Group #13135933 — subgroups

Subgroups — #13135933

Order 1728.

495 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 161 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	C₂²×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#88538	C₂	1	2	yes	maximal
3	C₃×#4552	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₂×C₈×H(3,𝔽₃)	—	2	2	no	—
6	C₂×C₈×H(3,𝔽₃)	C₄	1	2	yes	—
7	#88538	—	2	2	no	—
8	#88538	C₄	1	4	yes	—
9	C₂×#6180	—	2	2	no	—
10	C₂²×C₃²×C₈	D₃	1	4	yes	—
11	C₂×C₃×#208	—	3	8	no	—
12	C₂³×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
13	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
14	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
16	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
17	C₈×H(3,𝔽₃)	—	2	2	no	—
18	#6180	—	2	4	no	—
19	C₃×#341	—	9	1	no	—
20	#4552	—	3	4	no	—
21	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
22	C₂×C₃²×C₈	—	2	8	no	—
23	C₂×C₃²×C₈	Dic₃	1	8	yes	—
24	C₃×#208	—	3	16	no	—
25	C₃×#208	—	6	8	no	—
26	C₂×C₃×C₃⋊C₈	—	6	8	no	—
27	C₂²×H(3,𝔽₃)	—	2	2	no	—
28	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	2	yes	—
29	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
30	C₄×H(3,𝔽₃)	C₄²	1	1	yes	characteristic
31	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	2	yes	—
32	C₄×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×M₁₆	—	9	2	no	—
34	C₂²×C₃×C₈	—	3	4	no	—
35	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
36	C₂×#208	—	3	8	no	—
37	C₂³×C₃²	Dic₆	1	4	yes	—
38	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
39	C₂×C₃²×C₄	D₁₂	1	4	yes	—
40	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
41	C₂×C₃²×C₄	#67	1	8	yes	—
42	C₃²×C₈	—	2	8	no	—
43	C₃×C₃⋊C₈	—	6	16	no	—
44	#341	—	9	1	no	Sylow
45	C₂×H(3,𝔽₃)	—	2	2	no	—
46	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
47	C₂×H(3,𝔽₃)	#108	1	2	yes	—
48	C₂²×C₃×C₄	—	3	4	no	—
49	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₃×M₁₆	—	9	4	no	—
51	C₃×M₁₆	—	18	2	no	—
52	C₂×C₃×C₈	—	2	2	no	—
53	C₂×C₃×C₈	—	3	8	no	—
54	C₂×C₃×C₈	—	6	8	no	—
55	C₂×C₃×C₈	—	18	2	no	—
56	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
57	#208	—	3	16	no	—
58	#208	—	6	8	no	—
59	C₂×C₃⋊C₈	—	6	8	no	—
60	C₂²×C₃²	—	2	8	no	—
61	C₂²×C₃²	Dic₃⋊C₄	1	8	yes	—
62	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
63	C₃²×C₄	C₄×Dic₃	1	4	yes	—
64	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
65	C₃²×C₄	#217	1	4	yes	—
66	C₂×M₁₆	—	9	2	no	—
67	C₂²×C₈	#272	1	1	yes	characteristic
68	H(3,𝔽₃)	#341	1	1	yes	characteristic, complemented, Sylow
69	C₂³×C₃	—	3	4	no	—
70	C₂³×C₃	#634	1	1	yes	characteristic
71	C₂×C₃×C₄	—	3	8	no	—
72	C₂×C₃×C₄	—	3	4	no	—
73	C₂×C₃×C₄	—	3	4	no	—
74	C₂×C₃×C₄	—	3	8	no	—
75	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
76	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
78	C₂×C₃×C₄	#638	1	2	yes	—
79	C₃×C₈	—	2	2	no	—
80	C₃×C₈	—	6	8	no	—
81	C₃×C₈	—	18	4	no	—
82	C₃⋊C₈	—	6	16	no	—
83	C₂×C₃²	—	2	8	no	—
84	C₂×C₃²	#819	1	8	yes	—
85	C₂×C₃²	C₃⋊#95	1	4	yes	—
86	C₂²×C₄	C₂×#272	1	1	yes	characteristic
87	M₁₆	—	9	4	no	—
88	M₁₆	—	18	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	18	2	no	—
91	C₂×C₈	#1084	1	2	yes	—
92	C₂²×C₃	—	2	2	no	—
93	C₂²×C₃	—	3	8	no	—
94	C₂²×C₃	—	3	4	no	—
95	C₂²×C₃	—	6	8	no	—
96	C₂²×C₃	#3750	1	2	yes	—
97	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
98	C₃×C₄	—	3	4	no	—
99	C₃×C₄	—	3	8	no	—
100	C₃×C₄	—	3	4	no	—
101	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
102	C₃×C₄	#3752	1	2	yes	—
103	C₃×C₄	#3757	1	1	yes	characteristic
104	C₃²	#4552	1	4	yes	complemented
105	C₂³	#6229	1	1	yes	characteristic
106	C₂×C₄	C₄×#272	1	2	yes	—
107	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
108	C₂×C₄	C₂×#1084	1	1	yes	characteristic
109	C₂×C₄	#6235	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	18	4	no	—
112	C₂×C₃	—	2	2	no	—
113	C₂×C₃	—	3	8	no	—
114	C₂×C₃	—	3	4	no	—
115	C₂×C₃	—	6	8	no	—
116	C₂×C₃	#63646	1	2	yes	—
117	C₂×C₃	#63658	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#88540	1	2	yes	—
120	C₂²	C₄⋊#1084	1	1	yes	characteristic
121	C₄	C₄×#1084	1	1	yes	characteristic
122	C₄	#88542	1	2	yes	—
123	C₄	#88552	1	1	yes	characteristic
124	C₃	—	3	4	no	—
125	C₃	#10589418	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	#11720552	1	2	yes	—
128	C₂	#11720574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₂×#88538

C₂

yes

maximal

C₃×#4552

—

maximal

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×C₈×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₄

yes

—

#88538

—

#88538

C₄

yes

—

C₂×#6180

—

C₂²×C₃²×C₈

D₃

yes

—

C₂×C₃×#208

—

C₂³×H(3,𝔽₃)

Q₈

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₈×H(3,𝔽₃)

—

#6180

—

C₃×#341

—

#4552

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

Dic₃

yes

—

C₃×#208

—

C₃×#208

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₄⋊C₄

yes

—

C₂²×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

C₄²

yes

characteristic

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

—

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

—

C₂²×C₃×C₈

C₃⋊D₃

yes

characteristic

C₂×#208

—

C₂³×C₃²

Dic₆

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃²×C₈

—

C₃×C₃⋊C₈

—

#341

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#95

yes

characteristic

C₂×H(3,𝔽₃)

#108

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

Dic₃⋊C₄

yes

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₃²×C₄

C₄×Dic₃

yes

—

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₃²×C₄

#217

yes

—

C₂×M₁₆

—

C₂²×C₈

yes

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

#634

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#819

yes

—

C₂×C₃²

C₃⋊#95

yes

—

C₂²×C₄

C₂×#272

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#1084

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#3750

yes

—

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

C₄×C₃⋊Dic₃

yes

characteristic

102

C₃×C₄

#3752

yes

—

103

C₃×C₄

#3757

yes

characteristic

104

C₃²

#4552

yes

complemented

105

C₂³

#6229

yes

characteristic

106

C₂×C₄

C₄×#272

yes

—

107

C₂×C₄

C₄⋊#272

yes

characteristic

108

C₂×C₄

C₂×#1084

yes

characteristic

109

C₂×C₄

#6235

yes

—

110

C₈

—

111

C₈

—

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

C₂×C₃

#63646

yes

—

117

C₂×C₃

#63658

yes

characteristic

118

C₂²

—

119

C₂²

#88540

yes

—

120

C₂²

C₄⋊#1084

yes

characteristic

121

C₄

C₄×#1084

yes

characteristic

122

C₄

#88542

yes

—

123

C₄

#88552

yes

characteristic

124

C₃

—

125

C₃

#10589418

yes

characteristic

126

C₂

—

127

C₂

#11720552

yes

—

128

C₂

#11720574

yes

characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)
commutator	C₄×H(3,𝔽₃)	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂²×C₈×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₄⋊(C₃⋊Dic₃)
prime radical	C₂³×H(3,𝔽₃)	Q₈
cosocle	C₂²×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#341
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

subgroup

G/H

center

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

commutator

C₄×H(3,𝔽₃)

C₄²

Frattini

C₂²×C₃×C₄

C₂×C₃⋊D₃

Fitting

C₂²×C₈×H(3,𝔽₃)

C₂

socle

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

prime radical

C₂³×H(3,𝔽₃)

Q₈

cosocle

C₂²×C₄×H(3,𝔽₃)

C₂²

hypercenter

C₂²×C₃×C₈

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

N	H ≅ G/N	type
68. H(3,𝔽₃)	#341	semidirect
104. C₃²	#4552	semidirect

H ≅ G/N

type

68. H(3,𝔽₃)

#341

semidirect

104. C₃²

#4552

semidirect

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

subgroup	index
C₂²×C₈×H(3,𝔽₃)	2
C₂×#88538	2
C₃×#4552	3

subgroup

index

C₂²×C₈×H(3,𝔽₃)

C₂×#88538

C₃×#4552

Subgroups — #13135933

Conjugacy classes of subgroups

Complements

Maximal subgroups