Group #41006387 — subgroups

Subgroups — #732×#66524 #41006387

Order 26973.

328 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 100 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	H(3,𝔽₃)×#66524	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)⋊#66524	C₃	1	6	yes	maximal, complemented
4	#22582472	C₃	1	2	yes	maximal, complemented
5	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	—	3	3	no	—
7	C₃²×#66524	C₃²	1	2	yes	—
8	C₃²×#66524	C₃²	1	1	yes	characteristic
9	C₃²⋊#66524	—	3	6	no	—
10	C₃²⋊#66524	C₃²	1	6	yes	—
11	C₉×#66524	—	3	2	no	—
12	C₉×#66524	—	3	1	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×#732	C₉	1	9	yes	complemented
16	C₃³×C₃₇	—	3	3	no	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₇	—	3	6	no	—
19	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	complemented
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	complemented
23	C₃×#66524	—	3	2	no	—
24	C₃×#66524	—	3	3	no	—
25	C₃×#66524	—	3	6	no	—
26	C₃×#66524	—	3	6	no	—
27	C₃×#66524	—	9	9	no	—
28	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
29	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
30	C₉×#732	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	9	3	no	—
35	C₃²×C₃₇	—	9	6	no	—
36	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃²×C₃₇	#737	1	2	yes	—
39	C₉×C₃₇	—	3	3	no	—
40	#66524	—	3	2	no	—
41	#66524	—	3	6	no	—
42	#66524	—	9	6	no	—
43	#66524	#732	1	3	yes	direct product factor
44	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic
45	C₉×H(3,𝔽₃)	—	37	3	no	—
46	C₃×C₉²	—	37	1	no	—
47	H(3,𝔽₃)⋊C₉	—	37	6	no	—
48	#6868	—	37	2	no	—
49	C₃×C₃₇	—	3	1	no	—
50	C₃×C₃₇	—	3	2	no	—
51	C₃×C₃₇	—	9	9	no	—
52	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
54	C₃×C₃₇	#6908	1	2	yes	—
55	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
56	C₃²×C₉	—	37	1	no	—
57	C₃²×C₉	—	37	2	no	—
58	C₃²×C₉	—	111	3	no	—
59	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	#732	#66524	1	9	yes	direct product factor
61	C₃²⋊C₉	—	37	6	no	—
62	C₃²⋊C₉	—	111	6	no	—
63	C₉²	—	111	1	no	—
64	C₉²	—	111	2	no	—
65	C₃₇	C₉×#732	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	3	3	no	—
67	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
68	H(3,𝔽₃)	—	3	6	no	—
69	H(3,𝔽₃)	C₃×#66524	1	9	yes	complemented
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	37	1	no	—
73	C₃×C₉	—	37	2	no	—
74	C₃×C₉	—	333	9	no	—
75	C₃×C₉	—	111	3	no	—
76	C₃×C₉	—	111	6	no	—
77	C₃×C₉	—	111	2	no	—
78	C₃×C₉	—	111	6	no	—
79	C₃×C₉	C₃×#66524	1	3	yes	complemented
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	9	no	—
83	C₃²	—	9	3	no	—
84	C₃²	—	9	6	no	—
85	C₃²	C₃²×#66524	1	1	yes	characteristic
86	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
87	C₃²	#15688530	1	2	yes	—
88	C₉	—	3	3	no	—
89	C₉	—	37	3	no	—
90	C₉	—	333	6	no	—
91	C₉	—	111	2	no	—
92	C₉	—	111	6	no	—
93	C₃	—	3	1	no	—
94	C₃	—	3	2	no	—
95	C₃	—	9	9	no	—
96	C₃	H(3,𝔽₃)×#66524	1	1	yes	characteristic
97	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃	#22582631	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#66524

C₃

yes

maximal, complemented

C₃×C₉×#66524

C₃

yes

maximal, characteristic, complemented

H(3,𝔽₃)⋊#66524

C₃

yes

maximal, complemented

#22582472

C₃

yes

maximal, complemented

C₃×C₃₇×#732

C₃

yes

maximal, characteristic

C₃²×#66524

—

C₃²×#66524

C₃²

yes

—

C₃²×#66524

C₃²

yes

characteristic

C₃²⋊#66524

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

—

C₉×#66524

—

C₃×H(3,𝔽₃)×C₃₇

C₃²

yes

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×#732

C₉

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

H(3,𝔽₃)×C₃₇

—

H(3,𝔽₃)×C₃₇

C₃×C₉

yes

complemented

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

—

—

—

—

—

yes

—

yes

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₃₇

#737

yes

—

C₉×C₃₇

—

—

—

—

yes

direct product factor

C₃×#732

C₃₇⋊C₃

yes

characteristic

C₉×H(3,𝔽₃)

—

C₃×C₉²

—

H(3,𝔽₃)⋊C₉

—

#6868

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₉×H(3,𝔽₃)

yes

C₃×C₃₇

yes

C₃×C₃₇

yes

—

C₃×H(3,𝔽₃)

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

#732

#66524

yes

direct product factor

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₉²

—

111

—

C₃₇

C₉×#732

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

C₃²×C₃₇⋊C₃

yes

—

yes

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66524

yes

complemented

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×#66524

yes

characteristic

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₃²

#15688530

yes

—

C₉

—

C₉

—

C₉

—

333

—

C₉

—

111

—

C₉

—

111

—

C₃

—

C₃

—

C₃

—

C₃

H(3,𝔽₃)×#66524

yes

characteristic

C₃

#732×C₃₇⋊C₃

yes

characteristic

C₃

#22582631

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	C₉×#732
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting, prime radical

C₃×C₃₇×#732

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

cosocle

C₃³×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

N	H ≅ G/N	type
1. H(3,𝔽₃)×#66524	C₃	semidirect
2. C₃×C₉×#66524	C₃	semidirect
3. H(3,𝔽₃)⋊#66524	C₃	semidirect
4. #22582472	C₃	semidirect
15. C₃₇×#732	C₉	semidirect
19. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
22. C₃×C₉×C₃₇	C₃×C₉	semidirect
43. #66524	#732	direct (N × H)
60. #732	#66524	direct (N × H)
65. C₃₇	C₉×#732	semidirect
69. H(3,𝔽₃)	C₃×#66524	semidirect
79. C₃×C₉	C₃×#66524	semidirect

H ≅ G/N

type

1. H(3,𝔽₃)×#66524

C₃

semidirect

2. C₃×C₉×#66524

C₃

semidirect

3. H(3,𝔽₃)⋊#66524

C₃

semidirect

4. #22582472

C₃

semidirect

15. C₃₇×#732

C₉

semidirect

19. H(3,𝔽₃)×C₃₇

C₃×C₉

semidirect

22. C₃×C₉×C₃₇

C₃×C₉

semidirect

43. #66524

#732

direct (N × H)

60. #732

#66524

direct (N × H)

65. C₃₇

C₉×#732

semidirect

69. H(3,𝔽₃)

C₃×#66524

semidirect

79. C₃×C₉

C₃×#66524

semidirect

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

subgroup	index
H(3,𝔽₃)×#66524	3
C₃×C₉×#66524	3
H(3,𝔽₃)⋊#66524	3
#22582472	3
C₃×C₃₇×#732	3
C₉×#732	37

subgroup

index

C₃×C₉×#66524

C₃×C₃₇×#732

Subgroups — #732×#66524 #41006387

Conjugacy classes of subgroups

Complements

Maximal subgroups