Group #15669586 — subgroups

Subgroups — C₂₃×#1967 #15669586

Order 2944.

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

normal subgroups only

This group has 312 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×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#101	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#100	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, 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	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#124	—	2	2	no	—
11	C₂₃×#124	—	2	1	no	—
12	C₂₃×#124	—	2	1	no	—
13	C₂×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₈	—	2	2	no	—
16	C₂₃×#101	—	2	4	no	—
17	C₂₃×#100	—	2	4	no	—
18	C₂₃×#97	—	2	1	no	—
19	C₄×C₈×C₂₃	—	2	1	no	—
20	C₂⁴×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₂₃	—	2	2	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×Q₈×C₂₃	—	2	1	no	—
24	C₂×Q₈×C₂₃	—	2	2	no	—
25	C₂×D₄×C₂₃	—	2	2	no	—
26	C₂×D₄×C₂₃	—	2	1	no	—
27	C₂×D₄×C₂₃	—	2	1	no	—
28	C₂×D₄×C₂₃	—	2	2	no	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₃	—	2	2	no	—
33	M₁₆×C₂₃	—	2	2	no	—
34	M₁₆×C₂₃	—	2	2	no	—
35	M₁₆×C₂₃	D₄	1	4	yes	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	2	no	—
38	C₂²⋊C₄×C₂₃	—	2	4	no	—
39	C₂²⋊C₄×C₂₃	—	2	4	no	—
40	C₄²×C₂₃	D₄	1	2	yes	—
41	C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	—	2	4	no	—
45	C₂³×C₂₃	—	4	4	no	—
46	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	4	2	no	—
48	Q₈×C₂₃	—	4	1	no	—
49	Q₈×C₂₃	—	4	1	no	—
50	D₄×C₂₃	—	4	2	no	—
51	D₄×C₂₃	—	4	1	no	—
52	D₄×C₂₃	—	4	1	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	4	2	no	—
56	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#41	1	2	yes	—
62	C₈×C₂₃	—	2	4	no	—
63	C₈×C₂₃	—	4	2	no	—
64	#1967	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	4	2	no	—
67	C₂²×C₂₃	—	4	4	no	—
68	C₂²×C₂₃	—	8	4	no	—
69	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₂₃	C₄×D₄	1	2	yes	—
71	C₄×C₂₃	—	2	2	no	—
72	C₄×C₂₃	—	2	2	no	—
73	C₄×C₂₃	—	4	2	no	—
74	C₄×C₂₃	#121	1	2	yes	—
75	C₄×C₂₃	#124	1	1	yes	characteristic
76	C₄×C₂₃	#127	1	1	yes	characteristic
77	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
78	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
79	C₂×#101	C₂×C₂₃	1	2	yes	complemented
80	C₂×#100	C₂×C₂₃	1	2	yes	—
81	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
82	C₂×C₂₃	—	2	2	no	—
83	C₂×C₂₃	—	8	2	no	—
84	C₂×C₂₃	#390	1	1	yes	characteristic
85	C₂×C₂₃	#413	1	2	yes	—
86	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
87	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
88	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
89	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
90	#124	—	2	1	no	—
91	#124	—	2	1	no	—
92	#124	—	2	2	no	—
93	C₂×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
94	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	#101	—	2	4	no	—
97	#100	—	2	4	no	—
98	#97	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₂₃	#1967	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁴	C₂×C₄×C₂₃	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
111	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
112	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	D₄×C₂₃	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	D₄×C₂₃	1	1	yes	characteristic
122	C₄²	D₄×C₂₃	1	1	yes	characteristic
123	C₄²	D₄×C₂₃	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
128	Q₈	—	4	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
138	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
141	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₂×C₄	#41×C₂₃	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	4	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
150	C₂²	C₄×D₄×C₂₃	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	C₂₃×#121	1	2	yes	—
155	C₄	C₂₃×#124	1	1	yes	characteristic
156	C₄	C₂₃×#127	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	C₂₃×#390	1	1	yes	characteristic
160	C₂	C₂₃×#413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#124

C₂

yes

maximal, characteristic

C₂₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#101

C₂

yes

maximal, complemented

C₂×C₂₃×#100

C₂

yes

maximal

C₄×M₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂₃×#124

—

C₂₃×#124

—

C₂₃×#124

—

C₂×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×#101

—

C₂₃×#100

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1967

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×C₂₃

yes

characteristic

#423

C₂×C₂₃

yes

characteristic, complemented

C₂×#101

C₂×C₂₃

yes

complemented

C₂×#100

C₂×C₂₃

yes

—

C₄×M₁₆

C₂×C₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#413

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×D₄

C₂²×C₂₃

yes

characteristic

C₂×M₁₆

C₂²×C₂₃

yes

—

C₂×M₁₆

C₂²×C₂₃

yes

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

C₄×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₄⋊C₈

—

#101

—

#100

—

#97

—

C₄×C₈

—

100

C₂₃

#1967

yes

characteristic, direct product factor, Sylow

101

C₂⁴

C₂×C₄×C₂₃

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×C₄×C₂₃

yes

—

111

C₂²×C₄

C₂×C₄×C₂₃

yes

—

112

C₂²×C₄

C₂³×C₂₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

D₄×C₂₃

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

D₄×C₂₃

yes

characteristic

122

C₄²

D₄×C₂₃

yes

characteristic

123

C₄²

D₄×C₂₃

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×C₄×C₂₃

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

138

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

139

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

140

C₂×C₄

C₂×D₄×C₂₃

yes

—

141

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

142

C₂×C₄

#41×C₂₃

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

150

C₂²

C₄×D₄×C₂₃

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

C₂₃×#121

yes

—

155

C₄

C₂₃×#124

yes

characteristic

156

C₄

C₂₃×#127

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

C₂₃×#390

yes

characteristic

160

C₂

C₂₃×#413

yes

—

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

subgroup

G/H

center, socle

C₂²×C₂₃

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

prime radical

C₂²×D₄×C₂₃

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#423	C₂	semidirect
3. C₂×C₂₃×#101	C₂	semidirect
5. C₄×M₁₆×C₂₃	C₂	semidirect
64. #1967	C₂₃	direct (N × H)
78. #423	C₂×C₂₃	semidirect
79. C₂×#101	C₂×C₂₃	semidirect
81. C₄×M₁₆	C₂×C₂₃	semidirect
100. C₂₃	#1967	direct (N × H)

H ≅ G/N

type

2. C₂₃×#423

C₂

semidirect

3. C₂×C₂₃×#101

C₂

semidirect

5. C₄×M₁₆×C₂₃

C₂

semidirect

64. #1967

C₂₃

direct (N × H)

78. #423

C₂×C₂₃

semidirect

79. C₂×#101

C₂×C₂₃

semidirect

81. C₄×M₁₆

C₂×C₂₃

semidirect

100. C₂₃

#1967

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×#423	2
C₂×C₂₃×#101	2
C₂×C₂₃×#100	2
C₄×M₁₆×C₂₃	2
#1967	23

subgroup

index

C₂×C₂₃×#124

C₂₃×#423

C₂×C₂₃×#101

C₂×C₂₃×#100

C₄×M₁₆×C₂₃

#1967

Subgroups — C₂₃×#1967 #15669586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1967 #15669586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1967 #15669586