Group #14518053 — subgroups

Subgroups — #14518053

Order 1944.

262 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 30 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	#11757829	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊#6258	C₃	1	1	yes	maximal, characteristic
3	C₉×#638	—	3	1	no	maximal
4	(C₃⋊S₄)⋊C₉	—	3	3	no	maximal
5	#92305	—	4	1	no	maximal
6	C₂²×C₃²×C₉	D₃	1	1	yes	characteristic, complemented
7	C₂×C₉×C₃⋊D₃	—	3	1	no	—
8	C₃×C₂²⋊H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₃×#1094	—	2	1	no	—
10	C₉×C₃⋊Dic₃	—	3	1	no	—
11	(C₃×A₄)⋊C₉	—	2	3	no	—
12	(C₃×A₄)⋊C₉	D₃	1	3	yes	complemented
13	#6893	—	4	1	no	Sylow
14	C₃×C₃⋊S₄	—	3	3	no	—
15	C₃×#638	—	3	1	no	—
16	#6258	—	9	3	no	—
17	C₉×#67	—	9	3	no	—
18	C₉×#67	—	9	1	no	—
19	C₂×C₃²×C₉	—	3	1	no	—
20	C₃⋊#272	—	4	1	no	—
21	C₉×C₃⋊D₃	—	6	1	no	—
22	(C₃⋊D₃)⋊C₉	—	12	3	no	—
23	C₂²×C₃³	C₃×D₃	1	1	yes	characteristic
24	C₂×C₃×C₃⋊D₃	—	3	1	no	—
25	C₃²×A₄	C₃×D₃	1	3	yes	—
26	C₃×C₃⋊Dic₃	—	3	1	no	—
27	C₂²×C₃×C₉	—	2	1	no	—
28	C₂²×C₃×C₉	—	3	3	no	—
29	C₂²×C₃×C₉	C₃⋊D₃	1	1	yes	characteristic
30	C₂×D₃×C₉	—	9	3	no	—
31	C₂×D₃×C₉	—	9	1	no	—
32	C₂²⋊H(3,𝔽₃)	—	3	3	no	—
33	#1094	—	6	3	no	—
34	C₃×C₂²⋊C₉	—	2	3	no	—
35	C₉×Dic₃	—	9	3	no	—
36	C₉×Dic₃	—	9	1	no	—
37	C₃×3₋¹⁺²	—	8	1	no	—
38	C₃×H(3,𝔽₃)	—	4	1	no	—
39	C₃²×C₉	S₄	1	1	yes	characteristic, complemented
40	C₃²⋊C₉	—	4	3	no	—
41	C₃²⋊C₉	—	8	3	no	—
42	C₃⋊S₄	—	3	3	no	—
43	C₃×S₄	—	27	3	no	—
44	#638	—	3	1	no	—
45	C₃×#67	—	9	3	no	—
46	C₃×#67	—	9	1	no	—
47	D₄×C₉	—	27	1	no	—
48	C₂×C₃³	—	3	1	no	—
49	C₃×C₃⋊D₃	—	6	1	no	—
50	C₃×C₃⋊D₃	—	12	3	no	—
51	C₂×C₃×C₉	—	3	3	no	—
52	C₂×C₃×C₉	—	3	1	no	—
53	C₂×C₃×C₉	—	6	1	no	—
54	C₂×C₃×C₉	—	6	3	no	—
55	#272	—	36	3	no	—
56	D₃×C₉	—	18	3	no	—
57	D₃×C₉	—	18	1	no	—
58	C₂²×C₃²	—	3	1	no	—
59	C₂²×C₃²	—	6	1	no	—
60	C₂²×C₃²	(C₃⋊D₃)⋊C₃	1	3	yes	—
61	C₂²×C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
62	C₂×C₃⋊D₃	—	3	1	no	—
63	C₂×C₃×D₃	—	9	3	no	—
64	C₂×C₃×D₃	—	9	1	no	—
65	C₃×A₄	—	3	3	no	—
66	C₃×A₄	—	3	3	no	—
67	C₃×A₄	—	6	3	no	—
68	C₃⋊Dic₃	—	3	1	no	—
69	C₃×Dic₃	—	9	3	no	—
70	C₃×Dic₃	—	9	1	no	—
71	C₂²×C₉	—	3	1	no	—
72	C₂²×C₉	—	6	1	no	—
73	C₂²×C₉	—	27	1	no	—
74	C₂²⋊C₉	—	6	3	no	—
75	C₄×C₉	—	27	1	no	—
76	C₃³	—	4	3	no	—
77	C₃³	C₃×S₄	1	1	yes	characteristic
78	3₋¹⁺²	—	24	3	no	—
79	H(3,𝔽₃)	—	12	3	no	—
80	C₃×C₉	—	2	1	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	8	3	no	—
83	C₃×C₉	C₃⋊S₄	1	1	yes	characteristic
84	S₄	—	27	3	no	—
85	C₃×D₄	—	27	1	no	—
86	#67	—	9	3	no	—
87	#67	—	9	1	no	—
88	C₂×C₃²	—	3	1	no	—
89	C₂×C₃²	—	3	3	no	—
90	C₂×C₃²	—	3	1	no	—
91	C₂×C₃²	—	6	1	no	—
92	C₂×C₃²	—	6	3	no	—
93	C₃⋊D₃	—	6	1	no	—
94	C₃⋊D₃	—	12	3	no	—
95	C₃×D₃	—	18	3	no	—
96	C₃×D₃	—	18	1	no	—
97	C₃×D₃	—	108	3	no	—
98	C₂×C₉	—	3	1	no	—
99	C₂×C₉	—	6	1	no	—
100	C₂×C₉	—	6	3	no	—
101	C₂×C₉	—	54	1	no	—
102	C₂²×C₃	—	2	1	no	—
103	C₂²×C₃	—	3	3	no	—
104	C₂²×C₃	—	27	1	no	—
105	C₂²×C₃	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
106	C₂²×C₃	#4208	1	1	yes	characteristic
107	C₂×D₃	—	9	3	no	—
108	C₂×D₃	—	9	1	no	—
109	A₄	—	9	3	no	—
110	C₃×C₄	—	27	1	no	—
111	Dic₃	—	9	3	no	—
112	Dic₃	—	9	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	6	1	no	—
115	C₃²	—	12	3	no	—
116	C₃²	—	12	3	no	—
117	C₃²	—	24	3	no	—
118	C₃²	(C₃⋊S₄)⋊C₃	1	3	yes	—
119	C₃²	C₃×C₃⋊S₄	1	1	yes	characteristic
120	C₉	—	3	1	no	—
121	C₉	—	6	1	no	—
122	C₉	—	24	3	no	—
123	D₄	—	27	1	no	Sylow
124	C₂×C₃	—	3	3	no	—
125	C₂×C₃	—	3	1	no	—
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	—	6	1	no	—
128	C₂×C₃	—	6	3	no	—
129	C₂×C₃	—	54	1	no	—
130	D₃	—	18	3	no	—
131	D₃	—	18	1	no	—
132	D₃	—	108	3	no	—
133	C₂²	—	27	1	no	—
134	C₂²	#92305	1	1	yes	characteristic, complemented
135	C₄	—	27	1	no	—
136	C₃	—	2	1	no	—
137	C₃	—	3	3	no	—
138	C₃	—	36	3	no	—
139	C₃	#10620044	1	1	yes	characteristic
140	C₃	C₃⋊((C₃⋊S₄)⋊C₃)	1	1	yes	characteristic
141	C₂	—	3	1	no	—
142	C₂	—	54	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11757829

C₂

yes

maximal, characteristic, complemented

C₃⋊#6258

C₃

yes

maximal, characteristic

C₉×#638

—

maximal

(C₃⋊S₄)⋊C₉

—

maximal

#92305

—

maximal

C₂²×C₃²×C₉

D₃

yes

characteristic, complemented

C₂×C₉×C₃⋊D₃

—

C₃×C₂²⋊H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₃×#1094

—

C₉×C₃⋊Dic₃

—

(C₃×A₄)⋊C₉

—

(C₃×A₄)⋊C₉

D₃

yes

complemented

#6893

—

Sylow

C₃×C₃⋊S₄

—

C₃×#638

—

#6258

—

C₉×#67

—

C₉×#67

—

C₂×C₃²×C₉

—

C₃⋊#272

—

C₉×C₃⋊D₃

—

(C₃⋊D₃)⋊C₉

—

C₂²×C₃³

C₃×D₃

yes

characteristic

C₂×C₃×C₃⋊D₃

—

C₃²×A₄

C₃×D₃

yes

—

C₃×C₃⋊Dic₃

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₃⋊D₃

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₂²⋊H(3,𝔽₃)

—

#1094

—

C₃×C₂²⋊C₉

—

C₉×Dic₃

—

C₉×Dic₃

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

S₄

yes

characteristic, complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃⋊S₄

—

C₃×S₄

—

#638

—

C₃×#67

—

C₃×#67

—

D₄×C₉

—

C₂×C₃³

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

#272

—

D₃×C₉

—

D₃×C₉

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

(C₃⋊D₃)⋊C₃

yes

—

C₂²×C₃²

C₃×C₃⋊D₃

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃×A₄

—

C₃×A₄

—

C₃×A₄

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²⋊C₉

—

C₄×C₉

—

C₃³

—

C₃³

C₃×S₄

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃⋊S₄

yes

characteristic

S₄

—

C₃×D₄

—

#67

—

#67

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃⋊D₃

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

108

—

C₂×C₉

—

C₂×C₉

—

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

C₃⋊((C₃⋊D₃)⋊C₃)

yes

characteristic

106

C₂²×C₃

#4208

yes

characteristic

107

C₂×D₃

—

108

C₂×D₃

—

109

A₄

—

110

C₃×C₄

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

(C₃⋊S₄)⋊C₃

yes

—

119

C₃²

C₃×C₃⋊S₄

yes

characteristic

120

C₉

—

121

C₉

—

122

C₉

—

123

D₄

—

Sylow

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

D₃

—

131

D₃

—

132

D₃

—

108

—

133

C₂²

—

134

C₂²

#92305

yes

characteristic, complemented

135

C₄

—

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

#10620044

yes

characteristic

140

C₃

C₃⋊((C₃⋊S₄)⋊C₃)

yes

characteristic

141

C₂

—

142

C₂

—

	subgroup	G/H
center, hypercenter	C₃	C₃⋊((C₃⋊S₄)⋊C₃)
commutator, cosocle, hypocenter	C₃×C₂²⋊H(3,𝔽₃)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊S₄
Fitting	C₂²×C₃²×C₉	D₃
socle	C₂²×C₃²	C₃×C₃⋊D₃
prime radical	C₃⋊#6258	C₃
LCS	G ▷ C₃×C₂²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃

subgroup

G/H

center, hypercenter

C₃

C₃⋊((C₃⋊S₄)⋊C₃)

commutator, cosocle, hypocenter

C₃×C₂²⋊H(3,𝔽₃)

C₂×C₃

Frattini

C₃²

C₃×C₃⋊S₄

Fitting

C₂²×C₃²×C₉

D₃

socle

C₂²×C₃²

C₃×C₃⋊D₃

C₃

G ▷ C₃×C₂²⋊H(3,𝔽₃)

C₁ ◁ C₃

N	H ≅ G/N	type
1. #11757829	C₂	semidirect
6. C₂²×C₃²×C₉	D₃	semidirect
12. (C₃×A₄)⋊C₉	D₃	semidirect
39. C₃²×C₉	S₄	semidirect
134. C₂²	#92305	semidirect

H ≅ G/N

type

1. #11757829

C₂

semidirect

6. C₂²×C₃²×C₉

D₃

semidirect

12. (C₃×A₄)⋊C₉

D₃

semidirect

39. C₃²×C₉

S₄

semidirect

134. C₂²

#92305

semidirect

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

subgroup	index
#11757829	2
C₃⋊#6258	3
C₉×#638	3
(C₃⋊S₄)⋊C₉	3
#92305	4

subgroup

index

(C₃⋊S₄)⋊C₉

Subgroups — #14518053

Conjugacy classes of subgroups

Complements

Maximal subgroups