Group #41006215 — subgroups

Subgroups — #41006215

Order 26973.

332 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₃×#15688531	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688511	C₃	1	2	yes	maximal, complemented
3	#22582465	C₃	1	9	yes	maximal, complemented
4	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	#15688531	—	3	3	no	—
6	#15688531	—	3	3	no	—
7	#15688531	—	3	3	no	—
8	C₃×#11758608	C₉	1	2	yes	—
9	C₃×#11758605	C₉	1	3	yes	—
10	C₃×#11758606	C₉	1	3	yes	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
12	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
13	#15688511	—	3	6	no	—
14	C₃×#11758601	C₃²	1	6	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
16	C₃×C₂₇×C₃₇	—	3	3	no	—
17	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
18	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
21	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
22	C₃×C₉×C₃₇	H(3,𝔽₃)	1	3	yes	—
23	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
27	#11758608	—	3	6	no	—
28	C₃×#66524	C₃×C₉	1	6	yes	—
29	#11758606	—	3	9	no	—
30	#11758605	—	3	9	no	—
31	C₉×C₃₇⋊C₃	—	3	3	no	—
32	C₉×C₃₇⋊C₃	—	3	3	no	—
33	C₉×C₃₇⋊C₃	—	3	3	no	—
34	C₂₇×C₃₇	—	3	3	no	—
35	#11758601	—	3	6	no	—
36	#10624684	—	37	1	no	maximal, Sylow
37	C₃²×C₃₇	—	3	3	no	—
38	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
39	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
40	C₃×C₃₇⋊C₃	—	3	3	no	—
41	C₃×C₃₇⋊C₃	—	3	9	no	—
42	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
43	C₉×C₃₇	C₃²⋊C₉	1	3	yes	—
44	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
47	#66524	—	3	6	no	—
48	C₃×#737	—	37	1	no	—
49	C₃×C₂₇⋊C₃	—	37	2	no	—
50	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
51	C₃²⋊C₂₇	—	37	9	no	—
52	C₃×C₃₇	—	3	3	no	—
53	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
54	C₃×C₃₇	#6903	1	3	yes	—
55	C₃₇⋊C₃	—	9	3	no	—
56	#737	—	111	3	no	—
57	#737	—	111	3	no	—
58	#737	—	111	3	no	—
59	C₃×3₋¹⁺²	—	37	2	no	—
60	C₃×3₋¹⁺²	—	37	3	no	—
61	C₃×3₋¹⁺²	—	37	3	no	—
62	C₃×H(3,𝔽₃)	—	37	1	no	—
63	C₃²×C₉	—	37	3	no	—
64	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
65	C₂₇⋊C₃	—	111	6	no	—
66	C₃×C₂₇	—	3	3	no	—
67	C₃×C₂₇	—	37	6	no	—
68	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
69	C₃₇	#10624684	1	1	yes	characteristic, complemented, Sylow
70	C₃³	—	37	3	no	—
71	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	111	6	no	—
73	3₋¹⁺²	—	111	9	no	—
74	3₋¹⁺²	—	111	9	no	—
75	H(3,𝔽₃)	—	111	3	no	—
76	C₃×C₉	—	37	6	no	—
77	C₃×C₉	—	111	3	no	—
78	C₃×C₉	—	111	3	no	—
79	C₃×C₉	—	111	3	no	—
80	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
82	C₃×C₉	#11758605	1	3	yes	—
83	C₃×C₉	#11758606	1	3	yes	—
84	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
85	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
86	C₂₇	—	3	3	no	—
87	C₂₇	—	111	6	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	111	3	no	—
90	C₃²	—	111	9	no	—
91	C₃²	#15688502	1	3	yes	—
92	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
93	C₉	—	111	6	no	—
94	C₉	#15688502	1	3	yes	—
95	C₉	#15688502	1	3	yes	—
96	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
97	C₉	C₃×#11758606	1	1	yes	characteristic
98	C₉	C₃×#11758605	1	1	yes	characteristic
99	C₃	—	3	3	no	—
100	C₃	—	333	3	no	—
101	C₃	C₃×#15688502	1	1	yes	characteristic
102	C₃	#22582612	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic

C₃×#15688511

C₃

yes

maximal, complemented

#22582465

C₃

yes

maximal, complemented

C₃²×C₂₇×C₃₇

C₃

yes

maximal, characteristic, complemented

—

—

—

C₉

yes

—

C₃×#11758605

C₉

yes

—

C₃×#11758606

C₉

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

#15688511

—

C₃×#11758601

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃×C₂₇×C₃₇

—

C₃×C₂₇×C₃₇

C₃²

yes

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

3₋¹⁺²

yes

—

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃×C₉

yes

—

#11758606

—

#11758605

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

#11758601

—

#10624684

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

C₃²⋊C₉

yes

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃²⋊C₉

yes

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

characteristic

#66524

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃²×C₂₇

C₃₇⋊C₃

yes

characteristic, complemented

C₃²⋊C₂₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃×C₃₇

#6903

yes

—

C₃₇⋊C₃

—

#737

—

111

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624684

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₉×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

C₃×C₉

#11758605

yes

—

C₃×C₉

#11758606

yes

—

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

111

—

C₃²

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

#15688502

yes

—

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

C₉

—

111

—

C₉

#15688502

yes

—

C₉

#15688502

yes

—

C₉

C₃×C₃²⋊(C₃₇⋊C₃)

yes

C₉

yes

C₉

yes

C₃

—

100

C₃

—

333

—

101

C₃

C₃×#15688502

yes

characteristic

102

C₃

#22582612

yes

—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃²×C₂₇×C₃₇	C₃
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624684
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₂₇

subgroup

G/H

center, Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

commutator, socle

C₃²×C₃₇

C₃²×C₉

Fitting

C₃²×C₂₇×C₃₇

C₃

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

cosocle

C₃×C₉×C₃₇

C₃³

hypercenter

C₃²×C₂₇

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃₇

C₁ ◁ C₃×C₉ ◁ C₃²×C₂₇

N	H ≅ G/N	type
2. C₃×#15688511	C₃	semidirect
3. #22582465	C₃	semidirect
4. C₃²×C₂₇×C₃₇	C₃	semidirect
50. C₃²×C₂₇	C₃₇⋊C₃	semidirect
69. C₃₇	#10624684	semidirect

H ≅ G/N

type

2. C₃×#15688511

C₃

semidirect

3. #22582465

C₃

semidirect

4. C₃²×C₂₇×C₃₇

C₃

semidirect

50. C₃²×C₂₇

C₃₇⋊C₃

semidirect

69. C₃₇

#10624684

semidirect

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

subgroup	index
C₃×#15688531	3
C₃×#15688511	3
#22582465	3
C₃²×C₂₇×C₃₇	3
#10624684	37

subgroup

index

C₃²×C₂₇×C₃₇

Subgroups — #41006215

Conjugacy classes of subgroups

Complements

Maximal subgroups