Group #11766712 — subgroups

Subgroups — C₉×#1991 #11766712

Order 1152.

378 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 192 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×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#382	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	4	yes	maximal
5	C₃×#1991	C₃	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₉	C₂²	1	2	yes	—
7	C₉×#125	—	2	2	no	—
8	C₉×#125	C₄	1	4	yes	complemented
9	C₂×C₉×C₄⋊C₄	—	2	2	no	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₉	—	2	2	no	—
14	C₉×#95	—	2	2	no	—
15	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
17	C₃×#382	C₂×C₃	1	1	yes	characteristic
18	C₃×#340	C₂×C₃	1	4	yes	—
19	C₂²×C₄×C₉	—	2	2	no	—
20	C₂²×C₄×C₉	—	2	2	no	—
21	C₂²×C₄×C₉	D₄	1	2	yes	—
22	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₉	—	2	2	no	—
24	C₂×C₈×C₉	—	2	2	no	—
25	C₂×C₈×C₉	Q₈	1	4	yes	—
26	C₉×C₄⋊C₄	—	2	8	no	—
27	C₉×C₄⋊C₄	—	2	2	no	—
28	C₉×C₄⋊C₄	—	2	4	no	—
29	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₄²×C₉	C₂×C₄	1	2	yes	—
31	C₄²×C₉	D₄	1	2	yes	—
32	#1991	C₉	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
34	C₃×#125	—	2	2	no	—
35	C₃×#125	C₃×C₄	1	4	yes	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
38	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
40	C₃×C₄×C₈	—	2	2	no	—
41	C₃×#95	—	2	2	no	—
42	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉	—	2	4	no	—
44	C₂×C₄×C₉	—	2	4	no	—
45	C₂×C₄×C₉	—	4	4	no	—
46	C₂×C₄×C₉	—	4	2	no	—
47	C₂×C₄×C₉	—	4	4	no	—
48	C₂×C₄×C₉	C₂²⋊C₄	1	4	yes	—
49	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₉	#41	1	2	yes	—
53	C₈×C₉	—	2	4	no	—
54	C₂×#125	C₂×C₉	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic
56	#382	C₂×C₉	1	1	yes	characteristic
57	#340	C₂×C₉	1	4	yes	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
61	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	2	2	no	—
64	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
65	C₃×C₄⋊C₄	—	2	8	no	—
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
69	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
70	C₃×C₄²	C₃×D₄	1	2	yes	—
71	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₉	#124	1	1	yes	characteristic
73	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂²×C₉	#135	1	4	yes	—
75	C₄×C₉	—	2	4	no	—
76	C₄×C₉	—	4	4	no	—
77	C₄×C₉	—	8	2	no	—
78	C₄×C₉	C₄×Q₈	1	2	yes	—
79	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
80	C₂²×C₈	C₂²×C₉	1	2	yes	—
81	#125	—	2	2	no	—
82	#125	C₄×C₉	1	4	yes	complemented
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
86	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
87	C₄×C₈	—	2	2	no	—
88	#95	—	2	2	no	—
89	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	4	4	no	—
93	C₂×C₃×C₄	—	4	2	no	—
94	C₂×C₃×C₄	—	4	4	no	—
95	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
96	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
99	C₂×C₃×C₄	C₃×#41	1	2	yes	—
100	C₃×C₈	—	2	4	no	—
101	C₂×C₉	#391	1	1	yes	characteristic
102	C₂×C₉	#416	1	2	yes	—
103	C₂×C₉	#488	1	2	yes	—
104	C₂×C₉	#499	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	D₄×C₉	1	2	yes	—
108	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	Q₈×C₉	1	4	yes	—
112	C₄⋊C₄	—	2	8	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
116	C₄²	C₂×C₄×C₉	1	2	yes	—
117	C₄²	D₄×C₉	1	2	yes	—
118	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₃	C₃×#124	1	1	yes	characteristic
120	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₃	C₃×#135	1	4	yes	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	—	4	4	no	—
124	C₃×C₄	—	8	2	no	—
125	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
126	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
127	C₉	#1991	1	1	yes	characteristic, direct product factor, Sylow
128	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	C₉×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
138	C₂×C₄	C₉×#41	1	2	yes	—
139	C₈	—	2	4	no	—
140	C₂×C₃	C₃×#391	1	1	yes	characteristic
141	C₂×C₃	C₃×#416	1	2	yes	—
142	C₂×C₃	C₃×#488	1	2	yes	—
143	C₂×C₃	C₃×#499	1	2	yes	—
144	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²	C₉×#124	1	1	yes	characteristic
146	C₂²	C₉×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²	C₉×#135	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	8	2	no	—
151	C₄	C₄×Q₈×C₉	1	2	yes	—
152	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
153	C₃	C₃×#1991	1	1	yes	characteristic
154	C₂	C₉×#391	1	1	yes	characteristic
155	C₂	C₉×#416	1	2	yes	—
156	C₂	C₉×#488	1	2	yes	—
157	C₂	C₉×#499	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#125

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₉

C₂

yes

maximal, characteristic

C₉×#382

C₂

yes

maximal, characteristic

C₉×#340

C₂

yes

maximal

C₃×#1991

C₃

yes

maximal, characteristic

C₂²×C₈×C₉

C₂²

yes

—

C₉×#125

—

C₉×#125

C₄

yes

complemented

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₄×C₈×C₉

—

C₉×#95

—

C₂×C₃×#125

C₂×C₃

yes

characteristic

C₂×C₃×C₄×C₈

C₂×C₃

yes

characteristic

C₃×#382

C₂×C₃

yes

characteristic

C₃×#340

C₂×C₃

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

Q₈

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₉

C₂×C₄

yes

—

C₄²×C₉

D₄

yes

—

#1991

C₉

yes

characteristic, direct product factor, Sylow

C₂²×C₃×C₈

C₂²×C₃

yes

—

C₃×#125

—

C₃×#125

C₃×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

characteristic

C₃×C₄×C₈

—

C₃×#95

—

C₂³×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²⋊C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×Q₈

yes

—

C₂×C₄×C₉

#41

yes

—

C₈×C₉

—

C₂×#125

C₂×C₉

yes

characteristic

C₂×C₄×C₈

C₂×C₉

yes

characteristic

#382

C₂×C₉

yes

characteristic

#340

C₂×C₉

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Q₈

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₃×C₄

yes

—

C₃×C₄²

C₂×C₃×C₄

yes

—

C₃×C₄²

C₃×D₄

yes

—

C₂²×C₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉

#124

yes

characteristic

C₂²×C₉

C₄⋊Q₈

yes

characteristic

C₂²×C₉

#135

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₄×Q₈

yes

—

C₄×C₉

C₂²⋊Q₈

yes

—

C₂²×C₈

C₂²×C₉

yes

—

#125

—

#125

C₄×C₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₉

yes

—

C₂×C₄⋊C₄

C₂²×C₉

yes

—

C₂×C₄²

C₂²×C₉

yes

characteristic

C₄×C₈

—

#95

—

C₂³×C₃

C₂×C₃×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₃×C₄

yes

characteristic

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

C₂×C₃×C₄

C₂×C₃×Q₈

yes

—

C₂×C₃×C₄

C₃×#41

yes

—

100

C₃×C₈

—

101

C₂×C₉

#391

yes

characteristic

102

C₂×C₉

#416

yes

—

103

C₂×C₉

#488

yes

—

104

C₂×C₉

#499

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

D₄×C₉

yes

—

108

C₂²×C₄

C₂³×C₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

Q₈×C₉

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×C₄×C₉

yes

—

116

C₄²

C₂×C₄×C₉

yes

—

117

C₄²

D₄×C₉

yes

—

118

C₂²×C₃

C₂×C₃×C₂²⋊C₄

yes

characteristic

119

C₂²×C₃

C₃×#124

yes

characteristic

120

C₂²×C₃

C₃×C₄⋊Q₈

yes

characteristic

121

C₂²×C₃

C₃×#135

yes

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

C₃×C₄×Q₈

yes

—

126

C₃×C₄

C₃×C₂²⋊Q₈

yes

—

127

C₉

#1991

yes

characteristic, direct product factor, Sylow

128

C₂³

C₂×D₄×C₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₉×C₂²⋊C₄

yes

—

135

C₂×C₄

C₂²×C₄×C₉

yes

characteristic

136

C₂×C₄

C₂×D₄×C₉

yes

characteristic

137

C₂×C₄

C₂×Q₈×C₉

yes

—

138

C₂×C₄

C₉×#41

yes

—

139

C₈

—

140

C₂×C₃

C₃×#391

yes

characteristic

141

C₂×C₃

C₃×#416

yes

—

142

C₂×C₃

C₃×#488

yes

—

143

C₂×C₃

C₃×#499

yes

—

144

C₂²

C₂×C₉×C₂²⋊C₄

yes

characteristic

145

C₂²

C₉×#124

yes

characteristic

146

C₂²

C₉×C₄⋊Q₈

yes

characteristic

147

C₂²

C₉×#135

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

C₄×Q₈×C₉

yes

—

152

C₄

C₉×C₂²⋊Q₈

yes

—

153

C₃

C₃×#1991

yes

characteristic

154

C₂

C₉×#391

yes

characteristic

155

C₂

C₉×#416

yes

—

156

C₂

C₉×#488

yes

—

157

C₂

C₉×#499

yes

—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

subgroup

G/H

center

C₂³×C₉

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle, prime radical

C₂³×C₃

C₂×C₃×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

N	H ≅ G/N	type
8. C₉×#125	C₄	semidirect
32. #1991	C₉	direct (N × H)
82. #125	C₄×C₉	semidirect
127. C₉	#1991	direct (N × H)

H ≅ G/N

type

8. C₉×#125

C₄

semidirect

32. #1991

C₉

direct (N × H)

82. #125

C₄×C₉

semidirect

127. C₉

#1991

direct (N × H)

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

subgroup	index
C₂×C₉×#125	2
C₂×C₄×C₈×C₉	2
C₉×#382	2
C₉×#340	2
C₃×#1991	3

subgroup

index

C₂×C₉×#125

C₂×C₄×C₈×C₉

C₉×#382

C₉×#340

C₃×#1991

Subgroups — C₉×#1991 #11766712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#1991 #11766712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1991 #11766712