Group #13143739 — subgroups

Subgroups — #13143739

Order 1728.

1023 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 195 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×#88542	C₂	1	3	yes	maximal
2	C₂×C₄⋊#1084	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₃)×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4677	—	3	4	no	maximal
5	C₂³×#272	C₂²	1	1	yes	characteristic
6	C₂²×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
7	C₂²×#1084	C₂²	1	3	yes	—
8	#88542	—	2	6	no	—
9	C₄⋊#1084	—	2	6	no	—
10	C₂×C₃×(C₂×D₃)⋊C₄	—	3	12	no	—
11	C₂×C₃×C₄⋊Dic₃	—	3	12	no	—
12	C₃²×#95	D₃	1	4	yes	complemented
13	C₂³×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
14	C₂²×#272	—	2	3	no	—
15	C₂²×#272	—	2	3	no	—
16	C₂²×#272	Q₈	1	2	yes	—
17	C₂×C₄×H(3,𝔽₃)	—	2	6	no	—
18	C₂×C₄×H(3,𝔽₃)	D₄	1	6	yes	—
19	C₂×#1084	—	2	6	no	—
20	C₂×#1084	—	2	3	no	—
21	C₃×#399	—	9	1	no	—
22	#4677	—	3	4	no	—
23	C₂³×C₃×D₃	—	3	4	no	—
24	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
25	C₂²×C₃×Dic₃	—	3	12	no	—
26	C₃×(C₂×D₃)⋊C₄	—	6	24	no	—
27	C₃×C₄⋊Dic₃	—	6	24	no	—
28	C₂²×H(3,𝔽₃)	C₂×D₄	1	3	yes	—
29	C₂²×H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic
30	C₂²×H(3,𝔽₃)	#41	1	3	yes	—
31	C₂×#272	—	2	6	no	—
32	C₂×#272	—	4	3	no	—
33	C₂×#272	—	4	1	no	—
34	C₄×H(3,𝔽₃)	—	2	6	no	—
35	#1084	—	4	3	no	—
36	C₂×C₃×C₄⋊C₄	—	9	3	no	—
37	C₂×C₃×C₂²⋊C₄	—	9	3	no	—
38	C₂×(C₂×D₃)⋊C₄	—	3	12	no	—
39	C₂×C₄⋊Dic₃	—	3	12	no	—
40	C₃×#95	—	3	4	no	—
41	C₃×#95	C₃⋊D₃	1	1	yes	characteristic
42	C₂³×C₃²	C₂²×D₃	1	4	yes	—
43	C₂²×C₃×D₃	—	3	8	no	—
44	C₂²×C₃×D₃	—	6	12	no	—
45	C₂²×C₃×D₃	—	6	12	no	—
46	C₂×C₃²×C₄	—	2	24	no	—
47	C₂×C₃²×C₄	D₁₂	1	24	yes	—
48	C₂×C₃×Dic₃	—	6	24	no	—
49	C₂×C₃×Dic₃	—	6	12	no	—
50	#399	—	9	1	no	Sylow
51	C₂×H(3,𝔽₃)	C₂²⋊Q₈	1	3	yes	—
52	C₂×H(3,𝔽₃)	#123	1	3	yes	—
53	C₂×H(3,𝔽₃)	#127	1	1	yes	characteristic
54	#272	—	4	2	no	—
55	C₂⁴×C₃	—	9	1	no	—
56	C₂³×D₃	—	3	4	no	—
57	C₂²×C₃×C₄	—	3	12	no	—
58	C₂²×C₃×C₄	—	9	3	no	—
59	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
60	C₂²×Dic₃	—	3	12	no	—
61	C₃×C₄⋊C₄	—	18	6	no	—
62	C₃×C₂²⋊C₄	—	18	6	no	—
63	(C₂×D₃)⋊C₄	—	6	24	no	—
64	C₄⋊Dic₃	—	6	24	no	—
65	C₂²×C₃²	C₂×D₁₂	1	12	yes	—
66	C₂²×C₃²	#237	1	12	yes	—
67	C₂²×C₃²	D₃×Q₈	1	4	yes	—
68	C₂×C₃×D₃	—	6	24	no	—
69	C₂×C₃×D₃	—	12	4	no	—
70	C₂×C₃×D₃	—	12	12	no	—
71	C₃²×C₄	—	2	24	no	—
72	C₃×Dic₃	—	12	12	no	—
73	C₂×C₄⋊C₄	—	9	3	no	—
74	C₂×C₂²⋊C₄	—	9	3	no	—
75	#95	#272	1	1	yes	characteristic, complemented
76	H(3,𝔽₃)	#399	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₃	—	3	4	no	—
78	C₂³×C₃	—	9	2	no	—
79	C₂³×C₃	—	18	3	no	—
80	C₂³×C₃	—	18	3	no	—
81	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
82	C₂²×D₃	—	3	8	no	—
83	C₂²×D₃	—	6	12	no	—
84	C₂²×D₃	—	6	12	no	—
85	C₂×C₃×C₄	—	2	6	no	—
86	C₂×C₃×C₄	—	3	24	no	—
87	C₂×C₃×C₄	—	6	24	no	—
88	C₂×C₃×C₄	—	18	3	no	—
89	C₂×C₃×C₄	—	18	6	no	—
90	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	6	yes	—
91	C₂×Dic₃	—	6	24	no	—
92	C₂×Dic₃	—	6	12	no	—
93	C₂×C₃²	#874	1	4	yes	—
94	C₂×C₃²	#886	1	12	yes	—
95	C₂×C₃²	#897	1	12	yes	—
96	C₃×D₃	—	12	8	no	—
97	C₂⁴	—	9	1	no	—
98	C₂²×C₄	—	9	3	no	—
99	C₂²×C₄	C₂×#272	1	3	yes	—
100	C₄⋊C₄	—	18	6	no	—
101	C₂²⋊C₄	—	18	6	no	—
102	C₂²×C₃	—	3	12	no	—
103	C₂²×C₃	—	3	12	no	—
104	C₂²×C₃	—	3	4	no	—
105	C₂²×C₃	—	18	6	no	—
106	C₂²×C₃	—	36	1	no	—
107	C₂²×C₃	—	36	3	no	—
108	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
109	C₂²×C₃	#3830	1	3	yes	—
110	C₂²×C₃	Q₈×C₃⋊D₃	1	1	yes	characteristic
111	C₂×D₃	—	6	24	no	—
112	C₂×D₃	—	12	12	no	—
113	C₂×D₃	—	12	4	no	—
114	C₃×C₄	—	2	6	no	—
115	C₃×C₄	—	6	24	no	—
116	C₃×C₄	—	36	3	no	—
117	Dic₃	—	12	12	no	—
118	C₃²	#4677	1	4	yes	complemented
119	C₂³	—	9	2	no	—
120	C₂³	—	18	3	no	—
121	C₂³	—	18	3	no	—
122	C₂³	C₂²×#272	1	1	yes	characteristic
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	18	3	no	—
125	C₂×C₄	—	18	6	no	—
126	C₂×C₄	C₄⋊#272	1	6	yes	—
127	C₂×C₃	—	3	12	no	—
128	C₂×C₃	—	3	12	no	—
129	C₂×C₃	—	3	4	no	—
130	C₂×C₃	—	36	2	no	—
131	C₂×C₃	#64083	1	1	yes	characteristic
132	C₂×C₃	#64095	1	3	yes	—
133	C₂×C₃	#64106	1	3	yes	—
134	D₃	—	12	8	no	—
135	C₂²	—	18	6	no	—
136	C₂²	—	36	3	no	—
137	C₂²	—	36	1	no	—
138	C₂²	C₂×C₄⋊#272	1	3	yes	—
139	C₂²	#88744	1	3	yes	—
140	C₂²	Q₈×#272	1	1	yes	characteristic
141	C₄	—	2	6	no	—
142	C₄	—	36	3	no	—
143	C₃	—	3	4	no	—
144	C₃	#10592376	1	1	yes	characteristic
145	C₂	—	36	2	no	—
146	C₂	#11721720	1	1	yes	characteristic
147	C₂	#11721742	1	3	yes	—
148	C₂	#11721765	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88542

C₂

yes

maximal

C₂×C₄⋊#1084

C₂

yes

maximal, complemented

H(3,𝔽₃)×#95

C₂

yes

maximal, characteristic, complemented

C₃×#4677

—

maximal

C₂³×#272

C₂²

yes

characteristic

C₂²×C₄×H(3,𝔽₃)

C₂²

yes

—

C₂²×#1084

C₂²

yes

—

#88542

—

C₄⋊#1084

—

C₂×C₃×(C₂×D₃)⋊C₄

—

C₂×C₃×C₄⋊Dic₃

—

C₃²×#95

D₃

yes

complemented

C₂³×H(3,𝔽₃)

C₂³

yes

characteristic

C₂²×#272

—

C₂²×#272

—

C₂²×#272

Q₈

yes

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

—

—

—

—

C₂³×C₃×D₃

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₃×(C₂×D₃)⋊C₄

—

C₃×C₄⋊Dic₃

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

—

C₂²×H(3,𝔽₃)

C₂×Q₈

yes

characteristic

C₂²×H(3,𝔽₃)

#41

yes

—

C₂×#272

—

C₂×#272

—

C₂×#272

—

C₄×H(3,𝔽₃)

—

#1084

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₂²⋊C₄

—

C₂×(C₂×D₃)⋊C₄

—

C₂×C₄⋊Dic₃

—

C₃×#95

—

C₃×#95

C₃⋊D₃

yes

characteristic

C₂³×C₃²

C₂²×D₃

yes

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

#399

—

Sylow

C₂×H(3,𝔽₃)

C₂²⋊Q₈

yes

—

C₂×H(3,𝔽₃)

#123

yes

—

C₂×H(3,𝔽₃)

#127

yes

characteristic

#272

—

C₂⁴×C₃

—

C₂³×D₃

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂²×C₃²

#237

yes

—

C₂²×C₃²

D₃×Q₈

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃×Dic₃

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#95

#272

yes

characteristic, complemented

H(3,𝔽₃)

#399

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

#874

yes

—

C₂×C₃²

#886

yes

—

C₂×C₃²

#897

yes

—

C₃×D₃

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

C₂×#272

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²×C₃

—

103

C₂²×C₃

—

104

C₂²×C₃

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

—

109

C₂²×C₃

#3830

yes

—

110

C₂²×C₃

Q₈×C₃⋊D₃

yes

characteristic

111

C₂×D₃

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

—

117

Dic₃

—

118

C₃²

#4677

yes

complemented

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂²×#272

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄⋊#272

yes

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

#64083

yes

characteristic

132

C₂×C₃

#64095

yes

—

133

C₂×C₃

#64106

yes

—

134

D₃

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄⋊#272

yes

—

139

C₂²

#88744

yes

—

140

C₂²

Q₈×#272

yes

characteristic

141

C₄

—

142

C₄

—

143

C₃

—

144

C₃

#10592376

yes

characteristic

145

C₂

—

146

C₂

#11721720

yes

characteristic

147

C₂

#11721742

yes

—

148

C₂

#11721765

yes

—

	subgroup	G/H
center, Frattini, socle	C₂³×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂³×H(3,𝔽₃)	C₂³
Fitting	H(3,𝔽₃)×#95	C₂
prime radical	C₂³×#272	C₂²
hypercenter	C₃×#95	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#399
LCS	G ▷ C₂³×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

subgroup

G/H

center, Frattini, socle

C₂³×C₃

C₂²×C₃⋊D₃

commutator, cosocle

C₂³×H(3,𝔽₃)

C₂³

Fitting

H(3,𝔽₃)×#95

C₂

prime radical

C₂³×#272

C₂²

hypercenter

C₃×#95

C₃⋊D₃

G ▷ C₂³×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂³×C₃ ◁ C₃×#95

N	H ≅ G/N	type
2. C₂×C₄⋊#1084	C₂	semidirect
3. H(3,𝔽₃)×#95	C₂	semidirect
12. C₃²×#95	D₃	semidirect
75. #95	#272	semidirect
76. H(3,𝔽₃)	#399	semidirect
118. C₃²	#4677	semidirect

H ≅ G/N

type

2. C₂×C₄⋊#1084

C₂

semidirect

3. H(3,𝔽₃)×#95

C₂

semidirect

12. C₃²×#95

D₃

semidirect

75. #95

#272

semidirect

76. H(3,𝔽₃)

#399

semidirect

118. C₃²

#4677

semidirect

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

subgroup	index
C₂×#88542	2
C₂×C₄⋊#1084	2
H(3,𝔽₃)×#95	2
C₃×#4677	3

subgroup

index

C₂×#88542

C₂×C₄⋊#1084

H(3,𝔽₃)×#95

C₃×#4677

Subgroups — #13143739

Conjugacy classes of subgroups

Complements

Maximal subgroups