Group #41006256 — subgroups

Subgroups — #41006256

Order 26973.

312 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 88 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×#15688531	C₃	1	1	yes	maximal, characteristic
2	C₂₇⋊C₃×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	#22582474	C₃	1	2	yes	maximal, complemented
4	#22582473	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6869	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688531	—	3	3	no	—
7	#15688531	—	3	6	no	—
8	C₃×#11758608	C₉	1	2	yes	—
9	C₃×#11758605	C₉	1	1	yes	characteristic
10	C₃×#11758605	C₉	1	2	yes	—
11	C₃×#11758606	C₉	1	1	yes	characteristic
12	C₃×#11758606	C₉	1	2	yes	—
13	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
15	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
16	#15688511	C₃²	1	6	yes	—
17	#15688510	—	3	3	no	—
18	#15688509	—	3	3	no	—
19	C₂₇×C₃₇⋊C₃	—	3	3	no	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	—
22	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	C₃×C₉	1	2	yes	—
24	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
25	C₃×C₉×C₃₇	—	3	1	no	—
26	C₃×C₉×C₃₇	—	3	2	no	—
27	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
28	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
29	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
30	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
31	#11758608	—	3	6	no	—
32	C₃×#66524	C₃×C₉	1	2	yes	—
33	C₃×#66524	C₃×C₉	1	2	yes	—
34	C₃×#66524	C₃×C₉	1	2	yes	—
35	#11758606	—	3	3	no	—
36	#11758606	—	3	6	no	—
37	#11758605	—	3	3	no	—
38	#11758605	—	3	6	no	—
39	C₉×C₃₇⋊C₃	—	3	2	no	—
40	C₉×C₃₇⋊C₃	—	3	4	no	—
41	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
42	C₉×C₃₇⋊C₃	3₋¹⁺²	1	6	yes	—
43	C₂₇×C₃₇	—	3	3	no	—
44	#11758601	—	3	6	no	—
45	#10624694	—	37	1	no	maximal, Sylow
46	C₃²×C₃₇	—	3	1	no	—
47	C₃²×C₃₇	—	9	1	no	—
48	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
49	C₃×C₃₇⋊C₃	—	3	2	no	—
50	C₃×C₃₇⋊C₃	—	9	1	no	—
51	C₃×C₃₇⋊C₃	—	9	2	no	—
52	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	3	yes	—
53	C₉×C₃₇	—	3	1	no	—
54	C₉×C₃₇	—	3	1	no	—
55	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
57	#66524	—	3	2	no	—
58	#66524	—	3	2	no	—
59	#66524	C₃²⋊C₉	1	6	yes	—
60	C₃×#737	—	37	1	no	—
61	C₃×C₂₇⋊C₃	—	37	3	no	—
62	#6869	—	37	2	no	—
63	#6869	—	37	6	no	—
64	#6869	C₃₇⋊C₃	1	1	yes	characteristic, complemented
65	C₃×C₃₇	—	3	1	no	—
66	C₃×C₃₇	—	9	1	no	—
67	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
68	C₃₇⋊C₃	—	3	3	no	—
69	C₃₇⋊C₃	—	9	2	no	—
70	#737	—	111	3	no	—
71	#737	—	111	6	no	—
72	C₃×3₋¹⁺²	—	37	2	no	—
73	C₃×3₋¹⁺²	—	37	2	no	—
74	C₃×3₋¹⁺²	—	37	1	no	—
75	C₃×3₋¹⁺²	—	37	2	no	—
76	C₃×3₋¹⁺²	—	37	1	no	—
77	C₃×H(3,𝔽₃)	—	37	1	no	—
78	C₃²×C₉	—	37	1	no	—
79	C₃²×C₉	—	37	2	no	—
80	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
81	C₂₇⋊C₃	—	37	6	no	—
82	C₂₇⋊C₃	—	111	3	no	—
83	C₂₇⋊C₃	—	111	3	no	—
84	C₂₇⋊C₃	C₃×C₃₇⋊C₃	1	3	yes	—
85	C₃×C₂₇	—	111	3	no	—
86	C₃₇	#10624694	1	1	yes	characteristic, complemented, Sylow
87	C₃³	—	37	1	no	—
88	C₃³	—	37	2	no	—
89	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
90	3₋¹⁺²	—	111	6	no	—
91	3₋¹⁺²	—	111	6	no	—
92	3₋¹⁺²	—	111	3	no	—
93	3₋¹⁺²	—	111	3	no	—
94	3₋¹⁺²	—	111	6	no	—
95	H(3,𝔽₃)	—	111	3	no	—
96	C₃×C₉	—	3	1	no	—
97	C₃×C₉	—	3	2	no	—
98	C₃×C₉	—	37	2	no	—
99	C₃×C₉	—	37	3	no	—
100	C₃×C₉	—	37	6	no	—
101	C₃×C₉	—	37	2	no	—
102	C₃×C₉	—	37	2	no	—
103	C₃×C₉	—	111	2	no	—
104	C₃×C₉	—	111	4	no	—
105	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
106	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
107	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
108	C₂₇	—	3	3	no	—
109	C₂₇	—	111	6	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	9	1	no	—
112	C₃²	—	37	3	no	—
113	C₃²	—	333	2	no	—
114	C₃²	—	333	1	no	—
115	C₃²	—	111	2	no	—
116	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
117	C₉	—	3	1	no	—
118	C₉	—	3	1	no	—
119	C₉	—	37	6	no	—
120	C₉	—	111	2	no	—
121	C₉	—	111	2	no	—
122	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
123	C₉	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
124	C₃	—	3	1	no	—
125	C₃	—	9	1	no	—
126	C₃	—	333	2	no	—
127	C₃	—	111	3	no	—
128	C₃	C₃²⋊C₉×C₃₇⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#15688531

C₃

yes

maximal, characteristic

C₂₇⋊C₃×C₃₇⋊C₃

C₃

yes

maximal, complemented

#22582474

C₃

yes

maximal, complemented

#22582473

C₃

yes

maximal, complemented

C₃₇×#6869

C₃

yes

maximal, characteristic, complemented

#15688531

—

#15688531

—

C₃×#11758608

C₉

yes

—

C₃×#11758605

C₉

yes

characteristic

C₃×#11758605

C₉

yes

—

C₃×#11758606

C₉

yes

characteristic

C₃×#11758606

C₉

yes

—

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

yes

characteristic

C₃×C₉×C₃₇⋊C₃

C₃²

yes

—

C₃×C₉×C₃₇⋊C₃

C₃²

yes

characteristic

#15688511

C₃²

yes

—

#15688510

—

#15688509

—

C₂₇×C₃₇⋊C₃

—

C₃²×C₉×C₃₇

C₃²

yes

characteristic

C₃₇×C₂₇⋊C₃

C₃²

yes

—

C₃³×C₃₇

C₃×C₉

yes

characteristic

C₃²×C₃₇⋊C₃

C₃×C₉

yes

—

C₃²×C₃₇⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃×C₉

yes

characteristic

C₃×C₉×C₃₇

C₃³

yes

characteristic

C₃²⋊(C₃₇⋊C₃)

—

#11758608

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

C₃×#66524

C₃×C₉

yes

—

—

—

—

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

—

C₉×C₃₇⋊C₃

H(3,𝔽₃)

yes

—

C₉×C₃₇⋊C₃

3₋¹⁺²

yes

—

C₂₇×C₃₇

—

#11758601

—

#10624694

—

maximal, Sylow

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃²×C₉

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

C₃²⋊C₉

yes

—

C₉×C₃₇

—

C₉×C₃₇

—

C₉×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₃₇

C₃×3₋¹⁺²

yes

—

#66524

—

#66524

—

#66524

C₃²⋊C₉

yes

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

#6869

—

#6869

—

#6869

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₃×C₃²⋊C₉

yes

characteristic

C₃₇⋊C₃

—

C₃₇⋊C₃

—

#737

—

111

—

#737

—

111

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₃₇⋊C₃

yes

characteristic

C₂₇⋊C₃

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

—

111

—

C₂₇⋊C₃

C₃×C₃₇⋊C₃

yes

—

C₃×C₂₇

—

111

—

C₃₇

#10624694

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

C₉×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

3₋¹⁺²

—

111

—

H(3,𝔽₃)

—

111

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

111

—

104

C₃×C₉

—

111

—

105

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

106

C₃×C₉

C₉×C₃₇⋊C₃

yes

characteristic

107

C₃×C₉

C₃²×C₃₇⋊C₃

yes

characteristic

108

C₂₇

—

109

C₂₇

—

111

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

333

—

114

C₃²

—

333

—

115

C₃²

—

111

—

116

C₃²

C₃×C₉×C₃₇⋊C₃

yes

characteristic

117

C₉

—

118

C₉

—

119

C₉

—

120

C₉

—

111

—

121

C₉

—

111

—

122

C₉

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

123

C₉

3₋¹⁺²×C₃₇⋊C₃

yes

—

124

C₃

—

125

C₃

—

126

C₃

—

333

—

127

C₃

—

111

—

128

C₃

C₃²⋊C₉×C₃₇⋊C₃

yes

characteristic

	subgroup	G/H
center	C₉	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6869	C₃
socle	C₃×C₃₇	C₃×C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6869	C₃₇⋊C₃
hypocenter	C₃₇	#10624694
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6869

subgroup

G/H

center

C₉

H(3,𝔽₃)×C₃₇⋊C₃

commutator

C₃²×C₃₇

C₃²×C₉

Frattini

C₃×C₉

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6869

C₃

socle

C₃×C₃₇

C₃×C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₃₇⋊C₃)

C₉

cosocle

C₃×C₉×C₃₇

C₃³

C₃₇⋊C₃

C₃₇

G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ #6869

N	H ≅ G/N	type
2. C₂₇⋊C₃×C₃₇⋊C₃	C₃	semidirect
3. #22582474	C₃	semidirect
4. #22582473	C₃	semidirect
5. C₃₇×#6869	C₃	semidirect
64. #6869	C₃₇⋊C₃	semidirect
86. C₃₇	#10624694	semidirect

H ≅ G/N

type

2. C₂₇⋊C₃×C₃₇⋊C₃

C₃

semidirect

3. #22582474

C₃

semidirect

4. #22582473

C₃

semidirect

5. C₃₇×#6869

C₃

semidirect

64. #6869

C₃₇⋊C₃

semidirect

86. C₃₇

#10624694

semidirect

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

subgroup	index
C₃×#15688531	3
C₂₇⋊C₃×C₃₇⋊C₃	3
#22582474	3
#22582473	3
C₃₇×#6869	3
#10624694	37

subgroup

index

C₂₇⋊C₃×C₃₇⋊C₃

C₃₇×#6869

Subgroups — #41006256

Conjugacy classes of subgroups

Complements

Maximal subgroups