Group #36379716 — subgroups

Subgroups — #36379716

Order 22599.

242 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 31 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	#21419226	C₃	1	2	yes	maximal
3	C₃₁×#6866	C₃	1	1	yes	maximal, characteristic
4	C₃×#11719689	—	3	1	no	—
5	C₃×#11719689	—	3	1	no	—
6	C₃×#11719689	—	3	3	no	—
7	C₃×#11719689	—	3	6	no	—
8	C₃×#11719689	C₉	1	3	yes	complemented
9	C₃²×C₃₁⋊C₉	—	3	1	no	—
10	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
11	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
12	C₃₁×C₃²⋊C₉	—	3	1	no	—
13	C₃³×C₃₁	—	3	1	no	—
14	C₃³×C₃₁	—	3	2	no	—
15	C₃³×C₃₁	—	9	3	no	—
16	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
17	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
19	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃×C₉×C₃₁	—	9	1	no	—
21	#11719689	—	3	9	no	—
22	#11719689	—	3	3	no	—
23	#11719689	—	3	6	no	—
24	#11719689	—	3	9	no	—
25	C₃×C₃₁⋊C₉	—	3	9	no	—
26	C₃×C₃₁⋊C₉	—	3	3	no	—
27	C₃×C₃₁⋊C₉	—	9	2	no	—
28	#10624509	—	31	1	no	maximal, Sylow
29	C₃²×C₃₁	—	3	3	no	—
30	C₃²×C₃₁	—	3	1	no	—
31	C₃²×C₃₁	—	3	1	no	—
32	C₃²×C₃₁	—	3	1	no	—
33	C₃²×C₃₁	—	9	1	no	—
34	C₃²×C₃₁	—	9	6	no	—
35	C₃²×C₃₁	—	9	3	no	—
36	C₃²×C₃₁	—	9	1	no	—
37	C₃²×C₃₁	—	9	1	no	—
38	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
39	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
40	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
41	C₉×C₃₁	—	27	1	no	—
42	C₃₁⋊C₉	—	3	9	no	—
43	C₃₁⋊C₉	—	27	2	no	—
44	C₃²×3₋¹⁺²	—	31	1	no	—
45	#6866	—	31	2	no	—
46	#6866	C₃₁⋊C₃	1	1	yes	characteristic
47	C₃×C₃₁	—	3	2	no	—
48	C₃×C₃₁	—	3	1	no	—
49	C₃×C₃₁	—	3	3	no	—
50	C₃×C₃₁	—	3	1	no	—
51	C₃×C₃₁	—	9	1	no	—
52	C₃×C₃₁	—	9	1	no	—
53	C₃×C₃₁	#6866	1	1	yes	characteristic
54	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	31	3	no	—
56	C₃×3₋¹⁺²	—	93	1	no	—
57	C₃×3₋¹⁺²	—	93	1	no	—
58	C₃×3₋¹⁺²	—	93	3	no	—
59	C₃×3₋¹⁺²	—	93	6	no	—
60	C₃²×C₉	—	93	1	no	—
61	C₃²⋊C₉	—	3	1	no	—
62	C₃²⋊C₉	—	93	2	no	—
63	C₃₁	#10624509	1	1	yes	characteristic, complemented, Sylow
64	C₃³	—	3	1	no	—
65	C₃³	—	3	2	no	—
66	C₃³	—	9	3	no	—
67	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
68	C₃³	#11719686	1	1	yes	characteristic
69	C₃³	#11719687	1	1	yes	characteristic
70	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
71	3₋¹⁺²	—	93	9	no	—
72	3₋¹⁺²	—	93	3	no	—
73	3₋¹⁺²	—	93	6	no	—
74	3₋¹⁺²	—	93	9	no	—
75	C₃×C₉	—	9	1	no	—
76	C₃×C₉	—	279	2	no	—
77	C₃×C₉	—	93	9	no	—
78	C₃×C₉	—	93	3	no	—
79	C₃²	—	3	3	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	3	1	no	—
83	C₃²	—	9	1	no	—
84	C₃²	—	9	3	no	—
85	C₃²	—	9	6	no	—
86	C₃²	—	9	1	no	—
87	C₃²	—	9	1	no	—
88	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
89	C₃²	#14578893	1	1	yes	characteristic
90	C₃²	#14578895	1	2	yes	—
91	C₉	—	27	1	no	—
92	C₉	—	837	2	no	—
93	C₉	—	93	9	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	2	no	—
96	C₃	—	3	1	no	—
97	C₃	—	3	3	no	—
98	C₃	—	9	1	no	—
99	C₃	—	9	1	no	—
100	C₃	#21419225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11719689

C₃

yes

maximal, characteristic

#21419226

C₃

yes

maximal

C₃₁×#6866

C₃

yes

maximal, characteristic

—

—

—

—

C₉

yes

complemented

C₃²×C₃₁⋊C₉

—

C₃²⋊(C₃₁⋊C₉)

—

C₃⁴×C₃₁

C₃²

yes

characteristic

C₃₁×C₃²⋊C₉

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

—

C₃³×C₃₁

C₃×C₉

yes

characteristic

C₃³×C₃₁

H(3,𝔽₃)

yes

characteristic

C₃³×C₃₁

3₋¹⁺²

yes

characteristic

C₃³×C₃₁

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₁

—

—

—

—

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

C₃×C₃₁⋊C₉

—

#10624509

—

maximal, Sylow

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

—

C₃²×C₃₁

C₃²⋊C₉

yes

characteristic

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

characteristic

C₃²×C₃₁

AΣL(1,𝔽₂₇)

yes

—

C₉×C₃₁

—

C₃₁⋊C₉

—

C₃₁⋊C₉

—

C₃²×3₋¹⁺²

—

#6866

—

#6866

C₃₁⋊C₃

yes

characteristic

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

—

C₃×C₃₁

#6866

yes

characteristic

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

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃₁

#10624509

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₉×C₃₁⋊C₃

yes

C₃³

yes

C₃³

yes

C₃³

C₃²⋊(C₃₁⋊C₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

279

—

C₃×C₉

—

C₃×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

(C₃×C₃₁⋊C₃)⋊C₉

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

837

—

C₉

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

100

C₃

#21419225

yes

characteristic

	subgroup	G/H
center	C₃	#21419225
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6866	C₃
socle	C₃×C₃₁	#6866
prime radical, cosocle	C₃⁴×C₃₁	C₃²
hypercenter	#6866	C₃₁⋊C₃
hypocenter	C₃₁	#10624509
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

subgroup

G/H

center

C₃

#21419225

commutator

C₃³×C₃₁

C₃×C₉

Frattini

C₃⁴

C₃×C₃₁⋊C₃

Fitting

C₃₁×#6866

C₃

socle

C₃×C₃₁

#6866

prime radical, cosocle

C₃⁴×C₃₁

C₃²

C₃₁⋊C₃

C₃₁

G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

N	H ≅ G/N	type
8. C₃×#11719689	C₉	semidirect
63. C₃₁	#10624509	semidirect

H ≅ G/N

type

8. C₃×#11719689

C₉

semidirect

63. C₃₁

#10624509

semidirect

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

subgroup	index
C₃²×#11719689	3
#21419226	3
C₃₁×#6866	3
#10624509	31

subgroup

index

C₃²×#11719689

#21419226

C₃₁×#6866

#10624509

Subgroups — #36379716

Conjugacy classes of subgroups

Complements

Maximal subgroups