Group #13135919 — subgroups

Subgroups — #13135919

Order 1728.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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)×#127	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6180	C₂	1	2	yes	maximal, complemented
3	C₃×#4544	—	3	4	no	maximal
4	C₂×D₄×H(3,𝔽₃)	—	2	2	no	—
5	C₂×D₄×H(3,𝔽₃)	C₄	1	2	yes	—
6	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×#6180	—	2	2	no	—
8	C₃²×#127	D₃	1	4	yes	—
9	C₃×C₄⋊(C₃⋊C₈)	—	3	8	no	—
10	C₂³×H(3,𝔽₃)	—	2	2	no	—
11	D₄×H(3,𝔽₃)	—	4	2	no	—
12	D₄×H(3,𝔽₃)	—	4	2	no	—
13	D₄×H(3,𝔽₃)	—	4	2	no	—
14	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
16	#6180	—	4	2	no	—
17	C₃×#331	—	9	1	no	—
18	#4544	—	3	4	no	—
19	C₂×C₃²×D₄	—	2	8	no	—
20	C₂×C₃²×D₄	Dic₃	1	8	yes	—
21	C₃²×C₄²	C₂×D₃	1	4	yes	—
22	C₂×C₃×C₃⋊C₈	—	6	8	no	—
23	C₂²×H(3,𝔽₃)	—	4	2	no	—
24	C₂²×H(3,𝔽₃)	—	4	4	no	—
25	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×H(3,𝔽₃)	—	2	1	no	—
27	C₄×H(3,𝔽₃)	D₈	1	2	yes	—
28	C₄×H(3,𝔽₃)	SD₁₆	1	2	yes	—
29	C₃×#127	—	3	4	no	—
30	C₃×#127	C₃⋊D₃	1	1	yes	characteristic
31	C₃×C₄⋊C₈	—	9	2	no	—
32	C₄⋊(C₃⋊C₈)	—	3	8	no	—
33	C₂³×C₃²	—	2	8	no	—
34	C₃²×D₄	—	4	8	no	—
35	C₃²×D₄	—	4	8	no	—
36	C₃²×D₄	—	4	8	no	—
37	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
38	C₂×C₃²×C₄	#67	1	8	yes	—
39	C₃×C₃⋊C₈	—	12	8	no	—
40	#331	—	9	1	no	Sylow
41	C₂×H(3,𝔽₃)	—	8	2	no	—
42	C₂×H(3,𝔽₃)	#100	1	1	yes	characteristic
43	C₂×H(3,𝔽₃)	#102	1	2	yes	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	3	8	no	—
46	C₂×C₃×D₄	—	6	8	no	—
47	C₂×C₃×D₄	C₃⋊Dic₃	1	2	yes	—
48	C₂×C₃×C₈	—	18	2	no	—
49	C₃×C₄²	—	3	4	no	—
50	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₂×C₃⋊C₈	—	6	8	no	—
52	C₂²×C₃²	—	4	16	no	—
53	C₂²×C₃²	—	4	8	no	—
54	C₂²×C₃²	#217	1	4	yes	—
55	C₃²×C₄	—	2	4	no	—
56	C₃²×C₄	#213	1	8	yes	—
57	C₃²×C₄	#214	1	8	yes	—
58	#127	#272	1	1	yes	characteristic
59	C₄⋊C₈	—	9	2	no	—
60	H(3,𝔽₃)	#331	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₃	—	2	2	no	—
62	C₂³×C₃	—	6	8	no	—
63	C₃×D₄	—	4	2	no	—
64	C₃×D₄	—	4	2	no	—
65	C₃×D₄	—	4	2	no	—
66	C₃×D₄	—	12	8	no	—
67	C₃×D₄	—	12	8	no	—
68	C₃×D₄	—	12	8	no	—
69	C₂×C₃×C₄	—	3	4	no	—
70	C₂×C₃×C₄	—	3	8	no	—
71	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
72	C₂×C₃×C₄	#638	1	2	yes	—
73	C₃×C₈	—	36	2	no	—
74	C₃⋊C₈	—	12	8	no	—
75	C₂×C₃²	—	8	8	no	—
76	C₂×C₃²	#832	1	8	yes	—
77	C₂×C₃²	#833	1	4	yes	—
78	C₂×D₄	—	2	2	no	—
79	C₂×D₄	#1084	1	2	yes	—
80	C₂×C₈	—	18	2	no	—
81	C₄²	C₂×#272	1	1	yes	characteristic
82	C₂²×C₃	—	3	4	no	—
83	C₂²×C₃	—	4	4	no	—
84	C₂²×C₃	—	4	2	no	—
85	C₂²×C₃	—	12	16	no	—
86	C₂²×C₃	—	12	8	no	—
87	C₂²×C₃	#3757	1	1	yes	characteristic
88	C₃×C₄	—	2	1	no	—
89	C₃×C₄	—	3	8	no	—
90	C₃×C₄	—	3	8	no	—
91	C₃×C₄	—	6	4	no	—
92	C₃×C₄	#3753	1	2	yes	—
93	C₃×C₄	#3754	1	2	yes	—
94	C₃²	#4544	1	4	yes	complemented
95	C₂³	—	2	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	C₂×#1084	1	1	yes	characteristic
100	C₂×C₄	#6235	1	2	yes	—
101	C₈	—	36	2	no	—
102	C₂×C₃	—	3	8	no	—
103	C₂×C₃	—	3	4	no	—
104	C₂×C₃	—	8	2	no	—
105	C₂×C₃	—	24	8	no	—
106	C₂×C₃	#63659	1	2	yes	—
107	C₂×C₃	#63660	1	1	yes	characteristic
108	C₂²	—	4	4	no	—
109	C₂²	—	4	2	no	—
110	C₂²	#88552	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	#88545	1	2	yes	—
113	C₄	#88546	1	2	yes	—
114	C₃	—	3	4	no	—
115	C₃	#10589410	1	1	yes	characteristic
116	C₂	—	8	2	no	—
117	C₂	#11720578	1	2	yes	—
118	C₂	#11720579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#127

C₂

yes

maximal, characteristic

C₄⋊#6180

C₂

yes

maximal, complemented

C₃×#4544

—

maximal

C₂×D₄×H(3,𝔽₃)

—

C₂×D₄×H(3,𝔽₃)

C₄

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#6180

—

C₃²×#127

D₃

yes

—

C₃×C₄⋊(C₃⋊C₈)

—

C₂³×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

#6180

—

C₃×#331

—

#4544

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

D₈

yes

—

C₄×H(3,𝔽₃)

SD₁₆

yes

—

C₃×#127

—

C₃×#127

C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

#331

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#100

yes

characteristic

C₂×H(3,𝔽₃)

#102

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃⋊Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#217

yes

—

C₃²×C₄

—

C₃²×C₄

#213

yes

—

C₃²×C₄

#214

yes

—

#127

#272

yes

characteristic

C₄⋊C₈

—

H(3,𝔽₃)

#331

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#832

yes

—

C₂×C₃²

#833

yes

—

C₂×D₄

—

C₂×D₄

#1084

yes

—

C₂×C₈

—

C₄²

C₂×#272

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#3757

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3753

yes

—

C₃×C₄

#3754

yes

—

C₃²

#4544

yes

complemented

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

C₂×#1084

yes

characteristic

100

C₂×C₄

#6235

yes

—

101

C₈

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

#63659

yes

—

107

C₂×C₃

#63660

yes

characteristic

108

C₂²

—

109

C₂²

—

110

C₂²

#88552

yes

characteristic

111

C₄

—

112

C₄

#88545

yes

—

113

C₄

#88546

yes

—

114

C₃

—

115

C₃

#10589410

yes

characteristic

116

C₂

—

117

C₂

#11720578

yes

—

118

C₂

#11720579

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#3757
commutator	C₂×C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting, prime radical	H(3,𝔽₃)×#127	C₂
cosocle	C₄²×H(3,𝔽₃)	C₂²
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#331
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

subgroup

G/H

center, socle

C₂²×C₃

#3757

commutator

C₂×C₄×H(3,𝔽₃)

C₂×C₄

Frattini

C₃×C₄²

C₂×C₃⋊D₃

Fitting, prime radical

H(3,𝔽₃)×#127

C₂

cosocle

C₄²×H(3,𝔽₃)

C₂²

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

N	H ≅ G/N	type
2. C₄⋊#6180	C₂	semidirect
60. H(3,𝔽₃)	#331	semidirect
94. C₃²	#4544	semidirect

H ≅ G/N

type

2. C₄⋊#6180

C₂

semidirect

60. H(3,𝔽₃)

#331

semidirect

94. C₃²

#4544

semidirect

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

subgroup	index
H(3,𝔽₃)×#127	2
C₄⋊#6180	2
C₃×#4544	3

subgroup

index

H(3,𝔽₃)×#127

C₄⋊#6180

C₃×#4544

Subgroups — #13135919

Conjugacy classes of subgroups

Complements

Maximal subgroups