Group #14519263 — subgroups

Subgroups — #14519263

Order 1944.

444 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11758106	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4208	C₃	1	1	yes	maximal, characteristic, complemented
3	A₄×#272	C₃	1	1	yes	maximal, characteristic
4	#10620072	C₃	1	2	yes	maximal, complemented
5	D₃×C₉⋊A₄	—	3	4	no	maximal
6	#92394	—	4	1	no	maximal
7	C₂²×#737	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×#4208	—	3	1	no	—
9	C₂×#4208	—	3	1	no	—
10	#66440	—	2	8	no	—
11	A₄×3₋¹⁺²	—	2	4	no	—
12	A₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
13	#66437	C₂×C₃	1	2	yes	complemented
14	#66436	—	2	4	no	—
15	C₃×C₉⋊A₄	D₃	1	4	yes	complemented
16	3₋¹⁺⁴	—	4	1	no	Sylow
17	C₃×D₃×A₄	—	3	4	no	—
18	C₂²×#272	C₃²	1	1	yes	characteristic
19	C₂²×D₃×C₉	—	3	4	no	—
20	C₂×C₉⋊A₄	—	9	1	no	—
21	D₃×C₂²⋊C₉	—	3	8	no	—
22	C₂×#737	—	3	1	no	—
23	#4208	—	3	1	no	—
24	#4208	—	4	2	no	—
25	#4208	A₄	1	1	yes	characteristic, complemented
26	C₃×#272	—	4	1	no	—
27	D₃×3₋¹⁺²	—	12	4	no	—
28	C₃²×A₄	C₃×D₃	1	4	yes	—
29	C₂²×3₋¹⁺²	—	2	4	no	—
30	C₂²×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
31	C₂²×C₃×C₉	C₃×D₃	1	4	yes	complemented
32	C₂×#272	—	3	1	no	—
33	C₂×#272	—	3	1	no	—
34	C₂×D₃×C₉	—	9	4	no	—
35	C₂×D₃×C₉	—	9	4	no	—
36	C₂²⋊H(3,𝔽₃)	—	2	4	no	—
37	#1094	—	2	8	no	—
38	C₃×C₂²⋊C₉	C₃×D₃	1	8	yes	complemented
39	C₉⋊A₄	—	2	12	no	—
40	C₉⋊A₄	—	2	4	no	—
41	C₉⋊A₄	—	2	4	no	—
42	C₉⋊A₄	—	2	8	no	—
43	C₉⋊A₄	C₃⋊D₃	1	1	yes	characteristic
44	C₉×A₄	—	2	12	no	—
45	#737	—	4	2	no	—
46	#737	—	8	8	no	—
47	#737	—	8	4	no	—
48	#737	C₂×A₄	1	1	yes	characteristic, complemented
49	C₃×3₋¹⁺²	—	4	4	no	—
50	C₃×3₋¹⁺²	—	8	4	no	—
51	C₃×H(3,𝔽₃)	—	4	1	no	—
52	C₂²×C₃×D₃	—	3	4	no	—
53	C₂×C₃×A₄	—	9	1	no	—
54	D₃×A₄	—	9	4	no	—
55	C₂³×C₉	—	9	1	no	—
56	C₂×C₂²⋊C₉	—	9	2	no	—
57	C₃²×D₃	—	12	4	no	—
58	C₂×3₋¹⁺²	—	6	4	no	—
59	C₂×3₋¹⁺²	—	36	1	no	—
60	C₂×H(3,𝔽₃)	—	3	1	no	—
61	C₂×C₃×C₉	—	3	4	no	—
62	#272	—	3	1	no	—
63	#272	C₃×A₄	1	1	yes	characteristic
64	D₃×C₉	—	3	4	no	—
65	D₃×C₉	—	9	4	no	—
66	D₃×C₉	—	12	8	no	—
67	C₂²×C₃²	C₃²×D₃	1	4	yes	—
68	C₂×C₃×D₃	—	9	4	no	—
69	C₂×C₃×D₃	—	9	4	no	—
70	C₃×A₄	—	2	4	no	—
71	C₃×A₄	—	9	4	no	—
72	C₃×A₄	C₃×C₃⋊D₃	1	1	yes	characteristic
73	C₂²×C₉	—	2	4	no	—
74	C₂²×C₉	—	27	1	no	—
75	C₂²×C₉	—	27	1	no	—
76	C₂²×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
77	C₂²⋊C₉	—	2	8	no	—
78	C₂²⋊C₉	C₃×C₃⋊D₃	1	2	yes	—
79	C₃³	—	4	4	no	—
80	3₋¹⁺²	—	2	4	no	—
81	3₋¹⁺²	—	4	1	no	—
82	3₋¹⁺²	—	8	8	no	—
83	3₋¹⁺²	—	8	4	no	—
84	3₋¹⁺²	—	8	12	no	—
85	3₋¹⁺²	—	8	8	no	—
86	3₋¹⁺²	—	8	4	no	—
87	H(3,𝔽₃)	—	8	4	no	—
88	H(3,𝔽₃)	C₂×C₃×A₄	1	1	yes	characteristic
89	C₃×C₉	—	4	8	no	—
90	C₃×C₉	—	8	12	no	—
91	C₃×C₉	D₃×A₄	1	4	yes	complemented
92	C₂³×C₃	—	9	1	no	—
93	C₂²×D₃	—	3	4	no	—
94	C₂×A₄	—	27	1	no	—
95	C₂×C₃²	—	3	4	no	—
96	C₂×C₃²	—	36	1	no	—
97	C₃×D₃	—	3	4	no	—
98	C₃×D₃	—	9	4	no	—
99	C₃×D₃	—	36	4	no	—
100	C₂×C₉	—	3	1	no	—
101	C₂×C₉	—	6	4	no	—
102	C₂×C₉	—	9	1	no	—
103	C₂×C₉	—	27	1	no	—
104	C₂×C₉	—	36	2	no	—
105	C₂²×C₃	—	3	4	no	—
106	C₂²×C₃	—	27	1	no	—
107	C₂²×C₃	—	27	1	no	—
108	C₂²×C₃	C₃²×C₃⋊D₃	1	1	yes	characteristic
109	C₂×D₃	—	9	4	no	—
110	C₂×D₃	—	9	4	no	—
111	A₄	—	3	1	no	—
112	A₄	—	6	4	no	—
113	C₃²	—	4	1	no	—
114	C₃²	—	8	4	no	—
115	C₃²	—	36	4	no	—
116	C₃²	C₃×D₃×A₄	1	4	yes	—
117	C₉	—	2	4	no	—
118	C₉	—	4	2	no	—
119	C₉	—	8	8	no	—
120	C₉	A₄×C₃⋊D₃	1	1	yes	characteristic
121	C₂³	—	9	1	no	Sylow
122	C₂×C₃	—	3	1	no	—
123	C₂×C₃	—	9	4	no	—
124	C₂×C₃	—	9	1	no	—
125	C₂×C₃	—	27	1	no	—
126	C₂×C₃	—	108	1	no	—
127	D₃	—	3	4	no	—
128	D₃	—	9	4	no	—
129	C₂²	—	27	1	no	—
130	C₂²	—	27	1	no	—
131	C₂²	#92394	1	1	yes	characteristic, complemented
132	C₃	—	3	4	no	—
133	C₃	—	12	1	no	—
134	C₃	—	24	4	no	—
135	C₃	C₃×A₄×C₃⋊D₃	1	1	yes	characteristic
136	C₂	—	3	1	no	—
137	C₂	—	9	1	no	—
138	C₂	—	27	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11758106

C₂

yes

maximal, characteristic, complemented

C₂²×#4208

C₃

yes

maximal, characteristic, complemented

A₄×#272

C₃

yes

maximal, characteristic

#10620072

C₃

yes

maximal, complemented

D₃×C₉⋊A₄

—

maximal

#92394

—

maximal

C₂²×#737

C₂×C₃

yes

characteristic, complemented

C₂×#4208

—

C₂×#4208

—

#66440

—

A₄×3₋¹⁺²

—

A₄×H(3,𝔽₃)

C₂×C₃

yes

characteristic

#66437

C₂×C₃

yes

complemented

#66436

—

C₃×C₉⋊A₄

D₃

yes

complemented

3₋¹⁺⁴

—

Sylow

C₃×D₃×A₄

—

C₂²×#272

C₃²

yes

characteristic

C₂²×D₃×C₉

—

C₂×C₉⋊A₄

—

D₃×C₂²⋊C₉

—

—

—

—

A₄

yes

characteristic, complemented

C₃×#272

—

D₃×3₋¹⁺²

—

C₃²×A₄

C₃×D₃

yes

—

C₂²×3₋¹⁺²

—

C₂²×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₂²×C₃×C₉

C₃×D₃

yes

complemented

C₂×#272

—

C₂×#272

—

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₂²⋊H(3,𝔽₃)

—

#1094

—

C₃×C₂²⋊C₉

C₃×D₃

yes

complemented

C₉⋊A₄

—

C₉⋊A₄

—

C₉⋊A₄

—

C₉⋊A₄

—

C₉⋊A₄

C₃⋊D₃

yes

characteristic

C₉×A₄

—

—

—

—

C₂×A₄

yes

characteristic, complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₂²×C₃×D₃

—

C₂×C₃×A₄

—

D₃×A₄

—

C₂³×C₉

—

C₂×C₂²⋊C₉

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

#272

—

#272

C₃×A₄

yes

characteristic

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₂²×C₃²

C₃²×D₃

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃×A₄

—

C₃×A₄

—

C₃×A₄

C₃×C₃⋊D₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₃⋊D₃

yes

characteristic

C₂²⋊C₉

—

C₂²⋊C₉

C₃×C₃⋊D₃

yes

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₂×C₃×A₄

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

D₃×A₄

yes

complemented

C₂³×C₃

—

C₂²×D₃

—

C₂×A₄

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

C₃²×C₃⋊D₃

yes

characteristic

109

C₂×D₃

—

110

C₂×D₃

—

111

A₄

—

112

A₄

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

C₃×D₃×A₄

yes

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

A₄×C₃⋊D₃

yes

characteristic

121

C₂³

—

Sylow

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

108

—

127

D₃

—

128

D₃

—

129

C₂²

—

130

C₂²

—

131

C₂²

#92394

yes

characteristic, complemented

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

C₃×A₄×C₃⋊D₃

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

—

	subgroup	G/H
center, Frattini	C₃	C₃×A₄×C₃⋊D₃
commutator, cosocle, hypocenter	C₂²×H(3,𝔽₃)	C₂×C₃²
Fitting	C₂²×#737	C₂×C₃
socle	C₂²×C₃	C₃²×C₃⋊D₃
prime radical	A₄×#272	C₃
hypercenter	C₉	A₄×C₃⋊D₃
LCS	G ▷ C₂²×H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ C₉

subgroup

G/H

center, Frattini

C₃

C₃×A₄×C₃⋊D₃

commutator, cosocle, hypocenter

C₂²×H(3,𝔽₃)

C₂×C₃²

Fitting

C₂²×#737

C₂×C₃

socle

C₂²×C₃

C₃²×C₃⋊D₃

prime radical

A₄×#272

C₃

hypercenter

C₉

A₄×C₃⋊D₃

LCS

G ▷ C₂²×H(3,𝔽₃)

UCS

C₁ ◁ C₃ ◁ C₉

N	H ≅ G/N	type
1. #11758106	C₂	semidirect
2. C₂²×#4208	C₃	semidirect
4. #10620072	C₃	semidirect
7. C₂²×#737	C₂×C₃	semidirect
13. #66437	C₂×C₃	semidirect
15. C₃×C₉⋊A₄	D₃	semidirect
25. #4208	A₄	semidirect
31. C₂²×C₃×C₉	C₃×D₃	semidirect
38. C₃×C₂²⋊C₉	C₃×D₃	semidirect
48. #737	C₂×A₄	semidirect
91. C₃×C₉	D₃×A₄	semidirect
131. C₂²	#92394	semidirect

H ≅ G/N

type

1. #11758106

C₂

semidirect

2. C₂²×#4208

C₃

semidirect

4. #10620072

C₃

semidirect

7. C₂²×#737

C₂×C₃

semidirect

13. #66437

C₂×C₃

semidirect

15. C₃×C₉⋊A₄

D₃

semidirect

25. #4208

A₄

semidirect

31. C₂²×C₃×C₉

C₃×D₃

semidirect

38. C₃×C₂²⋊C₉

C₃×D₃

semidirect

48. #737

C₂×A₄

semidirect

91. C₃×C₉

D₃×A₄

semidirect

131. C₂²

#92394

semidirect

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

subgroup	index
#11758106	2
C₂²×#4208	3
A₄×#272	3
#10620072	3
D₃×C₉⋊A₄	3
#92394	4

subgroup

index

C₂²×#4208

D₃×C₉⋊A₄

Subgroups — #14519263

Conjugacy classes of subgroups

Complements

Maximal subgroups