Group #41008551 — subgroups

Subgroups — #41008551

Order 26973.

198 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 84 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	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal
2	#22582722	C₃	1	1	yes	maximal, characteristic
3	#22582711	C₃	1	3	yes	maximal, complemented
4	#22582709	C₃	1	3	yes	maximal, complemented
5	#22582707	C₃	1	3	yes	maximal, complemented
6	C₃×#15688510	C₃	1	1	yes	maximal, characteristic
7	#15688543	—	3	2	no	—
8	#15688542	C₃²	1	6	yes	—
9	#15688541	—	3	2	no	—
10	(C₃₇⋊C₃)⋊C₂₇	—	3	2	no	—
11	#15688538	—	3	1	no	—
12	#15688537	C₃²	1	3	yes	—
13	C₃²⋊#66526	—	3	1	no	—
14	C₉×#66526	—	3	1	no	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
16	#15688510	C₉	1	2	yes	complemented
17	#15688510	C₉	1	2	yes	complemented
18	#15688510	C₉	1	2	yes	complemented
19	#15688510	C₃²	1	1	yes	characteristic
20	#15688510	C₃²	1	1	yes	characteristic
21	#15688510	C₃²	1	1	yes	characteristic
22	C₃×#11758601	C₉	1	2	yes	complemented
23	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic, complemented
24	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
25	C₃×#66526	—	3	3	no	—
26	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
27	C₃×#66524	3₋¹⁺²	1	2	yes	—
28	C₉×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
29	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
30	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
31	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
32	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
33	C₉×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
34	#11758603	—	3	6	no	—
35	C₂₇×C₃₇	C₃×C₉	1	1	yes	characteristic
36	C₂₇×C₃₇	C₃×C₉	1	1	yes	characteristic
37	C₂₇×C₃₇	C₃×C₉	1	1	yes	characteristic
38	#11758601	C₃×C₉	1	2	yes	—
39	#11758601	C₃×C₉	1	2	yes	—
40	#11758601	C₃×C₉	1	2	yes	—
41	#10624709	—	37	1	no	maximal, Sylow
42	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
43	C₃×C₃₇⋊C₃	—	9	1	no	—
44	C₃×C₃₇⋊C₃	C₉⋊C₉	1	2	yes	—
45	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
46	#66526	—	9	3	no	—
47	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
48	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
49	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
50	#66524	C₉⋊C₉	1	2	yes	—
51	#66524	C₉⋊C₉	1	2	yes	—
52	#66524	C₃×3₋¹⁺²	1	2	yes	—
53	C₃×C₂₇⋊C₃	—	37	1	no	—
54	C₃×C₂₇⋊C₃	—	37	2	no	—
55	#6887	—	37	1	no	—
56	#6875	—	37	3	no	—
57	#6875	—	37	3	no	—
58	#6875	—	37	3	no	—
59	C₃×C₃₇	—	3	1	no	—
60	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
61	C₃₇⋊C₃	—	3	1	no	—
62	C₃₇⋊C₃	—	3	2	no	—
63	C₃²×C₉	—	37	1	no	—
64	C₂₇⋊C₃	—	37	2	no	—
65	C₂₇⋊C₃	—	37	2	no	—
66	C₂₇⋊C₃	—	37	2	no	—
67	C₂₇⋊C₃	—	37	1	no	—
68	C₂₇⋊C₃	—	37	1	no	—
69	C₂₇⋊C₃	—	37	1	no	—
70	C₂₇⋊C₃	—	37	6	no	—
71	C₂₇⋊C₃	—	111	2	no	—
72	C₂₇⋊C₃	—	111	2	no	—
73	C₃×C₂₇	—	37	2	no	—
74	C₃×C₂₇	—	111	2	no	—
75	C₃×C₂₇	#66526	1	1	yes	characteristic, complemented
76	C₉⋊C₉	—	37	3	no	—
77	C₉⋊C₉	—	111	1	no	—
78	C₃²⋊C₉	—	111	1	no	—
79	C₉²	—	111	1	no	—
80	C₃₇	#10624709	1	1	yes	characteristic, complemented, Sylow
81	C₃³	—	37	1	no	—
82	C₃×C₉	—	37	2	no	—
83	C₃×C₉	—	37	1	no	—
84	C₃×C₉	—	37	2	no	—
85	C₃×C₉	—	37	1	no	—
86	C₃×C₉	—	37	2	no	—
87	C₃×C₉	—	37	2	no	—
88	C₃×C₉	—	37	1	no	—
89	C₃×C₉	—	111	3	no	—
90	C₃×C₉	C₃×#66526	1	1	yes	characteristic
91	C₂₇	—	37	2	no	—
92	C₂₇	—	37	2	no	—
93	C₂₇	—	37	2	no	—
94	C₂₇	—	111	6	no	—
95	C₂₇	C₃×#66526	1	1	yes	characteristic
96	C₂₇	C₃×#66526	1	1	yes	characteristic
97	C₂₇	C₃×#66526	1	1	yes	characteristic
98	C₃²	—	37	1	no	—
99	C₃²	—	37	2	no	—
100	C₃²	—	333	1	no	—
101	C₃²	#15688537	1	1	yes	characteristic
102	C₉	—	37	2	no	—
103	C₉	—	37	2	no	—
104	C₉	—	37	2	no	—
105	C₉	—	333	3	no	—
106	C₉	#15688537	1	1	yes	characteristic
107	C₉	#15688537	1	1	yes	characteristic
108	C₉	C₃²×#66526	1	1	yes	characteristic
109	C₃	—	3	1	no	—
110	C₃	—	111	1	no	—
111	C₃	—	111	2	no	—
112	C₃	C₃×#15688537	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal

#22582722

C₃

yes

maximal, characteristic

#22582711

C₃

yes

maximal, complemented

#22582709

C₃

yes

maximal, complemented

#22582707

C₃

yes

maximal, complemented

C₃×#15688510

C₃

yes

maximal, characteristic

#15688543

—

#15688542

C₃²

yes

—

#15688541

—

(C₃₇⋊C₃)⋊C₂₇

—

#15688538

—

#15688537

C₃²

yes

—

C₃²⋊#66526

—

C₉×#66526

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688510

C₉

yes

complemented

#15688510

C₉

yes

complemented

#15688510

C₉

yes

complemented

#15688510

C₃²

yes

characteristic

#15688510

C₃²

yes

characteristic

#15688510

C₃²

yes

characteristic

C₃×#11758601

C₉

yes

complemented

C₃×C₂₇×C₃₇

C₉

yes

characteristic, complemented

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃×#66526

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×#66524

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

C₃×C₉

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

C₃³

yes

characteristic

#11758603

—

C₂₇×C₃₇

C₃×C₉

yes

characteristic

C₂₇×C₃₇

C₃×C₉

yes

characteristic

C₂₇×C₃₇

C₃×C₉

yes

characteristic

#11758601

C₃×C₉

yes

—

#11758601

C₃×C₉

yes

—

#11758601

C₃×C₉

yes

—

#10624709

—

maximal, Sylow

C₃²×C₃₇

C₉⋊C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

—

C₃×C₃₇⋊C₃

C₃×3₋¹⁺²

yes

characteristic

#66526

—

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₉⋊C₉

yes

characteristic

C₉×C₃₇

C₃²×C₉

yes

characteristic

#66524

C₉⋊C₉

yes

—

#66524

C₉⋊C₉

yes

—

#66524

C₃×3₋¹⁺²

yes

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

#66526

yes

characteristic, complemented

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624709

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

C₃×#66526

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

—

111

—

C₂₇

yes

C₂₇

yes

C₂₇

yes

C₃²

—

C₃²

—

100

C₃²

—

333

—

101

C₃²

#15688537

yes

characteristic

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

333

—

106

C₉

#15688537

yes

characteristic

107

C₉

#15688537

yes

characteristic

108

C₉

C₃²×#66526

yes

characteristic

109

C₃

—

110

C₃

—

111

—

111

C₃

—

111

—

112

C₃

C₃×#15688537

yes

characteristic

	subgroup	G/H
center	C₉	#15688537
commutator	C₉×C₃₇	C₃²×C₉
Frattini	C₉	C₃²×#66526
Fitting	C₃×C₂₇×C₃₇	C₉
socle	C₃×C₃₇	C₃×C₉⋊C₉
prime radical	C₃²×C₃₇⋊C₃	3₋¹⁺²
cosocle	C₉×C₃₇⋊C₃	C₃³
hypercenter	C₃×C₂₇	#66526
hypocenter	C₃₇	#10624709
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

subgroup

G/H

center

C₉

#15688537

commutator

C₉×C₃₇

C₃²×C₉

Frattini

C₉

C₃²×#66526

Fitting

C₃×C₂₇×C₃₇

C₉

socle

C₃×C₃₇

C₃×C₉⋊C₉

prime radical

C₃²×C₃₇⋊C₃

3₋¹⁺²

cosocle

C₉×C₃₇⋊C₃

C₃³

C₃×C₂₇

C₃₇

G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

N	H ≅ G/N	type
3. #22582711	C₃	semidirect
4. #22582709	C₃	semidirect
5. #22582707	C₃	semidirect
16. #15688510	C₉	semidirect
17. #15688510	C₉	semidirect
18. #15688510	C₉	semidirect
22. C₃×#11758601	C₉	semidirect
23. C₃×C₂₇×C₃₇	C₉	semidirect
75. C₃×C₂₇	#66526	semidirect
80. C₃₇	#10624709	semidirect

H ≅ G/N

type

3. #22582711

C₃

semidirect

4. #22582709

C₃

semidirect

5. #22582707

C₃

semidirect

16. #15688510

C₉

semidirect

17. #15688510

C₉

semidirect

18. #15688510

C₉

semidirect

22. C₃×#11758601

C₉

semidirect

23. C₃×C₂₇×C₃₇

C₉

semidirect

75. C₃×C₂₇

#66526

semidirect

80. C₃₇

#10624709

semidirect

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

subgroup	index
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
#22582722	3
#22582711	3
#22582709	3
#22582707	3
C₃×#15688510	3
#10624709	37

subgroup

index

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

Subgroups — #41008551

Conjugacy classes of subgroups

Complements

Maximal subgroups