Group #13135861 — subgroups

Subgroups — #13135861

Order 1728.

319 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#88527	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720545	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4514	—	3	4	no	maximal
5	C₂×C₄×#272	C₄	1	1	yes	characteristic
6	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×#6180	C₄	1	1	yes	characteristic
8	#88527	—	2	1	no	—
9	#88527	—	2	1	no	—
10	C₁₆×H(3,𝔽₃)	—	2	1	no	—
11	#88386	—	2	1	no	—
12	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
13	C₃²×M₃₂	D₃	1	4	yes	complemented
14	C₃×#812	—	3	4	no	—
15	C₂²×#272	C₈	1	1	yes	characteristic
16	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
17	C₂×#1084	C₈	1	1	yes	characteristic
18	C₄×#272	—	2	1	no	—
19	C₄×#272	—	2	1	no	—
20	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
21	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
22	#6180	—	2	1	no	—
23	C₃×#349	—	9	1	no	—
24	#4514	—	3	4	no	—
25	C₂×C₃×C₄×D₃	—	3	4	no	—
26	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
27	C₂×C₃×C₃⋊C₈	—	3	4	no	—
28	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
29	C₃×ΓL(1,𝔽₂₅)	—	6	4	no	—
30	C₃²×C₁₆	—	2	4	no	—
31	C₃×C₃⋊C₁₆	—	6	4	no	—
32	C₂²×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
33	C₂×#272	—	2	1	no	—
34	C₂×#272	—	4	1	no	—
35	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
37	#1084	—	2	1	no	—
38	C₂×C₃×M₁₆	—	9	1	no	—
39	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
40	C₃×M₃₂	—	3	4	no	—
41	C₃×M₃₂	—	9	1	no	—
42	C₃×M₃₂	C₃⋊D₃	1	1	yes	characteristic
43	#812	—	3	4	no	—
44	C₂²×C₃×D₃	—	3	4	no	—
45	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
46	C₂×C₃×Dic₃	—	3	4	no	—
47	C₃×C₄×D₃	—	6	4	no	—
48	C₃×C₄×D₃	—	6	4	no	—
49	C₃²×C₈	D₁₂	1	4	yes	—
50	C₃²×C₈	#67	1	4	yes	—
51	C₃×C₃⋊C₈	—	6	4	no	—
52	#349	—	9	1	no	Sylow
53	C₂×H(3,𝔽₃)	—	2	1	no	—
54	C₂×H(3,𝔽₃)	C₂²⋊C₈	1	1	yes	characteristic
55	#272	—	4	1	no	—
56	C₂²×C₃×C₄	—	9	1	no	—
57	C₂×C₄×D₃	—	3	4	no	—
58	C₃×M₁₆	—	18	1	no	—
59	C₃×M₁₆	—	18	1	no	—
60	C₂×C₃×C₈	—	3	4	no	—
61	C₂×C₃×C₈	—	9	1	no	—
62	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
63	C₂×C₃⋊C₈	—	3	4	no	—
64	ΓL(1,𝔽₂₅)	—	6	4	no	—
65	ΓL(1,𝔽₂₅)	—	6	4	no	—
66	C₃×C₁₆	—	2	1	no	—
67	C₃×C₁₆	—	6	4	no	—
68	C₃×C₁₆	—	18	1	no	—
69	C₃⋊C₁₆	—	6	4	no	—
70	C₂²×C₃²	D₃×C₈	1	4	yes	—
71	C₂×C₃×D₃	—	6	4	no	—
72	C₂×C₃×D₃	—	12	4	no	—
73	C₃²×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
74	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
75	C₃×Dic₃	—	6	4	no	—
76	C₂×M₁₆	—	9	1	no	—
77	M₃₂	—	9	1	no	—
78	M₃₂	#272	1	1	yes	characteristic, complemented
79	H(3,𝔽₃)	#349	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₃	—	9	1	no	—
81	C₂²×D₃	—	3	4	no	—
82	C₂×C₃×C₄	—	3	4	no	—
83	C₂×C₃×C₄	—	9	1	no	—
84	C₂×C₃×C₄	—	18	1	no	—
85	C₂×C₃×C₄	—	18	1	no	—
86	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
87	C₂×Dic₃	—	3	4	no	—
88	C₄×D₃	—	6	4	no	—
89	C₄×D₃	—	6	4	no	—
90	C₃×C₈	—	3	4	no	—
91	C₃×C₈	—	3	4	no	—
92	C₃×C₈	—	18	1	no	—
93	C₃×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
94	C₃×C₈	#638	1	1	yes	characteristic
95	C₃⋊C₈	—	6	4	no	—
96	C₂×C₃²	—	2	4	no	—
97	C₂×C₃²	(C₂×D₃)⋊C₈	1	4	yes	—
98	C₃×D₃	—	12	4	no	—
99	C₂²×C₄	—	9	1	no	—
100	M₁₆	—	18	1	no	—
101	M₁₆	—	18	1	no	—
102	C₂×C₈	—	9	1	no	—
103	C₂×C₈	C₂×#272	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	18	1	no	—
106	C₂²×C₃	—	3	4	no	—
107	C₂²×C₃	—	18	1	no	—
108	C₂²×C₃	—	36	1	no	—
109	C₂²×C₃	C₈×C₃⋊D₃	1	1	yes	characteristic
110	C₂×D₃	—	6	4	no	—
111	C₂×D₃	—	12	4	no	—
112	C₃×C₄	—	3	4	no	—
113	C₃×C₄	—	3	4	no	—
114	C₃×C₄	—	18	1	no	—
115	C₃×C₄	#3743	1	1	yes	characteristic
116	C₃×C₄	#3752	1	1	yes	characteristic
117	Dic₃	—	6	4	no	—
118	C₃²	#4514	1	4	yes	complemented
119	C₂³	—	9	1	no	—
120	C₂×C₄	—	9	1	no	—
121	C₂×C₄	—	18	1	no	—
122	C₂×C₄	—	18	1	no	—
123	C₂×C₄	C₄×#272	1	1	yes	characteristic
124	C₈	—	18	1	no	—
125	C₈	C₄⋊#272	1	1	yes	characteristic
126	C₈	#6235	1	1	yes	characteristic
127	C₂×C₃	—	2	1	no	—
128	C₂×C₃	—	3	4	no	—
129	C₂×C₃	—	6	4	no	—
130	C₂×C₃	—	36	1	no	—
131	C₂×C₃	#63647	1	1	yes	characteristic
132	D₃	—	12	4	no	—
133	C₂²	—	18	1	no	—
134	C₂²	—	36	1	no	—
135	C₂²	C₈×#272	1	1	yes	characteristic
136	C₄	—	18	1	no	—
137	C₄	#88527	1	1	yes	characteristic
138	C₄	#88542	1	1	yes	characteristic
139	C₃	—	3	4	no	—
140	C₃	#10589380	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	36	1	no	—
143	C₂	#11720553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88527

C₂

yes

maximal, characteristic

H(3,𝔽₃)×M₃₂

C₂

yes

maximal, characteristic, complemented

#11720545

C₂

yes

maximal, characteristic, complemented

C₃×#4514

—

maximal

C₂×C₄×#272

C₄

yes

characteristic

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#6180

C₄

yes

characteristic

#88527

—

#88527

—

C₁₆×H(3,𝔽₃)

—

#88386

—

C₂×C₃×ΓL(1,𝔽₂₅)

—

C₃²×M₃₂

D₃

yes

complemented

C₃×#812

—

C₂²×#272

C₈

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×#1084

C₈

yes

characteristic

C₄×#272

—

C₄×#272

—

C₈×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

D₄

yes

—

—

—

C₂×C₃×C₄×D₃

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₂×C₃×C₃⋊C₈

—

C₃×ΓL(1,𝔽₂₅)

—

C₃×ΓL(1,𝔽₂₅)

—

C₃²×C₁₆

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₂×#272

—

C₂×#272

—

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

M₁₆

yes

characteristic

#1084

—

C₂×C₃×M₁₆

—

C₂×ΓL(1,𝔽₂₅)

—

C₃×M₃₂

—

C₃×M₃₂

—

C₃×M₃₂

C₃⋊D₃

yes

characteristic

#812

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃×Dic₃

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

C₃²×C₈

D₁₂

yes

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

#349

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²⋊C₈

yes

characteristic

#272

—

C₂²×C₃×C₄

—

C₂×C₄×D₃

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

ΓL(1,𝔽₂₅)

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

D₃×C₈

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

ΓL(1,𝔽₂₅)

yes

—

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₃×Dic₃

—

C₂×M₁₆

—

M₃₂

—

M₃₂

#272

yes

characteristic, complemented

H(3,𝔽₃)

#349

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₃×C₈

#638

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

(C₂×D₃)⋊C₈

yes

—

C₃×D₃

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

C₂×#272

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

C₈×C₃⋊D₃

yes

characteristic

110

C₂×D₃

—

111

C₂×D₃

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

#3743

yes

characteristic

116

C₃×C₄

#3752

yes

characteristic

117

Dic₃

—

118

C₃²

#4514

yes

complemented

119

C₂³

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₄×#272

yes

characteristic

124

C₈

—

125

C₈

C₄⋊#272

yes

characteristic

126

C₈

#6235

yes

characteristic

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

#63647

yes

characteristic

132

D₃

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₈×#272

yes

characteristic

136

C₄

—

137

C₄

#88527

yes

characteristic

138

C₄

#88542

yes

characteristic

139

C₃

—

140

C₃

#10589380

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

#11720553

yes

characteristic

	subgroup	G/H
center	C₃×C₄	#3752
commutator	C₂²×H(3,𝔽₃)	C₂×C₈
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×M₃₂	C₂
socle	C₂×C₃	#63647
prime radical	C₂²×#272	C₈
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#349
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

subgroup

G/H

center

C₃×C₄

#3752

commutator

C₂²×H(3,𝔽₃)

C₂×C₈

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting

H(3,𝔽₃)×M₃₂

C₂

socle

C₂×C₃

#63647

prime radical

C₂²×#272

C₈

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₃×M₃₂

C₃⋊D₃

G ▷ C₂²×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

N	H ≅ G/N	type
2. H(3,𝔽₃)×M₃₂	C₂	semidirect
3. #11720545	C₂	semidirect
13. C₃²×M₃₂	D₃	semidirect
78. M₃₂	#272	semidirect
79. H(3,𝔽₃)	#349	semidirect
118. C₃²	#4514	semidirect

H ≅ G/N

type

2. H(3,𝔽₃)×M₃₂

C₂

semidirect

3. #11720545

C₂

semidirect

13. C₃²×M₃₂

D₃

semidirect

78. M₃₂

#272

semidirect

79. H(3,𝔽₃)

#349

semidirect

118. C₃²

#4514

semidirect

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

subgroup	index
C₂×#88527	2
H(3,𝔽₃)×M₃₂	2
#11720545	2
C₃×#4514	3

subgroup

index

C₂×#88527

H(3,𝔽₃)×M₃₂

#11720545

C₃×#4514

Subgroups — #13135861

Conjugacy classes of subgroups

Complements

Maximal subgroups