Group #44532104 — subgroups

Subgroups — #44532104

Order 31347.

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×#11923958	C₃	1	1	yes	maximal, characteristic
2	#23782712	C₃	1	2	yes	maximal
3	C₄₃×#6866	C₃	1	1	yes	maximal, characteristic
4	C₃×#11923958	—	3	3	no	—
5	C₃×#11923958	—	3	6	no	—
6	C₃×#11923958	—	3	1	no	—
7	C₃×#11923958	—	3	1	no	—
8	C₃×#11923958	—	3	1	no	—
9	C₃²×C₄₃⋊C₉	C₉	1	3	yes	complemented
10	C₃²⋊(C₄₃⋊C₉)	—	3	2	no	—
11	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
12	C₄₃×C₃²⋊C₉	—	3	1	no	—
13	#11923958	—	3	9	no	—
14	#11923958	—	3	9	no	—
15	#11923958	—	3	3	no	—
16	#11923958	—	3	6	no	—
17	C₃×C₄₃⋊C₉	—	3	3	no	—
18	C₃×C₄₃⋊C₉	—	3	9	no	—
19	C₃×C₄₃⋊C₉	—	9	2	no	—
20	C₃³×C₄₃	—	3	1	no	—
21	C₃³×C₄₃	—	3	2	no	—
22	C₃³×C₄₃	—	9	3	no	—
23	C₃³×C₄₃	C₃×C₉	1	1	yes	characteristic
24	C₃³×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃³×C₄₃	3₋¹⁺²	1	1	yes	characteristic
26	C₃³×C₄₃	3₋¹⁺²	1	1	yes	characteristic
27	C₃×C₉×C₄₃	—	9	1	no	—
28	#10624506	—	43	1	no	maximal, Sylow
29	C₃²×C₄₃	—	3	1	no	—
30	C₃²×C₄₃	—	3	3	no	—
31	C₃²×C₄₃	—	3	1	no	—
32	C₃²×C₄₃	—	3	1	no	—
33	C₃²×C₄₃	—	9	1	no	—
34	C₃²×C₄₃	—	9	6	no	—
35	C₃²×C₄₃	—	9	3	no	—
36	C₃²×C₄₃	—	9	1	no	—
37	C₃²×C₄₃	—	9	1	no	—
38	C₃²×C₄₃	C₃²⋊C₉	1	1	yes	characteristic
39	C₃²×C₄₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
40	C₃²×C₄₃	AΣL(1,𝔽₂₇)	1	2	yes	—
41	C₉×C₄₃	—	27	1	no	—
42	C₄₃⋊C₉	—	3	9	no	—
43	C₄₃⋊C₉	—	27	2	no	—
44	C₃²×3₋¹⁺²	—	43	1	no	—
45	#6866	—	43	2	no	—
46	#6866	C₄₃⋊C₃	1	1	yes	characteristic
47	C₃×C₄₃	—	3	2	no	—
48	C₃×C₄₃	—	3	1	no	—
49	C₃×C₄₃	—	3	3	no	—
50	C₃×C₄₃	—	3	1	no	—
51	C₃×C₄₃	—	9	1	no	—
52	C₃×C₄₃	—	9	1	no	—
53	C₃×C₄₃	#6866	1	1	yes	characteristic
54	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	129	1	no	—
56	C₃×3₋¹⁺²	—	129	1	no	—
57	C₃×3₋¹⁺²	—	129	1	no	—
58	C₃×3₋¹⁺²	—	129	3	no	—
59	C₃×3₋¹⁺²	—	129	6	no	—
60	C₃²×C₉	—	43	3	no	—
61	C₃²⋊C₉	—	3	1	no	—
62	C₃²⋊C₉	—	129	2	no	—
63	C₄₃	#10624506	1	1	yes	characteristic, complemented, Sylow
64	C₃³	—	3	1	no	—
65	C₃³	—	3	2	no	—
66	C₃³	—	9	3	no	—
67	C₃³	C₉×C₄₃⋊C₃	1	1	yes	characteristic
68	C₃³	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
69	C₃³	#11923956	1	1	yes	characteristic
70	C₃³	#11923957	1	1	yes	characteristic
71	3₋¹⁺²	—	129	9	no	—
72	3₋¹⁺²	—	129	9	no	—
73	3₋¹⁺²	—	129	3	no	—
74	3₋¹⁺²	—	129	6	no	—
75	C₃×C₉	—	9	1	no	—
76	C₃×C₉	—	129	3	no	—
77	C₃×C₉	—	129	9	no	—
78	C₃×C₉	—	387	2	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×C₄₃⋊C₃)⋊C₉	1	1	yes	characteristic
89	C₃²	#16812506	1	1	yes	characteristic
90	C₃²	#16812508	1	2	yes	—
91	C₉	—	27	1	no	—
92	C₉	—	129	9	no	—
93	C₉	—	1161	2	no	—
94	C₃	—	3	1	no	—
95	C₃	—	3	2	no	—
96	C₃	—	3	3	no	—
97	C₃	—	3	1	no	—
98	C₃	—	9	1	no	—
99	C₃	—	9	1	no	—
100	C₃	#23782711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11923958

C₃

yes

maximal, characteristic

#23782712

C₃

yes

maximal

C₄₃×#6866

C₃

yes

maximal, characteristic

—

—

—

—

—

C₃²×C₄₃⋊C₉

C₉

yes

complemented

C₃²⋊(C₄₃⋊C₉)

—

C₃⁴×C₄₃

C₃²

yes

characteristic

C₄₃×C₃²⋊C₉

—

—

—

—

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃×C₄₃⋊C₉

—

C₃³×C₄₃

—

C₃³×C₄₃

—

C₃³×C₄₃

—

C₃³×C₄₃

C₃×C₉

yes

characteristic

C₃³×C₄₃

H(3,𝔽₃)

yes

characteristic

C₃³×C₄₃

3₋¹⁺²

yes

characteristic

C₃³×C₄₃

3₋¹⁺²

yes

characteristic

C₃×C₉×C₄₃

—

#10624506

—

maximal, Sylow

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

—

C₃²×C₄₃

C₃²⋊C₉

yes

characteristic

C₃²×C₄₃

AΣL(1,𝔽₂₇)

yes

characteristic

C₃²×C₄₃

AΣL(1,𝔽₂₇)

yes

—

C₉×C₄₃

—

C₄₃⋊C₉

—

C₄₃⋊C₉

—

C₃²×3₋¹⁺²

—

#6866

—

#6866

C₄₃⋊C₃

yes

characteristic

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

—

C₃×C₄₃

#6866

yes

characteristic

C₃⁴

C₃×C₄₃⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

129

—

C₃×3₋¹⁺²

—

129

—

C₃×3₋¹⁺²

—

129

—

C₃×3₋¹⁺²

—

129

—

C₃×3₋¹⁺²

—

129

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

129

—

C₄₃

#10624506

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

C₉×C₄₃⋊C₃

yes

characteristic

C₃³

C₃²⋊(C₄₃⋊C₃)

yes

C₃³

yes

C₃³

yes

3₋¹⁺²

—

129

—

3₋¹⁺²

—

129

—

3₋¹⁺²

—

129

—

3₋¹⁺²

—

129

—

C₃×C₉

—

C₃×C₉

—

129

—

C₃×C₉

—

129

—

C₃×C₉

—

387

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

(C₃×C₄₃⋊C₃)⋊C₉

yes

C₃²

yes

C₃²

yes

—

C₉

—

C₉

—

129

—

C₉

—

1161

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

100

C₃

#23782711

yes

characteristic

	subgroup	G/H
center	C₃	#23782711
commutator	C₃³×C₄₃	C₃×C₉
Frattini	C₃⁴	C₃×C₄₃⋊C₃
Fitting	C₄₃×#6866	C₃
socle	C₃×C₄₃	#6866
prime radical, cosocle	C₃⁴×C₄₃	C₃²
hypercenter	#6866	C₄₃⋊C₃
hypocenter	C₄₃	#10624506
LCS	G ▷ C₃³×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

subgroup

G/H

center

C₃

#23782711

commutator

C₃³×C₄₃

C₃×C₉

Frattini

C₃⁴

C₃×C₄₃⋊C₃

Fitting

C₄₃×#6866

C₃

socle

C₃×C₄₃

#6866

prime radical, cosocle

C₃⁴×C₄₃

C₃²

C₄₃⋊C₃

C₄₃

G ▷ C₃³×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

N	H ≅ G/N	type
9. C₃²×C₄₃⋊C₉	C₉	semidirect
63. C₄₃	#10624506	semidirect

H ≅ G/N

type

9. C₃²×C₄₃⋊C₉

C₉

semidirect

63. C₄₃

#10624506

semidirect

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

subgroup	index
C₃²×#11923958	3
#23782712	3
C₄₃×#6866	3
#10624506	43

subgroup

index

C₃²×#11923958

#23782712

C₄₃×#6866

#10624506

Subgroups — #44532104

Conjugacy classes of subgroups

Complements

Maximal subgroups