Group #11771501 — subgroups

Subgroups — #11771501

Order 1152.

288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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	#10592817	C₂	1	1	yes	maximal, characteristic
2	C₃²×#349	C₂	1	1	yes	maximal, characteristic
3	#10589419	C₂	1	1	yes	maximal, characteristic
4	#67532	—	3	4	no	maximal
5	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂×#3748	C₄	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	1	no	—
9	#63641	—	2	1	no	—
10	#63641	—	2	1	no	—
11	#63639	—	2	1	no	—
12	#5118	—	3	4	no	—
13	C₃×#349	D₃	1	4	yes	—
14	#4553	—	3	4	no	—
15	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃⋊Dic₃	—	2	1	no	—
17	C₃²×M₁₆	—	4	1	no	—
18	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
19	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
20	C₄×C₃⋊Dic₃	—	2	1	no	—
21	C₄×C₃⋊Dic₃	C₈	1	1	yes	characteristic
22	C₄×C₃⋊Dic₃	C₈	1	1	yes	characteristic
23	#3748	—	4	1	no	—
24	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
25	C₃²×C₁₆	—	4	1	no	—
26	C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
27	#1327	—	9	1	no	Sylow
28	C₂×C₃×M₁₆	C₂×D₃	1	4	yes	—
29	C₂×C₄×Dic₃	—	3	4	no	—
30	C₂×#208	—	3	4	no	—
31	C₃×M₃₂	—	2	4	no	—
32	Dic₃⋊C₈	—	6	4	no	—
33	Dic₃⋊C₈	—	6	4	no	—
34	#812	—	6	4	no	—
35	C₂³×C₃²	M₁₆	1	1	yes	characteristic
36	C₂×C₃²×C₄	—	2	1	no	—
37	C₂×C₃²×C₄	—	2	1	no	—
38	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃⋊Dic₃	—	2	1	no	—
41	C₂×C₃⋊Dic₃	—	2	1	no	—
42	C₂×C₃⋊Dic₃	—	4	2	no	—
43	C₃²×C₈	—	2	1	no	—
44	C₃²×C₈	—	2	1	no	—
45	C₃⋊(C₃⋊C₈)	—	4	1	no	—
46	#423	—	9	1	no	—
47	#349	—	9	1	no	—
48	#349	C₃⋊D₃	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
50	C₂²×Dic₃	—	6	4	no	—
51	C₃×M₁₆	—	4	4	no	—
52	C₂×C₃×C₈	D₁₂	1	4	yes	—
53	C₂×C₃×C₈	#67	1	4	yes	—
54	C₄×Dic₃	—	3	4	no	—
55	C₄×Dic₃	—	3	4	no	—
56	C₄×Dic₃	—	6	4	no	—
57	#208	—	12	4	no	—
58	C₂×C₃⋊C₈	—	6	4	no	—
59	C₃×C₁₆	—	4	4	no	—
60	C₃⋊C₁₆	—	12	4	no	—
61	C₂²×C₃²	—	2	1	no	—
62	C₂²×C₃²	—	4	1	no	—
63	C₂²×C₃²	C₂²⋊C₈	1	1	yes	characteristic
64	C₃²×C₄	—	2	1	no	—
65	C₃²×C₄	#99	1	1	yes	characteristic
66	C₃²×C₄	#101	1	1	yes	characteristic
67	C₃⋊Dic₃	—	2	2	no	—
68	C₃⋊Dic₃	—	4	1	no	—
69	C₂×M₁₆	—	9	1	no	—
70	C₂×M₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₂×C₄²	—	9	1	no	—
72	M₃₂	—	2	1	no	—
73	M₃₂	—	18	1	no	—
74	C₄⋊C₈	—	18	1	no	—
75	C₄⋊C₈	—	18	1	no	—
76	C₂³×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
77	C₂×C₃×C₄	—	2	4	no	—
78	C₂×C₃×C₄	—	2	4	no	—
79	C₂×C₃×C₄	D₃×C₈	1	4	yes	—
80	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
81	C₂×Dic₃	—	6	4	no	—
82	C₂×Dic₃	—	6	4	no	—
83	C₂×Dic₃	—	12	8	no	—
84	C₃×C₈	—	2	4	no	—
85	C₃×C₈	—	2	4	no	—
86	C₃⋊C₈	—	12	4	no	—
87	C₂×C₃²	—	2	1	no	—
88	C₂×C₃²	—	4	1	no	—
89	C₂×C₃²	#324	1	1	yes	characteristic
90	C₂²×C₄	—	18	1	no	—
91	C₂²×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
92	M₁₆	—	4	1	no	—
93	M₁₆	—	36	1	no	—
94	C₂×C₈	—	18	1	no	—
95	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
96	C₂×C₈	#638	1	1	yes	characteristic
97	C₄²	—	9	1	no	—
98	C₄²	—	9	1	no	—
99	C₄²	—	18	1	no	—
100	C₁₆	—	4	1	no	—
101	C₁₆	—	36	1	no	—
102	C₂²×C₃	—	2	4	no	—
103	C₂²×C₃	—	4	4	no	—
104	C₂²×C₃	(C₂×D₃)⋊C₈	1	4	yes	—
105	C₃×C₄	—	2	4	no	—
106	C₃×C₄	#806	1	4	yes	—
107	C₃×C₄	#824	1	4	yes	—
108	Dic₃	—	6	8	no	—
109	Dic₃	—	12	4	no	—
110	C₃²	#1327	1	1	yes	characteristic, complemented, Sylow
111	C₂³	#3743	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	18	1	no	—
115	C₂×C₄	—	18	1	no	—
116	C₂×C₄	—	36	2	no	—
117	C₂×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
118	C₂×C₄	#3752	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	—	36	1	no	—
122	C₂×C₃	—	2	4	no	—
123	C₂×C₃	—	4	4	no	—
124	C₂×C₃	#4469	1	4	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	4	1	no	—
127	C₂²	#63647	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	18	2	no	—
130	C₄	—	36	1	no	—
131	C₄	#63633	1	1	yes	characteristic
132	C₄	#63651	1	1	yes	characteristic
133	C₃	#67532	1	4	yes	complemented
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#10589335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592817

C₂

yes

maximal, characteristic

C₃²×#349

C₂

yes

maximal, characteristic

#10589419

C₂

yes

maximal, characteristic

#67532

—

maximal

C₂×C₃²×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₃⋊Dic₃

C₄

yes

characteristic

C₂×#3748

C₄

yes

characteristic

C₃²×M₃₂

—

—

—

—

—

D₃

yes

—

#4553

—

C₂²×C₃²×C₄

C₂×C₄

yes

characteristic

C₂²×C₃⋊Dic₃

—

C₃²×M₁₆

—

C₂×C₃²×C₈

D₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

C₈

yes

characteristic

C₄×C₃⋊Dic₃

C₈

yes

characteristic

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#1327

—

Sylow

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₄×Dic₃

—

C₂×#208

—

C₃×M₃₂

—

Dic₃⋊C₈

—

Dic₃⋊C₈

—

#812

—

C₂³×C₃²

M₁₆

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃²×C₄

C₂×C₈

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₃²×C₈

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#423

—

#349

—

#349

C₃⋊D₃

yes

characteristic

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

D₁₂

yes

—

C₂×C₃×C₈

#67

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂³×C₃

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×C₈

yes

—

C₂×C₃×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#324

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×C₃⋊D₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₈

#638

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

(C₂×D₃)⋊C₈

yes

—

105

C₃×C₄

—

106

C₃×C₄

#806

yes

—

107

C₃×C₄

#824

yes

—

108

Dic₃

—

109

Dic₃

—

110

C₃²

#1327

yes

characteristic, complemented, Sylow

111

C₂³

#3743

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₈×C₃⋊D₃

yes

characteristic

118

C₂×C₄

#3752

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

#4469

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

#63647

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#63633

yes

characteristic

132

C₄

#63651

yes

characteristic

133

C₃

#67532

yes

complemented

134

C₂

—

135

C₂

—

136

C₂

#10589335

yes

characteristic

	subgroup	G/H
center	C₄	#63633
commutator	C₂×C₃²×C₄	C₂×C₈
Frattini	C₂×M₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#349	C₂
socle	C₂×C₃²	#324
prime radical	C₂³×C₃²	M₁₆
cosocle	C₂×C₃²×M₁₆	C₂²
hypercenter	#349	C₃⋊D₃
hypocenter	C₃²	#1327
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₄

#63633

commutator

C₂×C₃²×C₄

C₂×C₈

Frattini

C₂×M₁₆

C₂×C₃⋊D₃

Fitting

C₃²×#349

C₂

socle

C₂×C₃²

#324

prime radical

C₂³×C₃²

M₁₆

cosocle

C₂×C₃²×M₁₆

C₂²

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
110. C₃²	#1327	semidirect
133. C₃	#67532	semidirect

H ≅ G/N

type

110. C₃²

#1327

semidirect

133. C₃

#67532

semidirect

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

subgroup	index
#10592817	2
C₃²×#349	2
#10589419	2
#67532	3

subgroup

index

#10592817

C₃²×#349

#10589419

#67532

Subgroups — #11771501

Conjugacy classes of subgroups

Complements

Maximal subgroups