Group #41007835 — subgroups

Subgroups — #41007835

Order 26973.

836 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 298 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×#11758608	C₃	1	2	yes	maximal
2	#22582601	C₃	1	9	yes	maximal, complemented
3	#22582579	C₃	1	9	yes	maximal, complemented
4	#22582574	C₃	1	6	yes	maximal, complemented
5	#22582573	C₃	1	6	yes	maximal, complemented
6	#22582554	C₃	1	1	yes	maximal, characteristic
7	#22582548	C₃	1	6	yes	maximal, complemented
8	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
9	C₃×#11758608	—	3	18	no	—
10	C₃×#11758608	C₃²	1	2	yes	—
11	C₃×#11758608	C₃²	1	2	yes	—
12	C₃×#11758608	C₃²	1	6	yes	—
13	C₃×#11758608	C₃²	1	6	yes	—
14	C₃×#11758608	C₃²	1	3	yes	—
15	C₃×#11758608	C₃²	1	2	yes	—
16	C₃²×#66524	C₃²	1	6	yes	—
17	#15688506	C₃²	1	9	yes	complemented
18	#15688505	C₃²	1	9	yes	complemented
19	#15688504	C₃²	1	18	yes	complemented
20	#15688504	C₃²	1	18	yes	complemented
21	#15688504	C₃²	1	18	yes	complemented
22	C₃²⋊#66524	C₃²	1	6	yes	—
23	C₃²⋊#66524	C₃²	1	3	yes	—
24	C₉×#66524	C₃²	1	9	yes	complemented
25	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
26	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
27	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
28	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
29	C₃³×C₃₇	—	3	6	no	—
30	C₃³×C₃₇	—	9	1	no	—
31	C₃³×C₃₇	—	9	1	no	—
32	C₃³×C₃₇	C₃³	1	1	yes	characteristic
33	C₃³×C₃₇	C₃³	1	1	yes	characteristic
34	C₃³×C₃₇	C₃³	1	2	yes	—
35	3₋¹⁺²×C₃₇	—	9	3	no	—
36	C₃×C₉×C₃₇	C₃³	1	9	yes	—
37	#11758608	—	3	6	no	—
38	#11758608	—	3	6	no	—
39	#11758608	—	3	18	no	—
40	#11758608	—	3	18	no	—
41	#11758608	—	9	3	no	—
42	#11758608	3₋¹⁺²	1	18	yes	complemented
43	C₃×#66524	—	3	18	no	—
44	C₃×#66524	C₃³	1	9	yes	—
45	C₃×#66524	C₃³	1	6	yes	—
46	C₃×#66524	C₃³	1	6	yes	—
47	C₃×#66524	C₃³	1	6	yes	—
48	3₋¹⁺²²	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	—	3	2	no	—
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	3	2	no	—
52	C₃²×C₃₇	—	3	2	no	—
53	C₃²×C₃₇	—	3	2	no	—
54	C₃²×C₃₇	—	3	1	no	—
55	C₃²×C₃₇	—	3	1	no	—
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	9	9	no	—
59	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
60	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
61	C₉×C₃₇	—	3	9	no	—
62	#66524	—	3	6	no	—
63	#66524	—	3	6	no	—
64	#66524	—	3	9	no	—
65	#66524	C₃×3₋¹⁺²	1	18	yes	complemented
66	C₃²×3₋¹⁺²	—	37	2	no	—
67	#6900	—	37	9	no	—
68	#6896	—	37	9	no	—
69	#6894	—	37	6	no	—
70	#6894	—	37	6	no	—
71	#6891	—	37	1	no	—
72	#6891	C₃₇⋊C₃	1	1	yes	characteristic
73	C₉×3₋¹⁺²	—	37	6	no	—
74	C₃×C₃₇	—	3	6	no	—
75	C₃×C₃₇	—	9	1	no	—
76	C₃×C₃₇	—	9	1	no	—
77	C₃×C₃₇	C₃²×3₋¹⁺²	1	2	yes	—
78	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
79	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
80	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃×3₋¹⁺²	—	37	2	no	—
82	C₃×3₋¹⁺²	—	37	3	no	—
83	C₃×3₋¹⁺²	—	37	2	no	—
84	C₃×3₋¹⁺²	—	37	2	no	—
85	C₃×3₋¹⁺²	—	37	6	no	—
86	C₃×3₋¹⁺²	—	37	6	no	—
87	C₃×3₋¹⁺²	—	111	18	no	—
88	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
89	C₃²×C₉	—	37	6	no	—
90	C₉⋊C₉	—	37	18	no	—
91	C₉⋊C₉	—	37	18	no	—
92	C₉⋊C₉	—	37	18	no	—
93	C₉⋊C₉	—	37	9	no	—
94	C₉⋊C₉	—	37	9	no	—
95	C₃²⋊C₉	—	37	6	no	—
96	C₃²⋊C₉	—	37	3	no	—
97	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	—
98	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
99	C₉²	—	37	9	no	—
100	C₃₇	3₋¹⁺²²	1	1	yes	characteristic, complemented, Sylow
101	C₃³	—	3	6	no	—
102	C₃³	—	9	1	no	—
103	C₃³	—	9	1	no	—
104	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
105	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
106	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
107	3₋¹⁺²	—	9	3	no	—
108	3₋¹⁺²	—	37	18	no	—
109	3₋¹⁺²	—	333	3	no	—
110	3₋¹⁺²	—	111	18	no	—
111	3₋¹⁺²	—	111	18	no	—
112	3₋¹⁺²	—	111	6	no	—
113	3₋¹⁺²	—	111	6	no	—
114	C₃×C₉	—	37	6	no	—
115	C₃×C₉	—	37	9	no	—
116	C₃×C₉	—	37	6	no	—
117	C₃×C₉	—	37	6	no	—
118	C₃×C₉	—	111	18	no	—
119	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
120	C₃²	—	3	2	no	—
121	C₃²	—	3	2	no	—
122	C₃²	—	3	2	no	—
123	C₃²	—	3	2	no	—
124	C₃²	—	3	2	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	9	9	no	—
130	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	6	yes	—
131	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
132	C₉	—	3	9	no	—
133	C₉	—	37	18	no	—
134	C₉	—	111	6	no	—
135	C₉	—	111	6	no	—
136	C₉	—	111	9	no	—
137	C₃	—	3	6	no	—
138	C₃	—	9	1	no	—
139	C₃	—	9	1	no	—
140	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
141	C₃	#22582679	1	1	yes	characteristic
142	C₃	#22582680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#11758608

C₃

yes

maximal

#22582601

C₃

yes

maximal, complemented

#22582579

C₃

yes

maximal, complemented

#22582574

C₃

yes

maximal, complemented

#22582573

C₃

yes

maximal, complemented

#22582554

C₃

yes

maximal, characteristic

#22582548

C₃

yes

maximal, complemented

C₃₇×#6891

C₃

yes

maximal, characteristic

C₃×#11758608

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

—

C₃²×#66524

C₃²

yes

—

#15688506

C₃²

yes

complemented

#15688505

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

#15688504

C₃²

yes

complemented

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₉×#66524

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

3₋¹⁺²×C₃₇

—

C₃×C₉×C₃₇

C₃³

yes

—

—

—

—

—

—

3₋¹⁺²

yes

complemented

C₃×#66524

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

—

3₋¹⁺²²

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×3₋¹⁺²

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₉×C₃₇

—

—

—

—

C₃×3₋¹⁺²

yes

complemented

C₃²×3₋¹⁺²

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic

C₉×3₋¹⁺²

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃²×3₋¹⁺²

yes

—

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃×C₃₇

3₋¹⁺⁴

yes

characteristic

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₉²

—

100

C₃₇

3₋¹⁺²²

yes

characteristic, complemented, Sylow

101

C₃³

—

102

C₃³

—

103

C₃³

—

104

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

105

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

106

C₃³

C₃²×C₃₇⋊C₃

yes

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

333

—

110

3₋¹⁺²

—

111

—

111

3₋¹⁺²

—

111

—

112

3₋¹⁺²

—

111

—

113

3₋¹⁺²

—

111

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

—

111

—

119

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

3₋¹⁺²×C₃₇⋊C₃

yes

—

131

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

132

C₉

—

133

C₉

—

134

C₉

—

111

—

135

C₉

—

111

—

136

C₉

—

111

—

137

C₃

—

138

C₃

—

139

C₃

—

140

C₃

C₃×3₋¹⁺²×C₃₇⋊C₃

yes

—

141

C₃

#22582679

yes

characteristic

142

C₃

#22582680

yes

characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6891	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	3₋¹⁺²²
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6891

subgroup

G/H

center, Frattini

C₃²

C₃³×C₃₇⋊C₃

commutator, socle, cosocle

C₃²×C₃₇

C₃⁴

Fitting

C₃₇×#6891

C₃

prime radical

C₃⁴×C₃₇

C₃²

hypercenter

#6891

C₃₇⋊C₃

hypocenter

C₃₇

3₋¹⁺²²

LCS

G ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ #6891

N	H ≅ G/N	type
2. #22582601	C₃	semidirect
3. #22582579	C₃	semidirect
4. #22582574	C₃	semidirect
5. #22582573	C₃	semidirect
7. #22582548	C₃	semidirect
17. #15688506	C₃²	semidirect
18. #15688505	C₃²	semidirect
19. #15688504	C₃²	semidirect
20. #15688504	C₃²	semidirect
21. #15688504	C₃²	semidirect
24. C₉×#66524	C₃²	semidirect
42. #11758608	3₋¹⁺²	semidirect
65. #66524	C₃×3₋¹⁺²	semidirect
100. C₃₇	3₋¹⁺²²	semidirect

H ≅ G/N

type

2. #22582601

C₃

semidirect

3. #22582579

C₃

semidirect

4. #22582574

C₃

semidirect

5. #22582573

C₃

semidirect

7. #22582548

C₃

semidirect

17. #15688506

C₃²

semidirect

18. #15688505

C₃²

semidirect

19. #15688504

C₃²

semidirect

20. #15688504

C₃²

semidirect

21. #15688504

C₃²

semidirect

24. C₉×#66524

C₃²

semidirect

42. #11758608

3₋¹⁺²

semidirect

65. #66524

C₃×3₋¹⁺²

semidirect

100. C₃₇

3₋¹⁺²²

semidirect

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

subgroup	index
C₃²×#11758608	3
#22582601	3
#22582579	3
#22582574	3
#22582573	3
#22582554	3
#22582548	3
C₃₇×#6891	3
3₋¹⁺²²	37

subgroup

index

C₃₇×#6891

3₋¹⁺²²

Subgroups — #41007835

Conjugacy classes of subgroups

Complements

Maximal subgroups