Group #13093106 — subgroups

Subgroups — #13093106

Order 1472.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂²⋊D₂₃	C₂	1	1	yes	maximal, characteristic
3	#10625095	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625068	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625061	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625058	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625054	C₂	1	2	yes	maximal
9	#10625050	C₂	1	2	yes	maximal, complemented
10	#10625043	C₂	1	2	yes	maximal, complemented
11	#10625041	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₃	C₂²	1	2	yes	—
15	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
23	(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₂₃	C₂²	1	2	yes	—
26	C₄⋊Dic₂₃	C₂²	1	2	yes	—
27	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	Dic₂₃⋊C₄	C₂²	1	2	yes	—
31	Dic₂₃⋊C₄	C₂²	1	2	yes	—
32	Dic₂₃⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₂₃	C₂²	1	2	yes	—
34	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂³×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×D₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²⋊D₂₃	—	2	2	no	—
43	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₃	C₂³	1	2	yes	—
46	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₃	C₂³	1	2	yes	—
48	C₄×D₂₃	—	2	4	no	—
49	C₄×D₂₃	—	2	2	no	—
50	Dic₄₆	—	2	2	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₂₃	—	2	1	no	—
56	C₂×D₂₃	—	2	1	no	—
57	C₂×D₂₃	#41	1	2	yes	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	2	1	no	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	#41	1	2	yes	—
62	Dic₂₃	—	2	2	no	—
63	Dic₂₃	—	2	1	no	—
64	Dic₂₃	—	2	2	no	—
65	Dic₂₃	—	2	1	no	—
66	Dic₂₃	—	2	1	no	—
67	#552	—	23	1	no	maximal, Sylow
68	C₂×C₂₃	—	4	1	no	—
69	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
72	D₂₃	—	2	2	no	—
73	#126	—	23	2	no	—
74	#125	—	23	1	no	—
75	#125	—	23	2	no	—
76	#123	—	23	2	no	—
77	C₂²⋊Q₈	—	23	1	no	—
78	C₂²⋊Q₈	—	23	1	no	—
79	C₄×Q₈	—	23	1	no	—
80	C₄×D₄	—	23	1	no	—
81	#117	—	23	2	no	—
82	#117	D₂₃	1	1	yes	characteristic, complemented
83	C₂×C₄⋊C₄	—	23	1	no	—
84	C₂₃	#552	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	23	1	no	—
86	C₂×D₄	—	23	1	no	—
87	C₂²×C₄	—	23	2	no	—
88	C₂²×C₄	—	23	1	no	—
89	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
90	C₄⋊C₄	—	23	1	no	—
91	C₄⋊C₄	—	23	2	no	—
92	C₄⋊C₄	—	23	2	no	—
93	C₄⋊C₄	—	23	2	no	—
94	C₄⋊C₄	—	23	1	no	—
95	C₄⋊C₄	—	23	2	no	—
96	C₄⋊C₄	—	23	2	no	—
97	C₄⋊C₄	—	23	2	no	—
98	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
100	C₂²⋊C₄	—	23	1	no	—
101	C₂²⋊C₄	—	23	2	no	—
102	C₂²⋊C₄	—	23	1	no	—
103	C₂²⋊C₄	—	23	2	no	—
104	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
105	C₄²	—	23	2	no	—
106	C₄²	—	23	1	no	—
107	C₄²	C₂×D₂₃	1	2	yes	—
108	C₂³	—	23	1	no	—
109	C₂³	C₂²×D₂₃	1	1	yes	characteristic
110	Q₈	—	46	2	no	—
111	D₄	—	46	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	23	1	no	—
114	C₂×C₄	—	23	2	no	—
115	C₂×C₄	—	23	1	no	—
116	C₂×C₄	—	23	1	no	—
117	C₂×C₄	—	23	2	no	—
118	C₂×C₄	—	46	4	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	C₂²×D₂₃	1	2	yes	—
121	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₂₃	1	2	yes	—
123	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	23	2	no	—
128	C₂²	—	46	1	no	—
129	C₂²	—	46	1	no	—
130	C₂²	C₂³×D₂₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	46	1	no	—
135	C₄	—	46	2	no	—
136	C₄	—	46	2	no	—
137	C₄	—	46	1	no	—
138	C₄	—	46	1	no	—
139	C₄	#67288	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	46	2	no	—
142	C₂	C₂×#67288	1	1	yes	characteristic
143	C₂	#41×D₂₃	1	1	yes	characteristic
144	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#117

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal, characteristic

#10625095

C₂

yes

maximal, characteristic, complemented

#10625068

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₂₃

C₂

yes

maximal, characteristic, complemented

#10625061

C₂

yes

maximal, characteristic, complemented

#10625058

C₂

yes

maximal, characteristic, complemented

#10625054

C₂

yes

maximal

#10625050

C₂

yes

maximal, complemented

#10625043

C₂

yes

maximal, complemented

#10625041

C₂

yes

maximal, complemented

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×C₄×D₂₃

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

#41

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#552

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

D₂₃

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂₃

#552

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

characteristic

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₂₃

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×D₂₃

yes

—

108

C₂³

—

109

C₂³

C₂²×D₂₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₂₃

yes

—

121

C₂×C₄

C₂²×D₂₃

yes

characteristic

122

C₂×C₄

C₂²×D₂₃

yes

—

123

C₂×C₄

C₂²×D₂₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂³×D₂₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#67288

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#67288

yes

characteristic

143

C₂

#41×D₂₃

yes

characteristic

144

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#117	C₂
prime radical	C₂×C₂²⋊D₂₃	C₂²
hypercenter	#117	D₂₃
hypocenter	C₂₃	#552
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#117

C₂

prime radical

C₂×C₂²⋊D₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂₃×#117	C₂	semidirect
3. #10625095	C₂	semidirect
4. #10625068	C₂	semidirect
5. C₄⋊C₄×D₂₃	C₂	semidirect
6. #10625061	C₂	semidirect
7. #10625058	C₂	semidirect
9. #10625050	C₂	semidirect
10. #10625043	C₂	semidirect
11. #10625041	C₂	semidirect
82. #117	D₂₃	semidirect
84. C₂₃	#552	semidirect

H ≅ G/N

type

1. C₂₃×#117

C₂

semidirect

3. #10625095

C₂

semidirect

4. #10625068

C₂

semidirect

5. C₄⋊C₄×D₂₃

C₂

semidirect

6. #10625061

C₂

semidirect

7. #10625058

C₂

semidirect

9. #10625050

C₂

semidirect

10. #10625043

C₂

semidirect

11. #10625041

C₂

semidirect

82. #117

D₂₃

semidirect

84. C₂₃

#552

semidirect

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

subgroup	index
C₂₃×#117	2
C₄×C₂²⋊D₂₃	2
#10625095	2
#10625068	2
C₄⋊C₄×D₂₃	2
#10625061	2
#10625058	2
#10625054	2
#10625050	2
#10625043	2
#10625041	2
#552	23

subgroup

index

C₂₃×#117

C₄×C₂²⋊D₂₃

C₄⋊C₄×D₂₃

Subgroups — #13093106

Conjugacy classes of subgroups

Complements

Maximal subgroups