Group #11776998 — subgroups

Subgroups — #11776998

Order 1152.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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	#10596484	C₂	1	1	yes	maximal, characteristic
2	C₂×#63744	C₂	1	1	yes	maximal, characteristic
3	#10593328	C₂	1	1	yes	maximal, characteristic
4	AGL(1,𝔽₉)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×#157	C₂²	1	1	yes	characteristic
6	C₂×#3787	—	2	1	no	—
7	C₂²⋊AGL(1,𝔽₉)	C₄	1	2	yes	—
8	C₄⋊AGL(1,𝔽₉)	C₄	1	2	yes	—
9	C₄×AGL(1,𝔽₉)	C₂²	1	2	yes	—
10	#63818	—	2	1	no	—
11	#63744	—	2	1	no	—
12	#63744	C₂²	1	2	yes	—
13	#63732	—	2	1	no	—
14	#63732	C₂²	1	2	yes	—
15	C₂²×#157	C₂×C₄	1	1	yes	characteristic
16	C₂²×#157	Q₈	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₉)	C₂×C₄	1	4	yes	—
18	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
19	C₂×#622	—	4	1	no	—
20	C₄×#157	C₂×C₄	1	1	yes	characteristic
21	C₄×#157	D₄	1	1	yes	characteristic
22	C₄×#157	Q₈	1	1	yes	characteristic
23	C₄×#157	C₂³	1	1	yes	characteristic
24	#3787	—	2	2	no	—
25	#3787	—	4	1	no	—
26	C₃×#208	—	4	1	no	—
27	#3712	—	4	1	no	—
28	#3710	—	2	2	no	—
29	#1570	—	9	1	no	maximal, Sylow
30	D₃×M₁₆	—	12	1	no	—
31	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×#157	—	2	2	no	—
33	C₂×#157	—	4	1	no	—
34	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
36	C₂×#157	C₂²×C₄	1	1	yes	characteristic
37	C₂×#157	C₂×Q₈	1	1	yes	characteristic
38	AGL(1,𝔽₉)	—	2	4	no	—
39	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
40	C₂×C₃⋊Dic₃	SD₁₆	1	1	yes	characteristic
41	C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
43	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
44	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
45	#622	—	4	2	no	—
46	C₃×C₃⋊C₈	—	4	2	no	—
47	#431	—	9	1	no	—
48	#423	—	9	1	no	—
49	C₂×C₄⋊C₈	—	9	1	no	—
50	#334	—	9	4	no	—
51	C₂×C₄×D₃	—	12	1	no	—
52	C₃×M₁₆	—	12	1	no	—
53	#208	—	4	1	no	—
54	ΓL(1,𝔽₂₅)	—	12	2	no	—
55	D₃×C₈	—	12	2	no	—
56	C₂²×C₃²	#106	1	1	yes	characteristic
57	C₂×C₃⋊D₃	—	2	1	no	—
58	C₂×C₃⋊D₃	—	2	1	no	—
59	C₂×C₃⋊D₃	#106	1	1	yes	characteristic
60	C₂×C₃⋊D₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	#157	—	2	1	no	—
62	#157	—	2	1	no	—
63	#157	—	2	1	no	—
64	#157	#131	1	2	yes	—
65	C₃²×C₄	#106	1	1	yes	characteristic
66	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
67	C₃⋊Dic₃	#106	1	1	yes	characteristic
68	C₃⋊Dic₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×M₁₆	—	18	1	no	—
70	C₂²×C₈	—	18	1	no	—
71	C₂×C₄²	—	9	1	no	—
72	C₄⋊C₈	—	9	2	no	—
73	C₄⋊C₈	—	9	2	no	—
74	C₄⋊C₈	—	9	2	no	—
75	C₄⋊C₈	—	18	1	no	—
76	C₄⋊C₈	—	18	1	no	—
77	C₂²⋊C₈	—	9	2	no	—
78	C₄×C₈	—	9	2	no	—
79	C₂²×D₃	—	12	1	no	—
80	C₂×C₃×C₄	—	4	1	no	—
81	C₂×Dic₃	—	12	1	no	—
82	C₄×D₃	—	12	1	no	—
83	C₄×D₃	—	12	1	no	—
84	C₄×D₃	—	12	1	no	—
85	C₄×D₃	—	12	1	no	—
86	C₃×C₈	—	12	2	no	—
87	C₃⋊C₈	—	4	2	no	—
88	C₂×C₃²	—	2	1	no	—
89	C₂×C₃²	C₂×#106	1	1	yes	characteristic
90	C₃⋊D₃	—	2	1	no	—
91	C₃⋊D₃	#424	1	1	yes	characteristic
92	C₃⋊D₃	#430	1	1	yes	characteristic
93	C₂²×C₄	—	9	1	no	—
94	C₂²×C₄	—	9	1	no	—
95	C₂²×C₄	—	9	1	no	—
96	M₁₆	—	12	1	no	—
97	M₁₆	—	36	1	no	—
98	C₂×C₈	—	9	4	no	—
99	C₂×C₈	—	18	2	no	—
100	C₂×C₈	—	18	2	no	—
101	C₂×C₈	—	36	1	no	—
102	C₂×C₈	—	36	1	no	—
103	C₄²	—	9	1	no	—
104	C₄²	—	9	1	no	—
105	C₄²	—	9	1	no	—
106	C₄²	—	9	1	no	—
107	C₂²×C₃	—	4	1	no	—
108	C₂×D₃	—	12	1	no	—
109	C₂×D₃	—	12	1	no	—
110	C₂×D₃	—	24	1	no	—
111	C₂×D₃	—	24	1	no	—
112	C₃×C₄	—	4	1	no	—
113	C₃×C₄	—	4	1	no	—
114	Dic₃	—	12	1	no	—
115	Dic₃	—	12	1	no	—
116	C₃²	#1570	1	1	yes	characteristic, complemented, Sylow
117	C₂³	—	9	1	no	—
118	C₂×C₄	—	9	1	no	—
119	C₂×C₄	—	9	1	no	—
120	C₂×C₄	—	9	1	no	—
121	C₂×C₄	—	9	1	no	—
122	C₂×C₄	—	9	1	no	—
123	C₂×C₄	—	9	1	no	—
124	C₂×C₄	—	9	1	no	—
125	C₂×C₄	—	9	1	no	—
126	C₂×C₄	—	9	1	no	—
127	C₂×C₄	—	18	2	no	—
128	C₂×C₄	—	36	1	no	—
129	C₂×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
130	C₈	—	12	2	no	—
131	C₈	—	18	4	no	—
132	C₈	—	36	2	no	—
133	C₂×C₃	—	4	1	no	—
134	C₂×C₃	—	8	1	no	—
135	D₃	—	12	1	no	—
136	D₃	—	12	1	no	—
137	D₃	—	24	1	no	—
138	C₂²	—	9	1	no	—
139	C₂²	—	9	1	no	—
140	C₂²	—	18	1	no	—
141	C₂²	—	18	1	no	—
142	C₂²	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
143	C₄	—	9	2	no	—
144	C₄	—	9	1	no	—
145	C₄	—	9	1	no	—
146	C₄	—	18	1	no	—
147	C₄	—	18	1	no	—
148	C₄	—	18	1	no	—
149	C₄	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
150	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
151	C₃	—	4	1	no	—
152	C₂	—	2	1	no	—
153	C₂	—	9	1	no	—
154	C₂	—	9	1	no	—
155	C₂	—	18	1	no	—
156	C₂	C₂×AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10596484

C₂

yes

maximal, characteristic

C₂×#63744

C₂

yes

maximal, characteristic

#10593328

C₂

yes

maximal, characteristic

AGL(1,𝔽₉)⋊C₈

C₂

yes

maximal, complemented

C₂×C₄×#157

C₂²

yes

characteristic

C₂×#3787

—

C₂²⋊AGL(1,𝔽₉)

C₄

yes

—

C₄⋊AGL(1,𝔽₉)

C₄

yes

—

C₄×AGL(1,𝔽₉)

C₂²

yes

—

#63818

—

#63744

—

#63744

C₂²

yes

—

#63732

—

#63732

C₂²

yes

—

C₂²×#157

C₂×C₄

yes

characteristic

C₂²×#157

Q₈

yes

characteristic

C₂×AGL(1,𝔽₉)

C₂×C₄

yes

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂×#622

—

C₄×#157

C₂×C₄

yes

characteristic

C₄×#157

D₄

yes

characteristic

C₄×#157

Q₈

yes

characteristic

C₄×#157

C₂³

yes

—

—

—

—

—

—

D₃×M₁₆

—

C₂²×C₃⋊D₃

C₄⋊C₄

yes

—

—

C₄⋊C₄

yes

characteristic

C₂×#157

C₄⋊C₄

yes

characteristic

C₂×#157

C₂²×C₄

yes

characteristic

C₂×#157

C₂×Q₈

yes

characteristic

AGL(1,𝔽₉)

—

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₄⋊C₄

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

#622

—

C₃×C₃⋊C₈

—

#431

—

#423

—

C₂×C₄⋊C₈

—

#334

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

#106

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#106

yes

characteristic

C₂×C₃⋊D₃

C₂×C₄⋊C₄

yes

—

—

—

yes

—

C₃²×C₄

#106

yes

characteristic

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

#106

yes

characteristic

C₃⋊Dic₃

C₂×SD₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×#106

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

yes

C₃⋊D₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₂²×C₃

—

108

C₂×D₃

—

109

C₂×D₃

—

110

C₂×D₃

—

111

C₂×D₃

—

112

C₃×C₄

—

113

C₃×C₄

—

114

Dic₃

—

115

Dic₃

—

116

C₃²

#1570

yes

characteristic, complemented, Sylow

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

AΓL(1,𝔽₉)

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂×C₃

—

134

C₂×C₃

—

135

D₃

—

136

D₃

—

137

D₃

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

AGL(1,𝔽₉)⋊C₄

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

AGL(1,𝔽₉)⋊C₄

yes

characteristic

150

C₄

C₂×AΓL(1,𝔽₉)

yes

characteristic

151

C₃

—

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

C₂×AGL(1,𝔽₉)⋊C₄

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×AΓL(1,𝔽₉)
commutator	C₂×#157	C₂²×C₄
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	C₂×#106
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₄×#157	C₂³
hypercenter	C₂×C₄	AΓL(1,𝔽₉)
hypocenter	C₃²	#1570
LCS	G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

C₄

C₂²×C₄

C₂×C₃²×C₄

SD₁₆

socle

C₂×C₃²

C₂×#106

prime radical

C₂²×C₃⋊D₃

C₄⋊C₄

C₂³

C₂×C₄

C₃²

G ▷ C₂×#157 ▷ C₃⋊D₃ ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
4. AGL(1,𝔽₉)⋊C₈	C₂	semidirect
116. C₃²	#1570	semidirect

H ≅ G/N

type

4. AGL(1,𝔽₉)⋊C₈

C₂

semidirect

116. C₃²

#1570

semidirect

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

subgroup	index
#10596484	2
C₂×#63744	2
#10593328	2
AGL(1,𝔽₉)⋊C₈	2
#1570	9

subgroup

index

Subgroups — #11776998

Conjugacy classes of subgroups

Complements

Maximal subgroups