Group #15669507 — subgroups

Subgroups — C₂₃×#1888 #15669507

Order 2944.

344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 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×#124	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#418	C₂	1	2	yes	maximal
4	C₂×C₂₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#339	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#124	—	2	2	no	—
10	C₂₃×#124	C₄	1	4	yes	complemented
11	C₄×Q₈×C₂₃	—	2	4	no	—
12	C₂₃×#117	—	2	2	no	—
13	C₂₃×#117	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#102	—	2	4	no	—
17	C₂₃×#101	—	2	2	no	—
18	C₂⁴×C₂₃	—	2	1	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×Q₈×C₂₃	D₄	1	2	yes	—
23	C₂×D₄×C₂₃	—	2	2	no	—
24	C₂×D₄×C₂₃	—	2	1	no	—
25	C₂×D₄×C₂₃	—	2	2	no	—
26	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄×C₂₃	D₄	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₂₃	—	2	2	no	—
31	M₁₆×C₂₃	—	2	2	no	—
32	C₂×C₈×C₂₃	—	2	2	no	—
33	C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
35	C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₂²⋊C₄×C₂₃	—	2	8	no	—
37	C₄²×C₂₃	—	2	4	no	—
38	C₄²×C₂₃	—	2	1	no	—
39	C₄²×C₂₃	—	2	2	no	—
40	C₄²×C₂₃	C₂×C₄	1	2	yes	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	—	4	4	no	—
43	C₂³×C₂₃	—	4	1	no	—
44	C₂³×C₂₃	—	4	1	no	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₂₃	—	2	4	no	—
47	Q₈×C₂₃	—	2	4	no	—
48	D₄×C₂₃	—	2	4	no	—
49	D₄×C₂₃	—	4	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	4	no	—
54	C₂×C₄×C₂₃	—	4	2	no	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	#41	1	2	yes	—
60	C₈×C₂₃	—	4	2	no	—
61	#1888	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	4	4	no	—
64	C₂²×C₂₃	—	4	2	no	—
65	C₂²×C₂₃	—	8	2	no	—
66	C₂²×C₂₃	—	8	2	no	—
67	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₃	#120	1	1	yes	characteristic
69	C₂²×C₂₃	#123	1	1	yes	characteristic
70	C₄×C₂₃	—	2	4	no	—
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	—	4	4	no	—
73	C₄×C₂₃	C₄×D₄	1	2	yes	—
74	C₄×C₂₃	#121	1	2	yes	—
75	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
76	#519	C₂×C₂₃	1	1	yes	characteristic, complemented
77	#418	C₂×C₂₃	1	2	yes	—
78	C₂×#101	C₂×C₂₃	1	1	yes	characteristic, complemented
79	#339	C₂×C₂₃	1	2	yes	complemented
80	C₂×C₂₃	—	2	1	no	—
81	C₂×C₂₃	—	2	1	no	—
82	C₂×C₂₃	—	8	2	no	—
83	C₂×C₂₃	#386	1	1	yes	characteristic
84	C₂×C₂₃	#454	1	2	yes	—
85	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
86	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
87	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
88	#124	—	2	2	no	—
89	#124	C₄×C₂₃	1	4	yes	complemented
90	C₄×Q₈	—	2	4	no	—
91	#117	—	2	2	no	—
92	#117	C₂²×C₂₃	1	2	yes	—
93	C₂×C₂²⋊C₄	—	2	2	no	—
94	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
95	#102	—	2	4	no	—
96	#101	—	2	2	no	—
97	C₂₃	#1888	1	1	yes	characteristic, direct product factor, Sylow
98	C₂⁴	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
102	C₂×Q₈	D₄×C₂₃	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	D₄×C₂₃	1	2	yes	—
109	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	2	no	—
120	C₄²	C₂×C₄×C₂₃	1	2	yes	—
121	C₂³	—	2	2	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	2	4	no	—
128	D₄	—	2	4	no	—
129	D₄	—	4	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
136	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
139	C₂×C₄	#41×C₂₃	1	2	yes	—
140	C₈	—	4	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	2	no	—
146	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂²	C₂₃×#120	1	1	yes	characteristic
148	C₂²	C₂₃×#123	1	1	yes	characteristic
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	4	no	—
152	C₄	C₄×D₄×C₂₃	1	2	yes	—
153	C₄	C₂₃×#121	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	8	2	no	—
157	C₂	C₂₃×#386	1	1	yes	characteristic
158	C₂	C₂₃×#454	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#124

C₂

yes

maximal, characteristic

C₂₃×#519

C₂

yes

maximal, characteristic, complemented

C₂₃×#418

C₂

yes

maximal

C₂×C₂₃×#101

C₂

yes

maximal, characteristic, complemented

C₂₃×#339

C₂

yes

maximal, complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂₃×#124

—

C₂₃×#124

C₄

yes

complemented

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂₃×#117

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#102

—

C₂₃×#101

—

C₂⁴×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

#1888

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

C₂×#124

C₂×C₂₃

yes

characteristic

#519

C₂×C₂₃

yes

characteristic, complemented

#418

C₂×C₂₃

yes

—

C₂×#101

C₂×C₂₃

yes

characteristic, complemented

#339

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#386

yes

characteristic

C₂×C₂₃

#454

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×D₄

C₂²×C₂₃

yes

characteristic

C₂×M₁₆

C₂²×C₂₃

yes

—

#124

—

#124

C₄×C₂₃

yes

complemented

C₄×Q₈

—

#117

—

#117

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#102

—

#101

—

C₂₃

#1888

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×C₄×C₂₃

yes

—

102

C₂×Q₈

D₄×C₂₃

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂×C₄×C₂₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×C₂₃

yes

—

109

C₂²×C₄

C₂³×C₂₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

D₄×C₂₃

yes

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×C₄×C₂₃

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×D₄×C₂₃

yes

characteristic

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

136

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

137

C₂×C₄

C₂×D₄×C₂₃

yes

—

138

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

139

C₂×C₄

#41×C₂₃

yes

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

147

C₂²

C₂₃×#120

yes

characteristic

148

C₂²

C₂₃×#123

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₄×D₄×C₂₃

yes

—

153

C₄

C₂₃×#121

yes

—

154

C₂

—

155

C₂

—

156

C₂

—

157

C₂

C₂₃×#386

yes

characteristic

158

C₂

C₂₃×#454

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#120
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#120

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

prime radical

C₂²×D₄×C₂₃

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#519	C₂	semidirect
4. C₂×C₂₃×#101	C₂	semidirect
5. C₂₃×#339	C₂	semidirect
10. C₂₃×#124	C₄	semidirect
61. #1888	C₂₃	direct (N × H)
76. #519	C₂×C₂₃	semidirect
78. C₂×#101	C₂×C₂₃	semidirect
79. #339	C₂×C₂₃	semidirect
89. #124	C₄×C₂₃	semidirect
97. C₂₃	#1888	direct (N × H)

H ≅ G/N

type

2. C₂₃×#519

C₂

semidirect

4. C₂×C₂₃×#101

C₂

semidirect

5. C₂₃×#339

C₂

semidirect

10. C₂₃×#124

C₄

semidirect

61. #1888

C₂₃

direct (N × H)

76. #519

C₂×C₂₃

semidirect

78. C₂×#101

C₂×C₂₃

semidirect

79. #339

C₂×C₂₃

semidirect

89. #124

C₄×C₂₃

semidirect

97. C₂₃

#1888

direct (N × H)

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

subgroup	index
C₂×C₂₃×#124	2
C₂₃×#519	2
C₂₃×#418	2
C₂×C₂₃×#101	2
C₂₃×#339	2
#1888	23

subgroup

index

C₂×C₂₃×#124

C₂₃×#519

C₂₃×#418

C₂×C₂₃×#101

C₂₃×#339

#1888

Subgroups — C₂₃×#1888 #15669507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1888 #15669507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1888 #15669507