Group #19161146 — subgroups

Subgroups — C₂₃×#13520 #19161146

Order 5888.

238 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2161	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1953	C₂	1	2	yes	maximal
4	C₂₃×#1385	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1381	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#430	—	2	2	no	—
9	C₂×C₂₃×#108	C₂²	1	2	yes	—
10	C₂×C₂₃×#108	C₂²	1	2	yes	—
11	C₂₃×#405	C₂²	1	1	yes	characteristic
12	C₂₃×#368	—	2	2	no	—
13	C₂₃×C₄⋊C₁₆	—	2	2	no	—
14	C₂₃×#344	—	2	2	no	—
15	C₂×M₁₆×C₂₃	—	2	2	no	—
16	C₂×M₁₆×C₂₃	—	2	2	no	—
17	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×#117	D₄	1	1	yes	characteristic
20	C₂₃×M₃₂	—	2	2	no	—
21	C₂×C₁₆×C₂₃	—	2	1	no	—
22	C₂×C₁₆×C₂₃	—	2	2	no	—
23	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
24	C₂₃×#108	—	2	2	no	—
25	C₂₃×#108	—	2	2	no	—
26	C₂₃×#108	—	2	2	no	—
27	C₂₃×#108	—	4	4	no	—
28	C₂₃×#108	Q₈	1	4	yes	—
29	C₂₃×#97	—	2	1	no	—
30	C₄×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₂₃	—	2	4	no	—
33	M₁₆×C₂₃	—	2	2	no	—
34	M₁₆×C₂₃	—	4	2	no	—
35	M₁₆×C₂₃	—	4	2	no	—
36	M₁₆×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₂×C₈×C₂₃	—	4	2	no	—
40	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₂₃	#41	1	2	yes	—
43	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₂²⋊C₄×C₂₃	Q₁₆	1	2	yes	—
46	C₄²×C₂₃	—	2	1	no	—
47	C₁₆×C₂₃	—	2	2	no	—
48	C₁₆×C₂₃	—	4	1	no	—
49	#13520	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
55	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
56	C₈×C₂₃	—	4	1	no	—
57	C₈×C₂₃	—	4	4	no	—
58	C₈×C₂₃	—	8	2	no	—
59	C₈×C₂₃	#123	1	1	yes	characteristic
60	C₈×C₂₃	#125	1	2	yes	—
61	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
62	#2161	C₂×C₂₃	1	1	yes	characteristic
63	#2151	C₂×C₂₃	1	1	yes	characteristic
64	#1953	C₂×C₂₃	1	2	yes	—
65	#1385	C₂×C₂₃	1	1	yes	characteristic
66	#1381	C₂×C₂₃	1	2	yes	—
67	C₂²×C₂₃	—	4	1	no	—
68	C₂²×C₂₃	#467	1	1	yes	characteristic
69	C₂²×C₂₃	#477	1	2	yes	—
70	C₄×C₂₃	—	4	2	no	—
71	C₄×C₂₃	#398	1	1	yes	characteristic
72	C₄×C₂₃	#469	1	1	yes	characteristic
73	C₄×C₂₃	#478	1	2	yes	—
74	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
75	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
76	#430	—	2	2	no	—
77	C₂×#108	C₂²×C₂₃	1	2	yes	—
78	C₂×#108	C₂²×C₂₃	1	2	yes	—
79	#405	C₂²×C₂₃	1	1	yes	characteristic
80	#368	—	2	2	no	—
81	C₄⋊C₁₆	—	2	2	no	—
82	#344	—	2	2	no	—
83	C₂×C₂₃	—	2	1	no	—
84	C₂×C₂₃	—	2	2	no	—
85	C₂×C₂₃	#2087	1	1	yes	characteristic
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
89	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
90	#117	D₄×C₂₃	1	1	yes	characteristic
91	M₃₂	—	2	2	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
95	#108	—	2	2	no	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	4	4	no	—
99	#108	Q₈×C₂₃	1	4	yes	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₂₃	#13520	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
114	C₂×C₈	#41×C₂₃	1	2	yes	—
115	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Q₁₆×C₂₃	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
124	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
126	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	4	4	no	—
129	C₈	—	8	2	no	—
130	C₈	C₂₃×#123	1	1	yes	characteristic
131	C₈	C₂₃×#125	1	2	yes	—
132	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	C₂₃×#467	1	1	yes	characteristic
135	C₂²	C₂₃×#477	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	C₂₃×#398	1	1	yes	characteristic
138	C₄	C₂₃×#469	1	1	yes	characteristic
139	C₄	C₂₃×#478	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	C₂₃×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2161

C₂

yes

maximal, characteristic

C₂₃×#2151

C₂

yes

maximal, characteristic

C₂₃×#1953

C₂

yes

maximal

C₂₃×#1385

C₂

yes

maximal, characteristic

C₂₃×#1381

C₂

yes

maximal

C₂×C₂₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#430

—

C₂×C₂₃×#108

C₂²

yes

—

C₂×C₂₃×#108

C₂²

yes

—

C₂₃×#405

C₂²

yes

characteristic

C₂₃×#368

—

C₂₃×C₄⋊C₁₆

—

C₂₃×#344

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂₃×#117

D₄

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

D₄

yes

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×#108

Q₈

yes

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

—

C₂×C₈×C₂₃

#41

yes

—

C₂×C₈×C₂₃

#41

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

Q₁₆

yes

—

C₄²×C₂₃

—

C₁₆×C₂₃

—

C₁₆×C₂₃

—

#13520

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#121

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₃

#135

yes

characteristic

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#123

yes

characteristic

C₈×C₂₃

#125

yes

—

C₈×C₂₃

C₄⋊Q₈

yes

characteristic

#2161

C₂×C₂₃

yes

characteristic

#2151

C₂×C₂₃

yes

characteristic

#1953

C₂×C₂₃

yes

—

#1385

C₂×C₂₃

yes

characteristic

#1381

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

#467

yes

characteristic

C₂²×C₂₃

#477

yes

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×M₃₂

C₂²×C₂₃

yes

characteristic

C₂²×C₁₆

C₂²×C₂₃

yes

characteristic

#430

—

C₂×#108

C₂²×C₂₃

yes

—

C₂×#108

C₂²×C₂₃

yes

—

#405

C₂²×C₂₃

yes

characteristic

#368

—

C₄⋊C₁₆

—

#344

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#2087

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₂₃

yes

characteristic

C₂²×C₈

C₂³×C₂₃

yes

characteristic

#117

D₄×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₂₃

yes

—

—

—

—

—

Q₈×C₂₃

yes

—

100

#97

—

101

C₄×C₈

—

102

C₂₃

#13520

yes

characteristic, direct product factor, Sylow

103

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₂₃

yes

—

114

C₂×C₈

#41×C₂₃

yes

—

115

C₂×C₈

#41×C₂₃

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Q₁₆×C₂₃

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Q₁₆×C₂₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

C₂₃×#121

yes

characteristic

124

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

125

C₂×C₄

C₂₃×#135

yes

characteristic

126

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

C₂₃×#123

yes

characteristic

131

C₈

C₂₃×#125

yes

—

132

C₈

C₂₃×C₄⋊Q₈

yes

characteristic

133

C₂²

—

134

C₂²

C₂₃×#467

yes

characteristic

135

C₂²

C₂₃×#477

yes

—

136

C₄

—

137

C₄

C₂₃×#398

yes

characteristic

138

C₄

C₂₃×#469

yes

characteristic

139

C₄

C₂₃×#478

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂₃×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₂₃	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#2087
prime radical	C₂³×C₂₃	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

subgroup

G/H

center

C₄×C₂₃

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

socle

C₂×C₂₃

#2087

prime radical

C₂³×C₂₃

C₂×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

N	H ≅ G/N	type
49. #13520	C₂₃	direct (N × H)
102. C₂₃	#13520	direct (N × H)

H ≅ G/N

type

49. #13520

C₂₃

direct (N × H)

102. C₂₃

#13520

direct (N × H)

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

subgroup	index
C₂₃×#2161	2
C₂₃×#2151	2
C₂₃×#1953	2
C₂₃×#1385	2
C₂₃×#1381	2
#13520	23

subgroup

index

C₂₃×#2161

C₂₃×#2151

C₂₃×#1953

C₂₃×#1385

C₂₃×#1381

#13520

Subgroups — C₂₃×#13520 #19161146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#13520 #19161146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13520 #19161146