Group #13147024 — subgroups

Subgroups — #13147024

Order 1728.

320 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11722389	C₂	1	2	yes	maximal
2	C₃×#131⋊C₉	C₂	1	1	yes	maximal, characteristic
3	#10592842	—	3	1	no	maximal
4	#10589505	—	3	3	no	maximal
5	C₃×#41⋊C₉	C₂²	1	1	yes	characteristic
6	C₃⋊#3688	—	2	2	no	—
7	#88521	—	4	1	no	maximal
8	C₃²×#131	D₃	1	1	yes	characteristic
9	#64160	—	3	2	no	—
10	C₂×#3746	—	3	1	no	—
11	#63690	—	3	6	no	—
12	#63651	—	3	2	no	—
13	#63646	—	3	1	no	—
14	#131⋊C₉	D₃	1	3	yes	—
15	#6226	—	4	2	no	—
16	C₃×Q₈⋊C₉	D₄	1	1	yes	characteristic
17	C₃×C₈×C₉	—	4	1	no	—
18	#5143	—	9	3	no	—
19	#5143	—	9	1	no	—
20	C₃²×#41	C₂×D₃	1	1	yes	characteristic
21	C₂×#634	—	3	2	no	—
22	C₃²×M₁₆	—	3	1	no	—
23	C₂×C₃²×C₈	—	3	1	no	—
24	#3756	—	6	2	no	—
25	#3754	—	6	2	no	—
26	#3748	—	3	2	no	—
27	#3746	—	6	1	no	—
28	#3746	—	6	1	no	—
29	#41⋊C₉	C₂×D₃	1	3	yes	—
30	#3688	—	6	6	no	—
31	Dic₃₆	—	12	3	no	—
32	C₃×C₄×C₉	—	4	1	no	—
33	C₃⋊Dic₉	—	8	2	no	—
34	C₃×#131	D₉	1	3	yes	—
35	C₃×#131	C₃⋊D₃	1	1	yes	characteristic
36	#951	—	9	6	no	—
37	#951	—	9	2	no	—
38	C₂×Dic₁₂	—	9	3	no	—
39	C₂×Dic₁₂	—	9	1	no	—
40	#824	—	9	6	no	—
41	#824	—	9	2	no	—
42	#819	—	9	3	no	—
43	#819	—	9	1	no	—
44	C₃²×Q₈	D₁₂	1	1	yes	characteristic
45	C₃²×D₄	—	3	1	no	—
46	C₂×C₃²×C₄	—	3	1	no	—
47	C₂×C₃⋊Dic₃	—	6	2	no	—
48	#634	—	6	2	no	—
49	#634	—	6	2	no	—
50	C₃²×C₈	—	3	1	no	—
51	C₃²×C₈	S₄	1	1	yes	characteristic
52	C₃⋊(C₃⋊C₈)	—	6	2	no	—
53	Dic₁₈	—	12	6	no	—
54	Q₈⋊C₉	D₁₂	1	3	yes	—
55	C₈×C₉	—	4	3	no	—
56	#473	—	27	1	no	Sylow
57	C₂×C₃×C₉	—	4	1	no	—
58	C₃×#41	C₂×D₉	1	3	yes	—
59	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
60	C₂×Dic₆	—	9	2	no	—
61	C₂×Dic₆	—	9	6	no	—
62	C₃×M₁₆	—	3	3	no	—
63	C₃×M₁₆	—	3	1	no	—
64	C₂×C₃×C₈	—	3	3	no	—
65	C₂×C₃×C₈	—	3	1	no	—
66	#216	—	18	6	no	—
67	#216	—	18	2	no	—
68	#214	—	18	6	no	—
69	#214	—	18	2	no	—
70	#208	—	9	2	no	—
71	#208	—	9	6	no	—
72	Dic₁₂	—	18	1	no	—
73	Dic₁₂	—	18	3	no	—
74	Dic₁₂	—	18	3	no	—
75	Dic₁₂	—	18	1	no	—
76	C₂²×C₃²	—	3	1	no	—
77	C₃²×C₄	—	3	1	no	—
78	C₃²×C₄	C₂×S₄	1	1	yes	characteristic
79	C₃⋊Dic₃	—	12	2	no	—
80	C₄×C₉	—	4	3	no	—
81	Dic₉	—	24	6	no	—
82	#137	—	27	2	no	—
83	C₂×Q₁₆	—	27	1	no	—
84	#131	C₃⋊D₉	1	1	yes	characteristic
85	#108	—	27	1	no	—
86	#101	—	27	2	no	—
87	C₃×C₉	—	4	1	no	Sylow
88	C₃×Q₈	D₃₆	1	3	yes	—
89	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
90	C₃×D₄	—	3	3	no	—
91	C₃×D₄	—	3	1	no	—
92	C₂×C₃×C₄	—	3	1	no	—
93	C₂×C₃×C₄	—	3	3	no	—
94	C₂×Dic₃	—	18	2	no	—
95	C₂×Dic₃	—	18	6	no	—
96	Dic₆	—	18	2	no	—
97	Dic₆	—	18	6	no	—
98	Dic₆	—	18	2	no	—
99	Dic₆	—	18	6	no	—
100	C₃×C₈	—	3	1	no	—
101	C₃×C₈	—	3	3	no	—
102	C₃×C₈	(C₂²⋊C₉)⋊C₂	1	3	yes	—
103	C₃×C₈	C₃⋊S₄	1	1	yes	characteristic
104	C₃⋊C₈	—	18	2	no	—
105	C₃⋊C₈	—	18	6	no	—
106	C₂×C₃²	—	6	1	no	—
107	C₂×C₃²	C₄⋊S₄	1	1	yes	characteristic
108	C₂×C₉	—	4	3	no	—
109	#41	C₂×C₃⋊D₉	1	1	yes	characteristic
110	C₂×Q₈	—	27	2	no	—
111	Q₁₆	—	54	1	no	—
112	Q₁₆	—	54	2	no	—
113	Q₁₆	—	54	1	no	—
114	SD₁₆	—	54	2	no	—
115	M₁₆	—	3	1	no	—
116	M₁₆	—	27	2	no	—
117	C₂×C₈	—	3	1	no	—
118	C₂²×C₃	—	3	1	no	—
119	C₂²×C₃	—	3	3	no	—
120	C₃×C₄	—	3	1	no	—
121	C₃×C₄	—	3	3	no	—
122	C₃×C₄	C₂×(C₂²⋊C₉)⋊C₂	1	3	yes	—
123	C₃×C₄	C₂×C₃⋊S₄	1	1	yes	characteristic
124	Dic₃	—	36	2	no	—
125	Dic₃	—	36	6	no	—
126	C₃²	#5403	1	1	yes	characteristic
127	C₉	—	4	3	no	—
128	Q₈	—	54	2	no	—
129	Q₈	—	54	2	no	—
130	Q₈	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
131	D₄	—	3	1	no	—
132	C₂×C₄	—	3	1	no	—
133	C₂×C₄	—	54	2	no	—
134	C₈	—	3	1	no	—
135	C₈	—	54	2	no	—
136	C₈	C₃⋊((C₂²⋊C₉)⋊C₂)	1	1	yes	characteristic
137	C₂×C₃	—	6	3	no	—
138	C₂×C₃	—	6	1	no	—
139	C₂×C₃	C₄⋊((C₂²⋊C₉)⋊C₂)	1	3	yes	—
140	C₂×C₃	C₄⋊(C₃⋊S₄)	1	1	yes	characteristic
141	C₂²	—	3	1	no	—
142	C₄	—	3	1	no	—
143	C₄	—	108	2	no	—
144	C₄	C₂×C₃⋊((C₂²⋊C₉)⋊C₂)	1	1	yes	characteristic
145	C₃	#10589505	1	3	yes	complemented
146	C₃	#10593637	1	1	yes	characteristic
147	C₂	—	6	1	no	—
148	C₂	C₄⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11722389

C₂

yes

maximal

C₃×#131⋊C₉

C₂

yes

maximal, characteristic

—

—

C₃×#41⋊C₉

C₂²

yes

—

—

C₃²×#131

D₃

yes

—

—

—

—

—

D₃

yes

—

#6226

—

C₃×Q₈⋊C₉

D₄

yes

characteristic

C₃×C₈×C₉

—

#5143

—

#5143

—

C₃²×#41

C₂×D₃

yes

characteristic

C₂×#634

—

C₃²×M₁₆

—

C₂×C₃²×C₈

—

—

—

—

—

—

C₂×D₃

yes

—

#3688

—

Dic₃₆

—

C₃×C₄×C₉

—

C₃⋊Dic₉

—

C₃×#131

D₉

yes

—

C₃×#131

C₃⋊D₃

yes

characteristic

#951

—

#951

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

—

—

—

—

C₃²×Q₈

D₁₂

yes

characteristic

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

—

C₃²×C₈

S₄

yes

characteristic

C₃⋊(C₃⋊C₈)

—

Dic₁₈

—

Q₈⋊C₉

D₁₂

yes

—

C₈×C₉

—

#473

—

Sylow

C₂×C₃×C₉

—

C₃×#41

C₂×D₉

yes

—

C₃×#41

C₂×C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

—

—

—

—

—

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₂²×C₃²

—

C₃²×C₄

—

C₃²×C₄

C₂×S₄

yes

characteristic

C₃⋊Dic₃

—

C₄×C₉

—

Dic₉

—

#137

—

C₂×Q₁₆

—

#131

C₃⋊D₉

yes

characteristic

#108

—

#101

—

C₃×C₉

—

Sylow

C₃×Q₈

D₃₆

yes

—

C₃×Q₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

Dic₆

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

(C₂²⋊C₉)⋊C₂

yes

—

103

C₃×C₈

C₃⋊S₄

yes

characteristic

104

C₃⋊C₈

—

105

C₃⋊C₈

—

106

C₂×C₃²

—

107

C₂×C₃²

C₄⋊S₄

yes

characteristic

108

C₂×C₉

—

109

#41

C₂×C₃⋊D₉

yes

characteristic

110

C₂×Q₈

—

111

Q₁₆

—

112

Q₁₆

—

113

Q₁₆

—

114

SD₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

C₂×(C₂²⋊C₉)⋊C₂

yes

—

123

C₃×C₄

C₂×C₃⋊S₄

yes

characteristic

124

Dic₃

—

125

Dic₃

—

126

C₃²

#5403

yes

characteristic

127

C₉

—

128

Q₈

—

129

Q₈

—

130

Q₈

C₄⋊(C₃⋊D₉)

yes

characteristic

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₈

—

135

C₈

—

136

C₈

C₃⋊((C₂²⋊C₉)⋊C₂)

yes

characteristic

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

C₄⋊((C₂²⋊C₉)⋊C₂)

yes

—

140

C₂×C₃

C₄⋊(C₃⋊S₄)

yes

characteristic

141

C₂²

—

142

C₄

—

143

C₄

—

108

—

144

C₄

C₂×C₃⋊((C₂²⋊C₉)⋊C₂)

yes

characteristic

145

C₃

#10589505

yes

complemented

146

C₃

#10593637

yes

characteristic

147

C₂

—

148

C₂

C₄⋊(C₃⋊((C₂²⋊C₉)⋊C₂))

yes

characteristic

	subgroup	G/H
center	C₂	C₄⋊(C₃⋊((C₂²⋊C₉)⋊C₂))
commutator, cosocle	C₃×#41⋊C₉	C₂²
Frattini	C₃×C₄	C₂×C₃⋊S₄
Fitting	C₃²×#131	D₃
socle	C₂×C₃²	C₄⋊S₄
prime radical	C₃²×#41	C₂×D₃
hypercenter	C₈	C₃⋊((C₂²⋊C₉)⋊C₂)
hypocenter	C₃×Q₈⋊C₉	D₄
LCS	G ▷ C₃×#41⋊C₉ ▷ C₃×Q₈⋊C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

center

C₂

C₄⋊(C₃⋊((C₂²⋊C₉)⋊C₂))

commutator, cosocle

C₃×#41⋊C₉

C₂²

Frattini

C₃×C₄

C₂×C₃⋊S₄

Fitting

C₃²×#131

D₃

socle

C₂×C₃²

C₄⋊S₄

prime radical

C₃²×#41

C₂×D₃

hypercenter

C₈

C₃⋊((C₂²⋊C₉)⋊C₂)

hypocenter

C₃×Q₈⋊C₉

D₄

LCS

G ▷ C₃×#41⋊C₉ ▷ C₃×Q₈⋊C₉

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
145. C₃	#10589505	semidirect

H ≅ G/N

type

145. C₃

#10589505

semidirect

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

subgroup	index
#11722389	2
C₃×#131⋊C₉	2
#10592842	3
#10589505	3
#88521	4

subgroup

index

C₃×#131⋊C₉

Subgroups — #13147024

Conjugacy classes of subgroups

Complements

Maximal subgroups