Group #13135942 — subgroups

Subgroups — #13135942

Order 1728.

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

normal subgroups only

This group has 319 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	C₂×M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11720545	C₂	1	2	yes	maximal, complemented
3	C₃×#4553	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	M₁₆×H(3,𝔽₃)	—	2	2	no	—
6	C₂×C₈×H(3,𝔽₃)	C₄	1	1	yes	characteristic
7	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	#88386	—	2	2	no	—
9	C₂×C₃²×M₁₆	D₃	1	4	yes	—
10	C₃×#812	—	3	8	no	—
11	C₂³×H(3,𝔽₃)	C₈	1	1	yes	characteristic
12	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
13	C₂×C₄×H(3,𝔽₃)	—	2	1	no	—
14	C₂×C₄×H(3,𝔽₃)	C₈	1	1	yes	characteristic
15	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₈×H(3,𝔽₃)	—	2	1	no	—
17	C₈×H(3,𝔽₃)	D₄	1	2	yes	—
18	C₃×#349	—	9	1	no	—
19	#4553	—	3	4	no	—
20	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
21	C₃²×M₁₆	—	2	8	no	—
22	C₂×C₃²×C₈	Dic₃	1	4	yes	—
23	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
24	C₃×C₃⋊C₁₆	—	6	8	no	—
25	C₂²×H(3,𝔽₃)	—	2	1	no	—
26	C₂²×H(3,𝔽₃)	—	4	1	no	—
27	C₂²×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
28	C₄×H(3,𝔽₃)	—	2	1	no	—
29	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
31	C₂×C₃×M₁₆	—	3	4	no	—
32	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
33	C₃×M₃₂	—	9	2	no	—
34	#812	—	3	8	no	—
35	C₂³×C₃²	C₃⋊C₈	1	4	yes	—
36	C₂×C₃²×C₄	—	2	4	no	—
37	C₂×C₃²×C₄	—	2	4	no	—
38	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
39	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
40	C₃²×C₈	—	2	4	no	—
41	C₃²×C₈	#67	1	8	yes	—
42	#349	—	9	1	no	Sylow
43	C₂×H(3,𝔽₃)	—	2	1	no	—
44	C₂×H(3,𝔽₃)	—	4	1	no	—
45	C₂×H(3,𝔽₃)	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₃×C₄	—	3	4	no	—
47	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
48	C₃×M₁₆	—	2	2	no	—
49	C₃×M₁₆	—	6	8	no	—
50	C₂×C₃×C₈	—	3	4	no	—
51	C₂×C₃×C₈	—	3	4	no	—
52	C₂×C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
53	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
54	C₃×C₁₆	—	18	2	no	—
55	C₃⋊C₁₆	—	6	8	no	—
56	C₂²×C₃²	—	2	4	no	—
57	C₂²×C₃²	—	4	4	no	—
58	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
59	C₃²×C₄	—	2	4	no	—
60	C₃²×C₄	#208	1	4	yes	—
61	C₃²×C₄	#217	1	4	yes	—
62	C₂×M₁₆	#272	1	1	yes	characteristic
63	M₃₂	—	9	2	no	—
64	H(3,𝔽₃)	#349	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₃	—	3	4	no	—
66	C₂³×C₃	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
67	C₂×C₃×C₄	—	2	1	no	—
68	C₂×C₃×C₄	—	2	1	no	—
69	C₂×C₃×C₄	—	3	4	no	—
70	C₂×C₃×C₄	—	3	4	no	—
71	C₂×C₃×C₄	—	6	4	no	—
72	C₂×C₃×C₄	—	6	4	no	—
73	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
74	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
75	C₃×C₈	—	2	1	no	—
76	C₃×C₈	—	3	8	no	—
77	C₃×C₈	—	6	4	no	—
78	C₃×C₈	#638	1	2	yes	—
79	C₂×C₃²	—	2	4	no	—
80	C₂×C₃²	—	4	4	no	—
81	C₂×C₃²	#830	1	4	yes	—
82	C₂²×C₄	#1084	1	1	yes	characteristic
83	M₁₆	—	2	2	no	—
84	C₂×C₈	#1084	1	1	yes	characteristic
85	C₂×C₈	C₂×#272	1	1	yes	characteristic
86	C₁₆	—	18	2	no	—
87	C₂²×C₃	—	2	1	no	—
88	C₂²×C₃	—	3	4	no	—
89	C₂²×C₃	—	4	1	no	—
90	C₂²×C₃	—	6	4	no	—
91	C₂²×C₃	—	12	4	no	—
92	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
93	C₃×C₄	—	2	1	no	—
94	C₃×C₄	—	3	4	no	—
95	C₃×C₄	—	3	4	no	—
96	C₃×C₄	—	6	4	no	—
97	C₃×C₄	#3748	1	1	yes	characteristic
98	C₃×C₄	#3757	1	1	yes	characteristic
99	C₃²	#4553	1	4	yes	complemented
100	C₂³	#6180	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	#6180	1	1	yes	characteristic
104	C₂×C₄	C₂×#1084	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	#6235	1	2	yes	—
107	C₂×C₃	—	2	1	no	—
108	C₂×C₃	—	3	4	no	—
109	C₂×C₃	—	4	1	no	—
110	C₂×C₃	—	6	4	no	—
111	C₂×C₃	—	12	4	no	—
112	C₂×C₃	#63657	1	1	yes	characteristic
113	C₂²	—	2	1	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#6180	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	#88538	1	1	yes	characteristic
118	C₄	#88552	1	1	yes	characteristic
119	C₃	—	3	4	no	—
120	C₃	#10589419	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	4	1	no	—
123	C₂	#11720573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11720545

C₂

yes

maximal, complemented

C₃×#4553

—

maximal

C₂²×C₄×H(3,𝔽₃)

C₄

yes

characteristic

M₁₆×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₄

yes

characteristic

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#88386

—

C₂×C₃²×M₁₆

D₃

yes

—

C₃×#812

—

C₂³×H(3,𝔽₃)

C₈

yes

characteristic

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₈

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

—

C₈×H(3,𝔽₃)

D₄

yes

—

C₃×#349

—

#4553

—

C₂²×C₃²×C₄

Dic₃

yes

—

C₃²×M₁₆

—

C₂×C₃²×C₈

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

M₁₆

yes

characteristic

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃⋊D₃

yes

characteristic

C₃×M₃₂

—

#812

—

C₂³×C₃²

C₃⋊C₈

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃⋊C₈

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₃²×C₈

—

C₃²×C₈

#67

yes

—

#349

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²⋊C₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃⋊Dic₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

—

C₃²×C₄

—

C₃²×C₄

#208

yes

—

C₃²×C₄

#217

yes

—

C₂×M₁₆

#272

yes

characteristic

M₃₂

—

H(3,𝔽₃)

#349

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₃⋊(C₃⋊C₈)

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃⋊(C₃⋊C₈)

yes

characteristic

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#638

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#830

yes

—

C₂²×C₄

#1084

yes

characteristic

M₁₆

—

C₂×C₈

yes

C₂×C₈

yes

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₃⋊(C₃⋊C₈)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃²

yes

100

C₂³

#6180

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#6180

yes

characteristic

104

C₂×C₄

C₂×#1084

yes

characteristic

105

C₈

—

106

C₈

#6235

yes

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

#63657

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×#6180

yes

characteristic

116

C₄

—

117

C₄

#88538

yes

characteristic

118

C₄

#88552

yes

characteristic

119

C₃

—

120

C₃

#10589419

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

#11720573

yes

characteristic

	subgroup	G/H
center	C₃×C₄	#3757
commutator	C₂²×H(3,𝔽₃)	C₂×C₈
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	C₂×M₁₆×H(3,𝔽₃)	C₂
socle	C₂×C₃	#63657
prime radical	C₂³×H(3,𝔽₃)	C₈
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×M₁₆	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 ◁ C₂×C₃×M₁₆

subgroup

G/H

center

C₃×C₄

#3757

commutator

C₂²×H(3,𝔽₃)

C₂×C₈

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting

C₂×M₁₆×H(3,𝔽₃)

C₂

socle

C₂×C₃

#63657

prime radical

C₂³×H(3,𝔽₃)

C₈

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₂×C₃×M₁₆

C₃⋊D₃

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

UCS

C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₂×C₃×M₁₆

N	H ≅ G/N	type
2. #11720545	C₂	semidirect
64. H(3,𝔽₃)	#349	semidirect
99. C₃²	#4553	semidirect

H ≅ G/N

type

2. #11720545

C₂

semidirect

64. H(3,𝔽₃)

#349

semidirect

99. C₃²

#4553

semidirect

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

subgroup	index
C₂×M₁₆×H(3,𝔽₃)	2
#11720545	2
C₃×#4553	3

subgroup

index

C₂×M₁₆×H(3,𝔽₃)

#11720545

C₃×#4553

Subgroups — #13135942

Conjugacy classes of subgroups

Complements

Maximal subgroups