Group #13143767 — subgroups

Subgroups — #13143767

Order 1728.

539 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 165 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	#11721715	C₂	1	2	yes	maximal
2	C₄×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#11720551	C₂	1	4	yes	maximal
4	C₃×#4682	—	3	4	no	maximal
5	C₂×#6229	—	2	2	no	—
6	C₂×C₈×H(3,𝔽₃)	C₂²	1	2	yes	—
7	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₄⋊#1084	—	2	2	no	—
9	C₄⋊#1084	C₂²	1	4	yes	—
10	C₃×#869	—	3	8	no	—
11	C₃²×C₄×C₈	D₃	1	4	yes	—
12	C₃×#818	—	3	16	no	—
13	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
14	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
15	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
16	C₂×#1084	—	2	4	no	—
17	#6229	—	4	4	no	—
18	C₈×H(3,𝔽₃)	Q₈	1	4	yes	complemented
19	C₃×#500	—	9	1	no	—
20	#4682	—	3	4	no	—
21	C₂×C₃×Dic₆	—	6	8	no	—
22	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
23	C₃²×C₄²	C₂×D₃	1	4	yes	—
24	C₃×C₄⋊Dic₃	—	3	16	no	—
25	C₃×C₄⋊Dic₃	—	6	8	no	—
26	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
27	C₄×H(3,𝔽₃)	D₈	1	2	yes	—
28	C₄×H(3,𝔽₃)	Q₁₆	1	2	yes	—
29	C₄×H(3,𝔽₃)	C₂×Q₈	1	2	yes	—
30	#1084	—	4	4	no	—
31	C₃×C₄⋊Q₈	—	9	2	no	—
32	#869	—	3	8	no	—
33	C₃×#107	—	9	4	no	—
34	C₃×C₄×C₈	—	3	4	no	—
35	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
36	#818	—	3	16	no	—
37	C₂×C₃²×C₄	D₁₂	1	4	yes	—
38	C₂×C₃²×C₄	D₁₂	1	4	yes	—
39	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
40	C₂×C₃×Dic₃	—	6	16	no	—
41	C₃×Dic₆	—	12	16	no	—
42	C₃²×C₈	Dic₆	1	16	yes	complemented
43	#500	—	9	1	no	Sylow
44	C₂×H(3,𝔽₃)	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×H(3,𝔽₃)	C₂×D₈	1	1	yes	characteristic
46	C₂×H(3,𝔽₃)	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₃×Q₈	—	18	2	no	—
48	C₂×Dic₆	—	6	8	no	—
49	C₂×C₃×C₈	—	3	8	no	—
50	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
51	C₃×C₄⋊C₄	—	9	4	no	—
52	C₃×C₄⋊C₄	—	18	2	no	—
53	C₃×C₄²	—	3	4	no	—
54	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
55	C₄⋊Dic₃	—	3	16	no	—
56	C₄⋊Dic₃	—	6	8	no	—
57	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
58	C₃²×C₄	D₂₄	1	8	yes	—
59	C₃²×C₄	Dic₁₂	1	8	yes	—
60	C₃²×C₄	C₂×Dic₆	1	8	yes	—
61	C₃×Dic₃	—	12	16	no	—
62	C₄⋊Q₈	—	9	2	no	—
63	#107	—	9	4	no	—
64	C₄×C₈	#272	1	1	yes	characteristic
65	H(3,𝔽₃)	#500	1	1	yes	characteristic, complemented, Sylow
66	C₃×Q₈	—	36	4	no	—
67	C₂×C₃×C₄	—	3	4	no	—
68	C₂×C₃×C₄	—	3	4	no	—
69	C₂×C₃×C₄	—	3	4	no	—
70	C₂×C₃×C₄	—	18	4	no	—
71	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
72	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
74	C₂×Dic₃	—	6	16	no	—
75	Dic₆	—	12	16	no	—
76	C₃×C₈	—	3	16	no	—
77	C₃×C₈	#634	1	4	yes	—
78	C₂×C₃²	#869	1	4	yes	—
79	C₂×C₃²	C₂×D₂₄	1	4	yes	—
80	C₂×C₃²	C₂×Dic₁₂	1	4	yes	—
81	C₂×Q₈	—	18	2	no	—
82	C₂×C₈	C₂×#272	1	2	yes	—
83	C₄⋊C₄	—	9	4	no	—
84	C₄⋊C₄	—	18	2	no	—
85	C₄²	C₂×#272	1	1	yes	characteristic
86	C₂²×C₃	—	3	4	no	—
87	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
88	C₃×C₄	—	3	8	no	—
89	C₃×C₄	—	3	8	no	—
90	C₃×C₄	—	3	8	no	—
91	C₃×C₄	—	36	4	no	—
92	C₃×C₄	#3745	1	2	yes	—
93	C₃×C₄	#3746	1	2	yes	—
94	C₃×C₄	C₂×#634	1	2	yes	—
95	Dic₃	—	12	16	no	—
96	C₃²	#4682	1	4	yes	complemented
97	Q₈	—	36	4	no	—
98	C₂×C₄	—	18	4	no	—
99	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
100	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
101	C₂×C₄	C₂²×#272	1	1	yes	characteristic
102	C₈	#6229	1	4	yes	complemented
103	C₂×C₃	—	3	4	no	—
104	C₂×C₃	—	3	4	no	—
105	C₂×C₃	—	3	4	no	—
106	C₂×C₃	#64078	1	1	yes	characteristic
107	C₂×C₃	C₂×#3745	1	1	yes	characteristic
108	C₂×C₃	C₂×#3746	1	1	yes	characteristic
109	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
110	C₄	—	36	4	no	—
111	C₄	#88529	1	2	yes	—
112	C₄	#88530	1	2	yes	—
113	C₄	C₂×#6229	1	2	yes	—
114	C₃	—	3	4	no	—
115	C₃	#10592381	1	1	yes	characteristic
116	C₂	#11721715	1	1	yes	characteristic
117	C₂	C₂×#88529	1	1	yes	characteristic
118	C₂	C₂×#88530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721715

C₂

yes

maximal

C₄×C₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11720551

C₂

yes

—

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

—

C₄⋊#1084

C₂²

yes

—

C₃×#869

—

C₃²×C₄×C₈

D₃

yes

—

C₃×#818

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×#1084

—

#6229

—

C₈×H(3,𝔽₃)

Q₈

yes

complemented

C₃×#500

—

#4682

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

D₈

yes

—

C₄×H(3,𝔽₃)

Q₁₆

yes

—

C₄×H(3,𝔽₃)

C₂×Q₈

yes

—

#1084

—

C₃×C₄⋊Q₈

—

#869

—

C₃×#107

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

#818

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

Dic₆

yes

complemented

#500

—

Sylow

C₂×H(3,𝔽₃)

C₄⋊Q₈

yes

characteristic

C₂×H(3,𝔽₃)

C₂×D₈

yes

characteristic

C₂×H(3,𝔽₃)

C₂×Q₁₆

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

D₂₄

yes

—

C₃²×C₄

Dic₁₂

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃×Dic₃

—

C₄⋊Q₈

—

#107

—

C₄×C₈

yes

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#634

yes

—

C₂×C₃²

#869

yes

—

C₂×C₃²

C₂×D₂₄

yes

—

C₂×C₃²

C₂×Dic₁₂

yes

—

C₂×Q₈

—

C₂×C₈

C₂×#272

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×#272

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3745

yes

—

C₃×C₄

#3746

yes

—

C₃×C₄

C₂×#634

yes

—

Dic₃

—

C₃²

#4682

yes

complemented

Q₈

—

C₂×C₄

—

C₂×C₄

C₄⋊#272

yes

characteristic

100

C₂×C₄

C₄⋊#272

yes

characteristic

101

C₂×C₄

C₂²×#272

yes

characteristic

102

C₈

#6229

yes

complemented

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

#64078

yes

characteristic

107

C₂×C₃

C₂×#3745

yes

characteristic

108

C₂×C₃

C₂×#3746

yes

characteristic

109

C₂²

C₂×C₄⋊#272

yes

characteristic

110

C₄

—

111

C₄

#88529

yes

—

112

C₄

#88530

yes

—

113

C₄

C₂×#6229

yes

—

114

C₃

—

115

C₃

#10592381

yes

characteristic

116

C₂

#11721715

yes

characteristic

117

C₂

C₂×#88529

yes

characteristic

118

C₂

C₂×#88530

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₄×C₈×H(3,𝔽₃)	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
hypercenter	C₃×C₄×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#500
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

subgroup

G/H

center, socle

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₄×C₈×H(3,𝔽₃)

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

hypercenter

C₃×C₄×C₈

C₃⋊D₃

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

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

N	H ≅ G/N	type
18. C₈×H(3,𝔽₃)	Q₈	semidirect
42. C₃²×C₈	Dic₆	semidirect
65. H(3,𝔽₃)	#500	semidirect
96. C₃²	#4682	semidirect
102. C₈	#6229	semidirect

H ≅ G/N

type

18. C₈×H(3,𝔽₃)

Q₈

semidirect

42. C₃²×C₈

Dic₆

semidirect

65. H(3,𝔽₃)

#500

semidirect

96. C₃²

#4682

semidirect

102. C₈

#6229

semidirect

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

subgroup	index
#11721715	2
C₄×C₈×H(3,𝔽₃)	2
#11720551	2
C₃×#4682	3

subgroup

index

#11721715

C₄×C₈×H(3,𝔽₃)

#11720551

C₃×#4682

Subgroups — #13143767

Conjugacy classes of subgroups

Complements

Maximal subgroups