Group #11766539 — subgroups

Subgroups — C₉×#1818 #11766539

Order 1152.

438 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 180 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×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₉×#336	C₂	1	4	yes	maximal, complemented
4	C₃×#1818	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₉	C₂²	1	4	yes	—
7	C₉×#117	—	2	4	no	—
8	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₉×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₄²×C₉	C₂²	1	2	yes	—
11	C₉×C₂²⋊C₈	—	2	4	no	—
12	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	2	yes	—
14	C₃×#336	C₂×C₃	1	4	yes	—
15	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₉	—	2	4	no	—
17	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉	D₄	1	4	yes	—
20	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₉	—	2	4	no	—
22	C₂×C₈×C₉	—	2	8	no	—
23	C₉×C₄⋊C₄	—	2	4	no	—
24	C₉×C₂²⋊C₄	—	2	4	no	—
25	C₄²×C₉	—	2	4	no	—
26	#1818	C₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
29	C₃×#117	—	2	4	no	—
30	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
31	C₂×C₃×C₂²⋊C₄	C₃×C₄	1	2	yes	—
32	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
33	C₃×C₂²⋊C₈	—	2	4	no	—
34	C₂³×C₉	—	2	2	no	—
35	C₂³×C₉	—	2	4	no	—
36	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉	—	2	2	no	—
39	C₂×C₄×C₉	—	2	8	no	—
40	C₂×C₄×C₉	—	2	4	no	—
41	C₂×C₄×C₉	—	2	8	no	—
42	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₉	#41	1	4	yes	—
45	C₈×C₉	—	4	4	no	—
46	C₂×#117	C₂×C₉	1	1	yes	characteristic
47	C₂×C₂²⋊C₈	C₂×C₉	1	2	yes	—
48	#336	C₂×C₉	1	4	yes	complemented
49	C₂⁴×C₃	C₂×C₃×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	—	2	4	no	—
51	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
52	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
54	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
55	C₂×C₃×C₈	—	2	4	no	—
56	C₂×C₃×C₈	—	2	8	no	—
57	C₃×C₄⋊C₄	—	2	4	no	—
58	C₃×C₂²⋊C₄	—	2	4	no	—
59	C₃×C₄²	—	2	4	no	—
60	C₂²×C₉	—	2	2	no	—
61	C₂²×C₉	—	2	4	no	—
62	C₂²×C₉	—	4	4	no	—
63	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₉	#117	1	2	yes	—
65	C₂²×C₉	#131	1	4	yes	—
66	C₄×C₉	—	2	2	no	—
67	C₄×C₉	—	4	4	no	—
68	C₄×C₉	#123	1	4	yes	—
69	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₉	1	4	yes	—
71	#117	—	2	4	no	—
72	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
73	C₂×C₂²⋊C₄	C₄×C₉	1	2	yes	—
74	C₂×C₄²	C₂²×C₉	1	2	yes	—
75	C₂²⋊C₈	—	2	4	no	—
76	C₂³×C₃	—	2	2	no	—
77	C₂³×C₃	—	2	4	no	—
78	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
79	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	8	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	—	2	8	no	—
84	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₃×#41	1	4	yes	—
87	C₃×C₈	—	4	4	no	—
88	C₂×C₉	—	4	2	no	—
89	C₂×C₉	#381	1	1	yes	characteristic
90	C₂×C₉	#408	1	2	yes	—
91	C₂×C₉	#433	1	4	yes	—
92	C₂⁴	C₂×C₄×C₉	1	1	yes	characteristic
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	C₂×C₄×C₉	1	4	yes	—
95	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
96	C₂²×C₄	D₄×C₉	1	4	yes	—
97	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₂²⋊C₄	—	2	4	no	—
102	C₄²	—	2	4	no	—
103	C₂²×C₃	—	2	2	no	—
104	C₂²×C₃	—	2	4	no	—
105	C₂²×C₃	—	4	4	no	—
106	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃	C₃×#117	1	2	yes	—
108	C₂²×C₃	C₃×#131	1	4	yes	—
109	C₃×C₄	—	2	2	no	—
110	C₃×C₄	—	4	4	no	—
111	C₃×C₄	C₃×#123	1	4	yes	—
112	C₉	#1818	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³	—	2	2	no	—
114	C₂³	—	2	4	no	—
115	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
116	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	8	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
123	C₂×C₄	C₉×#41	1	4	yes	—
124	C₈	—	4	4	no	—
125	C₂×C₃	—	4	2	no	—
126	C₂×C₃	C₃×#381	1	1	yes	characteristic
127	C₂×C₃	C₃×#408	1	2	yes	—
128	C₂×C₃	C₃×#433	1	4	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	4	4	no	—
132	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²	C₉×#117	1	2	yes	—
134	C₂²	C₉×#131	1	4	yes	—
135	C₄	—	2	2	no	—
136	C₄	—	4	4	no	—
137	C₄	C₉×#123	1	4	yes	—
138	C₃	C₃×#1818	1	1	yes	characteristic
139	C₂	—	4	2	no	—
140	C₂	C₉×#381	1	1	yes	characteristic
141	C₂	C₉×#408	1	2	yes	—
142	C₂	C₉×#433	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#117

C₂

yes

maximal, characteristic

C₂×C₉×C₂²⋊C₈

C₂

yes

maximal

C₉×#336

C₂

yes

maximal, complemented

C₃×#1818

C₃

yes

maximal, characteristic

C₂³×C₄×C₉

C₂²

yes

characteristic

C₂²×C₈×C₉

C₂²

yes

—

C₉×#117

—

C₂×C₉×C₄⋊C₄

C₄

yes

—

C₂×C₉×C₂²⋊C₄

C₄

yes

—

C₂×C₄²×C₉

C₂²

yes

—

C₉×C₂²⋊C₈

—

C₂×C₃×#117

C₂×C₃

yes

characteristic

C₂×C₃×C₂²⋊C₈

C₂×C₃

yes

—

C₃×#336

C₂×C₃

yes

—

C₂⁴×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂×C₄

yes

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

#1818

C₉

yes

characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₂²×C₃

yes

characteristic

C₂²×C₃×C₈

C₂²×C₃

yes

—

C₃×#117

—

C₂×C₃×C₄⋊C₄

C₃×C₄

yes

—

C₂×C₃×C₂²⋊C₄

C₃×C₄

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

—

C₃×C₂²⋊C₈

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂²⋊C₄

yes

—

C₂³×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₂×#117

C₂×C₉

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₉

yes

—

#336

C₂×C₉

yes

complemented

C₂⁴×C₃

C₂×C₃×C₄

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×C₄

yes

—

C₂²×C₃×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉

#117

yes

—

C₂²×C₉

#131

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#123

yes

—

C₂³×C₄

C₂²×C₉

yes

characteristic

C₂²×C₈

C₂²×C₉

yes

—

#117

—

C₂×C₄⋊C₄

C₄×C₉

yes

—

C₂×C₂²⋊C₄

C₄×C₉

yes

—

C₂×C₄²

C₂²×C₉

yes

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₃×C₂²⋊C₄

yes

—

C₂³×C₃

C₂²×C₃×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

C₂×C₃×C₄

C₃×#41

yes

—

C₃×C₈

—

C₂×C₉

—

C₂×C₉

#381

yes

characteristic

C₂×C₉

#408

yes

—

C₂×C₉

#433

yes

—

C₂⁴

C₂×C₄×C₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₉

yes

—

C₂²×C₄

C₂×C₄×C₉

yes

characteristic

C₂²×C₄

D₄×C₉

yes

—

C₂²×C₄

C₂³×C₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

C₂²×C₃

C₂×C₃×C₂²⋊C₄

yes

characteristic

107

C₂²×C₃

C₃×#117

yes

—

108

C₂²×C₃

C₃×#131

yes

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

C₃×#123

yes

—

112

C₉

#1818

yes

characteristic, direct product factor, Sylow

113

C₂³

—

114

C₂³

—

115

C₂³

C₉×C₂²⋊C₄

yes

—

116

C₂³

C₂²×C₄×C₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₉×C₂²⋊C₄

yes

—

122

C₂×C₄

C₂×D₄×C₉

yes

—

123

C₂×C₄

C₉×#41

yes

—

124

C₈

—

125

C₂×C₃

—

126

C₂×C₃

C₃×#381

yes

characteristic

127

C₂×C₃

C₃×#408

yes

—

128

C₂×C₃

C₃×#433

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₉×C₂²⋊C₄

yes

characteristic

133

C₂²

C₉×#117

yes

—

134

C₂²

C₉×#131

yes

—

135

C₄

—

136

C₄

—

137

C₄

C₉×#123

yes

—

138

C₃

C₃×#1818

yes

characteristic

139

C₂

—

140

C₂

C₉×#381

yes

characteristic

141

C₂

C₉×#408

yes

—

142

C₂

C₉×#433

yes

—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₂²×C₄×C₉

C₂³

commutator

C₂³

C₂²×C₄×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle

C₂³×C₃

C₂²×C₃×C₄

prime radical

C₂⁴×C₃

C₂×C₃×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
3. C₉×#336	C₂	semidirect
26. #1818	C₉	direct (N × H)
48. #336	C₂×C₉	semidirect
112. C₉	#1818	direct (N × H)

H ≅ G/N

type

3. C₉×#336

C₂

semidirect

26. #1818

C₉

direct (N × H)

48. #336

C₂×C₉

semidirect

112. C₉

#1818

direct (N × H)

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

subgroup	index
C₂×C₉×#117	2
C₂×C₉×C₂²⋊C₈	2
C₉×#336	2
C₃×#1818	3

subgroup

index

C₂×C₉×#117

C₂×C₉×C₂²⋊C₈

C₉×#336

C₃×#1818

Subgroups — C₉×#1818 #11766539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1818 #11766539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1818 #11766539