Group #13165128 — subgroups

Subgroups — #13165128

Order 1728.

158 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 39 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	#11724365	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic
3	#11722786	C₂	1	1	yes	maximal, characteristic
4	#10593157	—	3	1	no	maximal
5	C₂×C₃⋊PSU(3,𝔽₂)	—	2	1	no	—
6	C₂×C₃×AGL(1,𝔽₉)	—	2	1	no	—
7	C₄⋊(C₃⋊#157)	—	2	1	no	—
8	C₃×C₄×#157	C₂²	1	1	yes	characteristic
9	#88946	C₄	1	2	yes	—
10	#88808	—	2	1	no	—
11	C₄⋊PSU(3,𝔽₂)	—	3	1	no	—
12	C₄⋊AGL(1,𝔽₉)	D₃	1	1	yes	characteristic
13	#63732	—	3	1	no	—
14	C₂×C₃⋊#157	—	2	2	no	—
15	C₂×C₃×#157	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×#157	D₄	1	1	yes	characteristic
17	C₃⋊PSU(3,𝔽₂)	—	4	1	no	—
18	C₃⋊PSU(3,𝔽₂)	—	4	1	no	—
19	C₃×AGL(1,𝔽₉)	—	4	1	no	—
20	C₃×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
21	C₃×C₃⋊(C₃⋊C₈)	—	4	1	no	—
22	#4482	—	9	1	no	maximal
23	C₂×PSU(3,𝔽₂)	—	6	1	no	—
24	C₂×AGL(1,𝔽₉)	—	2	1	no	—
25	C₄⋊#157	—	6	1	no	—
26	C₄×#157	C₂×D₃	1	1	yes	characteristic
27	#3777	—	3	2	no	—
28	#3710	—	6	1	no	—
29	D₃×C₃⋊C₈	—	12	1	no	—
30	C₂×C₃×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₃⋊#157	—	4	2	no	—
32	C₃×#157	—	2	1	no	—
33	C₃×#157	D₈	1	1	yes	characteristic
34	C₃×#157	SD₁₆	1	1	yes	characteristic
35	C₃³×C₄	SD₁₆	1	1	yes	characteristic
36	C₃×C₃⋊Dic₃	Q₁₆	1	1	yes	characteristic
37	#869	—	9	1	no	—
38	C₃×C₄⋊C₈	—	9	1	no	—
39	C₄⋊(C₃⋊C₈)	—	9	1	no	—
40	C₂×#157	—	6	2	no	—
41	C₂×#157	C₄×D₃	1	1	yes	characteristic
42	C₂×#157	#67	1	1	yes	characteristic
43	PSU(3,𝔽₂)	—	12	1	no	—
44	PSU(3,𝔽₂)	—	12	1	no	—
45	AGL(1,𝔽₉)	—	4	1	no	—
46	C₄×C₃⋊D₃	D₁₂	1	1	yes	characteristic
47	C₃×C₄×D₃	—	12	1	no	—
48	C₃⋊(C₃⋊C₈)	—	4	1	no	—
49	C₃×C₃⋊C₈	—	12	1	no	—
50	C₃×C₃⋊C₈	—	12	1	no	—
51	C₃×C₃⋊C₈	—	24	1	no	—
52	#332	—	27	1	no	Sylow
53	C₂×C₃³	#103	1	1	yes	characteristic
54	C₃×C₃⋊D₃	#101	1	1	yes	characteristic
55	C₃×C₃⋊D₃	#102	1	1	yes	characteristic
56	C₂×Dic₆	—	18	1	no	—
57	C₂×C₃×C₈	—	18	1	no	—
58	C₃×C₄²	—	9	1	no	—
59	C₄⋊Dic₃	—	9	2	no	—
60	C₄⋊Dic₃	—	18	1	no	—
61	C₂×C₃⋊C₈	—	18	1	no	—
62	D₃×C₈	—	36	1	no	—
63	C₂×C₃⋊D₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
64	C₂×C₃×D₃	—	12	1	no	—
65	#157	—	2	1	no	—
66	#157	—	12	2	no	—
67	#157	#213	1	1	yes	characteristic
68	#157	#215	1	1	yes	characteristic
69	C₃²×C₄	—	4	1	no	—
70	C₃²×C₄	—	8	1	no	—
71	C₃²×C₄	#204	1	1	yes	characteristic
72	C₃⋊Dic₃	Dic₁₂	1	1	yes	characteristic
73	C₃×Dic₃	—	12	1	no	—
74	C₄⋊Q₈	—	27	1	no	—
75	C₄⋊C₈	—	9	1	no	—
76	C₄⋊C₈	—	27	1	no	—
77	C₃³	#332	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₃×C₄	—	9	1	no	—
79	C₂×C₃×C₄	—	9	1	no	—
80	C₂×C₃×C₄	—	9	1	no	—
81	C₂×Dic₃	—	18	2	no	—
82	C₄×D₃	—	12	1	no	—
83	Dic₆	—	36	1	no	—
84	Dic₆	—	36	1	no	—
85	C₃×C₈	—	36	1	no	—
86	C₃×C₈	—	36	1	no	—
87	C₃⋊C₈	—	12	1	no	—
88	C₃⋊C₈	—	12	1	no	—
89	C₃⋊C₈	—	24	1	no	—
90	C₂×C₃²	—	4	1	no	—
91	C₂×C₃²	—	8	1	no	—
92	C₂×C₃²	#816	1	1	yes	characteristic
93	C₃⋊D₃	#809	1	1	yes	characteristic
94	C₃⋊D₃	#824	1	1	yes	characteristic
95	C₃×D₃	—	12	1	no	—
96	C₃×D₃	—	12	1	no	—
97	C₂×Q₈	—	54	1	no	—
98	C₂×C₈	—	18	1	no	—
99	C₂×C₈	—	54	1	no	—
100	C₄⋊C₄	—	27	2	no	—
101	C₄⋊C₄	—	54	1	no	—
102	C₄²	—	9	1	no	—
103	C₂²×C₃	—	9	1	no	—
104	C₂×D₃	—	12	1	no	—
105	C₃×C₄	—	4	1	no	—
106	C₃×C₄	—	8	1	no	—
107	C₃×C₄	—	9	1	no	—
108	C₃×C₄	—	9	1	no	—
109	C₃×C₄	—	9	1	no	—
110	C₃×C₄	—	18	1	no	—
111	C₃×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
112	Dic₃	—	12	1	no	—
113	Dic₃	—	36	2	no	—
114	C₃²	—	4	1	no	—
115	C₃²	—	8	1	no	—
116	C₃²	#4482	1	1	yes	characteristic, complemented
117	Q₈	—	108	1	no	—
118	Q₈	—	108	1	no	—
119	C₂×C₄	—	9	1	no	—
120	C₂×C₄	—	9	1	no	—
121	C₂×C₄	—	9	1	no	—
122	C₂×C₄	—	54	2	no	—
123	C₈	—	36	1	no	—
124	C₈	—	36	1	no	—
125	C₂×C₃	—	4	1	no	—
126	C₂×C₃	—	8	1	no	—
127	C₂×C₃	—	9	1	no	—
128	C₂×C₃	—	9	1	no	—
129	C₂×C₃	PSU(3,𝔽₂)⋊C₄	1	1	yes	characteristic
130	D₃	—	12	1	no	—
131	D₃	—	12	1	no	—
132	C₂²	—	9	1	no	—
133	C₄	—	9	1	no	—
134	C₄	—	9	1	no	—
135	C₄	—	9	1	no	—
136	C₄	—	18	1	no	—
137	C₄	—	108	2	no	—
138	C₄	AGL(1,𝔽₉)⋊D₃	1	1	yes	characteristic
139	C₃	—	4	1	no	—
140	C₃	—	8	1	no	—
141	C₃	#10593157	1	1	yes	characteristic, complemented
142	C₂	—	9	1	no	—
143	C₂	—	9	1	no	—
144	C₂	#11724105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11724365

C₂

yes

maximal, characteristic

C₃×C₄⋊AGL(1,𝔽₉)

C₂

yes

maximal, characteristic

#11722786

C₂

yes

maximal, characteristic

#10593157

—

maximal

C₂×C₃⋊PSU(3,𝔽₂)

—

C₂×C₃×AGL(1,𝔽₉)

—

C₄⋊(C₃⋊#157)

—

C₃×C₄×#157

C₂²

yes

characteristic

#88946

C₄

yes

—

#88808

—

C₄⋊PSU(3,𝔽₂)

—

C₄⋊AGL(1,𝔽₉)

D₃

yes

characteristic

#63732

—

C₂×C₃⋊#157

—

C₂×C₃×#157

C₂×C₄

yes

characteristic

C₂×C₃×#157

D₄

yes

—

—

—

C₃×C₄×C₃⋊D₃

D₄

yes

characteristic

C₃×C₃⋊(C₃⋊C₈)

—

—

—

—

—

C₂×D₃

yes

characteristic

#3777

—

#3710

—

D₃×C₃⋊C₈

—

C₂×C₃×C₃⋊D₃

C₂²⋊C₄

yes

—

—

D₈

yes

characteristic

C₃×#157

SD₁₆

yes

characteristic

C₃³×C₄

SD₁₆

yes

characteristic

C₃×C₃⋊Dic₃

Q₁₆

yes

characteristic

#869

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂×#157

—

C₂×#157

C₄×D₃

yes

yes

—

—

—

C₄×C₃⋊D₃

D₁₂

yes

characteristic

C₃×C₄×D₃

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

—

C₂×C₃³

yes

C₃×C₃⋊D₃

yes

C₃×C₃⋊D₃

yes

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

D₃×C₈

—

C₂×C₃⋊D₃

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₃×D₃

—

—

—

yes

yes

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#204

yes

characteristic

C₃⋊Dic₃

Dic₁₂

yes

characteristic

C₃×Dic₃

—

C₄⋊Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₃³

#332

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

yes

C₃⋊D₃

yes

C₃⋊D₃

yes

C₃×D₃

—

C₃×D₃

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₂²×C₃

—

104

C₂×D₃

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

AΓL(1,𝔽₉)

yes

characteristic

112

Dic₃

—

113

Dic₃

—

114

C₃²

—

115

C₃²

—

116

C₃²

#4482

yes

characteristic, complemented

117

Q₈

—

108

—

118

Q₈

—

108

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₈

—

124

C₈

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

PSU(3,𝔽₂)⋊C₄

yes

characteristic

130

D₃

—

131

D₃

—

132

C₂²

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

108

—

138

C₄

AGL(1,𝔽₉)⋊D₃

yes

characteristic

139

C₃

—

140

C₃

—

141

C₃

#10593157

yes

characteristic, complemented

142

C₂

—

143

C₂

—

144

C₂

#11724105

yes

characteristic

	subgroup	G/H
center	C₂	#11724105
commutator	C₂×C₃×#157	C₂×C₄
Frattini, hypercenter	C₄	AGL(1,𝔽₉)⋊D₃
Fitting	C₃³×C₄	SD₁₆
socle	C₂×C₃³	#103
prime radical	C₂×C₃×C₃⋊D₃	C₂²⋊C₄
cosocle	C₃×C₄×#157	C₂²
hypocenter	C₃³	#332
LCS	G ▷ C₂×C₃×#157 ▷ C₂×C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center

C₂

#11724105

commutator

C₂×C₃×#157

C₂×C₄

Frattini, hypercenter

C₄

AGL(1,𝔽₉)⋊D₃

Fitting

C₃³×C₄

SD₁₆

socle

C₂×C₃³

#103

prime radical

C₂×C₃×C₃⋊D₃

C₂²⋊C₄

cosocle

C₃×C₄×#157

C₂²

hypocenter

C₃³

#332

LCS

G ▷ C₂×C₃×#157 ▷ C₂×C₃×C₃⋊D₃ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
77. C₃³	#332	semidirect
116. C₃²	#4482	semidirect
141. C₃	#10593157	semidirect

H ≅ G/N

type

77. C₃³

#332

semidirect

116. C₃²

#4482

semidirect

141. C₃

#10593157

semidirect

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

subgroup	index
#11724365	2
C₃×C₄⋊AGL(1,𝔽₉)	2
#11722786	2
#10593157	3
#4482	9

subgroup

index

C₃×C₄⋊AGL(1,𝔽₉)

Subgroups — #13165128

Conjugacy classes of subgroups

Complements

Maximal subgroups