Group #14548321 — subgroups

Subgroups — #14548321

Order 2187.

537 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 53 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	#10624880	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624805	C₃	1	4	yes	maximal, complemented
3	#10624778	C₃	1	2	yes	maximal, complemented
4	#10624742	C₃	1	2	yes	maximal, complemented
5	#10624742	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624632	C₃	1	2	yes	maximal, complemented
7	C₃×#6856	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
9	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
13	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
14	#6891	C₃²	1	4	yes	—
15	#6891	C₃²	1	2	yes	—
16	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
17	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
18	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
19	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
20	H(3,𝔽₃)⋊C₉	—	3	6	no	—
21	#6866	—	3	12	no	—
22	#6859	—	3	6	no	—
23	#6856	—	3	3	no	—
24	C₃⁴	—	3	1	no	—
25	C₃⁴	—	3	1	no	—
26	C₃⁴	—	3	1	no	—
27	C₃⁴	H(3,𝔽₃)	1	6	yes	—
28	C₃⁴	C₃³	1	1	yes	characteristic
29	C₃×3₋¹⁺²	—	3	4	no	—
30	C₃×3₋¹⁺²	—	3	2	no	—
31	C₃×H(3,𝔽₃)	—	3	2	no	—
32	C₃×H(3,𝔽₃)	—	3	2	no	—
33	C₃×H(3,𝔽₃)	—	3	2	no	—
34	C₃×H(3,𝔽₃)	—	3	2	no	—
35	C₃×H(3,𝔽₃)	—	3	2	no	—
36	C₃×H(3,𝔽₃)	—	3	2	no	—
37	C₃×H(3,𝔽₃)	—	3	1	no	—
38	C₃×H(3,𝔽₃)	—	3	9	no	—
39	C₃×H(3,𝔽₃)	—	3	18	no	—
40	C₃×H(3,𝔽₃)	—	9	3	no	—
41	C₃×H(3,𝔽₃)	—	9	3	no	—
42	C₃×H(3,𝔽₃)	—	9	3	no	—
43	C₃×H(3,𝔽₃)	—	9	6	no	—
44	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
45	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
46	C₃²×C₉	—	3	2	no	—
47	AΣL(1,𝔽₂₇)	—	9	18	no	—
48	AΣL(1,𝔽₂₇)	—	9	9	no	—
49	C₃²⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	4	no	—
51	C₃²⋊C₉	—	3	2	no	—
52	C₃²⋊C₉	—	3	4	no	—
53	C₃²⋊C₉	—	3	4	no	—
54	C₃²⋊C₉	—	3	4	no	—
55	C₃²⋊C₉	—	3	4	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	3	6	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	9	1	no	—
61	C₃³	—	9	2	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	9	3	no	—
64	C₃³	—	9	6	no	—
65	C₃³	—	9	6	no	—
66	C₃³	—	9	12	no	—
67	C₃³	—	9	1	no	—
68	C₃³	—	9	3	no	—
69	C₃³	—	9	3	no	—
70	C₃³	—	27	1	no	—
71	C₃³	—	27	2	no	—
72	C₃³	—	27	1	no	—
73	C₃³	—	27	2	no	—
74	C₃³	—	27	2	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	—	9	12	no	—
79	3₋¹⁺²	—	9	6	no	—
80	H(3,𝔽₃)	—	3	6	no	—
81	H(3,𝔽₃)	—	3	3	no	—
82	H(3,𝔽₃)	—	3	6	no	—
83	H(3,𝔽₃)	—	9	3	no	—
84	H(3,𝔽₃)	—	9	3	no	—
85	H(3,𝔽₃)	—	9	6	no	—
86	H(3,𝔽₃)	—	9	6	no	—
87	H(3,𝔽₃)	—	9	6	no	—
88	H(3,𝔽₃)	—	9	6	no	—
89	H(3,𝔽₃)	—	9	4	no	—
90	H(3,𝔽₃)	—	9	6	no	—
91	H(3,𝔽₃)	—	27	9	no	—
92	H(3,𝔽₃)	—	27	18	no	—
93	C₃×C₉	—	9	2	no	—
94	C₃×C₉	—	9	2	no	—
95	C₃×C₉	—	9	4	no	—
96	C₃×C₉	—	27	2	no	—
97	C₃²	—	3	1	no	—
98	C₃²	—	3	2	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	2	no	—
102	C₃²	—	3	1	no	—
103	C₃²	—	3	4	no	—
104	C₃²	—	3	2	no	—
105	C₃²	—	9	3	no	—
106	C₃²	—	9	6	no	—
107	C₃²	—	9	3	no	—
108	C₃²	—	9	3	no	—
109	C₃²	—	9	6	no	—
110	C₃²	—	9	3	no	—
111	C₃²	—	9	3	no	—
112	C₃²	—	9	12	no	—
113	C₃²	—	9	6	no	—
114	C₃²	—	9	6	no	—
115	C₃²	—	27	3	no	—
116	C₃²	—	27	9	no	—
117	C₃²	—	27	9	no	—
118	C₃²	—	27	18	no	—
119	C₃²	—	27	9	no	—
120	C₃²	—	27	1	no	—
121	C₃²	—	27	2	no	—
122	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
123	C₃²	#6909	1	3	yes	—
124	C₉	—	27	2	no	—
125	C₉	—	27	4	no	—
126	C₉	—	27	2	no	—
127	C₃	—	3	3	no	—
128	C₃	—	9	2	no	—
129	C₃	—	9	1	no	—
130	C₃	—	27	6	no	—
131	C₃	—	27	3	no	—
132	C₃	—	27	3	no	—
133	C₃	—	27	3	no	—
134	C₃	#10624735	1	1	yes	characteristic
135	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
136	C₃	#10624748	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10624880

C₃

yes

maximal, characteristic, complemented

#10624805

C₃

yes

maximal, complemented

#10624778

C₃

yes

maximal, complemented

#10624742

C₃

yes

maximal, complemented

#10624742

C₃

yes

maximal, characteristic, complemented

#10624632

C₃

yes

maximal, complemented

C₃×#6856

C₃

yes

maximal, characteristic, complemented

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×AΣL(1,𝔽₂₇)

—

C₃×AΣL(1,𝔽₂₇)

—

#6891

C₃²

yes

—

#6891

C₃²

yes

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃²⋊H(3,𝔽₃)

—

C₃×C₃²⋊C₉

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

#6866

—

#6859

—

#6856

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

H(3,𝔽₃)

yes

—

C₃⁴

C₃³

yes

characteristic

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)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

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

—

C₃×H(3,𝔽₃)

H(3,𝔽₃)

yes

—

C₃²×C₉

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

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

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

123

C₃²

#6909

yes

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₃

—

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

#10624735

yes

characteristic

135

C₃

UT(4,𝔽₃)

yes

characteristic

136

C₃

#10624748

yes

—

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

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

N	H ≅ G/N	type
1. #10624880	C₃	semidirect
2. #10624805	C₃	semidirect
3. #10624778	C₃	semidirect
4. #10624742	C₃	semidirect
5. #10624742	C₃	semidirect
6. #10624632	C₃	semidirect
7. C₃×#6856	C₃	semidirect

H ≅ G/N

type

1. #10624880

C₃

semidirect

2. #10624805

C₃

semidirect

3. #10624778

C₃

semidirect

4. #10624742

C₃

semidirect

5. #10624742

C₃

semidirect

6. #10624632

C₃

semidirect

7. C₃×#6856

C₃

semidirect

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

subgroup	index
#10624880	3
#10624805	3
#10624778	3
#10624742	3
#10624632	3
C₃×#6856	3

subgroup

index

Subgroups — #14548321

Conjugacy classes of subgroups

Complements

Maximal subgroups