Group #11781948 — subgroups

Subgroups — #11781948

Order 1152.

200 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 38 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	#10596409	C₂	1	2	yes	maximal
2	#10593536	C₂	1	1	yes	maximal, characteristic, complemented
3	#10593283	C₂	1	1	yes	maximal, characteristic
4	#10593252	C₂	1	1	yes	maximal, characteristic, complemented
5	#10593157	C₂	1	2	yes	maximal, complemented
6	C₂×AΓL(1,𝔽₉)	—	2	2	no	—
7	C₄⋊PSU(3,𝔽₂)	C₂²	1	2	yes	—
8	C₄×GO⁺(4,𝔽₂)	C₂²	1	1	yes	characteristic
9	C₄⋊AGL(1,𝔽₉)	C₂²	1	2	yes	—
10	PSU(3,𝔽₂)⋊C₄	—	2	2	no	—
11	#63791	—	2	1	no	—
12	#63769	—	2	1	no	—
13	C₈×#157	C₂²	1	1	yes	characteristic
14	#63746	—	2	1	no	—
15	#63732	C₂²	1	1	yes	characteristic
16	D₃²⋊C₈	—	2	1	no	—
17	C₂×PSU(3,𝔽₂)	—	2	2	no	—
18	C₂×GO⁺(4,𝔽₂)	D₄	1	1	yes	characteristic
19	C₂×AGL(1,𝔽₉)	—	2	2	no	—
20	AΓL(1,𝔽₉)	—	4	4	no	—
21	C₄×D₃²	—	2	1	no	—
22	C₄⋊#157	—	2	2	no	—
23	C₄×#157	C₂³	1	1	yes	characteristic
24	#3787	D₄	1	1	yes	characteristic
25	#3777	—	2	2	no	—
26	#3773	D₄	1	1	yes	characteristic
27	D₃²⋊C₄	—	2	1	no	—
28	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
29	C₃×ΓL(1,𝔽₂₅)	—	4	1	no	—
30	#3711	—	4	1	no	—
31	#3710	—	2	1	no	—
32	#1693	—	9	1	no	maximal, Sylow
33	D₃×M₁₆	—	12	1	no	—
34	C₂×D₃²	—	2	1	no	—
35	C₂×#157	—	4	2	no	—
36	C₂×#157	C₂×D₄	1	1	yes	characteristic
37	C₂×#157	#41	1	1	yes	characteristic
38	PSU(3,𝔽₂)	—	4	2	no	—
39	PSU(3,𝔽₂)	—	4	2	no	—
40	GO⁺(4,𝔽₂)	—	2	2	no	—
41	AGL(1,𝔽₉)	—	4	2	no	—
42	C₄×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄×D₃	—	4	1	no	—
44	#624	—	2	1	no	—
45	D₃×Dic₃	—	4	1	no	—
46	#622	—	2	1	no	—
47	C₃²×C₈	SD₁₆	1	1	yes	characteristic, complemented
48	C₃⋊(C₃⋊C₈)	SD₁₆	1	1	yes	characteristic, complemented
49	C₃×C₃⋊C₈	—	4	1	no	—
50	#500	—	9	1	no	—
51	#461	—	9	2	no	—
52	#436	—	9	1	no	—
53	#334	—	9	1	no	—
54	#332	—	9	2	no	—
55	C₂×C₄×D₃	—	12	1	no	—
56	C₃×M₁₆	—	12	1	no	—
57	#208	—	12	1	no	—
58	ΓL(1,𝔽₂₅)	—	4	1	no	—
59	ΓL(1,𝔽₂₅)	—	12	1	no	—
60	D₃×C₈	—	12	1	no	—
61	D₃×C₈	—	12	1	no	—
62	C₂×C₃⋊D₃	#121	1	1	yes	characteristic
63	C₂×C₃×D₃	—	4	1	no	—
64	D₃²	—	4	1	no	—
65	D₃²	—	4	1	no	—
66	#157	—	2	1	no	—
67	#157	—	8	2	no	—
68	#157	AGL(1,ℤ/8)	1	2	yes	—
69	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
70	C₃⋊Dic₃	#137	1	1	yes	characteristic
71	C₃×Dic₃	—	4	1	no	—
72	C₂×SD₁₆	—	18	2	no	—
73	C₂×M₁₆	—	18	1	no	—
74	C₄⋊Q₈	—	9	2	no	—
75	C₄×D₄	—	9	1	no	—
76	#107	—	18	1	no	—
77	#107	—	18	1	no	—
78	C₄⋊C₈	—	9	1	no	—
79	C₄⋊C₈	—	9	2	no	—
80	#103	—	18	2	no	—
81	C₂²⋊C₈	—	18	1	no	—
82	C₄×C₈	—	9	1	no	—
83	C₂²×D₃	—	12	1	no	—
84	C₂×C₃×C₄	—	12	1	no	—
85	C₂×Dic₃	—	12	1	no	—
86	C₄×D₃	—	4	1	no	—
87	C₄×D₃	—	12	1	no	—
88	C₄×D₃	—	12	1	no	—
89	C₄×D₃	—	12	1	no	—
90	C₃×C₈	—	4	1	no	—
91	C₃×C₈	—	12	1	no	—
92	C₃⋊C₈	—	4	1	no	—
93	C₃⋊C₈	—	12	1	no	—
94	C₂×C₃²	#461	1	1	yes	characteristic
95	C₃⋊D₃	#469	1	1	yes	characteristic
96	C₃⋊D₃	#473	1	1	yes	characteristic
97	C₃×D₃	—	8	1	no	—
98	C₂×Q₈	—	18	2	no	—
99	C₂×D₄	—	9	1	no	—
100	C₂²×C₄	—	18	1	no	—
101	SD₁₆	—	36	4	no	—
102	M₁₆	—	12	1	no	—
103	M₁₆	—	36	1	no	—
104	C₂×C₈	—	9	1	no	—
105	C₂×C₈	—	9	1	no	—
106	C₂×C₈	—	18	1	no	—
107	C₂×C₈	—	18	2	no	—
108	C₄⋊C₄	—	9	1	no	—
109	C₄⋊C₄	—	18	2	no	—
110	C₄⋊C₄	—	18	2	no	—
111	C₂²⋊C₄	—	18	1	no	—
112	C₄²	—	9	1	no	—
113	C₂²×C₃	—	12	1	no	—
114	C₂×D₃	—	4	1	no	—
115	C₂×D₃	—	12	1	no	—
116	C₂×D₃	—	24	1	no	—
117	C₂×D₃	—	24	1	no	—
118	C₃×C₄	—	4	1	no	—
119	C₃×C₄	—	12	1	no	—
120	Dic₃	—	4	1	no	—
121	Dic₃	—	12	1	no	—
122	C₃²	#1693	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	18	1	no	—
124	Q₈	—	36	2	no	—
125	Q₈	—	36	2	no	—
126	D₄	—	18	2	no	—
127	C₂×C₄	—	9	1	no	—
128	C₂×C₄	—	9	1	no	—
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	12	1	no	—
131	C₂×C₄	—	18	1	no	—
132	C₂×C₄	—	36	1	no	—
133	C₂×C₄	—	36	2	no	—
134	C₈	—	9	1	no	—
135	C₈	—	12	1	no	—
136	C₈	—	18	1	no	—
137	C₈	—	36	2	no	—
138	C₈	AΓL(1,𝔽₉)	1	1	yes	characteristic, complemented
139	C₂×C₃	—	4	1	no	—
140	C₂×C₃	—	24	1	no	—
141	D₃	—	8	1	no	—
142	D₃	—	12	1	no	—
143	D₃	—	12	1	no	—
144	C₂²	—	9	1	no	—
145	C₂²	—	12	1	no	—
146	C₂²	—	36	1	no	—
147	C₂²	—	36	1	no	—
148	C₄	—	9	2	no	—
149	C₄	—	9	1	no	—
150	C₄	—	12	1	no	—
151	C₄	—	18	1	no	—
152	C₄	—	72	2	no	—
153	C₄	C₂×AΓL(1,𝔽₉)	1	1	yes	characteristic
154	C₃	—	4	1	no	—
155	C₂	—	9	1	no	—
156	C₂	—	9	1	no	—
157	C₂	—	24	1	no	—
158	C₂	#10596409	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10596409

C₂

yes

maximal

#10593536

C₂

yes

maximal, characteristic, complemented

#10593283

C₂

yes

maximal, characteristic

#10593252

C₂

yes

maximal, characteristic, complemented

#10593157

C₂

yes

maximal, complemented

C₂×AΓL(1,𝔽₉)

—

C₄⋊PSU(3,𝔽₂)

C₂²

yes

—

C₄×GO⁺(4,𝔽₂)

C₂²

yes

characteristic

C₄⋊AGL(1,𝔽₉)

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#63746

—

#63732

C₂²

yes

characteristic

D₃²⋊C₈

—

C₂×PSU(3,𝔽₂)

—

C₂×GO⁺(4,𝔽₂)

D₄

yes

characteristic

C₂×AGL(1,𝔽₉)

—

AΓL(1,𝔽₉)

—

C₄×D₃²

—

C₄⋊#157

—

C₄×#157

C₂³

yes

characteristic

#3787

D₄

yes

characteristic

#3777

—

#3773

D₄

yes

characteristic

D₃²⋊C₄

—

C₈×C₃⋊D₃

D₄

yes

characteristic

C₃×ΓL(1,𝔽₂₅)

—

—

—

—

D₃×M₁₆

—

C₂×D₃²

—

C₂×#157

—

C₂×#157

C₂×D₄

yes

yes

—

—

GO⁺(4,𝔽₂)

—

AGL(1,𝔽₉)

—

C₄×C₃⋊D₃

C₂×D₄

yes

characteristic

C₃×C₄×D₃

—

#624

—

D₃×Dic₃

—

#622

—

C₃²×C₈

SD₁₆

yes

characteristic, complemented

C₃⋊(C₃⋊C₈)

SD₁₆

yes

characteristic, complemented

C₃×C₃⋊C₈

—

—

—

—

—

—

C₂×C₄×D₃

—

C₃×M₁₆

—

#208

—

ΓL(1,𝔽₂₅)

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

D₃×C₈

—

C₂×C₃⋊D₃

#121

yes

characteristic

C₂×C₃×D₃

—

D₃²

—

D₃²

—

—

—

yes

—

C₃²×C₄

C₂×SD₁₆

yes

characteristic

C₃⋊Dic₃

#137

yes

characteristic

C₃×Dic₃

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

yes

C₃⋊D₃

yes

C₃⋊D₃

yes

C₃×D₃

—

C₂×Q₈

—

C₂×D₄

—

100

C₂²×C₄

—

101

SD₁₆

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₂²×C₃

—

114

C₂×D₃

—

115

C₂×D₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₃×C₄

—

119

C₃×C₄

—

120

Dic₃

—

121

Dic₃

—

122

C₃²

#1693

yes

characteristic, complemented, Sylow

123

C₂³

—

124

Q₈

—

125

Q₈

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

AΓL(1,𝔽₉)

yes

characteristic, complemented

139

C₂×C₃

—

140

C₂×C₃

—

141

D₃

—

142

D₃

—

143

D₃

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₂×AΓL(1,𝔽₉)

yes

characteristic

154

C₃

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

#10596409

yes

characteristic

	subgroup	G/H
center	C₂	#10596409
commutator, cosocle	C₄×#157	C₂³
Frattini	C₄	C₂×AΓL(1,𝔽₉)
Fitting	C₃²×C₈	SD₁₆
socle	C₂×C₃²	#461
prime radical	C₂×GO⁺(4,𝔽₂)	D₄
hypercenter	C₈	AΓL(1,𝔽₉)
hypocenter	C₃²	#1693
LCS	G ▷ C₄×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

C₂

C₂³

C₄

C₃²×C₈

SD₁₆

socle

C₂×C₃²

#461

prime radical

C₂×GO⁺(4,𝔽₂)

D₄

C₈

C₃²

G ▷ C₄×#157 ▷ C₂×C₃⋊D₃ ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
2. #10593536	C₂	semidirect
4. #10593252	C₂	semidirect
5. #10593157	C₂	semidirect
47. C₃²×C₈	SD₁₆	semidirect
48. C₃⋊(C₃⋊C₈)	SD₁₆	semidirect
122. C₃²	#1693	semidirect
138. C₈	AΓL(1,𝔽₉)	semidirect

H ≅ G/N

type

2. #10593536

C₂

semidirect

4. #10593252

C₂

semidirect

5. #10593157

C₂

semidirect

47. C₃²×C₈

SD₁₆

semidirect

48. C₃⋊(C₃⋊C₈)

SD₁₆

semidirect

122. C₃²

#1693

semidirect

138. C₈

AΓL(1,𝔽₉)

semidirect

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

subgroup	index
#10596409	2
#10593536	2
#10593283	2
#10593252	2
#10593157	2
#1693	9

subgroup

index

Subgroups — #11781948

Conjugacy classes of subgroups

Complements

Maximal subgroups