Group #19161169 — subgroups

Subgroups — C₂₃×#13543 #19161169

Order 5888.

268 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1945	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1386	C₂	1	2	yes	maximal
4	C₂₃×#1383	C₂	1	2	yes	maximal
5	C₂₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
8	C₂₃×#427	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#106	—	2	1	no	—
10	C₂×C₂₃×#106	C₂²	1	2	yes	—
11	C₂₃×#424	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#384	—	2	1	no	—
14	C₂₃×#339	—	2	1	no	—
15	C₂×M₁₆×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₂₃×#117	—	2	1	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
21	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	2	no	—
23	C₂₃×M₃₂	—	2	2	no	—
24	C₂×C₁₆×C₂₃	—	2	2	no	—
25	C₂₃×#108	—	2	2	no	—
26	C₂₃×#107	Q₈	1	2	yes	—
27	C₂₃×#106	—	2	1	no	—
28	C₂₃×#106	—	2	4	no	—
29	C₂₃×#106	—	4	2	no	—
30	C₂₃×C₄⋊C₈	—	4	2	no	—
31	C₄×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	—	4	2	no	—
34	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₂₃	—	4	1	no	—
36	M₁₆×C₂₃	—	4	1	no	—
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	4	1	no	—
40	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₂₃	#41	1	2	yes	—
43	C₂×C₈×C₂₃	#41	1	2	yes	—
44	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₄⋊C₄×C₂₃	—	4	2	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	8	4	no	—
49	C₂²⋊C₄×C₂₃	—	4	1	no	—
50	C₄²×C₂₃	—	2	1	no	—
51	C₄²×C₂₃	—	4	1	no	—
52	C₄²×C₂₃	Q₁₆	1	2	yes	—
53	C₁₆×C₂₃	—	4	2	no	—
54	#13543	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	4	2	no	—
59	C₂×C₄×C₂₃	—	8	4	no	—
60	C₂×C₄×C₂₃	—	8	2	no	—
61	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₈×C₂₃	—	2	2	no	—
68	C₈×C₂₃	—	8	1	no	—
69	C₈×C₂₃	#125	1	2	yes	—
70	C₈×C₂₃	#126	1	2	yes	—
71	#1945	C₂×C₂₃	1	1	yes	characteristic
72	#1844	C₂×C₂₃	1	1	yes	characteristic
73	#1386	C₂×C₂₃	1	2	yes	—
74	#1383	C₂×C₂₃	1	2	yes	—
75	#1376	C₂×C₂₃	1	1	yes	characteristic
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	#476	1	1	yes	characteristic
78	C₂²×C₂₃	#484	1	1	yes	characteristic
79	C₂²×C₂₃	#490	1	1	yes	characteristic
80	C₄×C₂₃	—	2	4	no	—
81	C₄×C₂₃	—	8	2	no	—
82	C₄×C₂₃	—	16	2	no	—
83	C₄×C₂₃	#400	1	1	yes	characteristic
84	C₄×C₂₃	#477	1	1	yes	characteristic
85	C₄×C₂₃	#481	1	1	yes	characteristic
86	C₄×C₂₃	#487	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
88	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
89	#427	C₂²×C₂₃	1	1	yes	characteristic
90	C₂×#106	—	2	1	no	—
91	C₂×#106	C₂²×C₂₃	1	2	yes	—
92	#424	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	#384	—	2	1	no	—
95	#339	—	2	1	no	—
96	C₂×C₂₃	—	2	1	no	—
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	#2100	1	1	yes	characteristic
99	C₂×C₂₃	#2230	1	2	yes	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
103	#117	—	2	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	4	2	no	—
106	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	#108	—	2	2	no	—
111	#107	Q₈×C₂₃	1	2	yes	—
112	#106	—	2	1	no	—
113	#106	—	2	4	no	—
114	#106	—	4	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₂₃	#13543	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
127	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
128	C₂×C₈	#41×C₂₃	1	2	yes	—
129	C₂×C₈	#41×C₂₃	1	2	yes	—
130	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	8	4	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	Q₁₆×C₂₃	1	2	yes	—
139	C₁₆	—	4	2	no	—
140	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
148	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
149	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
150	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	C₂₃×#125	1	2	yes	—
155	C₈	C₂₃×#126	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂₃×#476	1	1	yes	characteristic
158	C₂²	C₂₃×#484	1	1	yes	characteristic
159	C₂²	C₂₃×#490	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	2	no	—
163	C₄	C₂₃×#400	1	1	yes	characteristic
164	C₄	C₂₃×#477	1	1	yes	characteristic
165	C₄	C₂₃×#481	1	1	yes	characteristic
166	C₄	C₂₃×#487	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	C₂₃×#2100	1	1	yes	characteristic
170	C₂	C₂₃×#2230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1945

C₂

yes

maximal, characteristic

C₂₃×#1844

C₂

yes

maximal, characteristic

C₂₃×#1386

C₂

yes

maximal

C₂₃×#1383

C₂

yes

maximal

C₂₃×#1376

C₂

yes

maximal, characteristic

C₂×C₂₃×M₃₂

C₂²

yes

—

C₂×C₂₃×#108

C₂²

yes

characteristic

C₂₃×#427

C₂²

yes

characteristic

C₂×C₂₃×#106

—

C₂×C₂₃×#106

C₂²

yes

—

C₂₃×#424

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#384

—

C₂₃×#339

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂₃×#117

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

D₄

yes

characteristic

C₂₃×M₃₂

—

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂₃×#108

—

C₂₃×#107

Q₈

yes

—

C₂₃×#106

—

C₂₃×#106

—

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₂×C₈×C₂₃

#41

yes

—

C₂×C₈×C₂₃

#41

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

Q₁₆

yes

—

C₁₆×C₂₃

—

#13543

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#125

yes

—

C₈×C₂₃

#126

yes

—

#1945

C₂×C₂₃

yes

characteristic

#1844

C₂×C₂₃

yes

characteristic

#1386

C₂×C₂₃

yes

—

#1383

C₂×C₂₃

yes

—

#1376

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×M₃₂

C₂²×C₂₃

yes

—

C₂×#108

C₂²×C₂₃

yes

characteristic

#427

C₂²×C₂₃

yes

characteristic

C₂×#106

—

C₂×#106

C₂²×C₂₃

yes

—

#424

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#384

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2100

yes

characteristic

C₂×C₂₃

#2230

yes

—

100

C₂×M₁₆

—

101

C₂²×C₈

D₄×C₂₃

yes

characteristic

102

C₂²×C₈

C₂³×C₂₃

yes

characteristic

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

D₄×C₂₃

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

#108

—

111

#107

Q₈×C₂₃

yes

—

112

#106

—

113

#106

—

114

#106

—

115

C₄⋊C₈

—

116

C₄×C₈

—

117

C₂₃

#13543

yes

characteristic, direct product factor, Sylow

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

SD₁₆×C₂₃

yes

—

127

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

128

C₂×C₈

#41×C₂₃

yes

—

129

C₂×C₈

#41×C₂₃

yes

—

130

C₂×C₈

#41×C₂₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

Q₁₆×C₂₃

yes

—

139

C₁₆

—

140

C₂³

C₂₃×AGL(1,ℤ/8)

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

characteristic

147

C₂×C₄

C₂₃×#123

yes

characteristic

148

C₂×C₄

C₂₃×#124

yes

characteristic

149

C₂×C₄

C₂×SD₁₆×C₂₃

yes

characteristic

150

C₂×C₄

C₂×Q₁₆×C₂₃

yes

characteristic

151

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

C₂₃×#125

yes

—

155

C₈

C₂₃×#126

yes

—

156

C₂²

—

157

C₂²

C₂₃×#476

yes

characteristic

158

C₂²

C₂₃×#484

yes

characteristic

159

C₂²

C₂₃×#490

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₂₃×#400

yes

characteristic

164

C₄

C₂₃×#477

yes

characteristic

165

C₄

C₂₃×#481

yes

characteristic

166

C₄

C₂₃×#487

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

C₂₃×#2100

yes

characteristic

170

C₂

C₂₃×#2230

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

prime radical

C₂³×C₂₃

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
54. #13543	C₂₃	direct (N × H)
117. C₂₃	#13543	direct (N × H)

H ≅ G/N

type

54. #13543

C₂₃

direct (N × H)

117. C₂₃

#13543

direct (N × H)

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

subgroup	index
C₂₃×#1945	2
C₂₃×#1844	2
C₂₃×#1386	2
C₂₃×#1383	2
C₂₃×#1376	2
#13543	23

subgroup

index

C₂₃×#1945

C₂₃×#1844

C₂₃×#1386

C₂₃×#1383

C₂₃×#1376

#13543

Subgroups — C₂₃×#13543 #19161169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#13543 #19161169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13543 #19161169