Group #13135516 — subgroups

Subgroups — #10588188⋊C₃ #13135516

Order 1728.

151 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 64 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	3₋¹⁺²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#63371⋊C₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4511	—	3	1	no	maximal
5	#10588188	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₄×3₋¹⁺²⋊C₄	C₄	1	1	yes	characteristic
8	C₂×3₋¹⁺²⋊C₈	C₄	1	1	yes	characteristic
9	C₁₆×3₋¹⁺²	—	2	1	no	—
10	3₋¹⁺²⋊C₁₆	—	2	1	no	—
11	C₃²×M₃₂	D₃	1	1	yes	characteristic
12	C₃×C₈×Dic₃	—	3	1	no	—
13	C₃×#812	—	3	1	no	—
14	C₉×M₃₂	—	2	1	no	—
15	C₉×M₃₂	C₂×C₃	1	1	yes	characteristic
16	C₈×Dic₉	C₂×C₃	1	1	yes	characteristic
17	#63371	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
20	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
21	C₈×3₋¹⁺²	Q₈	1	1	yes	characteristic
22	3₋¹⁺²⋊C₈	C₈	1	2	yes	—
23	C₃×#364	—	9	1	no	—
24	#4511	—	3	1	no	—
25	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
26	C₃×C₄×Dic₃	—	3	1	no	—
27	C₂×C₃×C₃⋊C₈	—	3	1	no	—
28	C₃²×C₁₆	—	2	1	no	—
29	C₃×C₃⋊C₁₆	—	6	1	no	—
30	C₂×C₈×C₉	—	2	1	no	—
31	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
32	C₄×Dic₉	C₃×C₄	1	1	yes	characteristic
33	C₂×C₉⋊C₈	C₃×C₄	1	1	yes	characteristic
34	C₉×C₁₆	—	2	2	no	—
35	C₉×C₁₆	—	2	1	no	—
36	C₉⋊C₁₆	—	2	1	no	—
37	C₂²×3₋¹⁺²	M₁₆	1	1	yes	characteristic
38	C₄×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
39	C₄×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
40	3₋¹⁺²⋊C₄	—	2	2	no	—
41	C₃×M₃₂	—	3	1	no	—
42	C₃×M₃₂	—	9	1	no	—
43	C₃×M₃₂	C₃×D₃	1	1	yes	characteristic
44	C₃×C₄×C₈	—	9	1	no	—
45	C₈×Dic₃	—	3	1	no	—
46	#812	—	3	1	no	—
47	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
48	C₂×C₃×Dic₃	—	6	1	no	—
49	C₃²×C₈	Dic₆	1	1	yes	characteristic
50	C₃²×C₈	#67	1	1	yes	characteristic
51	C₃×C₃⋊C₈	—	3	2	no	—
52	C₂×C₄×C₉	—	2	1	no	—
53	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×Dic₉	—	2	1	no	—
55	C₈×C₉	—	2	1	no	—
56	C₈×C₉	—	2	1	no	—
57	C₈×C₉	C₃×D₄	1	1	yes	characteristic
58	C₈×C₉	C₃×Q₈	1	1	yes	characteristic
59	C₉⋊C₈	C₃×C₈	1	2	yes	—
60	#364	—	9	1	no	Sylow
61	C₂×3₋¹⁺²	—	2	1	no	—
62	C₂×3₋¹⁺²	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₃×C₈	—	3	1	no	—
64	C₂×C₃×C₈	—	9	1	no	—
65	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
66	C₃×C₄²	—	9	1	no	—
67	C₄×Dic₃	—	3	1	no	—
68	C₂×C₃⋊C₈	—	3	1	no	—
69	C₃×C₁₆	—	2	1	no	—
70	C₃×C₁₆	—	6	1	no	—
71	C₃×C₁₆	—	18	1	no	—
72	C₃⋊C₁₆	—	6	1	no	—
73	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
74	C₃²×C₄	D₃×C₈	1	1	yes	characteristic
75	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
76	C₃×Dic₃	—	6	2	no	—
77	C₂²×C₉	—	2	1	no	—
78	C₂²×C₉	C₃×M₁₆	1	1	yes	characteristic
79	C₄×C₉	—	2	1	no	—
80	C₄×C₉	—	2	1	no	—
81	C₄×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₉	C₂×C₃×C₈	1	1	yes	characteristic
83	Dic₉	—	2	2	no	—
84	M₃₂	—	9	1	no	—
85	M₃₂	AGL(1,ℤ/9)	1	1	yes	characteristic
86	C₄×C₈	—	9	1	no	—
87	3₋¹⁺²	#364	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃×C₄	—	3	1	no	—
89	C₂×C₃×C₄	—	18	1	no	—
90	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
91	C₂×Dic₃	—	6	1	no	—
92	C₃×C₈	—	3	1	no	—
93	C₃×C₈	—	3	1	no	—
94	C₃×C₈	—	9	2	no	—
95	C₃×C₈	C₃×Dic₆	1	1	yes	characteristic
96	C₃×C₈	C₃×#67	1	1	yes	characteristic
97	C₃⋊C₈	—	3	2	no	—
98	C₂×C₃²	—	2	1	no	—
99	C₂×C₃²	Dic₃⋊C₈	1	1	yes	characteristic
100	C₂×C₉	—	2	1	no	—
101	C₂×C₉	—	2	1	no	—
102	C₂×C₉	—	4	1	no	—
103	C₂×C₉	C₃×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₈	—	9	1	no	—
105	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
106	C₄²	—	9	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	18	1	no	—
109	C₂²×C₃	—	3	1	no	—
110	C₂²×C₃	C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
111	C₃×C₄	—	3	1	no	—
112	C₃×C₄	—	3	1	no	—
113	C₃×C₄	—	18	2	no	—
114	C₃×C₄	C₃×D₃×C₈	1	1	yes	characteristic
115	C₃×C₄	C₃×Dic₃⋊C₄	1	1	yes	characteristic
116	Dic₃	—	6	2	no	—
117	C₃²	#4511	1	1	yes	characteristic
118	C₉	—	2	1	no	—
119	C₉	C₃×#364	1	1	yes	characteristic, complemented
120	C₂×C₄	—	18	1	no	—
121	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
122	C₈	—	9	2	no	—
123	C₈	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
124	C₈	#611⋊C₃	1	1	yes	characteristic
125	C₂×C₃	—	2	1	no	—
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	—	6	1	no	—
128	C₂×C₃	C₃×Dic₃⋊C₈	1	1	yes	characteristic
129	C₂²	#3663⋊C₃	1	1	yes	characteristic
130	C₄	—	18	2	no	—
131	C₄	C₈×AGL(1,ℤ/9)	1	1	yes	characteristic
132	C₄	(3₋¹⁺²⋊C₄)⋊C₄	1	1	yes	characteristic
133	C₃	—	3	1	no	—
134	C₃	C₃×#4511	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	(3₋¹⁺²⋊C₄)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×M₃₂

C₂

yes

maximal, characteristic

C₈×3₋¹⁺²⋊C₄

C₂

yes

maximal, characteristic

#63371⋊C₃

C₂

yes

maximal, characteristic

C₃×#4511

—

maximal

#10588188

C₃

yes

maximal, characteristic, complemented

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₄×3₋¹⁺²⋊C₄

C₄

yes

characteristic

C₂×3₋¹⁺²⋊C₈

C₄

yes

characteristic

C₁₆×3₋¹⁺²

—

3₋¹⁺²⋊C₁₆

—

C₃²×M₃₂

D₃

yes

characteristic

C₃×C₈×Dic₃

—

C₃×#812

—

C₉×M₃₂

—

C₉×M₃₂

C₂×C₃

yes

characteristic

C₈×Dic₉

C₂×C₃

yes

characteristic

#63371

C₂×C₃

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

C₈×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

Q₈

yes

characteristic

3₋¹⁺²⋊C₈

C₈

yes

—

C₃×#364

—

#4511

—

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₃×C₄×Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₃²×C₁₆

—

C₃×C₃⋊C₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

characteristic

C₄×Dic₉

C₃×C₄

yes

characteristic

C₂×C₉⋊C₈

C₃×C₄

yes

characteristic

C₉×C₁₆

—

C₉×C₁₆

—

C₉⋊C₁₆

—

C₂²×3₋¹⁺²

M₁₆

yes

characteristic

C₄×3₋¹⁺²

C₄⋊C₄

yes

characteristic

C₄×3₋¹⁺²

C₂×C₈

yes

characteristic

3₋¹⁺²⋊C₄

—

C₃×M₃₂

—

C₃×M₃₂

—

C₃×M₃₂

C₃×D₃

yes

characteristic

C₃×C₄×C₈

—

C₈×Dic₃

—

#812

—

C₂×C₃²×C₄

C₄×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃²×C₈

Dic₆

yes

characteristic

C₃²×C₈

#67

yes

characteristic

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×Dic₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

characteristic

C₈×C₉

C₃×Q₈

yes

characteristic

C₉⋊C₈

C₃×C₈

yes

—

#364

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₄⋊C₈

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

characteristic

C₃×C₄²

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

ΓL(1,𝔽₂₅)

yes

characteristic

C₃²×C₄

D₃×C₈

yes

characteristic

C₃²×C₄

Dic₃⋊C₄

yes

characteristic

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₃×M₁₆

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×C₄⋊C₄

yes

characteristic

C₄×C₉

C₂×C₃×C₈

yes

characteristic

Dic₉

—

M₃₂

—

M₃₂

AGL(1,ℤ/9)

yes

characteristic

C₄×C₈

—

3₋¹⁺²

#364

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₄×D₃

yes

characteristic

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₃×Dic₆

yes

characteristic

C₃×C₈

C₃×#67

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

Dic₃⋊C₈

yes

characteristic

100

C₂×C₉

—

101

C₂×C₉

—

102

C₂×C₉

—

103

C₂×C₉

C₃×C₄⋊C₈

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

characteristic

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

109

C₂²×C₃

—

110

C₂²×C₃

C₃×ΓL(1,𝔽₂₅)

yes

characteristic

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

C₃×D₃×C₈

yes

characteristic

115

C₃×C₄

C₃×Dic₃⋊C₄

yes

characteristic

116

Dic₃

—

117

C₃²

#4511

yes

characteristic

118

C₉

—

119

C₉

C₃×#364

yes

characteristic, complemented

120

C₂×C₄

—

121

C₂×C₄

C₄×AGL(1,ℤ/9)

yes

characteristic

122

C₈

—

123

C₈

3₋¹⁺²⋊Q₈

yes

characteristic

124

C₈

#611⋊C₃

yes

characteristic

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

C₃×Dic₃⋊C₈

yes

characteristic

129

C₂²

#3663⋊C₃

yes

characteristic

130

C₄

—

131

C₄

C₈×AGL(1,ℤ/9)

yes

characteristic

132

C₄

(3₋¹⁺²⋊C₄)⋊C₄

yes

characteristic

133

C₃

—

134

C₃

C₃×#4511

yes

characteristic

135

C₂

—

136

C₂

(3₋¹⁺²⋊C₄)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	#611⋊C₃
commutator	C₄×C₉	C₂×C₃×C₈
Frattini	C₂×C₃×C₈	C₂×C₃×D₃
Fitting	3₋¹⁺²×M₃₂	C₂
socle	C₂×C₃	C₃×Dic₃⋊C₈
prime radical	C₂²×C₃²	ΓL(1,𝔽₂₅)
cosocle	C₂×C₈×C₉	C₂²×C₃
hypercenter	M₃₂	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#364
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₈

#611⋊C₃

commutator

C₄×C₉

C₂×C₃×C₈

Frattini

C₂×C₃×C₈

C₂×C₃×D₃

Fitting

3₋¹⁺²×M₃₂

C₂

socle

C₂×C₃

C₃×Dic₃⋊C₈

prime radical

C₂²×C₃²

ΓL(1,𝔽₂₅)

cosocle

C₂×C₈×C₉

C₂²×C₃

M₃₂

C₉

G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
5. #10588188	C₃	semidirect
87. 3₋¹⁺²	#364	semidirect
119. C₉	C₃×#364	semidirect

H ≅ G/N

type

5. #10588188

C₃

semidirect

87. 3₋¹⁺²

#364

semidirect

119. C₉

C₃×#364

semidirect

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

subgroup	index
3₋¹⁺²×M₃₂	2
C₈×3₋¹⁺²⋊C₄	2
#63371⋊C₃	2
C₃×#4511	3
#10588188	3

subgroup

index

3₋¹⁺²×M₃₂

C₈×3₋¹⁺²⋊C₄

#63371⋊C₃

C₃×#4511

#10588188

Subgroups — #10588188⋊C₃ #13135516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588188⋊C3 #13135516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588188⋊C₃ #13135516