Group #41007300 — subgroups

Subgroups — #41007300

Order 26973.

142 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582495	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582495	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582494	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582494	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582493	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582492	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582491	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582491	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582473	C₃	1	1	yes	maximal, characteristic
11	#22582473	C₃	1	1	yes	maximal, characteristic
12	#22582473	C₃	1	1	yes	maximal, characteristic
13	C₃₇×#6875	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
16	#15688511	C₃²	1	1	yes	characteristic
17	#15688511	C₃²	1	1	yes	characteristic
18	#15688511	C₃²	1	1	yes	characteristic
19	#15688511	C₃²	1	1	yes	characteristic
20	#15688511	C₃²	1	1	yes	characteristic
21	#15688511	C₃²	1	1	yes	characteristic
22	#15688506	C₃²	1	1	yes	characteristic
23	#15688505	C₃²	1	1	yes	characteristic
24	#15688504	—	3	2	no	—
25	C₃₇×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
26	C₃₇×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
27	C₃₇×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
28	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
32	C₃×#66524	—	3	1	no	—
33	C₃×#66524	—	3	1	no	—
34	C₃×#66524	3₋¹⁺²	1	2	yes	—
35	C₉×C₃₇⋊C₃	—	3	1	no	—
36	C₉×C₃₇⋊C₃	—	3	1	no	—
37	C₉×C₃₇⋊C₃	—	3	1	no	—
38	C₉×C₃₇⋊C₃	—	9	3	no	—
39	C₂₇×C₃₇	—	3	1	no	—
40	C₂₇×C₃₇	—	3	1	no	—
41	C₂₇×C₃₇	—	3	1	no	—
42	#11758601	—	3	1	no	—
43	#11758601	—	3	1	no	—
44	#11758601	—	3	1	no	—
45	#11758601	—	3	1	no	—
46	#11758601	—	3	1	no	—
47	#11758601	—	3	1	no	—
48	#10624828	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
50	C₃×C₃₇⋊C₃	—	3	1	no	—
51	C₃×C₃₇⋊C₃	—	3	3	no	—
52	C₉×C₃₇	—	9	1	no	—
53	C₉×C₃₇	#737	1	1	yes	characteristic
54	C₉×C₃₇	#737	1	1	yes	characteristic
55	C₉×C₃₇	#737	1	1	yes	characteristic
56	#66524	—	3	2	no	—
57	#66524	—	9	1	no	—
58	#66524	—	9	1	no	—
59	C₃×C₉⋊C₉	—	37	1	no	—
60	#6875	—	37	1	no	—
61	#6875	—	37	1	no	—
62	#6875	—	37	1	no	—
63	#6875	—	37	1	no	—
64	#6875	—	37	1	no	—
65	#6875	—	37	1	no	—
66	#6875	—	37	1	no	—
67	#6875	—	37	1	no	—
68	#6875	C₃₇⋊C₃	1	1	yes	characteristic, complemented
69	#6869	—	37	1	no	—
70	#6869	—	37	1	no	—
71	#6869	—	37	1	no	—
72	C₃×C₃₇	—	3	1	no	—
73	C₃×C₃₇	#6895	1	1	yes	characteristic
74	C₃₇⋊C₃	—	3	3	no	—
75	C₃²×C₉	—	37	1	no	—
76	C₃²×C₉	—	111	1	no	—
77	C₂₇⋊C₃	—	37	1	no	—
78	C₂₇⋊C₃	—	37	1	no	—
79	C₂₇⋊C₃	—	37	1	no	—
80	C₂₇⋊C₃	—	37	1	no	—
81	C₂₇⋊C₃	—	37	1	no	—
82	C₂₇⋊C₃	—	37	1	no	—
83	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	1	yes	characteristic
84	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	1	yes	characteristic
85	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	1	yes	characteristic
86	C₉⋊C₉	—	37	1	no	—
87	C₉⋊C₉	—	37	1	no	—
88	C₉⋊C₉	—	111	2	no	—
89	C₉⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
90	C₃₇	#10624828	1	1	yes	characteristic, complemented, Sylow
91	C₃³	—	37	1	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	37	2	no	—
94	C₃×C₉	—	333	3	no	—
95	C₃×C₉	—	111	1	no	—
96	C₃×C₉	—	111	1	no	—
97	C₃×C₉	—	111	1	no	—
98	C₃×C₉	—	111	1	no	—
99	C₃×C₉	—	111	1	no	—
100	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
101	C₂₇	—	3	1	no	—
102	C₂₇	—	3	1	no	—
103	C₂₇	—	3	1	no	—
104	C₂₇	—	111	1	no	—
105	C₂₇	—	111	1	no	—
106	C₂₇	—	111	1	no	—
107	C₂₇	—	111	1	no	—
108	C₂₇	—	111	1	no	—
109	C₂₇	—	111	1	no	—
110	C₃²	—	111	3	no	—
111	C₃²	—	111	1	no	—
112	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
113	C₉	—	9	1	no	—
114	C₉	—	333	1	no	—
115	C₉	—	333	1	no	—
116	C₉	—	111	2	no	—
117	C₉	#15688531	1	1	yes	characteristic
118	C₉	#15688532	1	1	yes	characteristic
119	C₉	#15688532	1	1	yes	characteristic
120	C₃	—	3	1	no	—
121	C₃	—	111	3	no	—
122	C₃	#22582577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉⋊C₉×C₃₇⋊C₃

C₃

yes

maximal, characteristic

#22582495

C₃

yes

maximal, characteristic, complemented

#22582495

C₃

yes

maximal, characteristic, complemented

#22582494

C₃

yes

maximal, characteristic, complemented

#22582494

C₃

yes

maximal, characteristic, complemented

#22582493

C₃

yes

maximal, characteristic, complemented

#22582492

C₃

yes

maximal, characteristic, complemented

#22582491

C₃

yes

maximal, characteristic, complemented

#22582491

C₃

yes

maximal, characteristic, complemented

#22582473

C₃

yes

maximal, characteristic

#22582473

C₃

yes

maximal, characteristic

#22582473

C₃

yes

maximal, characteristic

C₃₇×#6875

C₃

yes

maximal, characteristic, complemented

C₃×C₉×C₃₇⋊C₃

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688511

C₃²

yes

characteristic

#15688506

C₃²

yes

characteristic

#15688505

C₃²

yes

characteristic

#15688504

—

C₃₇×C₂₇⋊C₃

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

C₃²

yes

characteristic

C₃²×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

3₋¹⁺²

yes

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₂₇×C₃₇

—

C₂₇×C₃₇

—

C₂₇×C₃₇

—

—

—

—

—

—

—

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

yes

C₉×C₃₇

yes

C₉×C₃₇

yes

—

—

—

C₃×C₉⋊C₉

—

—

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

#6869

—

#6869

—

#6869

—

C₃×C₃₇

—

C₃×C₃₇

#6895

yes

characteristic

C₃₇⋊C₃

—

C₃²×C₉

—

C₃²×C₉

—

111

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₉⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

#10624828

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

333

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

100

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

101

C₂₇

—

102

C₂₇

—

103

C₂₇

—

104

C₂₇

—

111

—

105

C₂₇

—

111

—

106

C₂₇

—

111

—

107

C₂₇

—

111

—

108

C₂₇

—

111

—

109

C₂₇

—

111

—

110

C₃²

—

111

—

111

C₃²

—

111

—

112

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

characteristic

113

C₉

—

114

C₉

—

333

—

115

C₉

—

333

—

116

C₉

—

111

—

117

C₉

#15688531

yes

characteristic

118

C₉

#15688532

yes

characteristic

119

C₉

#15688532

yes

characteristic

120

C₃

—

121

C₃

—

111

—

122

C₃

#22582577

yes

characteristic

	subgroup	G/H
center	C₃	#22582577
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6875	C₃
socle	C₃×C₃₇	#6895
prime radical	C₃²×C₃₇⋊C₃	3₋¹⁺²
hypercenter	#6875	C₃₇⋊C₃
hypocenter	C₃₇	#10624828
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6875

subgroup

G/H

center

C₃

#22582577

commutator, cosocle

C₃×C₉×C₃₇

C₃³

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6875

C₃

socle

C₃×C₃₇

#6895

prime radical

C₃²×C₃₇⋊C₃

3₋¹⁺²

C₃₇⋊C₃

C₃₇

G ▷ C₃×C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6875

N	H ≅ G/N	type
2. #22582495	C₃	semidirect
3. #22582495	C₃	semidirect
4. #22582494	C₃	semidirect
5. #22582494	C₃	semidirect
6. #22582493	C₃	semidirect
7. #22582492	C₃	semidirect
8. #22582491	C₃	semidirect
9. #22582491	C₃	semidirect
13. C₃₇×#6875	C₃	semidirect
68. #6875	C₃₇⋊C₃	semidirect
90. C₃₇	#10624828	semidirect

H ≅ G/N

type

2. #22582495

C₃

semidirect

3. #22582495

C₃

semidirect

4. #22582494

C₃

semidirect

5. #22582494

C₃

semidirect

6. #22582493

C₃

semidirect

7. #22582492

C₃

semidirect

8. #22582491

C₃

semidirect

9. #22582491

C₃

semidirect

13. C₃₇×#6875

C₃

semidirect

68. #6875

C₃₇⋊C₃

semidirect

90. C₃₇

#10624828

semidirect

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

subgroup	index
C₉⋊C₉×C₃₇⋊C₃	3
#22582495	3
#22582494	3
#22582493	3
#22582492	3
#22582491	3
#22582473	3
C₃₇×#6875	3
#10624828	37

subgroup

index

C₉⋊C₉×C₃₇⋊C₃

C₃₇×#6875

Subgroups — #41007300

Conjugacy classes of subgroups

Complements

Maximal subgroups