Group #19160464 — subgroups

Subgroups — C₂₃×#12838 #19160464

Order 5888.

374 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#368	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1810	C₂	1	2	yes	maximal
4	C₂₃×#1381	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₂₃×M₃₂	—	2	4	no	—
7	C₂×C₂₃×M₃₂	C₄	1	4	yes	—
8	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#108	C₂²	1	4	yes	—
11	C₂₃×#407	—	2	2	no	—
12	C₂₃×#368	—	2	4	no	—
13	C₂₃×#341	—	2	2	no	—
14	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₂₃	—	2	4	no	—
16	C₂²×C₈×C₂₃	—	2	2	no	—
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
21	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	8	no	—
23	C₂×C₁₆×C₂₃	—	2	2	no	—
24	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
27	C₂₃×#108	—	2	4	no	—
28	C₂₃×#108	—	2	4	no	—
29	C₂₃×C₂²⋊C₈	—	4	4	no	—
30	C₂⁴×C₂₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	D₈	1	2	yes	—
35	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₂₃	—	4	4	no	—
38	M₁₆×C₂₃	—	4	4	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	4	4	no	—
44	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₂₃	#41	1	2	yes	—
49	C₁₆×C₂₃	—	2	2	no	—
50	C₁₆×C₂₃	C₂²⋊C₄	1	4	yes	—
51	#12838	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	—	2	2	no	—
53	C₂³×C₂₃	—	4	2	no	—
54	C₂³×C₂₃	#107	1	2	yes	—
55	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	4	no	—
60	C₂×C₄×C₂₃	#107	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#121	1	2	yes	—
63	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#135	1	2	yes	—
65	C₈×C₂₃	—	2	2	no	—
66	C₈×C₂₃	—	8	4	no	—
67	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₂₃	#117	1	1	yes	characteristic
69	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
70	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
71	C₂×#368	C₂×C₂₃	1	2	yes	complemented
72	#1810	C₂×C₂₃	1	2	yes	—
73	#1381	C₂×C₂₃	1	2	yes	complemented
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	2	4	no	—
77	C₂²×C₂₃	—	4	4	no	—
78	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₂₃	#467	1	2	yes	—
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	#380	1	1	yes	characteristic
82	C₄×C₂₃	#427	1	1	yes	characteristic
83	C₄×C₂₃	#466	1	2	yes	—
84	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
85	C₂×M₃₂	—	2	4	no	—
86	C₂×M₃₂	C₄×C₂₃	1	4	yes	—
87	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
88	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
89	C₂×#108	C₂²×C₂₃	1	4	yes	—
90	#407	—	2	2	no	—
91	#368	—	2	4	no	—
92	#341	—	2	2	no	—
93	C₂×C₂₃	—	2	2	no	—
94	C₂×C₂₃	—	4	2	no	—
95	C₂×C₂₃	#1808	1	1	yes	characteristic
96	C₂×C₂₃	#2161	1	2	yes	—
97	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂²×C₈	—	2	2	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
102	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
104	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
105	M₃₂	—	2	8	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
108	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
109	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
110	#108	—	2	4	no	—
111	#108	—	2	4	no	—
112	C₂²⋊C₈	—	4	4	no	—
113	C₂₃	#12838	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	Q₁₆×C₂₃	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
118	C₂²×C₄	D₈×C₂₃	1	2	yes	—
119	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
121	M₁₆	—	4	4	no	—
122	M₁₆	—	4	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
131	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
132	C₂×C₈	#41×C₂₃	1	2	yes	—
133	C₁₆	—	2	2	no	—
134	C₁₆	C₂²⋊C₄×C₂₃	1	4	yes	—
135	C₂³	—	2	2	no	—
136	C₂³	—	4	2	no	—
137	C₂³	C₂₃×#107	1	2	yes	—
138	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	C₂₃×#107	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#121	1	2	yes	—
146	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
147	C₂×C₄	C₂₃×#135	1	2	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	8	4	no	—
150	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
151	C₈	C₂₃×#117	1	1	yes	characteristic
152	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
158	C₂²	C₂₃×#467	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	C₂₃×#380	1	1	yes	characteristic
161	C₄	C₂₃×#427	1	1	yes	characteristic
162	C₄	C₂₃×#466	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂₃×#1808	1	1	yes	characteristic
166	C₂	C₂₃×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂₃×M₃₂

C₂

yes

maximal, characteristic

C₂×C₂₃×#368

C₂

yes

maximal, complemented

C₂₃×#1810

C₂

yes

maximal

C₂₃×#1381

C₂

yes

maximal, complemented

C₂³×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×M₃₂

—

C₂×C₂₃×M₃₂

C₄

yes

—

C₂²×C₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₁₆×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×#108

C₂²

yes

—

C₂₃×#407

—

C₂₃×#368

—

C₂₃×#341

—

C₂³×C₄×C₂₃

D₄

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

Q₈

yes

—

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₁₆×C₂₃

D₄

yes

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×C₂²⋊C₈

—

C₂⁴×C₂₃

Q₁₆

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

Q₁₆

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₁₆×C₂₃

—

C₁₆×C₂₃

C₂²⋊C₄

yes

—

#12838

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

#107

yes

—

C₂³×C₂₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#107

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

C₂×D₈

yes

characteristic

C₂×C₄×C₂₃

#135

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₂₃

#117

yes

characteristic

C₈×C₂₃

C₂²⋊Q₈

yes

—

C₂²×M₃₂

C₂×C₂₃

yes

characteristic

C₂×#368

C₂×C₂₃

yes

complemented

#1810

C₂×C₂₃

yes

—

#1381

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×#107

yes

characteristic

C₂²×C₂₃

#467

yes

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂³×C₈

C₂²×C₂₃

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₄×C₂₃

yes

—

C₂²×C₁₆

C₂²×C₂₃

yes

characteristic

C₂²×C₁₆

C₂²×C₂₃

yes

characteristic

C₂×#108

C₂²×C₂₃

yes

—

#407

—

#368

—

#341

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1808

yes

characteristic

C₂×C₂₃

#2161

yes

—

C₂³×C₄

D₄×C₂₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×C₄×C₂₃

yes

—

102

C₂²×C₈

D₄×C₂₃

yes

characteristic

103

C₂²×C₈

Q₈×C₂₃

yes

—

104

C₂²×C₈

C₂³×C₂₃

yes

characteristic

105

M₃₂

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₂×C₄×C₂₃

yes

—

108

C₂×C₁₆

C₂×C₄×C₂₃

yes

—

109

C₂×C₁₆

D₄×C₂₃

yes

—

110

#108

—

111

#108

—

112

C₂²⋊C₈

—

113

C₂₃

#12838

yes

characteristic, direct product factor, Sylow

114

C₂⁴

Q₁₆×C₂₃

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

118

C₂²×C₄

D₈×C₂₃

yes

—

119

C₂²×C₄

Q₁₆×C₂₃

yes

characteristic

120

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

129

C₂×C₈

C₂²×C₄×C₂₃

yes

characteristic

130

C₂×C₈

C₂×D₄×C₂₃

yes

—

131

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

132

C₂×C₈

#41×C₂₃

yes

—

133

C₁₆

—

134

C₁₆

C₂²⋊C₄×C₂₃

yes

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂₃×#107

yes

—

138

C₂³

C₂×Q₁₆×C₂₃

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂₃×#107

yes

—

144

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

145

C₂×C₄

C₂₃×#121

yes

—

146

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

147

C₂×C₄

C₂₃×#135

yes

—

148

C₈

—

149

C₈

—

150

C₈

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

151

C₈

C₂₃×#117

yes

characteristic

152

C₈

C₂₃×C₂²⋊Q₈

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂₃×#107

yes

characteristic

158

C₂²

C₂₃×#467

yes

—

159

C₄

—

160

C₄

C₂₃×#380

yes

characteristic

161

C₄

C₂₃×#427

yes

characteristic

162

C₄

C₂₃×#466

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂₃×#1808

yes

characteristic

166

C₂

C₂₃×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂⁴×C₂₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

socle

C₂²×C₂₃

C₂×#107

prime radical

C₂⁴×C₂₃

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₂×C₂₃×#368	C₂	semidirect
4. C₂₃×#1381	C₂	semidirect
51. #12838	C₂₃	direct (N × H)
71. C₂×#368	C₂×C₂₃	semidirect
73. #1381	C₂×C₂₃	semidirect
113. C₂₃	#12838	direct (N × H)

H ≅ G/N

type

2. C₂×C₂₃×#368

C₂

semidirect

4. C₂₃×#1381

C₂

semidirect

51. #12838

C₂₃

direct (N × H)

71. C₂×#368

C₂×C₂₃

semidirect

73. #1381

C₂×C₂₃

semidirect

113. C₂₃

#12838

direct (N × H)

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

subgroup	index
C₂²×C₂₃×M₃₂	2
C₂×C₂₃×#368	2
C₂₃×#1810	2
C₂₃×#1381	2
#12838	23

subgroup

index

C₂²×C₂₃×M₃₂

C₂×C₂₃×#368

C₂₃×#1810

C₂₃×#1381

#12838

Subgroups — C₂₃×#12838 #19160464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12838 #19160464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12838 #19160464