Group #13163491 — subgroups

Subgroups — #13163491

Order 1728.

1512 conjugacy classes of subgroups; only the 461 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 461 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	C₂×#88853	C₂	1	2	yes	maximal
2	#11723910	C₂	1	1	yes	maximal, characteristic
3	C₄×#6308	C₂	1	1	yes	maximal, characteristic
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11722186	C₂	1	2	yes	maximal
6	#10592548	—	3	13	no	maximal
7	C₂×#6308	C₂²	1	2	yes	—
8	C₂×#6308	C₂²	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃³×C₄²	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
14	#88857	—	2	1	no	—
15	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
16	#88853	—	2	4	no	—
17	C₂×#3746	—	3	26	no	—
18	#64078	—	3	13	no	—
19	C₄×#634	—	3	13	no	—
20	C₃²×C₄⋊C₈	D₃	1	13	yes	—
21	#63643	—	3	26	no	—
22	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
26	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
27	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
28	#6308	—	2	1	no	—
29	#6308	—	2	2	no	—
30	#6308	—	2	2	no	—
31	#6308	D₄	1	2	yes	—
32	C₃³×C₈	—	2	2	no	—
33	#4849	—	9	13	no	—
34	C₂×#634	—	3	13	no	—
35	C₂×#634	—	3	26	no	—
36	C₂×C₃²×C₈	C₂×D₃	1	26	yes	—
37	C₃²×C₄²	C₂×D₃	1	13	yes	—
38	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	13	no	—
40	C₄⋊(C₃⋊Dic₃)	—	6	13	no	—
41	#3750	—	6	13	no	—
42	C₄×C₃⋊Dic₃	—	6	13	no	—
43	#3746	—	6	52	no	—
44	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
45	C₃³×C₄	—	2	1	no	—
46	C₃³×C₄	Q₁₆	1	2	yes	complemented
47	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
48	C₃³×C₄	#41	1	1	yes	characteristic
49	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
50	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
51	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
52	C₂×Dic₁₂	—	9	26	no	—
53	#869	—	9	13	no	—
54	C₄×Dic₆	—	9	13	no	—
55	C₃×C₄⋊C₈	C₃⋊D₃	1	13	yes	—
56	#816	—	9	26	no	—
57	C₂×C₃²×C₄	D₁₂	1	13	yes	—
58	C₂×C₃²×C₄	D₁₂	1	13	yes	—
59	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
60	C₂×C₃⋊Dic₃	—	6	13	no	—
61	C₂×C₃⋊Dic₃	—	6	13	no	—
62	C₂×C₃⋊Dic₃	—	6	26	no	—
63	#634	—	3	26	no	—
64	#634	—	6	13	no	—
65	#634	—	6	26	no	—
66	#634	—	6	26	no	—
67	C₃²×C₈	—	2	26	no	—
68	#462	—	27	1	no	Sylow
69	C₂×C₃³	#121	1	1	yes	characteristic
70	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₃³	#137	1	1	yes	characteristic
72	C₂×Dic₆	—	9	13	no	—
73	C₂×Dic₆	—	9	26	no	—
74	C₂×C₃×C₈	C₂×C₃⋊D₃	1	26	yes	—
75	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
76	C₄⋊Dic₃	—	9	13	no	—
77	C₄⋊Dic₃	—	18	13	no	—
78	C₄⋊Dic₃	—	18	13	no	—
79	Dic₃⋊C₄	—	18	13	no	—
80	C₄×Dic₃	—	18	13	no	—
81	Dic₁₂	—	18	52	no	—
82	C₂²×C₃²	C₂×D₁₂	1	13	yes	—
83	C₃²×C₄	—	2	13	no	—
84	C₃²×C₄	Dic₁₂	1	26	yes	complemented
85	C₃²×C₄	D₃×D₄	1	13	yes	—
86	C₃²×C₄	#239	1	13	yes	—
87	C₃⋊Dic₃	—	6	26	no	—
88	C₃⋊Dic₃	—	12	26	no	—
89	C₃⋊Dic₃	—	12	13	no	—
90	C₂×Q₁₆	—	27	2	no	—
91	C₄⋊Q₈	—	27	1	no	—
92	C₄×Q₈	—	27	1	no	—
93	C₄⋊C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
94	#103	—	27	2	no	—
95	C₃³	#462	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
97	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
98	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
99	C₂×Dic₃	—	18	13	no	—
100	C₂×Dic₃	—	18	13	no	—
101	C₂×Dic₃	—	18	26	no	—
102	Dic₆	—	9	26	no	—
103	Dic₆	—	18	13	no	—
104	Dic₆	—	18	26	no	—
105	Dic₆	—	18	26	no	—
106	C₃×C₈	—	2	26	no	—
107	C₂×C₃²	#895	1	13	yes	—
108	C₂×C₃²	C₂×Dic₁₂	1	13	yes	—
109	C₂×C₃²	#909	1	13	yes	—
110	C₂×Q₈	—	27	1	no	—
111	C₂×Q₈	—	27	2	no	—
112	Q₁₆	—	54	4	no	—
113	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
114	C₄⋊C₄	—	27	1	no	—
115	C₄⋊C₄	—	54	1	no	—
116	C₄⋊C₄	—	54	1	no	—
117	C₄⋊C₄	—	54	1	no	—
118	C₄²	—	54	1	no	—
119	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
120	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
121	C₃×C₄	—	2	13	no	—
122	C₃×C₄	#3746	1	26	yes	complemented
123	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
124	C₃×C₄	#3832	1	13	yes	—
125	Dic₃	—	18	26	no	—
126	Dic₃	—	36	26	no	—
127	Dic₃	—	36	13	no	—
128	C₃²	#4849	1	13	yes	complemented
129	Q₈	—	27	2	no	—
130	Q₈	—	54	1	no	—
131	Q₈	—	54	2	no	—
132	Q₈	—	54	2	no	—
133	C₂×C₄	—	54	1	no	—
134	C₂×C₄	—	54	1	no	—
135	C₂×C₄	—	54	2	no	—
136	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
137	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₂×C₃	#64104	1	13	yes	—
141	C₂×C₃	C₂×#3746	1	13	yes	—
142	C₂×C₃	#64118	1	13	yes	—
143	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	54	2	no	—
146	C₄	—	108	2	no	—
147	C₄	—	108	1	no	—
148	C₄	#88853	1	2	yes	complemented
149	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
150	C₄	#89080	1	1	yes	characteristic
151	C₃	#10592548	1	13	yes	complemented
152	C₂	#11723936	1	1	yes	characteristic
153	C₂	C₂×#88853	1	1	yes	characteristic
154	C₂	#11723950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88853

C₂

yes

maximal

#11723910

C₂

yes

maximal, characteristic

C₄×#6308

C₂

yes

maximal, characteristic

C₃³×C₄⋊C₈

C₂

yes

maximal, characteristic

#11722186

C₂

yes

—

C₂²

yes

—

C₂×#6308

C₂²

yes

characteristic

C₂×C₃³×C₈

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

characteristic

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

—

—

—

—

—

C₃²×C₄⋊C₈

D₃

yes

—

#63643

—

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

—

—

—

D₄

yes

—

C₃³×C₈

—

#4849

—

C₂×#634

—

C₂×#634

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

#3746

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

Q₁₆

yes

complemented

C₃³×C₄

C₂×D₄

yes

characteristic

C₃³×C₄

#41

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₂×Dic₁₂

—

#869

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₃⋊D₃

yes

—

#816

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

—

—

—

—

C₃²×C₈

—

—

C₂×C₃³

yes

C₂×C₃³

C₂×Q₁₆

yes

characteristic

C₂×C₃³

#137

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

Dic₁₂

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

—

C₃²×C₄

Dic₁₂

yes

complemented

C₃²×C₄

D₃×D₄

yes

—

C₃²×C₄

#239

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄⋊C₈

C₃⋊(C₃⋊D₃)

yes

characteristic

#103

—

C₃³

#462

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

100

C₂×Dic₃

—

101

C₂×Dic₃

—

102

Dic₆

—

103

Dic₆

—

104

Dic₆

—

105

Dic₆

—

106

C₃×C₈

—

107

C₂×C₃²

#895

yes

—

108

C₂×C₃²

C₂×Dic₁₂

yes

—

109

C₂×C₃²

#909

yes

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

Q₁₆

—

113

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

120

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

—

121

C₃×C₄

—

122

C₃×C₄

#3746

yes

complemented

123

C₃×C₄

D₄×C₃⋊D₃

yes

—

124

C₃×C₄

#3832

yes

—

125

Dic₃

—

126

Dic₃

—

127

Dic₃

—

128

C₃²

#4849

yes

complemented

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

137

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

138

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

139

C₈

—

140

C₂×C₃

#64104

yes

—

141

C₂×C₃

C₂×#3746

yes

—

142

C₂×C₃

#64118

yes

—

143

C₂²

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

108

—

147

C₄

—

108

—

148

C₄

#88853

yes

complemented

149

C₄

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

150

C₄

#89080

yes

characteristic

151

C₃

#10592548

yes

complemented

152

C₂

#11723936

yes

characteristic

153

C₂

C₂×#88853

yes

characteristic

154

C₂

#11723950

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₄⋊C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#462
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×C₄⋊C₈

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

hypercenter

C₄⋊C₈

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#462

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
46. C₃³×C₄	Q₁₆	semidirect
84. C₃²×C₄	Dic₁₂	semidirect
95. C₃³	#462	semidirect
122. C₃×C₄	#3746	semidirect
128. C₃²	#4849	semidirect
148. C₄	#88853	semidirect
151. C₃	#10592548	semidirect

H ≅ G/N

type

46. C₃³×C₄

Q₁₆

semidirect

84. C₃²×C₄

Dic₁₂

semidirect

95. C₃³

#462

semidirect

122. C₃×C₄

#3746

semidirect

128. C₃²

#4849

semidirect

148. C₄

#88853

semidirect

151. C₃

#10592548

semidirect

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

subgroup	index
C₂×#88853	2
#11723910	2
C₄×#6308	2
C₃³×C₄⋊C₈	2
#11722186	2
#10592548	3

subgroup

index

C₃³×C₄⋊C₈

Subgroups — #13163491

Conjugacy classes of subgroups

Complements

Maximal subgroups