Group #11798021 — subgroups

Subgroups — #11798021

Order 1152.

189 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 124 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	C₈×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₉⋊C₁₆)	C₂	1	2	yes	maximal
3	C₄⋊(C₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#10588138	C₂	1	2	yes	maximal
5	C₄×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#70949	—	3	1	no	maximal
7	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
8	C₉×C₄⋊C₈	C₄	1	2	yes	—
9	C₉×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₉	C₂²	1	2	yes	—
12	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₉⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₃×C₈×Q₈	D₃	1	1	yes	characteristic
16	C₄⋊(C₃⋊C₁₆)	—	3	2	no	—
17	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
18	#4461	—	3	2	no	—
19	C₄×C₃⋊C₁₆	—	3	1	no	—
20	C₂×Q₈×C₉	C₈	1	1	yes	characteristic
21	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
24	C₉×C₄⋊C₄	C₈	1	2	yes	—
25	C₉×C₄⋊C₄	C₈	1	1	yes	characteristic
26	C₄²×C₉	C₂×C₄	1	2	yes	—
27	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
28	C₉⋊C₁₆	—	2	1	no	—
29	C₉⋊C₁₆	—	2	2	no	—
30	C₉⋊C₁₆	Q₈	1	2	yes	complemented
31	#2183	—	9	1	no	Sylow
32	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
33	C₃×C₄⋊C₈	Dic₃	1	2	yes	—
34	C₃×C₄⋊C₈	Dic₃	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
36	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₃⋊C₁₆	—	3	1	no	—
38	C₂×C₃⋊C₁₆	—	3	1	no	—
39	C₂×C₃⋊C₁₆	—	3	2	no	—
40	Q₈×C₉	—	2	2	no	—
41	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₉	—	2	1	no	—
47	C₈×C₉	—	2	2	no	—
48	C₈×C₉	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₉	#41	1	1	yes	characteristic
50	C₈×Q₈	D₉	1	1	yes	characteristic
51	C₄⋊C₁₆	—	9	1	no	—
52	C₄⋊C₁₆	—	9	2	no	—
53	#346	—	9	2	no	—
54	C₄×C₁₆	—	9	1	no	—
55	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
58	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
59	C₃×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
60	C₃×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
61	C₃×C₄²	C₂×Dic₃	1	2	yes	—
62	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
63	C₃⋊C₁₆	—	3	2	no	—
64	C₃⋊C₁₆	—	6	1	no	—
65	C₃⋊C₁₆	—	6	2	no	—
66	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₉	—	2	2	no	—
68	C₄×C₉	—	2	2	no	—
69	C₄×C₉	—	2	1	no	—
70	C₄×C₉	M₃₂	1	2	yes	—
71	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
72	C₄×C₉	#131	1	1	yes	characteristic
73	C₄×Q₈	Dic₉	1	1	yes	characteristic
74	C₂×C₁₆	—	9	1	no	—
75	C₂×C₁₆	—	9	1	no	—
76	C₂×C₁₆	—	9	2	no	—
77	C₄⋊C₈	Dic₉	1	2	yes	—
78	C₄⋊C₈	Dic₉	1	1	yes	characteristic
79	C₄×C₈	C₂×D₉	1	1	yes	characteristic
80	C₄×C₈	C₂×D₉	1	2	yes	—
81	C₃×Q₈	—	2	2	no	—
82	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
83	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
85	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
87	C₃×C₈	—	2	1	no	—
88	C₃×C₈	—	2	2	no	—
89	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
90	C₃×C₈	#239	1	1	yes	characteristic
91	C₂×C₉	C₈×Q₈	1	1	yes	characteristic
92	C₂×C₉	C₂×M₃₂	1	1	yes	characteristic
93	C₂×C₉	#504	1	1	yes	characteristic
94	C₂×Q₈	C₉⋊C₈	1	1	yes	characteristic
95	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₉	1	2	yes	—
97	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
98	C₄⋊C₄	C₉⋊C₈	1	2	yes	—
99	C₄⋊C₄	C₉⋊C₈	1	1	yes	characteristic
100	C₄²	C₂×Dic₉	1	2	yes	—
101	C₄²	C₂×Dic₉	1	1	yes	characteristic
102	C₁₆	—	9	2	no	—
103	C₁₆	—	18	1	no	—
104	C₁₆	—	18	2	no	—
105	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
106	C₃×C₄	—	2	2	no	—
107	C₃×C₄	—	2	1	no	—
108	C₃×C₄	—	2	2	no	—
109	C₃×C₄	#812	1	2	yes	—
110	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
111	C₃×C₄	#948	1	1	yes	characteristic
112	C₉	#2183	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	2	2	no	—
114	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
115	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
116	C₂×C₄	C₂×C₉⋊C₈	1	2	yes	—
117	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	2	no	—
121	C₈	Q₈×D₉	1	1	yes	characteristic
122	C₈	#3701	1	1	yes	characteristic
123	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
124	C₂×C₃	C₂×#812	1	1	yes	characteristic
125	C₂×C₃	#5140	1	1	yes	characteristic
126	C₂²	C₂²×C₉⋊C₈	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	#63371	1	2	yes	—
131	C₄	Q₈×Dic₉	1	1	yes	characteristic
132	C₄	#63510	1	1	yes	characteristic
133	C₃	#70949	1	1	yes	characteristic
134	C₂	Q₈×C₉⋊C₈	1	1	yes	characteristic
135	C₂	C₂×#63371	1	1	yes	characteristic
136	C₂	#10588820	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₉

C₂

yes

maximal, characteristic

C₄⋊(C₉⋊C₁₆)

C₂

yes

maximal

C₄⋊(C₉⋊C₁₆)

C₂

yes

maximal, characteristic

#10588138

C₂

yes

maximal

C₄×C₉⋊C₁₆

C₂

yes

maximal, characteristic

#70949

—

maximal

C₄×Q₈×C₉

C₄

yes

characteristic

C₉×C₄⋊C₈

C₄

yes

—

C₉×C₄⋊C₈

C₄

yes

characteristic

C₄×C₈×C₉

C₂²

yes

characteristic

C₄×C₈×C₉

C₂²

yes

—

C₂×C₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₉⋊C₁₆

C₂²

yes

—

C₂×C₉⋊C₁₆

C₂²

yes

characteristic

C₃×C₈×Q₈

D₃

yes

characteristic

C₄⋊(C₃⋊C₁₆)

—

C₄⋊(C₃⋊C₁₆)

—

#4461

—

C₄×C₃⋊C₁₆

—

C₂×Q₈×C₉

C₈

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

—

C₂×C₈×C₉

C₂³

yes

characteristic

C₉×C₄⋊C₄

C₈

yes

—

C₉×C₄⋊C₄

C₈

yes

characteristic

C₄²×C₉

C₂×C₄

yes

—

C₄²×C₉

C₂×C₄

yes

characteristic

C₉⋊C₁₆

—

C₉⋊C₁₆

—

C₉⋊C₁₆

Q₈

yes

complemented

#2183

—

Sylow

C₃×C₄×Q₈

Dic₃

yes

characteristic

C₃×C₄⋊C₈

Dic₃

yes

—

C₃×C₄⋊C₈

Dic₃

yes

characteristic

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₃×C₄×C₈

C₂×D₃

yes

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

Q₈×C₉

—

C₂×C₄×C₉

C₂×C₈

yes

—

C₂×C₄×C₉

C₂×C₈

yes

characteristic

C₂×C₄×C₉

C₂×C₈

yes

—

C₂×C₄×C₉

C₂×C₈

yes

characteristic

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₂×Q₈

yes

characteristic

C₈×C₉

#41

yes

characteristic

C₈×Q₈

D₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂×C₃×Q₈

C₃⋊C₈

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

—

C₂×C₃×C₈

C₂²×D₃

yes

characteristic

C₃×C₄⋊C₄

C₃⋊C₈

yes

—

C₃×C₄⋊C₄

C₃⋊C₈

yes

characteristic

C₃×C₄²

C₂×Dic₃

yes

—

C₃×C₄²

C₂×Dic₃

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₉

C₂²×C₈

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

M₃₂

yes

—

C₄×C₉

C₄×Q₈

yes

characteristic

C₄×C₉

#131

yes

characteristic

C₄×Q₈

Dic₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

Dic₉

yes

—

C₄⋊C₈

Dic₉

yes

characteristic

C₄×C₈

C₂×D₉

yes

characteristic

C₄×C₈

C₂×D₉

yes

—

C₃×Q₈

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

characteristic

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

characteristic

C₂×C₃×C₄

C₂²×Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×Q₈

yes

characteristic

C₃×C₈

#239

yes

characteristic

C₂×C₉

C₈×Q₈

yes

characteristic

C₂×C₉

C₂×M₃₂

yes

characteristic

C₂×C₉

#504

yes

characteristic

C₂×Q₈

C₉⋊C₈

yes

characteristic

C₂×C₈

C₂×Dic₉

yes

characteristic

C₂×C₈

C₂×Dic₉

yes

—

C₂×C₈

C₂²×D₉

yes

characteristic

C₄⋊C₄

C₉⋊C₈

yes

—

C₄⋊C₄

C₉⋊C₈

yes

characteristic

100

C₄²

C₂×Dic₉

yes

—

101

C₄²

C₂×Dic₉

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

—

105

C₂²×C₃

C₂²×C₃⋊C₈

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

#812

yes

—

110

C₃×C₄

Q₈×Dic₃

yes

characteristic

111

C₃×C₄

#948

yes

characteristic

112

C₉

#2183

yes

characteristic, complemented, Sylow

113

Q₈

—

114

C₂×C₄

C₂×C₉⋊C₈

yes

—

115

C₂×C₄

C₂×C₉⋊C₈

yes

characteristic

116

C₂×C₄

C₂×C₉⋊C₈

yes

—

117

C₂×C₄

C₂×C₉⋊C₈

yes

characteristic

118

C₂×C₄

C₂²×Dic₉

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

Q₈×D₉

yes

characteristic

122

C₈

#3701

yes

characteristic

123

C₂×C₃

Q₈×C₃⋊C₈

yes

characteristic

124

C₂×C₃

C₂×#812

yes

characteristic

125

C₂×C₃

#5140

yes

characteristic

126

C₂²

C₂²×C₉⋊C₈

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#63371

yes

—

131

C₄

Q₈×Dic₉

yes

characteristic

132

C₄

#63510

yes

characteristic

133

C₃

#70949

yes

characteristic

134

C₂

Q₈×C₉⋊C₈

yes

characteristic

135

C₂

C₂×#63371

yes

characteristic

136

C₂

#10588820

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂×C₃×C₈	C₂²×D₃
Fitting	C₈×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃	C₂²×C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂³
hypercenter	C₈×Q₈	D₉
hypocenter	C₉	#2183
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₈

C₂²×D₉

commutator

C₂²×C₉

C₂²×C₈

Frattini

C₂×C₃×C₈

C₂²×D₃

Fitting

C₈×Q₈×C₉

C₂

socle, prime radical

C₂²×C₃

C₂²×C₃⋊C₈

cosocle

C₂×C₈×C₉

C₂³

hypercenter

C₈×Q₈

D₉

C₉

G ▷ C₂²×C₉ ▷ C₉

C₁ ◁ C₂×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
30. C₉⋊C₁₆	Q₈	semidirect
112. C₉	#2183	semidirect

H ≅ G/N

type

30. C₉⋊C₁₆

Q₈

semidirect

112. C₉

#2183

semidirect

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

subgroup	index
C₈×Q₈×C₉	2
C₄⋊(C₉⋊C₁₆)	2
#10588138	2
C₄×C₉⋊C₁₆	2
#70949	3

subgroup

index

C₈×Q₈×C₉

C₄⋊(C₉⋊C₁₆)

#10588138

C₄×C₉⋊C₁₆

#70949

Subgroups — #11798021

Conjugacy classes of subgroups

Complements

Maximal subgroups