Group #11772308 — subgroups

Subgroups — #11772308

Order 1152.

210 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 118 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×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#63371	C₂	1	2	yes	maximal
3	#67664	—	3	1	no	maximal
4	C₂²×C₈×C₉	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₉	C₄	1	1	yes	characteristic
7	C₉×#97	—	2	2	no	—
8	#63371	—	2	2	no	—
9	#63371	C₄	1	4	yes	complemented
10	C₂×C₉⋊C₁₆	—	2	2	no	—
11	C₂×C₃×#97	D₃	1	1	yes	characteristic
12	C₂×#812	—	3	2	no	—
13	C₂²×C₄×C₉	C₈	1	2	yes	—
14	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₉	—	2	1	no	—
16	C₂×C₈×C₉	—	2	1	no	—
17	C₂×C₈×C₉	C₈	1	2	yes	—
18	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₉	D₄	1	2	yes	—
21	C₂×C₈×C₉	Q₈	1	1	yes	characteristic
22	C₄²×C₉	—	2	1	no	—
23	C₄²×C₉	—	2	1	no	—
24	C₉⋊C₁₆	—	2	4	no	—
25	#1377	—	9	1	no	Sylow
26	C₂²×C₃×C₈	Dic₃	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄²	Dic₃	1	1	yes	characteristic
29	C₃×#97	—	2	2	no	—
30	#812	—	3	4	no	—
31	#812	—	6	2	no	—
32	C₂×C₃⋊C₁₆	—	6	2	no	—
33	C₂³×C₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₉	—	2	2	no	—
35	C₂×C₄×C₉	—	4	2	no	—
36	C₂×C₄×C₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₉	M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₉	M₁₆	1	1	yes	characteristic
42	C₈×C₉	—	2	2	no	—
43	C₈×C₉	C₂²⋊C₄	1	2	yes	—
44	C₈×C₉	C₄⋊C₄	1	2	yes	—
45	C₂×M₃₂	—	9	2	no	—
46	C₂×#97	D₉	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₃⋊C₈	1	2	yes	—
48	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
49	C₂×C₃×C₈	—	2	1	no	—
50	C₂×C₃×C₈	—	2	1	no	—
51	C₂×C₃×C₈	C₃⋊C₈	1	2	yes	—
52	C₂×C₃×C₈	Dic₆	1	1	yes	characteristic
53	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
54	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
55	C₂×C₃×C₈	#67	1	2	yes	—
56	C₃×C₄²	—	2	1	no	—
57	C₃×C₄²	—	2	1	no	—
58	C₃⋊C₁₆	—	6	4	no	—
59	C₂²×C₉	—	2	2	no	—
60	C₂²×C₉	C₄×C₈	1	1	yes	characteristic
61	C₂²×C₉	C₂²⋊C₈	1	1	yes	characteristic
62	C₂²×C₉	C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₉	—	4	2	no	—
64	C₄×C₉	#95	1	1	yes	characteristic
65	C₄×C₉	#97	1	1	yes	characteristic
66	C₄×C₉	C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₉	C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₈	Dic₉	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
70	C₂×C₄²	Dic₉	1	1	yes	characteristic
71	M₃₂	—	9	4	no	—
72	M₃₂	—	18	2	no	—
73	C₂×C₁₆	—	18	2	no	—
74	#97	—	2	2	no	—
75	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	4	2	no	—
78	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
79	C₂×C₃×C₄	#208	1	1	yes	characteristic
80	C₂×C₃×C₄	#208	1	1	yes	characteristic
81	C₂×C₃×C₄	C₄×Dic₃	1	1	yes	characteristic
82	C₂×C₃×C₄	C₄⋊Dic₃	1	1	yes	characteristic
83	C₂×C₃×C₄	#217	1	1	yes	characteristic
84	C₃×C₈	—	2	2	no	—
85	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
86	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
87	C₂×C₉	—	2	1	no	—
88	C₂×C₉	—	2	1	no	—
89	C₂×C₉	#336	1	1	yes	characteristic
90	C₂×C₉	#349	1	2	yes	—
91	C₂²×C₄	C₉⋊C₈	1	2	yes	—
92	C₂²×C₄	C₂×Dic₉	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	C₉⋊C₈	1	2	yes	—
96	C₂×C₈	Dic₁₈	1	1	yes	characteristic
97	C₂×C₈	C₄×D₉	1	2	yes	—
98	C₂×C₈	D₃₆	1	1	yes	characteristic
99	C₂×C₈	#611	1	2	yes	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₁₆	—	18	4	no	—
103	C₂²×C₃	—	2	2	no	—
104	C₂²×C₃	C₄×C₃⋊C₈	1	1	yes	characteristic
105	C₂²×C₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
106	C₂²×C₃	#830	1	1	yes	characteristic
107	C₃×C₄	—	4	2	no	—
108	C₃×C₄	#803	1	1	yes	characteristic
109	C₃×C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
110	C₃×C₄	#830	1	1	yes	characteristic
111	C₃×C₄	C₃⋊#95	1	1	yes	characteristic
112	C₉	#1377	1	1	yes	characteristic, complemented, Sylow
113	C₂³	C₂×C₉⋊C₈	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
117	C₂×C₄	#3667	1	1	yes	characteristic
118	C₂×C₄	#3667	1	1	yes	characteristic
119	C₂×C₄	C₄×Dic₉	1	1	yes	characteristic
120	C₂×C₄	C₄⋊Dic₉	1	1	yes	characteristic
121	C₂×C₄	#3676	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	Dic₉⋊C₄	1	2	yes	—
124	C₈	(C₂×D₉)⋊C₄	1	2	yes	—
125	C₂×C₃	—	2	1	no	—
126	C₂×C₃	—	2	1	no	—
127	C₂×C₃	#4524	1	1	yes	characteristic
128	C₂×C₃	#4553	1	2	yes	—
129	C₂²	—	2	2	no	—
130	C₂²	C₄×C₉⋊C₈	1	1	yes	characteristic
131	C₂²	C₄⋊(C₉⋊C₈)	1	1	yes	characteristic
132	C₂²	#63389	1	1	yes	characteristic
133	C₄	—	4	2	no	—
134	C₄	#63362	1	1	yes	characteristic
135	C₄	C₄⋊(C₉⋊C₈)	1	1	yes	characteristic
136	C₄	#63389	1	1	yes	characteristic
137	C₄	C₉⋊#95	1	1	yes	characteristic
138	C₃	#67664	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	#10588200	1	1	yes	characteristic
142	C₂	#10588230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#97

C₂

yes

maximal, characteristic

C₂×#63371

C₂

yes

maximal

#67664

—

maximal

C₂²×C₈×C₉

C₄

yes

characteristic

C₂²×C₈×C₉

C₂²

yes

characteristic

C₂×C₄²×C₉

C₄

yes

characteristic

C₉×#97

—

#63371

—

#63371

C₄

yes

complemented

C₂×C₉⋊C₁₆

—

C₂×C₃×#97

D₃

yes

characteristic

C₂×#812

—

C₂²×C₄×C₉

C₈

yes

—

C₂²×C₄×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₈

yes

—

C₂×C₈×C₉

C₂×C₄

yes

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

D₄

yes

—

C₂×C₈×C₉

Q₈

yes

characteristic

C₄²×C₉

—

C₄²×C₉

—

C₉⋊C₁₆

—

#1377

—

Sylow

C₂²×C₃×C₈

Dic₃

yes

characteristic

C₂²×C₃×C₈

C₂×D₃

yes

characteristic

C₂×C₃×C₄²

Dic₃

yes

characteristic

C₃×#97

—

#812

—

#812

—

C₂×C₃⋊C₁₆

—

C₂³×C₉

C₂×C₈

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄²

yes

characteristic

C₂×C₄×C₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉

C₂×C₈

yes

characteristic

C₂×C₄×C₉

M₁₆

yes

characteristic

C₂×C₄×C₉

M₁₆

yes

characteristic

C₈×C₉

—

C₈×C₉

C₂²⋊C₄

yes

—

C₈×C₉

C₄⋊C₄

yes

—

C₂×M₃₂

—

C₂×#97

D₉

yes

characteristic

C₂²×C₃×C₄

C₃⋊C₈

yes

—

C₂²×C₃×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃⋊C₈

yes

—

C₂×C₃×C₈

Dic₆

yes

characteristic

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

D₁₂

yes

characteristic

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃⋊C₁₆

—

C₂²×C₉

—

C₂²×C₉

C₄×C₈

yes

characteristic

C₂²×C₉

C₂²⋊C₈

yes

characteristic

C₂²×C₉

C₄⋊C₈

yes

characteristic

C₄×C₉

—

C₄×C₉

yes

C₄×C₉

yes

C₄×C₉

C₂²⋊C₈

yes

characteristic

C₄×C₉

C₄⋊C₈

yes

characteristic

C₂²×C₈

Dic₉

yes

characteristic

C₂²×C₈

C₂×D₉

yes

characteristic

C₂×C₄²

Dic₉

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₂³×C₃

C₂×C₃⋊C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊C₈

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

C₄×Dic₃

yes

characteristic

C₂×C₃×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#217

yes

characteristic

C₃×C₈

—

C₃×C₈

Dic₃⋊C₄

yes

—

C₃×C₈

(C₂×D₃)⋊C₄

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#336

yes

characteristic

C₂×C₉

#349

yes

—

C₂²×C₄

C₉⋊C₈

yes

—

C₂²×C₄

C₂×Dic₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₉⋊C₈

yes

—

C₂×C₈

Dic₁₈

yes

characteristic

C₂×C₈

C₄×D₉

yes

—

C₂×C₈

D₃₆

yes

characteristic

C₂×C₈

#611

yes

—

100

C₄²

—

101

C₄²

—

102

C₁₆

—

103

C₂²×C₃

—

104

C₂²×C₃

C₄×C₃⋊C₈

yes

characteristic

105

C₂²×C₃

C₄⋊(C₃⋊C₈)

yes

characteristic

106

C₂²×C₃

#830

yes

characteristic

107

C₃×C₄

—

108

C₃×C₄

#803

yes

characteristic

109

C₃×C₄

C₄⋊(C₃⋊C₈)

yes

characteristic

110

C₃×C₄

#830

yes

characteristic

111

C₃×C₄

C₃⋊#95

yes

characteristic

112

C₉

#1377

yes

characteristic, complemented, Sylow

113

C₂³

C₂×C₉⋊C₈

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₉⋊C₈

yes

characteristic

117

C₂×C₄

#3667

yes

characteristic

118

C₂×C₄

#3667

yes

characteristic

119

C₂×C₄

C₄×Dic₉

yes

characteristic

120

C₂×C₄

C₄⋊Dic₉

yes

characteristic

121

C₂×C₄

#3676

yes

characteristic

122

C₈

—

123

C₈

Dic₉⋊C₄

yes

—

124

C₈

(C₂×D₉)⋊C₄

yes

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

#4524

yes

characteristic

128

C₂×C₃

#4553

yes

—

129

C₂²

—

130

C₂²

C₄×C₉⋊C₈

yes

characteristic

131

C₂²

C₄⋊(C₉⋊C₈)

yes

characteristic

132

C₂²

#63389

yes

characteristic

133

C₄

—

134

C₄

#63362

yes

characteristic

135

C₄

C₄⋊(C₉⋊C₈)

yes

characteristic

136

C₄

#63389

yes

characteristic

137

C₄

C₉⋊#95

yes

characteristic

138

C₃

#67664

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#10588200

yes

characteristic

142

C₂

#10588230

yes

—

	subgroup	G/H
center	C₂×C₄	#3676
commutator	C₂²×C₉	C₄×C₈
Frattini	C₂²×C₃×C₈	C₂×D₃
Fitting	C₂×C₉×#97	C₂
socle	C₂²×C₃	#830
prime radical	C₂³×C₃	C₂×C₃⋊C₈
cosocle	C₂²×C₈×C₉	C₂²
hypercenter	C₂×#97	D₉
hypocenter	C₉	#1377
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

#3676

commutator

C₂²×C₉

C₄×C₈

Frattini

C₂²×C₃×C₈

C₂×D₃

Fitting

C₂×C₉×#97

C₂

socle

C₂²×C₃

#830

prime radical

C₂³×C₃

C₂×C₃⋊C₈

cosocle

C₂²×C₈×C₉

C₂²

hypercenter

C₂×#97

D₉

hypocenter

C₉

#1377

LCS

G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

N	H ≅ G/N	type
9. #63371	C₄	semidirect
112. C₉	#1377	semidirect

H ≅ G/N

type

9. #63371

C₄

semidirect

112. C₉

#1377

semidirect

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

subgroup	index
C₂×C₉×#97	2
C₂×#63371	2
#67664	3

subgroup

index

C₂×C₉×#97

C₂×#63371

#67664

Subgroups — #11772308

Conjugacy classes of subgroups

Complements

Maximal subgroups