Group #41008553 — subgroups

Subgroups — #41008553

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	C₃×#15688542	C₃	1	1	yes	maximal, characteristic
2	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#22582722	C₃	1	1	yes	maximal, characteristic
4	#22582711	C₃	1	3	yes	maximal, complemented
5	#22582709	C₃	1	3	yes	maximal, complemented
6	#22582707	C₃	1	3	yes	maximal, complemented
7	C₂₇⋊C₃×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
8	#15688543	—	3	1	no	—
9	#15688542	—	3	1	no	—
10	#15688542	—	3	1	no	—
11	#15688542	C₃²	1	1	yes	characteristic
12	#15688542	C₃²	1	3	yes	—
13	#15688542	C₃²	1	1	yes	characteristic
14	#15688542	C₃²	1	1	yes	characteristic
15	#15688541	—	3	1	no	—
16	(C₃₇⋊C₃)⋊C₂₇	—	3	1	no	—
17	C₃×#11758603	—	3	1	no	—
18	#15688538	—	3	1	no	—
19	#15688537	C₃²	1	3	yes	—
20	C₃²⋊#66526	—	3	1	no	—
21	C₉×#66526	—	3	1	no	—
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
23	#15688511	C₉	1	2	yes	complemented
24	#15688510	C₃²	1	3	yes	—
25	#15688509	C₉	1	3	yes	complemented
26	C₂₇×C₃₇⋊C₃	C₉	1	3	yes	complemented
27	C₃₇×C₂₇⋊C₃	C₉	1	1	yes	characteristic, complemented
28	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
29	C₃×#66526	—	3	3	no	—
30	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
31	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
32	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
33	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
34	C₉×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
35	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
36	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
37	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
38	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
39	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
40	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
41	C₉×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
42	#11758603	—	3	1	no	—
43	#11758603	—	3	1	no	—
44	#11758603	—	3	1	no	—
45	#11758603	—	3	3	no	—
46	C₂₇×C₃₇	C₃×C₉	1	3	yes	—
47	#11758601	C₃×C₉	1	6	yes	—
48	#10624709	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
50	C₃×C₃₇⋊C₃	—	9	1	no	—
51	C₃×C₃₇⋊C₃	C₉⋊C₉	1	1	yes	characteristic
52	C₃×C₃₇⋊C₃	C₉⋊C₉	1	1	yes	characteristic
53	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
54	#66526	—	9	3	no	—
55	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
56	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
57	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
58	#66524	C₉⋊C₉	1	1	yes	characteristic
59	#66524	C₉⋊C₉	1	1	yes	characteristic
60	#66524	C₉⋊C₉	1	1	yes	characteristic
61	#66524	C₉⋊C₉	1	1	yes	characteristic
62	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
63	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×C₂₇⋊C₃	—	37	1	no	—
65	C₃×C₂₇⋊C₃	—	37	1	no	—
66	C₃×C₂₇⋊C₃	—	37	1	no	—
67	#6887	—	37	1	no	—
68	#6875	—	37	3	no	—
69	#6875	—	37	3	no	—
70	#6875	—	37	3	no	—
71	C₃×C₃₇	—	3	1	no	—
72	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
73	C₃₇⋊C₃	—	3	1	no	—
74	C₃₇⋊C₃	—	3	1	no	—
75	C₃₇⋊C₃	—	3	1	no	—
76	C₃²×C₉	—	37	1	no	—
77	C₂₇⋊C₃	—	37	2	no	—
78	C₂₇⋊C₃	—	37	3	no	—
79	C₂₇⋊C₃	—	37	3	no	—
80	C₂₇⋊C₃	—	37	1	no	—
81	C₂₇⋊C₃	—	37	3	no	—
82	C₂₇⋊C₃	—	37	1	no	—
83	C₂₇⋊C₃	—	37	1	no	—
84	C₂₇⋊C₃	—	111	1	no	—
85	C₂₇⋊C₃	—	111	1	no	—
86	C₂₇⋊C₃	—	111	1	no	—
87	C₂₇⋊C₃	—	111	1	no	—
88	C₂₇⋊C₃	#66526	1	1	yes	characteristic, complemented
89	C₃×C₂₇	—	37	3	no	—
90	C₃×C₂₇	—	111	1	no	—
91	C₃×C₂₇	—	111	1	no	—
92	C₉⋊C₉	—	37	3	no	—
93	C₉⋊C₉	—	111	1	no	—
94	C₃²⋊C₉	—	111	1	no	—
95	C₉²	—	111	1	no	—
96	C₃₇	#10624709	1	1	yes	characteristic, complemented, Sylow
97	C₃³	—	37	1	no	—
98	C₃×C₉	—	37	1	no	—
99	C₃×C₉	—	37	1	no	—
100	C₃×C₉	—	37	1	no	—
101	C₃×C₉	—	37	1	no	—
102	C₃×C₉	—	37	1	no	—
103	C₃×C₉	—	37	1	no	—
104	C₃×C₉	—	37	1	no	—
105	C₃×C₉	—	37	1	no	—
106	C₃×C₉	—	37	1	no	—
107	C₃×C₉	—	37	1	no	—
108	C₃×C₉	—	37	1	no	—
109	C₃×C₉	—	111	3	no	—
110	C₃×C₉	C₃×#66526	1	1	yes	characteristic
111	C₂₇	—	37	6	no	—
112	C₂₇	—	111	1	no	—
113	C₂₇	—	111	1	no	—
114	C₂₇	—	111	1	no	—
115	C₂₇	—	111	3	no	—
116	C₂₇	C₃×#66526	1	3	yes	—
117	C₃²	—	37	1	no	—
118	C₃²	—	37	1	no	—
119	C₃²	—	37	1	no	—
120	C₃²	—	333	1	no	—
121	C₃²	#15688537	1	1	yes	characteristic
122	C₉	—	37	1	no	—
123	C₉	—	37	1	no	—
124	C₉	—	37	1	no	—
125	C₉	—	37	1	no	—
126	C₉	—	37	1	no	—
127	C₉	—	37	1	no	—
128	C₉	—	333	3	no	—
129	C₉	#15688537	1	1	yes	characteristic
130	C₉	#15688537	1	1	yes	characteristic
131	C₉	C₃²×#66526	1	1	yes	characteristic
132	C₃	—	3	1	no	—
133	C₃	—	111	1	no	—
134	C₃	—	111	1	no	—
135	C₃	—	111	1	no	—
136	C₃	C₃×#15688537	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688542

C₃

yes

maximal, characteristic

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal, characteristic

#22582722

C₃

yes

maximal, characteristic

#22582711

C₃

yes

maximal, complemented

#22582709

C₃

yes

maximal, complemented

#22582707

C₃

yes

maximal, complemented

C₂₇⋊C₃×C₃₇⋊C₃

C₃

yes

maximal, characteristic

—

—

—

C₃²

yes

characteristic

#15688542

C₃²

yes

—

#15688542

C₃²

yes

characteristic

#15688542

C₃²

yes

characteristic

#15688541

—

(C₃₇⋊C₃)⋊C₂₇

—

C₃×#11758603

—

#15688538

—

#15688537

C₃²

yes

—

C₃²⋊#66526

—

C₉×#66526

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688511

C₉

yes

complemented

#15688510

C₃²

yes

—

#15688509

C₉

yes

complemented

C₂₇×C₃₇⋊C₃

C₉

yes

complemented

C₃₇×C₂₇⋊C₃

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

characteristic

C₃×#66524

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₃₇⋊C₃

C₃³

yes

—

—

—

—

C₂₇×C₃₇

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

characteristic

C₃×C₃₇⋊C₃

C₉⋊C₉

yes

characteristic

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

characteristic

#66524

C₉⋊C₉

yes

characteristic

#66524

C₉⋊C₉

yes

characteristic

#66524

C₉⋊C₉

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

#66524

C₃×3₋¹⁺²

yes

characteristic

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

—

—

—

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₉⋊C₉

yes

characteristic

C₃₇⋊C₃

—

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

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

#66526

yes

characteristic, complemented

C₃×C₂₇

—

C₃×C₂₇

—

111

—

C₃×C₂₇

—

111

—

C₉⋊C₉

—

C₉⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₉²

—

111

—

C₃₇

#10624709

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

111

—

110

C₃×C₉

C₃×#66526

yes

characteristic

111

C₂₇

—

112

C₂₇

—

111

—

113

C₂₇

—

111

—

114

C₂₇

—

111

—

115

C₂₇

—

111

—

116

C₂₇

C₃×#66526

yes

—

117

C₃²

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

333

—

121

C₃²

#15688537

yes

characteristic

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

126

C₉

—

127

C₉

—

128

C₉

—

333

—

129

C₉

#15688537

yes

characteristic

130

C₉

#15688537

yes

characteristic

131

C₉

C₃²×#66526

yes

characteristic

132

C₃

—

133

C₃

—

111

—

134

C₃

—

111

—

135

C₃

—

111

—

136

C₃

C₃×#15688537

yes

characteristic

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

subgroup

G/H

center

C₃

C₃×#15688537

commutator

C₉×C₃₇

C₃²×C₉

Frattini

C₉

C₃²×#66526

Fitting

C₃₇×C₂₇⋊C₃

C₉

socle

C₃×C₃₇

C₃×C₉⋊C₉

prime radical

C₃²×C₃₇⋊C₃

3₋¹⁺²

cosocle

C₉×C₃₇⋊C₃

C₃³

C₂₇⋊C₃

C₃₇

G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₂₇⋊C₃

N	H ≅ G/N	type
4. #22582711	C₃	semidirect
5. #22582709	C₃	semidirect
6. #22582707	C₃	semidirect
23. #15688511	C₉	semidirect
25. #15688509	C₉	semidirect
26. C₂₇×C₃₇⋊C₃	C₉	semidirect
27. C₃₇×C₂₇⋊C₃	C₉	semidirect
88. C₂₇⋊C₃	#66526	semidirect
96. C₃₇	#10624709	semidirect

H ≅ G/N

type

4. #22582711

C₃

semidirect

5. #22582709

C₃

semidirect

6. #22582707

C₃

semidirect

23. #15688511

C₉

semidirect

25. #15688509

C₉

semidirect

26. C₂₇×C₃₇⋊C₃

C₉

semidirect

27. C₃₇×C₂₇⋊C₃

C₉

semidirect

88. C₂₇⋊C₃

#66526

semidirect

96. C₃₇

#10624709

semidirect

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

subgroup	index
C₃×#15688542	3
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
#22582722	3
#22582711	3
#22582709	3
#22582707	3
C₂₇⋊C₃×C₃₇⋊C₃	3
#10624709	37

subgroup

index

(C₃₇⋊C₃)⋊(C₂₇⋊C₃)

C₂₇⋊C₃×C₃₇⋊C₃

Subgroups — #41008553

Conjugacy classes of subgroups

Complements

Maximal subgroups