Group #36382083 — subgroups

Subgroups — #36382083

Order 22599.

226 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 52 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	#21419391	C₃	1	2	yes	maximal, complemented
2	C₃×C₉×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419267	C₃	1	1	yes	maximal, characteristic, complemented
4	#21419266	C₃	1	1	yes	maximal, characteristic, complemented
5	#21419264	C₃	1	1	yes	maximal, characteristic, complemented
6	#21419263	C₃	1	2	yes	maximal, complemented
7	#21419262	C₃	1	2	yes	maximal, complemented
8	#21419259	C₃	1	2	yes	maximal, complemented
9	C₃₁×#6908	C₃	1	1	yes	maximal, characteristic, complemented
10	#14578912	—	3	2	no	—
11	C₃²×C₃₁⋊C₉	—	3	1	no	—
12	#14578902	C₃²	1	2	yes	—
13	#14578902	C₃²	1	2	yes	—
14	#14578900	C₃²	1	2	yes	—
15	#14578897	—	3	2	no	—
16	#14578894	—	3	2	no	—
17	C₉×C₃₁⋊C₉	—	3	1	no	—
18	C₉×C₃₁⋊C₉	—	3	1	no	—
19	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
20	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
21	C₃₁×#737	—	3	1	no	—
22	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
23	C₃₁×#733	C₃²	1	1	yes	characteristic
24	C₃₁×#733	C₃²	1	1	yes	characteristic
25	C₃₁×#732	C₃²	1	1	yes	characteristic
26	C₃₁×#731	—	3	1	no	—
27	C₃₁×AΣL(1,𝔽₂₇)	—	3	1	no	—
28	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
29	3₋¹⁺²×C₃₁	—	3	1	no	—
30	3₋¹⁺²×C₃₁	—	3	1	no	—
31	3₋¹⁺²×C₃₁	—	9	2	no	—
32	H(3,𝔽₃)×C₃₁	—	3	1	no	—
33	C₃×C₉×C₃₁	—	3	2	no	—
34	C₃×C₉×C₃₁	—	3	1	no	—
35	C₃×C₉×C₃₁	—	9	1	no	—
36	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
39	C₃²⋊(C₃₁⋊C₃)	—	3	2	no	—
40	#11719689	—	3	2	no	—
41	#11719689	—	3	2	no	—
42	C₃×C₃₁⋊C₉	—	3	9	no	—
43	C₃×C₃₁⋊C₉	—	3	1	no	—
44	C₃×C₃₁⋊C₉	—	3	2	no	—
45	#11719687	—	9	2	no	—
46	#11719686	—	9	2	no	—
47	C₉×C₃₁⋊C₃	—	9	2	no	—
48	#10624837	—	31	1	no	maximal, Sylow
49	C₃²×C₃₁	—	3	3	no	—
50	C₃²×C₃₁	—	3	1	no	—
51	C₃²×C₃₁	—	9	1	no	—
52	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×C₃₁⋊C₃	—	9	2	no	—
54	C₉×C₃₁	—	3	1	no	—
55	C₉×C₃₁	—	3	1	no	—
56	C₉×C₃₁	—	9	1	no	—
57	C₉×C₃₁	—	9	1	no	—
58	C₉×C₃₁	#732	1	2	yes	—
59	C₉×C₃₁	#732	1	1	yes	characteristic
60	C₃₁⋊C₉	—	3	3	no	—
61	C₃₁⋊C₉	—	3	6	no	—
62	C₃₁⋊C₉	—	9	2	no	—
63	C₃₁⋊C₉	—	9	2	no	—
64	#6908	—	31	2	no	—
65	#6908	C₃₁⋊C₃	1	1	yes	characteristic, complemented
66	C₃×C₉²	—	31	1	no	—
67	#6880	—	31	1	no	—
68	#6880	—	31	1	no	—
69	#6879	—	31	1	no	—
70	#6878	—	31	2	no	—
71	#6878	—	31	2	no	—
72	#6878	—	31	2	no	—
73	C₃×C₃₁	—	3	1	no	—
74	C₃×C₃₁	—	3	3	no	—
75	C₃×C₃₁	—	27	1	no	—
76	C₃×C₃₁	C₃×#732	1	1	yes	characteristic
77	C₃₁⋊C₃	—	27	2	no	—
78	#737	—	3	1	no	—
79	#737	—	93	2	no	—
80	C₃²×C₉	—	93	1	no	—
81	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
82	#733	—	31	2	no	—
83	#733	—	31	2	no	—
84	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
85	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
86	#732	—	31	2	no	—
87	#732	C₃×C₃₁⋊C₃	1	1	yes	characteristic
88	#731	—	3	1	no	—
89	#731	—	93	2	no	—
90	AΣL(1,𝔽₂₇)	—	3	1	no	—
91	AΣL(1,𝔽₂₇)	—	93	2	no	—
92	C₉²	—	31	2	no	—
93	C₉²	—	31	1	no	—
94	C₉²	—	93	1	no	—
95	C₉²	—	93	1	no	—
96	C₃₁	#10624837	1	1	yes	characteristic, complemented, Sylow
97	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
98	3₋¹⁺²	—	3	1	no	—
99	3₋¹⁺²	—	3	1	no	—
100	3₋¹⁺²	—	9	2	no	—
101	3₋¹⁺²	—	279	2	no	—
102	3₋¹⁺²	—	279	2	no	—
103	3₋¹⁺²	—	93	2	no	—
104	3₋¹⁺²	—	93	2	no	—
105	H(3,𝔽₃)	—	3	1	no	—
106	H(3,𝔽₃)	—	93	2	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	9	1	no	—
110	C₃×C₉	—	279	2	no	—
111	C₃×C₉	—	93	9	no	—
112	C₃×C₉	—	93	2	no	—
113	C₃×C₉	—	93	1	no	—
114	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
115	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
116	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
117	C₃²	—	3	3	no	—
118	C₃²	—	3	1	no	—
119	C₃²	—	9	1	no	—
120	C₃²	—	279	2	no	—
121	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
122	C₉	—	3	1	no	—
123	C₉	—	3	1	no	—
124	C₉	—	9	1	no	—
125	C₉	—	9	1	no	—
126	C₉	—	279	2	no	—
127	C₉	—	279	2	no	—
128	C₉	—	93	6	no	—
129	C₉	—	93	3	no	—
130	C₉	#14578899	1	2	yes	—
131	C₉	#14578899	1	1	yes	characteristic
132	C₃	—	3	1	no	—
133	C₃	—	3	3	no	—
134	C₃	—	27	1	no	—
135	C₃	—	837	2	no	—
136	C₃	C₃×#14578899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21419391

C₃

yes

maximal, complemented

C₃×C₉×C₃₁⋊C₉

C₃

yes

maximal, characteristic, complemented

#21419267

C₃

yes

maximal, characteristic, complemented

#21419266

C₃

yes

maximal, characteristic, complemented

#21419264

C₃

yes

maximal, characteristic, complemented

#21419263

C₃

yes

maximal, complemented

#21419262

C₃

yes

maximal, complemented

#21419259

C₃

yes

maximal, complemented

C₃₁×#6908

C₃

yes

maximal, characteristic, complemented

#14578912

—

C₃²×C₃₁⋊C₉

—

#14578902

C₃²

yes

—

#14578902

C₃²

yes

—

#14578900

C₃²

yes

—

#14578897

—

#14578894

—

C₉×C₃₁⋊C₉

—

C₉×C₃₁⋊C₉

—

C₉×C₃₁⋊C₉

C₃²

yes

characteristic

C₉×C₃₁⋊C₉

C₃²

yes

—

C₃₁×#737

—

C₃²×C₉×C₃₁

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

characteristic

C₃₁×#733

C₃²

yes

characteristic

C₃₁×#732

C₃²

yes

characteristic

C₃₁×#731

—

C₃₁×AΣL(1,𝔽₂₇)

—

C₃³×C₃₁

H(3,𝔽₃)

yes

characteristic

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

H(3,𝔽₃)×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

yes

C₃×C₉×C₃₁

yes

C₃×C₉×C₃₁

C₃³

yes

characteristic

C₃²⋊(C₃₁⋊C₃)

—

#11719689

—

#11719689

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

#11719687

—

#11719686

—

C₉×C₃₁⋊C₃

—

#10624837

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₁⋊C₃

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

—

C₉×C₃₁

#732

yes

—

C₉×C₃₁

#732

yes

characteristic

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

#6908

—

#6908

C₃₁⋊C₃

yes

characteristic, complemented

C₃×C₉²

—

—

—

—

—

—

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

C₃×#732

yes

characteristic

C₃₁⋊C₃

—

#737

—

#737

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₁⋊C₃

yes

—

—

C₃×C₃₁⋊C₃

yes

characteristic

#733

C₃×C₃₁⋊C₃

yes

characteristic

#732

—

#732

C₃×C₃₁⋊C₃

yes

characteristic

#731

—

#731

—

AΣL(1,𝔽₂₇)

—

AΣL(1,𝔽₂₇)

—

C₉²

—

C₉²

—

C₉²

—

C₉²

—

C₃₁

#10624837

yes

characteristic, complemented, Sylow

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

279

—

102

3₋¹⁺²

—

279

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

H(3,𝔽₃)

—

106

H(3,𝔽₃)

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

279

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

115

C₃×C₉

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

116

C₃×C₉

C₃²×C₃₁⋊C₃

yes

characteristic

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

279

—

121

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

279

—

127

C₉

—

279

—

128

C₉

—

129

C₉

—

130

C₉

#14578899

yes

—

131

C₉

#14578899

yes

characteristic

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

837

—

136

C₃

C₃×#14578899

yes

characteristic

	subgroup	G/H
center	C₉	#14578899
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6908	C₃
socle	C₃×C₃₁	C₃×#732
hypercenter	#6908	C₃₁⋊C₃
hypocenter	C₃₁	#10624837
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

subgroup

G/H

center

C₉

#14578899

commutator, cosocle

C₃×C₉×C₃₁

C₃³

Frattini

C₃×C₉

C₃²×C₃₁⋊C₃

Fitting

C₃₁×#6908

C₃

C₃×C₃₁

C₃₁⋊C₃

C₃₁

G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6908

N	H ≅ G/N	type
1. #21419391	C₃	semidirect
2. C₃×C₉×C₃₁⋊C₉	C₃	semidirect
3. #21419267	C₃	semidirect
4. #21419266	C₃	semidirect
5. #21419264	C₃	semidirect
6. #21419263	C₃	semidirect
7. #21419262	C₃	semidirect
8. #21419259	C₃	semidirect
9. C₃₁×#6908	C₃	semidirect
65. #6908	C₃₁⋊C₃	semidirect
96. C₃₁	#10624837	semidirect

H ≅ G/N

type

1. #21419391

C₃

semidirect

2. C₃×C₉×C₃₁⋊C₉

C₃

semidirect

3. #21419267

C₃

semidirect

4. #21419266

C₃

semidirect

5. #21419264

C₃

semidirect

6. #21419263

C₃

semidirect

7. #21419262

C₃

semidirect

8. #21419259

C₃

semidirect

9. C₃₁×#6908

C₃

semidirect

65. #6908

C₃₁⋊C₃

semidirect

96. C₃₁

#10624837

semidirect

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

subgroup	index
#21419391	3
C₃×C₉×C₃₁⋊C₉	3
#21419267	3
#21419266	3
#21419264	3
#21419263	3
#21419262	3
#21419259	3
C₃₁×#6908	3
#10624837	31

subgroup

index

C₃×C₉×C₃₁⋊C₉

C₃₁×#6908

Subgroups — #36382083

Conjugacy classes of subgroups

Complements

Maximal subgroups