Group #72275939 — subgroups

Subgroups — C₂×C₃×C₉×C₂²⋊A₄ #72275939

Order 2592.

1900 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1900 conjugacy classes of subgroups — too many to draw. Only its 220 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	C₃×C₉×C₂²⋊A₄	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×C₃²×C₂²⋊A₄	C₃	1	1	yes	maximal, characteristic
3	C₂⁵×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	C₂×C₉×C₂²⋊A₄	C₃	1	9	yes	maximal, direct product factor
6	C₂×C₃×C₉×A₄	—	4	5	no	maximal
7	C₃²×C₂²⋊A₄	C₂×C₃	1	1	yes	characteristic
8	C₂⁴×C₃×C₉	—	3	5	no	—
9	C₂⁴×C₃×C₉	—	3	5	no	—
10	C₂⁴×C₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×C₂²⋊(C₂²⋊C₉)	C₂×C₃	1	2	yes	complemented
12	C₉×C₂²⋊A₄	C₂×C₃	1	9	yes	direct product factor
13	C₃×C₉×A₄	—	4	5	no	—
14	C₂⁵×C₃²	C₃²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊A₄	C₉	1	9	yes	direct product factor
16	C₂×C₃×C₂²⋊A₄	C₃²	1	3	yes	—
17	C₂⁵×C₉	C₃²	1	3	yes	complemented
18	C₂×C₂²⋊(C₂²⋊C₉)	C₃²	1	6	yes	complemented
19	C₂×C₃²×A₄	—	4	5	no	—
20	C₂³×C₃×C₉	—	3	30	no	—
21	C₂³×C₃×C₉	—	3	10	no	—
22	C₂³×C₃×C₉	—	3	5	no	—
23	C₂³×C₃×C₉	—	3	5	no	—
24	C₂³×C₃×C₉	A₄	1	5	yes	complemented
25	C₂×C₃×C₂²⋊C₉	—	4	10	no	—
26	C₂×C₉×A₄	—	4	45	no	—
27	C₂×C₃²×C₉	—	16	1	no	—
28	C₂⁴×C₃²	—	3	5	no	—
29	C₂⁴×C₃²	—	3	5	no	—
30	C₂⁴×C₃²	C₂×C₃²	1	1	yes	characteristic
31	C₃×C₂²⋊A₄	C₂×C₉	1	9	yes	direct product factor
32	C₃×C₂²⋊A₄	C₂×C₃²	1	3	yes	—
33	C₂⁴×C₉	—	3	15	no	—
34	C₂⁴×C₉	—	3	15	no	—
35	C₂⁴×C₉	C₂×C₃²	1	3	yes	complemented
36	C₂²⋊(C₂²⋊C₉)	C₂×C₃²	1	6	yes	complemented
37	C₃²×A₄	—	4	5	no	—
38	C₂²×C₃×C₉	—	3	30	no	—
39	C₂²×C₃×C₉	—	3	10	no	—
40	C₂²×C₃×C₉	—	3	5	no	—
41	C₂²×C₃×C₉	—	3	5	no	—
42	C₂²×C₃×C₉	C₂×A₄	1	5	yes	complemented
43	C₃×C₂²⋊C₉	—	4	10	no	—
44	C₉×A₄	—	4	45	no	—
45	C₂⁵×C₃	C₃×C₉	1	3	yes	complemented
46	C₂⁵×C₃	C₃³	1	1	yes	characteristic
47	C₂×C₂²⋊A₄	C₃×C₉	1	9	yes	direct product factor
48	C₃²×C₉	—	16	1	no	Sylow
49	C₂³×C₃²	—	3	30	no	—
50	C₂³×C₃²	—	3	10	no	—
51	C₂³×C₃²	—	3	5	no	—
52	C₂³×C₃²	—	3	5	no	—
53	C₂³×C₃²	C₃×A₄	1	5	yes	—
54	C₂×C₃×A₄	—	4	45	no	—
55	C₂×C₃×A₄	—	4	15	no	—
56	C₂³×C₉	—	3	90	no	—
57	C₂³×C₉	—	3	30	no	—
58	C₂³×C₉	—	3	15	no	—
59	C₂³×C₉	—	3	15	no	—
60	C₂³×C₉	C₃×A₄	1	15	yes	complemented
61	C₂×C₂²⋊C₉	—	4	30	no	—
62	C₂×C₃³	—	16	1	no	—
63	C₂×C₃×C₉	—	3	5	no	—
64	C₂×C₃×C₉	—	3	5	no	—
65	C₂×C₃×C₉	—	16	2	no	—
66	C₂×C₃×C₉	—	16	9	no	—
67	C₂×C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic, direct product factor
68	C₂⁴×C₃	—	3	15	no	—
69	C₂⁴×C₃	—	3	5	no	—
70	C₂⁴×C₃	—	3	15	no	—
71	C₂⁴×C₃	—	3	5	no	—
72	C₂⁴×C₃	C₂×C₃×C₉	1	3	yes	complemented
73	C₂⁴×C₃	C₂×C₃³	1	1	yes	characteristic
74	C₂²⋊A₄	C₂×C₃×C₉	1	9	yes	direct product factor
75	C₂²×C₃²	—	3	5	no	—
76	C₂²×C₃²	—	3	30	no	—
77	C₂²×C₃²	—	3	10	no	—
78	C₂²×C₃²	—	3	5	no	—
79	C₂²×C₃²	C₂×C₃×A₄	1	5	yes	—
80	C₃×A₄	—	4	45	no	—
81	C₃×A₄	—	4	15	no	—
82	C₂²×C₉	—	3	15	no	—
83	C₂²×C₉	—	3	90	no	—
84	C₂²×C₉	—	3	30	no	—
85	C₂²×C₉	—	3	15	no	—
86	C₂²×C₉	C₂×C₃×A₄	1	15	yes	complemented
87	C₂²⋊C₉	—	4	30	no	—
88	C₂⁵	C₃²×C₉	1	1	yes	characteristic, complemented, Sylow
89	C₃³	—	16	1	no	—
90	C₃×C₉	—	16	2	no	—
91	C₃×C₉	—	16	9	no	—
92	C₃×C₉	C₂×C₂²⋊A₄	1	1	yes	characteristic, direct product factor
93	C₂³×C₃	—	3	30	no	—
94	C₂³×C₃	—	3	10	no	—
95	C₂³×C₃	—	3	90	no	—
96	C₂³×C₃	—	3	30	no	—
97	C₂³×C₃	—	3	15	no	—
98	C₂³×C₃	—	3	15	no	—
99	C₂³×C₃	—	3	5	no	—
100	C₂³×C₃	—	3	5	no	—
101	C₂³×C₃	C₉×A₄	1	15	yes	complemented
102	C₂³×C₃	C₃²×A₄	1	5	yes	—
103	C₂×A₄	—	4	45	no	—
104	C₂×C₃²	—	3	5	no	—
105	C₂×C₃²	—	3	5	no	—
106	C₂×C₃²	—	16	3	no	—
107	C₂×C₃²	—	16	9	no	—
108	C₂×C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
109	C₂×C₉	—	3	15	no	—
110	C₂×C₉	—	3	15	no	—
111	C₂×C₉	—	16	6	no	—
112	C₂×C₉	C₃×C₂²⋊A₄	1	3	yes	direct product factor
113	C₂⁴	—	3	5	no	—
114	C₂⁴	—	3	5	no	—
115	C₂⁴	C₂×C₃²×C₉	1	1	yes	characteristic, complemented
116	C₂²×C₃	—	3	15	no	—
117	C₂²×C₃	—	3	5	no	—
118	C₂²×C₃	—	3	10	no	—
119	C₂²×C₃	—	3	30	no	—
120	C₂²×C₃	—	3	30	no	—
121	C₂²×C₃	—	3	90	no	—
122	C₂²×C₃	—	3	5	no	—
123	C₂²×C₃	—	3	15	no	—
124	C₂²×C₃	C₂×C₉×A₄	1	15	yes	complemented
125	C₂²×C₃	C₂×C₃²×A₄	1	5	yes	—
126	A₄	—	4	45	no	—
127	C₃²	—	16	9	no	—
128	C₃²	—	16	3	no	—
129	C₃²	C₂×C₃×C₂²⋊A₄	1	1	yes	characteristic
130	C₉	—	16	6	no	—
131	C₉	C₂×C₃×C₂²⋊A₄	1	3	yes	direct product factor
132	C₂³	—	3	30	no	—
133	C₂³	—	3	10	no	—
134	C₂³	—	3	5	no	—
135	C₂³	—	3	5	no	—
136	C₂³	C₃×C₉×A₄	1	5	yes	complemented
137	C₂×C₃	—	3	15	no	—
138	C₂×C₃	—	3	15	no	—
139	C₂×C₃	—	3	5	no	—
140	C₂×C₃	—	3	5	no	—
141	C₂×C₃	—	16	9	no	—
142	C₂×C₃	C₉×C₂²⋊A₄	1	3	yes	direct product factor
143	C₂×C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
144	C₂²	—	3	5	no	—
145	C₂²	—	3	10	no	—
146	C₂²	—	3	30	no	—
147	C₂²	—	3	5	no	—
148	C₂²	C₂×C₃×C₉×A₄	1	5	yes	complemented
149	C₃	—	16	9	no	—
150	C₃	C₂×C₉×C₂²⋊A₄	1	3	yes	direct product factor
151	C₃	C₂×C₃²×C₂²⋊A₄	1	1	yes	characteristic
152	C₂	—	3	5	no	—
153	C₂	—	3	5	no	—
154	C₂	C₃×C₉×C₂²⋊A₄	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×C₂²⋊A₄

C₂

yes

maximal, characteristic, direct product factor

C₂×C₃²×C₂²⋊A₄

C₃

yes

maximal, characteristic

C₂⁵×C₃×C₉

C₃

yes

maximal, characteristic, complemented

C₂×C₃×C₂²⋊(C₂²⋊C₉)

C₃

yes

maximal, complemented

C₂×C₉×C₂²⋊A₄

C₃

yes

maximal, direct product factor

C₂×C₃×C₉×A₄

—

maximal

C₃²×C₂²⋊A₄

C₂×C₃

yes

characteristic

C₂⁴×C₃×C₉

—

C₂⁴×C₃×C₉

—

C₂⁴×C₃×C₉

C₂×C₃

yes

characteristic, complemented

C₃×C₂²⋊(C₂²⋊C₉)

C₂×C₃

yes

complemented

C₉×C₂²⋊A₄

C₂×C₃

yes

direct product factor

C₃×C₉×A₄

—

C₂⁵×C₃²

C₃²

yes

characteristic

C₂×C₃×C₂²⋊A₄

C₉

yes

direct product factor

C₂×C₃×C₂²⋊A₄

C₃²

yes

—

C₂⁵×C₉

C₃²

yes

complemented

C₂×C₂²⋊(C₂²⋊C₉)

C₃²

yes

complemented

C₂×C₃²×A₄

—

C₂³×C₃×C₉

—

C₂³×C₃×C₉

—

C₂³×C₃×C₉

—

C₂³×C₃×C₉

—

C₂³×C₃×C₉

A₄

yes

complemented

C₂×C₃×C₂²⋊C₉

—

C₂×C₉×A₄

—

C₂×C₃²×C₉

—

C₂⁴×C₃²

—

C₂⁴×C₃²

—

C₂⁴×C₃²

C₂×C₃²

yes

characteristic

C₃×C₂²⋊A₄

C₂×C₉

yes

direct product factor

C₃×C₂²⋊A₄

C₂×C₃²

yes

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

C₂×C₃²

yes

complemented

C₂²⋊(C₂²⋊C₉)

C₂×C₃²

yes

complemented

C₃²×A₄

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×A₄

yes

complemented

C₃×C₂²⋊C₉

—

C₉×A₄

—

C₂⁵×C₃

C₃×C₉

yes

complemented

C₂⁵×C₃

C₃³

yes

characteristic

C₂×C₂²⋊A₄

C₃×C₉

yes

direct product factor

C₃²×C₉

—

Sylow

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

C₃×A₄

yes

—

C₂×C₃×A₄

—

C₂×C₃×A₄

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₃×A₄

yes

complemented

C₂×C₂²⋊C₉

—

C₂×C₃³

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²⋊A₄

yes

characteristic, direct product factor

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

C₂×C₃×C₉

yes

complemented

C₂⁴×C₃

C₂×C₃³

yes

characteristic

C₂²⋊A₄

C₂×C₃×C₉

yes

direct product factor

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×A₄

yes

—

C₃×A₄

—

C₃×A₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×A₄

yes

complemented

C₂²⋊C₉

—

C₂⁵

C₃²×C₉

yes

characteristic, complemented, Sylow

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×C₂²⋊A₄

yes

characteristic, direct product factor

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

100

C₂³×C₃

—

101

C₂³×C₃

C₉×A₄

yes

complemented

102

C₂³×C₃

C₃²×A₄

yes

—

103

C₂×A₄

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

C₃×C₂²⋊A₄

yes

characteristic

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂×C₉

—

112

C₂×C₉

C₃×C₂²⋊A₄

yes

direct product factor

113

C₂⁴

—

114

C₂⁴

—

115

C₂⁴

C₂×C₃²×C₉

yes

characteristic, complemented

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂²×C₃

C₂×C₉×A₄

yes

complemented

125

C₂²×C₃

C₂×C₃²×A₄

yes

—

126

A₄

—

127

C₃²

—

128

C₃²

—

129

C₃²

C₂×C₃×C₂²⋊A₄

yes

characteristic

130

C₉

—

131

C₉

C₂×C₃×C₂²⋊A₄

yes

direct product factor

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

C₃×C₉×A₄

yes

complemented

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

C₉×C₂²⋊A₄

yes

direct product factor

143

C₂×C₃

C₃²×C₂²⋊A₄

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₃×C₉×A₄

yes

complemented

149

C₃

—

150

C₃

C₂×C₉×C₂²⋊A₄

yes

direct product factor

151

C₃

C₂×C₃²×C₂²⋊A₄

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

C₃×C₉×C₂²⋊A₄

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₉	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₂×C₃²×C₉
Frattini	C₃	C₂×C₃²×C₂²⋊A₄
Fitting	C₂⁵×C₃×C₉	C₃
socle	C₂⁵×C₃²	C₃²
prime radical	C₂×C₃²×C₂²⋊A₄	C₃
cosocle	C₂⁴×C₃	C₂×C₃³
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₂×C₃×C₉

subgroup

G/H

center, hypercenter

C₂×C₃×C₉

C₂²⋊A₄

commutator, hypocenter

C₂⁴

C₂×C₃²×C₉

Frattini

C₃

C₂×C₃²×C₂²⋊A₄

Fitting

C₂⁵×C₃×C₉

C₃

socle

C₂⁵×C₃²

C₃²

prime radical

C₂×C₃²×C₂²⋊A₄

C₃

cosocle

C₂⁴×C₃

C₂×C₃³

LCS

G ▷ C₂⁴

UCS

C₁ ◁ C₂×C₃×C₉

N	H ≅ G/N	type
1. C₃×C₉×C₂²⋊A₄	C₂	direct (N × H)
3. C₂⁵×C₃×C₉	C₃	semidirect
4. C₂×C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
5. C₂×C₉×C₂²⋊A₄	C₃	direct (N × H)
10. C₂⁴×C₃×C₉	C₂×C₃	semidirect
11. C₃×C₂²⋊(C₂²⋊C₉)	C₂×C₃	semidirect
12. C₉×C₂²⋊A₄	C₂×C₃	direct (N × H)
15. C₂×C₃×C₂²⋊A₄	C₉	direct (N × H)
17. C₂⁵×C₉	C₃²	semidirect
18. C₂×C₂²⋊(C₂²⋊C₉)	C₃²	semidirect
24. C₂³×C₃×C₉	A₄	semidirect
31. C₃×C₂²⋊A₄	C₂×C₉	direct (N × H)
35. C₂⁴×C₉	C₂×C₃²	semidirect
36. C₂²⋊(C₂²⋊C₉)	C₂×C₃²	semidirect
42. C₂²×C₃×C₉	C₂×A₄	semidirect
45. C₂⁵×C₃	C₃×C₉	semidirect
47. C₂×C₂²⋊A₄	C₃×C₉	direct (N × H)
60. C₂³×C₉	C₃×A₄	semidirect
67. C₂×C₃×C₉	C₂²⋊A₄	direct (N × H)
72. C₂⁴×C₃	C₂×C₃×C₉	semidirect
74. C₂²⋊A₄	C₂×C₃×C₉	direct (N × H)
86. C₂²×C₉	C₂×C₃×A₄	semidirect
88. C₂⁵	C₃²×C₉	semidirect
92. C₃×C₉	C₂×C₂²⋊A₄	direct (N × H)
101. C₂³×C₃	C₉×A₄	semidirect
112. C₂×C₉	C₃×C₂²⋊A₄	direct (N × H)
115. C₂⁴	C₂×C₃²×C₉	semidirect
124. C₂²×C₃	C₂×C₉×A₄	semidirect
131. C₉	C₂×C₃×C₂²⋊A₄	direct (N × H)
136. C₂³	C₃×C₉×A₄	semidirect
142. C₂×C₃	C₉×C₂²⋊A₄	direct (N × H)
148. C₂²	C₂×C₃×C₉×A₄	semidirect
150. C₃	C₂×C₉×C₂²⋊A₄	direct (N × H)
154. C₂	C₃×C₉×C₂²⋊A₄	direct (N × H)

H ≅ G/N

type

1. C₃×C₉×C₂²⋊A₄

C₂

direct (N × H)

3. C₂⁵×C₃×C₉

C₃

semidirect

4. C₂×C₃×C₂²⋊(C₂²⋊C₉)

C₃

semidirect

5. C₂×C₉×C₂²⋊A₄

C₃

direct (N × H)

10. C₂⁴×C₃×C₉

C₂×C₃

semidirect

11. C₃×C₂²⋊(C₂²⋊C₉)

C₂×C₃

semidirect

12. C₉×C₂²⋊A₄

C₂×C₃

direct (N × H)

15. C₂×C₃×C₂²⋊A₄

C₉

direct (N × H)

17. C₂⁵×C₉

C₃²

semidirect

18. C₂×C₂²⋊(C₂²⋊C₉)

C₃²

semidirect

24. C₂³×C₃×C₉

A₄

semidirect

31. C₃×C₂²⋊A₄

C₂×C₉

direct (N × H)

35. C₂⁴×C₉

C₂×C₃²

semidirect

36. C₂²⋊(C₂²⋊C₉)

C₂×C₃²

semidirect

42. C₂²×C₃×C₉

C₂×A₄

semidirect

45. C₂⁵×C₃

C₃×C₉

semidirect

47. C₂×C₂²⋊A₄

C₃×C₉

direct (N × H)

60. C₂³×C₉

C₃×A₄

semidirect

67. C₂×C₃×C₉

C₂²⋊A₄

direct (N × H)

72. C₂⁴×C₃

C₂×C₃×C₉

semidirect

74. C₂²⋊A₄

C₂×C₃×C₉

direct (N × H)

86. C₂²×C₉

C₂×C₃×A₄

semidirect

88. C₂⁵

C₃²×C₉

semidirect

92. C₃×C₉

C₂×C₂²⋊A₄

direct (N × H)

101. C₂³×C₃

C₉×A₄

semidirect

112. C₂×C₉

C₃×C₂²⋊A₄

direct (N × H)

115. C₂⁴

C₂×C₃²×C₉

semidirect

124. C₂²×C₃

C₂×C₉×A₄

semidirect

131. C₉

C₂×C₃×C₂²⋊A₄

direct (N × H)

136. C₂³

C₃×C₉×A₄

semidirect

142. C₂×C₃

C₉×C₂²⋊A₄

direct (N × H)

148. C₂²

C₂×C₃×C₉×A₄

semidirect

150. C₃

C₂×C₉×C₂²⋊A₄

direct (N × H)

154. C₂

C₃×C₉×C₂²⋊A₄

direct (N × H)

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

subgroup	index
C₃×C₉×C₂²⋊A₄	2
C₂×C₃²×C₂²⋊A₄	3
C₂⁵×C₃×C₉	3
C₂×C₃×C₂²⋊(C₂²⋊C₉)	3
C₂×C₉×C₂²⋊A₄	3
C₂×C₃×C₉×A₄	4

subgroup

index

C₃×C₉×C₂²⋊A₄

C₂×C₃²×C₂²⋊A₄

C₂⁵×C₃×C₉

C₂×C₃×C₂²⋊(C₂²⋊C₉)

C₂×C₉×C₂²⋊A₄

C₂×C₃×C₉×A₄

Subgroups — C₂×C₃×C₉×C₂²⋊A₄ #72275939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C3×C9×C22⋊A4 #72275939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₉×C₂²⋊A₄ #72275939