Group #41008257 — subgroups

Subgroups — C₃×#22582679 #41008257

Order 26973.

5588 conjugacy classes of subgroups; only the 2884 normal subgroups are shown.

normal subgroups only

This group has 5588 conjugacy classes of subgroups — too many to draw. Only its 2884 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	#22582679	C₃	1	81	yes	maximal, direct product factor
2	C₃×#15688532	C₃	1	18	yes	maximal, complemented
3	C₃×#15688531	C₃	1	9	yes	maximal, complemented
4	C₃²×#11758608	C₃	1	2	yes	maximal, complemented
5	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	9	yes	maximal, complemented
6	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
7	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	3	yes	—
9	#15688532	C₃²	1	486	yes	complemented
10	#15688531	C₃²	1	243	yes	complemented
11	C₃×#11758608	C₃²	1	6	yes	—
12	C₃×#11758608	C₃²	1	6	yes	complemented
13	C₃×#11758608	C₃²	1	6	yes	complemented
14	C₃×#11758608	C₃²	1	54	yes	complemented
15	C₃×#11758605	C₃²	1	27	yes	complemented
16	C₃×#11758606	C₃²	1	27	yes	complemented
17	3₋¹⁺²×C₃₇⋊C₃	C₃²	1	243	yes	complemented
18	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
19	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
20	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	27	yes	—
21	C₃²×#66524	C₃²	1	6	yes	complemented
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	27	yes	complemented
23	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
24	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
25	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	complemented
26	C₃×3₋¹⁺²×C₃₇	C₃²	1	27	yes	complemented
27	C₃²×C₉×C₃₇	C₃²	1	3	yes	complemented
28	C₃³×C₃₇	—	9	3	no	—
29	C₃³×C₃₇	C₃³	1	1	yes	characteristic
30	C₃³×C₃₇	C₃³	1	1	yes	characteristic
31	C₃³×C₃₇	C₃³	1	2	yes	—
32	C₃³×C₃₇	C₃³	1	9	yes	—
33	C₃²×C₃₇⋊C₃	—	9	9	no	—
34	C₃²×C₃₇⋊C₃	C₃³	1	9	yes	—
35	C₃²×C₃₇⋊C₃	C₃³	1	27	yes	—
36	3₋¹⁺²×C₃₇	C₃³	1	27	yes	—
37	3₋¹⁺²×C₃₇	C₃³	1	54	yes	complemented
38	C₃×C₉×C₃₇	C₃³	1	9	yes	—
39	C₃×C₉×C₃₇	C₃³	1	27	yes	complemented
40	C₃²⋊(C₃₇⋊C₃)	C₃³	1	27	yes	—
41	C₃²⋊(C₃₇⋊C₃)	C₃³	1	54	yes	—
42	#11758608	C₃³	1	54	yes	—
43	#11758608	C₃³	1	54	yes	complemented
44	#11758608	C₃³	1	54	yes	complemented
45	C₃×#66524	C₃³	1	18	yes	—
46	C₃×#66524	C₃³	1	54	yes	complemented
47	#11758606	C₃³	1	243	yes	complemented
48	#11758605	C₃³	1	243	yes	complemented
49	C₉×C₃₇⋊C₃	C₃³	1	243	yes	complemented
50	C₃×3₋¹⁺⁴	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	3	no	—
52	C₃²×C₃₇	—	3	3	no	—
53	C₃²×C₃₇	—	3	6	no	—
54	C₃²×C₃₇	—	9	9	no	—
55	C₃²×C₃₇	C₃⁴	1	6	yes	—
56	C₃²×C₃₇	C₃⁴	1	3	yes	—
57	C₃²×C₃₇	C₃⁴	1	3	yes	—
58	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
59	C₃×C₃₇⋊C₃	—	3	27	no	—
60	C₃×C₃₇⋊C₃	—	9	27	no	—
61	C₃×C₃₇⋊C₃	C₃⁴	1	27	yes	—
62	C₉×C₃₇	C₃⁴	1	27	yes	—
63	#66524	C₃⁴	1	54	yes	—
64	3₋¹⁺⁴	—	37	81	no	—
65	C₃×#737	—	37	18	no	—
66	C₃×#737	—	37	9	no	—
67	C₃²×3₋¹⁺²	—	37	2	no	—
68	C₃²×3₋¹⁺²	—	37	9	no	—
69	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
70	C₃²×H(3,𝔽₃)	—	37	1	no	—
71	C₃×C₃₇	—	3	3	no	—
72	C₃×C₃₇	—	3	6	no	—
73	C₃×C₃₇	—	3	3	no	—
74	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	complemented
75	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
76	C₃₇⋊C₃	—	3	27	no	—
77	C₃⁴	—	37	3	no	—
78	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
79	#737	—	37	486	no	—
80	#737	—	37	243	no	—
81	C₃×3₋¹⁺²	—	37	6	no	—
82	C₃×3₋¹⁺²	—	37	6	no	—
83	C₃×3₋¹⁺²	—	37	6	no	—
84	C₃×3₋¹⁺²	—	37	54	no	—
85	C₃×3₋¹⁺²	—	37	243	no	—
86	C₃×3₋¹⁺²	—	37	27	no	—
87	C₃×3₋¹⁺²	—	37	27	no	—
88	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
89	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	6	yes	complemented
90	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	27	yes	complemented
91	C₃×H(3,𝔽₃)	—	37	6	no	—
92	C₃×H(3,𝔽₃)	—	37	3	no	—
93	C₃×H(3,𝔽₃)	—	37	27	no	—
94	C₃²×C₉	—	37	6	no	—
95	C₃²×C₉	—	37	27	no	—
96	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
97	C₃₇	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
98	C₃³	—	9	3	no	—
99	C₃³	—	37	27	no	—
100	C₃³	—	37	9	no	—
101	C₃³	—	333	9	no	—
102	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
103	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
105	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
106	3₋¹⁺²	—	37	54	no	—
107	3₋¹⁺²	—	37	54	no	—
108	3₋¹⁺²	—	37	54	no	—
109	3₋¹⁺²	—	37	243	no	—
110	3₋¹⁺²	—	37	243	no	—
111	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	27	yes	—
112	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	54	yes	complemented
113	H(3,𝔽₃)	—	37	54	no	—
114	H(3,𝔽₃)	—	37	27	no	—
115	C₃×C₉	—	37	54	no	—
116	C₃×C₉	—	37	18	no	—
117	C₃×C₉	—	37	243	no	—
118	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
119	C₃×C₉	C₃²×C₃₇⋊C₃	1	27	yes	complemented
120	C₃²	—	3	3	no	—
121	C₃²	—	3	6	no	—
122	C₃²	—	3	3	no	—
123	C₃²	—	9	9	no	—
124	C₃²	—	37	27	no	—
125	C₃²	—	333	27	no	—
126	C₃²	—	111	27	no	—
127	C₃²	C₃³×C₃₇⋊C₃	1	6	yes	—
128	C₃²	C₃³×C₃₇⋊C₃	1	3	yes	—
129	C₃²	C₃³×C₃₇⋊C₃	1	3	yes	—
130	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
131	C₉	—	37	54	no	—
132	C₉	C₃³×C₃₇⋊C₃	1	27	yes	—
133	C₃	—	3	3	no	—
134	C₃	—	3	6	no	—
135	C₃	—	3	3	no	—
136	C₃	—	111	27	no	—
137	C₃	#22582679	1	3	yes	direct product factor
138	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582679

C₃

yes

maximal, direct product factor

C₃×#15688532

C₃

yes

maximal, complemented

C₃×#15688531

C₃

yes

maximal, complemented

C₃²×#11758608

C₃

yes

maximal, complemented

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

yes

maximal, complemented

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, characteristic

C₃²×3₋¹⁺²×C₃₇

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

#15688532

C₃²

486

yes

complemented

#15688531

C₃²

243

yes

complemented

C₃×#11758608

C₃²

yes

—

C₃×#11758608

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

complemented

C₃×#11758608

C₃²

yes

complemented

C₃×#11758605

C₃²

yes

complemented

C₃×#11758606

C₃²

yes

complemented

3₋¹⁺²×C₃₇⋊C₃

C₃²

243

yes

complemented

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃²×#66524

C₃²

yes

complemented

C₃×C₉×C₃₇⋊C₃

C₃²

yes

complemented

C₃⁴×C₃₇

C₃²

yes

characteristic

C₃×3₋¹⁺²×C₃₇

C₃²

yes

—

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃×3₋¹⁺²×C₃₇

C₃²

yes

complemented

C₃²×C₉×C₃₇

C₃²

yes

complemented

C₃³×C₃₇

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

characteristic

C₃³×C₃₇

C₃³

yes

—

C₃³×C₃₇

C₃³

yes

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

C₃²×C₃₇⋊C₃

C₃³

yes

—

3₋¹⁺²×C₃₇

C₃³

yes

—

3₋¹⁺²×C₃₇

C₃³

yes

complemented

C₃×C₉×C₃₇

C₃³

yes

—

C₃×C₉×C₃₇

C₃³

yes

complemented

C₃²⋊(C₃₇⋊C₃)

C₃³

yes

—

C₃²⋊(C₃₇⋊C₃)

C₃³

yes

—

#11758608

C₃³

yes

—

#11758608

C₃³

yes

complemented

#11758608

C₃³

yes

complemented

C₃×#66524

C₃³

yes

—

C₃×#66524

C₃³

yes

complemented

#11758606

C₃³

243

yes

complemented

#11758605

C₃³

243

yes

complemented

C₉×C₃₇⋊C₃

C₃³

243

yes

complemented

C₃×3₋¹⁺⁴

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃⁴

yes

—

C₃²×C₃₇

C₃⁴

yes

—

C₃²×C₃₇

C₃⁴

yes

—

C₃²×C₃₇

C₃⁴

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃⁴

yes

—

C₉×C₃₇

C₃⁴

yes

—

#66524

C₃⁴

yes

—

3₋¹⁺⁴

—

C₃×#737

—

C₃×#737

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃₇⋊C₃

yes

characteristic, complemented

C₃²×H(3,𝔽₃)

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

3₋¹⁺⁴

yes

complemented

C₃×C₃₇

C₃⁵

yes

characteristic

C₃₇⋊C₃

—

C₃⁴

—

C₃⁴

C₃×C₃₇⋊C₃

yes

characteristic

#737

—

486

—

#737

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

243

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

—

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

complemented

C₃₇

C₃×3₋¹⁺⁴

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

333

—

102

C₃³

C₃²×C₃₇⋊C₃

yes

—

103

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

104

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

105

C₃³

C₃²×C₃₇⋊C₃

yes

—

106

3₋¹⁺²

—

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

243

—

110

3₋¹⁺²

—

243

—

111

3₋¹⁺²

C₃²×C₃₇⋊C₃

yes

—

112

3₋¹⁺²

C₃²×C₃₇⋊C₃

yes

complemented

113

H(3,𝔽₃)

—

114

H(3,𝔽₃)

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

243

—

118

C₃×C₉

C₃²×C₃₇⋊C₃

yes

—

119

C₃×C₉

C₃²×C₃₇⋊C₃

yes

complemented

120

C₃²

—

121

C₃²

—

122

C₃²

—

123

C₃²

—

124

C₃²

—

125

C₃²

—

333

—

126

C₃²

—

111

—

127

C₃²

C₃³×C₃₇⋊C₃

yes

—

128

C₃²

C₃³×C₃₇⋊C₃

yes

—

129

C₃²

C₃³×C₃₇⋊C₃

yes

—

130

C₃²

C₃³×C₃₇⋊C₃

yes

characteristic

131

C₉

—

132

C₉

C₃³×C₃₇⋊C₃

yes

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

111

—

137

C₃

#22582679

yes

direct product factor

138

C₃

C₃⁴×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₃²	C₃³×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃²×C₃₇	C₃⁴
prime radical	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

subgroup

G/H

center

C₃²

C₃³×C₃₇⋊C₃

commutator, cosocle

C₃×C₃₇

C₃⁵

Frattini

C₃

C₃⁴×C₃₇⋊C₃

Fitting

C₃²×3₋¹⁺²×C₃₇

C₃

socle

C₃²×C₃₇

C₃⁴

prime radical

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃

hypercenter

C₃²×3₋¹⁺²

C₃₇⋊C₃

hypocenter

C₃₇

C₃×3₋¹⁺⁴

LCS

G ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×3₋¹⁺²

N	H ≅ G/N	type
1. #22582679	C₃	direct (N × H)
2. C₃×#15688532	C₃	semidirect
3. C₃×#15688531	C₃	semidirect
4. C₃²×#11758608	C₃	semidirect
5. C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	semidirect
7. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
9. #15688532	C₃²	semidirect
10. #15688531	C₃²	semidirect
12. C₃×#11758608	C₃²	semidirect
13. C₃×#11758608	C₃²	semidirect
14. C₃×#11758608	C₃²	semidirect
15. C₃×#11758605	C₃²	semidirect
16. C₃×#11758606	C₃²	semidirect
17. 3₋¹⁺²×C₃₇⋊C₃	C₃²	semidirect
21. C₃²×#66524	C₃²	semidirect
22. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
25. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
26. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
27. C₃²×C₉×C₃₇	C₃²	semidirect
37. 3₋¹⁺²×C₃₇	C₃³	semidirect
39. C₃×C₉×C₃₇	C₃³	semidirect
43. #11758608	C₃³	semidirect
44. #11758608	C₃³	semidirect
46. C₃×#66524	C₃³	semidirect
47. #11758606	C₃³	semidirect
48. #11758605	C₃³	semidirect
49. C₉×C₃₇⋊C₃	C₃³	semidirect
69. C₃²×3₋¹⁺²	C₃₇⋊C₃	semidirect
74. C₃×C₃₇	3₋¹⁺⁴	semidirect
89. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
90. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
96. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
97. C₃₇	C₃×3₋¹⁺⁴	semidirect
112. 3₋¹⁺²	C₃²×C₃₇⋊C₃	semidirect
119. C₃×C₉	C₃²×C₃₇⋊C₃	semidirect
137. C₃	#22582679	direct (N × H)

H ≅ G/N

type

1. #22582679

C₃

direct (N × H)

2. C₃×#15688532

C₃

semidirect

3. C₃×#15688531

C₃

semidirect

4. C₃²×#11758608

C₃

semidirect

5. C₃×3₋¹⁺²×C₃₇⋊C₃

C₃

semidirect

7. C₃²×3₋¹⁺²×C₃₇

C₃

semidirect

9. #15688532

C₃²

semidirect

10. #15688531

C₃²

semidirect

12. C₃×#11758608

C₃²

semidirect

13. C₃×#11758608

C₃²

semidirect

14. C₃×#11758608

C₃²

semidirect

15. C₃×#11758605

C₃²

semidirect

16. C₃×#11758606

C₃²

semidirect

17. 3₋¹⁺²×C₃₇⋊C₃

C₃²

semidirect

21. C₃²×#66524

C₃²

semidirect

22. C₃×C₉×C₃₇⋊C₃

C₃²

semidirect

25. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

26. C₃×3₋¹⁺²×C₃₇

C₃²

semidirect

27. C₃²×C₉×C₃₇

C₃²

semidirect

37. 3₋¹⁺²×C₃₇

C₃³

semidirect

39. C₃×C₉×C₃₇

C₃³

semidirect

43. #11758608

C₃³

semidirect

44. #11758608

C₃³

semidirect

46. C₃×#66524

C₃³

semidirect

47. #11758606

C₃³

semidirect

48. #11758605

C₃³

semidirect

49. C₉×C₃₇⋊C₃

C₃³

semidirect

69. C₃²×3₋¹⁺²

C₃₇⋊C₃

semidirect

74. C₃×C₃₇

3₋¹⁺⁴

semidirect

89. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

90. C₃×3₋¹⁺²

C₃×C₃₇⋊C₃

semidirect

96. C₃²×C₉

C₃×C₃₇⋊C₃

semidirect

97. C₃₇

C₃×3₋¹⁺⁴

semidirect

112. 3₋¹⁺²

C₃²×C₃₇⋊C₃

semidirect

119. C₃×C₉

C₃²×C₃₇⋊C₃

semidirect

137. C₃

#22582679

direct (N × H)

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

subgroup	index
#22582679	3
C₃×#15688532	3
C₃×#15688531	3
C₃²×#11758608	3
C₃×3₋¹⁺²×C₃₇⋊C₃	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃²×3₋¹⁺²×C₃₇	3
C₃×3₋¹⁺⁴	37

subgroup

index

C₃×3₋¹⁺²×C₃₇⋊C₃

C₃²×C₃²⋊(C₃₇⋊C₃)

C₃²×3₋¹⁺²×C₃₇

C₃×3₋¹⁺⁴

Subgroups — C₃×#22582679 #41008257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#22582679 #41008257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582679 #41008257