Group #13163881 — subgroups

Subgroups — #13163881

Order 1728.

2240 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 2240 conjugacy classes of subgroups — too many to draw. Only its 459 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	C₃³×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#11723976	C₂	1	3	yes	maximal, complemented
3	#11722204	C₂	1	3	yes	maximal, complemented
4	#10592942	—	3	13	no	maximal
5	C₃³×#41	—	2	3	no	—
6	C₂×C₃³×D₄	—	2	3	no	—
7	C₂×C₃³×D₄	C₂²	1	3	yes	—
8	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic, complemented
9	#88864	—	2	3	no	—
10	#88864	C₂²	1	3	yes	complemented
11	#88857	—	2	3	no	—
12	C₃²×H(5,𝔽₂)	D₃	1	13	yes	—
13	#64144	—	3	39	no	—
14	#63661	—	3	39	no	—
15	C₂³×C₃³	—	2	1	no	—
16	C₂³×C₃³	D₄	1	3	yes	—
17	C₂³×C₃³	C₂³	1	1	yes	characteristic
18	C₃³×Q₈	—	2	1	no	—
19	C₃³×D₄	—	2	3	no	—
20	C₃³×D₄	—	2	6	no	—
21	C₂×C₃³×C₄	—	2	3	no	—
22	C₂×C₃³×C₄	D₄	1	3	yes	—
23	C₂×C₃⋊(C₃⋊Dic₃)	—	2	3	no	—
24	C₂×C₃⋊(C₃⋊Dic₃)	—	2	3	no	—
25	#5243	—	9	13	no	—
26	C₃²×#41	—	2	39	no	—
27	C₂×C₃²×D₄	—	2	39	no	—
28	C₂×C₃²×D₄	C₂×D₃	1	39	yes	—
29	C₂²×C₃⋊Dic₃	—	3	13	no	—
30	#3757	—	3	39	no	—
31	#3757	—	6	39	no	—
32	#3750	—	6	39	no	—
33	C₂²×C₃³	—	2	3	no	—
34	C₂²×C₃³	—	4	1	no	—
35	C₂²×C₃³	—	4	3	no	—
36	C₂²×C₃³	—	8	1	no	—
37	C₂²×C₃³	C₂×D₄	1	3	yes	—
38	C₃³×C₄	—	2	3	no	—
39	C₃⋊(C₃⋊Dic₃)	—	4	3	no	—
40	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
41	C₃×H(5,𝔽₂)	C₃⋊D₃	1	13	yes	—
42	#935	—	9	39	no	—
43	#834	—	9	39	no	—
44	C₂³×C₃²	—	2	13	no	—
45	C₂³×C₃²	#67	1	39	yes	—
46	C₂³×C₃²	C₂²×D₃	1	13	yes	—
47	C₃²×Q₈	—	2	13	no	—
48	C₃²×D₄	—	2	39	no	—
49	C₃²×D₄	—	2	78	no	—
50	C₂×C₃²×C₄	—	2	39	no	—
51	C₂×C₃²×C₄	#67	1	39	yes	—
52	C₂×C₃⋊Dic₃	—	6	39	no	—
53	C₂×C₃⋊Dic₃	—	6	39	no	—
54	#458	—	27	1	no	Sylow
55	C₂×C₃³	—	2	3	no	—
56	C₂×C₃³	—	4	3	no	—
57	C₂×C₃³	#120	1	1	yes	characteristic
58	C₃×#41	—	2	39	no	—
59	C₂×C₃×D₄	—	2	39	no	—
60	C₂×C₃×D₄	C₂×C₃⋊D₃	1	39	yes	—
61	C₂²×Dic₃	—	9	13	no	—
62	#217	—	9	39	no	—
63	#217	—	18	39	no	—
64	Dic₃⋊C₄	—	18	39	no	—
65	C₂²×C₃²	—	2	39	no	—
66	C₂²×C₃²	—	4	13	no	—
67	C₂²×C₃²	—	4	39	no	—
68	C₂²×C₃²	—	8	13	no	—
69	C₂²×C₃²	C₂×#67	1	39	yes	—
70	C₃²×C₄	—	2	39	no	—
71	C₃⋊Dic₃	—	12	39	no	—
72	C₃⋊Dic₃	—	12	13	no	—
73	H(5,𝔽₂)	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
74	#123	—	27	3	no	—
75	#99	—	27	3	no	—
76	C₃³	#458	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₃	—	2	13	no	—
78	C₂³×C₃	#638	1	39	yes	—
79	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
80	C₃×Q₈	—	2	13	no	—
81	C₃×D₄	—	2	39	no	—
82	C₃×D₄	—	2	78	no	—
83	C₂×C₃×C₄	—	2	39	no	—
84	C₂×C₃×C₄	#638	1	39	yes	—
85	C₂×Dic₃	—	18	39	no	—
86	C₂×Dic₃	—	18	39	no	—
87	C₂×C₃²	—	2	39	no	—
88	C₂×C₃²	—	4	39	no	—
89	C₂×C₃²	#953	1	13	yes	—
90	#41	—	2	3	no	—
91	C₂×D₄	—	2	3	no	—
92	C₂×D₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
93	C₂²×C₄	—	27	1	no	—
94	C₄⋊C₄	—	54	3	no	—
95	C₂²⋊C₄	—	27	3	no	—
96	C₂²⋊C₄	—	54	3	no	—
97	C₂²×C₃	—	2	39	no	—
98	C₂²×C₃	—	4	13	no	—
99	C₂²×C₃	—	4	39	no	—
100	C₂²×C₃	—	8	13	no	—
101	C₂²×C₃	C₂×#638	1	39	yes	—
102	C₃×C₄	—	2	39	no	—
103	Dic₃	—	36	39	no	—
104	Dic₃	—	36	13	no	—
105	C₃²	#5243	1	13	yes	complemented
106	C₂³	—	2	1	no	—
107	C₂³	#6312	1	3	yes	—
108	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	D₄	—	2	3	no	—
111	D₄	—	2	6	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	54	3	no	—
114	C₂×C₄	—	54	3	no	—
115	C₂×C₄	#6312	1	3	yes	—
116	C₂×C₃	—	2	39	no	—
117	C₂×C₃	—	4	39	no	—
118	C₂×C₃	#64162	1	13	yes	—
119	C₂²	—	2	3	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	3	no	—
122	C₂²	—	8	1	no	—
123	C₂²	C₂×#6312	1	3	yes	—
124	C₄	—	2	3	no	—
125	C₄	—	108	3	no	—
126	C₄	—	108	1	no	—
127	C₃	#10592942	1	13	yes	complemented
128	C₂	—	2	3	no	—
129	C₂	—	4	3	no	—
130	C₂	#11723994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×H(5,𝔽₂)

C₂

yes

maximal, characteristic

#11723976

C₂

yes

maximal, complemented

#11722204

C₂

yes

maximal, complemented

#10592942

—

maximal

C₃³×#41

—

C₂×C₃³×D₄

—

C₂×C₃³×D₄

C₂²

yes

—

C₂²×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic, complemented

#88864

—

#88864

C₂²

yes

complemented

#88857

—

C₃²×H(5,𝔽₂)

D₃

yes

—

#64144

—

#63661

—

C₂³×C₃³

—

C₂³×C₃³

D₄

yes

—

C₂³×C₃³

C₂³

yes

characteristic

C₃³×Q₈

—

C₃³×D₄

—

C₃³×D₄

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

#5243

—

C₃²×#41

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

#3757

—

#3757

—

#3750

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

C₂×D₄

yes

—

C₃³×C₄

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃×H(5,𝔽₂)

C₃⋊D₃

yes

—

#935

—

#834

—

C₂³×C₃²

—

C₂³×C₃²

#67

yes

—

C₂³×C₃²

C₂²×D₃

yes

—

C₃²×Q₈

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#458

—

Sylow

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

#120

yes

characteristic

C₃×#41

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

#217

—

#217

—

Dic₃⋊C₄

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

H(5,𝔽₂)

C₃⋊(C₃⋊D₃)

yes

characteristic

#123

—

#99

—

C₃³

#458

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

#638

yes

—

C₂³×C₃

C₂²×C₃⋊D₃

yes

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#638

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#953

yes

—

#41

—

C₂×D₄

—

C₂×D₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

—

101

C₂²×C₃

C₂×#638

yes

—

102

C₃×C₄

—

103

Dic₃

—

104

Dic₃

—

105

C₃²

#5243

yes

complemented

106

C₂³

—

107

C₂³

#6312

yes

—

108

C₂³

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

109

Q₈

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#6312

yes

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

#64162

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂×#6312

yes

—

124

C₄

—

125

C₄

—

108

—

126

C₄

—

108

—

127

C₃

#10592942

yes

complemented

128

C₂

—

129

C₂

—

130

C₂

#11723994

yes

characteristic

	subgroup	G/H
center	C₂	#11723994
commutator, cosocle	C₂³×C₃³	C₂³
Frattini	C₂³	C₂²×C₃⋊(C₃⋊D₃)
Fitting, prime radical	C₃³×H(5,𝔽₂)	C₂
socle	C₂×C₃³	#120
hypercenter	H(5,𝔽₂)	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#458
LCS	G ▷ C₂³×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

#11723994

commutator, cosocle

C₂³×C₃³

C₂³

Frattini

C₂³

C₂²×C₃⋊(C₃⋊D₃)

Fitting, prime radical

C₃³×H(5,𝔽₂)

C₂

C₂×C₃³

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂³×C₃³ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
2. #11723976	C₂	semidirect
3. #11722204	C₂	semidirect
8. C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
10. #88864	C₂²	semidirect
76. C₃³	#458	semidirect
105. C₃²	#5243	semidirect
127. C₃	#10592942	semidirect

H ≅ G/N

type

2. #11723976

C₂

semidirect

3. #11722204

C₂

semidirect

8. C₂²×C₃⋊(C₃⋊Dic₃)

C₂²

semidirect

10. #88864

C₂²

semidirect

76. C₃³

#458

semidirect

105. C₃²

#5243

semidirect

127. C₃

#10592942

semidirect

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

subgroup	index
C₃³×H(5,𝔽₂)	2
#11723976	2
#11722204	2
#10592942	3

subgroup

index

C₃³×H(5,𝔽₂)

#11723976

#11722204

#10592942

Subgroups — #13163881

Conjugacy classes of subgroups

Complements

Maximal subgroups