Group #13179641 — subgroups

Subgroups — #13179641

Order 1728.

3724 conjugacy classes of subgroups; only the 931 normal subgroups are shown.

normal subgroups only

This group has 3724 conjugacy classes of subgroups — too many to draw. Only its 931 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	C₂²×#6308	C₂	1	1	yes	maximal, characteristic
2	C₃³×#117	C₂	1	1	yes	maximal, characteristic
3	#11723967	C₂	1	1	yes	maximal, characteristic
4	#11723928	C₂	1	4	yes	maximal, complemented
5	#11723918	C₂	1	4	yes	maximal
6	C₄×#6308	C₂	1	4	yes	maximal, complemented
7	#10595987	—	3	13	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×#6308	C₄	1	8	yes	complemented
11	C₂×#6308	C₂²	1	2	yes	—
12	C₂×#6308	C₂²	1	2	yes	—
13	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₃³×C₄²	C₂²	1	2	yes	—
16	#88864	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
18	#88857	C₂²	1	8	yes	—
19	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	8	yes	—
20	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
21	C₂²×#634	—	3	13	no	—
22	C₃²×#117	D₃	1	13	yes	—
23	#64135	—	3	13	no	—
24	#64096	—	3	52	no	—
25	#64086	—	3	52	no	—
26	C₄×#634	—	3	52	no	—
27	C₂³×C₃³	C₂³	1	1	yes	characteristic
28	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
29	C₂×C₃³×C₄	C₂³	1	4	yes	—
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	8	yes	—
33	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
34	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
35	#6308	C₂×C₄	1	8	yes	—
36	#6308	C₂×C₄	1	8	yes	—
37	#5516	—	9	13	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
39	C₂²×C₃⋊Dic₃	—	3	26	no	—
40	C₂×#634	—	3	104	no	—
41	C₂×#634	—	3	26	no	—
42	C₂×#634	—	3	26	no	—
43	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
44	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
45	C₃²×C₄²	C₂×D₃	1	26	yes	—
46	#3757	—	3	26	no	—
47	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
48	#3750	—	3	104	no	—
49	C₄×C₃⋊Dic₃	—	3	104	no	—
50	C₄×C₃⋊Dic₃	—	3	26	no	—
51	C₂²×C₃³	—	2	2	no	—
52	C₂²×C₃³	C₂²×C₄	1	2	yes	—
53	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
54	C₃³×C₄	—	2	4	no	—
55	C₃³×C₄	C₂²×C₄	1	2	yes	—
56	C₃³×C₄	C₂²×C₄	1	2	yes	—
57	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
58	C₃⋊(C₃⋊Dic₃)	C₂²×C₄	1	8	yes	—
59	C₂²×Dic₆	—	9	13	no	—
60	C₃×#117	C₃⋊D₃	1	13	yes	—
61	#926	—	9	13	no	—
62	#887	—	9	52	no	—
63	#877	—	9	52	no	—
64	C₄×Dic₆	—	9	52	no	—
65	C₂³×C₃²	C₂²×D₃	1	13	yes	—
66	C₂×C₃²×C₄	C₄×D₃	1	52	yes	—
67	C₂×C₃²×C₄	C₂²×D₃	1	52	yes	—
68	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
69	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
70	C₂×C₃⋊Dic₃	—	3	104	no	—
71	C₂×C₃⋊Dic₃	—	3	52	no	—
72	C₂×C₃⋊Dic₃	—	3	52	no	—
73	#634	—	3	104	no	—
74	#634	—	3	104	no	—
75	#519	—	27	1	no	Sylow
76	C₂×C₃³	—	2	2	no	—
77	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₃³	2₋¹⁺⁴	1	2	yes	—
79	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
80	C₂²×Dic₃	—	9	26	no	—
81	C₂×Dic₆	—	9	104	no	—
82	C₂×Dic₆	—	9	26	no	—
83	C₂×Dic₆	—	9	26	no	—
84	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
85	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
86	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
87	#217	—	9	26	no	—
88	C₄⋊Dic₃	—	9	26	no	—
89	Dic₃⋊C₄	—	9	104	no	—
90	C₄×Dic₃	—	9	104	no	—
91	C₄×Dic₃	—	9	26	no	—
92	C₂²×C₃²	—	2	26	no	—
93	C₂²×C₃²	C₂×C₄×D₃	1	26	yes	—
94	C₂²×C₃²	C₂³×D₃	1	13	yes	—
95	C₃²×C₄	—	2	52	no	—
96	C₃²×C₄	C₂×C₄×D₃	1	26	yes	—
97	C₃²×C₄	C₂×C₄×D₃	1	26	yes	—
98	C₃⋊Dic₃	—	3	104	no	—
99	C₃⋊Dic₃	—	6	52	no	—
100	C₂²×Q₈	—	27	1	no	—
101	C₄×Q₈	—	27	4	no	—
102	C₄×Q₈	—	27	4	no	—
103	#117	—	27	1	no	—
104	#117	—	27	4	no	—
105	#117	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
106	C₃³	#519	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
108	C₂×C₃×C₄	C₄×C₃⋊D₃	1	52	yes	—
109	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	52	yes	—
110	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
111	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
112	C₂×Dic₃	—	9	52	no	—
113	C₂×Dic₃	—	9	104	no	—
114	C₂×Dic₃	—	9	52	no	—
115	Dic₆	—	9	104	no	—
116	Dic₆	—	9	104	no	—
117	C₂×C₃²	—	2	26	no	—
118	C₂×C₃²	C₂²×C₄×D₃	1	13	yes	—
119	C₂×C₃²	#1010	1	26	yes	—
120	C₂×Q₈	—	27	8	no	—
121	C₂×Q₈	—	27	2	no	—
122	C₂×Q₈	—	27	2	no	—
123	C₂²×C₄	—	27	2	no	—
124	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
125	C₄⋊C₄	—	27	8	no	—
126	C₄⋊C₄	—	27	2	no	—
127	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
128	C₂²⋊C₄	—	27	2	no	—
129	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
130	C₄²	—	27	8	no	—
131	C₄²	—	27	2	no	—
132	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
133	C₂²×C₃	—	2	26	no	—
134	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	26	yes	—
135	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
136	C₃×C₄	—	2	52	no	—
137	C₃×C₄	C₂×C₄×C₃⋊D₃	1	26	yes	—
138	C₃×C₄	C₂×C₄×C₃⋊D₃	1	26	yes	—
139	Dic₃	—	9	104	no	—
140	Dic₃	—	18	52	no	—
141	C₃²	#5516	1	13	yes	complemented
142	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
143	Q₈	—	27	8	no	—
144	Q₈	—	27	8	no	—
145	C₂×C₄	—	27	4	no	—
146	C₂×C₄	—	27	8	no	—
147	C₂×C₄	—	27	4	no	—
148	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	4	yes	—
149	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	4	yes	—
150	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
152	C₂×C₃	—	2	26	no	—
153	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	13	yes	—
154	C₂×C₃	#64367	1	26	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
157	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	27	8	no	—
160	C₄	—	54	4	no	—
161	C₄	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
162	C₄	C₂×C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
163	C₃	#10595987	1	13	yes	complemented
164	C₂	—	2	2	no	—
165	C₂	C₂²×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
166	C₂	#11724685	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6308

C₂

yes

maximal, characteristic

C₃³×#117

C₂

yes

maximal, characteristic

#11723967

C₂

yes

maximal, characteristic

#11723928

C₂

yes

maximal, complemented

#11723918

C₂

yes

maximal

C₄×#6308

C₂

yes

maximal, complemented

#10595987

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂²×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₂×#6308

C₄

yes

complemented

C₂×#6308

C₂²

yes

—

C₂×#6308

C₂²

yes

—

C₃³×C₄⋊C₄

C₂²

yes

—

C₃³×C₂²⋊C₄

C₂²

yes

—

C₃³×C₄²

C₂²

yes

—

#88864

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₂²×#634

—

C₃²×#117

D₃

yes

—

—

—

—

—

C₂³×C₃³

C₂³

yes

characteristic

C₂×C₃³×C₄

C₂×C₄

yes

—

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

#6308

C₂×C₄

yes

—

#6308

C₂×C₄

yes

—

#5516

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

C₂×#634

—

104

—

C₂×#634

—

C₂×#634

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₂²⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

#3757

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

104

—

C₄×C₃⋊Dic₃

—

104

—

C₄×C₃⋊Dic₃

—

C₂²×C₃³

—

C₂²×C₃³

C₂²×C₄

yes

—

C₂²×C₃³

C₂⁴

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂²×C₄

yes

—

C₃³×C₄

C₂²×C₄

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

C₂²×C₄

yes

—

C₂²×Dic₆

—

C₃×#117

C₃⋊D₃

yes

—

#926

—

#887

—

#877

—

C₄×Dic₆

—

C₂³×C₃²

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

104

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

104

—

#634

—

104

—

#519

—

Sylow

C₂×C₃³

—

C₂×C₃³

C₂³×C₄

yes

characteristic

C₂×C₃³

2₋¹⁺⁴

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

104

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₂²⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

104

—

C₄×Dic₃

—

104

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄×D₃

yes

—

C₂²×C₃²

C₂³×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃⋊Dic₃

—

104

—

C₃⋊Dic₃

—

100

C₂²×Q₈

—

101

C₄×Q₈

—

102

C₄×Q₈

—

103

#117

—

104

#117

—

105

#117

C₃⋊(C₃⋊D₃)

yes

characteristic

106

C₃³

#519

yes

characteristic, complemented, Sylow

107

C₂³×C₃

C₂²×C₃⋊D₃

yes

—

108

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

109

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

110

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

111

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

112

C₂×Dic₃

—

113

C₂×Dic₃

—

104

—

114

C₂×Dic₃

—

115

Dic₆

—

104

—

116

Dic₆

—

104

—

117

C₂×C₃²

—

118

C₂×C₃²

C₂²×C₄×D₃

yes

—

119

C₂×C₃²

#1010

yes

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

—

133

C₂²×C₃

—

134

C₂²×C₃

C₂×C₄×C₃⋊D₃

yes

—

135

C₂²×C₃

C₂³×C₃⋊D₃

yes

—

136

C₃×C₄

—

137

C₃×C₄

C₂×C₄×C₃⋊D₃

yes

—

138

C₃×C₄

C₂×C₄×C₃⋊D₃

yes

—

139

Dic₃

—

104

—

140

Dic₃

—

141

C₃²

#5516

yes

complemented

142

C₂³

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₄×C₃⋊(C₃⋊D₃)

yes

—

149

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

150

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

151

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

152

C₂×C₃

—

153

C₂×C₃

C₂²×C₄×C₃⋊D₃

yes

—

154

C₂×C₃

#64367

yes

—

155

C₂²

—

156

C₂²

C₂×C₄×C₃⋊(C₃⋊D₃)

yes

—

157

C₂²

C₂³×C₃⋊(C₃⋊D₃)

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂×C₄×C₃⋊(C₃⋊D₃)

yes

—

162

C₄

C₂×C₄×C₃⋊(C₃⋊D₃)

yes

—

163

C₃

#10595987

yes

complemented

164

C₂

—

165

C₂

C₂²×C₄×C₃⋊(C₃⋊D₃)

yes

characteristic

166

C₂

#11724685

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂³×C₄
Fitting	C₃³×#117	C₂
socle, cosocle	C₂²×C₃³	C₂⁴
prime radical	C₂³×C₃³	C₂³
hypercenter	#117	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#519
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊(C₃⋊D₃)

commutator

C₂×C₃³

C₂³×C₄

Fitting

C₃³×#117

C₂

socle, cosocle

C₂²×C₃³

C₂⁴

prime radical

C₂³×C₃³

C₂³

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂×C₃³ ▷ C₃³

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
4. #11723928	C₂	semidirect
6. C₄×#6308	C₂	semidirect
10. C₂×#6308	C₄	semidirect
106. C₃³	#519	semidirect
141. C₃²	#5516	semidirect
163. C₃	#10595987	semidirect

H ≅ G/N

type

4. #11723928

C₂

semidirect

6. C₄×#6308

C₂

semidirect

10. C₂×#6308

C₄

semidirect

106. C₃³

#519

semidirect

141. C₃²

#5516

semidirect

163. C₃

#10595987

semidirect

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

subgroup	index
C₂²×#6308	2
C₃³×#117	2
#11723967	2
#11723928	2
#11723918	2
C₄×#6308	2
#10595987	3

subgroup

index

C₂²×#6308

C₃³×#117

Subgroups — #13179641

Conjugacy classes of subgroups

Complements

Maximal subgroups