Group #41005589 — subgroups

Subgroups — #41005589

Order 26973.

236 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 52 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	#22582559	C₃	1	1	yes	maximal, characteristic, complemented
2	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#22582456	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582456	C₃	1	2	yes	maximal, complemented
5	#22582452	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582452	C₃	1	2	yes	maximal, complemented
7	#22582443	C₃	1	2	yes	maximal, complemented
8	#22582443	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758606	—	3	1	no	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₃₇⋊C₃	—	3	2	no	—
14	C₃²⋊#66524	C₃²	1	2	yes	—
15	C₃²⋊#66524	C₃²	1	2	yes	—
16	C₃²⋊#66524	C₃²	1	1	yes	characteristic
17	C₃²⋊#66524	C₃²	1	1	yes	characteristic
18	#15688502	—	3	2	no	—
19	#15688502	—	3	2	no	—
20	#15688502	—	3	2	no	—
21	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
22	C₃₇×C₉⋊C₉	—	3	2	no	—
23	C₃₇×C₉⋊C₉	—	3	1	no	—
24	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	2	no	—
28	C₃²×C₃₇⋊C₃	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
30	C₃×C₉×C₃₇	—	3	2	no	—
31	C₃×C₉×C₃₇	—	9	1	no	—
32	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
33	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
35	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	1	no	—
40	#11758606	—	9	3	no	—
41	C₉×C₃₇⋊C₃	—	9	6	no	—
42	#10624622	—	37	1	no	maximal, Sylow
43	C₃²×C₃₇	—	3	1	no	—
44	C₃²×C₃₇	—	3	2	no	—
45	C₃²×C₃₇	—	3	1	no	—
46	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₃₇⋊C₃	—	3	6	no	—
48	C₃×C₃₇⋊C₃	—	9	2	no	—
49	C₃×C₃₇⋊C₃	—	9	2	no	—
50	C₃×C₃₇⋊C₃	—	9	2	no	—
51	C₃×C₃₇⋊C₃	—	9	1	no	—
52	C₃×C₃₇⋊C₃	—	9	2	no	—
53	C₃×C₃₇⋊C₃	—	9	1	no	—
54	C₉×C₃₇	—	3	2	no	—
55	C₉×C₃₇	—	3	1	no	—
56	C₉×C₃₇	—	9	1	no	—
57	C₉×C₃₇	—	9	2	no	—
58	#66524	—	9	2	no	—
59	#66524	—	9	2	no	—
60	#66524	—	9	1	no	—
61	#66524	—	9	1	no	—
62	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
63	C₉⋊H(3,𝔽₃)	—	37	1	no	—
64	C₉×H(3,𝔽₃)	—	37	2	no	—
65	#6861	—	37	1	no	—
66	#6861	—	37	2	no	—
67	#6859	—	37	1	no	—
68	#6859	—	37	2	no	—
69	#6856	—	37	2	no	—
70	#6856	—	37	1	no	—
71	C₃×C₃₇	—	9	1	no	—
72	C₃×C₃₇	#6908	1	2	yes	—
73	C₃×C₃₇	#6911	1	1	yes	characteristic
74	C₃×C₃₇	#6913	1	1	yes	characteristic
75	C₃₇⋊C₃	—	9	6	no	—
76	C₃₇⋊C₃	—	27	1	no	—
77	C₃×3₋¹⁺²	—	111	1	no	—
78	C₃×H(3,𝔽₃)	—	37	1	no	—
79	C₃×H(3,𝔽₃)	—	37	2	no	—
80	C₃²×C₉	—	111	2	no	—
81	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
82	C₉⋊C₉	—	3	2	no	—
83	C₉⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	37	2	no	—
85	C₃²⋊C₉	—	37	1	no	—
86	C₃²⋊C₉	—	37	2	no	—
87	C₃²⋊C₉	—	37	1	no	—
88	C₃²⋊C₉	—	111	2	no	—
89	C₃²⋊C₉	—	111	2	no	—
90	C₃²⋊C₉	—	111	2	no	—
91	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
92	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
93	C₃₇	#10624622	1	1	yes	characteristic, complemented, Sylow
94	C₃³	—	111	1	no	—
95	C₃³	—	111	2	no	—
96	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
97	3₋¹⁺²	—	333	3	no	—
98	H(3,𝔽₃)	—	333	1	no	—
99	H(3,𝔽₃)	—	111	6	no	—
100	C₃×C₉	—	3	2	no	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	9	1	no	—
103	C₃×C₉	—	333	6	no	—
104	C₃×C₉	—	111	2	no	—
105	C₃×C₉	—	111	2	no	—
106	C₃×C₉	—	111	1	no	—
107	C₃×C₉	—	111	1	no	—
108	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
109	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
110	C₃²	—	3	2	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	333	2	no	—
114	C₃²	—	333	2	no	—
115	C₃²	—	333	2	no	—
116	C₃²	—	333	2	no	—
117	C₃²	—	333	1	no	—
118	C₃²	—	333	1	no	—
119	C₃²	—	111	6	no	—
120	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
121	C₉	—	3	2	no	—
122	C₉	—	3	1	no	—
123	C₉	—	9	2	no	—
124	C₉	—	9	1	no	—
125	C₉	—	333	2	no	—
126	C₉	—	333	2	no	—
127	C₉	—	333	1	no	—
128	C₉	—	333	1	no	—
129	C₃	—	9	1	no	—
130	C₃	—	333	6	no	—
131	C₃	—	999	1	no	—
132	C₃	#22582630	1	2	yes	—
133	C₃	#22582648	1	1	yes	characteristic
134	C₃	#22582656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22582559

C₃

yes

maximal, characteristic, complemented

C₉×C₃²⋊(C₃₇⋊C₃)

C₃

yes

maximal, complemented

#22582456

C₃

yes

maximal, characteristic, complemented

#22582456

C₃

yes

maximal, complemented

#22582452

C₃

yes

maximal, characteristic, complemented

#22582452

C₃

yes

maximal, complemented

#22582443

C₃

yes

maximal, complemented

#22582443

C₃

yes

maximal, characteristic, complemented

C₃₇×#6895

C₃

yes

maximal, characteristic, complemented

C₃×#11758606

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₃₇⋊C₃

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

characteristic

C₃²⋊#66524

C₃²

yes

—

—

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₉⋊C₉

—

C₃₇×C₉⋊C₉

—

C₃₇×C₃²⋊C₉

C₃²

yes

characteristic

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

—

C₃×C₉×C₃₇

H(3,𝔽₃)

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

C₃²⋊(C₃₇⋊C₃)

—

—

—

—

—

—

C₉×C₃₇⋊C₃

—

#10624622

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

—

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

#6908

yes

—

C₃×C₃₇

yes

C₃×C₃₇

yes

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃×3₋¹⁺²

—

111

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

111

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₃₇

#10624622

yes

characteristic, complemented, Sylow

C₃³

—

111

—

C₃³

—

111

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

H(3,𝔽₃)

—

333

—

H(3,𝔽₃)

—

111

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

333

—

104

C₃×C₉

—

111

—

105

C₃×C₉

—

111

—

106

C₃×C₉

—

111

—

107

C₃×C₉

—

111

—

108

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

—

109

C₃×C₉

C₃²⋊(C₃₇⋊C₃)

yes

characteristic

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

333

—

114

C₃²

—

333

—

115

C₃²

—

333

—

116

C₃²

—

333

—

117

C₃²

—

333

—

118

C₃²

—

333

—

119

C₃²

—

111

—

120

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

333

—

126

C₉

—

333

—

127

C₉

—

333

—

128

C₉

—

333

—

129

C₃

—

130

C₃

—

333

—

131

C₃

—

999

—

132

C₃

#22582630

yes

—

133

C₃

#22582648

yes

characteristic

134

C₃

#22582656

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6895	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624622
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₃₇⋊C₃)

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6895

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

N	H ≅ G/N	type
1. #22582559	C₃	semidirect
2. C₉×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
3. #22582456	C₃	semidirect
4. #22582456	C₃	semidirect
5. #22582452	C₃	semidirect
6. #22582452	C₃	semidirect
7. #22582443	C₃	semidirect
8. #22582443	C₃	semidirect
9. C₃₇×#6895	C₃	semidirect
62. #6895	C₃₇⋊C₃	semidirect
93. C₃₇	#10624622	semidirect

H ≅ G/N

type

1. #22582559

C₃

semidirect

2. C₉×C₃²⋊(C₃₇⋊C₃)

C₃

semidirect

3. #22582456

C₃

semidirect

4. #22582456

C₃

semidirect

5. #22582452

C₃

semidirect

6. #22582452

C₃

semidirect

7. #22582443

C₃

semidirect

8. #22582443

C₃

semidirect

9. C₃₇×#6895

C₃

semidirect

62. #6895

C₃₇⋊C₃

semidirect

93. C₃₇

#10624622

semidirect

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

subgroup	index
#22582559	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582456	3
#22582452	3
#22582443	3
C₃₇×#6895	3
#10624622	37

subgroup

index

C₉×C₃²⋊(C₃₇⋊C₃)

C₃₇×#6895

Subgroups — #41005589

Conjugacy classes of subgroups

Complements

Maximal subgroups