Group #13143131 — subgroups

Subgroups — C₂²×3₋¹⁺²⋊C₁₆ #13143131

Order 1728.

332 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 212 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	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×3₋¹⁺²⋊C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₃⋊C₁₆	—	3	1	no	maximal
4	C₂²×C₉⋊C₁₆	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₂×C₈×3₋¹⁺²	C₄	1	3	yes	—
7	C₂×C₈×3₋¹⁺²	C₂²	1	3	yes	—
8	3₋¹⁺²⋊C₁₆	C₂²	1	4	yes	direct product factor
9	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
10	C₂×C₃×C₃⋊C₁₆	—	3	6	no	—
11	C₂²×C₈×C₉	—	2	1	no	—
12	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
13	C₂×C₉⋊C₁₆	C₂×C₃	1	6	yes	complemented
14	C₂³×3₋¹⁺²	C₈	1	1	yes	characteristic
15	C₂×C₄×3₋¹⁺²	C₈	1	3	yes	—
16	C₂×C₄×3₋¹⁺²	C₂×C₄	1	3	yes	—
17	C₈×3₋¹⁺²	C₂×C₄	1	3	yes	—
18	C₈×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	C₂²×C₃×C₁₆	—	9	1	no	—
20	C₂²×C₃⋊C₁₆	—	3	1	no	—
21	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
22	C₂×C₃²×C₈	Dic₃	1	3	yes	—
23	C₂×C₃²×C₈	C₂×D₃	1	3	yes	—
24	C₂²×C₄×C₉	—	2	1	no	—
25	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
26	C₃×C₃⋊C₁₆	—	3	4	no	—
27	C₂×C₈×C₉	—	2	3	no	—
28	C₂×C₈×C₉	—	2	3	no	—
29	C₂×C₈×C₉	C₃×C₄	1	3	yes	—
30	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
31	C₉⋊C₁₆	C₂²×C₃	1	4	yes	complemented
32	C₂²×3₋¹⁺²	C₁₆	1	4	yes	complemented
33	C₂²×3₋¹⁺²	C₂×C₈	1	3	yes	—
34	C₄×3₋¹⁺²	C₂×C₈	1	3	yes	—
35	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	—	3	1	no	—
37	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
38	C₂×C₃×C₁₆	—	9	6	no	—
39	C₂×C₃⋊C₁₆	—	3	6	no	—
40	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₃⋊C₈	1	3	yes	—
42	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
43	C₂³×C₉	—	2	1	no	—
44	C₂³×C₉	C₃×C₈	1	1	yes	characteristic
45	C₃²×C₈	C₂×Dic₃	1	3	yes	—
46	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
47	C₂×C₄×C₉	—	2	3	no	—
48	C₂×C₄×C₉	—	2	3	no	—
49	C₂×C₄×C₉	C₃×C₈	1	3	yes	—
50	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
51	C₈×C₉	—	2	3	no	—
52	C₈×C₉	—	2	1	no	—
53	C₈×C₉	C₂×C₃×C₄	1	3	yes	—
54	C₈×C₉	C₂³×C₃	1	1	yes	characteristic
55	C₂²×C₁₆	—	9	1	no	Sylow
56	C₂×3₋¹⁺²	C₂×C₁₆	1	6	yes	complemented
57	C₂×3₋¹⁺²	C₂²×C₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	3	1	no	—
59	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
60	C₂×C₃×C₈	—	3	3	no	—
61	C₂×C₃×C₈	—	3	3	no	—
62	C₂×C₃×C₈	C₃×Dic₃	1	3	yes	—
63	C₂×C₃×C₈	C₂×C₃×D₃	1	3	yes	—
64	C₃×C₁₆	—	9	4	no	—
65	C₃⋊C₁₆	—	3	4	no	—
66	C₂²×C₃²	C₃⋊C₁₆	1	4	yes	—
67	C₂²×C₃²	C₂×C₃⋊C₈	1	3	yes	—
68	C₃²×C₄	C₂×C₃⋊C₈	1	3	yes	—
69	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
70	C₂²×C₉	—	2	4	no	—
71	C₂²×C₉	—	2	3	no	—
72	C₂²×C₉	C₃×C₁₆	1	4	yes	complemented
73	C₂²×C₉	C₂×C₃×C₈	1	3	yes	—
74	C₄×C₉	—	2	3	no	—
75	C₄×C₉	—	2	1	no	—
76	C₄×C₉	C₂×C₃×C₈	1	3	yes	—
77	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
78	C₂²×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
79	C₂×C₁₆	—	9	6	no	—
80	3₋¹⁺²	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₃	—	3	1	no	—
82	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
83	C₂×C₃×C₄	—	3	3	no	—
84	C₂×C₃×C₄	—	3	3	no	—
85	C₂×C₃×C₄	C₃×C₃⋊C₈	1	3	yes	—
86	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
87	C₃×C₈	—	3	3	no	—
88	C₃×C₈	—	3	1	no	—
89	C₃×C₈	C₂×C₃×Dic₃	1	3	yes	—
90	C₃×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
91	C₂×C₃²	C₂×C₃⋊C₁₆	1	6	yes	—
92	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
93	C₂×C₉	—	2	1	no	—
94	C₂×C₉	—	2	6	no	—
95	C₂×C₉	C₂×C₃×C₁₆	1	6	yes	complemented
96	C₂×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
97	C₂²×C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
98	C₂×C₈	3₋¹⁺²⋊C₄	1	3	yes	—
99	C₂×C₈	C₂×AGL(1,ℤ/9)	1	3	yes	—
100	C₁₆	—	9	4	no	—
101	C₂²×C₃	—	3	4	no	—
102	C₂²×C₃	—	3	3	no	—
103	C₂²×C₃	C₃×C₃⋊C₁₆	1	4	yes	—
104	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	3	yes	—
105	C₃×C₄	—	3	1	no	—
106	C₃×C₄	—	3	3	no	—
107	C₃×C₄	C₂×C₃×C₃⋊C₈	1	3	yes	—
108	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
109	C₃²	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
110	C₉	—	2	1	no	—
111	C₉	C₂²×C₃×C₁₆	1	1	yes	characteristic, complemented
112	C₂³	3₋¹⁺²⋊C₈	1	1	yes	characteristic
113	C₂×C₄	3₋¹⁺²⋊C₈	1	3	yes	—
114	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	3	yes	—
115	C₈	C₂×3₋¹⁺²⋊C₄	1	3	yes	—
116	C₈	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
117	C₂×C₃	—	3	1	no	—
118	C₂×C₃	—	3	6	no	—
119	C₂×C₃	C₂×C₃×C₃⋊C₁₆	1	6	yes	—
120	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
121	C₂²	3₋¹⁺²⋊C₁₆	1	4	yes	direct product factor
122	C₂²	C₂×3₋¹⁺²⋊C₈	1	3	yes	—
123	C₄	C₂×3₋¹⁺²⋊C₈	1	3	yes	—
124	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
125	C₃	—	3	1	no	—
126	C₃	C₂²×C₃×C₃⋊C₁₆	1	1	yes	characteristic
127	C₂	C₂×3₋¹⁺²⋊C₁₆	1	6	yes	direct product factor
128	C₂	C₂²×3₋¹⁺²⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×3₋¹⁺²⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₃×C₃⋊C₁₆

—

maximal

C₂²×C₉⋊C₁₆

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

C₄

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

3₋¹⁺²⋊C₁₆

C₂²

yes

direct product factor

C₂²×C₃²×C₈

D₃

yes

characteristic

C₂×C₃×C₃⋊C₁₆

—

C₂²×C₈×C₉

—

C₂²×C₈×C₉

C₂×C₃

yes

characteristic

C₂×C₉⋊C₁₆

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₈

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₈×3₋¹⁺²

C₂×C₄

yes

—

C₈×3₋¹⁺²

C₂³

yes

characteristic

C₂²×C₃×C₁₆

—

C₂²×C₃⋊C₁₆

—

C₂²×C₃²×C₄

Dic₃

yes

characteristic

C₂×C₃²×C₈

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₃×C₄

yes

characteristic

C₃×C₃⋊C₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×C₄

yes

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉⋊C₁₆

C₂²×C₃

yes

complemented

C₂²×3₋¹⁺²

C₁₆

yes

complemented

C₂²×3₋¹⁺²

C₂×C₈

yes

—

C₄×3₋¹⁺²

C₂×C₈

yes

—

C₄×3₋¹⁺²

C₂²×C₄

yes

characteristic

C₂²×C₃×C₈

—

C₂²×C₃×C₈

C₃×D₃

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃⋊C₁₆

—

C₂³×C₃²

C₃⋊C₈

yes

characteristic

C₂×C₃²×C₄

C₃⋊C₈

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂³×C₉

—

C₂³×C₉

C₃×C₈

yes

characteristic

C₃²×C₈

C₂×Dic₃

yes

—

C₃²×C₈

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×C₈

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

C₂³×C₃

yes

characteristic

C₂²×C₁₆

—

Sylow

C₂×3₋¹⁺²

C₂×C₁₆

yes

complemented

C₂×3₋¹⁺²

C₂²×C₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×Dic₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Dic₃

yes

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₃⋊C₁₆

yes

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

—

C₃²×C₄

C₂×C₃⋊C₈

yes

—

C₃²×C₄

C₂²×Dic₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₁₆

yes

complemented

C₂²×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

C₂²×C₃×C₄

yes

characteristic

C₂²×C₈

AGL(1,ℤ/9)

yes

characteristic

C₂×C₁₆

—

3₋¹⁺²

C₂²×C₁₆

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₃×C₃⋊C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₃⋊C₈

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃×Dic₃

yes

—

C₃×C₈

C₂²×C₃×D₃

yes

characteristic

C₂×C₃²

C₂×C₃⋊C₁₆

yes

—

C₂×C₃²

C₂²×C₃⋊C₈

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₃×C₁₆

yes

complemented

C₂×C₉

C₂²×C₃×C₈

yes

characteristic

C₂²×C₄

3₋¹⁺²⋊C₄

yes

characteristic

C₂×C₈

3₋¹⁺²⋊C₄

yes

—

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

100

C₁₆

—

101

C₂²×C₃

—

102

C₂²×C₃

—

103

C₂²×C₃

C₃×C₃⋊C₁₆

yes

—

104

C₂²×C₃

C₂×C₃×C₃⋊C₈

yes

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

C₂×C₃×C₃⋊C₈

yes

—

108

C₃×C₄

C₂²×C₃×Dic₃

yes

characteristic

109

C₃²

C₂²×C₃⋊C₁₆

yes

characteristic

110

C₉

—

111

C₉

C₂²×C₃×C₁₆

yes

characteristic, complemented

112

C₂³

3₋¹⁺²⋊C₈

yes

characteristic

113

C₂×C₄

3₋¹⁺²⋊C₈

yes

—

114

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

115

C₈

C₂×3₋¹⁺²⋊C₄

yes

—

116

C₈

C₂²×AGL(1,ℤ/9)

yes

characteristic

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

C₂×C₃×C₃⋊C₁₆

yes

—

120

C₂×C₃

C₂²×C₃×C₃⋊C₈

yes

characteristic

121

C₂²

3₋¹⁺²⋊C₁₆

yes

direct product factor

122

C₂²

C₂×3₋¹⁺²⋊C₈

yes

—

123

C₄

C₂×3₋¹⁺²⋊C₈

yes

—

124

C₄

C₂²×3₋¹⁺²⋊C₄

yes

characteristic

125

C₃

—

126

C₃

C₂²×C₃×C₃⋊C₁₆

yes

characteristic

127

C₂

C₂×3₋¹⁺²⋊C₁₆

yes

direct product factor

128

C₂

C₂²×3₋¹⁺²⋊C₈

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂²×C₃×C₁₆
Frattini	C₃×C₈	C₂²×C₃×D₃
Fitting	C₂²×C₈×3₋¹⁺²	C₂
socle	C₂³×C₃	C₃×C₃⋊C₈
prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₈×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂²×C₈

subgroup

G/H

center, hypercenter

C₂²×C₈

AGL(1,ℤ/9)

commutator, hypocenter

C₉

C₂²×C₃×C₁₆

Frattini

C₃×C₈

C₂²×C₃×D₃

Fitting

C₂²×C₈×3₋¹⁺²

C₂

socle

C₂³×C₃

C₃×C₃⋊C₈

prime radical

C₂³×C₃²

C₃⋊C₈

cosocle

C₈×C₉

C₂³×C₃

LCS

G ▷ C₉

UCS

C₁ ◁ C₂²×C₈

N	H ≅ G/N	type
2. C₂×3₋¹⁺²⋊C₁₆	C₂	direct (N × H)
4. C₂²×C₉⋊C₁₆	C₃	semidirect
8. 3₋¹⁺²⋊C₁₆	C₂²	direct (N × H)
13. C₂×C₉⋊C₁₆	C₂×C₃	semidirect
31. C₉⋊C₁₆	C₂²×C₃	semidirect
32. C₂²×3₋¹⁺²	C₁₆	semidirect
56. C₂×3₋¹⁺²	C₂×C₁₆	semidirect
72. C₂²×C₉	C₃×C₁₆	semidirect
80. 3₋¹⁺²	C₂²×C₁₆	semidirect
95. C₂×C₉	C₂×C₃×C₁₆	semidirect
111. C₉	C₂²×C₃×C₁₆	semidirect
121. C₂²	3₋¹⁺²⋊C₁₆	direct (N × H)
127. C₂	C₂×3₋¹⁺²⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×3₋¹⁺²⋊C₁₆

C₂

direct (N × H)

4. C₂²×C₉⋊C₁₆

C₃

semidirect

8. 3₋¹⁺²⋊C₁₆

C₂²

direct (N × H)

13. C₂×C₉⋊C₁₆

C₂×C₃

semidirect

31. C₉⋊C₁₆

C₂²×C₃

semidirect

32. C₂²×3₋¹⁺²

C₁₆

semidirect

56. C₂×3₋¹⁺²

C₂×C₁₆

semidirect

72. C₂²×C₉

C₃×C₁₆

semidirect

80. 3₋¹⁺²

C₂²×C₁₆

semidirect

95. C₂×C₉

C₂×C₃×C₁₆

semidirect

111. C₉

C₂²×C₃×C₁₆

semidirect

121. C₂²

3₋¹⁺²⋊C₁₆

direct (N × H)

127. C₂

C₂×3₋¹⁺²⋊C₁₆

direct (N × H)

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

subgroup	index
C₂²×C₈×3₋¹⁺²	2
C₂×3₋¹⁺²⋊C₁₆	2
C₂²×C₃×C₃⋊C₁₆	3
C₂²×C₉⋊C₁₆	3

subgroup

index

C₂²×C₈×3₋¹⁺²

C₂×3₋¹⁺²⋊C₁₆

C₂²×C₃×C₃⋊C₁₆

C₂²×C₉⋊C₁₆

Subgroups — C₂²×3₋¹⁺²⋊C₁₆ #13143131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×3−1+2⋊C16 #13143131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×3₋¹⁺²⋊C₁₆ #13143131