Group #13143818 — subgroups

Subgroups — #13143818

Order 1728.

594 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 131 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×#88530	C₂	1	1	yes	maximal, characteristic
2	C₄×#6229	C₂	1	2	yes	maximal
3	H(3,𝔽₃)×#97	C₂	1	1	yes	maximal, characteristic
4	#11720550	C₂	1	1	yes	maximal, characteristic
5	#11720549	C₂	1	2	yes	maximal
6	C₃×#4716	—	3	4	no	maximal
7	C₂×#6229	C₂²	1	2	yes	—
8	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	C₄⋊#1084	C₂²	1	2	yes	—
12	#88540	—	2	2	no	—
13	C₄×#1084	—	2	2	no	—
14	#88530	C₄	1	4	yes	complemented
15	C₂×C₃×Dic₁₂	—	3	4	no	—
16	C₃×C₄×Dic₆	—	3	8	no	—
17	C₃²×#97	D₃	1	4	yes	—
18	C₃×#817	—	3	4	no	—
19	C₃×#816	—	3	8	no	—
20	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
21	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
22	C₂×#1084	—	2	2	no	—
23	C₂×#1084	—	2	2	no	—
24	#6229	—	2	2	no	—
25	#6229	C₂×C₄	1	4	yes	—
26	C₈×H(3,𝔽₃)	—	2	1	no	—
27	C₈×H(3,𝔽₃)	C₂×C₄	1	2	yes	—
28	C₃×#441	—	9	1	no	—
29	#4716	—	3	4	no	—
30	C₂×C₃×Dic₆	—	3	8	no	—
31	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
32	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
33	C₃²×C₄²	C₂×D₃	1	4	yes	—
34	C₃×C₄⋊Dic₃	—	3	8	no	—
35	C₃×Dic₃⋊C₄	—	6	8	no	—
36	C₃×C₄×Dic₃	—	6	8	no	—
37	C₃×Dic₁₂	—	3	16	no	—
38	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
39	C₄×H(3,𝔽₃)	—	2	2	no	—
40	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
41	C₄×H(3,𝔽₃)	#41	1	1	yes	characteristic
42	#1084	—	2	4	no	—
43	#1084	—	4	2	no	—
44	C₂×C₃×Q₁₆	—	9	1	no	—
45	C₃×C₄×Q₈	—	9	2	no	—
46	C₂×Dic₁₂	—	3	4	no	—
47	C₄×Dic₆	—	3	8	no	—
48	C₃×#106	—	9	1	no	—
49	C₃×#103	—	9	2	no	—
50	C₃×#97	—	3	4	no	—
51	C₃×#97	C₃⋊D₃	1	1	yes	characteristic
52	#817	—	3	4	no	—
53	#816	—	3	8	no	—
54	C₂×C₃²×C₄	D₁₂	1	8	yes	—
55	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
56	C₂×C₃×Dic₃	—	6	8	no	—
57	C₂×C₃×Dic₃	—	6	8	no	—
58	C₃×Dic₆	—	3	16	no	—
59	C₃×Dic₆	—	6	8	no	—
60	C₃²×C₈	—	2	4	no	—
61	C₃²×C₈	C₄×D₃	1	8	yes	—
62	#441	—	9	1	no	Sylow
63	C₂×H(3,𝔽₃)	C₄×D₄	1	1	yes	characteristic
64	C₂×H(3,𝔽₃)	#137	1	2	yes	—
65	C₂×C₃×Q₈	—	9	2	no	—
66	C₂×Dic₆	—	3	8	no	—
67	C₃×Q₁₆	—	9	4	no	—
68	C₂×C₃×C₈	—	3	4	no	—
69	C₂×C₃×C₈	—	3	4	no	—
70	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
72	C₃×C₄⋊C₄	—	9	2	no	—
73	C₃×C₄⋊C₄	—	18	2	no	—
74	C₃×C₄²	—	3	4	no	—
75	C₃×C₄²	—	18	2	no	—
76	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
77	C₄⋊Dic₃	—	3	8	no	—
78	Dic₃⋊C₄	—	6	8	no	—
79	C₄×Dic₃	—	6	8	no	—
80	Dic₁₂	—	3	16	no	—
81	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
82	C₃²×C₄	—	2	8	no	—
83	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
84	C₃²×C₄	#235	1	4	yes	—
85	C₃×Dic₃	—	6	16	no	—
86	C₃×Dic₃	—	12	8	no	—
87	C₂×Q₁₆	—	9	1	no	—
88	C₄×Q₈	—	9	2	no	—
89	#106	—	9	1	no	—
90	#103	—	9	2	no	—
91	#97	#272	1	1	yes	characteristic
92	H(3,𝔽₃)	#441	1	1	yes	characteristic, complemented, Sylow
93	C₃×Q₈	—	9	4	no	—
94	C₃×Q₈	—	18	2	no	—
95	C₂×C₃×C₄	—	3	8	no	—
96	C₂×C₃×C₄	—	3	4	no	—
97	C₂×C₃×C₄	—	18	2	no	—
98	C₂×C₃×C₄	—	18	2	no	—
99	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
100	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
101	C₂×Dic₃	—	6	8	no	—
102	C₂×Dic₃	—	6	8	no	—
103	Dic₆	—	3	16	no	—
104	Dic₆	—	6	8	no	—
105	C₃×C₈	—	2	1	no	—
106	C₃×C₈	—	3	8	no	—
107	C₃×C₈	—	6	4	no	—
108	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
109	C₂×C₃²	C₄×D₁₂	1	4	yes	—
110	C₂×C₃²	#909	1	8	yes	—
111	C₂×Q₈	—	9	2	no	—
112	Q₁₆	—	9	4	no	—
113	C₂×C₈	C₂×#272	1	1	yes	characteristic
114	C₂×C₈	C₂×#272	1	1	yes	characteristic
115	C₄⋊C₄	—	9	2	no	—
116	C₄⋊C₄	—	18	2	no	—
117	C₄²	—	18	2	no	—
118	C₄²	C₂×#272	1	1	yes	characteristic
119	C₂²×C₃	—	3	4	no	—
120	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	3	4	no	—
123	C₃×C₄	—	3	4	no	—
124	C₃×C₄	—	6	8	no	—
125	C₃×C₄	—	18	4	no	—
126	C₃×C₄	—	36	2	no	—
127	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
128	C₃×C₄	#3828	1	1	yes	characteristic
129	Dic₃	—	6	16	no	—
130	Dic₃	—	12	8	no	—
131	C₃²	#4716	1	4	yes	complemented
132	Q₈	—	9	4	no	—
133	Q₈	—	18	2	no	—
134	C₂×C₄	—	18	2	no	—
135	C₂×C₄	—	18	2	no	—
136	C₂×C₄	C₄⋊#272	1	2	yes	—
137	C₂×C₄	C₂²×#272	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	C₄×#272	1	2	yes	—
140	C₂×C₃	—	3	8	no	—
141	C₂×C₃	—	3	4	no	—
142	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₂×C₃	#64118	1	2	yes	—
144	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	18	4	no	—
147	C₄	—	36	2	no	—
148	C₄	C₂×C₄×#272	1	1	yes	characteristic
149	C₄	#88740	1	1	yes	characteristic
150	C₃	—	3	4	no	—
151	C₃	#10592415	1	1	yes	characteristic
152	C₂	C₄×C₄⋊#272	1	1	yes	characteristic
153	C₂	#11721788	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88530

C₂

yes

maximal, characteristic

C₄×#6229

C₂

yes

maximal

H(3,𝔽₃)×#97

C₂

yes

maximal, characteristic

#11720550

C₂

yes

maximal, characteristic

#11720549

C₂

yes

—

C₂²

yes

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

C₂²

yes

—

#88540

—

C₄×#1084

—

#88530

C₄

yes

complemented

C₂×C₃×Dic₁₂

—

C₃×C₄×Dic₆

—

C₃²×#97

D₃

yes

—

C₃×#817

—

C₃×#816

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₈×H(3,𝔽₃)

—

C₈×H(3,𝔽₃)

C₂×C₄

yes

—

C₃×#441

—

#4716

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₃×Dic₁₂

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂²×C₄

yes

C₄×H(3,𝔽₃)

yes

—

—

C₂×C₃×Q₁₆

—

C₃×C₄×Q₈

—

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×#106

—

C₃×#103

—

C₃×#97

—

C₃×#97

C₃⋊D₃

yes

characteristic

#817

—

#816

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

—

C₃²×C₈

C₄×D₃

yes

—

#441

—

Sylow

C₂×H(3,𝔽₃)

C₄×D₄

yes

characteristic

C₂×H(3,𝔽₃)

#137

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

Dic₁₂

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃²×C₄

#235

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

—

—

yes

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

100

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

101

C₂×Dic₃

—

102

C₂×Dic₃

—

103

Dic₆

—

104

Dic₆

—

105

C₃×C₈

—

106

C₃×C₈

—

107

C₃×C₈

—

108

C₃×C₈

C₄×C₃⋊D₃

yes

—

109

C₂×C₃²

C₄×D₁₂

yes

—

110

C₂×C₃²

#909

yes

—

111

C₂×Q₈

—

112

Q₁₆

—

113

C₂×C₈

C₂×#272

yes

characteristic

114

C₂×C₈

C₂×#272

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

C₂×#272

yes

characteristic

119

C₂²×C₃

—

120

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

128

C₃×C₄

#3828

yes

characteristic

129

Dic₃

—

130

Dic₃

—

131

C₃²

#4716

yes

complemented

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₄⋊#272

yes

—

137

C₂×C₄

C₂²×#272

yes

characteristic

138

C₈

—

139

C₈

C₄×#272

yes

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

C₄×C₄⋊(C₃⋊D₃)

yes

characteristic

143

C₂×C₃

#64118

yes

—

144

C₂²

C₂×C₄⋊#272

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₂×C₄×#272

yes

characteristic

149

C₄

#88740

yes

characteristic

150

C₃

—

151

C₃

#10592415

yes

characteristic

152

C₂

C₄×C₄⋊#272

yes

characteristic

153

C₂

#11721788

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator	C₄×H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	H(3,𝔽₃)×#97	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
cosocle	C₂×C₄×H(3,𝔽₃)	C₂³
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#441
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

subgroup

G/H

center, socle

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

commutator

C₄×H(3,𝔽₃)

C₂²×C₄

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

H(3,𝔽₃)×#97

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

hypercenter

C₃×#97

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

N	H ≅ G/N	type
14. #88530	C₄	semidirect
92. H(3,𝔽₃)	#441	semidirect
131. C₃²	#4716	semidirect

H ≅ G/N

type

14. #88530

C₄

semidirect

92. H(3,𝔽₃)

#441

semidirect

131. C₃²

#4716

semidirect

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

subgroup	index
C₂×#88530	2
C₄×#6229	2
H(3,𝔽₃)×#97	2
#11720550	2
#11720549	2
C₃×#4716	3

subgroup

index

Subgroups — #13143818

Conjugacy classes of subgroups

Complements

Maximal subgroups