Group #14548027 — subgroups

Subgroups — #14548027

Order 2187.

300 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624889	C₃	1	1	yes	maximal, characteristic
2	#10624826	C₃	1	2	yes	maximal
3	#10624815	C₃	1	2	yes	maximal
4	#10624775	C₃	1	1	yes	maximal, characteristic
5	#10624744	C₃	1	1	yes	maximal, characteristic
6	#10624675	C₃	1	1	yes	maximal, characteristic
7	#10624668	C₃	1	2	yes	maximal
8	#10624617	C₃	1	1	yes	maximal, characteristic
9	#10624592	C₃	1	2	yes	maximal
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃×#733	—	3	1	no	—
12	C₃×#733	—	3	2	no	—
13	#6897	—	3	6	no	—
14	#6896	—	3	3	no	—
15	#6894	—	3	2	no	—
16	#6891	C₃²	1	2	yes	—
17	#6891	C₃²	1	1	yes	characteristic
18	#6891	C₃²	1	2	yes	—
19	#6891	C₃²	1	1	yes	characteristic
20	#6891	C₃²	1	1	yes	characteristic
21	#6891	C₃²	1	2	yes	—
22	C₉×3₋¹⁺²	—	3	1	no	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	3₋¹⁺²⋊C₉	—	3	2	no	—
26	3₋¹⁺²⋊C₉	—	3	4	no	—
27	3₋¹⁺²⋊C₉	—	3	6	no	—
28	#6867	—	3	3	no	—
29	#6862	—	3	3	no	—
30	#6861	—	3	3	no	—
31	#6861	—	3	6	no	—
32	#6858	—	3	6	no	—
33	C₃⁴	C₃³	1	1	yes	characteristic
34	C₃×3₋¹⁺²	—	3	1	no	—
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	—	3	1	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	1	no	—
41	C₃×3₋¹⁺²	—	3	1	no	—
42	C₃×3₋¹⁺²	—	9	3	no	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
47	#733	—	9	3	no	—
48	#733	—	9	6	no	—
49	C₉⋊C₉	—	3	6	no	—
50	C₉⋊C₉	—	3	18	no	—
51	C₃²⋊C₉	—	3	1	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	3	2	no	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	2	no	—
58	C₃²⋊C₉	—	3	2	no	—
59	C₃²⋊C₉	—	3	1	no	—
60	C₃²⋊C₉	—	3	2	no	—
61	C₃²⋊C₉	—	3	2	no	—
62	C₃²⋊C₉	—	3	2	no	—
63	C₃²⋊C₉	—	3	1	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
70	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
71	C₉²	—	3	3	no	—
72	C₃³	—	9	1	no	—
73	C₃³	—	9	2	no	—
74	C₃³	—	9	1	no	—
75	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
77	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	3₋¹⁺²	—	3	3	no	—
79	3₋¹⁺²	—	9	3	no	—
80	3₋¹⁺²	—	9	2	no	—
81	3₋¹⁺²	—	9	6	no	—
82	3₋¹⁺²	—	9	1	no	—
83	3₋¹⁺²	—	9	2	no	—
84	3₋¹⁺²	—	9	1	no	—
85	3₋¹⁺²	—	9	2	no	—
86	3₋¹⁺²	—	9	6	no	—
87	3₋¹⁺²	—	9	6	no	—
88	3₋¹⁺²	—	27	2	no	—
89	C₃×C₉	—	3	3	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	6	no	—
92	C₃×C₉	—	3	3	no	—
93	C₃×C₉	—	3	2	no	—
94	C₃×C₉	—	9	2	no	—
95	C₃×C₉	—	9	1	no	—
96	C₃×C₉	—	9	1	no	—
97	C₃×C₉	—	9	2	no	—
98	C₃×C₉	—	9	2	no	—
99	C₃×C₉	—	9	1	no	—
100	C₃×C₉	—	9	1	no	—
101	C₃×C₉	—	9	2	no	—
102	C₃×C₉	—	9	2	no	—
103	C₃×C₉	—	27	2	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	2	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	27	3	no	—
115	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
116	C₉	—	9	1	no	—
117	C₉	—	9	2	no	—
118	C₉	—	9	3	no	—
119	C₉	—	9	6	no	—
120	C₉	—	9	3	no	—
121	C₉	—	27	2	no	—
122	C₉	—	27	1	no	—
123	C₉	—	27	1	no	—
124	C₉	—	27	2	no	—
125	C₉	—	27	2	no	—
126	C₃	—	9	1	no	—
127	C₃	—	9	1	no	—
128	C₃	—	9	2	no	—
129	C₃	#10624749	1	1	yes	characteristic
130	C₃	#10624751	1	2	yes	—
131	C₃	#10624758	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624889

C₃

yes

maximal, characteristic

#10624826

C₃

yes

maximal

#10624815

C₃

yes

maximal

#10624775

C₃

yes

maximal, characteristic

#10624744

C₃

yes

maximal, characteristic

#10624675

C₃

yes

maximal, characteristic

#10624668

C₃

yes

maximal

#10624617

C₃

yes

maximal, characteristic

#10624592

C₃

yes

maximal

C₃²×3₋¹⁺²

C₃²

yes

—

—

—

—

—

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

—

C₉×3₋¹⁺²

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

#733

—

#733

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

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

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

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

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

116

C₉

—

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₃

—

127

C₃

—

128

C₃

—

129

C₃

#10624749

yes

characteristic

130

C₃

#10624751

yes

—

131

C₃

#10624758

yes

characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, prime radical, cosocle	C₃⁴	C₃³
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, prime radical, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

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

subgroup	index
#10624889	3
#10624826	3
#10624815	3
#10624775	3
#10624744	3
#10624675	3
#10624668	3
#10624617	3
#10624592	3

subgroup

index

Subgroups — #14548027

Conjugacy classes of subgroups

Maximal subgroups