Group #11772481 — subgroups

Subgroups — #11772481

Order 1152.

178 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 49 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	#10593768	C₂	1	1	yes	maximal, characteristic
2	C₂×#63768	C₂	1	1	yes	maximal, characteristic
3	#10593762	C₂	1	1	yes	maximal, characteristic
4	#64286	—	2	1	no	—
5	C₂×C₄⋊#157	—	2	1	no	—
6	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	#63773	C₄	1	2	yes	complemented
8	#63769	C₄	1	2	yes	complemented
9	#63768	—	2	1	no	—
10	#63768	—	2	1	no	—
11	#63768	C₄	1	2	yes	complemented
12	#63765	—	2	1	no	—
13	#63764	—	2	1	no	—
14	C₂²×#157	—	4	1	no	—
15	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
16	C₂²⋊#157	—	4	1	no	—
17	C₄⋊#157	—	2	2	no	—
18	C₄⋊#157	—	2	2	no	—
19	C₄⋊#157	—	4	1	no	—
20	C₄×#157	—	4	1	no	—
21	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
23	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
26	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
27	#3708	—	2	2	no	—
28	#1386	—	9	1	no	maximal, Sylow
29	C₂×D₃×C₈	—	12	1	no	—
30	C₂²×C₃⋊D₃	D₈	1	1	yes	characteristic
31	C₂×#157	—	4	2	no	—
32	C₂×#157	—	4	2	no	—
33	C₂×#157	—	8	2	no	—
34	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	Q₁₆	1	1	yes	characteristic
36	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
39	C₄×C₃⋊D₃	SD₁₆	1	1	yes	characteristic
40	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₃⋊(C₃⋊C₈)	C₄²	1	1	yes	characteristic
43	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×M₃₂	—	9	1	no	—
45	#427	—	9	1	no	—
46	C₂×#106	—	9	1	no	—
47	C₂×C₄×D₃	—	12	1	no	—
48	C₂×C₃×C₈	—	4	1	no	—
49	C₂×C₃⋊C₈	—	12	1	no	—
50	D₃×C₈	—	12	1	no	—
51	D₃×C₈	—	12	1	no	—
52	D₃×C₈	—	12	1	no	—
53	D₃×C₈	—	12	1	no	—
54	C₂²×C₃²	#107	1	1	yes	characteristic
55	C₂×C₃⋊D₃	—	2	2	no	—
56	C₂×C₃⋊D₃	#102	1	1	yes	characteristic
57	C₂×C₃⋊D₃	#102	1	1	yes	characteristic
58	#157	—	8	2	no	—
59	#157	—	8	2	no	—
60	C₃²×C₄	#95	1	1	yes	characteristic
61	C₃²×C₄	#106	1	1	yes	characteristic
62	C₃⋊Dic₃	#103	1	1	yes	characteristic
63	C₃⋊Dic₃	#103	1	1	yes	characteristic
64	C₂²×C₈	—	9	1	no	—
65	#117	—	18	1	no	—
66	C₂×C₄⋊C₄	—	18	1	no	—
67	M₃₂	—	9	2	no	—
68	M₃₂	—	18	1	no	—
69	C₂×C₁₆	—	18	1	no	—
70	#107	—	9	2	no	—
71	#106	—	9	2	no	—
72	#106	—	18	1	no	—
73	#106	—	18	1	no	—
74	C₂²×D₃	—	12	1	no	—
75	C₂×C₃×C₄	—	4	1	no	—
76	C₂×Dic₃	—	12	1	no	—
77	C₄×D₃	—	12	1	no	—
78	C₄×D₃	—	12	1	no	—
79	C₄×D₃	—	12	1	no	—
80	C₄×D₃	—	12	1	no	—
81	C₃×C₈	—	4	1	no	—
82	C₃×C₈	—	4	1	no	—
83	C₃⋊C₈	—	12	1	no	—
84	C₃⋊C₈	—	12	1	no	—
85	C₂×C₃²	—	2	1	no	—
86	C₂×C₃²	#340	1	1	yes	characteristic
87	C₃⋊D₃	—	2	1	no	—
88	C₃⋊D₃	#360	1	2	yes	—
89	C₂²×C₄	—	9	1	no	—
90	C₂²×C₄	—	36	1	no	—
91	C₂×C₈	—	9	1	no	—
92	C₂×C₈	—	9	1	no	—
93	C₂×C₈	—	9	1	no	—
94	C₂×C₈	—	9	1	no	—
95	C₂×C₈	—	9	1	no	—
96	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
97	C₄⋊C₄	—	18	2	no	—
98	C₄⋊C₄	—	18	2	no	—
99	C₄⋊C₄	—	36	1	no	—
100	C₂²⋊C₄	—	36	1	no	—
101	C₄²	—	36	1	no	—
102	C₁₆	—	18	2	no	—
103	C₂²×C₃	—	4	1	no	—
104	C₂×D₃	—	12	1	no	—
105	C₂×D₃	—	12	1	no	—
106	C₂×D₃	—	12	4	no	—
107	C₃×C₄	—	4	1	no	—
108	C₃×C₄	—	4	1	no	—
109	Dic₃	—	12	1	no	—
110	Dic₃	—	12	1	no	—
111	C₃²	#1386	1	1	yes	characteristic, complemented, Sylow
112	C₂³	—	9	1	no	—
113	C₂×C₄	—	9	1	no	—
114	C₂×C₄	—	9	1	no	—
115	C₂×C₄	—	9	1	no	—
116	C₂×C₄	—	9	1	no	—
117	C₂×C₄	—	9	1	no	—
118	C₂×C₄	—	36	2	no	—
119	C₂×C₄	—	36	2	no	—
120	C₂×C₄	—	72	2	no	—
121	C₂×C₄	#3777	1	1	yes	characteristic
122	C₈	—	9	1	no	—
123	C₈	—	9	1	no	—
124	C₈	#3777	1	1	yes	characteristic
125	C₈	#3777	1	1	yes	characteristic
126	C₂×C₃	—	4	2	no	—
127	C₂×C₃	—	4	1	no	—
128	D₃	—	12	2	no	—
129	D₃	—	12	2	no	—
130	C₂²	—	9	1	no	—
131	C₂²	—	9	1	no	—
132	C₂²	—	18	2	no	—
133	C₂²	#63746	1	1	yes	characteristic
134	C₄	—	9	1	no	—
135	C₄	—	9	1	no	—
136	C₄	—	72	2	no	—
137	C₄	—	72	2	no	—
138	C₄	#63745	1	1	yes	characteristic
139	C₄	#63747	1	1	yes	characteristic
140	C₃	—	4	1	no	—
141	C₂	—	2	1	no	—
142	C₂	—	9	2	no	—
143	C₂	—	18	1	no	—
144	C₂	#10590093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593768

C₂

yes

maximal, characteristic

C₂×#63768

C₂

yes

maximal, characteristic

#10593762

C₂

yes

maximal, characteristic

#64286

—

C₂×C₄⋊#157

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63773

C₄

yes

complemented

#63769

C₄

yes

—

—

C₄

yes

complemented

#63765

—

#63764

—

C₂²×#157

—

C₂×C₄×C₃⋊D₃

D₄

yes

characteristic

C₂²⋊#157

—

—

—

—

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

characteristic

C₈×C₃⋊D₃

D₄

yes

characteristic

C₈×C₃⋊D₃

D₄

yes

—

—

C₂×D₃×C₈

—

C₂²×C₃⋊D₃

D₈

yes

—

—

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

Q₁₆

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₄×C₃⋊D₃

SD₁₆

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₄²

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₂×M₃₂

—

#427

—

C₂×#106

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

#107

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

yes

C₂×C₃⋊D₃

yes

—

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

C₃⋊Dic₃

yes

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

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

—

C₂×C₃²

#340

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

#360

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₁₆

—

103

C₂²×C₃

—

104

C₂×D₃

—

105

C₂×D₃

—

106

C₂×D₃

—

107

C₃×C₄

—

108

C₃×C₄

—

109

Dic₃

—

110

Dic₃

—

111

C₃²

#1386

yes

characteristic, complemented, Sylow

112

C₂³

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#3777

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#3777

yes

characteristic

125

C₈

#3777

yes

characteristic

126

C₂×C₃

—

127

C₂×C₃

—

128

D₃

—

129

D₃

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#63746

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#63745

yes

characteristic

139

C₄

#63747

yes

characteristic

140

C₃

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#10590093

yes

characteristic

	subgroup	G/H
center	C₂	#10590093
commutator	C₃⋊(C₃⋊C₈)	C₄²
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂×C₃²	#340
prime radical	C₂²×C₃⋊D₃	D₈
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1386
LCS	G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂

#10590093

commutator

C₃⋊(C₃⋊C₈)

C₄²

Frattini, hypercenter

C₂×C₈

PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂×C₃²

#340

prime radical

C₂²×C₃⋊D₃

D₈

cosocle

C₂×C₈×C₃⋊D₃

C₂²

hypocenter

C₃²

#1386

LCS

G ▷ C₃⋊(C₃⋊C₈) ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
7. #63773	C₄	semidirect
8. #63769	C₄	semidirect
11. #63768	C₄	semidirect
111. C₃²	#1386	semidirect

H ≅ G/N

type

7. #63773

C₄

semidirect

8. #63769

C₄

semidirect

11. #63768

C₄

semidirect

111. C₃²

#1386

semidirect

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

subgroup	index
#10593768	2
C₂×#63768	2
#10593762	2
#1386	9

subgroup

index

Subgroups — #11772481

Conjugacy classes of subgroups

Complements

Maximal subgroups