Group #11771213 — subgroups

Subgroups — #11771213

Order 1152.

210 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 48 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	C₄×#3777	C₂	1	1	yes	maximal, characteristic
3	#10593328	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#157	—	2	1	no	—
5	C₂×C₄×#157	C₂²	1	1	yes	characteristic
6	C₂×#3777	—	2	1	no	—
7	C₂²⋊AGL(1,𝔽₉)	—	2	1	no	—
8	C₄⋊AGL(1,𝔽₉)	C₄	1	2	yes	complemented
9	C₄×AGL(1,𝔽₉)	—	2	1	no	—
10	#63818	—	2	1	no	—
11	#63747	C₄	1	2	yes	—
12	#63732	—	2	1	no	—
13	#63732	C₄	1	2	yes	complemented
14	C₂²×#157	—	2	2	no	—
15	C₂²×#157	C₂×C₄	1	1	yes	characteristic
16	C₂²×#157	D₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₉)	—	2	2	no	—
18	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
19	C₄×#157	—	4	1	no	—
20	C₄×#157	—	4	1	no	—
21	C₄×#157	C₂×C₄	1	1	yes	characteristic
22	C₄×#157	C₂×C₄	1	1	yes	characteristic
23	C₄×#157	D₄	1	1	yes	characteristic
24	C₄×#157	Q₈	1	1	yes	characteristic
25	#3777	—	2	4	no	—
26	C₃×#208	—	4	1	no	—
27	#3712	—	4	1	no	—
28	#3710	—	2	2	no	—
29	#1303	—	9	1	no	maximal, Sylow
30	D₃×M₁₆	—	12	1	no	—
31	C₂²×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×#157	—	2	2	no	—
33	C₂×#157	—	2	4	no	—
34	C₂×#157	—	4	1	no	—
35	C₂×#157	—	4	2	no	—
36	C₂×#157	—	4	2	no	—
37	C₂×#157	C₄²	1	1	yes	characteristic
38	C₂×#157	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#157	C₄⋊C₄	1	1	yes	characteristic
41	AGL(1,𝔽₉)	—	4	2	no	—
42	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
43	C₂×C₃⋊Dic₃	Q₁₆	1	1	yes	characteristic
44	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃⋊D₃	D₈	1	1	yes	characteristic
47	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
48	C₃×C₃⋊C₈	—	4	2	no	—
49	#431	—	9	1	no	—
50	#423	—	9	1	no	—
51	C₄×C₄⋊C₄	—	9	1	no	—
52	C₂×C₄×D₃	—	12	1	no	—
53	C₃×M₁₆	—	12	1	no	—
54	#208	—	4	1	no	—
55	ΓL(1,𝔽₂₅)	—	12	2	no	—
56	D₃×C₈	—	12	2	no	—
57	C₂²×C₃²	#103	1	1	yes	characteristic
58	C₂×C₃⋊D₃	—	2	1	no	—
59	C₂×C₃⋊D₃	—	2	1	no	—
60	C₂×C₃⋊D₃	#95	1	1	yes	characteristic
61	C₂×C₃⋊D₃	#102	1	1	yes	characteristic
62	#157	—	2	1	no	—
63	#157	—	2	1	no	—
64	#157	—	2	1	no	—
65	#157	—	4	4	no	—
66	#157	#104	1	2	yes	—
67	C₃²×C₄	#102	1	1	yes	characteristic
68	C₃²×C₄	#106	1	1	yes	characteristic
69	C₃⋊Dic₃	#103	1	1	yes	characteristic
70	C₃⋊Dic₃	#107	1	1	yes	characteristic
71	C₂×M₁₆	—	18	1	no	—
72	C₂×C₄⋊C₄	—	18	1	no	—
73	C₂×C₄²	—	9	1	no	—
74	C₂×C₄²	—	18	1	no	—
75	C₄⋊C₈	—	9	2	no	—
76	C₄⋊C₈	—	9	2	no	—
77	C₄⋊C₈	—	18	1	no	—
78	C₂²⋊C₈	—	18	1	no	—
79	C₄×C₈	—	18	1	no	—
80	#95	—	9	2	no	—
81	C₂²×D₃	—	12	1	no	—
82	C₂×C₃×C₄	—	4	1	no	—
83	C₂×Dic₃	—	12	1	no	—
84	C₄×D₃	—	12	1	no	—
85	C₄×D₃	—	12	1	no	—
86	C₄×D₃	—	12	1	no	—
87	C₄×D₃	—	12	1	no	—
88	C₃×C₈	—	12	2	no	—
89	C₃⋊C₈	—	4	2	no	—
90	C₂×C₃²	—	2	1	no	—
91	C₂×C₃²	#340	1	1	yes	characteristic
92	C₃⋊D₃	—	2	1	no	—
93	C₃⋊D₃	#339	1	1	yes	characteristic
94	C₃⋊D₃	#344	1	1	yes	characteristic
95	C₂²×C₄	—	9	1	no	—
96	C₂²×C₄	—	9	1	no	—
97	C₂²×C₄	—	9	1	no	—
98	C₂²×C₄	—	18	2	no	—
99	M₁₆	—	12	1	no	—
100	M₁₆	—	36	1	no	—
101	C₂×C₈	—	18	2	no	—
102	C₂×C₈	—	18	2	no	—
103	C₄⋊C₄	—	18	4	no	—
104	C₄²	—	9	1	no	—
105	C₄²	—	9	1	no	—
106	C₄²	—	9	1	no	—
107	C₄²	—	9	1	no	—
108	C₄²	—	36	1	no	—
109	C₄²	—	36	1	no	—
110	C₂²×C₃	—	4	1	no	—
111	C₂×D₃	—	12	1	no	—
112	C₂×D₃	—	12	1	no	—
113	C₂×D₃	—	24	1	no	—
114	C₂×D₃	—	24	1	no	—
115	C₃×C₄	—	4	1	no	—
116	C₃×C₄	—	4	1	no	—
117	Dic₃	—	12	1	no	—
118	Dic₃	—	12	1	no	—
119	C₃²	#1303	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	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	—	9	1	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	18	2	no	—
131	C₂×C₄	—	18	4	no	—
132	C₂×C₄	—	36	1	no	—
133	C₂×C₄	—	36	2	no	—
134	C₂×C₄	—	36	2	no	—
135	C₂×C₄	AΓL(1,𝔽₉)	1	1	yes	characteristic
136	C₈	—	12	2	no	—
137	C₈	—	36	2	no	—
138	C₂×C₃	—	4	1	no	—
139	C₂×C₃	—	8	1	no	—
140	D₃	—	12	1	no	—
141	D₃	—	12	1	no	—
142	D₃	—	24	1	no	—
143	C₂²	—	9	1	no	—
144	C₂²	—	9	1	no	—
145	C₂²	—	18	1	no	—
146	C₂²	—	18	1	no	—
147	C₂²	PSU(3,𝔽₂)⋊C₄	1	1	yes	characteristic
148	C₄	—	9	2	no	—
149	C₄	—	9	1	no	—
150	C₄	—	9	1	no	—
151	C₄	—	18	1	no	—
152	C₄	—	18	1	no	—
153	C₄	—	18	1	no	—
154	C₄	—	36	4	no	—
155	C₄	GO⁺(4,𝔽₂)⋊C₄	1	1	yes	characteristic
156	C₄	AGL(1,𝔽₉)⋊C₄	1	1	yes	characteristic
157	C₃	—	4	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	9	1	no	—
160	C₂	—	9	1	no	—
161	C₂	—	18	1	no	—
162	C₂	#10593166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10596484

C₂

yes

maximal, characteristic

C₄×#3777

C₂

yes

maximal, characteristic

#10593328

C₂

yes

maximal, characteristic

C₂×C₄×#157

—

C₂×C₄×#157

C₂²

yes

characteristic

C₂×#3777

—

C₂²⋊AGL(1,𝔽₉)

—

C₄⋊AGL(1,𝔽₉)

C₄

yes

—

—

C₄

yes

—

#63732

—

#63732

C₄

yes

complemented

C₂²×#157

—

C₂²×#157

C₂×C₄

yes

characteristic

C₂²×#157

D₄

yes

characteristic

C₂×AGL(1,𝔽₉)

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

—

C₂×C₄

yes

characteristic

C₄×#157

C₂×C₄

yes

characteristic

C₄×#157

D₄

yes

characteristic

C₄×#157

Q₈

yes

—

—

—

—

—

D₃×M₁₆

—

C₂²×C₃⋊D₃

C₂²⋊C₄

yes

—

—

—

—

—

C₄²

yes

characteristic

C₂×#157

C₂²⋊C₄

yes

characteristic

C₂×#157

C₄⋊C₄

yes

characteristic

C₂×#157

C₄⋊C₄

yes

characteristic

AGL(1,𝔽₉)

—

C₂×C₃²×C₄

SD₁₆

yes

characteristic

C₂×C₃⋊Dic₃

Q₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₄⋊C₄

yes

characteristic

C₄×C₃⋊D₃

D₈

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₃×C₃⋊C₈

—

#431

—

#423

—

C₄×C₄⋊C₄

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₃²

#103

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

—

—

—

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#340

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

yes

C₃⋊D₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₂²×C₃

—

111

C₂×D₃

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₂×D₃

—

115

C₃×C₄

—

116

C₃×C₄

—

117

Dic₃

—

118

Dic₃

—

119

C₃²

#1303

yes

characteristic, complemented, Sylow

120

C₂³

—

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

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

AΓL(1,𝔽₉)

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂×C₃

—

139

C₂×C₃

—

140

D₃

—

141

D₃

—

142

D₃

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

PSU(3,𝔽₂)⋊C₄

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

GO⁺(4,𝔽₂)⋊C₄

yes

characteristic

156

C₄

AGL(1,𝔽₉)⋊C₄

yes

characteristic

157

C₃

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#10593166

yes

characteristic

	subgroup	G/H
center	C₄	GO⁺(4,𝔽₂)⋊C₄
commutator	C₂×#157	C₄²
Frattini, hypercenter	C₂×C₄	AΓL(1,𝔽₉)
Fitting	C₂×C₃²×C₄	SD₁₆
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	C₂²⋊C₄
cosocle	C₂×C₄×#157	C₂²
hypocenter	C₃²	#1303
LCS	G ▷ C₂×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center

C₄

GO⁺(4,𝔽₂)⋊C₄

commutator

C₂×#157

C₄²

Frattini, hypercenter

C₂×C₄

AΓL(1,𝔽₉)

Fitting

C₂×C₃²×C₄

SD₁₆

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

C₂²⋊C₄

cosocle

C₂×C₄×#157

C₂²

hypocenter

C₃²

#1303

LCS

G ▷ C₂×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
8. C₄⋊AGL(1,𝔽₉)	C₄	semidirect
13. #63732	C₄	semidirect
119. C₃²	#1303	semidirect

H ≅ G/N

type

8. C₄⋊AGL(1,𝔽₉)

C₄

semidirect

13. #63732

C₄

semidirect

119. C₃²

#1303

semidirect

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

subgroup	index
#10596484	2
C₄×#3777	2
#10593328	2
#1303	9

subgroup

index

Subgroups — #11771213

Conjugacy classes of subgroups

Complements

Maximal subgroups