Group #14554348 — subgroups

Subgroups — #14554348

Order 2187.

1965 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 1965 conjugacy classes of subgroups — too many to draw. Only its 311 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	#10624943	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624929	C₃	1	8	yes	maximal, complemented
3	C₃×#6913	C₃	1	4	yes	maximal, complemented
4	#10624764	C₃	1	8	yes	maximal, complemented
5	#10624763	C₃	1	8	yes	maximal, complemented
6	#10624762	C₃	1	8	yes	maximal, complemented
7	#10624761	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624760	C₃	1	2	yes	maximal, complemented
9	3₋¹⁺⁴	—	3	24	no	—
10	3₋¹⁺⁴	C₃²	1	8	yes	—
11	H(5,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×#737	—	3	12	no	—
13	C₃×#737	C₃²	1	4	yes	—
14	C₃×#737	C₃²	1	8	yes	complemented
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
16	#6913	—	3	12	no	—
17	#6913	—	3	24	no	—
18	#6913	—	3	24	no	—
19	#6913	—	3	24	no	—
20	#6912	C₃²	1	16	yes	complemented
21	#6911	C₃²	1	8	yes	—
22	#6910	C₃²	1	8	yes	—
23	#6910	C₃²	1	16	yes	complemented
24	#6909	C₃²	1	8	yes	—
25	#6909	C₃²	1	16	yes	complemented
26	#6908	—	3	24	no	—
27	C₃×#733	C₃²	1	8	yes	complemented
28	C₃×#732	C₃²	1	4	yes	—
29	C₃×#731	C₃²	1	4	yes	—
30	C₃×#731	C₃²	1	8	yes	complemented
31	C₃×AΣL(1,𝔽₂₇)	C₃²	1	4	yes	—
32	C₃×AΣL(1,𝔽₂₇)	C₃²	1	8	yes	complemented
33	C₃⁴	C₃³	1	1	yes	characteristic
34	#737	—	3	18	no	—
35	#737	—	3	144	no	—
36	#737	—	3	48	no	—
37	#737	—	3	24	no	—
38	#737	—	3	8	no	—
39	#737	—	3	16	no	—
40	#737	—	3	8	no	—
41	#737	—	3	16	no	—
42	#737	—	3	16	no	—
43	#737	—	3	16	no	—
44	#737	—	3	16	no	—
45	#737	H(3,𝔽₃)	1	12	yes	—
46	#737	H(3,𝔽₃)	1	24	yes	—
47	C₃×3₋¹⁺²	—	3	72	no	—
48	C₃×3₋¹⁺²	—	3	24	no	—
49	C₃×3₋¹⁺²	—	3	8	no	—
50	C₃×3₋¹⁺²	—	3	4	no	—
51	C₃×3₋¹⁺²	—	3	8	no	—
52	C₃×3₋¹⁺²	—	3	8	no	—
53	C₃×3₋¹⁺²	—	3	8	no	—
54	C₃×3₋¹⁺²	C₃³	1	8	yes	—
55	C₃×3₋¹⁺²	C₃³	1	8	yes	—
56	C₃×3₋¹⁺²	C₃³	1	8	yes	—
57	C₃×H(3,𝔽₃)	—	3	12	no	—
58	C₃×H(3,𝔽₃)	C₃³	1	8	yes	—
59	C₃×H(3,𝔽₃)	C₃³	1	4	yes	—
60	C₃²×C₉	—	3	9	no	—
61	C₃²×C₉	—	3	4	no	—
62	C₃²×C₉	—	3	8	no	—
63	C₃²×C₉	C₃³	1	2	yes	—
64	C₃²×C₉	C₃³	1	1	yes	characteristic
65	#733	—	3	48	no	—
66	#733	—	3	24	no	—
67	#732	—	3	12	no	—
68	#732	—	3	24	no	—
69	#731	—	3	24	no	—
70	#731	—	3	48	no	—
71	#731	—	3	12	no	—
72	#731	—	3	24	no	—
73	AΣL(1,𝔽₂₇)	—	3	24	no	—
74	AΣL(1,𝔽₂₇)	—	3	48	no	—
75	AΣL(1,𝔽₂₇)	—	3	12	no	—
76	AΣL(1,𝔽₂₇)	—	3	24	no	—
77	C₃³	—	3	9	no	—
78	C₃³	—	3	4	no	—
79	C₃³	—	3	8	no	—
80	C₃³	—	3	4	no	—
81	C₃³	—	27	1	no	—
82	C₃³	C₃⁴	1	1	yes	characteristic
83	3₋¹⁺²	—	3	144	no	—
84	3₋¹⁺²	—	3	16	no	—
85	3₋¹⁺²	—	3	48	no	—
86	3₋¹⁺²	—	3	24	no	—
87	3₋¹⁺²	—	3	8	no	—
88	3₋¹⁺²	—	3	16	no	—
89	3₋¹⁺²	—	3	8	no	—
90	3₋¹⁺²	—	3	16	no	—
91	3₋¹⁺²	—	9	1	no	—
92	3₋¹⁺²	—	9	8	no	—
93	3₋¹⁺²	—	9	8	no	—
94	3₋¹⁺²	—	9	16	no	—
95	3₋¹⁺²	—	9	16	no	—
96	3₋¹⁺²	—	9	8	no	—
97	3₋¹⁺²	C₃×H(3,𝔽₃)	1	24	yes	—
98	H(3,𝔽₃)	—	3	18	no	—
99	H(3,𝔽₃)	—	3	8	no	—
100	H(3,𝔽₃)	—	3	8	no	—
101	H(3,𝔽₃)	—	3	16	no	—
102	H(3,𝔽₃)	C₃×H(3,𝔽₃)	1	12	yes	—
103	C₃×C₉	—	3	72	no	—
104	C₃×C₉	—	3	8	no	—
105	C₃×C₉	—	3	12	no	—
106	C₃×C₉	—	3	24	no	—
107	C₃×C₉	—	3	8	no	—
108	C₃×C₉	—	3	4	no	—
109	C₃×C₉	—	3	8	no	—
110	C₃×C₉	—	3	4	no	—
111	C₃×C₉	—	9	8	no	—
112	C₃×C₉	—	9	16	no	—
113	C₃×C₉	C₃×H(3,𝔽₃)	1	4	yes	—
114	C₃×C₉	C₃×H(3,𝔽₃)	1	8	yes	—
115	C₃²	—	3	12	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	9	8	no	—
118	C₃²	—	9	1	no	—
119	C₃²	—	27	4	no	—
120	C₃²	—	27	8	no	—
121	C₃²	—	27	4	no	—
122	C₃²	—	27	9	no	—
123	C₃²	C₃²×H(3,𝔽₃)	1	4	yes	—
124	C₉	—	3	24	no	—
125	C₉	—	9	2	no	—
126	C₉	—	9	8	no	—
127	C₉	—	9	8	no	—
128	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
129	C₉	C₃²⋊H(3,𝔽₃)	1	8	yes	—
130	C₃	—	3	4	no	—
131	C₃	—	9	12	no	—
132	C₃	—	27	8	no	—
133	C₃	—	27	1	no	—
134	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624943

C₃

yes

maximal, characteristic, complemented

#10624929

C₃

yes

maximal, complemented

C₃×#6913

C₃

yes

maximal, complemented

#10624764

C₃

yes

maximal, complemented

#10624763

C₃

yes

maximal, complemented

#10624762

C₃

yes

maximal, complemented

#10624761

C₃

yes

maximal, characteristic, complemented

#10624760

C₃

yes

maximal, complemented

3₋¹⁺⁴

—

3₋¹⁺⁴

C₃²

yes

—

H(5,𝔽₃)

C₃²

yes

characteristic

C₃×#737

—

C₃×#737

C₃²

yes

—

C₃×#737

C₃²

yes

complemented

C₃²×3₋¹⁺²

C₃²

yes

characteristic, complemented

—

—

—

—

C₃²

yes

complemented

#6911

C₃²

yes

—

#6910

C₃²

yes

—

#6910

C₃²

yes

complemented

#6909

C₃²

yes

—

#6909

C₃²

yes

complemented

#6908

—

C₃×#733

C₃²

yes

complemented

C₃×#732

C₃²

yes

—

C₃×#731

C₃²

yes

—

C₃×#731

C₃²

yes

complemented

C₃×AΣL(1,𝔽₂₇)

C₃²

yes

—

C₃×AΣL(1,𝔽₂₇)

C₃²

yes

complemented

C₃⁴

C₃³

yes

characteristic

#737

—

#737

—

144

—

—

—

—

—

—

—

—

—

—

yes

—

#737

H(3,𝔽₃)

yes

—

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

C₃³

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×3₋¹⁺²

C₃³

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃×H(3,𝔽₃)

C₃³

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃³

yes

—

C₃²×C₉

C₃³

yes

—

—

—

—

—

—

—

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃⁴

yes

characteristic

3₋¹⁺²

—

144

—

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

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×H(3,𝔽₃)

yes

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

101

H(3,𝔽₃)

—

102

H(3,𝔽₃)

C₃×H(3,𝔽₃)

yes

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

114

C₃×C₉

C₃×H(3,𝔽₃)

yes

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

C₃²×H(3,𝔽₃)

yes

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

C₃²⋊H(3,𝔽₃)

yes

characteristic

129

C₉

C₃²⋊H(3,𝔽₃)

yes

—

130

C₃

—

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

C₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, socle	C₃	C₃×C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃

C₃×C₃²⋊H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃⁴

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G

LECS

G ▷ C₃³ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
1. #10624943	C₃	semidirect
2. #10624929	C₃	semidirect
3. C₃×#6913	C₃	semidirect
4. #10624764	C₃	semidirect
5. #10624763	C₃	semidirect
6. #10624762	C₃	semidirect
7. #10624761	C₃	semidirect
8. #10624760	C₃	semidirect
14. C₃×#737	C₃²	semidirect
15. C₃²×3₋¹⁺²	C₃²	semidirect
20. #6912	C₃²	semidirect
23. #6910	C₃²	semidirect
25. #6909	C₃²	semidirect
27. C₃×#733	C₃²	semidirect
30. C₃×#731	C₃²	semidirect
32. C₃×AΣL(1,𝔽₂₇)	C₃²	semidirect

H ≅ G/N

type

1. #10624943

C₃

semidirect

2. #10624929

C₃

semidirect

3. C₃×#6913

C₃

semidirect

4. #10624764

C₃

semidirect

5. #10624763

C₃

semidirect

6. #10624762

C₃

semidirect

7. #10624761

C₃

semidirect

8. #10624760

C₃

semidirect

14. C₃×#737

C₃²

semidirect

15. C₃²×3₋¹⁺²

C₃²

semidirect

20. #6912

C₃²

semidirect

23. #6910

C₃²

semidirect

25. #6909

C₃²

semidirect

27. C₃×#733

C₃²

semidirect

30. C₃×#731

C₃²

semidirect

32. C₃×AΣL(1,𝔽₂₇)

C₃²

semidirect

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

subgroup	index
#10624943	3
#10624929	3
C₃×#6913	3
#10624764	3
#10624763	3
#10624762	3
#10624761	3
#10624760	3

subgroup

index

Subgroups — #14554348

Conjugacy classes of subgroups

Complements

Maximal subgroups