Group #13143766 — subgroups

Subgroups — #13143766

Order 1728.

539 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 137 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	#11721715	C₂	1	1	yes	maximal, characteristic
2	#11721715	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
4	#11720549	C₂	1	4	yes	maximal
5	C₃×#4681	—	3	4	no	maximal
6	C₂×#6229	—	2	1	no	—
7	C₂×#6229	C₂²	1	2	yes	—
8	C₂×C₈×H(3,𝔽₃)	C₂²	1	2	yes	—
9	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₄⋊#1084	—	2	1	no	—
11	C₄⋊#1084	—	2	1	no	—
12	C₄⋊#1084	—	2	1	no	—
13	C₄⋊#1084	C₂²	1	2	yes	—
14	C₃×#869	—	3	4	no	—
15	C₃×#869	—	3	4	no	—
16	C₃²×C₄×C₈	D₃	1	4	yes	—
17	C₃×#816	—	3	16	no	—
18	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
19	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
20	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
21	C₂×#1084	—	2	2	no	—
22	C₂×#1084	—	2	2	no	—
23	#6229	—	2	4	no	—
24	#6229	—	4	2	no	—
25	C₈×H(3,𝔽₃)	—	2	2	no	—
26	C₃×#487	—	9	1	no	—
27	#4681	—	3	4	no	—
28	C₂×C₃×Dic₆	—	3	8	no	—
29	C₂×C₃×Dic₆	—	6	4	no	—
30	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
31	C₃²×C₄²	C₂×D₃	1	4	yes	—
32	C₃×C₄⋊Dic₃	—	3	8	no	—
33	C₃×C₄⋊Dic₃	—	6	4	no	—
34	C₃×C₄⋊Dic₃	—	6	4	no	—
35	C₃×C₄⋊Dic₃	—	6	4	no	—
36	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
37	C₄×H(3,𝔽₃)	SD₁₆	1	2	yes	—
38	C₄×H(3,𝔽₃)	Q₁₆	1	2	yes	—
39	C₄×H(3,𝔽₃)	#41	1	2	yes	—
40	#1084	—	4	2	no	—
41	#1084	—	4	2	no	—
42	C₃×C₄⋊Q₈	—	9	1	no	—
43	C₃×C₄⋊Q₈	—	9	1	no	—
44	#869	—	3	4	no	—
45	#869	—	3	4	no	—
46	C₃×#103	—	9	4	no	—
47	C₃×C₄×C₈	—	3	4	no	—
48	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
49	#816	—	3	16	no	—
50	C₂×C₃²×C₄	D₁₂	1	4	yes	—
51	C₂×C₃²×C₄	D₁₂	1	4	yes	—
52	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
53	C₂×C₃×Dic₃	—	6	8	no	—
54	C₂×C₃×Dic₃	—	6	8	no	—
55	C₃×Dic₆	—	6	16	no	—
56	C₃×Dic₆	—	12	8	no	—
57	C₃²×C₈	—	2	8	no	—
58	#487	—	9	1	no	Sylow
59	C₂×H(3,𝔽₃)	#124	1	1	yes	characteristic
60	C₂×H(3,𝔽₃)	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×H(3,𝔽₃)	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₃×Q₈	—	9	2	no	—
63	C₂×C₃×Q₈	—	18	1	no	—
64	C₂×Dic₆	—	3	8	no	—
65	C₂×Dic₆	—	6	4	no	—
66	C₂×C₃×C₈	—	3	8	no	—
67	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
68	C₃×C₄⋊C₄	—	9	2	no	—
69	C₃×C₄⋊C₄	—	18	1	no	—
70	C₃×C₄⋊C₄	—	18	1	no	—
71	C₃×C₄⋊C₄	—	18	1	no	—
72	C₃×C₄²	—	3	4	no	—
73	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
74	C₄⋊Dic₃	—	3	8	no	—
75	C₄⋊Dic₃	—	6	4	no	—
76	C₄⋊Dic₃	—	6	4	no	—
77	C₄⋊Dic₃	—	6	4	no	—
78	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
79	C₃²×C₄	#204	1	8	yes	—
80	C₃²×C₄	Dic₁₂	1	8	yes	—
81	C₃²×C₄	#235	1	8	yes	—
82	C₃×Dic₃	—	12	8	no	—
83	C₃×Dic₃	—	12	8	no	—
84	C₄⋊Q₈	—	9	1	no	—
85	C₄⋊Q₈	—	9	1	no	—
86	#103	—	9	4	no	—
87	C₄×C₈	#272	1	1	yes	characteristic
88	H(3,𝔽₃)	#487	1	1	yes	characteristic, complemented, Sylow
89	C₃×Q₈	—	18	4	no	—
90	C₃×Q₈	—	36	2	no	—
91	C₂×C₃×C₄	—	3	4	no	—
92	C₂×C₃×C₄	—	3	4	no	—
93	C₂×C₃×C₄	—	3	4	no	—
94	C₂×C₃×C₄	—	18	2	no	—
95	C₂×C₃×C₄	—	18	2	no	—
96	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
97	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
99	C₂×Dic₃	—	6	8	no	—
100	C₂×Dic₃	—	6	8	no	—
101	Dic₆	—	6	16	no	—
102	Dic₆	—	12	8	no	—
103	C₃×C₈	—	2	2	no	—
104	C₃×C₈	—	6	8	no	—
105	C₂×C₃²	#875	1	4	yes	—
106	C₂×C₃²	C₂×#204	1	4	yes	—
107	C₂×C₃²	C₂×Dic₁₂	1	4	yes	—
108	C₂×Q₈	—	9	2	no	—
109	C₂×Q₈	—	18	1	no	—
110	C₂×C₈	C₂×#272	1	2	yes	—
111	C₄⋊C₄	—	9	2	no	—
112	C₄⋊C₄	—	18	1	no	—
113	C₄⋊C₄	—	18	1	no	—
114	C₄⋊C₄	—	18	1	no	—
115	C₄²	C₂×#272	1	1	yes	characteristic
116	C₂²×C₃	—	3	4	no	—
117	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
118	C₃×C₄	—	3	8	no	—
119	C₃×C₄	—	3	8	no	—
120	C₃×C₄	—	3	8	no	—
121	C₃×C₄	—	36	2	no	—
122	C₃×C₄	—	36	2	no	—
123	C₃×C₄	#3744	1	2	yes	—
124	C₃×C₄	#3746	1	2	yes	—
125	C₃×C₄	#3828	1	2	yes	—
126	Dic₃	—	12	8	no	—
127	Dic₃	—	12	8	no	—
128	C₃²	#4681	1	4	yes	complemented
129	Q₈	—	18	4	no	—
130	Q₈	—	36	2	no	—
131	C₂×C₄	—	18	2	no	—
132	C₂×C₄	—	18	2	no	—
133	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
134	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
135	C₂×C₄	C₂²×#272	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₂×C₃	—	3	4	no	—
138	C₂×C₃	—	3	4	no	—
139	C₂×C₃	—	3	4	no	—
140	C₂×C₃	#64084	1	1	yes	characteristic
141	C₂×C₃	C₂×#3744	1	1	yes	characteristic
142	C₂×C₃	C₂×#3746	1	1	yes	characteristic
143	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
144	C₄	—	36	2	no	—
145	C₄	—	36	2	no	—
146	C₄	#88528	1	2	yes	—
147	C₄	#88530	1	2	yes	—
148	C₄	#88740	1	2	yes	—
149	C₃	—	3	4	no	—
150	C₃	#10592380	1	1	yes	characteristic
151	C₂	#11721721	1	1	yes	characteristic
152	C₂	C₂×#88528	1	1	yes	characteristic
153	C₂	C₂×#88530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721715

C₂

yes

maximal, characteristic

#11721715

C₂

yes

maximal, characteristic

C₄×C₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

#11720549

C₂

yes

—

—

C₂²

yes

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

—

—

—

C₂²

yes

—

C₃×#869

—

C₃×#869

—

C₃²×C₄×C₈

D₃

yes

—

C₃×#816

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

—

—

—

C₈×H(3,𝔽₃)

—

C₃×#487

—

#4681

—

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

SD₁₆

yes

—

C₄×H(3,𝔽₃)

Q₁₆

yes

—

C₄×H(3,𝔽₃)

#41

yes

—

#1084

—

#1084

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

#869

—

#869

—

C₃×#103

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃⋊D₃

yes

characteristic

#816

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

C₂×SD₁₆

yes

characteristic

C₂×H(3,𝔽₃)

C₂×Q₁₆

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

#204

yes

—

C₃²×C₄

Dic₁₂

yes

—

C₃²×C₄

#235

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#103

—

C₄×C₈

yes

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

100

C₂×Dic₃

—

101

Dic₆

—

102

Dic₆

—

103

C₃×C₈

—

104

C₃×C₈

—

105

C₂×C₃²

#875

yes

—

106

C₂×C₃²

C₂×#204

yes

—

107

C₂×C₃²

C₂×Dic₁₂

yes

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×C₈

C₂×#272

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂×#272

yes

characteristic

116

C₂²×C₃

—

117

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

#3744

yes

—

124

C₃×C₄

#3746

yes

—

125

C₃×C₄

#3828

yes

—

126

Dic₃

—

127

Dic₃

—

128

C₃²

#4681

yes

complemented

129

Q₈

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊#272

yes

characteristic

134

C₂×C₄

C₄⋊#272

yes

characteristic

135

C₂×C₄

C₂²×#272

yes

characteristic

136

C₈

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

#64084

yes

characteristic

141

C₂×C₃

C₂×#3744

yes

characteristic

142

C₂×C₃

C₂×#3746

yes

characteristic

143

C₂²

C₂×C₄⋊#272

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#88528

yes

—

147

C₄

#88530

yes

—

148

C₄

#88740

yes

—

149

C₃

—

150

C₃

#10592380

yes

characteristic

151

C₂

#11721721

yes

characteristic

152

C₂

C₂×#88528

yes

characteristic

153

C₂

C₂×#88530

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₄×C₈×H(3,𝔽₃)	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
hypercenter	C₃×C₄×C₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#487
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

subgroup

G/H

center, socle

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₄×C₈×H(3,𝔽₃)

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

hypercenter

C₃×C₄×C₈

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

N	H ≅ G/N	type
88. H(3,𝔽₃)	#487	semidirect
128. C₃²	#4681	semidirect

H ≅ G/N

type

88. H(3,𝔽₃)

#487

semidirect

128. C₃²

#4681

semidirect

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

subgroup	index
#11721715	2
C₄×C₈×H(3,𝔽₃)	2
#11720549	2
C₃×#4681	3

subgroup

index

#11721715

C₄×C₈×H(3,𝔽₃)

#11720549

C₃×#4681

Subgroups — #13143766

Conjugacy classes of subgroups

Complements

Maximal subgroups