Group #13141470 — subgroups

Subgroups — #10588363⋊C₃ #13141470

Order 1728.

280 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 112 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	#63432⋊C₃	C₂	1	2	yes	maximal
2	C₄×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	#63378⋊C₃	C₂	1	2	yes	maximal
4	#63377⋊C₃	C₂	1	2	yes	maximal
5	C₃×#4683	—	3	1	no	maximal
6	#10588363	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	4	yes	—
11	(3₋¹⁺²⋊C₄)⋊C₄	—	2	4	no	—
12	C₃×#870	—	3	2	no	—
13	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
14	C₃×#818	—	3	2	no	—
15	C₃×#817	—	3	2	no	—
16	#63432	C₂×C₃	1	2	yes	—
17	C₄×C₈×C₉	—	2	1	no	—
18	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic
19	#63378	C₂×C₃	1	2	yes	—
20	#63377	C₂×C₃	1	2	yes	—
21	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
22	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
23	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
24	C₈×3₋¹⁺²	Q₈	1	2	yes	—
25	C₈×3₋¹⁺²	Q₈	1	2	yes	—
26	C₃×#499	—	9	1	no	—
27	#4683	—	3	1	no	—
28	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
30	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	—	3	4	no	—
32	C₃×Dic₃⋊C₄	—	6	4	no	—
33	C₂×C₈×C₉	—	2	1	no	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₄²×C₉	—	2	1	no	—
38	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
39	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
40	Dic₉⋊C₄	—	2	4	no	—
41	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
42	C₄×3₋¹⁺²	—	2	2	no	—
43	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
44	C₄×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
45	3₋¹⁺²⋊C₄	—	4	4	no	—
46	C₃×#125	—	9	2	no	—
47	#870	—	3	2	no	—
48	C₃×#107	—	9	2	no	—
49	C₃×#106	—	9	2	no	—
50	C₃×C₄×C₈	—	3	1	no	—
51	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
52	#818	—	3	2	no	—
53	#817	—	3	2	no	—
54	C₂×C₃²×C₄	D₁₂	1	2	yes	—
55	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₃×Dic₃	—	6	4	no	—
57	C₃²×C₈	Dic₆	1	2	yes	—
58	C₃²×C₈	Dic₆	1	2	yes	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
62	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
63	C₂×Dic₉	—	2	4	no	—
64	C₈×C₉	—	2	2	no	—
65	C₈×C₉	—	2	2	no	—
66	C₈×C₉	C₃×Q₈	1	2	yes	—
67	C₈×C₉	C₃×Q₈	1	2	yes	—
68	#499	—	9	1	no	Sylow
69	C₂×3₋¹⁺²	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×3₋¹⁺²	#135	1	2	yes	—
71	C₂×C₃×C₈	—	3	1	no	—
72	C₂×C₃×C₈	—	3	1	no	—
73	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
75	C₃×C₄⋊C₄	—	9	4	no	—
76	C₃×C₄⋊C₄	—	18	4	no	—
77	C₃×C₄²	—	3	1	no	—
78	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
79	C₄⋊Dic₃	—	3	4	no	—
80	Dic₃⋊C₄	—	6	4	no	—
81	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
82	C₃²×C₄	—	2	2	no	—
83	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
84	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
85	C₃×Dic₃	—	12	4	no	—
86	C₂²×C₉	—	2	1	no	—
87	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
88	C₄×C₉	—	2	4	no	—
89	C₄×C₉	—	2	2	no	—
90	C₄×C₉	—	2	1	no	—
91	C₄×C₉	—	2	1	no	—
92	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
93	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
94	Dic₉	—	4	4	no	—
95	#125	—	9	2	no	—
96	#107	—	9	2	no	—
97	#106	—	9	2	no	—
98	C₄×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
99	3₋¹⁺²	#499	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₃×C₄	—	3	2	no	—
101	C₂×C₃×C₄	—	3	1	no	—
102	C₂×C₃×C₄	—	18	4	no	—
103	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
104	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
105	C₂×Dic₃	—	6	4	no	—
106	C₃×C₈	—	3	2	no	—
107	C₃×C₈	—	3	2	no	—
108	C₃×C₈	C₃×Dic₆	1	2	yes	—
109	C₃×C₈	C₃×Dic₆	1	2	yes	—
110	C₂×C₃²	#869	1	1	yes	characteristic
111	C₂×C₃²	#904	1	2	yes	—
112	C₂×C₉	—	2	2	no	—
113	C₂×C₉	—	2	1	no	—
114	C₂×C₉	C₃×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₉	C₃×#135	1	2	yes	—
116	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
117	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
118	C₄⋊C₄	—	9	4	no	—
119	C₄⋊C₄	—	18	4	no	—
120	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
121	C₂²×C₃	—	3	1	no	—
122	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	3	1	no	—
125	C₃×C₄	—	3	1	no	—
126	C₃×C₄	—	6	2	no	—
127	C₃×C₄	—	36	4	no	—
128	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
129	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
130	Dic₃	—	12	4	no	—
131	C₃²	#4683	1	1	yes	characteristic
132	C₉	—	2	1	no	—
133	C₉	C₃×#499	1	1	yes	characteristic, complemented
134	C₂×C₄	—	18	4	no	—
135	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
136	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
137	C₈	3₋¹⁺²⋊Q₈	1	2	yes	—
138	C₈	3₋¹⁺²⋊Q₈	1	2	yes	—
139	C₂×C₃	—	3	2	no	—
140	C₂×C₃	—	3	1	no	—
141	C₂×C₃	C₃×#869	1	1	yes	characteristic
142	C₂×C₃	C₃×#904	1	2	yes	—
143	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	36	4	no	—
146	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
147	C₄	C₂×3₋¹⁺²⋊Q₈	1	1	yes	characteristic
148	C₃	—	3	1	no	—
149	C₃	C₃×#4683	1	1	yes	characteristic
150	C₂	#63431⋊C₃	1	1	yes	characteristic
151	C₂	#63467⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#63432⋊C₃

C₂

yes

maximal

C₄×C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

#63378⋊C₃

C₂

yes

maximal

#63377⋊C₃

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

—

C₃×#870

—

C₃²×C₄×C₈

D₃

yes

—

—

C₂×C₃

yes

—

C₄×C₈×C₉

—

C₄×C₈×C₉

C₂×C₃

yes

characteristic

#63378

C₂×C₃

yes

—

#63377

C₂×C₃

yes

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

C₈×3₋¹⁺²

Q₈

yes

—

C₈×3₋¹⁺²

Q₈

yes

—

C₃×#499

—

#4683

—

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

characteristic

C₂×C₈×C₉

C₂²×C₃

yes

characteristic

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

C₂²×C₃

yes

—

Dic₉⋊C₄

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×Q₈

yes

characteristic

C₄×3₋¹⁺²

C₂×Q₈

yes

characteristic

3₋¹⁺²⋊C₄

—

—

—

—

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃×D₃

yes

characteristic

#818

—

#817

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃²×C₈

Dic₆

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₃×Q₈

yes

—

C₈×C₉

C₃×Q₈

yes

—

#499

—

Sylow

C₂×3₋¹⁺²

C₄⋊Q₈

yes

characteristic

C₂×3₋¹⁺²

#135

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃×D₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

characteristic

C₃²×C₄

C₂×Dic₆

yes

characteristic

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃×Q₈

yes

characteristic

C₄×C₉

C₂×C₃×Q₈

yes

characteristic

Dic₉

—

#125

—

#107

—

#106

—

C₄×C₈

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

#499

yes

characteristic, complemented, Sylow

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

C₃×D₁₂

yes

—

104

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

105

C₂×Dic₃

—

106

C₃×C₈

—

107

C₃×C₈

—

108

C₃×C₈

C₃×Dic₆

yes

—

109

C₃×C₈

C₃×Dic₆

yes

—

110

C₂×C₃²

#869

yes

characteristic

111

C₂×C₃²

#904

yes

—

112

C₂×C₉

—

113

C₂×C₉

—

114

C₂×C₉

C₃×C₄⋊Q₈

yes

characteristic

115

C₂×C₉

C₃×#135

yes

—

116

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

characteristic

117

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

121

C₂²×C₃

—

122

C₂²×C₃

C₂×C₃×D₁₂

yes

characteristic

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

C₂×C₃×Dic₆

yes

characteristic

129

C₃×C₄

C₂×C₃×Dic₆

yes

characteristic

130

Dic₃

—

131

C₃²

#4683

yes

characteristic

132

C₉

—

133

C₉

C₃×#499

yes

characteristic, complemented

134

C₂×C₄

—

135

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

136

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

137

C₈

3₋¹⁺²⋊Q₈

yes

—

138

C₈

3₋¹⁺²⋊Q₈

yes

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

C₃×#869

yes

characteristic

142

C₂×C₃

C₃×#904

yes

—

143

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

C₂×3₋¹⁺²⋊Q₈

yes

characteristic

147

C₄

C₂×3₋¹⁺²⋊Q₈

yes

characteristic

148

C₃

—

149

C₃

C₃×#4683

yes

characteristic

150

C₂

#63431⋊C₃

yes

characteristic

151

C₂

#63467⋊C₃

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×C₈×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₁₂
prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	C₄×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#499
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₄×C₈×3₋¹⁺²

C₂

socle

C₂²×C₃

C₂×C₃×D₁₂

prime radical

C₂²×C₃²

C₂×D₁₂

C₄×C₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
6. #10588363	C₃	semidirect
99. 3₋¹⁺²	#499	semidirect
133. C₉	C₃×#499	semidirect

H ≅ G/N

type

6. #10588363

C₃

semidirect

99. 3₋¹⁺²

#499

semidirect

133. C₉

C₃×#499

semidirect

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

subgroup	index
#63432⋊C₃	2
C₄×C₈×3₋¹⁺²	2
#63378⋊C₃	2
#63377⋊C₃	2
C₃×#4683	3
#10588363	3

subgroup

index

C₄×C₈×3₋¹⁺²

Subgroups — #10588363⋊C₃ #13141470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588363⋊C3 #13141470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588363⋊C₃ #13141470