Group #13165785 — subgroups

Subgroups — #13165785

Order 1728.

202 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 50 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	#11722808	C₂	1	1	yes	maximal, characteristic
3	C₃×#63746	C₂	1	1	yes	maximal, characteristic
4	#11722714	C₂	1	2	yes	maximal
5	#11722703	C₂	1	1	yes	maximal, characteristic
6	#10593533	—	3	1	no	maximal
7	C₄⋊(C₃⋊#157)	C₂²	1	2	yes	—
8	C₄×C₃⋊#157	C₂²	1	1	yes	characteristic
9	C₃×C₄⋊#157	C₂²	1	2	yes	—
10	C₃×#3787	C₂²	1	1	yes	characteristic
11	#88946	—	2	2	no	—
12	#88945	—	2	2	no	—
13	C₃⋊D₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#64243	—	3	2	no	—
15	#63768	—	3	2	no	—
16	C₈×#157	—	3	1	no	—
17	#63746	—	3	1	no	—
18	#63746	D₃	1	1	yes	characteristic
19	C₂×C₃⋊#157	—	2	2	no	—
20	C₂×C₃⋊#157	D₄	1	1	yes	characteristic
21	C₂×C₃⋊#157	D₄	1	1	yes	characteristic
22	C₂×C₃×#157	—	2	2	no	—
23	C₃×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
24	C₃×#622	Q₈	1	2	yes	—
25	C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	1	1	yes	characteristic
26	C₃²×C₃⋊C₈	Q₈	1	1	yes	characteristic
27	#4878	—	9	1	no	maximal
28	C₄⋊#157	—	3	2	no	—
29	C₄⋊#157	C₂×D₃	1	2	yes	—
30	C₄×#157	—	3	1	no	—
31	#3787	C₂×D₃	1	1	yes	characteristic
32	#3777	—	6	2	no	—
33	#3777	—	6	2	no	—
34	C₈×C₃⋊D₃	—	3	1	no	—
35	D₃×C₃⋊C₈	—	12	1	no	—
36	C₂×C₃×C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
37	C₃⋊#157	—	2	1	no	—
38	C₃⋊#157	—	2	1	no	—
39	C₃⋊#157	—	4	2	no	—
40	C₃×#157	—	4	2	no	—
41	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
42	C₃×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
43	#890	—	9	2	no	—
44	C₃×#107	—	9	1	no	—
45	#818	—	9	1	no	—
46	C₈×Dic₃	—	9	1	no	—
47	#808	—	9	2	no	—
48	C₂×#157	—	2	2	no	—
49	C₂×#157	—	3	1	no	—
50	C₂×#157	—	3	1	no	—
51	C₂×#157	—	6	2	no	—
52	C₄×C₃⋊D₃	C₂²×D₃	1	1	yes	characteristic
53	C₃×C₄×D₃	—	12	1	no	—
54	#622	Dic₆	1	2	yes	—
55	C₃²×C₈	—	3	1	no	—
56	C₃⋊(C₃⋊C₈)	—	3	1	no	—
57	C₃⋊(C₃⋊C₈)	—	12	1	no	—
58	C₃⋊(C₃⋊C₈)	—	24	1	no	—
59	C₃×C₃⋊C₈	—	4	1	no	—
60	#499	—	27	1	no	Sylow
61	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
62	C₃×C₃⋊D₃	#135	1	1	yes	characteristic
63	C₃×C₃⋊D₃	#135	1	1	yes	characteristic
64	C₂×C₃×C₈	—	9	1	no	—
65	C₃×C₄⋊C₄	—	9	2	no	—
66	C₄⋊Dic₃	—	9	2	no	—
67	C₄⋊Dic₃	—	18	2	no	—
68	Dic₃⋊C₄	—	18	2	no	—
69	C₄×Dic₃	—	9	1	no	—
70	C₂×C₃⋊C₈	—	9	1	no	—
71	D₃×C₈	—	36	1	no	—
72	C₂×C₃⋊D₃	D₃×D₄	1	1	yes	characteristic
73	C₂×C₃×D₃	—	12	1	no	—
74	#157	—	4	2	no	—
75	#157	—	6	1	no	—
76	#157	—	6	1	no	—
77	#157	—	12	2	no	—
78	C₃²×C₄	—	4	1	no	—
79	C₃²×C₄	—	8	1	no	—
80	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
81	C₃⋊Dic₃	C₂×Dic₆	1	1	yes	characteristic
82	C₃×Dic₃	—	12	1	no	—
83	#125	—	27	2	no	—
84	#107	—	9	1	no	—
85	#107	—	27	1	no	—
86	#106	—	27	2	no	—
87	C₄×C₈	—	27	1	no	—
88	C₃³	#499	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₃×C₄	—	9	1	no	—
90	C₂×C₃×C₄	—	18	2	no	—
91	C₂×Dic₃	—	9	1	no	—
92	C₂×Dic₃	—	9	1	no	—
93	C₂×Dic₃	—	18	2	no	—
94	C₄×D₃	—	12	1	no	—
95	C₃×C₈	—	9	2	no	—
96	C₃×C₈	—	12	1	no	—
97	C₃⋊C₈	—	9	1	no	—
98	C₃⋊C₈	—	36	1	no	—
99	C₃⋊C₈	—	72	1	no	—
100	C₃⋊C₈	PSU(3,𝔽₂)	1	1	yes	characteristic
101	C₂×C₃²	—	4	1	no	—
102	C₂×C₃²	—	8	1	no	—
103	C₂×C₃²	#888	1	1	yes	characteristic
104	C₃⋊D₃	#912	1	1	yes	characteristic
105	C₃⋊D₃	#919	1	1	yes	characteristic
106	C₃×D₃	—	12	1	no	—
107	C₃×D₃	—	12	1	no	—
108	C₂×C₈	—	9	1	no	—
109	C₂×C₈	—	27	1	no	—
110	C₄⋊C₄	—	9	2	no	—
111	C₄⋊C₄	—	27	2	no	—
112	C₄⋊C₄	—	54	2	no	—
113	C₄⋊C₄	—	54	2	no	—
114	C₄²	—	27	1	no	—
115	C₂²×C₃	—	9	1	no	—
116	C₂×D₃	—	12	1	no	—
117	C₃×C₄	—	4	1	no	—
118	C₃×C₄	—	8	1	no	—
119	C₃×C₄	—	9	1	no	—
120	C₃×C₄	—	36	2	no	—
121	C₃×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
122	Dic₃	—	12	1	no	—
123	Dic₃	—	18	1	no	—
124	Dic₃	—	18	1	no	—
125	Dic₃	—	36	2	no	—
126	C₃²	—	4	1	no	—
127	C₃²	—	8	1	no	—
128	C₃²	#4878	1	1	yes	characteristic, complemented
129	C₂×C₄	—	9	1	no	—
130	C₂×C₄	—	18	2	no	—
131	C₂×C₄	—	27	1	no	—
132	C₂×C₄	—	27	1	no	—
133	C₂×C₄	—	54	2	no	—
134	C₈	—	3	1	no	—
135	C₈	—	9	2	no	—
136	C₈	—	27	1	no	—
137	C₂×C₃	—	4	1	no	—
138	C₂×C₃	—	8	1	no	—
139	C₂×C₃	—	9	1	no	—
140	C₂×C₃	—	9	1	no	—
141	C₂×C₃	C₄⋊PSU(3,𝔽₂)	1	1	yes	characteristic
142	D₃	—	12	1	no	—
143	D₃	—	12	1	no	—
144	C₂²	—	9	1	no	—
145	C₄	—	9	1	no	—
146	C₄	—	36	2	no	—
147	C₄	—	54	1	no	—
148	C₄	—	54	1	no	—
149	C₄	—	108	2	no	—
150	C₄	D₃×PSU(3,𝔽₂)	1	1	yes	characteristic
151	C₃	—	4	1	no	—
152	C₃	—	8	1	no	—
153	C₃	#10593533	1	1	yes	characteristic, complemented
154	C₂	—	9	1	no	—
155	C₂	—	9	1	no	—
156	C₂	#11724154	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11724362

C₂

yes

maximal

#11722808

C₂

yes

maximal, characteristic

C₃×#63746

C₂

yes

maximal, characteristic

#11722714

C₂

yes

maximal

#11722703

C₂

yes

maximal, characteristic

#10593533

—

maximal

C₄⋊(C₃⋊#157)

C₂²

yes

—

C₄×C₃⋊#157

C₂²

yes

characteristic

C₃×C₄⋊#157

C₂²

yes

—

C₃×#3787

C₂²

yes

characteristic

#88946

—

#88945

—

C₃⋊D₃×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

characteristic

C₂×C₃⋊#157

—

C₂×C₃⋊#157

D₄

yes

characteristic

C₂×C₃⋊#157

D₄

yes

characteristic

C₂×C₃×#157

—

C₃×C₄×C₃⋊D₃

C₂³

yes

characteristic

C₃×#622

Q₈

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

Q₈

yes

characteristic

C₃²×C₃⋊C₈

Q₈

yes

—

—

C₂×D₃

yes

—

C₄×#157

—

#3787

C₂×D₃

yes

characteristic

#3777

—

#3777

—

C₈×C₃⋊D₃

—

D₃×C₃⋊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

Dic₆

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

#499

—

Sylow

C₂×C₃³

C₄⋊Q₈

yes

C₃×C₃⋊D₃

yes

C₃×C₃⋊D₃

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

—

—

—

—

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×Q₈

yes

characteristic

C₃⋊Dic₃

C₂×Dic₆

yes

characteristic

C₃×Dic₃

—

—

—

—

—

C₄×C₈

—

C₃³

#499

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

100

C₃⋊C₈

PSU(3,𝔽₂)

yes

characteristic

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

#888

yes

characteristic

104

C₃⋊D₃

#912

yes

characteristic

105

C₃⋊D₃

#919

yes

characteristic

106

C₃×D₃

—

107

C₃×D₃

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₂²×C₃

—

116

C₂×D₃

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

122

Dic₃

—

123

Dic₃

—

124

Dic₃

—

125

Dic₃

—

126

C₃²

—

127

C₃²

—

128

C₃²

#4878

yes

characteristic, complemented

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

C₄⋊PSU(3,𝔽₂)

yes

characteristic

142

D₃

—

143

D₃

—

144

C₂²

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

108

—

150

C₄

D₃×PSU(3,𝔽₂)

yes

characteristic

151

C₃

—

152

C₃

—

153

C₃

#10593533

yes

characteristic, complemented

154

C₂

—

155

C₂

—

156

C₂

#11724154

yes

characteristic

	subgroup	G/H
center	C₂	#11724154
commutator, cosocle	C₃×C₄×C₃⋊D₃	C₂³
Frattini, hypercenter	C₄	D₃×PSU(3,𝔽₂)
Fitting	C₃³×C₄	C₂×Q₈
socle	C₂×C₃³	C₄⋊Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂×D₄
hypocenter	C₃³	#499
LCS	G ▷ C₃×C₄×C₃⋊D₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center

C₂

#11724154

commutator, cosocle

C₃×C₄×C₃⋊D₃

C₂³

Frattini, hypercenter

C₄

D₃×PSU(3,𝔽₂)

Fitting

C₃³×C₄

C₂×Q₈

socle

C₂×C₃³

C₄⋊Q₈

prime radical

C₂×C₃×C₃⋊D₃

C₂×D₄

hypocenter

C₃³

#499

LCS

G ▷ C₃×C₄×C₃⋊D₃ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
88. C₃³	#499	semidirect
128. C₃²	#4878	semidirect
153. C₃	#10593533	semidirect

H ≅ G/N

type

88. C₃³

#499

semidirect

128. C₃²

#4878

semidirect

153. C₃

#10593533

semidirect

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

subgroup	index
#11724362	2
#11722808	2
C₃×#63746	2
#11722714	2
#11722703	2
#10593533	3
#4878	9

subgroup

index

Subgroups — #13165785

Conjugacy classes of subgroups

Complements

Maximal subgroups