Group #36380037 — subgroups

Subgroups — C₃×#21419212 #36380037

Order 22599.

476 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₃²×#11719689	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	#21419212	C₃	1	9	yes	maximal, direct product factor
5	C₃²×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic
6	C₃×#11719689	—	3	2	no	—
7	C₃×#11719689	—	3	9	no	—
8	C₃×#11719689	C₃²	1	3	yes	—
9	C₃²×C₃₁⋊C₉	—	3	1	no	—
10	C₃²×C₃₁⋊C₉	—	3	1	no	—
11	C₃²×C₃₁⋊C₉	—	3	1	no	—
12	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
13	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
14	C₃²⋊(C₃₁⋊C₉)	C₃²	1	3	yes	—
15	C₃²⋊(C₃₁⋊C₉)	C₃²	1	3	yes	—
16	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
17	C₃×3₋¹⁺²×C₃₁	—	3	2	no	—
18	C₃×3₋¹⁺²×C₃₁	—	3	9	no	—
19	C₃×3₋¹⁺²×C₃₁	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₁	—	3	1	no	—
21	C₃³×C₃₁	—	3	3	no	—
22	C₃³×C₃₁	—	3	3	no	—
23	C₃³×C₃₁	—	3	6	no	—
24	C₃³×C₃₁	H(3,𝔽₃)	1	2	yes	—
25	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃³×C₃₁	C₃³	1	1	yes	characteristic
27	3₋¹⁺²×C₃₁	—	3	18	no	—
28	3₋¹⁺²×C₃₁	—	3	9	no	—
29	C₃×C₉×C₃₁	—	3	9	no	—
30	C₃×C₉×C₃₁	—	3	3	no	—
31	#11719689	—	3	18	no	—
32	#11719689	—	3	9	no	—
33	C₃×C₃₁⋊C₉	—	3	9	no	—
34	C₃×C₃₁⋊C₉	—	3	3	no	—
35	C₃×C₃₁⋊C₉	—	3	3	no	—
36	C₃×C₃₁⋊C₉	—	3	3	no	—
37	C₃×C₃₁⋊C₉	—	9	3	no	—
38	C₃×C₃₁⋊C₉	—	9	3	no	—
39	C₃×#6860	—	31	1	no	maximal, Sylow
40	C₃²×C₃₁	—	3	2	no	—
41	C₃²×C₃₁	—	3	1	no	—
42	C₃²×C₃₁	—	3	2	no	—
43	C₃²×C₃₁	—	3	1	no	—
44	C₃²×C₃₁	—	3	4	no	—
45	C₃²×C₃₁	—	3	2	no	—
46	C₃²×C₃₁	—	9	9	no	—
47	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
48	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
49	C₃²×C₃₁	#733	1	6	yes	—
50	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₉×C₃₁	—	3	9	no	—
52	C₃₁⋊C₉	—	3	9	no	—
53	C₃₁⋊C₉	—	9	3	no	—
54	C₃₁⋊C₉	—	9	3	no	—
55	C₃²×3₋¹⁺²	—	31	1	no	—
56	C₃²×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic
57	C₃×C₃²⋊C₉	—	31	1	no	—
58	C₃×C₃²⋊C₉	—	31	1	no	—
59	#6860	—	31	9	no	—
60	C₃×C₃₁	—	9	2	no	—
61	C₃×C₃₁	—	9	1	no	—
62	C₃×C₃₁	#6860	1	9	yes	complemented
63	C₃×C₃₁	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
64	C₃×C₃₁	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
65	C₃×C₃₁	C₃×#733	1	2	yes	—
66	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
67	C₃×3₋¹⁺²	—	3	2	no	—
68	C₃×3₋¹⁺²	—	3	9	no	—
69	C₃×3₋¹⁺²	—	31	3	no	—
70	C₃×3₋¹⁺²	—	93	2	no	—
71	C₃×3₋¹⁺²	—	93	9	no	—
72	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	3	yes	—
73	C₃²×C₉	—	3	1	no	—
74	C₃²×C₉	—	93	1	no	—
75	C₃²×C₉	—	93	1	no	—
76	C₃²×C₉	—	93	1	no	—
77	C₃²⋊C₉	—	31	3	no	—
78	C₃²⋊C₉	—	31	3	no	—
79	C₃²⋊C₉	—	93	2	no	—
80	C₃²⋊C₉	—	93	2	no	—
81	C₃₁	C₃×#6860	1	1	yes	characteristic, complemented, Sylow
82	C₃³	—	3	3	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	6	no	—
85	C₃³	C₃²⋊(C₃₁⋊C₃)	1	2	yes	—
86	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
87	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
88	3₋¹⁺²	—	3	18	no	—
89	3₋¹⁺²	—	3	9	no	—
90	3₋¹⁺²	—	93	18	no	—
91	3₋¹⁺²	—	93	9	no	—
92	C₃×C₉	—	3	9	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	279	3	no	—
95	C₃×C₉	—	279	3	no	—
96	C₃×C₉	—	93	3	no	—
97	C₃×C₉	—	93	3	no	—
98	C₃×C₉	—	93	9	no	—
99	C₃×C₉	—	93	3	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	2	no	—
103	C₃²	—	3	1	no	—
104	C₃²	—	3	4	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	9	9	no	—
107	C₃²	#14578894	1	3	yes	—
108	C₃²	#14578895	1	3	yes	—
109	C₃²	#14578901	1	6	yes	—
110	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
111	C₉	—	3	9	no	—
112	C₉	—	279	3	no	—
113	C₉	—	279	3	no	—
114	C₉	—	93	9	no	—
115	C₃	—	9	2	no	—
116	C₃	—	9	1	no	—
117	C₃	#21419212	1	9	yes	direct product factor
118	C₃	C₃×#14578894	1	1	yes	characteristic
119	C₃	C₃×#14578895	1	1	yes	characteristic
120	C₃	C₃×#14578901	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11719689

C₃

yes

maximal, characteristic

C₃×C₃²⋊(C₃₁⋊C₉)

C₃

yes

maximal, characteristic

C₃×C₃²⋊(C₃₁⋊C₉)

C₃

yes

maximal, characteristic

#21419212

C₃

yes

maximal, direct product factor

C₃²×3₋¹⁺²×C₃₁

C₃

yes

maximal, characteristic

C₃×#11719689

—

C₃×#11719689

—

C₃×#11719689

C₃²

yes

—

C₃²×C₃₁⋊C₉

—

C₃²×C₃₁⋊C₉

—

C₃²×C₃₁⋊C₉

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

—

C₃²⋊(C₃₁⋊C₉)

C₃²

yes

—

C₃²⋊(C₃₁⋊C₉)

C₃²

yes

—

C₃⁴×C₃₁

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

—

C₃×3₋¹⁺²×C₃₁

C₃²

yes

—

C₃²×C₉×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

H(3,𝔽₃)

yes

—

C₃³×C₃₁

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₁

C₃³

yes

characteristic

3₋¹⁺²×C₃₁

—

3₋¹⁺²×C₃₁

—

C₃×C₉×C₃₁

—

C₃×C₉×C₃₁

—

#11719689

—

#11719689

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×#6860

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₃₁

#733

yes

—

C₃²×C₃₁

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₁⋊C₃

yes

characteristic

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

#6860

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

#6860

yes

complemented

C₃×C₃₁

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₁

C₃×AΣL(1,𝔽₂₇)

yes

characteristic

C₃×C₃₁

C₃×#733

yes

—

C₃⁴

C₃×C₃₁⋊C₃

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×C₃₁⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃₁

C₃×#6860

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

—

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

C₃³

C₃²×C₃₁⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

279

—

C₃×C₉

—

279

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

#14578894

yes

—

108

C₃²

#14578895

yes

—

109

C₃²

#14578901

yes

—

110

C₃²

C₃×C₃²⋊(C₃₁⋊C₃)

yes

characteristic

111

C₉

—

112

C₉

—

279

—

113

C₉

—

279

—

114

C₉

—

115

C₃

—

116

C₃

—

117

C₃

#21419212

yes

direct product factor

118

C₃

C₃×#14578894

yes

characteristic

119

C₃

C₃×#14578895

yes

characteristic

120

C₃

C₃×#14578901

yes

—

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃³×C₃₁	C₃³
Frattini	C₃³	C₃²×C₃₁⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₁	C₃
socle	C₃³×C₃₁	H(3,𝔽₃)
prime radical	C₃⁴×C₃₁	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6860
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃³

C₃²⋊(C₃₁⋊C₃)

commutator, cosocle

C₃³×C₃₁

C₃³

Frattini

C₃³

C₃²×C₃₁⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₁

C₃

socle

C₃³×C₃₁

H(3,𝔽₃)

prime radical

C₃⁴×C₃₁

C₃²

hypercenter

C₃²×3₋¹⁺²

C₃₁⋊C₃

hypocenter

C₃₁

C₃×#6860

LCS

G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
4. #21419212	C₃	direct (N × H)
62. C₃×C₃₁	#6860	semidirect
81. C₃₁	C₃×#6860	semidirect
117. C₃	#21419212	direct (N × H)

H ≅ G/N

type

4. #21419212

C₃

direct (N × H)

62. C₃×C₃₁

#6860

semidirect

81. C₃₁

C₃×#6860

semidirect

117. C₃

#21419212

direct (N × H)

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

subgroup	index
C₃²×#11719689	3
C₃×C₃²⋊(C₃₁⋊C₉)	3
#21419212	3
C₃²×3₋¹⁺²×C₃₁	3
C₃×#6860	31

subgroup

index

C₃²×#11719689

C₃×C₃²⋊(C₃₁⋊C₉)

#21419212

C₃²×3₋¹⁺²×C₃₁

C₃×#6860

Subgroups — C₃×#21419212 #36380037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#21419212 #36380037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419212 #36380037