Group #13143768 — subgroups

Subgroups — #13143768

Order 1728.

473 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 137 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	#11721716	C₂	1	2	yes	maximal
2	C₄×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#11720551	C₂	1	2	yes	maximal
4	#11720550	C₂	1	2	yes	maximal
5	C₃×#4683	—	3	4	no	maximal
6	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₄⋊#1084	C₂²	1	4	yes	—
10	#88540	—	2	4	no	—
11	C₃×#870	—	3	8	no	—
12	C₃²×C₄×C₈	D₃	1	4	yes	—
13	C₃×#818	—	3	8	no	—
14	C₃×#817	—	3	8	no	—
15	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
16	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
17	C₂×#1084	—	2	4	no	—
18	C₈×H(3,𝔽₃)	Q₈	1	2	yes	—
19	C₈×H(3,𝔽₃)	Q₈	1	2	yes	—
20	C₃×#499	—	9	1	no	—
21	#4683	—	3	4	no	—
22	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
23	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
24	C₃²×C₄²	C₂×D₃	1	4	yes	—
25	C₃×C₄⋊Dic₃	—	3	16	no	—
26	C₃×Dic₃⋊C₄	—	6	16	no	—
27	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
28	C₄×H(3,𝔽₃)	—	2	2	no	—
29	C₄×H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
30	C₄×H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
31	#1084	—	4	4	no	—
32	C₃×#125	—	9	2	no	—
33	#870	—	3	8	no	—
34	C₃×#107	—	9	2	no	—
35	C₃×#106	—	9	2	no	—
36	C₃×C₄×C₈	—	3	4	no	—
37	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
38	#818	—	3	8	no	—
39	#817	—	3	8	no	—
40	C₂×C₃²×C₄	D₁₂	1	8	yes	—
41	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
42	C₂×C₃×Dic₃	—	6	16	no	—
43	C₃²×C₈	Dic₆	1	8	yes	—
44	C₃²×C₈	Dic₆	1	8	yes	—
45	#499	—	9	1	no	Sylow
46	C₂×H(3,𝔽₃)	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×H(3,𝔽₃)	#135	1	2	yes	—
48	C₂×C₃×C₈	—	3	4	no	—
49	C₂×C₃×C₈	—	3	4	no	—
50	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	9	4	no	—
53	C₃×C₄⋊C₄	—	18	4	no	—
54	C₃×C₄²	—	3	4	no	—
55	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₄⋊Dic₃	—	3	16	no	—
57	Dic₃⋊C₄	—	6	16	no	—
58	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
59	C₃²×C₄	—	2	8	no	—
60	C₃²×C₄	C₂×Dic₆	1	4	yes	—
61	C₃²×C₄	C₂×Dic₆	1	4	yes	—
62	C₃×Dic₃	—	12	16	no	—
63	#125	—	9	2	no	—
64	#107	—	9	2	no	—
65	#106	—	9	2	no	—
66	C₄×C₈	#272	1	1	yes	characteristic
67	H(3,𝔽₃)	#499	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃×C₄	—	3	8	no	—
69	C₂×C₃×C₄	—	3	4	no	—
70	C₂×C₃×C₄	—	18	4	no	—
71	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
72	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
73	C₂×Dic₃	—	6	16	no	—
74	C₃×C₈	—	3	8	no	—
75	C₃×C₈	—	3	8	no	—
76	C₃×C₈	#634	1	2	yes	—
77	C₃×C₈	#634	1	2	yes	—
78	C₂×C₃²	#869	1	4	yes	—
79	C₂×C₃²	#904	1	8	yes	—
80	C₂×C₈	C₂×#272	1	1	yes	characteristic
81	C₂×C₈	C₂×#272	1	1	yes	characteristic
82	C₄⋊C₄	—	9	4	no	—
83	C₄⋊C₄	—	18	4	no	—
84	C₄²	C₂×#272	1	1	yes	characteristic
85	C₂²×C₃	—	3	4	no	—
86	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	—	3	4	no	—
89	C₃×C₄	—	3	4	no	—
90	C₃×C₄	—	6	8	no	—
91	C₃×C₄	—	36	4	no	—
92	C₃×C₄	C₂×#634	1	1	yes	characteristic
93	C₃×C₄	C₂×#634	1	1	yes	characteristic
94	Dic₃	—	12	16	no	—
95	C₃²	#4683	1	4	yes	complemented
96	C₂×C₄	—	18	4	no	—
97	C₂×C₄	C₄⋊#272	1	2	yes	—
98	C₂×C₄	C₂²×#272	1	1	yes	characteristic
99	C₈	#6229	1	2	yes	—
100	C₈	#6229	1	2	yes	—
101	C₂×C₃	—	3	8	no	—
102	C₂×C₃	—	3	4	no	—
103	C₂×C₃	#64078	1	1	yes	characteristic
104	C₂×C₃	#64113	1	2	yes	—
105	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	36	4	no	—
108	C₄	C₂×#6229	1	1	yes	characteristic
109	C₄	C₂×#6229	1	1	yes	characteristic
110	C₃	—	3	4	no	—
111	C₃	#10592382	1	1	yes	characteristic
112	C₂	#11721715	1	1	yes	characteristic
113	C₂	#11721779	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11721716

C₂

yes

maximal

C₄×C₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11720551

C₂

yes

maximal

#11720550

C₂

yes

maximal

C₃×#4683

—

maximal

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

C₂²

yes

—

#88540

—

C₃×#870

—

C₃²×C₄×C₈

D₃

yes

—

C₃×#818

—

C₃×#817

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×#1084

—

C₈×H(3,𝔽₃)

Q₈

yes

—

C₈×H(3,𝔽₃)

Q₈

yes

—

C₃×#499

—

#4683

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×Q₈

yes

characteristic

C₄×H(3,𝔽₃)

C₂×Q₈

yes

—

—

—

—

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

#818

—

#817

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃²×C₈

Dic₆

yes

—

#499

—

Sylow

C₂×H(3,𝔽₃)

C₄⋊Q₈

yes

characteristic

C₂×H(3,𝔽₃)

#135

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃×Dic₃

—

#125

—

#107

—

#106

—

C₄×C₈

yes

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#634

yes

—

C₃×C₈

#634

yes

—

C₂×C₃²

#869

yes

—

C₂×C₃²

#904

yes

—

C₂×C₈

yes

C₂×C₈

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

—

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

C₃²

#4683

yes

complemented

C₂×C₄

—

C₂×C₄

C₄⋊#272

yes

—

C₂×C₄

C₂²×#272

yes

characteristic

C₈

#6229

yes

—

100

C₈

#6229

yes

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

#64078

yes

characteristic

104

C₂×C₃

#64113

yes

—

105

C₂²

C₂×C₄⋊#272

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×#6229

yes

characteristic

109

C₄

C₂×#6229

yes

characteristic

110

C₃

—

111

C₃

#10592382

yes

characteristic

112

C₂

#11721715

yes

characteristic

113

C₂

#11721779

yes

—

	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)	#499
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
67. H(3,𝔽₃)	#499	semidirect
95. C₃²	#4683	semidirect

H ≅ G/N

type

67. H(3,𝔽₃)

#499

semidirect

95. C₃²

#4683

semidirect

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

subgroup	index
#11721716	2
C₄×C₈×H(3,𝔽₃)	2
#11720551	2
#11720550	2
C₃×#4683	3

subgroup

index

C₄×C₈×H(3,𝔽₃)

Subgroups — #13143768

Conjugacy classes of subgroups

Complements

Maximal subgroups