Group #13147028 — subgroups

Subgroups — #13147028

Order 1728.

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

normal subgroups only

This group has 352 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	#11722391	C₂	1	2	yes	maximal, complemented
2	C₃×#131⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#10592841	—	3	1	no	maximal
4	#10589509	—	3	3	no	maximal
5	C₃×#41⋊C₉	C₂²	1	1	yes	characteristic
6	C₃⋊#3689	—	2	2	no	—
7	#88525	—	4	1	no	maximal
8	C₃²×#131	D₃	1	1	yes	characteristic
9	#64158	—	3	2	no	—
10	C₂×#3745	—	3	1	no	—
11	#63692	—	3	6	no	—
12	#63650	—	3	2	no	—
13	#63646	—	3	1	no	—
14	#131⋊C₉	D₃	1	3	yes	complemented
15	C₄⋊(C₃⋊D₉)	—	4	2	no	—
16	C₃×Q₈⋊C₉	D₄	1	1	yes	characteristic
17	C₃×C₈×C₉	—	4	1	no	—
18	#5142	—	9	3	no	—
19	#5142	—	9	1	no	—
20	C₃²×#41	C₂×D₃	1	1	yes	characteristic
21	C₂×C₄⋊(C₃⋊D₃)	—	3	2	no	—
22	C₃²×M₁₆	—	3	1	no	—
23	C₂×C₃²×C₈	—	3	1	no	—
24	#3755	—	6	2	no	—
25	#3753	—	6	2	no	—
26	#3748	—	3	2	no	—
27	#3745	—	6	1	no	—
28	#3745	—	6	1	no	—
29	#41⋊C₉	C₂×D₃	1	3	yes	—
30	#3689	—	6	6	no	—
31	D₇₂	—	12	3	no	—
32	C₂×C₃⋊D₉	—	8	2	no	—
33	C₃×C₄×C₉	—	4	1	no	—
34	C₃×#131	D₉	1	3	yes	complemented
35	C₃×#131	C₃⋊D₃	1	1	yes	characteristic
36	#949	—	9	6	no	—
37	#949	—	9	2	no	—
38	C₂×D₂₄	—	9	3	no	—
39	C₂×D₂₄	—	9	1	no	—
40	#823	—	9	6	no	—
41	#823	—	9	2	no	—
42	#819	—	9	3	no	—
43	#819	—	9	1	no	—
44	C₂²×C₃⋊D₃	—	6	2	no	—
45	C₃²×Q₈	D₁₂	1	1	yes	characteristic
46	C₃²×D₄	—	3	1	no	—
47	C₂×C₃²×C₄	—	3	1	no	—
48	C₄⋊(C₃⋊D₃)	—	6	2	no	—
49	C₄⋊(C₃⋊D₃)	—	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	D₃₆	—	12	6	no	—
54	Q₈⋊C₉	D₁₂	1	3	yes	—
55	C₈×C₉	—	4	3	no	—
56	#472	—	27	1	no	Sylow
57	C₂×C₃×C₉	—	4	1	no	—
58	C₃⋊D₉	—	16	2	no	—
59	C₃×#41	C₂×D₉	1	3	yes	—
60	C₃×#41	C₂×C₃⋊D₃	1	1	yes	characteristic
61	C₂×D₁₂	—	9	2	no	—
62	C₂×D₁₂	—	9	6	no	—
63	C₃×M₁₆	—	3	3	no	—
64	C₃×M₁₆	—	3	1	no	—
65	C₂×C₃×C₈	—	3	3	no	—
66	C₂×C₃×C₈	—	3	1	no	—
67	#215	—	18	6	no	—
68	#215	—	18	2	no	—
69	#213	—	18	6	no	—
70	#213	—	18	2	no	—
71	#208	—	9	2	no	—
72	#208	—	9	6	no	—
73	D₂₄	—	18	1	no	—
74	D₂₄	—	18	3	no	—
75	D₂₄	—	18	3	no	—
76	D₂₄	—	18	1	no	—
77	C₂²×C₃²	—	3	1	no	—
78	C₂×C₃⋊D₃	—	12	2	no	—
79	C₂×C₃⋊D₃	—	24	2	no	—
80	C₃²×C₄	—	3	1	no	—
81	C₃²×C₄	C₂×S₄	1	1	yes	characteristic
82	C₂×D₉	—	24	6	no	—
83	C₄×C₉	—	4	3	no	—
84	AGL(1,ℤ/8)	—	27	2	no	—
85	C₂×D₈	—	27	1	no	—
86	#131	C₃⋊D₉	1	1	yes	characteristic, complemented
87	#108	—	27	1	no	—
88	#100	—	27	2	no	—
89	C₃×C₉	—	4	1	no	Sylow
90	C₂²×D₃	—	18	2	no	—
91	C₂²×D₃	—	18	6	no	—
92	C₃×Q₈	D₃₆	1	3	yes	—
93	C₃×Q₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
94	C₃×D₄	—	3	3	no	—
95	C₃×D₄	—	3	1	no	—
96	C₂×C₃×C₄	—	3	1	no	—
97	C₂×C₃×C₄	—	3	3	no	—
98	D₁₂	—	18	2	no	—
99	D₁₂	—	18	6	no	—
100	D₁₂	—	18	2	no	—
101	D₁₂	—	18	6	no	—
102	C₃×C₈	—	3	1	no	—
103	C₃×C₈	—	3	3	no	—
104	C₃×C₈	(C₂²⋊C₉)⋊C₂	1	3	yes	—
105	C₃×C₈	C₃⋊S₄	1	1	yes	characteristic
106	C₃⋊C₈	—	18	2	no	—
107	C₃⋊C₈	—	18	6	no	—
108	C₂×C₃²	—	6	1	no	—
109	C₂×C₃²	C₄⋊S₄	1	1	yes	characteristic
110	C₃⋊D₃	—	24	2	no	—
111	C₂×C₉	—	4	3	no	—
112	D₉	—	48	6	no	—
113	#41	C₂×C₃⋊D₉	1	1	yes	characteristic
114	C₂×D₄	—	27	2	no	—
115	SD₁₆	—	54	2	no	—
116	D₈	—	54	1	no	—
117	D₈	—	54	2	no	—
118	D₈	—	54	1	no	—
119	M₁₆	—	3	1	no	—
120	M₁₆	—	27	2	no	—
121	C₂×C₈	—	3	1	no	—
122	C₂²×C₃	—	3	1	no	—
123	C₂²×C₃	—	3	3	no	—
124	C₂×D₃	—	36	2	no	—
125	C₂×D₃	—	36	6	no	—
126	C₂×D₃	—	72	6	no	—
127	C₂×D₃	—	72	2	no	—
128	C₃×C₄	—	3	1	no	—
129	C₃×C₄	—	3	3	no	—
130	C₃×C₄	C₂×(C₂²⋊C₉)⋊C₂	1	3	yes	—
131	C₃×C₄	C₂×C₃⋊S₄	1	1	yes	characteristic
132	C₃²	#5407	1	1	yes	characteristic
133	C₉	—	4	3	no	—
134	C₂³	—	54	2	no	—
135	Q₈	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
136	D₄	—	3	1	no	—
137	D₄	—	54	2	no	—
138	D₄	—	54	2	no	—
139	C₂×C₄	—	3	1	no	—
140	C₈	—	3	1	no	—
141	C₈	—	54	2	no	—
142	C₈	C₃⋊((C₂²⋊C₉)⋊C₂)	1	1	yes	characteristic
143	C₂×C₃	—	6	3	no	—
144	C₂×C₃	—	6	1	no	—
145	C₂×C₃	C₄⋊((C₂²⋊C₉)⋊C₂)	1	3	yes	—
146	C₂×C₃	C₄⋊(C₃⋊S₄)	1	1	yes	characteristic
147	D₃	—	72	6	no	—
148	D₃	—	72	2	no	—
149	C₂²	—	3	1	no	—
150	C₂²	—	108	2	no	—
151	C₂²	—	216	2	no	—
152	C₄	—	3	1	no	—
153	C₄	C₂×C₃⋊((C₂²⋊C₉)⋊C₂)	1	1	yes	characteristic
154	C₃	#10589509	1	3	yes	complemented
155	C₃	#10593641	1	1	yes	characteristic
156	C₂	—	6	1	no	—
157	C₂	—	216	2	no	—
158	C₂	C₄⋊(C₃⋊((C₂²⋊C₉)⋊C₂))	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11722391

C₂

yes

maximal, complemented

C₃×#131⋊C₉

C₂

yes

maximal, characteristic, complemented

—

—

C₃×#41⋊C₉

C₂²

yes

—

—

C₃²×#131

D₃

yes

—

—

—

—

—

D₃

yes

complemented

C₄⋊(C₃⋊D₉)

—

C₃×Q₈⋊C₉

D₄

yes

characteristic

C₃×C₈×C₉

—

#5142

—

#5142

—

C₃²×#41

C₂×D₃

yes

characteristic

C₂×C₄⋊(C₃⋊D₃)

—

C₃²×M₁₆

—

C₂×C₃²×C₈

—

—

—

—

—

—

C₂×D₃

yes

—

#3689

—

D₇₂

—

C₂×C₃⋊D₉

—

C₃×C₄×C₉

—

C₃×#131

D₉

yes

complemented

C₃×#131

C₃⋊D₃

yes

characteristic

#949

—

#949

—

C₂×D₂₄

—

C₂×D₂₄

—

—

—

—

—

C₂²×C₃⋊D₃

—

C₃²×Q₈

D₁₂

yes

characteristic

C₃²×D₄

—

C₂×C₃²×C₄

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

—

C₃²×C₈

S₄

yes

characteristic

C₃⋊(C₃⋊C₈)

—

D₃₆

—

Q₈⋊C₉

D₁₂

yes

—

C₈×C₉

—

#472

—

Sylow

C₂×C₃×C₉

—

C₃⋊D₉

—

C₃×#41

C₂×D₉

yes

—

C₃×#41

C₂×C₃⋊D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

—

—

—

—

—

—

D₂₄

—

D₂₄

—

D₂₄

—

D₂₄

—

C₂²×C₃²

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

C₂×S₄

yes

characteristic

C₂×D₉

—

C₄×C₉

—

AGL(1,ℤ/8)

—

C₂×D₈

—

#131

C₃⋊D₉

yes

characteristic, complemented

#108

—

#100

—

C₃×C₉

—

Sylow

C₂²×D₃

—

C₂²×D₃

—

C₃×Q₈

D₃₆

yes

—

C₃×Q₈

C₄⋊(C₃⋊D₃)

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

100

D₁₂

—

101

D₁₂

—

102

C₃×C₈

—

103

C₃×C₈

—

104

C₃×C₈

(C₂²⋊C₉)⋊C₂

yes

—

105

C₃×C₈

C₃⋊S₄

yes

characteristic

106

C₃⋊C₈

—

107

C₃⋊C₈

—

108

C₂×C₃²

—

109

C₂×C₃²

C₄⋊S₄

yes

characteristic

110

C₃⋊D₃

—

111

C₂×C₉

—

112

D₉

—

113

#41

C₂×C₃⋊D₉

yes

characteristic

114

C₂×D₄

—

115

SD₁₆

—

116

D₈

—

117

D₈

—

118

D₈

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂×D₃

—

125

C₂×D₃

—

126

C₂×D₃

—

127

C₂×D₃

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

C₂×(C₂²⋊C₉)⋊C₂

yes

—

131

C₃×C₄

C₂×C₃⋊S₄

yes

characteristic

132

C₃²

#5407

yes

characteristic

133

C₉

—

134

C₂³

—

135

Q₈

C₄⋊(C₃⋊D₉)

yes

characteristic

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₈

—

141

C₈

—

142

C₈

C₃⋊((C₂²⋊C₉)⋊C₂)

yes

characteristic

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

C₄⋊((C₂²⋊C₉)⋊C₂)

yes

—

146

C₂×C₃

C₄⋊(C₃⋊S₄)

yes

characteristic

147

D₃

—

148

D₃

—

149

C₂²

—

150

C₂²

—

108

—

151

C₂²

—

216

—

152

C₄

—

153

C₄

C₂×C₃⋊((C₂²⋊C₉)⋊C₂)

yes

characteristic

154

C₃

#10589509

yes

complemented

155

C₃

#10593641

yes

characteristic

156

C₂

—

157

C₂

—

216

—

158

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
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

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
1. #11722391	C₂	semidirect
2. C₃×#131⋊C₉	C₂	semidirect
14. #131⋊C₉	D₃	semidirect
34. C₃×#131	D₉	semidirect
86. #131	C₃⋊D₉	semidirect
154. C₃	#10589509	semidirect

H ≅ G/N

type

1. #11722391

C₂

semidirect

2. C₃×#131⋊C₉

C₂

semidirect

14. #131⋊C₉

D₃

semidirect

34. C₃×#131

D₉

semidirect

86. #131

C₃⋊D₉

semidirect

154. C₃

#10589509

semidirect

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

subgroup	index
#11722391	2
C₃×#131⋊C₉	2
#10592841	3
#10589509	3
#88525	4

subgroup

index

C₃×#131⋊C₉

Subgroups — #13147028

Conjugacy classes of subgroups

Complements

Maximal subgroups