Group #13143805 — subgroups

Subgroups — #13143805

Order 1728.

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

normal subgroups only

This group has 495 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	#11721716	C₂	1	1	yes	maximal, characteristic
2	#11721715	C₂	1	1	yes	maximal, characteristic
3	H(3,𝔽₃)×#97	C₂	1	1	yes	maximal, characteristic
4	#11720551	C₂	1	2	yes	maximal
5	#11720550	C₂	1	2	yes	maximal
6	C₃×#4702	—	3	4	no	maximal
7	C₂×#6229	—	2	1	no	—
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	C₂²	1	1	yes	characteristic
12	C₄⋊#1084	C₂²	1	2	yes	—
13	C₄⋊#1084	C₂²	1	1	yes	characteristic
14	#88540	—	2	1	no	—
15	#88540	—	2	1	no	—
16	C₃×#870	—	3	4	no	—
17	C₃×#869	—	3	4	no	—
18	C₃²×#97	D₃	1	4	yes	—
19	C₃×#818	—	3	8	no	—
20	C₃×#817	—	3	8	no	—
21	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
22	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
23	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
24	C₂×#1084	—	2	2	no	—
25	C₂×#1084	—	2	1	no	—
26	C₂×#1084	—	2	1	no	—
27	#6229	—	4	2	no	—
28	C₈×H(3,𝔽₃)	Q₈	1	4	yes	complemented
29	C₃×#501	—	9	1	no	—
30	#4702	—	3	4	no	—
31	C₂×C₃×Dic₆	—	6	4	no	—
32	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
33	C₃²×C₄²	C₂×D₃	1	4	yes	—
34	C₃×C₄⋊Dic₃	—	3	4	no	—
35	C₃×C₄⋊Dic₃	—	3	8	no	—
36	C₃×C₄⋊Dic₃	—	3	4	no	—
37	C₃×C₄⋊Dic₃	—	6	4	no	—
38	C₃×Dic₃⋊C₄	—	6	4	no	—
39	C₃×Dic₃⋊C₄	—	6	4	no	—
40	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
41	C₄×H(3,𝔽₃)	—	2	1	no	—
42	C₄×H(3,𝔽₃)	—	2	1	no	—
43	C₄×H(3,𝔽₃)	C₂×Q₈	1	2	yes	—
44	#1084	—	4	2	no	—
45	#1084	—	4	1	no	—
46	#1084	—	4	1	no	—
47	C₃×C₄⋊Q₈	—	9	1	no	—
48	C₃×#125	—	9	1	no	—
49	#870	—	3	4	no	—
50	#869	—	3	4	no	—
51	C₃×#107	—	9	2	no	—
52	C₃×#106	—	9	2	no	—
53	C₃×#97	—	3	4	no	—
54	C₃×#97	C₃⋊D₃	1	1	yes	characteristic
55	#818	—	3	8	no	—
56	#817	—	3	8	no	—
57	C₂×C₃²×C₄	D₁₂	1	4	yes	—
58	C₂×C₃²×C₄	D₁₂	1	4	yes	—
59	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
60	C₂×C₃×Dic₃	—	6	4	no	—
61	C₂×C₃×Dic₃	—	6	4	no	—
62	C₂×C₃×Dic₃	—	6	8	no	—
63	C₃×Dic₆	—	12	8	no	—
64	C₃²×C₈	Dic₆	1	16	yes	complemented
65	#501	—	9	1	no	Sylow
66	C₂×H(3,𝔽₃)	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×H(3,𝔽₃)	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×H(3,𝔽₃)	#137	1	1	yes	characteristic
69	C₂×C₃×Q₈	—	18	1	no	—
70	C₂×Dic₆	—	6	4	no	—
71	C₂×C₃×C₈	—	3	8	no	—
72	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
73	C₃×C₄⋊C₄	—	9	1	no	—
74	C₃×C₄⋊C₄	—	9	2	no	—
75	C₃×C₄⋊C₄	—	9	1	no	—
76	C₃×C₄⋊C₄	—	18	1	no	—
77	C₃×C₄⋊C₄	—	18	1	no	—
78	C₃×C₄⋊C₄	—	18	1	no	—
79	C₃×C₄²	—	3	4	no	—
80	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
81	C₄⋊Dic₃	—	3	4	no	—
82	C₄⋊Dic₃	—	3	8	no	—
83	C₄⋊Dic₃	—	3	4	no	—
84	C₄⋊Dic₃	—	6	4	no	—
85	Dic₃⋊C₄	—	6	4	no	—
86	Dic₃⋊C₄	—	6	4	no	—
87	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
88	C₃²×C₄	—	2	4	no	—
89	C₃²×C₄	—	2	4	no	—
90	C₃²×C₄	C₂×Dic₆	1	8	yes	—
91	C₃×Dic₃	—	12	8	no	—
92	C₃×Dic₃	—	12	4	no	—
93	C₃×Dic₃	—	12	4	no	—
94	C₄⋊Q₈	—	9	1	no	—
95	#125	—	9	1	no	—
96	#107	—	9	2	no	—
97	#106	—	9	2	no	—
98	#97	#272	1	1	yes	characteristic
99	H(3,𝔽₃)	#501	1	1	yes	characteristic, complemented, Sylow
100	C₃×Q₈	—	36	2	no	—
101	C₂×C₃×C₄	—	3	4	no	—
102	C₂×C₃×C₄	—	3	4	no	—
103	C₂×C₃×C₄	—	3	4	no	—
104	C₂×C₃×C₄	—	18	1	no	—
105	C₂×C₃×C₄	—	18	1	no	—
106	C₂×C₃×C₄	—	18	2	no	—
107	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
108	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
110	C₂×Dic₃	—	6	4	no	—
111	C₂×Dic₃	—	6	4	no	—
112	C₂×Dic₃	—	6	8	no	—
113	Dic₆	—	12	8	no	—
114	C₃×C₈	—	3	16	no	—
115	C₃×C₈	#634	1	4	yes	—
116	C₂×C₃²	#869	1	4	yes	—
117	C₂×C₃²	#908	1	4	yes	—
118	C₂×C₃²	#909	1	4	yes	—
119	C₂×Q₈	—	18	1	no	—
120	C₂×C₈	C₂×#272	1	2	yes	—
121	C₄⋊C₄	—	9	1	no	—
122	C₄⋊C₄	—	9	2	no	—
123	C₄⋊C₄	—	9	1	no	—
124	C₄⋊C₄	—	18	1	no	—
125	C₄⋊C₄	—	18	1	no	—
126	C₄⋊C₄	—	18	1	no	—
127	C₄²	C₂×#272	1	1	yes	characteristic
128	C₂²×C₃	—	3	4	no	—
129	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
130	C₃×C₄	—	2	1	no	—
131	C₃×C₄	—	2	1	no	—
132	C₃×C₄	—	3	8	no	—
133	C₃×C₄	—	6	4	no	—
134	C₃×C₄	—	6	4	no	—
135	C₃×C₄	—	36	2	no	—
136	C₃×C₄	—	36	1	no	—
137	C₃×C₄	—	36	1	no	—
138	C₃×C₄	C₂×#634	1	2	yes	—
139	Dic₃	—	12	8	no	—
140	Dic₃	—	12	4	no	—
141	Dic₃	—	12	4	no	—
142	C₃²	#4702	1	4	yes	complemented
143	Q₈	—	36	2	no	—
144	C₂×C₄	—	18	1	no	—
145	C₂×C₄	—	18	1	no	—
146	C₂×C₄	—	18	2	no	—
147	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
148	C₂×C₄	C₄⋊#272	1	1	yes	characteristic
149	C₂×C₄	C₂²×#272	1	1	yes	characteristic
150	C₈	#6229	1	4	yes	complemented
151	C₂×C₃	—	3	4	no	—
152	C₂×C₃	—	3	4	no	—
153	C₂×C₃	—	3	4	no	—
154	C₂×C₃	#64078	1	1	yes	characteristic
155	C₂×C₃	#64117	1	1	yes	characteristic
156	C₂×C₃	#64118	1	1	yes	characteristic
157	C₂²	C₂×C₄⋊#272	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	36	2	no	—
161	C₄	—	36	1	no	—
162	C₄	—	36	1	no	—
163	C₄	C₂×#6229	1	2	yes	—
164	C₃	—	3	4	no	—
165	C₃	#10592401	1	1	yes	characteristic
166	C₂	#11721715	1	1	yes	characteristic
167	C₂	#11721787	1	1	yes	characteristic
168	C₂	#11721788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721716

C₂

yes

maximal, characteristic

#11721715

C₂

yes

maximal, characteristic

H(3,𝔽₃)×#97

C₂

yes

maximal, characteristic

#11720551

C₂

yes

maximal

#11720550

C₂

yes

—

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

—

C₄⋊#1084

C₂²

yes

characteristic

C₄⋊#1084

C₂²

yes

—

C₄⋊#1084

C₂²

yes

—

—

—

—

C₃²×#97

D₃

yes

—

C₃×#818

—

C₃×#817

—

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)

Q₈

yes

complemented

C₃×#501

—

#4702

—

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

—

C₃×Dic₃⋊C₄

—

C₃×Dic₃⋊C₄

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×Q₈

yes

—

#1084

—

#1084

—

#1084

—

C₃×C₄⋊Q₈

—

—

—

—

—

—

C₃×#97

—

C₃×#97

C₃⋊D₃

yes

characteristic

#818

—

#817

—

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

—

C₃×Dic₆

—

C₃²×C₈

Dic₆

yes

complemented

#501

—

Sylow

C₂×H(3,𝔽₃)

C₄⋊Q₈

yes

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

C₂×C₃×Q₈

—

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×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

—

—

—

yes

yes

characteristic, complemented, Sylow

100

C₃×Q₈

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

108

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

characteristic

109

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

110

C₂×Dic₃

—

111

C₂×Dic₃

—

112

C₂×Dic₃

—

113

Dic₆

—

114

C₃×C₈

—

115

C₃×C₈

#634

yes

—

116

C₂×C₃²

#869

yes

—

117

C₂×C₃²

#908

yes

—

118

C₂×C₃²

#909

yes

—

119

C₂×Q₈

—

120

C₂×C₈

C₂×#272

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

C₂×#272

yes

characteristic

128

C₂²×C₃

—

129

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

C₂×#634

yes

—

139

Dic₃

—

140

Dic₃

—

141

Dic₃

—

142

C₃²

#4702

yes

complemented

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₄⋊#272

yes

characteristic

148

C₂×C₄

C₄⋊#272

yes

characteristic

149

C₂×C₄

C₂²×#272

yes

characteristic

150

C₈

#6229

yes

complemented

151

C₂×C₃

—

152

C₂×C₃

—

153

C₂×C₃

—

154

C₂×C₃

#64078

yes

characteristic

155

C₂×C₃

#64117

yes

characteristic

156

C₂×C₃

#64118

yes

characteristic

157

C₂²

C₂×C₄⋊#272

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₂×#6229

yes

—

164

C₃

—

165

C₃

#10592401

yes

characteristic

166

C₂

#11721715

yes

characteristic

167

C₂

#11721787

yes

characteristic

168

C₂

#11721788

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	H(3,𝔽₃)×#97	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#501
LCS	G ▷ C₂×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, cosocle

C₂×C₄×H(3,𝔽₃)

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

H(3,𝔽₃)×#97

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

hypercenter

C₃×#97

C₃⋊D₃

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

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

N	H ≅ G/N	type
28. C₈×H(3,𝔽₃)	Q₈	semidirect
64. C₃²×C₈	Dic₆	semidirect
99. H(3,𝔽₃)	#501	semidirect
142. C₃²	#4702	semidirect
150. C₈	#6229	semidirect

H ≅ G/N

type

28. C₈×H(3,𝔽₃)

Q₈

semidirect

64. C₃²×C₈

Dic₆

semidirect

99. H(3,𝔽₃)

#501

semidirect

142. C₃²

#4702

semidirect

150. C₈

#6229

semidirect

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

subgroup	index
#11721716	2
#11721715	2
H(3,𝔽₃)×#97	2
#11720551	2
#11720550	2
C₃×#4702	3

subgroup

index

Subgroups — #13143805

Conjugacy classes of subgroups

Complements

Maximal subgroups