Group #13135951 — subgroups

Subgroups — #13135951

Order 1728.

308 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 101 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	H(3,𝔽₃)×#131	C₂	1	1	yes	maximal, characteristic
2	#11720545	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88386	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4561	—	3	4	no	maximal
5	#41×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	M₁₆×H(3,𝔽₃)	—	2	1	no	—
7	M₁₆×H(3,𝔽₃)	C₄	1	1	yes	characteristic
8	C₂×C₈×H(3,𝔽₃)	—	2	1	no	—
9	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	#88386	—	2	1	no	—
11	#88386	—	2	1	no	—
12	C₃²×#131	D₃	1	4	yes	—
13	C₃×#812	—	3	4	no	—
14	C₂×C₃×C₃⋊C₁₆	—	3	4	no	—
15	Q₈×H(3,𝔽₃)	C₈	1	1	yes	characteristic
16	D₄×H(3,𝔽₃)	—	2	1	no	—
17	D₄×H(3,𝔽₃)	C₈	1	1	yes	characteristic
18	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
19	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
20	C₈×H(3,𝔽₃)	—	2	1	no	—
21	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
22	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
23	C₃×#350	—	9	1	no	—
24	#4561	—	3	4	no	—
25	C₃²×#41	Dic₃	1	4	yes	—
26	C₃²×M₁₆	—	2	4	no	—
27	C₃²×M₁₆	Dic₃	1	4	yes	—
28	C₂×C₃²×C₈	—	2	4	no	—
29	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
30	C₃×C₃⋊C₁₆	—	6	4	no	—
31	C₃×C₃⋊C₁₆	—	6	4	no	—
32	C₂²×H(3,𝔽₃)	—	2	1	no	—
33	C₂²×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
34	C₄×H(3,𝔽₃)	—	2	1	no	—
35	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
37	C₃×#131	—	3	4	no	—
38	C₃×#131	C₃⋊D₃	1	1	yes	characteristic
39	C₃×M₃₂	—	9	1	no	—
40	C₂×C₃×C₁₆	—	9	1	no	—
41	#812	—	3	4	no	—
42	C₂×C₃⋊C₁₆	—	3	4	no	—
43	C₃²×Q₈	C₃⋊C₈	1	4	yes	—
44	C₃²×D₄	—	2	4	no	—
45	C₃²×D₄	C₃⋊C₈	1	4	yes	—
46	C₂×C₃²×C₄	—	2	4	no	—
47	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
48	C₃²×C₈	—	2	4	no	—
49	C₃²×C₈	#67	1	4	yes	—
50	C₃²×C₈	#67	1	4	yes	—
51	#350	—	9	1	no	Sylow
52	C₂×H(3,𝔽₃)	—	2	1	no	—
53	C₂×H(3,𝔽₃)	—	4	1	no	—
54	C₂×H(3,𝔽₃)	C₂²⋊C₈	1	1	yes	characteristic
55	C₃×#41	—	3	4	no	—
56	C₃×#41	C₃⋊Dic₃	1	1	yes	characteristic
57	C₃×M₁₆	—	2	1	no	—
58	C₃×M₁₆	—	3	4	no	—
59	C₃×M₁₆	—	6	4	no	—
60	C₃×M₁₆	C₃⋊Dic₃	1	1	yes	characteristic
61	C₂×C₃×C₈	—	2	1	no	—
62	C₂×C₃×C₈	—	3	4	no	—
63	C₂×C₃×C₈	—	6	4	no	—
64	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₃×C₁₆	—	18	1	no	—
66	C₃×C₁₆	—	18	1	no	—
67	C₃⋊C₁₆	—	6	4	no	—
68	C₃⋊C₁₆	—	6	4	no	—
69	C₂²×C₃²	—	2	4	no	—
70	C₂²×C₃²	#208	1	4	yes	—
71	C₃²×C₄	—	2	4	no	—
72	C₃²×C₄	C₂×C₃⋊C₈	1	4	yes	—
73	C₃²×C₄	#217	1	4	yes	—
74	#131	#272	1	1	yes	characteristic
75	M₃₂	—	9	1	no	—
76	C₂×C₁₆	—	9	1	no	—
77	H(3,𝔽₃)	#350	1	1	yes	characteristic, complemented, Sylow
78	C₃×Q₈	—	3	4	no	—
79	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
80	C₃×D₄	—	2	1	no	—
81	C₃×D₄	—	3	4	no	—
82	C₃×D₄	—	6	4	no	—
83	C₃×D₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
84	C₂×C₃×C₄	—	2	1	no	—
85	C₂×C₃×C₄	—	3	4	no	—
86	C₂×C₃×C₄	—	6	4	no	—
87	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
88	C₃×C₈	—	2	1	no	—
89	C₃×C₈	—	3	4	no	—
90	C₃×C₈	—	3	4	no	—
91	C₃×C₈	—	6	4	no	—
92	C₃×C₈	#638	1	1	yes	characteristic
93	C₃×C₈	#638	1	1	yes	characteristic
94	C₂×C₃²	—	2	4	no	—
95	C₂×C₃²	—	4	4	no	—
96	C₂×C₃²	#830	1	4	yes	—
97	#41	#1084	1	1	yes	characteristic
98	M₁₆	—	2	1	no	—
99	M₁₆	#1084	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	C₂×#272	1	1	yes	characteristic
102	C₁₆	—	18	1	no	—
103	C₁₆	—	18	1	no	—
104	C₂²×C₃	—	2	1	no	—
105	C₂²×C₃	—	3	4	no	—
106	C₂²×C₃	—	6	4	no	—
107	C₂²×C₃	#3748	1	1	yes	characteristic
108	C₃×C₄	—	2	1	no	—
109	C₃×C₄	—	3	4	no	—
110	C₃×C₄	—	3	4	no	—
111	C₃×C₄	—	6	4	no	—
112	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
113	C₃×C₄	#3757	1	1	yes	characteristic
114	C₃²	#4561	1	4	yes	complemented
115	Q₈	#6180	1	1	yes	characteristic
116	D₄	—	2	1	no	—
117	D₄	#6180	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	C₂×#1084	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	#6235	1	1	yes	characteristic
122	C₈	#6235	1	1	yes	characteristic
123	C₂×C₃	—	2	1	no	—
124	C₂×C₃	—	3	4	no	—
125	C₂×C₃	—	4	1	no	—
126	C₂×C₃	—	6	4	no	—
127	C₂×C₃	—	12	4	no	—
128	C₂×C₃	#63657	1	1	yes	characteristic
129	C₂²	—	2	1	no	—
130	C₂²	#88538	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	C₂×#6180	1	1	yes	characteristic
133	C₄	#88552	1	1	yes	characteristic
134	C₃	—	3	4	no	—
135	C₃	#10589427	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	4	1	no	—
138	C₂	#11720573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#131

C₂

yes

maximal, characteristic

#11720545

C₂

yes

maximal, characteristic, complemented

C₂×#88386

C₂

yes

maximal, characteristic, complemented

C₃×#4561

—

maximal

#41×H(3,𝔽₃)

C₄

yes

characteristic

M₁₆×H(3,𝔽₃)

—

M₁₆×H(3,𝔽₃)

C₄

yes

characteristic

C₂×C₈×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#88386

—

#88386

—

C₃²×#131

D₃

yes

—

C₃×#812

—

C₂×C₃×C₃⋊C₁₆

—

Q₈×H(3,𝔽₃)

C₈

yes

characteristic

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

C₈

yes

characteristic

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

—

C₈×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

D₄

yes

characteristic

C₃×#350

—

#4561

—

C₃²×#41

Dic₃

yes

—

C₃²×M₁₆

—

C₃²×M₁₆

Dic₃

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

M₁₆

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₃×#131

—

C₃×#131

C₃⋊D₃

yes

characteristic

C₃×M₃₂

—

C₂×C₃×C₁₆

—

#812

—

C₂×C₃⋊C₁₆

—

C₃²×Q₈

C₃⋊C₈

yes

—

C₃²×D₄

—

C₃²×D₄

C₃⋊C₈

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₃²×C₈

—

C₃²×C₈

#67

yes

—

C₃²×C₈

#67

yes

—

#350

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²⋊C₈

yes

characteristic

C₃×#41

—

C₃×#41

C₃⋊Dic₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

#208

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊C₈

yes

—

C₃²×C₄

#217

yes

—

#131

#272

yes

characteristic

M₃₂

—

C₂×C₁₆

—

H(3,𝔽₃)

#350

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

C₃⋊(C₃⋊C₈)

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₃⋊(C₃⋊C₈)

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#830

yes

—

#41

#1084

yes

characteristic

M₁₆

—

M₁₆

#1084

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂×#272

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

#3748

yes

characteristic

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

C₂×C₃⋊(C₃⋊C₈)

yes

characteristic

113

C₃×C₄

#3757

yes

characteristic

114

C₃²

#4561

yes

complemented

115

Q₈

#6180

yes

characteristic

116

D₄

—

117

D₄

#6180

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

C₂×#1084

yes

characteristic

120

C₈

—

121

C₈

#6235

yes

characteristic

122

C₈

#6235

yes

characteristic

123

C₂×C₃

—

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

#63657

yes

characteristic

129

C₂²

—

130

C₂²

#88538

yes

characteristic

131

C₄

—

132

C₄

C₂×#6180

yes

characteristic

133

C₄

#88552

yes

characteristic

134

C₃

—

135

C₃

#10589427

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#11720573

yes

characteristic

	subgroup	G/H
center	C₃×C₈	#638
commutator	C₄×H(3,𝔽₃)	C₂×C₈
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×#131	C₂
socle	C₂×C₃	#63657
prime radical	#41×H(3,𝔽₃)	C₄
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₃×#131	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#350
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×#131

subgroup

G/H

center

C₃×C₈

#638

commutator

C₄×H(3,𝔽₃)

C₂×C₈

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

C₂

C₂×C₃

C₄

C₂×C₈×H(3,𝔽₃)

C₂²

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×#131

N	H ≅ G/N	type
2. #11720545	C₂	semidirect
3. C₂×#88386	C₂	semidirect
77. H(3,𝔽₃)	#350	semidirect
114. C₃²	#4561	semidirect

H ≅ G/N

type

2. #11720545

C₂

semidirect

3. C₂×#88386

C₂

semidirect

77. H(3,𝔽₃)

#350

semidirect

114. C₃²

#4561

semidirect

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

subgroup	index
H(3,𝔽₃)×#131	2
#11720545	2
C₂×#88386	2
C₃×#4561	3

subgroup

index

Subgroups — #13135951

Conjugacy classes of subgroups

Complements

Maximal subgroups