Group #13135864 — subgroups

Subgroups — #13135864

Order 1728.

297 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88530	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#11720552	C₂	1	1	yes	maximal, characteristic
4	C₃×#4517	—	3	4	no	maximal
5	C₂×#6229	—	2	1	no	—
6	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#88538	—	2	1	no	—
8	#88530	—	2	1	no	—
9	#88530	C₄	1	2	yes	—
10	C₁₆×H(3,𝔽₃)	—	2	1	no	—
11	C₂×C₃×Dic₁₂	—	3	4	no	—
12	C₃²×M₃₂	D₃	1	4	yes	—
13	C₃×#819	—	3	4	no	—
14	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
15	C₂×#1084	—	4	1	no	—
16	#6229	—	2	2	no	—
17	#6229	—	4	1	no	—
18	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
19	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
20	#6180	—	4	1	no	—
21	C₃×#362	—	9	1	no	—
22	#4517	—	3	4	no	—
23	C₂×C₃×Dic₆	—	6	4	no	—
24	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
25	C₃×#208	—	6	4	no	—
26	C₃×Dic₁₂	—	3	8	no	—
27	C₃×Dic₁₂	—	6	4	no	—
28	C₃²×C₁₆	—	2	4	no	—
29	C₂²×H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic
30	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
32	#1084	—	4	2	no	—
33	C₂×C₃×Q₁₆	—	9	1	no	—
34	C₂×Dic₁₂	—	3	4	no	—
35	C₃×M₃₂	—	3	4	no	—
36	C₃×M₃₂	C₃⋊D₃	1	1	yes	characteristic
37	C₃×#108	—	9	1	no	—
38	#819	—	3	4	no	—
39	C₂×C₃²×C₄	D₁₂	1	4	yes	—
40	C₂×C₃×Dic₃	—	12	4	no	—
41	C₃×Dic₆	—	6	8	no	—
42	C₃×Dic₆	—	12	4	no	—
43	C₃²×C₈	C₄×D₃	1	4	yes	—
44	C₃²×C₈	#67	1	4	yes	—
45	C₃×C₃⋊C₈	—	12	4	no	—
46	#362	—	9	1	no	Sylow
47	C₂×H(3,𝔽₃)	—	2	1	no	—
48	C₂×H(3,𝔽₃)	#102	1	1	yes	characteristic
49	C₂×C₃×Q₈	—	18	1	no	—
50	C₂×Dic₆	—	6	4	no	—
51	C₃×Q₁₆	—	9	2	no	—
52	C₃×Q₁₆	—	18	1	no	—
53	C₃×M₁₆	—	18	1	no	—
54	C₂×C₃×C₈	—	3	4	no	—
55	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
56	#208	—	6	4	no	—
57	Dic₁₂	—	3	8	no	—
58	Dic₁₂	—	6	4	no	—
59	C₃×C₁₆	—	2	1	no	—
60	C₃×C₁₆	—	6	4	no	—
61	C₂²×C₃²	#204	1	4	yes	—
62	C₃²×C₄	D₂₄	1	4	yes	—
63	C₃²×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
64	C₃×Dic₃	—	12	8	no	—
65	C₂×Q₁₆	—	9	1	no	—
66	M₃₂	#272	1	1	yes	characteristic
67	#108	—	9	1	no	—
68	H(3,𝔽₃)	#362	1	1	yes	characteristic, complemented, Sylow
69	C₃×Q₈	—	18	2	no	—
70	C₃×Q₈	—	36	1	no	—
71	C₂×C₃×C₄	—	3	4	no	—
72	C₂×C₃×C₄	—	36	1	no	—
73	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
74	C₂×Dic₃	—	12	4	no	—
75	Dic₆	—	6	8	no	—
76	Dic₆	—	12	4	no	—
77	C₃×C₈	—	3	4	no	—
78	C₃×C₈	—	3	4	no	—
79	C₃×C₈	—	36	1	no	—
80	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
81	C₃×C₈	#638	1	1	yes	characteristic
82	C₃⋊C₈	—	12	4	no	—
83	C₂×C₃²	—	2	4	no	—
84	C₂×C₃²	#821	1	4	yes	—
85	C₂×Q₈	—	18	1	no	—
86	Q₁₆	—	9	2	no	—
87	Q₁₆	—	18	1	no	—
88	M₁₆	—	18	1	no	—
89	C₂×C₈	C₂×#272	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₂²×C₃	—	3	4	no	—
92	C₂²×C₃	#3744	1	1	yes	characteristic
93	C₃×C₄	—	3	4	no	—
94	C₃×C₄	—	3	4	no	—
95	C₃×C₄	—	36	2	no	—
96	C₃×C₄	#3745	1	1	yes	characteristic
97	C₃×C₄	#3752	1	1	yes	characteristic
98	Dic₃	—	12	8	no	—
99	C₃²	#4517	1	4	yes	complemented
100	Q₈	—	18	2	no	—
101	Q₈	—	36	1	no	—
102	C₂×C₄	—	36	1	no	—
103	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
104	C₈	—	36	1	no	—
105	C₈	C₄×#272	1	1	yes	characteristic
106	C₈	#6235	1	1	yes	characteristic
107	C₂×C₃	—	2	1	no	—
108	C₂×C₃	—	3	4	no	—
109	C₂×C₃	—	6	4	no	—
110	C₂×C₃	#63648	1	1	yes	characteristic
111	C₂²	#88528	1	1	yes	characteristic
112	C₄	—	36	2	no	—
113	C₄	#88529	1	1	yes	characteristic
114	C₄	#88542	1	1	yes	characteristic
115	C₃	—	3	4	no	—
116	C₃	#10589383	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	#11720554	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88530

C₂

yes

maximal, characteristic

H(3,𝔽₃)×M₃₂

C₂

yes

maximal, characteristic

#11720552

C₂

yes

maximal, characteristic

C₃×#4517

—

maximal

C₂×#6229

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

—

C₄

yes

—

C₁₆×H(3,𝔽₃)

—

C₂×C₃×Dic₁₂

—

C₃²×M₃₂

D₃

yes

—

C₃×#819

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

—

—

C₈×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

D₄

yes

—

—

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#208

—

C₃×Dic₁₂

—

C₃×Dic₁₂

—

C₃²×C₁₆

—

C₂²×H(3,𝔽₃)

SD₁₆

yes

characteristic

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

D₈

yes

characteristic

#1084

—

C₂×C₃×Q₁₆

—

C₂×Dic₁₂

—

C₃×M₃₂

—

C₃×M₃₂

C₃⋊D₃

yes

characteristic

C₃×#108

—

#819

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

C₄×D₃

yes

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

#362

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#102

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

#208

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃×C₁₆

—

C₂²×C₃²

#204

yes

—

C₃²×C₄

D₂₄

yes

—

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₃×Dic₃

—

C₂×Q₁₆

—

M₃₂

yes

—

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₄×C₃⋊D₃

yes

characteristic

C₃×C₈

#638

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#821

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

C₂×#272

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#3744

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

C₃²

#4517

yes

complemented

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

C₄⋊#272

yes

characteristic

104

C₈

—

105

C₈

C₄×#272

yes

characteristic

106

C₈

#6235

yes

characteristic

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

—

110

C₂×C₃

#63648

yes

characteristic

111

C₂²

#88528

yes

characteristic

112

C₄

—

113

C₄

#88529

yes

characteristic

114

C₄

#88542

yes

characteristic

115

C₃

—

116

C₃

#10589383

yes

characteristic

117

C₂

—

118

C₂

#11720554

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	#63648
commutator	C₈×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×M₃₂	C₂
prime radical	C₂²×H(3,𝔽₃)	SD₁₆
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#362
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

subgroup

G/H

center, socle

C₂×C₃

#63648

commutator

C₈×H(3,𝔽₃)

C₂×C₄

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting

H(3,𝔽₃)×M₃₂

C₂

prime radical

C₂²×H(3,𝔽₃)

SD₁₆

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₃×M₃₂

C₃⋊D₃

G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

N	H ≅ G/N	type
68. H(3,𝔽₃)	#362	semidirect
99. C₃²	#4517	semidirect

H ≅ G/N

type

68. H(3,𝔽₃)

#362

semidirect

99. C₃²

#4517

semidirect

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

subgroup	index
C₂×#88530	2
H(3,𝔽₃)×M₃₂	2
#11720552	2
C₃×#4517	3

subgroup

index

C₂×#88530

H(3,𝔽₃)×M₃₂

#11720552

C₃×#4517

Subgroups — #13135864

Conjugacy classes of subgroups

Complements

Maximal subgroups