Group #11772364 — subgroups

Subgroups — #11772364

Order 1152.

174 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 53 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	#10593765	C₂	1	1	yes	maximal, characteristic
2	#10593762	C₂	1	1	yes	maximal, characteristic
3	C₂×#63764	C₂	1	1	yes	maximal, characteristic
4	#64286	C₄	1	1	yes	characteristic
5	#64283	C₄	1	1	yes	characteristic
6	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	#63773	—	2	1	no	—
8	C₂×#3708	—	2	1	no	—
9	#63767	—	2	1	no	—
10	C₈×#157	—	2	1	no	—
11	#63765	C₄	1	2	yes	complemented
12	#63764	—	2	1	no	—
13	#63764	—	2	1	no	—
14	#63764	C₄	1	2	yes	complemented
15	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊#157	—	2	1	no	—
17	#3792	C₈	1	2	yes	—
18	C₄⋊#157	C₈	1	2	yes	—
19	C₄×#157	—	2	1	no	—
20	#3788	—	2	1	no	—
21	#3787	—	2	1	no	—
22	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
24	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
26	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
27	C₈×C₃⋊D₃	D₄	1	1	yes	characteristic
28	#3708	—	2	2	no	—
29	#3708	—	2	2	no	—
30	#1379	—	9	1	no	maximal, Sylow
31	C₂×D₃×C₈	—	12	1	no	—
32	C₂²×C₃⋊D₃	M₁₆	1	1	yes	characteristic
33	C₂×#157	—	2	2	no	—
34	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
36	C₄×C₃⋊D₃	C₄²	1	1	yes	characteristic
37	C₄×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
39	C₄×C₃⋊D₃	M₁₆	1	1	yes	characteristic
40	#622	—	2	2	no	—
41	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
44	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×M₃₂	—	9	1	no	—
46	C₂²×C₁₆	—	9	1	no	—
47	#405	—	9	1	no	—
48	C₂×C₄×D₃	—	12	1	no	—
49	C₂×C₃×C₈	—	4	1	no	—
50	C₂×C₃⋊C₈	—	12	1	no	—
51	D₃×C₈	—	12	1	no	—
52	D₃×C₈	—	12	1	no	—
53	D₃×C₈	—	12	1	no	—
54	D₃×C₈	—	12	1	no	—
55	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₃⋊D₃	—	2	2	no	—
57	C₂×C₃⋊D₃	#97	1	1	yes	characteristic
58	C₂×C₃⋊D₃	C₂²⋊C₈	1	1	yes	characteristic
59	#157	—	4	2	no	—
60	C₃²×C₄	#95	1	1	yes	characteristic
61	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
62	C₃⋊Dic₃	C₄×C₈	1	1	yes	characteristic
63	C₃⋊Dic₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×M₁₆	—	9	1	no	—
65	C₂²×C₈	—	9	1	no	—
66	#117	—	9	1	no	—
67	M₃₂	—	9	2	no	—
68	M₃₂	—	18	1	no	—
69	C₂×C₁₆	—	9	2	no	—
70	C₂×C₁₆	—	18	1	no	—
71	C₂×C₁₆	—	18	1	no	—
72	C₂×C₁₆	—	18	1	no	—
73	#97	—	18	1	no	—
74	C₄×C₈	—	18	1	no	—
75	C₂²×D₃	—	12	1	no	—
76	C₂×C₃×C₄	—	4	1	no	—
77	C₂×Dic₃	—	12	1	no	—
78	C₄×D₃	—	12	1	no	—
79	C₄×D₃	—	12	1	no	—
80	C₄×D₃	—	12	1	no	—
81	C₄×D₃	—	12	1	no	—
82	C₃×C₈	—	4	1	no	—
83	C₃×C₈	—	4	1	no	—
84	C₃⋊C₈	—	12	1	no	—
85	C₃⋊C₈	—	12	1	no	—
86	C₂×C₃²	—	2	1	no	—
87	C₂×C₃²	#336	1	1	yes	characteristic
88	C₃⋊D₃	—	2	1	no	—
89	C₃⋊D₃	#350	1	2	yes	—
90	C₂²×C₄	—	9	1	no	—
91	M₁₆	—	9	2	no	—
92	M₁₆	—	18	1	no	—
93	C₂×C₈	—	9	1	no	—
94	C₂×C₈	—	9	1	no	—
95	C₂×C₈	—	9	1	no	—
96	C₂×C₈	—	9	1	no	—
97	C₂×C₈	—	9	1	no	—
98	C₂×C₈	—	18	1	no	—
99	C₂×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
100	C₄⋊C₄	—	9	2	no	—
101	C₂²⋊C₄	—	18	1	no	—
102	C₄²	—	18	1	no	—
103	C₁₆	—	18	2	no	—
104	C₁₆	—	18	2	no	—
105	C₂²×C₃	—	4	1	no	—
106	C₂×D₃	—	12	1	no	—
107	C₂×D₃	—	12	1	no	—
108	C₂×D₃	—	12	4	no	—
109	C₃×C₄	—	4	1	no	—
110	C₃×C₄	—	4	1	no	—
111	Dic₃	—	12	1	no	—
112	Dic₃	—	12	1	no	—
113	C₃²	#1379	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	9	1	no	—
115	C₂×C₄	—	9	1	no	—
116	C₂×C₄	—	9	1	no	—
117	C₂×C₄	—	9	1	no	—
118	C₂×C₄	—	9	1	no	—
119	C₂×C₄	—	9	1	no	—
120	C₂×C₄	—	18	2	no	—
121	C₂×C₄	#3777	1	1	yes	characteristic
122	C₈	—	9	1	no	—
123	C₈	—	9	1	no	—
124	C₈	—	18	2	no	—
125	C₈	#3777	1	1	yes	characteristic
126	C₈	#3777	1	1	yes	characteristic
127	C₂×C₃	—	4	2	no	—
128	C₂×C₃	—	4	1	no	—
129	D₃	—	12	2	no	—
130	D₃	—	12	2	no	—
131	C₂²	—	9	1	no	—
132	C₂²	—	9	1	no	—
133	C₂²	—	18	2	no	—
134	C₂²	#63744	1	1	yes	characteristic
135	C₄	—	9	1	no	—
136	C₄	—	9	1	no	—
137	C₄	—	36	2	no	—
138	C₄	#63744	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	#10590090	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593765

C₂

yes

maximal, characteristic

#10593762

C₂

yes

maximal, characteristic

C₂×#63764

C₂

yes

maximal, characteristic

#64286

C₄

yes

characteristic

#64283

C₄

yes

characteristic

C₂×C₈×C₃⋊D₃

C₂²

yes

—

—

—

—

C₄

yes

—

—

C₄

yes

complemented

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

characteristic

C₂²⋊#157

—

#3792

C₈

yes

—

C₄⋊#157

C₈

yes

—

C₄×#157

—

#3788

—

#3787

—

C₂×C₃²×C₈

Q₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

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

M₁₆

yes

characteristic

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

C₄²

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

characteristic

C₄×C₃⋊D₃

C₂×C₈

yes

characteristic

C₄×C₃⋊D₃

M₁₆

yes

characteristic

#622

—

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃²×C₈

C₄⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₂²⋊C₄

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

D₃×C₈

—

C₂²×C₃²

C₄⋊C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#97

yes

characteristic

C₂×C₃⋊D₃

C₂²⋊C₈

yes

characteristic

#157

—

C₃²×C₄

#95

yes

characteristic

C₃²×C₄

C₄⋊C₈

yes

characteristic

C₃⋊Dic₃

C₄×C₈

yes

characteristic

C₃⋊Dic₃

C₂²⋊C₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

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

—

C₂×C₃²

#336

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

#350

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

PSU(3,𝔽₂)

yes

characteristic

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₂²×C₃

—

106

C₂×D₃

—

107

C₂×D₃

—

108

C₂×D₃

—

109

C₃×C₄

—

110

C₃×C₄

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

#1379

yes

characteristic, complemented, Sylow

114

C₂³

—

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

—

125

C₈

#3777

yes

characteristic

126

C₈

#3777

yes

characteristic

127

C₂×C₃

—

128

C₂×C₃

—

129

D₃

—

130

D₃

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#63744

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#63744

yes

characteristic

139

C₄

#63747

yes

characteristic

140

C₃

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#10590090

yes

characteristic

	subgroup	G/H
center	C₈	#3777
commutator	C₃⋊Dic₃	C₄×C₈
Frattini, hypercenter	C₂×C₈	PSU(3,𝔽₂)
Fitting	C₂×C₃²×C₈	Q₈
socle	C₂×C₃²	#336
prime radical	C₂²×C₃⋊D₃	M₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	#1379
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₈

subgroup

G/H

center

C₈

#3777

commutator

C₃⋊Dic₃

C₄×C₈

Frattini, hypercenter

C₂×C₈

PSU(3,𝔽₂)

Fitting

C₂×C₃²×C₈

Q₈

socle

C₂×C₃²

#336

prime radical

C₂²×C₃⋊D₃

M₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

hypocenter

C₃²

#1379

LCS

G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₈ ◁ C₂×C₈

N	H ≅ G/N	type
11. #63765	C₄	semidirect
14. #63764	C₄	semidirect
113. C₃²	#1379	semidirect

H ≅ G/N

type

11. #63765

C₄

semidirect

14. #63764

C₄

semidirect

113. C₃²

#1379

semidirect

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

subgroup	index
#10593765	2
#10593762	2
C₂×#63764	2
#1379	9

subgroup

index

Subgroups — #11772364

Conjugacy classes of subgroups

Complements

Maximal subgroups