Group #13135840 — subgroups

Subgroups — #13135840

Order 1728.

275 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₈×#1084	C₂	1	1	yes	maximal, characteristic
3	C₂×#88386	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃⋊C₁₆	—	3	4	no	maximal
5	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄×#1084	C₄	1	1	yes	characteristic
7	C₂×#6180	C₄	1	1	yes	characteristic
8	C₁₆×H(3,𝔽₃)	—	2	1	no	—
9	#88386	—	2	1	no	—
10	C₂×C₃²×C₁₆	D₃	1	4	yes	—
11	C₃×C₈×Dic₃	—	3	4	no	—
12	C₂×C₃×C₃⋊C₁₆	—	3	4	no	—
13	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₂×#1084	C₈	1	1	yes	characteristic
15	C₂×#1084	C₈	1	1	yes	characteristic
16	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
18	#6180	—	2	1	no	—
19	C₃×C₄⋊C₁₆	—	9	1	no	—
20	Dic₃⋊C₁₆	—	3	4	no	—
21	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
22	C₃×C₄×Dic₃	—	3	4	no	—
23	C₂×C₃×C₃⋊C₈	—	3	4	no	—
24	C₃²×C₁₆	—	2	4	no	—
25	C₃×C₃⋊C₁₆	—	6	4	no	—
26	C₂²×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
27	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
28	C₄×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
29	#1084	—	2	1	no	—
30	#1084	C₁₆	1	2	yes	complemented
31	C₂×C₃×C₁₆	—	3	4	no	—
32	C₂×C₃×C₁₆	—	9	1	no	—
33	C₂×C₃×C₁₆	C₃⋊D₃	1	1	yes	characteristic
34	C₃×C₄×C₈	—	9	1	no	—
35	C₈×Dic₃	—	3	4	no	—
36	C₂×C₃⋊C₁₆	—	3	4	no	—
37	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
38	C₂×C₃×Dic₃	—	3	4	no	—
39	C₂×C₃×Dic₃	—	3	4	no	—
40	C₃²×C₈	Dic₆	1	4	yes	—
41	C₃²×C₈	#67	1	4	yes	—
42	C₃×C₃⋊C₈	—	6	4	no	—
43	C₄⋊C₁₆	—	9	1	no	Sylow
44	C₂×H(3,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic
45	C₂×H(3,𝔽₃)	C₂×C₁₆	1	1	yes	characteristic
46	C₂×H(3,𝔽₃)	M₃₂	1	1	yes	characteristic
47	C₂×C₃×C₈	—	3	4	no	—
48	C₂×C₃×C₈	—	9	1	no	—
49	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₃×C₄²	—	9	1	no	—
51	C₄×Dic₃	—	3	4	no	—
52	C₂×C₃⋊C₈	—	3	4	no	—
53	C₃×C₁₆	—	2	1	no	—
54	C₃×C₁₆	—	6	4	no	—
55	C₃×C₁₆	—	18	1	no	—
56	C₃⋊C₁₆	—	6	4	no	—
57	C₂²×C₃²	D₃×C₈	1	4	yes	—
58	C₃²×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
59	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
60	C₃×Dic₃	—	3	8	no	—
61	C₃×Dic₃	—	6	4	no	—
62	C₂×C₁₆	—	9	1	no	—
63	C₂×C₁₆	#272	1	1	yes	characteristic
64	C₄×C₈	—	9	1	no	—
65	H(3,𝔽₃)	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃×C₄	—	3	4	no	—
67	C₂×C₃×C₄	—	9	1	no	—
68	C₂×C₃×C₄	—	9	1	no	—
69	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
70	C₂×Dic₃	—	3	4	no	—
71	C₂×Dic₃	—	3	4	no	—
72	C₃×C₈	—	3	4	no	—
73	C₃×C₈	—	3	4	no	—
74	C₃×C₈	—	18	1	no	—
75	C₃×C₈	#634	1	1	yes	characteristic
76	C₃×C₈	#638	1	1	yes	characteristic
77	C₃⋊C₈	—	6	4	no	—
78	C₂×C₃²	D₃×C₁₆	1	4	yes	—
79	C₂×C₃²	#798	1	4	yes	—
80	C₂×C₃²	Dic₃⋊C₈	1	4	yes	—
81	C₂×C₈	—	9	1	no	—
82	C₂×C₈	C₂×#272	1	1	yes	characteristic
83	C₄²	—	9	1	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	—	18	1	no	—
86	C₂²×C₃	—	3	4	no	—
87	C₂²×C₃	C₈×C₃⋊D₃	1	1	yes	characteristic
88	C₃×C₄	—	3	4	no	—
89	C₃×C₄	—	3	4	no	—
90	C₃×C₄	—	9	2	no	—
91	C₃×C₄	—	18	1	no	—
92	C₃×C₄	#3743	1	1	yes	characteristic
93	C₃×C₄	#3750	1	1	yes	characteristic
94	Dic₃	—	3	8	no	—
95	Dic₃	—	6	4	no	—
96	C₃²	Dic₃⋊C₁₆	1	4	yes	complemented
97	C₂×C₄	—	9	1	no	—
98	C₂×C₄	—	9	1	no	—
99	C₂×C₄	C₄×#272	1	1	yes	characteristic
100	C₈	—	18	1	no	—
101	C₈	#6229	1	1	yes	characteristic
102	C₈	#6235	1	1	yes	characteristic
103	C₂×C₃	—	3	4	no	—
104	C₂×C₃	—	3	4	no	—
105	C₂×C₃	—	3	4	no	—
106	C₂×C₃	C₁₆×C₃⋊D₃	1	1	yes	characteristic
107	C₂×C₃	#63625	1	1	yes	characteristic
108	C₂×C₃	#63641	1	1	yes	characteristic
109	C₂²	C₈×#272	1	1	yes	characteristic
110	C₄	—	9	2	no	—
111	C₄	—	18	1	no	—
112	C₄	#88527	1	1	yes	characteristic
113	C₄	#88540	1	1	yes	characteristic
114	C₃	—	3	4	no	—
115	C₃	#10589367	1	1	yes	characteristic
116	C₂	C₁₆×#272	1	1	yes	characteristic
117	C₂	#11720511	1	1	yes	characteristic
118	C₂	#11720547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₈×#1084

C₂

yes

maximal, characteristic

C₂×#88386

C₂

yes

maximal, characteristic

C₃×Dic₃⋊C₁₆

—

maximal

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₄×#1084

C₄

yes

characteristic

C₂×#6180

C₄

yes

characteristic

C₁₆×H(3,𝔽₃)

—

#88386

—

C₂×C₃²×C₁₆

D₃

yes

—

C₃×C₈×Dic₃

—

C₂×C₃×C₃⋊C₁₆

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×#1084

C₈

yes

characteristic

C₂×#1084

C₈

yes

characteristic

C₈×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

Q₈

yes

characteristic

#6180

—

C₃×C₄⋊C₁₆

—

Dic₃⋊C₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×C₄×Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₃²×C₁₆

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

M₁₆

yes

characteristic

#1084

—

#1084

C₁₆

yes

complemented

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₃⋊D₃

yes

characteristic

C₃×C₄×C₈

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

C₄⋊C₁₆

—

Sylow

C₂×H(3,𝔽₃)

C₄⋊C₈

yes

characteristic

C₂×H(3,𝔽₃)

C₂×C₁₆

yes

characteristic

C₂×H(3,𝔽₃)

M₃₂

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄²

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

D₃×C₈

yes

—

C₃²×C₄

ΓL(1,𝔽₂₅)

yes

—

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×C₁₆

—

C₂×C₁₆

#272

yes

characteristic

C₄×C₈

—

H(3,𝔽₃)

C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃⋊C₈

—

C₂×C₃²

D₃×C₁₆

yes

—

C₂×C₃²

#798

yes

—

C₂×C₃²

Dic₃⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

C₂×#272

yes

characteristic

C₄²

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

C₈×C₃⋊D₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

Dic₃

—

C₃²

Dic₃⋊C₁₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×#272

yes

characteristic

100

C₈

—

101

C₈

#6229

yes

characteristic

102

C₈

#6235

yes

characteristic

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

C₁₆×C₃⋊D₃

yes

characteristic

107

C₂×C₃

#63625

yes

characteristic

108

C₂×C₃

#63641

yes

characteristic

109

C₂²

C₈×#272

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#88527

yes

characteristic

113

C₄

#88540

yes

characteristic

114

C₃

—

115

C₃

#10589367

yes

characteristic

116

C₂

C₁₆×#272

yes

characteristic

117

C₂

#11720511

yes

characteristic

118

C₂

#11720547

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂×C₁₆
Fitting	C₂×C₁₆×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₈×C₃⋊D₃
prime radical	C₂²×H(3,𝔽₃)	C₂×C₈
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₄⋊C₁₆
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

subgroup

G/H

center, Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

commutator

C₂×H(3,𝔽₃)

C₂×C₁₆

Fitting

C₂×C₁₆×H(3,𝔽₃)

C₂

socle

C₂²×C₃

C₈×C₃⋊D₃

prime radical

C₂²×H(3,𝔽₃)

C₂×C₈

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₂×C₃×C₁₆

C₃⋊D₃

hypocenter

H(3,𝔽₃)

C₄⋊C₁₆

LCS

G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₁₆

N	H ≅ G/N	type
30. #1084	C₁₆	semidirect
65. H(3,𝔽₃)	C₄⋊C₁₆	semidirect
96. C₃²	Dic₃⋊C₁₆	semidirect

H ≅ G/N

type

30. #1084

C₁₆

semidirect

65. H(3,𝔽₃)

C₄⋊C₁₆

semidirect

96. C₃²

Dic₃⋊C₁₆

semidirect

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

subgroup	index
C₂×C₁₆×H(3,𝔽₃)	2
C₈×#1084	2
C₂×#88386	2
C₃×Dic₃⋊C₁₆	3

subgroup

index

C₂×C₁₆×H(3,𝔽₃)

C₈×#1084

C₂×#88386

C₃×Dic₃⋊C₁₆

Subgroups — #13135840

Conjugacy classes of subgroups

Complements

Maximal subgroups