Group #13135955 — subgroups

Subgroups — #13135955

Order 1728.

341 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 87 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×D₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11720552	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720545	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4563	—	3	4	no	maximal
5	C₂×D₄×H(3,𝔽₃)	—	2	1	no	—
6	D₈×H(3,𝔽₃)	—	2	1	no	—
7	D₈×H(3,𝔽₃)	C₄	1	2	yes	—
8	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	#88538	—	2	1	no	—
10	#88386	—	2	1	no	—
11	C₂×C₃²×D₈	D₃	1	4	yes	—
12	C₃×#819	—	3	4	no	—
13	C₃×#812	—	3	4	no	—
14	C₂³×H(3,𝔽₃)	—	4	1	no	—
15	D₄×H(3,𝔽₃)	—	2	2	no	—
16	D₄×H(3,𝔽₃)	—	4	1	no	—
17	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
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×#361	—	9	1	no	—
22	#4563	—	3	4	no	—
23	C₂×C₃²×D₄	—	2	4	no	—
24	C₃²×D₈	—	2	4	no	—
25	C₃²×D₈	Dic₃	1	8	yes	—
26	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
27	C₃×#208	—	6	4	no	—
28	C₃×C₃⋊C₁₆	—	6	4	no	—
29	C₂²×H(3,𝔽₃)	—	4	2	no	—
30	C₂²×H(3,𝔽₃)	—	8	2	no	—
31	C₂²×H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic
32	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
34	C₂×C₃×D₈	—	3	4	no	—
35	C₂×C₃×D₈	C₃⋊D₃	1	1	yes	characteristic
36	C₃×M₃₂	—	9	1	no	—
37	C₃×#108	—	9	1	no	—
38	#819	—	3	4	no	—
39	#812	—	3	4	no	—
40	C₂³×C₃²	—	4	4	no	—
41	C₃²×D₄	—	2	8	no	—
42	C₃²×D₄	—	4	4	no	—
43	C₂×C₃²×C₄	#67	1	4	yes	—
44	C₃²×C₈	C₂×Dic₃	1	4	yes	—
45	C₃²×C₈	#67	1	4	yes	—
46	C₃×C₃⋊C₈	—	12	4	no	—
47	#361	—	9	1	no	Sylow
48	C₂×H(3,𝔽₃)	—	2	1	no	—
49	C₂×H(3,𝔽₃)	—	8	2	no	—
50	C₂×H(3,𝔽₃)	#102	1	1	yes	characteristic
51	C₂×C₃×D₄	—	2	1	no	—
52	C₂×C₃×D₄	—	6	4	no	—
53	C₃×D₈	—	2	1	no	—
54	C₃×D₈	—	3	8	no	—
55	C₃×D₈	—	6	4	no	—
56	C₃×D₈	C₃⋊Dic₃	1	2	yes	—
57	C₃×M₁₆	—	18	1	no	—
58	C₂×C₃×C₈	—	3	4	no	—
59	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
60	#208	—	6	4	no	—
61	C₃×C₁₆	—	18	1	no	—
62	C₃⋊C₁₆	—	6	4	no	—
63	C₂²×C₃²	—	4	8	no	—
64	C₂²×C₃²	—	8	8	no	—
65	C₂²×C₃²	#214	1	4	yes	—
66	C₃²×C₄	#213	1	4	yes	—
67	C₃²×C₄	#217	1	4	yes	—
68	C₂×D₈	#272	1	1	yes	characteristic
69	M₃₂	—	9	1	no	—
70	#108	—	9	1	no	—
71	H(3,𝔽₃)	#361	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₃	—	4	1	no	—
73	C₂³×C₃	—	12	4	no	—
74	C₃×D₄	—	2	2	no	—
75	C₃×D₄	—	4	1	no	—
76	C₃×D₄	—	6	8	no	—
77	C₃×D₄	—	12	4	no	—
78	C₂×C₃×C₄	—	3	4	no	—
79	C₂×C₃×C₄	#638	1	1	yes	characteristic
80	C₃×C₈	—	3	4	no	—
81	C₃×C₈	—	3	4	no	—
82	C₃×C₈	—	36	1	no	—
83	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
84	C₃×C₈	#638	1	1	yes	characteristic
85	C₃⋊C₈	—	12	4	no	—
86	C₂×C₃²	—	2	4	no	—
87	C₂×C₃²	—	8	8	no	—
88	C₂×C₃²	#832	1	4	yes	—
89	C₂×D₄	—	2	1	no	—
90	D₈	—	2	1	no	—
91	D₈	#1084	1	2	yes	—
92	M₁₆	—	18	1	no	—
93	C₂×C₈	C₂×#272	1	1	yes	characteristic
94	C₁₆	—	18	1	no	—
95	C₂²×C₃	—	3	4	no	—
96	C₂²×C₃	—	4	2	no	—
97	C₂²×C₃	—	8	2	no	—
98	C₂²×C₃	—	12	8	no	—
99	C₂²×C₃	—	24	8	no	—
100	C₂²×C₃	#3754	1	1	yes	characteristic
101	C₃×C₄	—	3	4	no	—
102	C₃×C₄	—	3	4	no	—
103	C₃×C₄	#3753	1	1	yes	characteristic
104	C₃×C₄	#3757	1	1	yes	characteristic
105	C₃²	#4563	1	4	yes	complemented
106	C₂³	—	4	1	no	—
107	D₄	—	2	2	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	#6235	1	1	yes	characteristic
110	C₈	—	36	1	no	—
111	C₈	C₂×#1084	1	1	yes	characteristic
112	C₈	#6235	1	1	yes	characteristic
113	C₂×C₃	—	2	1	no	—
114	C₂×C₃	—	3	4	no	—
115	C₂×C₃	—	6	4	no	—
116	C₂×C₃	—	8	2	no	—
117	C₂×C₃	—	24	8	no	—
118	C₂×C₃	#63659	1	1	yes	characteristic
119	C₂²	—	4	2	no	—
120	C₂²	—	8	2	no	—
121	C₂²	#88546	1	1	yes	characteristic
122	C₄	#88545	1	1	yes	characteristic
123	C₄	#88552	1	1	yes	characteristic
124	C₃	—	3	4	no	—
125	C₃	#10589429	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	8	2	no	—
128	C₂	#11720578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11720552

C₂

yes

maximal, characteristic, complemented

#11720545

C₂

yes

maximal, characteristic, complemented

C₃×#4563

—

maximal

C₂×D₄×H(3,𝔽₃)

—

D₈×H(3,𝔽₃)

—

D₈×H(3,𝔽₃)

C₄

yes

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#88538

—

#88386

—

C₂×C₃²×D₈

D₃

yes

—

C₃×#819

—

C₃×#812

—

C₂³×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

D₄

yes

—

—

—

C₂×C₃²×D₄

—

C₃²×D₈

—

C₃²×D₈

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#208

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

SD₁₆

yes

characteristic

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

D₈

yes

characteristic

C₂×C₃×D₈

—

C₂×C₃×D₈

C₃⋊D₃

yes

characteristic

C₃×M₃₂

—

C₃×#108

—

#819

—

#812

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

#67

yes

—

C₃²×C₈

C₂×Dic₃

yes

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

#361

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#102

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

C₃⋊Dic₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

#208

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#214

yes

—

C₃²×C₄

#213

yes

—

C₃²×C₄

#217

yes

—

C₂×D₈

#272

yes

characteristic

M₃₂

—

#108

—

H(3,𝔽₃)

#361

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#638

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃⋊Dic₃

yes

characteristic

C₃×C₈

#638

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#832

yes

—

C₂×D₄

—

D₈

—

D₈

#1084

yes

—

M₁₆

—

C₂×C₈

C₂×#272

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

#3754

yes

characteristic

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

#3753

yes

characteristic

104

C₃×C₄

#3757

yes

characteristic

105

C₃²

#4563

yes

complemented

106

C₂³

—

107

D₄

—

108

D₄

—

109

C₂×C₄

#6235

yes

characteristic

110

C₈

—

111

C₈

C₂×#1084

yes

characteristic

112

C₈

#6235

yes

characteristic

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

#63659

yes

characteristic

119

C₂²

—

120

C₂²

—

121

C₂²

#88546

yes

characteristic

122

C₄

#88545

yes

characteristic

123

C₄

#88552

yes

characteristic

124

C₃

—

125

C₃

#10589429

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#11720578

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	#63659
commutator	C₈×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting, prime radical	C₂×D₈×H(3,𝔽₃)	C₂
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×D₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#361
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 ◁ C₂×C₃×D₈

subgroup

G/H

center, socle

C₂×C₃

#63659

commutator

C₈×H(3,𝔽₃)

C₂×C₄

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting, prime radical

C₂×D₈×H(3,𝔽₃)

C₂

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₂×C₃×D₈

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 ◁ C₂×C₃×D₈

N	H ≅ G/N	type
2. #11720552	C₂	semidirect
3. #11720545	C₂	semidirect
71. H(3,𝔽₃)	#361	semidirect
105. C₃²	#4563	semidirect

H ≅ G/N

type

2. #11720552

C₂

semidirect

3. #11720545

C₂

semidirect

71. H(3,𝔽₃)

#361

semidirect

105. C₃²

#4563

semidirect

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

subgroup	index
C₂×D₈×H(3,𝔽₃)	2
#11720552	2
#11720545	2
C₃×#4563	3

subgroup

index

C₂×D₈×H(3,𝔽₃)

#11720552

#11720545

C₃×#4563

Subgroups — #13135955

Conjugacy classes of subgroups

Complements

Maximal subgroups