Group #19156415 — subgroups

Subgroups — C₂₃×#8789 #19156415

Order 5888.

342 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 166 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×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1277	C₂	1	4	yes	maximal
5	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
6	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
9	C₄³×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#336	—	2	2	no	—
11	C₂₃×#336	C₄	1	2	yes	—
12	C₂₃×#336	C₄	1	2	yes	—
13	C₂²×C₈×C₂₃	—	2	2	no	—
14	C₂²×C₈×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	D₄	1	2	yes	—
19	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×C₄⋊C₈	—	2	4	no	—
22	C₂₃×C₄⋊C₈	—	2	4	no	—
23	C₂₃×C₄⋊C₈	—	4	4	no	—
24	C₄×C₈×C₂₃	—	2	2	no	—
25	C₄×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₂₃	—	2	4	no	—
32	C₂×C₈×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	—	2	4	no	—
34	C₂×C₈×C₂₃	—	4	2	no	—
35	C₂×C₈×C₂₃	—	4	2	no	—
36	C₂×C₈×C₂₃	—	4	4	no	—
37	C₂×C₈×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₄²×C₂₃	—	2	4	no	—
40	C₄²×C₂₃	—	2	2	no	—
41	C₄²×C₂₃	—	2	2	no	—
42	C₄²×C₂₃	—	2	2	no	—
43	C₄²×C₂₃	—	2	2	no	—
44	C₄²×C₂₃	#41	1	2	yes	—
45	C₄²×C₂₃	#41	1	2	yes	—
46	#8789	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	#108	1	4	yes	—
55	C₂×C₄×C₂₃	#117	1	2	yes	—
56	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#125	1	2	yes	—
59	C₂×C₄×C₂₃	#131	1	2	yes	—
60	C₂×C₄×C₂₃	#131	1	2	yes	—
61	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₂×C₄×C₂₃	#137	1	2	yes	—
63	C₈×C₂₃	—	4	4	no	—
64	C₈×C₂₃	—	8	2	no	—
65	C₈×C₂₃	—	8	2	no	—
66	#1842	C₂×C₂₃	1	1	yes	characteristic
67	#1831	C₂×C₂₃	1	1	yes	characteristic
68	#1744	C₂×C₂₃	1	1	yes	characteristic
69	#1277	C₂×C₂₃	1	4	yes	—
70	C₂²×C₂₃	#382	1	1	yes	characteristic
71	C₂²×C₂₃	#424	1	1	yes	characteristic
72	C₂²×C₂₃	#424	1	1	yes	characteristic
73	C₂²×C₂₃	#428	1	2	yes	—
74	C₂²×C₂₃	C₂×#108	1	2	yes	—
75	C₄×C₂₃	—	2	4	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	4	2	no	—
78	C₄×C₂₃	#433	1	4	yes	—
79	C₄×C₂₃	#489	1	2	yes	—
80	C₄×C₂₃	#490	1	1	yes	characteristic
81	C₄×C₂₃	#491	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	2	2	no	—
83	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
84	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
85	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
86	C₄³	C₂²×C₂₃	1	1	yes	characteristic
87	#336	—	2	2	no	—
88	#336	C₄×C₂₃	1	2	yes	—
89	#336	C₄×C₂₃	1	2	yes	—
90	C₂×C₂₃	#1573	1	2	yes	—
91	C₂×C₂₃	#1574	1	2	yes	—
92	C₂×C₂₃	#1832	1	1	yes	characteristic
93	C₂×C₂₃	#1836	1	1	yes	characteristic
94	C₂×C₂₃	#1837	1	1	yes	characteristic
95	C₂²×C₈	—	2	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
98	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₂₃	1	2	yes	—
101	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
102	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
103	C₄⋊C₈	—	2	4	no	—
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₂₃	#8789	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
110	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
111	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#41×C₂₃	1	2	yes	—
128	C₄²	#41×C₂₃	1	2	yes	—
129	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₂₃×#108	1	4	yes	—
137	C₂×C₄	C₂₃×#117	1	2	yes	—
138	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
139	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
140	C₂×C₄	C₂₃×#125	1	2	yes	—
141	C₂×C₄	C₂₃×#131	1	2	yes	—
142	C₂×C₄	C₂₃×#131	1	2	yes	—
143	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
144	C₂×C₄	C₂₃×#137	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₂²	C₂₃×#382	1	1	yes	characteristic
149	C₂²	C₂₃×#424	1	1	yes	characteristic
150	C₂²	C₂₃×#424	1	1	yes	characteristic
151	C₂²	C₂₃×#428	1	2	yes	—
152	C₂²	C₂×C₂₃×#108	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	C₂₃×#433	1	4	yes	—
157	C₄	C₂₃×#489	1	2	yes	—
158	C₄	C₂₃×#490	1	1	yes	characteristic
159	C₄	C₂₃×#491	1	1	yes	characteristic
160	C₂	C₂₃×#1573	1	2	yes	—
161	C₂	C₂₃×#1574	1	2	yes	—
162	C₂	C₂₃×#1832	1	1	yes	characteristic
163	C₂	C₂₃×#1836	1	1	yes	characteristic
164	C₂	C₂₃×#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1842

C₂

yes

maximal, characteristic

C₂₃×#1831

C₂

yes

maximal, characteristic

C₂₃×#1744

C₂

yes

maximal, characteristic

C₂₃×#1277

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#336

—

C₂₃×#336

C₄

yes

—

C₂₃×#336

C₄

yes

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

#41

yes

—

C₄²×C₂₃

#41

yes

—

#8789

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#108

yes

—

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×C₄×C₂₃

#131

yes

—

C₂×C₄×C₂₃

#131

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1842

C₂×C₂₃

yes

characteristic

#1831

C₂×C₂₃

yes

characteristic

#1744

C₂×C₂₃

yes

characteristic

#1277

C₂×C₂₃

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

C₂×#108

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#433

yes

—

C₄×C₂₃

#489

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₄³

C₂²×C₂₃

yes

characteristic

#336

—

#336

C₄×C₂₃

yes

—

#336

C₄×C₂₃

yes

—

C₂×C₂₃

#1573

yes

—

C₂×C₂₃

#1574

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

100

C₂×C₄²

D₄×C₂₃

yes

—

101

C₂×C₄²

Q₈×C₂₃

yes

—

102

C₂×C₄²

C₂³×C₂₃

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂₃

#8789

yes

characteristic, direct product factor, Sylow

109

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

110

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

111

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

112

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

113

C₂²×C₄

C₂×Q₈×C₂₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#41×C₂₃

yes

—

128

C₄²

#41×C₂₃

yes

—

129

C₂³

C₂×C₄⋊C₄×C₂₃

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂₃×#108

yes

—

137

C₂×C₄

C₂₃×#117

yes

—

138

C₂×C₄

C₂₃×#124

yes

characteristic

139

C₂×C₄

C₂₃×#124

yes

characteristic

140

C₂×C₄

C₂₃×#125

yes

—

141

C₂×C₄

C₂₃×#131

yes

—

142

C₂×C₄

C₂₃×#131

yes

—

143

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

144

C₂×C₄

C₂₃×#137

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

C₂₃×#382

yes

characteristic

149

C₂²

C₂₃×#424

yes

characteristic

150

C₂²

C₂₃×#424

yes

characteristic

151

C₂²

C₂₃×#428

yes

—

152

C₂²

C₂×C₂₃×#108

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂₃×#433

yes

—

157

C₄

C₂₃×#489

yes

—

158

C₄

C₂₃×#490

yes

characteristic

159

C₄

C₂₃×#491

yes

characteristic

160

C₂

C₂₃×#1573

yes

—

161

C₂

C₂₃×#1574

yes

—

162

C₂

C₂₃×#1832

yes

characteristic

163

C₂

C₂₃×#1836

yes

characteristic

164

C₂

C₂₃×#1837

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₃

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle, prime radical

C₂³×C₂₃

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

N	H ≅ G/N	type
46. #8789	C₂₃	direct (N × H)
108. C₂₃	#8789	direct (N × H)

H ≅ G/N

type

46. #8789

C₂₃

direct (N × H)

108. C₂₃

#8789

direct (N × H)

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

subgroup	index
C₂₃×#1842	2
C₂₃×#1831	2
C₂₃×#1744	2
C₂₃×#1277	2
#8789	23

subgroup

index

C₂₃×#1842

C₂₃×#1831

C₂₃×#1744

C₂₃×#1277

#8789

Subgroups — C₂₃×#8789 #19156415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8789 #19156415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8789 #19156415