Group #11777306 — subgroups

Subgroups — #11777306

Order 1152.

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

normal subgroups only

This group has 240 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	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₉⋊C₈	C₂	1	2	yes	maximal
3	#10588212	C₂	1	2	yes	maximal
4	#10588134	C₂	1	1	yes	maximal, characteristic
5	#10588132	C₂	1	1	yes	maximal, characteristic
6	#70902	—	3	1	no	maximal
7	C₂×C₉×Q₁₆	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₂²	1	2	yes	—
9	C₉×#107	C₄	1	1	yes	characteristic
10	C₉×#103	C₄	1	2	yes	—
11	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₉⋊C₈)	—	2	2	no	—
13	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
14	C₄×C₉⋊C₈	—	2	2	no	—
15	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×Q₁₆	D₃	1	1	yes	characteristic
18	Q₈×C₃⋊C₈	—	3	2	no	—
19	#4535	—	3	2	no	—
20	#4457	—	3	1	no	—
21	#4455	—	3	1	no	—
22	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
23	C₉×Q₁₆	C₈	1	4	yes	complemented
24	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉	C₂×C₄	1	1	yes	characteristic
26	C₉×C₄⋊C₄	—	2	2	no	—
27	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄²×C₉	—	2	2	no	—
29	C₄²×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉⋊C₈	—	2	2	no	—
31	C₂×C₉⋊C₈	—	2	1	no	—
32	C₂×C₉⋊C₈	D₄	1	2	yes	—
33	#1579	—	9	1	no	Sylow
34	C₂×C₃×Q₁₆	Dic₃	1	1	yes	characteristic
35	C₃×C₄×Q₈	C₂×D₃	1	2	yes	—
36	C₃×#107	Dic₃	1	1	yes	characteristic
37	C₃×#103	Dic₃	1	2	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
39	C₄⋊(C₃⋊C₈)	—	3	2	no	—
40	C₄⋊(C₃⋊C₈)	—	6	2	no	—
41	C₄×C₃⋊C₈	—	3	1	no	—
42	C₄×C₃⋊C₈	—	3	1	no	—
43	C₄×C₃⋊C₈	—	6	2	no	—
44	Q₈×C₉	—	2	2	no	—
45	Q₈×C₉	C₂×C₈	1	4	yes	—
46	C₂×C₄×C₉	—	2	2	no	—
47	C₂×C₄×C₉	—	2	2	no	—
48	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₉	#41	1	1	yes	characteristic
51	C₈×C₉	—	2	1	no	—
52	C₈×C₉	C₂×C₈	1	2	yes	—
53	C₉⋊C₈	—	2	2	no	—
54	C₉⋊C₈	—	4	1	no	—
55	C₉⋊C₈	—	4	2	no	—
56	C₈×Q₈	—	9	2	no	—
57	C₄×Q₁₆	D₉	1	1	yes	characteristic
58	#334	—	9	1	no	—
59	Q₈⋊C₈	—	9	2	no	—
60	#322	—	9	1	no	—
61	C₂×C₃×Q₈	C₂×Dic₃	1	2	yes	—
62	C₃×Q₁₆	C₃⋊C₈	1	4	yes	—
63	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₃⋊C₈	—	3	2	no	—
70	C₂×C₃⋊C₈	—	6	2	no	—
71	C₂×C₃⋊C₈	—	6	1	no	—
72	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₉	—	2	4	no	—
74	C₄×C₉	—	2	1	no	—
75	C₄×C₉	—	4	2	no	—
76	C₄×C₉	C₂²×C₈	1	1	yes	characteristic
77	C₄×C₉	#131	1	1	yes	characteristic
78	C₄×C₉	#137	1	2	yes	—
79	C₂×Q₁₆	Dic₉	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₉	1	2	yes	—
81	#107	Dic₉	1	1	yes	characteristic
82	C₄⋊C₈	—	9	2	no	—
83	C₄⋊C₈	—	18	2	no	—
84	#103	Dic₉	1	2	yes	—
85	C₄×C₈	—	9	1	no	—
86	C₄×C₈	—	9	1	no	—
87	C₄×C₈	—	18	2	no	—
88	C₄×C₈	C₂×D₉	1	1	yes	characteristic
89	C₃×Q₈	—	2	2	no	—
90	C₃×Q₈	C₂×C₃⋊C₈	1	4	yes	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄	#237	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	—
98	C₃⋊C₈	—	6	2	no	—
99	C₃⋊C₈	—	12	1	no	—
100	C₃⋊C₈	—	12	2	no	—
101	C₂×C₉	C₈×D₄	1	1	yes	characteristic
102	C₂×C₉	#441	1	1	yes	characteristic
103	C₂×C₉	#444	1	1	yes	characteristic
104	C₂×Q₈	C₂×Dic₉	1	2	yes	—
105	Q₁₆	C₉⋊C₈	1	4	yes	complemented
106	C₂×C₈	—	9	2	no	—
107	C₂×C₈	—	18	2	no	—
108	C₂×C₈	—	18	1	no	—
109	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
110	C₂×C₈	C₂×Dic₉	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	C₂×Dic₉	1	2	yes	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²×D₉	1	1	yes	characteristic
115	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
116	C₃×C₄	—	2	1	no	—
117	C₃×C₄	—	2	4	no	—
118	C₃×C₄	—	4	2	no	—
119	C₃×C₄	#918	1	2	yes	—
120	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
121	C₃×C₄	#948	1	1	yes	characteristic
122	C₉	#1579	1	1	yes	characteristic, complemented, Sylow
123	Q₈	—	2	2	no	—
124	Q₈	C₂×C₉⋊C₈	1	4	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	D₄×D₉	1	1	yes	characteristic
128	C₂×C₄	#3699	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	18	2	no	—
132	C₈	—	36	1	no	—
133	C₈	—	36	2	no	—
134	C₈	C₂×C₉⋊C₈	1	2	yes	—
135	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
136	C₂×C₃	#5184	1	1	yes	characteristic
137	C₂×C₃	#5197	1	1	yes	characteristic
138	C₂²	D₄×Dic₉	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₄	#63480	1	2	yes	—
143	C₄	C₂²×C₉⋊C₈	1	1	yes	characteristic
144	C₄	#63510	1	1	yes	characteristic
145	C₃	#70902	1	1	yes	characteristic
146	C₂	D₄×C₉⋊C₈	1	1	yes	characteristic
147	C₂	#10588864	1	1	yes	characteristic
148	C₂	#10588877	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉×Q₁₆

C₂

yes

maximal, characteristic

Q₈×C₉⋊C₈

C₂

yes

maximal

#10588212

C₂

yes

maximal

#10588134

C₂

yes

maximal, characteristic

#10588132

C₂

yes

maximal, characteristic

#70902

—

maximal

C₂×C₉×Q₁₆

C₄

yes

characteristic

C₄×Q₈×C₉

C₂²

yes

—

C₉×#107

C₄

yes

characteristic

C₉×#103

C₄

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

C₄⋊(C₉⋊C₈)

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₄×C₉⋊C₈

—

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₃×C₄×Q₁₆

D₃

yes

characteristic

Q₈×C₃⋊C₈

—

#4535

—

#4457

—

#4455

—

C₂×Q₈×C₉

C₂×C₄

yes

—

C₉×Q₁₆

C₈

yes

complemented

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₂×C₈×C₉

C₂×C₄

yes

characteristic

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₉

—

C₄²×C₉

C₂³

yes

characteristic

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

D₄

yes

—

#1579

—

Sylow

C₂×C₃×Q₁₆

Dic₃

yes

characteristic

C₃×C₄×Q₈

C₂×D₃

yes

—

C₃×#107

Dic₃

yes

characteristic

C₃×#103

Dic₃

yes

—

C₃×C₄×C₈

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

C₂×C₈

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

characteristic

C₈×C₉

—

C₈×C₉

C₂×C₈

yes

—

C₉⋊C₈

—

C₉⋊C₈

—

C₉⋊C₈

—

C₈×Q₈

—

C₄×Q₁₆

D₉

yes

characteristic

#334

—

Q₈⋊C₈

—

#322

—

C₂×C₃×Q₈

C₂×Dic₃

yes

—

C₃×Q₁₆

C₃⋊C₈

yes

—

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×Dic₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×Dic₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂²×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

C₄×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²×C₈

yes

C₄×C₉

yes

C₄×C₉

yes

—

C₂×Q₁₆

Dic₉

yes

characteristic

C₄×Q₈

C₂×D₉

yes

—

#107

Dic₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

Dic₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₉

yes

characteristic

C₃×Q₈

—

C₃×Q₈

C₂×C₃⋊C₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×D₄

yes

characteristic

C₂×C₃×C₄

#237

yes

characteristic

C₂×C₃×C₄

C₂²×Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

C₂×C₃⋊C₈

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

100

C₃⋊C₈

—

101

C₂×C₉

C₈×D₄

yes

characteristic

102

C₂×C₉

#441

yes

characteristic

103

C₂×C₉

#444

yes

characteristic

104

C₂×Q₈

C₂×Dic₉

yes

—

105

Q₁₆

C₉⋊C₈

yes

complemented

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×Dic₉

yes

characteristic

110

C₂×C₈

C₂×Dic₉

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×Dic₉

yes

—

113

C₄²

—

114

C₄²

C₂²×D₉

yes

characteristic

115

C₂²×C₃

D₄×Dic₃

yes

characteristic

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

#918

yes

—

120

C₃×C₄

C₂²×C₃⋊C₈

yes

characteristic

121

C₃×C₄

#948

yes

characteristic

122

C₉

#1579

yes

characteristic, complemented, Sylow

123

Q₈

—

124

Q₈

C₂×C₉⋊C₈

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×D₉

yes

characteristic

128

C₂×C₄

#3699

yes

characteristic

129

C₂×C₄

C₂²×Dic₉

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

C₂×C₉⋊C₈

yes

—

135

C₂×C₃

D₄×C₃⋊C₈

yes

characteristic

136

C₂×C₃

#5184

yes

characteristic

137

C₂×C₃

#5197

yes

characteristic

138

C₂²

D₄×Dic₉

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#63480

yes

—

143

C₄

C₂²×C₉⋊C₈

yes

characteristic

144

C₄

#63510

yes

characteristic

145

C₃

#70902

yes

characteristic

146

C₂

D₄×C₉⋊C₈

yes

characteristic

147

C₂

#10588864

yes

characteristic

148

C₂

#10588877

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator	C₄×C₉	C₂²×C₈
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₄×C₉×Q₁₆	C₂
socle, prime radical	C₂²×C₃	D₄×Dic₃
cosocle	C₄²×C₉	C₂³
hypercenter	C₄×Q₁₆	D₉
hypocenter	C₉	#1579
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₉

commutator

C₄×C₉

C₂²×C₈

Frattini

C₃×C₄²

C₂²×D₃

Fitting

C₄×C₉×Q₁₆

C₂

socle, prime radical

C₂²×C₃

D₄×Dic₃

cosocle

C₄²×C₉

C₂³

hypercenter

C₄×Q₁₆

D₉

hypocenter

C₉

#1579

LCS

G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
23. C₉×Q₁₆	C₈	semidirect
105. Q₁₆	C₉⋊C₈	semidirect
122. C₉	#1579	semidirect

H ≅ G/N

type

23. C₉×Q₁₆

C₈

semidirect

105. Q₁₆

C₉⋊C₈

semidirect

122. C₉

#1579

semidirect

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

subgroup	index
C₄×C₉×Q₁₆	2
Q₈×C₉⋊C₈	2
#10588212	2
#10588134	2
#10588132	2
#70902	3

subgroup

index

C₄×C₉×Q₁₆

Q₈×C₉⋊C₈

Subgroups — #11777306

Conjugacy classes of subgroups

Complements

Maximal subgroups