Group #13135916 — subgroups

Subgroups — #13135916

Order 1728.

420 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	2	yes	maximal, complemented
3	#10589410	—	3	1	no	maximal
4	#10588221	—	3	3	no	maximal
5	C₂×C₃×D₄×C₉	—	2	2	no	—
6	C₂×C₃×D₄×C₉	C₄	1	2	yes	—
7	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
9	C₃²×#127	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	2	no	—
11	C₉×#127	D₃	1	3	yes	—
12	C₄⋊(C₉⋊C₈)	—	3	6	no	—
13	C₂³×C₃×C₉	—	2	2	no	—
14	C₃×D₄×C₉	—	4	2	no	—
15	C₃×D₄×C₉	—	4	2	no	—
16	C₃×D₄×C₉	—	4	2	no	—
17	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
19	C₃⋊(C₉⋊C₈)	—	4	2	no	—
20	#4544	—	9	3	no	—
21	#4544	—	9	1	no	—
22	C₂×C₃²×D₄	—	2	2	no	—
23	C₂×C₃²×D₄	Dic₃	1	2	yes	—
24	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
26	C₂×D₄×C₉	—	2	6	no	—
27	C₂×D₄×C₉	Dic₃	1	6	yes	—
28	C₄²×C₉	C₂×D₃	1	3	yes	—
29	C₂×C₉⋊C₈	—	6	6	no	—
30	C₂²×C₃×C₉	—	4	2	no	—
31	C₂²×C₃×C₉	—	4	4	no	—
32	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₄×C₉	—	2	1	no	—
34	C₃×C₄×C₉	D₈	1	2	yes	—
35	C₃×C₄×C₉	SD₁₆	1	2	yes	—
36	C₃×#127	D₉	1	3	yes	—
37	C₃×#127	C₃⋊D₃	1	1	yes	characteristic
38	C₄⋊(C₃⋊C₈)	—	9	6	no	—
39	C₄⋊(C₃⋊C₈)	—	9	2	no	—
40	C₂³×C₃²	—	2	2	no	—
41	C₃²×D₄	—	4	2	no	—
42	C₃²×D₄	—	4	2	no	—
43	C₃²×D₄	—	4	2	no	—
44	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
45	C₂×C₃²×C₄	#67	1	2	yes	—
46	C₂³×C₉	—	2	6	no	—
47	C₃⋊(C₃⋊C₈)	—	12	2	no	—
48	D₄×C₉	—	4	6	no	—
49	D₄×C₉	—	4	6	no	—
50	D₄×C₉	—	4	6	no	—
51	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
52	C₂×C₄×C₉	#67	1	6	yes	—
53	C₉⋊C₈	—	12	6	no	—
54	#331	—	27	1	no	Sylow
55	C₂×C₃×C₉	—	8	2	no	—
56	C₂×C₃×C₉	#100	1	1	yes	characteristic
57	C₂×C₃×C₉	#102	1	2	yes	—
58	C₂×C₃×D₄	—	2	6	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	Dic₉	1	6	yes	—
61	C₂×C₃×D₄	C₃⋊Dic₃	1	2	yes	—
62	C₃×C₄²	C₂×D₉	1	3	yes	—
63	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
64	C₂×C₃⋊C₈	—	18	6	no	—
65	C₂×C₃⋊C₈	—	18	2	no	—
66	C₂²×C₃²	—	4	4	no	—
67	C₂²×C₃²	—	4	2	no	—
68	C₂²×C₃²	#217	1	1	yes	characteristic
69	C₃²×C₄	—	2	1	no	—
70	C₃²×C₄	#213	1	2	yes	—
71	C₃²×C₄	#214	1	2	yes	—
72	C₂²×C₉	—	4	12	no	—
73	C₂²×C₉	—	4	6	no	—
74	C₂²×C₉	#217	1	3	yes	—
75	C₄×C₉	—	2	3	no	—
76	C₄×C₉	#213	1	6	yes	—
77	C₄×C₉	#214	1	6	yes	—
78	#127	C₃⋊D₉	1	1	yes	characteristic
79	C₄⋊C₈	—	27	2	no	—
80	C₃×C₉	#331	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₃	—	2	6	no	—
82	C₂³×C₃	—	2	2	no	—
83	C₃×D₄	—	4	2	no	—
84	C₃×D₄	—	4	6	no	—
85	C₃×D₄	—	4	6	no	—
86	C₃×D₄	—	4	6	no	—
87	C₃×D₄	—	4	2	no	—
88	C₃×D₄	—	4	2	no	—
89	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
90	C₂×C₃×C₄	#611	1	6	yes	—
91	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
92	C₂×C₃×C₄	#638	1	2	yes	—
93	C₃⋊C₈	—	36	6	no	—
94	C₃⋊C₈	—	36	2	no	—
95	C₂×C₃²	—	8	2	no	—
96	C₂×C₃²	#832	1	2	yes	—
97	C₂×C₃²	#833	1	1	yes	characteristic
98	C₂×C₉	—	8	6	no	—
99	C₂×C₉	#832	1	6	yes	—
100	C₂×C₉	#833	1	3	yes	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	C₃⋊Dic₉	1	2	yes	—
103	C₂×C₈	—	54	2	no	—
104	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
105	C₂²×C₃	—	4	6	no	—
106	C₂²×C₃	—	4	12	no	—
107	C₂²×C₃	—	4	4	no	—
108	C₂²×C₃	—	4	2	no	—
109	C₂²×C₃	#3676	1	3	yes	—
110	C₂²×C₃	#3757	1	1	yes	characteristic
111	C₃×C₄	—	2	3	no	—
112	C₃×C₄	—	2	1	no	—
113	C₃×C₄	#3672	1	6	yes	—
114	C₃×C₄	#3673	1	6	yes	—
115	C₃×C₄	#3753	1	2	yes	—
116	C₃×C₄	#3754	1	2	yes	—
117	C₃²	#4544	1	1	yes	characteristic
118	C₉	#4544	1	3	yes	complemented
119	C₂³	—	2	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
124	C₂×C₄	#6233	1	2	yes	—
125	C₈	—	108	2	no	—
126	C₂×C₃	—	8	6	no	—
127	C₂×C₃	—	8	2	no	—
128	C₂×C₃	#63391	1	3	yes	—
129	C₂×C₃	#63392	1	6	yes	—
130	C₂×C₃	#63659	1	2	yes	—
131	C₂×C₃	#63660	1	1	yes	characteristic
132	C₂²	—	4	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	#88551	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	#88543	1	2	yes	—
137	C₄	#88544	1	2	yes	—
138	C₃	#10588221	1	3	yes	complemented
139	C₃	#10589410	1	1	yes	characteristic
140	C₂	—	8	2	no	—
141	C₂	#11720575	1	1	yes	characteristic
142	C₂	#11720576	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#127

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₉⋊C₈))

C₂

yes

maximal, complemented

—

—

C₂×C₃×D₄×C₉

—

C₂×C₃×D₄×C₉

C₄

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

—

C₃²×#127

D₃

yes

characteristic

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₉×#127

D₃

yes

—

C₄⋊(C₉⋊C₈)

—

C₂³×C₃×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₃×D₄×C₉

—

C₂×C₃×C₄×C₉

C₂×C₄

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

—

C₃⋊(C₉⋊C₈)

—

#4544

—

#4544

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₂×D₄×C₉

—

C₂×D₄×C₉

Dic₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂²⋊C₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

D₈

yes

—

C₃×C₄×C₉

SD₁₆

yes

—

C₃×#127

D₉

yes

—

C₃×#127

C₃⋊D₃

yes

characteristic

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₂³×C₉

—

C₃⋊(C₃⋊C₈)

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₂×Dic₃

yes

—

C₂×C₄×C₉

#67

yes

—

C₉⋊C₈

—

#331

—

Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

#100

yes

characteristic

C₂×C₃×C₉

#102

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

Dic₉

yes

—

C₂×C₃×D₄

C₃⋊Dic₃

yes

—

C₃×C₄²

C₂×D₉

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#213

yes

—

C₃²×C₄

#214

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

#217

yes

—

C₄×C₉

—

C₄×C₉

#213

yes

—

C₄×C₉

#214

yes

—

#127

C₃⋊D₉

yes

characteristic

C₄⋊C₈

—

C₃×C₉

#331

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂×Dic₉

yes

—

C₂×C₃×C₄

#611

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#832

yes

—

C₂×C₃²

#833

yes

characteristic

C₂×C₉

—

C₂×C₉

#832

yes

—

100

C₂×C₉

#833

yes

—

101

C₂×D₄

—

102

C₂×D₄

C₃⋊Dic₉

yes

—

103

C₂×C₈

—

104

C₄²

C₂×C₃⋊D₉

yes

characteristic

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

#3676

yes

—

110

C₂²×C₃

#3757

yes

characteristic

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

#3672

yes

—

114

C₃×C₄

#3673

yes

—

115

C₃×C₄

#3753

yes

—

116

C₃×C₄

#3754

yes

—

117

C₃²

#4544

yes

characteristic

118

C₉

#4544

yes

complemented

119

C₂³

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

C₂×C₃⋊Dic₉

yes

characteristic

124

C₂×C₄

#6233

yes

—

125

C₈

—

108

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

#63391

yes

—

129

C₂×C₃

#63392

yes

—

130

C₂×C₃

#63659

yes

—

131

C₂×C₃

#63660

yes

characteristic

132

C₂²

—

133

C₂²

—

134

C₂²

#88551

yes

characteristic

135

C₄

—

136

C₄

#88543

yes

—

137

C₄

#88544

yes

—

138

C₃

#10588221

yes

complemented

139

C₃

#10589410

yes

characteristic

140

C₂

—

141

C₂

#11720575

yes

characteristic

142

C₂

#11720576

yes

—

	subgroup	G/H
center	C₂²	#88551
commutator	C₂×C₃×C₄×C₉	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	C₃×C₉×#127	C₂
socle	C₂²×C₃²	#217
prime radical	C₃²×#127	D₃
cosocle	C₃×C₄²×C₉	C₂²
hypercenter	#127	C₃⋊D₉
hypocenter	C₃×C₉	#331
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

#88551

commutator

C₂×C₃×C₄×C₉

C₂×C₄

Frattini

C₃×C₄²

C₂×C₃⋊D₃

Fitting

C₃×C₉×#127

C₂

socle

C₂²×C₃²

#217

prime radical

C₃²×#127

D₃

cosocle

C₃×C₄²×C₉

C₂²

C₃⋊D₉

C₃×C₉

G ▷ C₂×C₃×C₄×C₉ ▷ C₂²×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
80. C₃×C₉	#331	semidirect
118. C₉	#4544	semidirect
138. C₃	#10588221	semidirect

H ≅ G/N

type

2. C₄⋊(C₃⋊(C₉⋊C₈))

C₂

semidirect

80. C₃×C₉

#331

semidirect

118. C₉

#4544

semidirect

138. C₃

#10588221

semidirect

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

subgroup	index
C₃×C₉×#127	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10589410	3
#10588221	3

subgroup

index

C₃×C₉×#127

C₄⋊(C₃⋊(C₉⋊C₈))

#10589410

#10588221

Subgroups — #13135916

Conjugacy classes of subgroups

Complements

Maximal subgroups