Group #13093020 — subgroups

Subgroups — #13093020

Order 1472.

240 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 95 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×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625111	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625110	C₂	1	2	yes	maximal, complemented
4	#10625109	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625101	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625097	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625064	C₂	1	1	yes	maximal, characteristic, complemented
9	#10625061	C₂	1	1	yes	maximal, characteristic, complemented
10	#10625057	C₂	1	2	yes	maximal, complemented
11	#10625054	C₂	1	2	yes	maximal, complemented
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
18	C₂²×Dic₂₃	C₂²	1	2	yes	—
19	C₂×D₉₂	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₂₃	C₂²	1	2	yes	—
21	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
28	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₂₃	C₂²	1	2	yes	—
31	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
32	Dic₂₃⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂³×C₂₃	C₂³	1	1	yes	characteristic
36	C₂³×C₂₃	C₂³	1	2	yes	—
37	C₂²×D₂₃	C₂³	1	2	yes	—
38	D₄×C₂₃	—	2	2	no	—
39	D₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²⋊D₂₃	—	4	2	no	—
45	C₂²⋊D₂₃	—	4	2	no	—
46	C₂²⋊D₂₃	—	4	2	no	—
47	C₂×Dic₂₃	—	4	2	no	—
48	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
51	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₃	C₂³	1	2	yes	—
53	D₉₂	—	2	2	no	—
54	C₄×D₂₃	—	4	2	no	—
55	C₂²×C₂₃	—	2	1	no	—
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	—	2	2	no	—
58	C₂²×C₂₃	—	2	1	no	—
59	C₂²×C₂₃	—	2	2	no	—
60	C₂²×C₂₃	—	2	1	no	—
61	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂×D₂₃	—	2	2	no	—
63	C₂×D₂₃	—	2	2	no	—
64	C₂×D₂₃	—	2	2	no	—
65	C₄×C₂₃	—	2	1	no	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	#41	1	2	yes	—
68	Dic₂₃	—	2	1	no	—
69	Dic₂₃	—	2	1	no	—
70	Dic₂₃	—	2	2	no	—
71	Dic₂₃	—	2	1	no	—
72	Dic₂₃	—	2	1	no	—
73	#540	—	23	1	no	maximal, Sylow
74	C₂×C₂₃	—	4	1	no	—
75	C₂×C₂₃	—	4	2	no	—
76	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
77	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
79	D₂₃	—	4	2	no	—
80	#127	—	23	1	no	—
81	#125	—	23	1	no	—
82	#123	—	23	2	no	—
83	#123	—	23	2	no	—
84	#121	—	23	2	no	—
85	#121	—	23	2	no	—
86	#121	—	23	1	no	—
87	#121	D₂₃	1	1	yes	characteristic, complemented
88	C₄×D₄	—	23	1	no	—
89	C₄×D₄	—	23	1	no	—
90	#117	—	23	1	no	—
91	C₂₃	#540	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₄	—	23	2	no	—
93	C₂×D₄	—	23	2	no	—
94	C₂×D₄	—	23	2	no	—
95	C₂×D₄	—	23	1	no	—
96	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
97	C₂×D₄	C₂×D₂₃	1	2	yes	—
98	C₂²×C₄	—	23	2	no	—
99	C₂²×C₄	—	23	2	no	—
100	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
101	C₄⋊C₄	—	23	1	no	—
102	C₄⋊C₄	—	23	1	no	—
103	C₄⋊C₄	—	23	1	no	—
104	C₄⋊C₄	—	23	2	no	—
105	C₄⋊C₄	—	23	2	no	—
106	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
107	C₂²⋊C₄	—	23	1	no	—
108	C₂²⋊C₄	—	23	1	no	—
109	C₂²⋊C₄	—	23	2	no	—
110	C₂²⋊C₄	—	23	2	no	—
111	C₂²⋊C₄	—	23	2	no	—
112	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
113	C₄²	—	23	1	no	—
114	C₄²	—	23	1	no	—
115	C₂³	—	23	2	no	—
116	C₂³	C₂²×D₂₃	1	1	yes	characteristic
117	C₂³	C₂²×D₂₃	1	2	yes	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	46	2	no	—
121	D₄	—	92	2	no	—
122	D₄	—	92	2	no	—
123	D₄	—	92	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	23	1	no	—
126	C₂×C₄	—	23	1	no	—
127	C₂×C₄	—	23	1	no	—
128	C₂×C₄	—	23	1	no	—
129	C₂×C₄	—	23	2	no	—
130	C₂×C₄	—	92	2	no	—
131	C₂×C₄	—	92	2	no	—
132	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₂₃	1	2	yes	—
134	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	46	2	no	—
142	C₂²	—	46	2	no	—
143	C₂²	—	46	2	no	—
144	C₂²	C₂³×D₂₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	2	no	—
147	C₄	—	46	1	no	—
148	C₄	—	46	1	no	—
149	C₄	—	46	1	no	—
150	C₄	—	46	2	no	—
151	C₄	—	46	1	no	—
152	C₄	#67290	1	2	yes	—
153	C₂	—	4	1	no	—
154	C₂	—	4	2	no	—
155	C₂	—	92	2	no	—
156	C₂	C₂×#67290	1	1	yes	characteristic
157	C₂	#10625155	1	1	yes	characteristic
158	C₂	#10625160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#121

C₂

yes

maximal, characteristic, complemented

#10625111

C₂

yes

maximal, characteristic, complemented

#10625110

C₂

yes

maximal, complemented

#10625109

C₂

yes

maximal, complemented

D₄×Dic₂₃

C₂

yes

maximal, characteristic, complemented

#10625101

C₂

yes

maximal, characteristic, complemented

#10625097

C₂

yes

maximal, characteristic, complemented

#10625064

C₂

yes

maximal, characteristic, complemented

#10625061

C₂

yes

maximal, characteristic, complemented

#10625057

C₂

yes

maximal, complemented

#10625054

C₂

yes

maximal, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₃

C₂²

yes

—

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#540

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

—

—

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

C₂₃

#540

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂×D₄

C₂×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×D₂₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×D₂₃

yes

characteristic

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

C₂×D₂₃

yes

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂³

C₂²×D₂₃

yes

characteristic

117

C₂³

C₂²×D₂₃

yes

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×D₂₃

yes

characteristic

133

C₂×C₄

C₂²×D₂₃

yes

—

134

C₂×C₄

C₂²×D₂₃

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂³×D₂₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#67290

yes

—

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

C₂×#67290

yes

characteristic

157

C₂

#10625155

yes

characteristic

158

C₂

#10625160

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#121	C₂
hypercenter	#121	D₂₃
hypocenter	C₂₃	#540
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#121

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂₃×#121	C₂	semidirect
2. #10625111	C₂	semidirect
3. #10625110	C₂	semidirect
4. #10625109	C₂	semidirect
5. D₄×Dic₂₃	C₂	semidirect
6. #10625101	C₂	semidirect
7. #10625097	C₂	semidirect
8. #10625064	C₂	semidirect
9. #10625061	C₂	semidirect
10. #10625057	C₂	semidirect
11. #10625054	C₂	semidirect
87. #121	D₂₃	semidirect
91. C₂₃	#540	semidirect

H ≅ G/N

type

1. C₂₃×#121

C₂

semidirect

2. #10625111

C₂

semidirect

3. #10625110

C₂

semidirect

4. #10625109

C₂

semidirect

5. D₄×Dic₂₃

C₂

semidirect

6. #10625101

C₂

semidirect

7. #10625097

C₂

semidirect

8. #10625064

C₂

semidirect

9. #10625061

C₂

semidirect

10. #10625057

C₂

semidirect

11. #10625054

C₂

semidirect

87. #121

D₂₃

semidirect

91. C₂₃

#540

semidirect

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

subgroup	index
C₂₃×#121	2
#10625111	2
#10625110	2
#10625109	2
D₄×Dic₂₃	2
#10625101	2
#10625097	2
#10625064	2
#10625061	2
#10625057	2
#10625054	2
#540	23

subgroup

index

C₂₃×#121

D₄×Dic₂₃

Subgroups — #13093020

Conjugacy classes of subgroups

Complements

Maximal subgroups