Group #14548249 — subgroups

Subgroups — #14548249

Order 2187.

479 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10624862	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624783	C₃	1	4	yes	maximal, complemented
3	#10624779	C₃	1	2	yes	maximal, complemented
4	#10624742	C₃	1	1	yes	maximal, characteristic
5	#10624741	C₃	1	2	yes	maximal, complemented
6	#10624643	C₃	1	2	yes	maximal, complemented
7	C₃×#6857	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃×#731	—	3	6	no	—
12	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
13	#6891	C₃²	1	4	yes	—
14	#6891	C₃²	1	2	yes	—
15	C₉×H(3,𝔽₃)	—	3	3	no	—
16	C₉×H(3,𝔽₃)	—	3	6	no	—
17	C₃×C₃²⋊C₉	—	3	3	no	—
18	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	3₋¹⁺²⋊C₉	—	3	12	no	—
21	H(3,𝔽₃)⋊C₉	—	3	6	no	—
22	#6861	—	3	6	no	—
23	#6857	—	3	3	no	—
24	C₃⁴	—	3	1	no	—
25	C₃⁴	—	3	1	no	—
26	C₃⁴	C₃³	1	1	yes	characteristic
27	C₃×3₋¹⁺²	—	3	4	no	—
28	C₃×3₋¹⁺²	—	3	2	no	—
29	C₃×3₋¹⁺²	—	3	2	no	—
30	C₃×3₋¹⁺²	—	3	4	no	—
31	C₃×3₋¹⁺²	—	9	6	no	—
32	C₃×H(3,𝔽₃)	—	3	2	no	—
33	C₃×H(3,𝔽₃)	—	3	9	no	—
34	C₃×H(3,𝔽₃)	—	3	2	no	—
35	C₃×H(3,𝔽₃)	—	3	1	no	—
36	C₃×H(3,𝔽₃)	—	9	3	no	—
37	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
38	C₃²×C₉	—	3	9	no	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	2	no	—
41	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
42	#731	—	9	18	no	—
43	AΣL(1,𝔽₂₇)	—	9	9	no	—
44	C₃²⋊C₉	—	3	18	no	—
45	C₃²⋊C₉	—	3	2	no	—
46	C₃²⋊C₉	—	3	2	no	—
47	C₃²⋊C₉	—	3	2	no	—
48	C₃²⋊C₉	—	3	2	no	—
49	C₃²⋊C₉	—	3	4	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	4	no	—
52	C₃²⋊C₉	—	3	4	no	—
53	C₃²⋊C₉	—	3	4	no	—
54	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
55	C₃³	—	3	9	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	3	3	no	—
58	C₃³	—	9	1	no	—
59	C₃³	—	9	2	no	—
60	C₃³	—	9	3	no	—
61	C₃³	—	9	1	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	27	1	no	—
64	C₃³	—	27	2	no	—
65	C₃³	—	27	2	no	—
66	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
68	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	3₋¹⁺²	—	3	12	no	—
70	3₋¹⁺²	—	9	8	no	—
71	3₋¹⁺²	—	9	6	no	—
72	3₋¹⁺²	—	9	12	no	—
73	3₋¹⁺²	—	9	6	no	—
74	H(3,𝔽₃)	—	3	18	no	—
75	H(3,𝔽₃)	—	3	6	no	—
76	H(3,𝔽₃)	—	3	6	no	—
77	H(3,𝔽₃)	—	9	3	no	—
78	H(3,𝔽₃)	—	9	4	no	—
79	H(3,𝔽₃)	#731	1	9	yes	complemented
80	C₃×C₉	—	3	6	no	—
81	C₃×C₉	—	3	3	no	—
82	C₃×C₉	—	9	2	no	—
83	C₃×C₉	—	9	6	no	—
84	C₃×C₉	—	9	6	no	—
85	C₃×C₉	—	9	2	no	—
86	C₃×C₉	—	9	4	no	—
87	C₃×C₉	—	27	1	no	—
88	C₃×C₉	—	27	9	no	—
89	C₃²	—	3	1	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	9	no	—
93	C₃²	—	3	1	no	—
94	C₃²	—	3	2	no	—
95	C₃²	—	3	1	no	—
96	C₃²	—	3	4	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	9	3	no	—
99	C₃²	—	9	6	no	—
100	C₃²	—	9	3	no	—
101	C₃²	—	27	3	no	—
102	C₃²	—	27	9	no	—
103	C₃²	—	27	9	no	—
104	C₃²	—	27	1	no	—
105	C₃²	—	27	2	no	—
106	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
107	C₃²	C₃×#731	1	3	yes	—
108	C₉	—	9	6	no	—
109	C₉	—	9	3	no	—
110	C₉	—	9	6	no	—
111	C₉	—	27	2	no	—
112	C₉	—	27	4	no	—
113	C₃	—	3	3	no	—
114	C₃	—	9	2	no	—
115	C₃	—	9	1	no	—
116	C₃	—	9	9	no	—
117	C₃	—	27	3	no	—
118	C₃	#10624731	1	1	yes	characteristic
119	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
120	C₃	#10624752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624862

C₃

yes

maximal, characteristic, complemented

#10624783

C₃

yes

maximal, complemented

#10624779

C₃

yes

maximal, complemented

#10624742

C₃

yes

maximal, characteristic

#10624741

C₃

yes

maximal, complemented

#10624643

C₃

yes

maximal, complemented

C₃×#6857

C₃

yes

maximal, characteristic, complemented

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×#731

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₉×H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

H(3,𝔽₃)⋊C₉

—

#6861

—

#6857

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

#731

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

—

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

—

—

—

—

—

yes

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

107

C₃²

C₃×#731

yes

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₃

—

114

C₃

—

115

C₃

—

116

C₃

—

117

C₃

—

118

C₃

#10624731

yes

characteristic

119

C₃

UT(4,𝔽₃)

yes

characteristic

120

C₃

#10624752

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624742	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃⁴

C₃³

C₃

G ▷ C₃⁴ ▷ C₃² ▷ C₁

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624862	C₃	semidirect
2. #10624783	C₃	semidirect
3. #10624779	C₃	semidirect
5. #10624741	C₃	semidirect
6. #10624643	C₃	semidirect
7. C₃×#6857	C₃	semidirect
79. H(3,𝔽₃)	#731	semidirect

H ≅ G/N

type

1. #10624862

C₃

semidirect

2. #10624783

C₃

semidirect

3. #10624779

C₃

semidirect

5. #10624741

C₃

semidirect

6. #10624643

C₃

semidirect

7. C₃×#6857

C₃

semidirect

79. H(3,𝔽₃)

#731

semidirect

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

subgroup	index
#10624862	3
#10624783	3
#10624779	3
#10624742	3
#10624741	3
#10624643	3
C₃×#6857	3

subgroup

index

Subgroups — #14548249

Conjugacy classes of subgroups

Complements

Maximal subgroups