Group #13152101 — subgroups

Subgroups — C₂²×C₉⋊(C₄²⋊C₃) #13152101

Order 1728.

354 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₂×C₉⋊(C₄²⋊C₃)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₄²⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄²⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂²×C₉⋊A₄	—	4	1	no	maximal
6	C₉⋊(C₄²⋊C₃)	C₂²	1	1	yes	characteristic, direct product factor
7	C₂×C₃×C₄²⋊C₃	C₂×C₃	1	3	yes	—
8	C₂³×C₄×C₉	—	3	1	no	—
9	C₂×C₄²×C₉	—	3	3	no	—
10	C₂×C₄²×C₉	C₂×C₃	1	3	yes	complemented
11	C₂×C₄²⋊C₉	C₂×C₃	1	6	yes	complemented
12	C₂×C₉⋊A₄	—	4	3	no	—
13	C₂²×C₃×C₄²	C₃²	1	1	yes	characteristic
14	C₂²×C₄²⋊C₃	—	3	1	no	—
15	C₂²×C₃×A₄	—	4	1	no	—
16	C₂⁴×C₉	A₄	1	1	yes	characteristic
17	C₂²×C₂²⋊C₉	—	4	2	no	—
18	C₃×C₄²⋊C₃	C₂²×C₃	1	1	yes	characteristic
19	C₂²×C₄×C₉	—	3	6	no	—
20	C₂²×C₄×C₉	—	3	2	no	—
21	C₂²×C₄×C₉	—	3	3	no	—
22	C₂²×C₄×C₉	—	3	3	no	—
23	C₄²×C₉	—	3	2	no	—
24	C₄²×C₉	—	3	3	no	—
25	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
26	C₄²⋊C₉	C₂²×C₃	1	2	yes	complemented
27	C₂²×3₋¹⁺²	—	16	1	no	—
28	C₉⋊A₄	—	4	1	no	—
29	C₂³×C₃×C₄	—	3	1	no	—
30	C₂×C₃×C₄²	—	3	3	no	—
31	C₂×C₃×C₄²	C₂×C₃²	1	3	yes	—
32	C₂×C₄²⋊C₃	—	3	3	no	—
33	C₂×C₃×A₄	—	4	3	no	—
34	C₂³×C₉	—	3	3	no	—
35	C₂³×C₉	—	3	1	no	—
36	C₂³×C₉	C₂×A₄	1	3	yes	—
37	C₂×C₂²⋊C₉	—	4	6	no	—
38	C₂×C₄×C₉	—	3	6	no	—
39	C₂×C₄×C₉	—	3	6	no	—
40	C₂×C₄×C₉	—	3	6	no	—
41	C₂×C₄×C₉	—	3	6	no	—
42	C₂×C₄×C₉	—	3	1	no	—
43	C₂×C₄×C₉	—	3	3	no	—
44	C₂²×C₄²	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
45	C₂×3₋¹⁺²	—	16	3	no	—
46	C₂⁴×C₃	C₃×A₄	1	1	yes	characteristic
47	C₂²×A₄	—	12	1	no	—
48	C₂²×C₃×C₄	—	3	6	no	—
49	C₂²×C₃×C₄	—	3	2	no	—
50	C₂²×C₃×C₄	—	3	3	no	—
51	C₂²×C₃×C₄	—	3	3	no	—
52	C₃×C₄²	—	3	3	no	—
53	C₃×C₄²	—	3	2	no	—
54	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
55	C₄²⋊C₃	—	3	1	no	—
56	C₂²×C₃²	—	16	1	no	—
57	C₃×A₄	—	4	1	no	—
58	C₂²×C₉	—	3	3	no	—
59	C₂²×C₉	—	3	2	no	—
60	C₂²×C₉	—	3	3	no	—
61	C₂²×C₉	—	3	3	no	—
62	C₂²×C₉	—	16	2	no	—
63	C₂²×C₉	C₄²⋊C₃	1	1	yes	characteristic, complemented
64	C₂²×C₉	C₂²×A₄	1	1	yes	characteristic
65	C₂²⋊C₉	—	4	2	no	—
66	C₄×C₉	—	3	2	no	—
67	C₄×C₉	—	3	6	no	—
68	C₂³×C₄	—	3	1	no	—
69	C₂×C₄²	—	3	3	no	—
70	C₂×C₄²	C₂×3₋¹⁺²	1	3	yes	complemented
71	3₋¹⁺²	—	16	1	no	Sylow
72	C₂³×C₃	—	3	3	no	—
73	C₂³×C₃	—	3	1	no	—
74	C₂³×C₃	C₂×C₃×A₄	1	3	yes	—
75	C₂×A₄	—	12	3	no	—
76	C₂×C₃×C₄	—	3	6	no	—
77	C₂×C₃×C₄	—	3	6	no	—
78	C₂×C₃×C₄	—	3	6	no	—
79	C₂×C₃×C₄	—	3	6	no	—
80	C₂×C₃×C₄	—	3	1	no	—
81	C₂×C₃×C₄	—	3	3	no	—
82	C₂×C₃²	—	16	3	no	—
83	C₂×C₉	—	3	1	no	—
84	C₂×C₉	—	3	3	no	—
85	C₂×C₉	—	16	6	no	—
86	C₂×C₉	C₂×C₄²⋊C₃	1	3	yes	complemented
87	C₂⁴	C₉⋊A₄	1	1	yes	characteristic
88	C₂²×C₄	—	3	6	no	—
89	C₂²×C₄	—	3	2	no	—
90	C₂²×C₄	—	3	3	no	—
91	C₂²×C₄	—	3	3	no	—
92	C₄²	—	3	3	no	—
93	C₄²	—	3	2	no	—
94	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic, complemented
95	C₂²×C₃	—	3	3	no	—
96	C₂²×C₃	—	3	2	no	—
97	C₂²×C₃	—	3	3	no	—
98	C₂²×C₃	—	3	3	no	—
99	C₂²×C₃	—	48	1	no	—
100	C₂²×C₃	C₃×C₄²⋊C₃	1	1	yes	characteristic
101	C₂²×C₃	C₂²×C₃×A₄	1	1	yes	characteristic
102	A₄	—	12	1	no	—
103	C₃×C₄	—	3	2	no	—
104	C₃×C₄	—	3	6	no	—
105	C₃²	—	16	1	no	—
106	C₉	—	16	2	no	—
107	C₉	C₂²×C₄²⋊C₃	1	1	yes	characteristic, complemented
108	C₂³	—	3	3	no	—
109	C₂³	—	3	1	no	—
110	C₂³	C₂×C₉⋊A₄	1	3	yes	—
111	C₂×C₄	—	3	6	no	—
112	C₂×C₄	—	3	6	no	—
113	C₂×C₄	—	3	6	no	—
114	C₂×C₄	—	3	6	no	—
115	C₂×C₄	—	3	1	no	—
116	C₂×C₄	—	3	3	no	—
117	C₂×C₃	—	3	1	no	—
118	C₂×C₃	—	3	3	no	—
119	C₂×C₃	—	48	3	no	—
120	C₂×C₃	C₂×C₃×C₄²⋊C₃	1	3	yes	—
121	C₂²	—	3	3	no	—
122	C₂²	—	3	2	no	—
123	C₂²	—	3	3	no	—
124	C₂²	—	3	3	no	—
125	C₂²	C₉⋊(C₄²⋊C₃)	1	1	yes	characteristic, direct product factor
126	C₂²	C₂²×C₉⋊A₄	1	1	yes	characteristic
127	C₄	—	3	2	no	—
128	C₄	—	3	6	no	—
129	C₃	—	48	1	no	—
130	C₃	C₂²×C₃×C₄²⋊C₃	1	1	yes	characteristic
131	C₂	—	3	1	no	—
132	C₂	—	3	3	no	—
133	C₂	C₂×C₉⋊(C₄²⋊C₃)	1	3	yes	direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉⋊(C₄²⋊C₃)

C₂

yes

maximal, direct product factor

C₂²×C₃×C₄²⋊C₃

C₃

yes

maximal, characteristic

C₂²×C₄²×C₉

C₃

yes

maximal, characteristic, complemented

C₂²×C₄²⋊C₉

C₃

yes

maximal, complemented

C₂²×C₉⋊A₄

—

maximal

C₉⋊(C₄²⋊C₃)

C₂²

yes

characteristic, direct product factor

C₂×C₃×C₄²⋊C₃

C₂×C₃

yes

—

C₂³×C₄×C₉

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

C₂×C₃

yes

complemented

C₂×C₄²⋊C₉

C₂×C₃

yes

complemented

C₂×C₉⋊A₄

—

C₂²×C₃×C₄²

C₃²

yes

characteristic

C₂²×C₄²⋊C₃

—

C₂²×C₃×A₄

—

C₂⁴×C₉

A₄

yes

characteristic

C₂²×C₂²⋊C₉

—

C₃×C₄²⋊C₃

C₂²×C₃

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₄²×C₉

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic, complemented

C₄²⋊C₉

C₂²×C₃

yes

complemented

C₂²×3₋¹⁺²

—

C₉⋊A₄

—

C₂³×C₃×C₄

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₃²

yes

—

C₂×C₄²⋊C₃

—

C₂×C₃×A₄

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂×A₄

yes

—

C₂×C₂²⋊C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂²×C₄²

3₋¹⁺²

yes

characteristic, complemented, Sylow

C₂×3₋¹⁺²

—

C₂⁴×C₃

C₃×A₄

yes

characteristic

C₂²×A₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₃²

yes

characteristic

C₄²⋊C₃

—

C₂²×C₃²

—

C₃×A₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₄²⋊C₃

yes

characteristic, complemented

C₂²×C₉

C₂²×A₄

yes

characteristic

C₂²⋊C₉

—

C₄×C₉

—

C₄×C₉

—

C₂³×C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×3₋¹⁺²

yes

complemented

3₋¹⁺²

—

Sylow

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₃×A₄

yes

—

C₂×A₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃²

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₄²⋊C₃

yes

complemented

C₂⁴

C₉⋊A₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×3₋¹⁺²

yes

characteristic, complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

C₃×C₄²⋊C₃

yes

characteristic

101

C₂²×C₃

C₂²×C₃×A₄

yes

characteristic

102

A₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃²

—

106

C₉

—

107

C₉

C₂²×C₄²⋊C₃

yes

characteristic, complemented

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×C₉⋊A₄

yes

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

C₂×C₃×C₄²⋊C₃

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₉⋊(C₄²⋊C₃)

yes

characteristic, direct product factor

126

C₂²

C₂²×C₉⋊A₄

yes

characteristic

127

C₄

—

128

C₄

—

129

C₃

—

130

C₃

C₂²×C₃×C₄²⋊C₃

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

C₂×C₉⋊(C₄²⋊C₃)

yes

direct product factor

	subgroup	G/H
center	C₂²×C₃	C₃×C₄²⋊C₃
commutator, cosocle	C₃×C₄²	C₂²×C₃²
Frattini	C₂²×C₃	C₂²×C₃×A₄
Fitting	C₂²×C₄²×C₉	C₃
socle	C₂⁴×C₃	C₃×A₄
prime radical	C₂²×C₃×C₄²⋊C₃	C₃
hypercenter	C₂²×C₉	C₄²⋊C₃
hypocenter	C₄²	C₂²×3₋¹⁺²
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

subgroup

G/H

center

C₂²×C₃

C₃×C₄²⋊C₃

commutator, cosocle

C₃×C₄²

C₂²×C₃²

Frattini

C₂²×C₃

C₂²×C₃×A₄

Fitting

C₂²×C₄²×C₉

C₃

socle

C₂⁴×C₃

C₃×A₄

prime radical

C₂²×C₃×C₄²⋊C₃

C₃

hypercenter

C₂²×C₉

C₄²⋊C₃

hypocenter

C₄²

C₂²×3₋¹⁺²

LCS

G ▷ C₃×C₄² ▷ C₄²

UCS

C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

N	H ≅ G/N	type
1. C₂×C₉⋊(C₄²⋊C₃)	C₂	direct (N × H)
3. C₂²×C₄²×C₉	C₃	semidirect
4. C₂²×C₄²⋊C₉	C₃	semidirect
6. C₉⋊(C₄²⋊C₃)	C₂²	direct (N × H)
10. C₂×C₄²×C₉	C₂×C₃	semidirect
11. C₂×C₄²⋊C₉	C₂×C₃	semidirect
25. C₄²×C₉	C₂²×C₃	semidirect
26. C₄²⋊C₉	C₂²×C₃	semidirect
44. C₂²×C₄²	3₋¹⁺²	semidirect
63. C₂²×C₉	C₄²⋊C₃	semidirect
70. C₂×C₄²	C₂×3₋¹⁺²	semidirect
86. C₂×C₉	C₂×C₄²⋊C₃	semidirect
94. C₄²	C₂²×3₋¹⁺²	semidirect
107. C₉	C₂²×C₄²⋊C₃	semidirect
125. C₂²	C₉⋊(C₄²⋊C₃)	direct (N × H)
133. C₂	C₂×C₉⋊(C₄²⋊C₃)	direct (N × H)

H ≅ G/N

type

1. C₂×C₉⋊(C₄²⋊C₃)

C₂

direct (N × H)

3. C₂²×C₄²×C₉

C₃

semidirect

4. C₂²×C₄²⋊C₉

C₃

semidirect

6. C₉⋊(C₄²⋊C₃)

C₂²

direct (N × H)

10. C₂×C₄²×C₉

C₂×C₃

semidirect

11. C₂×C₄²⋊C₉

C₂×C₃

semidirect

25. C₄²×C₉

C₂²×C₃

semidirect

26. C₄²⋊C₉

C₂²×C₃

semidirect

44. C₂²×C₄²

3₋¹⁺²

semidirect

63. C₂²×C₉

C₄²⋊C₃

semidirect

70. C₂×C₄²

C₂×3₋¹⁺²

semidirect

86. C₂×C₉

C₂×C₄²⋊C₃

semidirect

94. C₄²

C₂²×3₋¹⁺²

semidirect

107. C₉

C₂²×C₄²⋊C₃

semidirect

125. C₂²

C₉⋊(C₄²⋊C₃)

direct (N × H)

133. C₂

C₂×C₉⋊(C₄²⋊C₃)

direct (N × H)

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

subgroup	index
C₂×C₉⋊(C₄²⋊C₃)	2
C₂²×C₃×C₄²⋊C₃	3
C₂²×C₄²×C₉	3
C₂²×C₄²⋊C₉	3
C₂²×C₉⋊A₄	4

subgroup

index

C₂×C₉⋊(C₄²⋊C₃)

C₂²×C₃×C₄²⋊C₃

C₂²×C₄²×C₉

C₂²×C₄²⋊C₉

C₂²×C₉⋊A₄

Subgroups — C₂²×C₉⋊(C₄²⋊C₃) #13152101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C9⋊(C42⋊C3) #13152101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉⋊(C₄²⋊C₃) #13152101