Group #19154860 — subgroups

Subgroups — C₂₃×#7234 #19154860

Order 5888.

264 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2112	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1810	C₂	1	2	yes	maximal
3	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
4	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
5	C₂×C₂₃×#108	—	2	2	no	—
6	C₂₃×#407	—	2	2	no	—
7	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
8	C₂₃×#341	C₄	1	4	yes	—
9	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	—	2	4	no	—
11	C₂²×C₈×C₂₃	—	4	2	no	—
12	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₂₃	D₄	1	2	yes	—
15	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
16	C₂₃×M₃₂	—	4	2	no	—
17	C₂×C₁₆×C₂₃	—	2	2	no	—
18	C₂₃×#108	—	4	4	no	—
19	C₂₃×C₂²⋊C₈	—	4	2	no	—
20	C₂₃×C₂²⋊C₈	—	4	2	no	—
21	C₂⁴×C₂₃	Q₁₆	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	—	4	1	no	—
23	C₂²×C₄×C₂₃	—	4	1	no	—
24	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
28	M₁₆×C₂₃	—	4	4	no	—
29	M₁₆×C₂₃	—	4	4	no	—
30	C₂×C₈×C₂₃	—	2	4	no	—
31	C₂×C₈×C₂₃	—	2	2	no	—
32	C₂×C₈×C₂₃	—	4	2	no	—
33	C₂×C₈×C₂₃	—	4	1	no	—
34	C₂×C₈×C₂₃	—	4	4	no	—
35	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₂₃	—	4	2	no	—
39	#7234	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	—	4	1	no	—
41	C₂³×C₂₃	—	4	2	no	—
42	C₂³×C₂₃	#103	1	2	yes	—
43	C₂³×C₂₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	1	no	—
48	C₂×C₄×C₂₃	—	4	1	no	—
49	C₂×C₄×C₂₃	—	4	2	no	—
50	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#102	1	2	yes	—
52	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
53	C₈×C₂₃	—	2	1	no	—
54	C₈×C₂₃	—	2	2	no	—
55	C₈×C₂₃	—	8	4	no	—
56	C₈×C₂₃	#99	1	2	yes	—
57	#2112	C₂×C₂₃	1	1	yes	characteristic
58	#1810	C₂×C₂₃	1	2	yes	—
59	C₂²×C₂₃	—	2	2	no	—
60	C₂²×C₂₃	—	2	1	no	—
61	C₂²×C₂₃	—	2	4	no	—
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	4	4	no	—
64	C₂²×C₂₃	#340	1	1	yes	characteristic
65	C₄×C₂₃	—	2	2	no	—
66	C₄×C₂₃	—	2	1	no	—
67	C₄×C₂₃	H(3,ℤ/4)	1	1	yes	characteristic
68	C₄×C₂₃	#342	1	1	yes	characteristic
69	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
70	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
71	C₂×#108	—	2	2	no	—
72	#407	—	2	2	no	—
73	C₂²⋊C₁₆	—	2	2	no	—
74	#341	C₄×C₂₃	1	4	yes	—
75	C₂×C₂₃	—	4	2	no	—
76	C₂×C₂₃	—	4	1	no	—
77	C₂×C₂₃	#1289	1	1	yes	characteristic
78	C₂×C₂₃	#1388	1	2	yes	—
79	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
80	C₂×M₁₆	—	2	4	no	—
81	C₂²×C₈	—	4	2	no	—
82	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
84	C₂²×C₈	D₄×C₂₃	1	2	yes	—
85	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
86	M₃₂	—	4	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	#108	—	4	4	no	—
89	C₂²⋊C₈	—	4	2	no	—
90	C₂²⋊C₈	—	4	2	no	—
91	C₂₃	#7234	1	1	yes	characteristic, direct product factor, Sylow
92	C₂⁴	Q₁₆×C₂₃	1	1	yes	characteristic
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
96	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
97	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
98	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
99	M₁₆	—	4	4	no	—
100	M₁₆	—	4	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
109	C₁₆	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	C₂₃×#103	1	2	yes	—
113	C₂³	C₂₃×#107	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
121	C₂×C₄	C₂₃×#102	1	2	yes	—
122	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	2	no	—
125	C₈	—	8	4	no	—
126	C₈	C₂₃×#99	1	2	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	4	no	—
132	C₂²	C₂₃×#340	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	C₂₃×H(3,ℤ/4)	1	1	yes	characteristic
136	C₄	C₂₃×#342	1	1	yes	characteristic
137	C₂	—	4	2	no	—
138	C₂	—	4	1	no	—
139	C₂	C₂₃×#1289	1	1	yes	characteristic
140	C₂	C₂₃×#1388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2112

C₂

yes

maximal, characteristic

C₂₃×#1810

C₂

yes

maximal

C₂³×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×M₃₂

C₄

yes

—

C₂×C₂₃×#108

—

C₂₃×#407

—

C₂₃×C₂²⋊C₁₆

—

C₂₃×#341

C₄

yes

—

C₂³×C₄×C₂₃

D₄

yes

characteristic

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

Q₈

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂₃×#108

—

C₂₃×C₂²⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂⁴×C₂₃

Q₁₆

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

D₈

yes

characteristic

C₂²×C₄×C₂₃

SD₁₆

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄²

yes

characteristic

C₂×C₈×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₁₆×C₂₃

—

#7234

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

#103

yes

—

C₂³×C₂₃

#107

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#95

yes

characteristic

C₂×C₄×C₂₃

#102

yes

—

C₂×C₄×C₂₃

#106

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#99

yes

—

#2112

C₂×C₂₃

yes

characteristic

#1810

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#340

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂³×C₈

C₂²×C₂₃

yes

characteristic

C₂×M₃₂

C₄×C₂₃

yes

—

C₂×#108

—

#407

—

C₂²⋊C₁₆

—

#341

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1289

yes

characteristic

C₂×C₂₃

#1388

yes

—

C₂³×C₄

D₄×C₂₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂²×C₈

Q₈×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#7234

yes

characteristic, direct product factor, Sylow

C₂⁴

Q₁₆×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

C₂²×C₄

C₄⋊C₄×C₂₃

yes

characteristic

C₂²×C₄

D₈×C₂₃

yes

characteristic

C₂²×C₄

SD₁₆×C₂₃

yes

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₄²×C₂₃

yes

characteristic

107

C₂×C₈

C₂²⋊C₄×C₂₃

yes

characteristic

108

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

109

C₁₆

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂₃×#103

yes

—

113

C₂³

C₂₃×#107

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂₃×#95

yes

characteristic

121

C₂×C₄

C₂₃×#102

yes

—

122

C₂×C₄

C₂₃×#106

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

C₂₃×#99

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂₃×#340

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

C₂₃×H(3,ℤ/4)

yes

characteristic

136

C₄

C₂₃×#342

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

C₂₃×#1289

yes

characteristic

140

C₂

C₂₃×#1388

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#340
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂³×C₈	C₂²×C₂₃
prime radical	C₂⁴×C₂₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#340

commutator

C₂×C₈

C₄²×C₂₃

Frattini, cosocle

C₂³×C₈

C₂²×C₂₃

prime radical

C₂⁴×C₂₃

Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
39. #7234	C₂₃	direct (N × H)
91. C₂₃	#7234	direct (N × H)

H ≅ G/N

type

39. #7234

C₂₃

direct (N × H)

91. C₂₃

#7234

direct (N × H)

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

subgroup	index
C₂₃×#2112	2
C₂₃×#1810	2
#7234	23

subgroup

index

C₂₃×#2112

C₂₃×#1810

#7234

Subgroups — C₂₃×#7234 #19154860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7234 #19154860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7234 #19154860