Group #19157481 — subgroups

Subgroups — C₂₃×#9855 #19157481

Order 5888.

422 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 174 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×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1493	C₂	1	4	yes	maximal
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#533	C₂²	1	2	yes	—
6	C₂₃×#500	C₄	1	2	yes	—
7	C₂₃×#499	—	2	2	no	—
8	C₂₃×#498	C₄	1	2	yes	—
9	C₂×C₂₃×#108	—	2	2	no	—
10	C₂₃×#427	—	2	2	no	—
11	C₂₃×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×Q₈⋊C₈	—	2	8	no	—
15	C₈²×C₂₃	—	2	2	no	—
16	C₂×M₁₆×C₂₃	—	2	2	no	—
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	D₄	1	2	yes	—
19	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
20	C₂₃×#125	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	—	2	4	no	—
22	C₄×Q₈×C₂₃	—	2	4	no	—
23	C₂₃×#117	—	2	2	no	—
24	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	C₂₃×#108	—	4	4	no	—
26	C₂₃×#107	—	2	4	no	—
27	C₂₃×#106	—	2	4	no	—
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	D₄	1	4	yes	—
33	C₄×C₈×C₂₃	D₄	1	2	yes	—
34	C₂×Q₈×C₂₃	—	2	4	no	—
35	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	#41	1	2	yes	—
37	M₁₆×C₂₃	—	4	4	no	—
38	C₂×C₈×C₂₃	—	2	8	no	—
39	C₂×C₈×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	4	2	no	—
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
44	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₄⋊C₄×C₂₃	—	4	4	no	—
48	C₂²⋊C₄×C₂₃	—	4	2	no	—
49	C₄²×C₂₃	—	4	2	no	—
50	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₂₃	C₂×D₄	1	2	yes	—
53	#9855	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	#124	1	1	yes	characteristic
55	Q₈×C₂₃	—	4	8	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	4	4	no	—
58	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	#121	1	2	yes	—
61	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
64	C₈×C₂₃	—	2	8	no	—
65	C₈×C₂₃	—	8	2	no	—
66	C₈×C₂₃	#103	1	4	yes	—
67	C₈×C₂₃	#103	1	4	yes	—
68	#3160	C₂×C₂₃	1	1	yes	characteristic
69	#1848	C₂×C₂₃	1	1	yes	characteristic
70	#1493	C₂×C₂₃	1	4	yes	—
71	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	#390	1	1	yes	characteristic
74	C₂²×C₂₃	#487	1	2	yes	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	—	8	4	no	—
77	C₄×C₂₃	C₂×#103	1	2	yes	—
78	C₄×C₂₃	#443	1	4	yes	—
79	C₄×C₂₃	#492	1	1	yes	characteristic
80	C₄×C₂₃	#494	1	1	yes	characteristic
81	#533	C₂²×C₂₃	1	2	yes	—
82	#500	C₄×C₂₃	1	2	yes	—
83	#499	—	2	2	no	—
84	#498	C₄×C₂₃	1	2	yes	—
85	C₂×#108	—	2	2	no	—
86	#427	—	2	2	no	—
87	#423	C₂²×C₂₃	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
89	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
90	Q₈⋊C₈	—	2	8	no	—
91	C₈²	—	2	2	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	#1956	1	1	yes	characteristic
94	C₂×C₂₃	#1974	1	2	yes	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	D₄×C₂₃	1	2	yes	—
98	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
99	#125	—	2	2	no	—
100	C₂²⋊Q₈	—	2	4	no	—
101	C₄×Q₈	—	2	4	no	—
102	#117	—	2	2	no	—
103	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
104	#108	—	4	4	no	—
105	#107	—	2	4	no	—
106	#106	—	2	4	no	—
107	C₄⋊C₈	—	2	4	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₄×C₈	D₄×C₂₃	1	4	yes	—
112	C₄×C₈	D₄×C₂₃	1	2	yes	—
113	C₂₃	#9855	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₄	#41×C₂₃	1	2	yes	—
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	2	8	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
122	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
123	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
124	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
132	C₄²	C₂×D₄×C₂₃	1	2	yes	—
133	C₂³	C₂₃×#124	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
139	C₂×C₄	C₂₃×#121	1	2	yes	—
140	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
141	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
142	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
143	C₈	—	2	8	no	—
144	C₈	—	8	2	no	—
145	C₈	C₂₃×#103	1	4	yes	—
146	C₈	C₂₃×#103	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂₃×#390	1	1	yes	characteristic
149	C₂²	C₂₃×#487	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×C₂₃×#103	1	2	yes	—
153	C₄	C₂₃×#443	1	4	yes	—
154	C₄	C₂₃×#492	1	1	yes	characteristic
155	C₄	C₂₃×#494	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₂₃×#1956	1	1	yes	characteristic
158	C₂	C₂₃×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#3160

C₂

yes

maximal, characteristic

C₂₃×#1848

C₂

yes

maximal, characteristic

C₂₃×#1493

C₂

yes

maximal

C₂×C₈²×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#533

C₂²

yes

—

C₂₃×#500

C₄

yes

—

C₂₃×#499

—

C₂₃×#498

C₄

yes

—

C₂×C₂₃×#108

—

C₂₃×#427

—

C₂₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×Q₈⋊C₈

—

C₈²×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂₃×#125

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#108

—

C₂₃×#107

—

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂×Q₈×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

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

Q₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

—

#9855

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#124

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄×D₄

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

#127

yes

characteristic

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

C₂×Q₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#103

yes

—

C₈×C₂₃

#103

yes

—

#3160

C₂×C₂₃

yes

characteristic

#1848

C₂×C₂₃

yes

characteristic

#1493

C₂×C₂₃

yes

—

C₂×C₈²

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#390

yes

characteristic

C₂²×C₂₃

#487

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#103

yes

—

C₄×C₂₃

#443

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂²×C₂₃

yes

—

#500

C₄×C₂₃

yes

—

#499

—

#498

C₄×C₂₃

yes

—

C₂×#108

—

#427

—

#423

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂×C₂₃

—

C₂×C₂₃

#1956

yes

characteristic

C₂×C₂₃

#1974

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄×C₂₃

yes

—

C₄⋊Q₈

C₂×C₄×C₂₃

yes

—

#125

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

C₂³×C₂₃

yes

characteristic

104

#108

—

105

#107

—

106

#106

—

107

C₄⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

110

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

111

C₄×C₈

D₄×C₂₃

yes

—

112

C₄×C₈

D₄×C₂₃

yes

—

113

C₂₃

#9855

yes

characteristic, direct product factor, Sylow

114

C₂×Q₈

—

115

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

116

C₂²×C₄

#41×C₂₃

yes

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

122

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

123

C₂×C₈

SD₁₆×C₂₃

yes

—

124

C₂×C₈

Q₁₆×C₂₃

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

C₂²×C₄×C₂₃

yes

characteristic

131

C₄²

C₂×D₄×C₂₃

yes

characteristic

132

C₄²

C₂×D₄×C₂₃

yes

—

133

C₂³

C₂₃×#124

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

138

C₂×C₄

C₄×D₄×C₂₃

yes

—

139

C₂×C₄

C₂₃×#121

yes

—

140

C₂×C₄

C₂₃×#127

yes

characteristic

141

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

142

C₂×C₄

C₂×Q₁₆×C₂₃

yes

—

143

C₈

—

144

C₈

—

145

C₈

C₂₃×#103

yes

—

146

C₈

C₂₃×#103

yes

—

147

C₂²

—

148

C₂²

C₂₃×#390

yes

characteristic

149

C₂²

C₂₃×#487

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×C₂₃×#103

yes

—

153

C₄

C₂₃×#443

yes

—

154

C₄

C₂₃×#492

yes

characteristic

155

C₄

C₂₃×#494

yes

characteristic

156

C₂

—

157

C₂

C₂₃×#1956

yes

characteristic

158

C₂

C₂₃×#1974

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂³×C₂₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

#127

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

C₂²×C₂₃

C₂³×C₂₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
53. #9855	C₂₃	direct (N × H)
113. C₂₃	#9855	direct (N × H)

H ≅ G/N

type

53. #9855

C₂₃

direct (N × H)

113. C₂₃

#9855

direct (N × H)

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

subgroup	index
C₂₃×#3160	2
C₂₃×#1848	2
C₂₃×#1493	2
C₂×C₈²×C₂₃	2
#9855	23

subgroup

index

C₂₃×#3160

C₂₃×#1848

C₂₃×#1493

C₂×C₈²×C₂₃

#9855

Subgroups — C₂₃×#9855 #19157481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#9855 #19157481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9855 #19157481