Group #13135946 — subgroups

Subgroups — #13135946

Order 1728.

517 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 147 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×M₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11721826	C₂	1	2	yes	maximal
3	C₃×#4557	—	3	4	no	maximal
4	C₂²×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	M₁₆×H(3,𝔽₃)	—	2	2	no	—
6	C₂×C₈×H(3,𝔽₃)	C₄	1	2	yes	complemented
7	#88552	—	2	2	no	—
8	C₄⋊#1084	—	2	2	no	—
9	C₄×#1084	C₄	1	4	yes	complemented
10	C₂×C₃²×M₁₆	D₃	1	4	yes	—
11	C₃×#926	—	3	8	no	—
12	C₂³×H(3,𝔽₃)	D₄	1	1	yes	characteristic
13	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
15	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
16	C₂×C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
17	C₂×#1084	—	2	4	no	—
18	C₈×H(3,𝔽₃)	—	2	2	no	—
19	C₃×H(3,ℤ/4)	—	9	1	no	—
20	#4557	—	3	4	no	—
21	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
22	C₃²×M₁₆	—	2	8	no	—
23	C₂×C₃²×C₈	Dic₃	1	8	yes	complemented
24	C₃×#217	—	6	8	no	—
25	C₃×C₄⋊Dic₃	—	6	8	no	—
26	C₃×C₄×Dic₃	—	3	16	no	—
27	C₂²×H(3,𝔽₃)	—	4	1	no	—
28	C₂²×H(3,𝔽₃)	C₂²⋊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	1	yes	characteristic
32	C₄×H(3,𝔽₃)	C₄⋊C₄	1	2	yes	—
33	#1084	—	4	4	no	—
34	C₂×C₃×M₁₆	—	3	4	no	—
35	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
36	C₃×#117	—	9	2	no	—
37	#926	—	3	8	no	—
38	C₂³×C₃²	D₁₂	1	4	yes	—
39	C₂×C₃²×C₄	Dic₆	1	4	yes	—
40	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
41	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
42	C₂×C₃²×C₄	#67	1	8	yes	—
43	C₂×C₃×Dic₃	—	6	16	no	—
44	C₃²×C₈	—	2	8	no	—
45	H(3,ℤ/4)	—	9	1	no	Sylow
46	C₂×H(3,𝔽₃)	—	2	2	no	—
47	C₂×H(3,𝔽₃)	—	2	1	no	—
48	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
49	C₂²×C₃×C₄	—	3	4	no	—
50	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₃×M₁₆	—	2	2	no	—
52	C₃×M₁₆	—	6	8	no	—
53	C₂×C₃×C₈	—	3	8	no	—
54	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
55	C₃×C₄⋊C₄	—	18	2	no	—
56	C₃×C₂²⋊C₄	—	18	2	no	—
57	C₃×C₄²	—	9	4	no	—
58	#217	—	6	8	no	—
59	C₄⋊Dic₃	—	6	8	no	—
60	C₄×Dic₃	—	3	16	no	—
61	C₂²×C₃²	—	4	4	no	—
62	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
63	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
64	C₃²×C₄	C₄×Dic₃	1	4	yes	—
65	C₃²×C₄	Dic₃⋊C₄	1	8	yes	—
66	C₃²×C₄	#217	1	4	yes	—
67	C₃×Dic₃	—	12	16	no	—
68	C₂×M₁₆	#272	1	1	yes	characteristic
69	#117	—	9	2	no	—
70	H(3,𝔽₃)	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
71	C₂³×C₃	—	3	4	no	—
72	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₂×C₃×C₄	—	3	4	no	—
74	C₂×C₃×C₄	—	3	8	no	—
75	C₂×C₃×C₄	—	3	4	no	—
76	C₂×C₃×C₄	—	3	8	no	—
77	C₂×C₃×C₄	—	18	4	no	—
78	C₂×C₃×C₄	#634	1	1	yes	characteristic
79	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
80	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
81	C₂×C₃×C₄	#638	1	2	yes	—
82	C₂×Dic₃	—	6	16	no	—
83	C₃×C₈	—	2	2	no	—
84	C₃×C₈	—	6	8	no	—
85	C₂×C₃²	—	2	8	no	—
86	C₂×C₃²	—	2	4	no	—
87	C₂×C₃²	C₃⋊#95	1	4	yes	—
88	C₂²×C₄	C₂×#272	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	C₂×C₈	#1084	1	2	yes	complemented
91	C₄⋊C₄	—	18	2	no	—
92	C₂²⋊C₄	—	18	2	no	—
93	C₄²	—	9	4	no	—
94	C₂²×C₃	—	3	4	no	—
95	C₂²×C₃	—	3	8	no	—
96	C₂²×C₃	—	4	1	no	—
97	C₂²×C₃	—	12	4	no	—
98	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
99	C₂²×C₃	#3752	1	2	yes	—
100	C₃×C₄	—	3	4	no	—
101	C₃×C₄	—	3	8	no	—
102	C₃×C₄	—	3	4	no	—
103	C₃×C₄	—	36	4	no	—
104	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
105	C₃×C₄	#3750	1	2	yes	—
106	C₃×C₄	#3757	1	1	yes	characteristic
107	Dic₃	—	12	16	no	—
108	C₃²	#4557	1	4	yes	complemented
109	C₂³	C₄⋊#272	1	1	yes	characteristic
110	C₂×C₄	—	18	4	no	—
111	C₂×C₄	#6229	1	1	yes	characteristic
112	C₂×C₄	C₄×#272	1	2	yes	—
113	C₂×C₄	C₂×#1084	1	1	yes	characteristic
114	C₂×C₄	#6235	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₂×C₃	—	2	2	no	—
117	C₂×C₃	—	2	1	no	—
118	C₂×C₃	—	3	4	no	—
119	C₂×C₃	—	6	8	no	—
120	C₂×C₃	—	6	4	no	—
121	C₂×C₃	#63658	1	1	yes	characteristic
122	C₂²	—	4	1	no	—
123	C₂²	C₄⋊#1084	1	1	yes	characteristic
124	C₂²	#88542	1	2	yes	—
125	C₄	—	36	4	no	—
126	C₄	C₄×#1084	1	1	yes	characteristic
127	C₄	#88540	1	2	yes	—
128	C₄	#88552	1	1	yes	characteristic
129	C₃	—	3	4	no	—
130	C₃	#10589423	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	2	1	no	—
133	C₂	#11720574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11721826

C₂

yes

maximal

C₃×#4557

—

maximal

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

M₁₆×H(3,𝔽₃)

—

C₂×C₈×H(3,𝔽₃)

C₄

yes

—

—

C₄

yes

complemented

C₂×C₃²×M₁₆

D₃

yes

—

C₃×#926

—

C₂³×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

Q₈

yes

characteristic

C₂×#1084

—

C₈×H(3,𝔽₃)

—

C₃×H(3,ℤ/4)

—

#4557

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₃²×M₁₆

—

C₂×C₃²×C₈

Dic₃

yes

complemented

C₃×#217

—

C₃×C₄⋊Dic₃

—

C₃×C₄×Dic₃

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

—

C₂²×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

C₄²

yes

characteristic

C₄×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

—

#1084

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃⋊D₃

yes

characteristic

C₃×#117

—

#926

—

C₂³×C₃²

D₁₂

yes

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃×Dic₃

—

C₃²×C₈

—

H(3,ℤ/4)

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#95

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₂²×C₃²

(C₂×D₃)⋊C₄

yes

—

C₃²×C₄

C₄×Dic₃

yes

—

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃²×C₄

#217

yes

—

C₃×Dic₃

—

C₂×M₁₆

yes

—

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#634

yes

characteristic

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃⋊#95

yes

—

C₂²×C₄

C₂×#272

yes

characteristic

M₁₆

—

C₂×C₈

#1084

yes

complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

characteristic

C₂²×C₃

#3752

yes

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

C₄×C₃⋊Dic₃

yes

characteristic

105

C₃×C₄

#3750

yes

—

106

C₃×C₄

#3757

yes

characteristic

107

Dic₃

—

108

C₃²

#4557

yes

complemented

109

C₂³

C₄⋊#272

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

#6229

yes

characteristic

112

C₂×C₄

C₄×#272

yes

—

113

C₂×C₄

C₂×#1084

yes

characteristic

114

C₂×C₄

#6235

yes

—

115

C₈

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

—

121

C₂×C₃

#63658

yes

characteristic

122

C₂²

—

123

C₂²

C₄⋊#1084

yes

characteristic

124

C₂²

#88542

yes

—

125

C₄

—

126

C₄

C₄×#1084

yes

characteristic

127

C₄

#88540

yes

—

128

C₄

#88552

yes

characteristic

129

C₃

—

130

C₃

#10589423

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#11720574

yes

characteristic

	subgroup	G/H
center	C₃×C₄	C₄×C₃⋊Dic₃
commutator	C₄×H(3,𝔽₃)	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×M₁₆×H(3,𝔽₃)	C₂
socle	C₂×C₃	#63658
prime radical	C₂³×H(3,𝔽₃)	D₄
cosocle	C₂²×C₄×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	H(3,ℤ/4)
LCS	G ▷ C₄×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆

subgroup

G/H

center

C₃×C₄

C₄×C₃⋊Dic₃

commutator

C₄×H(3,𝔽₃)

C₄²

Frattini

C₂²×C₃×C₄

C₂×C₃⋊D₃

Fitting

C₂×M₁₆×H(3,𝔽₃)

C₂

socle

C₂×C₃

#63658

prime radical

C₂³×H(3,𝔽₃)

D₄

cosocle

C₂²×C₄×H(3,𝔽₃)

C₂²

hypercenter

C₂×C₃×M₁₆

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆

N	H ≅ G/N	type
6. C₂×C₈×H(3,𝔽₃)	C₄	semidirect
9. C₄×#1084	C₄	semidirect
23. C₂×C₃²×C₈	Dic₃	semidirect
70. H(3,𝔽₃)	H(3,ℤ/4)	semidirect
90. C₂×C₈	#1084	semidirect
108. C₃²	#4557	semidirect

H ≅ G/N

type

6. C₂×C₈×H(3,𝔽₃)

C₄

semidirect

9. C₄×#1084

C₄

semidirect

23. C₂×C₃²×C₈

Dic₃

semidirect

70. H(3,𝔽₃)

H(3,ℤ/4)

semidirect

90. C₂×C₈

#1084

semidirect

108. C₃²

#4557

semidirect

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

subgroup	index
C₂×M₁₆×H(3,𝔽₃)	2
#11721826	2
C₃×#4557	3

subgroup

index

C₂×M₁₆×H(3,𝔽₃)

#11721826

C₃×#4557

Subgroups — #13135946

Conjugacy classes of subgroups

Complements

Maximal subgroups