Group #14519253 — subgroups

Subgroups — #14519253

Order 1944.

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

normal subgroups only

This group has 508 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	#11758099	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#272	C₃	1	1	yes	maximal, characteristic
3	#10620072	C₃	1	3	yes	maximal, complemented
4	D₃×#1094	—	3	4	no	maximal
5	#92394	—	4	1	no	maximal
6	C₂²×C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×#272	—	3	1	no	—
8	C₂×C₃×#272	—	3	1	no	—
9	C₃×#1094	—	2	4	no	—
10	C₃×#1094	D₃	1	4	yes	complemented
11	#66437	—	2	12	no	—
12	#66437	C₂×C₃	1	3	yes	complemented
13	3₋¹⁺⁴	—	4	1	no	Sylow
14	C₂²×C₃²×D₃	—	3	4	no	—
15	C₂²×#272	C₃²	1	1	yes	characteristic
16	C₂×#1094	—	9	1	no	—
17	D₃×C₂²⋊C₉	—	3	12	no	—
18	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
19	#4208	—	4	3	no	—
20	C₃×#272	—	3	1	no	—
21	C₃×#272	A₄	1	1	yes	characteristic
22	D₃×3₋¹⁺²	—	12	4	no	—
23	C₂²×C₃³	C₃×D₃	1	4	yes	—
24	C₂×C₃²×D₃	—	9	4	no	—
25	C₂×C₃²×D₃	—	9	4	no	—
26	C₂²×H(3,𝔽₃)	—	2	4	no	—
27	C₂²×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
28	C₂×#272	—	3	1	no	—
29	C₂×#272	—	3	1	no	—
30	#1094	—	2	12	no	—
31	#1094	—	2	12	no	—
32	#1094	—	2	4	no	—
33	#1094	—	2	12	no	—
34	#1094	C₃⋊D₃	1	1	yes	characteristic
35	C₃×C₂²⋊C₉	—	2	12	no	—
36	C₃×C₂²⋊C₉	C₃×D₃	1	12	yes	complemented
37	#737	—	4	3	no	—
38	#737	—	8	12	no	—
39	C₃×3₋¹⁺²	—	4	4	no	—
40	C₃×3₋¹⁺²	—	8	4	no	—
41	C₃×H(3,𝔽₃)	C₂×A₄	1	1	yes	characteristic
42	C₂³×C₃²	—	9	1	no	—
43	C₂²×C₃×D₃	—	3	4	no	—
44	C₂²×C₃×D₃	—	9	4	no	—
45	C₂×C₂²⋊C₉	—	9	3	no	—
46	C₂×C₃³	—	3	4	no	—
47	C₃²×D₃	—	3	4	no	—
48	C₃²×D₃	—	9	4	no	—
49	C₂×3₋¹⁺²	—	36	1	no	—
50	C₂×H(3,𝔽₃)	—	3	1	no	—
51	C₂×H(3,𝔽₃)	—	6	4	no	—
52	#272	—	3	1	no	—
53	#272	C₃×A₄	1	1	yes	characteristic
54	D₃×C₉	—	12	12	no	—
55	C₂²×C₃²	—	2	4	no	—
56	C₂²×C₃²	—	9	4	no	—
57	C₂²×C₃²	—	27	1	no	—
58	C₂²×C₃²	—	27	1	no	—
59	C₂²×C₃²	C₃²×D₃	1	4	yes	—
60	C₂²×C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
61	C₂×C₃×D₃	—	9	4	no	—
62	C₂×C₃×D₃	—	9	12	no	—
63	C₂×C₃×D₃	—	9	12	no	—
64	C₂×C₃×D₃	—	9	4	no	—
65	C₂²⋊C₉	—	2	12	no	—
66	C₂²⋊C₉	C₃×C₃⋊D₃	1	3	yes	—
67	C₃³	D₃×A₄	1	4	yes	—
68	3₋¹⁺²	—	4	1	no	—
69	3₋¹⁺²	—	8	12	no	—
70	3₋¹⁺²	—	8	12	no	—
71	3₋¹⁺²	—	8	12	no	—
72	3₋¹⁺²	—	8	4	no	—
73	H(3,𝔽₃)	—	2	4	no	—
74	H(3,𝔽₃)	C₂×C₃×A₄	1	1	yes	characteristic
75	C₃×C₉	—	4	12	no	—
76	C₃×C₉	—	8	12	no	—
77	C₂³×C₃	—	9	1	no	—
78	C₂³×C₃	—	27	1	no	—
79	C₂²×D₃	—	3	4	no	—
80	C₂×C₃²	—	3	1	no	—
81	C₂×C₃²	—	3	4	no	—
82	C₂×C₃²	—	6	4	no	—
83	C₂×C₃²	—	9	12	no	—
84	C₂×C₃²	—	9	1	no	—
85	C₂×C₃²	—	27	1	no	—
86	C₃×D₃	—	3	4	no	—
87	C₃×D₃	—	9	4	no	—
88	C₃×D₃	—	9	12	no	—
89	C₃×D₃	—	9	4	no	—
90	C₂×C₉	—	36	3	no	—
91	C₂²×C₃	—	3	1	no	—
92	C₂²×C₃	—	3	4	no	—
93	C₂²×C₃	—	6	4	no	—
94	C₂²×C₃	—	27	1	no	—
95	C₂²×C₃	—	27	1	no	—
96	C₂²×C₃	—	27	3	no	—
97	C₂²×C₃	—	27	3	no	—
98	C₂²×C₃	C₃²×C₃⋊D₃	1	1	yes	characteristic
99	C₂×D₃	—	9	4	no	—
100	C₂×D₃	—	9	4	no	—
101	C₃²	—	2	4	no	—
102	C₃²	—	9	4	no	—
103	C₃²	C₃×D₃×A₄	1	4	yes	—
104	C₃²	A₄×C₃⋊D₃	1	1	yes	characteristic
105	C₉	—	4	3	no	—
106	C₉	—	8	12	no	—
107	C₂³	—	9	1	no	Sylow
108	C₂×C₃	—	3	3	no	—
109	C₂×C₃	—	3	1	no	—
110	C₂×C₃	—	9	4	no	—
111	C₂×C₃	—	9	1	no	—
112	C₂×C₃	—	18	4	no	—
113	C₂×C₃	—	27	1	no	—
114	C₂×C₃	—	27	3	no	—
115	C₂×C₃	—	27	1	no	—
116	D₃	—	3	4	no	—
117	D₃	—	9	4	no	—
118	C₂²	—	27	1	no	—
119	C₂²	—	27	1	no	—
120	C₂²	#92394	1	1	yes	characteristic, complemented
121	C₃	—	3	4	no	—
122	C₃	—	3	1	no	—
123	C₃	—	6	4	no	—
124	C₃	C₃×A₄×C₃⋊D₃	1	1	yes	characteristic
125	C₂	—	3	1	no	—
126	C₂	—	9	1	no	—
127	C₂	—	27	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11758099

C₂

yes

maximal, characteristic, complemented

C₂²×C₃×#272

C₃

yes

maximal, characteristic

#10620072

C₃

yes

maximal, complemented

—

—

C₂²×C₃×H(3,𝔽₃)

C₂×C₃

yes

characteristic

C₂×C₃×#272

—

C₂×C₃×#272

—

C₃×#1094

—

C₃×#1094

D₃

yes

complemented

#66437

—

#66437

C₂×C₃

yes

complemented

3₋¹⁺⁴

—

Sylow

C₂²×C₃²×D₃

—

C₂²×#272

C₃²

yes

characteristic

C₂×#1094

—

D₃×C₂²⋊C₉

—

C₂×C₃×H(3,𝔽₃)

—

#4208

—

C₃×#272

—

C₃×#272

A₄

yes

characteristic

D₃×3₋¹⁺²

—

C₂²×C₃³

C₃×D₃

yes

—

C₂×C₃²×D₃

—

C₂×C₃²×D₃

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂×C₃²

yes

—

—

—

—

—

—

C₃⋊D₃

yes

characteristic

C₃×C₂²⋊C₉

—

C₃×C₂²⋊C₉

C₃×D₃

yes

complemented

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

C₂×A₄

yes

characteristic

C₂³×C₃²

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂×C₂²⋊C₉

—

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

#272

—

#272

C₃×A₄

yes

characteristic

D₃×C₉

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃²×D₃

yes

—

C₂²×C₃²

C₃×C₃⋊D₃

yes

characteristic

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂²⋊C₉

—

C₂²⋊C₉

C₃×C₃⋊D₃

yes

—

C₃³

D₃×A₄

yes

—

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

—

C₂³×C₃

—

C₂³×C₃

—

C₂²×D₃

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃²×C₃⋊D₃

yes

characteristic

C₂×D₃

—

100

C₂×D₃

—

101

C₃²

—

102

C₃²

—

103

C₃²

C₃×D₃×A₄

yes

—

104

C₃²

A₄×C₃⋊D₃

yes

characteristic

105

C₉

—

106

C₉

—

107

C₂³

—

Sylow

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

D₃

—

117

D₃

—

118

C₂²

—

119

C₂²

—

120

C₂²

#92394

yes

characteristic, complemented

121

C₃

—

122

C₃

—

123

C₃

—

124

C₃

C₃×A₄×C₃⋊D₃

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

—

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

subgroup

G/H

center, Frattini

C₃

C₃×A₄×C₃⋊D₃

commutator, cosocle, hypocenter

C₂²×H(3,𝔽₃)

C₂×C₃²

Fitting

C₂²×C₃×H(3,𝔽₃)

C₂×C₃

socle

C₂²×C₃

C₃²×C₃⋊D₃

prime radical

C₂²×C₃×#272

C₃

hypercenter

C₃²

A₄×C₃⋊D₃

LCS

G ▷ C₂²×H(3,𝔽₃)

UCS

C₁ ◁ C₃ ◁ C₃²

N	H ≅ G/N	type
1. #11758099	C₂	semidirect
3. #10620072	C₃	semidirect
10. C₃×#1094	D₃	semidirect
12. #66437	C₂×C₃	semidirect
36. C₃×C₂²⋊C₉	C₃×D₃	semidirect
120. C₂²	#92394	semidirect

H ≅ G/N

type

1. #11758099

C₂

semidirect

3. #10620072

C₃

semidirect

10. C₃×#1094

D₃

semidirect

12. #66437

C₂×C₃

semidirect

36. C₃×C₂²⋊C₉

C₃×D₃

semidirect

120. C₂²

#92394

semidirect

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

subgroup	index
#11758099	2
C₂²×C₃×#272	3
#10620072	3
D₃×#1094	3
#92394	4

subgroup

index

C₂²×C₃×#272

Subgroups — #14519253

Conjugacy classes of subgroups

Complements

Maximal subgroups