Group #13161465 — subgroups

Subgroups — #13161465

Order 1728.

430 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 95 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	#11723651	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₆×C₃⋊D₃	C₂	1	3	yes	maximal, complemented
3	#11722066	C₂	1	3	yes	maximal, complemented
4	#10590227	—	3	3	no	maximal
5	#89042	C₄	1	1	yes	characteristic
6	C₃×C₈×C₃⋊D₃	C₂²	1	3	yes	—
7	#88809	C₄	1	3	yes	—
8	C₃³×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×C₃⋊(C₃⋊C₁₆)	C₂²	1	3	yes	—
10	#63803	—	3	3	no	—
11	C₁₆×C₃⋊D₃	D₃	1	3	yes	complemented
12	C₃×D₃×C₁₆	—	3	6	no	—
13	C₃×D₃×C₁₆	—	6	3	no	—
14	#63544	—	3	3	no	—
15	#63543	—	3	6	no	—
16	C₃×C₄×C₃⋊D₃	C₂×C₄	1	3	yes	—
17	#6296	C₈	1	1	yes	characteristic
18	#6295	C₈	1	3	yes	—
19	C₃³×C₈	C₂³	1	1	yes	characteristic
20	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	3	yes	—
21	#4901	—	9	3	no	—
22	#3798	—	3	3	no	—
23	C₈×C₃⋊D₃	C₂×D₃	1	3	yes	—
24	C₃×D₃×C₈	—	3	6	no	—
25	C₃×D₃×C₈	—	6	3	no	—
26	#3712	—	3	3	no	—
27	#3711	—	3	6	no	—
28	C₃²×C₁₆	—	2	3	no	—
29	C₃²×C₁₆	—	4	1	no	—
30	C₃²×C₁₆	C₂×D₃	1	3	yes	—
31	C₃⋊(C₃⋊C₁₆)	C₂×D₃	1	3	yes	—
32	C₃×C₃⋊C₁₆	—	3	6	no	—
33	C₃×C₃⋊C₁₆	—	6	3	no	—
34	C₂×C₃×C₃⋊D₃	C₂×C₈	1	3	yes	—
35	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₃×C₃⋊Dic₃	C₂×C₈	1	3	yes	—
37	#901	—	9	3	no	—
38	C₂×C₃×C₁₆	—	9	3	no	—
39	#812	—	9	3	no	—
40	#798	—	9	6	no	—
41	D₃×C₁₆	—	3	6	no	—
42	D₃×C₁₆	—	6	3	no	—
43	C₄×C₃⋊D₃	C₄×D₃	1	3	yes	—
44	C₃×C₄×D₃	—	3	6	no	—
45	C₃×C₄×D₃	—	6	3	no	—
46	#627	—	3	3	no	—
47	#626	—	3	6	no	—
48	#625	—	3	3	no	—
49	C₃²×C₈	—	2	3	no	—
50	C₃²×C₈	—	4	1	no	—
51	C₃²×C₈	C₂²×D₃	1	3	yes	—
52	C₃⋊(C₃⋊C₈)	C₄×D₃	1	3	yes	—
53	C₃×C₃⋊C₈	—	3	6	no	—
54	C₃×C₃⋊C₈	—	6	3	no	—
55	#504	—	27	1	no	Sylow
56	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
57	C₃×C₃⋊D₃	—	2	3	no	—
58	#235	—	9	3	no	—
59	C₂×C₃×C₈	—	9	3	no	—
60	#208	—	9	3	no	—
61	ΓL(1,𝔽₂₅)	—	9	6	no	—
62	D₃×C₈	—	3	6	no	—
63	D₃×C₈	—	6	3	no	—
64	C₃×C₁₆	—	2	3	no	—
65	C₃×C₁₆	—	4	1	no	—
66	C₃×C₁₆	—	9	3	no	—
67	C₃×C₁₆	D₃²	1	3	yes	—
68	C₃⋊C₁₆	—	3	6	no	—
69	C₃⋊C₁₆	—	6	3	no	—
70	C₂×C₃⋊D₃	D₃×C₈	1	3	yes	—
71	C₂×C₃×D₃	—	3	6	no	—
72	C₂×C₃×D₃	—	6	3	no	—
73	C₃²×C₄	—	2	3	no	—
74	C₃²×C₄	—	4	1	no	—
75	C₃²×C₄	C₂×C₄×D₃	1	3	yes	—
76	C₃⋊Dic₃	D₃×C₈	1	3	yes	—
77	C₃×Dic₃	—	3	6	no	—
78	C₃×Dic₃	—	6	3	no	—
79	#131	—	27	1	no	—
80	M₃₂	—	27	3	no	—
81	C₂×C₁₆	—	9	3	no	—
82	C₃³	#504	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃×C₄	—	9	3	no	—
84	#67	—	9	6	no	—
85	D₁₂	—	9	3	no	—
86	C₄×D₃	—	3	6	no	—
87	C₄×D₃	—	6	3	no	—
88	Dic₆	—	9	3	no	—
89	C₃×C₈	—	2	3	no	—
90	C₃×C₈	—	4	1	no	—
91	C₃×C₈	—	9	3	no	—
92	C₃×C₈	C₂×D₃²	1	3	yes	—
93	C₃⋊C₈	—	3	6	no	—
94	C₃⋊C₈	—	6	3	no	—
95	C₂×C₃²	—	2	3	no	—
96	C₂×C₃²	—	4	1	no	—
97	C₂×C₃²	C₂×D₃×C₈	1	3	yes	—
98	C₃⋊D₃	—	2	3	no	—
99	C₃×D₃	—	6	6	no	—
100	C₃×D₃	—	6	6	no	—
101	#41	—	27	1	no	—
102	M₁₆	—	27	3	no	—
103	C₂×C₈	—	9	3	no	—
104	C₁₆	—	9	3	no	—
105	C₁₆	#1113	1	1	yes	characteristic
106	C₂²×C₃	—	9	3	no	—
107	C₂×D₃	—	3	6	no	—
108	C₂×D₃	—	6	3	no	—
109	C₃×C₄	—	2	3	no	—
110	C₃×C₄	—	4	1	no	—
111	C₃×C₄	—	9	3	no	—
112	C₃×C₄	C₄×D₃²	1	3	yes	—
113	Dic₃	—	3	6	no	—
114	Dic₃	—	6	3	no	—
115	C₃²	—	2	3	no	—
116	C₃²	—	4	1	no	—
117	C₃²	#4901	1	3	yes	complemented
118	Q₈	—	27	1	no	—
119	D₄	—	27	3	no	—
120	C₂×C₄	—	9	3	no	—
121	C₈	—	9	3	no	—
122	C₈	C₂×#1113	1	1	yes	characteristic
123	C₂×C₃	—	2	3	no	—
124	C₂×C₃	—	4	1	no	—
125	C₂×C₃	—	18	3	no	—
126	C₂×C₃	D₃²×C₈	1	3	yes	—
127	D₃	—	6	6	no	—
128	D₃	—	6	6	no	—
129	C₂²	—	9	3	no	—
130	C₄	—	9	3	no	—
131	C₄	C₄×#1113	1	1	yes	characteristic
132	C₃	—	2	3	no	—
133	C₃	—	4	1	no	—
134	C₃	#10590227	1	3	yes	complemented
135	C₂	—	18	3	no	—
136	C₂	C₈×#1113	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11723651

C₂

yes

maximal, characteristic

C₃×C₁₆×C₃⋊D₃

C₂

yes

maximal, complemented

#11722066

C₂

yes

maximal, complemented

#10590227

—

maximal

#89042

C₄

yes

characteristic

C₃×C₈×C₃⋊D₃

C₂²

yes

—

#88809

C₄

yes

—

C₃³×C₁₆

C₂²

yes

characteristic

C₃×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

#63803

—

C₁₆×C₃⋊D₃

D₃

yes

complemented

C₃×D₃×C₁₆

—

C₃×D₃×C₁₆

—

#63544

—

#63543

—

C₃×C₄×C₃⋊D₃

C₂×C₄

yes

—

#6296

C₈

yes

characteristic

#6295

C₈

yes

—

C₃³×C₈

C₂³

yes

characteristic

C₃×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

—

#4901

—

#3798

—

C₈×C₃⋊D₃

C₂×D₃

yes

—

C₃×D₃×C₈

—

C₃×D₃×C₈

—

#3712

—

#3711

—

C₃²×C₁₆

—

C₃²×C₁₆

—

C₃²×C₁₆

C₂×D₃

yes

—

C₃⋊(C₃⋊C₁₆)

C₂×D₃

yes

—

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₂×C₃×C₃⋊D₃

C₂×C₈

yes

—

C₃³×C₄

C₂²×C₄

yes

characteristic

C₃×C₃⋊Dic₃

C₂×C₈

yes

—

#901

—

C₂×C₃×C₁₆

—

#812

—

#798

—

D₃×C₁₆

—

D₃×C₁₆

—

C₄×C₃⋊D₃

C₄×D₃

yes

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

#627

—

#626

—

#625

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

C₂²×D₃

yes

—

C₃⋊(C₃⋊C₈)

C₄×D₃

yes

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

#504

—

Sylow

C₂×C₃³

C₂²×C₈

yes

characteristic

C₃×C₃⋊D₃

—

#235

—

C₂×C₃×C₈

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

D₃²

yes

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂×C₃⋊D₃

D₃×C₈

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃⋊Dic₃

D₃×C₈

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

#131

—

M₃₂

—

C₂×C₁₆

—

C₃³

#504

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×D₃²

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×D₃×C₈

yes

—

C₃⋊D₃

—

C₃×D₃

—

100

C₃×D₃

—

101

#41

—

102

M₁₆

—

103

C₂×C₈

—

104

C₁₆

—

105

C₁₆

#1113

yes

characteristic

106

C₂²×C₃

—

107

C₂×D₃

—

108

C₂×D₃

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

C₄×D₃²

yes

—

113

Dic₃

—

114

Dic₃

—

115

C₃²

—

116

C₃²

—

117

C₃²

#4901

yes

complemented

118

Q₈

—

119

D₄

—

120

C₂×C₄

—

121

C₈

—

122

C₈

C₂×#1113

yes

characteristic

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

D₃²×C₈

yes

—

127

D₃

—

128

D₃

—

129

C₂²

—

130

C₄

—

131

C₄

C₄×#1113

yes

characteristic

132

C₃

—

133

C₃

—

134

C₃

#10590227

yes

complemented

135

C₂

—

136

C₂

C₈×#1113

yes

characteristic

	subgroup	G/H
center, hypercenter	C₁₆	#1113
commutator, socle	C₂×C₃³	C₂²×C₈
Frattini	C₈	C₂×#1113
Fitting	C₃³×C₁₆	C₂²
prime radical	#89042	C₄
cosocle	C₃³×C₈	C₂³
hypocenter	C₃³	#504
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₁₆

subgroup

G/H

center, hypercenter

C₁₆

#1113

commutator, socle

C₂×C₃³

C₂²×C₈

Frattini

C₈

C₂×#1113

Fitting

C₃³×C₁₆

C₂²

prime radical

#89042

C₄

cosocle

C₃³×C₈

C₂³

C₃³

G ▷ C₂×C₃³ ▷ C₃³

C₁ ◁ C₁₆

N	H ≅ G/N	type
2. C₃×C₁₆×C₃⋊D₃	C₂	semidirect
3. #11722066	C₂	semidirect
11. C₁₆×C₃⋊D₃	D₃	semidirect
82. C₃³	#504	semidirect
117. C₃²	#4901	semidirect
134. C₃	#10590227	semidirect

H ≅ G/N

type

2. C₃×C₁₆×C₃⋊D₃

C₂

semidirect

3. #11722066

C₂

semidirect

11. C₁₆×C₃⋊D₃

D₃

semidirect

82. C₃³

#504

semidirect

117. C₃²

#4901

semidirect

134. C₃

#10590227

semidirect

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

subgroup	index
#11723651	2
C₃×C₁₆×C₃⋊D₃	2
#11722066	2
#10590227	3

subgroup

index

#11723651

C₃×C₁₆×C₃⋊D₃

#11722066

#10590227

Subgroups — #13161465

Conjugacy classes of subgroups

Complements

Maximal subgroups