Group #13167650 — subgroups

Subgroups — #13167650

Order 1728.

202 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 51 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	#11724362	C₂	1	2	yes	maximal
2	#11723093	C₂	1	1	yes	maximal, characteristic
3	C₈×C₃⋊#157	C₂	1	1	yes	maximal, characteristic
4	C₃×#63768	C₂	1	1	yes	maximal, characteristic
5	#11722807	C₂	1	2	yes	maximal
6	#10593533	—	3	1	no	maximal
7	C₄⋊(C₃⋊#157)	C₂²	1	2	yes	—
8	C₄×C₃⋊#157	C₂²	1	1	yes	characteristic
9	#88988	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊#157	C₂²	1	2	yes	—
11	#88946	—	2	2	no	—
12	#88945	—	2	2	no	—
13	C₃×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
14	#64243	—	3	2	no	—
15	#63768	—	3	1	no	—
16	#63768	D₃	1	1	yes	characteristic
17	C₈×#157	—	3	1	no	—
18	#63746	—	3	2	no	—
19	C₂×C₃⋊#157	—	2	2	no	—
20	C₂×C₃⋊#157	D₄	1	2	yes	—
21	C₂×C₃×#157	—	2	2	no	—
22	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
23	C₃⋊#622	Q₈	1	2	yes	—
24	C₃³×C₈	Q₈	1	1	yes	characteristic
25	C₃×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
26	#4858	—	9	1	no	maximal
27	C₄⋊#157	—	3	2	no	—
28	C₄⋊#157	C₂×D₃	1	2	yes	—
29	C₄×#157	—	3	1	no	—
30	#3787	—	3	1	no	—
31	#3777	—	6	2	no	—
32	#3777	—	6	2	no	—
33	C₈×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
34	C₃×D₃×C₈	—	12	1	no	—
35	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
36	C₃⋊#157	—	2	2	no	—
37	C₃⋊#157	—	4	2	no	—
38	C₃×#157	—	4	2	no	—
39	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₃×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
41	#890	—	9	2	no	—
42	C₃×#106	—	9	1	no	—
43	#817	—	9	1	no	—
44	C₈×Dic₃	—	9	1	no	—
45	#807	—	9	2	no	—
46	C₂×#157	—	2	2	no	—
47	C₂×#157	—	3	2	no	—
48	C₂×#157	—	6	2	no	—
49	C₄×C₃⋊D₃	C₂²×D₃	1	1	yes	characteristic
50	C₃×C₄×D₃	—	12	1	no	—
51	#622	—	3	2	no	—
52	C₃²×C₈	—	4	1	no	—
53	C₃²×C₈	—	8	1	no	—
54	C₃²×C₈	Dic₆	1	1	yes	characteristic
55	C₃⋊(C₃⋊C₈)	Dic₆	1	1	yes	characteristic
56	C₃×C₃⋊C₈	—	12	1	no	—
57	#499	—	27	1	no	Sylow
58	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
59	C₃×C₃⋊D₃	#135	1	2	yes	—
60	C₂×C₃×C₈	—	9	1	no	—
61	C₃×C₄⋊C₄	—	9	2	no	—
62	C₄⋊Dic₃	—	9	2	no	—
63	C₄⋊Dic₃	—	18	2	no	—
64	Dic₃⋊C₄	—	18	2	no	—
65	C₄×Dic₃	—	9	1	no	—
66	C₂×C₃⋊C₈	—	9	1	no	—
67	D₃×C₈	—	12	1	no	—
68	C₂×C₃⋊D₃	D₃×D₄	1	1	yes	characteristic
69	C₂×C₃×D₃	—	12	1	no	—
70	#157	—	4	2	no	—
71	#157	—	6	2	no	—
72	#157	—	12	2	no	—
73	C₃²×C₄	—	4	1	no	—
74	C₃²×C₄	—	8	1	no	—
75	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
76	C₃⋊Dic₃	D₃×Q₈	1	1	yes	characteristic
77	C₃×Dic₃	—	12	1	no	—
78	#125	—	27	2	no	—
79	#107	—	27	2	no	—
80	#106	—	9	1	no	—
81	#106	—	27	1	no	—
82	C₄×C₈	—	27	1	no	—
83	C₃³	#499	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₃×C₄	—	9	1	no	—
85	C₂×C₃×C₄	—	18	2	no	—
86	C₂×Dic₃	—	9	2	no	—
87	C₂×Dic₃	—	18	2	no	—
88	C₄×D₃	—	12	1	no	—
89	C₃×C₈	—	4	1	no	—
90	C₃×C₈	—	8	1	no	—
91	C₃×C₈	—	9	1	no	—
92	C₃×C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
93	C₃⋊C₈	—	9	2	no	—
94	C₃⋊C₈	—	12	1	no	—
95	C₂×C₃²	—	4	1	no	—
96	C₂×C₃²	—	8	1	no	—
97	C₂×C₃²	#888	1	1	yes	characteristic
98	C₃⋊D₃	#916	1	2	yes	—
99	C₃×D₃	—	12	2	no	—
100	C₂×C₈	—	9	1	no	—
101	C₂×C₈	—	27	1	no	—
102	C₄⋊C₄	—	9	2	no	—
103	C₄⋊C₄	—	27	2	no	—
104	C₄⋊C₄	—	54	2	no	—
105	C₄⋊C₄	—	54	2	no	—
106	C₄²	—	27	1	no	—
107	C₂²×C₃	—	9	1	no	—
108	C₂×D₃	—	12	1	no	—
109	C₃×C₄	—	4	1	no	—
110	C₃×C₄	—	8	1	no	—
111	C₃×C₄	—	9	1	no	—
112	C₃×C₄	—	36	2	no	—
113	C₃×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
114	Dic₃	—	12	1	no	—
115	Dic₃	—	18	2	no	—
116	Dic₃	—	36	2	no	—
117	C₃²	—	4	1	no	—
118	C₃²	—	8	1	no	—
119	C₃²	#4858	1	1	yes	characteristic, complemented
120	C₂×C₄	—	9	1	no	—
121	C₂×C₄	—	18	2	no	—
122	C₂×C₄	—	27	2	no	—
123	C₂×C₄	—	54	2	no	—
124	C₈	—	9	1	no	—
125	C₈	—	27	2	no	—
126	C₈	C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic
127	C₂×C₃	—	4	1	no	—
128	C₂×C₃	—	8	1	no	—
129	C₂×C₃	—	9	2	no	—
130	C₂×C₃	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
131	D₃	—	12	2	no	—
132	C₂²	—	9	1	no	—
133	C₄	—	9	1	no	—
134	C₄	—	36	2	no	—
135	C₄	—	54	2	no	—
136	C₄	—	108	2	no	—
137	C₄	C₂×C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic
138	C₃	—	4	1	no	—
139	C₃	—	8	1	no	—
140	C₃	#10593533	1	1	yes	characteristic, complemented
141	C₂	—	9	2	no	—
142	C₂	#11724364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11724362

C₂

yes

maximal

#11723093

C₂

yes

maximal, characteristic

C₈×C₃⋊#157

C₂

yes

maximal, characteristic

C₃×#63768

C₂

yes

maximal, characteristic

#11722807

C₂

yes

—

C₂²

yes

—

C₄×C₃⋊#157

C₂²

yes

characteristic

#88988

C₂²

yes

characteristic

C₃×C₄⋊#157

C₂²

yes

—

#88946

—

#88945

—

C₃×C₈×C₃⋊D₃

C₂²

yes

—

—

D₃

yes

characteristic

C₈×#157

—

#63746

—

C₂×C₃⋊#157

—

C₂×C₃⋊#157

D₄

yes

—

C₂×C₃×#157

—

C₃×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₃⋊#622

Q₈

yes

—

C₃³×C₈

Q₈

yes

characteristic

C₃×C₃⋊(C₃⋊C₈)

Q₈

yes

—

—

C₂×D₃

yes

—

—

—

—

—

C₈×C₃⋊D₃

C₂×D₃

yes

characteristic

C₃×D₃×C₈

—

C₂×C₃×C₃⋊D₃

C₂×D₄

yes

—

—

—

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃×C₃⋊Dic₃

C₂×Q₈

yes

—

—

—

C₈×Dic₃

—

—

—

—

—

C₄×C₃⋊D₃

C₂²×D₃

yes

characteristic

C₃×C₄×D₃

—

#622

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

Dic₆

yes

characteristic

C₃⋊(C₃⋊C₈)

Dic₆

yes

characteristic

C₃×C₃⋊C₈

—

#499

—

Sylow

C₂×C₃³

C₄⋊Q₈

yes

characteristic

C₃×C₃⋊D₃

#135

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

D₃×C₈

—

C₂×C₃⋊D₃

D₃×D₄

yes

characteristic

C₂×C₃×D₃

—

#157

—

#157

—

#157

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

characteristic

C₃⋊Dic₃

D₃×Q₈

yes

characteristic

C₃×Dic₃

—

—

—

—

—

C₄×C₈

—

C₃³

#499

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

PSU(3,𝔽₂)

yes

characteristic

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#888

yes

characteristic

C₃⋊D₃

#916

yes

—

C₃×D₃

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₂²×C₃

—

108

C₂×D₃

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

114

Dic₃

—

115

Dic₃

—

116

Dic₃

—

117

C₃²

—

118

C₃²

—

119

C₃²

#4858

yes

characteristic, complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₈

—

125

C₈

—

126

C₈

C₃⋊PSU(3,𝔽₂)

yes

characteristic

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

C₄⋊PSU(3,𝔽₂)

yes

characteristic

131

D₃

—

132

C₂²

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

108

—

137

C₄

C₂×C₃⋊PSU(3,𝔽₂)

yes

characteristic

138

C₃

—

139

C₃

—

140

C₃

#10593533

yes

characteristic, complemented

141

C₂

—

142

C₂

#11724364

yes

characteristic

	subgroup	G/H
center	C₂	#11724364
commutator, cosocle	C₃×C₄×C₃⋊D₃	C₂³
Frattini	C₄	C₂×C₃⋊PSU(3,𝔽₂)
Fitting	C₃³×C₈	Q₈
socle	C₂×C₃³	C₄⋊Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂×D₄
hypercenter	C₈	C₃⋊PSU(3,𝔽₂)
hypocenter	C₃³	#499
LCS	G ▷ C₃×C₄×C₃⋊D₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

center

C₂

#11724364

commutator, cosocle

C₃×C₄×C₃⋊D₃

C₂³

Frattini

C₄

C₂×C₃⋊PSU(3,𝔽₂)

Fitting

C₃³×C₈

Q₈

socle

C₂×C₃³

C₄⋊Q₈

prime radical

C₂×C₃×C₃⋊D₃

C₂×D₄

C₈

C₃³

G ▷ C₃×C₄×C₃⋊D₃ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
83. C₃³	#499	semidirect
119. C₃²	#4858	semidirect
140. C₃	#10593533	semidirect

H ≅ G/N

type

83. C₃³

#499

semidirect

119. C₃²

#4858

semidirect

140. C₃

#10593533

semidirect

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

subgroup	index
#11724362	2
#11723093	2
C₈×C₃⋊#157	2
C₃×#63768	2
#11722807	2
#10593533	3
#4858	9

subgroup

index

C₈×C₃⋊#157

Subgroups — #13167650

Conjugacy classes of subgroups

Complements

Maximal subgroups