Group #19160511 — subgroups

Subgroups — C₂₃×#12885 #19160511

Order 5888.

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2153	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1379	C₂	1	2	yes	maximal
5	C₂₃×#1378	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#405	C₂²	1	2	yes	—
10	C₂₃×#405	C₂²	1	1	yes	characteristic
11	C₂₃×#364	—	2	2	no	—
12	C₂₃×C₄⋊C₁₆	—	2	2	no	—
13	C₂₃×#338	C₄	1	1	yes	characteristic
14	C₂₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	C₂₃×H(3,ℤ/4)	C₄	1	2	yes	—
16	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×#117	C₂×C₄	1	2	yes	—
20	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
21	C₂₃×M₃₂	—	2	2	no	—
22	C₂₃×M₃₂	—	2	2	no	—
23	C₂₃×M₃₂	—	2	2	no	—
24	C₂×C₁₆×C₂₃	—	2	1	no	—
25	C₂×C₁₆×C₂₃	—	2	1	no	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	—	2	2	no	—
28	C₂₃×#97	—	2	2	no	—
29	C₂₃×#97	Q₈	1	2	yes	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	D₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₂₃	—	2	2	no	—
34	M₁₆×C₂₃	—	2	2	no	—
35	M₁₆×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₂₃	#41	1	2	yes	—
41	C₂×C₈×C₂₃	#41	1	2	yes	—
42	C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂²⋊C₄×C₂₃	—	2	1	no	—
45	C₂²⋊C₄×C₂₃	—	2	2	no	—
46	C₄²×C₂₃	—	2	2	no	—
47	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₂₃	—	4	1	no	—
49	C₁₆×C₂₃	—	4	1	no	—
50	C₁₆×C₂₃	—	4	2	no	—
51	#12885	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	#131	1	1	yes	characteristic
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	4	2	no	—
55	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#117	1	2	yes	—
57	C₂×C₄×C₂₃	#131	1	2	yes	—
58	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
59	C₈×C₂₃	—	2	2	no	—
60	C₈×C₂₃	—	4	2	no	—
61	C₈×C₂₃	#124	1	1	yes	characteristic
62	C₈×C₂₃	#124	1	1	yes	characteristic
63	C₈×C₂₃	#125	1	2	yes	—
64	#2153	C₂×C₂₃	1	1	yes	characteristic
65	#2151	C₂×C₂₃	1	1	yes	characteristic
66	#1747	C₂×C₂₃	1	1	yes	characteristic
67	#1379	C₂×C₂₃	1	2	yes	—
68	#1378	C₂×C₂₃	1	2	yes	—
69	C₂²×C₂₃	—	4	1	no	—
70	C₂²×C₂₃	#424	1	1	yes	characteristic
71	C₂²×C₂₃	#433	1	2	yes	—
72	C₄×C₂₃	—	4	2	no	—
73	C₄×C₂₃	—	8	2	no	—
74	C₄×C₂₃	#382	1	1	yes	characteristic
75	C₄×C₂₃	#424	1	1	yes	characteristic
76	C₄×C₂₃	#433	1	2	yes	—
77	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
78	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
79	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
80	#405	C₂²×C₂₃	1	2	yes	—
81	#405	C₂²×C₂₃	1	1	yes	characteristic
82	#364	—	2	2	no	—
83	C₄⋊C₁₆	—	2	2	no	—
84	#338	C₄×C₂₃	1	1	yes	characteristic
85	H(3,ℤ/4)	C₄×C₂₃	1	1	yes	characteristic
86	H(3,ℤ/4)	C₄×C₂₃	1	2	yes	—
87	C₂×C₂₃	—	2	1	no	—
88	C₂×C₂₃	—	2	2	no	—
89	C₂×C₂₃	#1832	1	1	yes	characteristic
90	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
91	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
93	#117	C₂×C₄×C₂₃	1	2	yes	—
94	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
95	M₃₂	—	2	2	no	—
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	#97	—	2	2	no	—
103	#97	Q₈×C₂₃	1	2	yes	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	D₄×C₂₃	1	2	yes	—
106	C₂₃	#12885	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
113	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
115	C₂×C₈	#41×C₂₃	1	2	yes	—
116	C₂×C₈	#41×C₂₃	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	C₂₃×#131	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄	C₂₃×#117	1	2	yes	—
131	C₂×C₄	C₂₃×#131	1	2	yes	—
132	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	C₂₃×#124	1	1	yes	characteristic
136	C₈	C₂₃×#124	1	1	yes	characteristic
137	C₈	C₂₃×#125	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂₃×#424	1	1	yes	characteristic
140	C₂²	C₂₃×#433	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	C₂₃×#382	1	1	yes	characteristic
144	C₄	C₂₃×#424	1	1	yes	characteristic
145	C₄	C₂₃×#433	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	C₂₃×#1832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2153

C₂

yes

maximal, characteristic

C₂₃×#2151

C₂

yes

maximal, characteristic

C₂₃×#1747

C₂

yes

maximal, characteristic

C₂₃×#1379

C₂

yes

maximal

C₂₃×#1378

C₂

yes

maximal

C₂×C₂₃×M₃₂

C₂²

yes

characteristic

C₂×C₂₃×M₃₂

C₂²

yes

—

C₂²×C₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#405

C₂²

yes

—

C₂₃×#405

C₂²

yes

characteristic

C₂₃×#364

—

C₂₃×C₄⋊C₁₆

—

C₂₃×#338

C₄

yes

characteristic

C₂₃×H(3,ℤ/4)

C₄

yes

characteristic

C₂₃×H(3,ℤ/4)

C₄

yes

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂₃×#117

C₂×C₄

yes

—

C₂₃×#117

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂₃×M₃₂

—

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

—

C₂₃×#97

—

C₂₃×#97

Q₈

yes

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₂×C₈×C₂₃

#41

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₁₆×C₂₃

—

C₁₆×C₂₃

—

C₁₆×C₂₃

—

#12885

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#131

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#131

yes

—

C₂×C₄×C₂₃

#131

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

yes

C₈×C₂₃

yes

C₈×C₂₃

yes

—

#2153

C₂×C₂₃

yes

characteristic

#2151

C₂×C₂₃

yes

characteristic

#1747

C₂×C₂₃

yes

characteristic

#1379

C₂×C₂₃

yes

—

#1378

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

#424

yes

characteristic

C₂²×C₂₃

#433

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂×M₃₂

C₂²×C₂₃

yes

characteristic

C₂×M₃₂

C₂²×C₂₃

yes

—

C₂²×C₁₆

C₂²×C₂₃

yes

characteristic

#405

C₂²×C₂₃

yes

—

#405

C₂²×C₂₃

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

C₄×C₂₃

yes

characteristic

H(3,ℤ/4)

C₄×C₂₃

yes

characteristic

H(3,ℤ/4)

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1832

yes

characteristic

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

C₂×M₁₆

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂³×C₂₃

yes

characteristic

#117

C₂×C₄×C₂₃

yes

—

#117

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

#97

—

103

#97

Q₈×C₂₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

D₄×C₂₃

yes

—

106

C₂₃

#12885

yes

characteristic, direct product factor, Sylow

107

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

113

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

114

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

115

C₂×C₈

#41×C₂₃

yes

—

116

C₂×C₈

#41×C₂₃

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊C₄×C₂₃

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

C₂₃×#131

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

130

C₂×C₄

C₂₃×#117

yes

—

131

C₂×C₄

C₂₃×#131

yes

—

132

C₂×C₄

C₂₃×#131

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

C₂₃×#124

yes

characteristic

136

C₈

C₂₃×#124

yes

characteristic

137

C₈

C₂₃×#125

yes

—

138

C₂²

—

139

C₂²

C₂₃×#424

yes

characteristic

140

C₂²

C₂₃×#433

yes

—

141

C₄

—

142

C₄

—

143

C₄

C₂₃×#382

yes

characteristic

144

C₄

C₂₃×#424

yes

characteristic

145

C₄

C₂₃×#433

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₂₃×#1832

yes

characteristic

	subgroup	G/H
center	C₈×C₂₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1832
prime radical	C₂³×C₂₃	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₈×C₂₃

#124

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

C₂×C₂₃

C₂³×C₂₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
51. #12885	C₂₃	direct (N × H)
106. C₂₃	#12885	direct (N × H)

H ≅ G/N

type

51. #12885

C₂₃

direct (N × H)

106. C₂₃

#12885

direct (N × H)

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

subgroup	index
C₂₃×#2153	2
C₂₃×#2151	2
C₂₃×#1747	2
C₂₃×#1379	2
C₂₃×#1378	2
#12885	23

subgroup

index

C₂₃×#2153

C₂₃×#2151

C₂₃×#1747

C₂₃×#1379

C₂₃×#1378

#12885

Subgroups — C₂₃×#12885 #19160511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12885 #19160511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12885 #19160511