Group #13163683 — subgroups

Subgroups — #13163683

Order 1728.

1456 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 459 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	C₃³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	1	yes	maximal, characteristic
3	#11722205	C₂	1	2	yes	maximal, complemented
4	#11722200	C₂	1	1	yes	maximal, characteristic
5	#11722178	C₂	1	2	yes	maximal, complemented
6	#10592742	—	3	13	no	maximal
7	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
10	C₃³×C₄⋊C₄	—	2	1	no	—
11	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃³×C₂²⋊C₄	—	2	1	no	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
14	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
15	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
16	C₃²×C₂²⋊Q₈	D₃	1	13	yes	—
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
18	#63662	—	3	26	no	—
19	#63657	—	3	13	no	—
20	#63635	—	3	26	no	—
21	C₂³×C₃³	D₄	1	1	yes	characteristic
22	C₃³×Q₈	—	2	2	no	—
23	C₂×C₃³×C₄	—	2	2	no	—
24	C₂×C₃³×C₄	—	2	1	no	—
25	C₂×C₃³×C₄	—	2	1	no	—
26	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
29	C₂×C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
30	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
31	#5043	—	9	13	no	—
32	C₂×C₃²×Q₈	C₂×D₃	1	13	yes	—
33	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
34	C₂²×C₃⋊Dic₃	—	6	13	no	—
35	C₃²×C₄⋊C₄	—	2	13	no	—
36	C₃²×C₄⋊C₄	C₂×D₃	1	13	yes	—
37	C₃²×C₂²⋊C₄	—	2	13	no	—
38	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	13	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	3	26	no	—
41	C₂²×C₃³	—	2	1	no	—
42	C₂²×C₃³	—	2	1	no	—
43	C₂²×C₃³	SD₁₆	1	2	yes	—
44	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
45	C₃³×C₄	—	2	1	no	—
46	C₃³×C₄	—	4	1	no	—
47	C₃³×C₄	—	4	1	no	—
48	C₃³×C₄	#41	1	2	yes	—
49	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
50	C₃×C₂²⋊Q₈	C₃⋊D₃	1	13	yes	—
51	C₂×C₄⋊Dic₃	—	9	13	no	—
52	#835	—	9	26	no	—
53	#830	—	9	13	no	—
54	#808	—	9	26	no	—
55	C₂³×C₃²	#67	1	13	yes	—
56	C₃²×Q₈	—	2	26	no	—
57	C₂×C₃²×C₄	—	2	26	no	—
58	C₂×C₃²×C₄	—	2	13	no	—
59	C₂×C₃²×C₄	—	2	13	no	—
60	C₂×C₃²×C₄	#67	1	13	yes	—
61	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
62	C₂×C₃⋊Dic₃	—	6	26	no	—
63	C₂×C₃⋊Dic₃	—	12	26	no	—
64	C₃⋊(C₃⋊C₈)	—	6	26	no	—
65	#483	—	27	1	no	Sylow
66	C₂×C₃³	—	2	2	no	—
67	C₂×C₃³	#123	1	1	yes	characteristic
68	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₃³	#137	1	1	yes	characteristic
70	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	13	yes	—
71	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
72	C₂²×Dic₃	—	18	13	no	—
73	C₃×C₄⋊C₄	—	2	13	no	—
74	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	13	yes	—
75	C₃×C₂²⋊C₄	—	2	13	no	—
76	C₄⋊Dic₃	—	9	26	no	—
77	C₄⋊Dic₃	—	18	13	no	—
78	C₂×C₃⋊C₈	—	9	26	no	—
79	C₂²×C₃²	—	2	13	no	—
80	C₂²×C₃²	—	2	13	no	—
81	C₂²×C₃²	#215	1	26	yes	—
82	C₂²×C₃²	C₂×#67	1	13	yes	—
83	C₃²×C₄	—	2	13	no	—
84	C₃²×C₄	—	4	13	no	—
85	C₃²×C₄	—	4	13	no	—
86	C₃²×C₄	#237	1	26	yes	—
87	C₃⋊Dic₃	—	12	26	no	—
88	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
89	C₂×C₄⋊C₄	—	27	1	no	—
90	#106	—	27	2	no	—
91	#103	—	27	2	no	—
92	C₂²⋊C₈	—	27	1	no	—
93	C₃³	#483	1	1	yes	characteristic, complemented, Sylow
94	C₂³×C₃	#638	1	13	yes	—
95	C₃×Q₈	—	2	26	no	—
96	C₂×C₃×C₄	—	2	26	no	—
97	C₂×C₃×C₄	—	2	13	no	—
98	C₂×C₃×C₄	—	2	13	no	—
99	C₂×C₃×C₄	#638	1	13	yes	—
100	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
101	C₂×Dic₃	—	18	26	no	—
102	C₂×Dic₃	—	36	26	no	—
103	C₃⋊C₈	—	18	26	no	—
104	C₂×C₃²	—	2	26	no	—
105	C₂×C₃²	#935	1	13	yes	—
106	C₂×C₃²	C₂×#215	1	13	yes	—
107	C₂×C₃²	#951	1	13	yes	—
108	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
109	C₂²×C₄	—	54	1	no	—
110	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
111	C₂×C₈	—	27	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	27	2	no	—
114	C₄⋊C₄	—	54	1	no	—
115	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₂²×C₃	—	2	13	no	—
118	C₂²×C₃	—	2	13	no	—
119	C₂²×C₃	#3755	1	26	yes	—
120	C₂²×C₃	C₂×#638	1	13	yes	—
121	C₃×C₄	—	2	13	no	—
122	C₃×C₄	—	4	13	no	—
123	C₃×C₄	—	4	13	no	—
124	C₃×C₄	#3830	1	26	yes	—
125	Dic₃	—	36	26	no	—
126	C₃²	#5043	1	13	yes	complemented
127	C₂³	#6312	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	54	2	no	—
133	C₂×C₄	—	108	2	no	—
134	C₂×C₄	#6312	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
136	C₈	—	54	2	no	—
137	C₂×C₃	—	2	26	no	—
138	C₂×C₃	#64144	1	13	yes	—
139	C₂×C₃	C₂×#3755	1	13	yes	—
140	C₂×C₃	#64160	1	13	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#88862	1	2	yes	—
144	C₂²	C₂×#6312	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	108	2	no	—
149	C₄	#89078	1	2	yes	—
150	C₃	#10592742	1	13	yes	complemented
151	C₂	—	2	2	no	—
152	C₂	#11723976	1	1	yes	characteristic
153	C₂	C₂×#88862	1	1	yes	characteristic
154	C₂	#11723992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂

yes

maximal, characteristic

#11722205

C₂

yes

maximal, complemented

#11722200

C₂

yes

maximal, characteristic

#11722178

C₂

yes

maximal, complemented

#10592742

—

maximal

C₂×C₃³×Q₈

C₂²

yes

characteristic

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂²×C₃⋊(C₃⋊Dic₃)

—

C₃³×C₄⋊C₄

—

C₃³×C₄⋊C₄

C₂²

yes

characteristic

C₃³×C₂²⋊C₄

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₃²×C₂²⋊Q₈

D₃

yes

—

C₂×C₄⋊(C₃⋊Dic₃)

—

#63662

—

#63657

—

#63635

—

C₂³×C₃³

D₄

yes

characteristic

C₃³×Q₈

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#5043

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₂²⋊C₄

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

SD₁₆

yes

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

#41

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃×C₂²⋊Q₈

C₃⋊D₃

yes

—

C₂×C₄⋊Dic₃

—

#835

—

#830

—

#808

—

C₂³×C₃²

#67

yes

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₃⋊(C₃⋊C₈)

—

#483

—

Sylow

C₂×C₃³

—

C₂×C₃³

#123

yes

characteristic

C₂×C₃³

C₂×SD₁₆

yes

characteristic

C₂×C₃³

#137

yes

characteristic

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₂²⋊C₄

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#215

yes

—

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#237

yes

—

C₃⋊Dic₃

—

C₂²⋊Q₈

C₃⋊(C₃⋊D₃)

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

#103

—

C₂²⋊C₈

—

C₃³

#483

yes

characteristic, complemented, Sylow

C₂³×C₃

#638

yes

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#638

yes

—

100

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

101

C₂×Dic₃

—

102

C₂×Dic₃

—

103

C₃⋊C₈

—

104

C₂×C₃²

—

105

C₂×C₃²

#935

yes

—

106

C₂×C₃²

C₂×#215

yes

—

107

C₂×C₃²

#951

yes

—

108

C₂×Q₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

116

C₂²⋊C₄

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

#3755

yes

—

120

C₂²×C₃

C₂×#638

yes

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

#3830

yes

—

125

Dic₃

—

126

C₃²

#5043

yes

complemented

127

C₂³

#6312

yes

characteristic

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

108

—

134

C₂×C₄

#6312

yes

characteristic

135

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

#64144

yes

—

139

C₂×C₃

C₂×#3755

yes

—

140

C₂×C₃

#64160

yes

—

141

C₂²

—

142

C₂²

—

143

C₂²

#88862

yes

—

144

C₂²

C₂×#6312

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

108

—

149

C₄

#89078

yes

—

150

C₃

#10592742

yes

complemented

151

C₂

—

152

C₂

#11723976

yes

characteristic

153

C₂

C₂×#88862

yes

characteristic

154

C₂

#11723992

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6312
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₂²⋊Q₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂³×C₃³	D₄
hypercenter	C₂²⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#483
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×#6312

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×C₂²⋊Q₈

C₂

socle

C₂²×C₃³

C₂×D₄

prime radical

C₂³×C₃³

D₄

hypercenter

C₂²⋊Q₈

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#483

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

N	H ≅ G/N	type
3. #11722205	C₂	semidirect
5. #11722178	C₂	semidirect
93. C₃³	#483	semidirect
126. C₃²	#5043	semidirect
150. C₃	#10592742	semidirect

H ≅ G/N

type

3. #11722205

C₂

semidirect

5. #11722178

C₂

semidirect

93. C₃³

#483

semidirect

126. C₃²

#5043

semidirect

150. C₃

#10592742

semidirect

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

subgroup	index
C₃³×C₂²⋊Q₈	2
C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	2
#11722205	2
#11722200	2
#11722178	2
#10592742	3

subgroup

index

C₃³×C₂²⋊Q₈

C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))

Subgroups — #13163683

Conjugacy classes of subgroups

Complements

Maximal subgroups