Group #13163824 — subgroups

Subgroups — #13163824

Order 1728.

1820 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 407 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	C₂×C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#11723988	C₂	1	2	yes	maximal, complemented
3	C₂×#88862	C₂	1	2	yes	maximal, complemented
4	C₂×#88852	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592890	—	3	13	no	maximal
7	C₂×C₃³×Q₈	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	—
9	C₃³×Q₁₆	—	2	2	no	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	#88862	—	2	4	no	—
12	#88859	—	2	4	no	—
13	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
15	#88852	—	2	2	no	—
16	C₂×C₃²×Q₁₆	D₃	1	13	yes	complemented
17	#64156	—	3	26	no	—
18	C₂×#3755	—	3	26	no	—
19	C₂×#3745	—	3	13	no	—
20	C₈×C₃⋊Dic₃	—	3	13	no	—
21	C₂²×C₃⋊(C₃⋊D₃)	—	2	2	no	—
22	C₃³×Q₈	—	2	2	no	—
23	C₃³×Q₈	—	2	2	no	—
24	C₂×C₃³×C₄	—	2	2	no	—
25	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
27	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
28	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
29	C₃³×C₈	D₄	1	2	yes	—
30	C₃⋊(C₃⋊(C₃⋊C₈))	D₄	1	2	yes	—
31	#5191	—	9	13	no	—
32	C₂×C₃²×Q₈	C₂×D₃	1	26	yes	—
33	C₂×C₄⋊(C₃⋊D₃)	—	3	26	no	—
34	C₃²×Q₁₆	—	2	26	no	—
35	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
36	#3755	—	6	52	no	—
37	#3752	—	6	52	no	—
38	C₄×C₃⋊Dic₃	—	3	13	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
40	#3745	—	6	26	no	—
41	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
42	C₂×C₃⋊(C₃⋊D₃)	—	4	2	no	—
43	C₂×C₃⋊(C₃⋊D₃)	—	4	4	no	—
44	C₃³×C₄	—	4	2	no	—
45	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
46	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
47	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
48	C₂×C₃×Q₁₆	C₃⋊D₃	1	13	yes	complemented
49	#947	—	9	26	no	—
50	C₂×#215	—	9	26	no	—
51	C₂×D₂₄	—	9	13	no	—
52	C₈×Dic₃	—	9	13	no	—
53	C₂²×C₃⋊D₃	—	6	26	no	—
54	C₃²×Q₈	—	2	26	no	—
55	C₃²×Q₈	—	2	26	no	—
56	C₂×C₃²×C₄	—	2	26	no	—
57	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
58	C₂×C₃⋊Dic₃	—	3	26	no	—
59	C₄⋊(C₃⋊D₃)	—	6	26	no	—
60	C₄⋊(C₃⋊D₃)	—	6	26	no	—
61	C₃²×C₈	#67	1	26	yes	—
62	C₃⋊(C₃⋊C₈)	—	3	26	no	—
63	#495	—	27	1	no	Sylow
64	C₂×C₃³	#127	1	1	yes	characteristic
65	C₂×C₃³	#135	1	2	yes	—
66	C₃⋊(C₃⋊D₃)	—	8	2	no	—
67	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	26	yes	—
68	C₂×D₁₂	—	9	26	no	—
69	C₃×Q₁₆	—	2	26	no	—
70	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
71	#215	—	18	52	no	—
72	(C₂×D₃)⋊C₄	—	18	52	no	—
73	C₄×Dic₃	—	9	13	no	—
74	C₂×C₃⋊C₈	—	9	13	no	—
75	D₂₄	—	18	26	no	—
76	C₂²×C₃²	D₃×D₄	1	13	yes	—
77	C₂×C₃⋊D₃	—	12	52	no	—
78	C₂×C₃⋊D₃	—	12	26	no	—
79	C₃²×C₄	—	4	26	no	—
80	C₃²×C₄	D₃×D₄	1	13	yes	—
81	C₃²×C₄	C₂×#67	1	13	yes	—
82	C₃⋊Dic₃	—	6	26	no	—
83	C₂×Q₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
84	C₂×SD₁₆	—	27	2	no	—
85	C₂×D₈	—	27	1	no	—
86	#124	—	27	2	no	—
87	C₄×C₈	—	27	1	no	—
88	C₃³	#495	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₃	—	18	26	no	—
90	C₃×Q₈	—	2	26	no	—
91	C₃×Q₈	—	2	26	no	—
92	C₂×C₃×C₄	—	2	26	no	—
93	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
94	C₂×Dic₃	—	9	26	no	—
95	D₁₂	—	18	26	no	—
96	D₁₂	—	18	26	no	—
97	C₃×C₈	#638	1	26	yes	—
98	C₃⋊C₈	—	9	26	no	—
99	C₂×C₃²	#919	1	26	yes	—
100	C₂×C₃²	#940	1	13	yes	—
101	C₃⋊D₃	—	24	26	no	—
102	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
103	C₂×D₄	—	27	2	no	—
104	Q₁₆	—	2	2	no	—
105	SD₁₆	—	54	4	no	—
106	D₈	—	54	2	no	—
107	C₂×C₈	—	27	1	no	—
108	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
109	C₂²⋊C₄	—	54	4	no	—
110	C₄²	—	27	1	no	—
111	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
112	C₂×D₃	—	36	52	no	—
113	C₂×D₃	—	36	26	no	—
114	C₃×C₄	—	4	26	no	—
115	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
116	C₃×C₄	C₂×#638	1	13	yes	—
117	Dic₃	—	18	26	no	—
118	C₃²	#5191	1	13	yes	complemented
119	C₂³	—	54	2	no	—
120	Q₈	—	2	2	no	—
121	Q₈	—	2	2	no	—
122	D₄	—	54	2	no	—
123	D₄	—	54	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	27	2	no	—
126	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
127	C₈	—	27	2	no	—
128	C₈	#6312	1	2	yes	—
129	C₂×C₃	#64128	1	26	yes	—
130	C₂×C₃	#64149	1	13	yes	—
131	D₃	—	72	26	no	—
132	C₂²	—	108	4	no	—
133	C₂²	—	108	2	no	—
134	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	54	2	no	—
137	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
138	C₄	C₂×#6312	1	1	yes	characteristic
139	C₃	#10592890	1	13	yes	complemented
140	C₂	—	216	2	no	—
141	C₂	#11723960	1	2	yes	—
142	C₂	#11723981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃³×Q₁₆

C₂

yes

maximal, characteristic, complemented

#11723988

C₂

yes

maximal, complemented

C₂×#88862

C₂

yes

maximal, complemented

C₂×#88852

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic, complemented

#10592890

—

maximal

C₂×C₃³×Q₈

C₂²

yes

—

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

C₂²

yes

—

C₃³×Q₁₆

—

C₂×C₃³×C₈

C₂²

yes

characteristic

#88862

—

#88859

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

#88852

—

C₂×C₃²×Q₁₆

D₃

yes

—

—

—

C₈×C₃⋊Dic₃

—

C₂²×C₃⋊(C₃⋊D₃)

—

C₃³×Q₈

—

C₃³×Q₈

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

—

C₄⋊(C₃⋊(C₃⋊D₃))

—

C₄⋊(C₃⋊(C₃⋊D₃))

—

C₃³×C₈

D₄

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

D₄

yes

—

#5191

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₄⋊(C₃⋊D₃)

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

#3755

—

#3752

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

#3745

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₂×C₃⋊(C₃⋊D₃)

—

C₂×C₃⋊(C₃⋊D₃)

—

C₃³×C₄

—

C₃³×C₄

C₂×D₄

yes

characteristic

C₃³×C₄

C₂×D₄

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₂×C₃×Q₁₆

C₃⋊D₃

yes

complemented

#947

—

C₂×#215

—

C₂×D₂₄

—

C₈×Dic₃

—

C₂²×C₃⋊D₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

#67

yes

—

C₃⋊(C₃⋊C₈)

—

—

C₂×C₃³

yes

C₂×C₃³

yes

—

C₃⋊(C₃⋊D₃)

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×D₁₂

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

#215

—

(C₂×D₃)⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

D₂₄

—

C₂²×C₃²

D₃×D₄

yes

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃²×C₄

C₂×#67

yes

—

C₃⋊Dic₃

—

C₂×Q₁₆

C₃⋊(C₃⋊D₃)

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

#124

—

C₄×C₈

—

C₃³

#495

yes

characteristic, complemented, Sylow

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

#638

yes

—

C₃⋊C₈

—

C₂×C₃²

#919

yes

—

100

C₂×C₃²

#940

yes

—

101

C₃⋊D₃

—

102

C₂×Q₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

103

C₂×D₄

—

104

Q₁₆

—

105

SD₁₆

—

106

D₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

109

C₂²⋊C₄

—

110

C₄²

—

111

C₂²×C₃

D₄×C₃⋊D₃

yes

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₃×C₄

—

115

C₃×C₄

D₄×C₃⋊D₃

yes

—

116

C₃×C₄

C₂×#638

yes

—

117

Dic₃

—

118

C₃²

#5191

yes

complemented

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

127

C₈

—

128

C₈

#6312

yes

—

129

C₂×C₃

#64128

yes

—

130

C₂×C₃

#64149

yes

—

131

D₃

—

132

C₂²

—

108

—

133

C₂²

—

108

—

134

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

138

C₄

C₂×#6312

yes

characteristic

139

C₃

#10592890

yes

complemented

140

C₂

—

216

—

141

C₂

#11723960

yes

—

142

C₂

#11723981

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×Q₁₆	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×#88852	C₂
hypercenter	C₂×Q₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#495
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₂×C₃³×Q₁₆

C₂

socle

C₂²×C₃³

C₂×D₄

prime radical

C₂×#88852

C₂

hypercenter

C₂×Q₁₆

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#495

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. C₂×C₃³×Q₁₆	C₂	semidirect
2. #11723988	C₂	semidirect
3. C₂×#88862	C₂	semidirect
5. C₈×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
16. C₂×C₃²×Q₁₆	D₃	semidirect
48. C₂×C₃×Q₁₆	C₃⋊D₃	semidirect
83. C₂×Q₁₆	C₃⋊(C₃⋊D₃)	semidirect
88. C₃³	#495	semidirect
118. C₃²	#5191	semidirect
139. C₃	#10592890	semidirect

H ≅ G/N

type

1. C₂×C₃³×Q₁₆

C₂

semidirect

2. #11723988

C₂

semidirect

3. C₂×#88862

C₂

semidirect

5. C₈×C₃⋊(C₃⋊Dic₃)

C₂

semidirect

16. C₂×C₃²×Q₁₆

D₃

semidirect

48. C₂×C₃×Q₁₆

C₃⋊D₃

semidirect

83. C₂×Q₁₆

C₃⋊(C₃⋊D₃)

semidirect

88. C₃³

#495

semidirect

118. C₃²

#5191

semidirect

139. C₃

#10592890

semidirect

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

subgroup	index
C₂×C₃³×Q₁₆	2
#11723988	2
C₂×#88862	2
C₂×#88852	2
C₈×C₃⋊(C₃⋊Dic₃)	2
#10592890	3

subgroup

index

C₂×C₃³×Q₁₆

C₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13163824

Conjugacy classes of subgroups

Complements

Maximal subgroups