Group #19158294 — subgroups

Subgroups — C₂₃×#10668 #19158294

Order 5888.

316 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1843	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1571	C₂	1	2	yes	maximal
4	C₂₃×#1481	C₂	1	2	yes	maximal
5	C₂₃×#1274	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#533	C₂²	1	1	yes	characteristic
7	C₂₃×#431	C₂²	1	2	yes	—
8	C₂₃×#423	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂₃×#376	C₂²	1	1	yes	characteristic
12	C₂₃×#339	—	2	2	no	—
13	C₂₃×#335	—	2	4	no	—
14	C₂₃×Q₈⋊C₈	—	2	4	no	—
15	C₂×M₁₆×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	—	2	2	no	—
17	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
18	C₂₃×#125	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₂₃	—	2	2	no	—
21	C₂₃×#117	—	2	1	no	—
22	C₂×C₄²×C₂₃	D₄	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	4	4	no	—
29	C₂₃×C₂²⋊C₈	—	2	2	no	—
30	C₄×C₈×C₂₃	D₄	1	4	yes	—
31	C₂₃×#95	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	2	no	—
34	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	#41	1	2	yes	—
36	M₁₆×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	2	4	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₂×C₈×C₂₃	—	4	1	no	—
40	C₂×C₈×C₂₃	—	4	4	no	—
41	C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₃	—	4	2	no	—
44	C₂²⋊C₄×C₂₃	—	4	1	no	—
45	C₄²×C₂₃	—	2	2	no	—
46	C₄²×C₂₃	—	4	1	no	—
47	C₄²×C₂₃	Q₁₆	1	4	yes	—
48	C₄²×C₂₃	C₂×D₄	1	2	yes	—
49	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₂₃	#41	1	1	yes	characteristic
51	#10668	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	#123	1	1	yes	characteristic
53	Q₈×C₂₃	—	4	4	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	4	2	no	—
56	C₂×C₄×C₂₃	—	4	4	no	—
57	C₂×C₄×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#123	1	2	yes	—
61	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
63	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
64	C₈×C₂₃	—	4	4	no	—
65	C₈×C₂₃	—	8	2	no	—
66	C₈×C₂₃	—	8	1	no	—
67	#1995	C₂×C₂₃	1	1	yes	characteristic
68	#1843	C₂×C₂₃	1	1	yes	characteristic
69	#1571	C₂×C₂₃	1	2	yes	—
70	#1481	C₂×C₂₃	1	2	yes	—
71	#1274	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	#399	1	1	yes	characteristic
74	C₂²×C₂₃	#484	1	2	yes	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	4	4	no	—
78	C₄×C₂₃	—	8	2	no	—
79	C₄×C₂₃	#477	1	2	yes	—
80	C₄×C₂₃	#494	1	1	yes	characteristic
81	C₄×C₂₃	#496	1	1	yes	characteristic
82	#533	C₂²×C₂₃	1	1	yes	characteristic
83	#431	C₂²×C₂₃	1	2	yes	—
84	#423	C₂²×C₂₃	1	1	yes	characteristic
85	C₂×C₄⋊C₈	—	2	2	no	—
86	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
87	#376	C₂²×C₂₃	1	1	yes	characteristic
88	#339	—	2	2	no	—
89	#335	—	2	4	no	—
90	Q₈⋊C₈	—	2	4	no	—
91	C₂×C₂₃	—	2	2	no	—
92	C₂×C₂₃	#2095	1	1	yes	characteristic
93	C₂×C₂₃	#2098	1	2	yes	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂²×C₈	—	2	2	no	—
96	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
97	#125	—	2	1	no	—
98	C₂²⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	2	2	no	—
100	#117	—	2	1	no	—
101	C₂×C₄²	D₄×C₂₃	1	2	yes	—
102	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈	D₄×C₂₃	1	4	yes	—
110	#95	—	2	2	no	—
111	C₂₃	#10668	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₄	#41×C₂₃	1	2	yes	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	Q₁₆×C₂₃	1	4	yes	—
128	C₄²	C₂×D₄×C₂₃	1	2	yes	—
129	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
130	C₄²	#41×C₂₃	1	1	yes	characteristic
131	C₂³	C₂₃×#123	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄	C₂₃×#123	1	2	yes	—
140	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
141	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
142	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₂₃×#399	1	1	yes	characteristic
148	C₂²	C₂₃×#484	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	C₂₃×#477	1	2	yes	—
154	C₄	C₂₃×#494	1	1	yes	characteristic
155	C₄	C₂₃×#496	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₂₃×#2095	1	1	yes	characteristic
158	C₂	C₂₃×#2098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1995

C₂

yes

maximal, characteristic

C₂₃×#1843

C₂

yes

maximal, characteristic

C₂₃×#1571

C₂

yes

maximal

C₂₃×#1481

C₂

yes

maximal

C₂₃×#1274

C₂

yes

maximal, characteristic

C₂₃×#533

C₂²

yes

characteristic

C₂₃×#431

C₂²

yes

—

C₂₃×#423

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊C₈

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×#376

C₂²

yes

characteristic

C₂₃×#339

—

C₂₃×#335

—

C₂₃×Q₈⋊C₈

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

—

C₂₃×C₄⋊Q₈

Q₈

yes

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

D₄

yes

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

#41

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

Q₁₆

yes

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

C₄²×C₂₃

yes

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#123

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

#123

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₁₆

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1995

C₂×C₂₃

yes

characteristic

#1843

C₂×C₂₃

yes

characteristic

#1571

C₂×C₂₃

yes

—

#1481

C₂×C₂₃

yes

—

#1274

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#399

yes

characteristic

C₂²×C₂₃

#484

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#477

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂²×C₂₃

yes

characteristic

#431

C₂²×C₂₃

yes

—

#423

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

#376

C₂²×C₂₃

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

#2095

yes

characteristic

C₂×C₂₃

#2098

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Q₈×C₂₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

100

#117

—

101

C₂×C₄²

D₄×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

C₂²⋊C₈

—

109

C₄×C₈

D₄×C₂₃

yes

—

110

#95

—

111

C₂₃

#10668

yes

characteristic, direct product factor, Sylow

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

115

C₂²×C₄

#41×C₂₃

yes

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

Q₁₆×C₂₃

yes

—

128

C₄²

C₂×D₄×C₂₃

yes

—

129

C₄²

C₂×Q₈×C₂₃

yes

characteristic

130

C₄²

#41×C₂₃

yes

characteristic

131

C₂³

C₂₃×#123

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

138

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

characteristic

139

C₂×C₄

C₂₃×#123

yes

—

140

C₂×C₄

C₂₃×#127

yes

characteristic

141

C₂×C₄

C₂×Q₁₆×C₂₃

yes

—

142

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

C₂₃×#399

yes

characteristic

148

C₂²

C₂₃×#484

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₂₃×#477

yes

—

154

C₄

C₂₃×#494

yes

characteristic

155

C₄

C₂₃×#496

yes

characteristic

156

C₂

—

157

C₂

C₂₃×#2095

yes

characteristic

158

C₂

C₂₃×#2098

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#399
prime radical	C₂³×C₂₃	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

C₂²×C₂₃

C₂³×C₂₃

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
51. #10668	C₂₃	direct (N × H)
111. C₂₃	#10668	direct (N × H)

H ≅ G/N

type

51. #10668

C₂₃

direct (N × H)

111. C₂₃

#10668

direct (N × H)

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

subgroup	index
C₂₃×#1995	2
C₂₃×#1843	2
C₂₃×#1571	2
C₂₃×#1481	2
C₂₃×#1274	2
#10668	23

subgroup

index

C₂₃×#1995

C₂₃×#1843

C₂₃×#1571

C₂₃×#1481

C₂₃×#1274

#10668

Subgroups — C₂₃×#10668 #19158294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10668 #19158294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10668 #19158294