Group #14517306 — subgroups

Subgroups — 3₋¹⁺²×D₃₆ #14517306

Order 1944.

267 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 84 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×D₉×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	C₄×C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₂×3₋¹⁺²	—	3	1	no	maximal
4	C₃²×D₃₆	C₃	1	1	yes	maximal, characteristic
5	C₉×D₃₆	C₃	1	3	yes	maximal, complemented
6	C₂×C₉×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	D₉×3₋¹⁺²	—	2	2	no	—
8	C₂×D₃×3₋¹⁺²	—	3	2	no	—
9	C₂×C₃²×D₉	C₂×C₃	1	2	yes	—
10	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic
11	C₃²×C₄×C₉	C₂×C₃	1	1	yes	characteristic
12	C₂×C₉×D₉	C₂×C₃	1	6	yes	complemented
13	C₄×C₉⋊C₉	—	2	3	no	—
14	C₄×C₃²⋊C₉	—	2	1	no	—
15	C₄×C₉²	C₂×C₃	1	3	yes	complemented
16	C₉×3₋¹⁺²	D₄	1	1	yes	characteristic, complemented, Sylow
17	C₃²×D₁₂	—	3	1	no	—
18	D₄×3₋¹⁺²	—	9	1	no	—
19	C₉×D₁₂	—	3	3	no	—
20	C₃×D₃₆	—	3	1	no	—
21	C₃×D₃₆	C₃²	1	1	yes	characteristic
22	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
24	D₃×3₋¹⁺²	—	6	2	no	—
25	C₃²×D₉	—	2	2	no	—
26	C₂×C₉⋊C₉	—	2	3	no	—
27	C₂×C₃²⋊C₉	—	2	1	no	—
28	C₂×C₉²	C₂²×C₃	1	3	yes	—
29	C₉×D₉	—	2	6	no	—
30	C₂×C₃²×D₃	—	3	2	no	—
31	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
32	C₂²×3₋¹⁺²	—	9	2	no	—
33	C₂×D₃×C₉	—	3	6	no	—
34	C₂×C₃×D₉	—	3	2	no	—
35	C₂×C₃×D₉	C₂×C₃²	1	2	yes	—
36	C₄×3₋¹⁺²	—	2	1	no	—
37	C₄×3₋¹⁺²	—	2	3	no	—
38	C₄×3₋¹⁺²	D₉	1	1	yes	characteristic, complemented
39	C₃×C₄×C₉	—	2	3	no	—
40	C₃×C₄×C₉	—	2	1	no	—
41	C₃×C₄×C₉	—	3	1	no	—
42	C₃×C₄×C₉	—	6	1	no	—
43	C₃×C₄×C₉	C₃×D₃	1	3	yes	—
44	C₃×C₄×C₉	C₂×C₃²	1	1	yes	characteristic
45	C₃×3₋¹⁺²	D₁₂	1	1	yes	characteristic
46	C₃²×C₉	C₃×D₄	1	1	yes	characteristic
47	C₉⋊C₉	—	2	3	no	—
48	C₃²⋊C₉	—	2	1	no	—
49	C₉²	C₃×D₄	1	3	yes	complemented
50	C₃²×D₄	—	9	1	no	—
51	C₃×D₁₂	—	3	1	no	—
52	C₃×D₁₂	—	9	1	no	—
53	D₄×C₉	—	9	3	no	—
54	D₃₆	3₋¹⁺²	1	1	yes	characteristic, direct product factor
55	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
56	C₃²×D₃	—	6	2	no	—
57	C₂×3₋¹⁺²	—	2	1	no	—
58	C₂×3₋¹⁺²	—	2	3	no	—
59	C₂×3₋¹⁺²	—	18	2	no	—
60	C₂×3₋¹⁺²	C₂×D₉	1	1	yes	characteristic
61	C₂×C₃×C₉	—	2	3	no	—
62	C₂×C₃×C₉	—	2	1	no	—
63	C₂×C₃×C₉	—	3	1	no	—
64	C₂×C₃×C₉	—	6	1	no	—
65	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
66	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
67	D₃×C₉	—	6	6	no	—
68	C₃×D₉	—	2	2	no	—
69	C₃×D₉	—	6	2	no	—
70	C₂²×C₃²	—	9	2	no	—
71	C₂×C₃×D₃	—	3	2	no	—
72	C₂×C₃×D₃	—	9	2	no	—
73	C₃²×C₄	—	2	1	no	—
74	C₃²×C₄	—	3	1	no	—
75	C₃²×C₄	—	6	1	no	—
76	C₃²×C₄	C₃×D₉	1	1	yes	characteristic
77	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
78	C₂²×C₉	—	9	6	no	—
79	C₂×D₉	C₂×3₋¹⁺²	1	2	yes	complemented
80	C₄×C₉	—	2	3	no	—
81	C₄×C₉	—	2	1	no	—
82	C₄×C₉	—	6	3	no	—
83	C₄×C₉	—	6	1	no	—
84	C₄×C₉	C₃×D₉	1	3	yes	complemented
85	C₄×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
86	C₃³	C₃×D₁₂	1	1	yes	characteristic
87	3₋¹⁺²	—	2	1	no	—
88	3₋¹⁺²	—	2	3	no	—
89	3₋¹⁺²	D₃₆	1	1	yes	characteristic, direct product factor
90	C₃×C₉	—	2	3	no	—
91	C₃×C₉	—	2	1	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	6	1	no	—
94	C₃×C₉	C₃×D₁₂	1	3	yes	—
95	C₃×C₉	C₃²×D₄	1	1	yes	characteristic
96	C₃×D₄	—	9	1	no	—
97	C₃×D₄	—	27	1	no	—
98	D₁₂	—	3	1	no	—
99	C₂×C₃²	—	2	1	no	—
100	C₂×C₃²	—	3	1	no	—
101	C₂×C₃²	—	6	1	no	—
102	C₂×C₃²	—	18	2	no	—
103	C₂×C₃²	C₂×C₃×D₉	1	1	yes	characteristic
104	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
105	C₃×D₃	—	6	2	no	—
106	C₃×D₃	—	18	2	no	—
107	C₂×C₉	—	2	3	no	—
108	C₂×C₉	—	2	1	no	—
109	C₂×C₉	—	6	3	no	—
110	C₂×C₉	—	6	1	no	—
111	C₂×C₉	—	18	6	no	—
112	C₂×C₉	C₂×C₃×D₉	1	3	yes	—
113	C₂×C₉	C₂²×3₋¹⁺²	1	1	yes	characteristic
114	D₉	—	2	2	no	—
115	C₂²×C₃	—	9	2	no	—
116	C₂²×C₃	—	27	2	no	—
117	C₂×D₃	—	3	2	no	—
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	3	1	no	—
120	C₃×C₄	—	6	1	no	—
121	C₃×C₄	C₃²×D₉	1	1	yes	characteristic
122	C₃×C₄	D₃×3₋¹⁺²	1	1	yes	characteristic
123	C₃²	—	2	1	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	6	1	no	—
126	C₃²	C₃×D₃₆	1	1	yes	characteristic
127	C₃²	C₃²×D₁₂	1	1	yes	characteristic
128	C₉	—	2	3	no	—
129	C₉	—	2	1	no	—
130	C₉	—	6	3	no	—
131	C₉	—	6	1	no	—
132	C₉	C₃×D₃₆	1	3	yes	complemented
133	C₉	D₄×3₋¹⁺²	1	1	yes	characteristic, complemented
134	D₄	—	9	1	no	Sylow
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	—	3	1	no	—
137	C₂×C₃	—	6	1	no	—
138	C₂×C₃	—	18	2	no	—
139	C₂×C₃	—	54	2	no	—
140	C₂×C₃	C₂×C₃²×D₉	1	1	yes	characteristic
141	C₂×C₃	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic
142	D₃	—	6	2	no	—
143	C₂²	—	9	2	no	—
144	C₄	D₉×3₋¹⁺²	1	1	yes	characteristic, complemented
145	C₃	—	2	1	no	—
146	C₃	—	3	1	no	—
147	C₃	—	6	1	no	—
148	C₃	C₃²×D₃₆	1	1	yes	characteristic
149	C₃	D₁₂×3₋¹⁺²	1	1	yes	characteristic
150	C₂	—	18	2	no	—
151	C₂	C₂×D₉×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₉×3₋¹⁺²

C₂

yes

maximal, complemented

C₄×C₉×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

D₁₂×3₋¹⁺²

—

maximal

C₃²×D₃₆

C₃

yes

maximal, characteristic

C₉×D₃₆

C₃

yes

maximal, complemented

C₂×C₉×3₋¹⁺²

C₂²

yes

characteristic

D₉×3₋¹⁺²

—

C₂×D₃×3₋¹⁺²

—

C₂×C₃²×D₉

C₂×C₃

yes

—

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic

C₃²×C₄×C₉

C₂×C₃

yes

characteristic

C₂×C₉×D₉

C₂×C₃

yes

complemented

C₄×C₉⋊C₉

—

C₄×C₃²⋊C₉

—

C₄×C₉²

C₂×C₃

yes

complemented

C₉×3₋¹⁺²

D₄

yes

characteristic, complemented, Sylow

C₃²×D₁₂

—

D₄×3₋¹⁺²

—

C₉×D₁₂

—

C₃×D₃₆

—

C₃×D₃₆

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

D₃×3₋¹⁺²

—

C₃²×D₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

C₂²×C₃

yes

—

C₉×D₉

—

C₂×C₃²×D₃

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₂²×3₋¹⁺²

—

C₂×D₃×C₉

—

C₂×C₃×D₉

—

C₂×C₃×D₉

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

D₉

yes

characteristic, complemented

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

characteristic

C₃×3₋¹⁺²

D₁₂

yes

characteristic

C₃²×C₉

C₃×D₄

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

C₃×D₄

yes

complemented

C₃²×D₄

—

C₃×D₁₂

—

C₃×D₁₂

—

D₄×C₉

—

D₃₆

3₋¹⁺²

yes

characteristic, direct product factor

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×D₉

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

D₃×C₉

—

C₃×D₉

—

C₃×D₉

—

C₂²×C₃²

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃×D₉

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

characteristic

C₂²×C₉

—

C₂×D₉

C₂×3₋¹⁺²

yes

complemented

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×D₉

yes

complemented

C₄×C₉

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃³

C₃×D₁₂

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

D₃₆

yes

characteristic, direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×D₁₂

yes

—

C₃×C₉

C₃²×D₄

yes

characteristic

C₃×D₄

—

C₃×D₄

—

D₁₂

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

C₂×C₃×D₉

yes

characteristic

104

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

105

C₃×D₃

—

106

C₃×D₃

—

107

C₂×C₉

—

108

C₂×C₉

—

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂×C₉

—

112

C₂×C₉

C₂×C₃×D₉

yes

—

113

C₂×C₉

C₂²×3₋¹⁺²

yes

characteristic

114

D₉

—

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂×D₃

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

C₃²×D₉

yes

characteristic

122

C₃×C₄

D₃×3₋¹⁺²

yes

characteristic

123

C₃²

—

124

C₃²

—

125

C₃²

—

126

C₃²

C₃×D₃₆

yes

characteristic

127

C₃²

C₃²×D₁₂

yes

characteristic

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

C₃×D₃₆

yes

complemented

133

C₉

D₄×3₋¹⁺²

yes

characteristic, complemented

134

D₄

—

Sylow

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

C₂×C₃²×D₉

yes

characteristic

141

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

characteristic

142

D₃

—

143

C₂²

—

144

C₄

D₉×3₋¹⁺²

yes

characteristic, complemented

145

C₃

—

146

C₃

—

147

C₃

—

148

C₃

C₃²×D₃₆

yes

characteristic

149

C₃

D₁₂×3₋¹⁺²

yes

characteristic

150

C₂

—

151

C₂

C₂×D₉×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₂×C₃²×D₉
commutator, cosocle	C₂×C₃×C₉	C₂²×C₃²
Frattini, socle	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₄×C₉×3₋¹⁺²	C₂
prime radical	C₃²×D₃₆	C₃
hypercenter	C₄×3₋¹⁺²	D₉
hypocenter	C₉	D₄×3₋¹⁺²
LCS	G ▷ C₂×C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

subgroup

G/H

center

C₂×C₃

C₂×C₃²×D₉

commutator, cosocle

C₂×C₃×C₉

C₂²×C₃²

Frattini, socle

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₄×C₉×3₋¹⁺²

C₂

prime radical

C₃²×D₃₆

C₃

hypercenter

C₄×3₋¹⁺²

D₉

hypocenter

C₉

D₄×3₋¹⁺²

LCS

G ▷ C₂×C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

N	H ≅ G/N	type
1. C₂×D₉×3₋¹⁺²	C₂	semidirect
2. C₄×C₉×3₋¹⁺²	C₂	semidirect
5. C₉×D₃₆	C₃	semidirect
12. C₂×C₉×D₉	C₂×C₃	semidirect
15. C₄×C₉²	C₂×C₃	semidirect
16. C₉×3₋¹⁺²	D₄	semidirect
38. C₄×3₋¹⁺²	D₉	semidirect
49. C₉²	C₃×D₄	semidirect
54. D₃₆	3₋¹⁺²	direct (N × H)
79. C₂×D₉	C₂×3₋¹⁺²	semidirect
84. C₄×C₉	C₃×D₉	semidirect
85. C₄×C₉	C₂×3₋¹⁺²	semidirect
89. 3₋¹⁺²	D₃₆	direct (N × H)
132. C₉	C₃×D₃₆	semidirect
133. C₉	D₄×3₋¹⁺²	semidirect
144. C₄	D₉×3₋¹⁺²	semidirect

H ≅ G/N

type

1. C₂×D₉×3₋¹⁺²

C₂

semidirect

2. C₄×C₉×3₋¹⁺²

C₂

semidirect

5. C₉×D₃₆

C₃

semidirect

12. C₂×C₉×D₉

C₂×C₃

semidirect

15. C₄×C₉²

C₂×C₃

semidirect

16. C₉×3₋¹⁺²

D₄

semidirect

38. C₄×3₋¹⁺²

D₉

semidirect

49. C₉²

C₃×D₄

semidirect

54. D₃₆

3₋¹⁺²

direct (N × H)

79. C₂×D₉

C₂×3₋¹⁺²

semidirect

84. C₄×C₉

C₃×D₉

semidirect

85. C₄×C₉

C₂×3₋¹⁺²

semidirect

89. 3₋¹⁺²

D₃₆

direct (N × H)

132. C₉

C₃×D₃₆

semidirect

133. C₉

D₄×3₋¹⁺²

semidirect

144. C₄

D₉×3₋¹⁺²

semidirect

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

subgroup	index
C₂×D₉×3₋¹⁺²	2
C₄×C₉×3₋¹⁺²	2
D₁₂×3₋¹⁺²	3
C₃²×D₃₆	3
C₉×D₃₆	3

subgroup

index

C₂×D₉×3₋¹⁺²

C₄×C₉×3₋¹⁺²

D₁₂×3₋¹⁺²

C₃²×D₃₆

C₉×D₃₆

Subgroups — 3₋¹⁺²×D₃₆ #14517306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×D36 #14517306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×D₃₆ #14517306