊Q₈

C₂²×C₂₃

yes

characteristic

#431

C₂²×C₂₃

yes

—

#423

C₂²×C₂₃

yes

—

C₂×#103

—

C₂×#97

C₂²×C₂₃

yes

characteristic

#382

C₂²×C₂₃

yes

characteristic

#341

—

#332

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2073

yes

characteristic

C₂×C₂₃

#2082

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

100

C₂²×C₈

D₄×C₂₃

yes

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

D₄×C₂₃

yes

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂³×C₂₃

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#103

—

111

C₂²⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

#95

—

115

C₂₃

#10635

yes

characteristic, direct product factor, Sylow

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

123

C₂²×C₄

#41×C₂₃

yes

—

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₄×C₂₃

yes

characteristic

135

C₄²

#41×C₂₃

yes

characteristic

136

C₄²

#41×C₂₃

yes

—

137

C₂³

C₂₃×#124

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂₃×#121

yes

characteristic

147

C₂×C₄

C₂₃×#123

yes

—

148

C₂×C₄

C₂₃×#123

yes

characteristic

149

C₂×C₄

C₂₃×#126

yes

—

150

C₂×C₄

C₂₃×#135

yes

—

151

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₂²

—

156

C₂²

C₂₃×#397

yes

characteristic

157

C₂²

C₂₃×#488

yes

characteristic

158

C₂²

C₂₃×#490

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₂₃×#463

yes

—

164

C₄

C₂₃×#482

yes

—

165

C₂

—

166

C₂

—

167

C₂

C₂₃×#2073

yes

characteristic

168

C₂

C₂₃×#2082

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂³×C₂₃

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
52. #10635	C₂₃	direct (N × H)
115. C₂₃	#10635	direct (N × H)

H ≅ G/N

type

52. #10635

C₂₃

direct (N × H)

115. C₂₃

#10635

direct (N × H)

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

subgroup	index
C₂₃×#1960	2
C₂₃×#1838	2
C₂₃×#1546	2
C₂₃×#1286	2
C₂₃×#1281	2
#10635	23

subgroup

index

C₂₃×#1960

C₂₃×#1838

C₂₃×#1546

C₂₃×#1286

C₂₃×#1281

#10635

Subgroups — C₂₃×#10635 #19158261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10635 #19158261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10635 #19158261