Group #13179646 — subgroups

Subgroups — #13179646

Order 1728.

2688 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 2688 conjugacy classes of subgroups — too many to draw. Only its 667 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	C₃³×#117	C₂	1	1	yes	maximal, characteristic
2	#11723965	C₂	1	2	yes	maximal
3	#11723930	C₂	1	4	yes	maximal, complemented
4	#11723920	C₂	1	4	yes	maximal
5	#11723911	C₂	1	1	yes	maximal, characteristic, complemented
6	#11723910	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6308	C₂	1	2	yes	maximal, complemented
8	#10595989	—	3	13	no	maximal
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂×#6308	C₂²	1	2	yes	—
11	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃³×C₄²	C₂²	1	1	yes	characteristic
15	#88864	C₂²	1	4	yes	—
16	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	4	yes	—
18	#88857	C₂²	1	4	yes	—
19	#88857	C₂²	1	4	yes	—
20	#88857	C₂²	1	4	yes	—
21	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
22	C₃²×#117	D₃	1	13	yes	—
23	#64133	—	3	26	no	—
24	#64098	—	3	52	no	—
25	#64088	—	3	52	no	—
26	#64079	—	3	13	no	—
27	#64078	—	3	13	no	—
28	C₄×#634	—	3	26	no	—
29	C₂³×C₃³	C₂³	1	1	yes	characteristic
30	C₂×C₃³×C₄	—	2	2	no	—
31	C₂×C₃³×C₄	C₂³	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃³×C₄	C₂³	1	2	yes	—
34	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
36	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
37	#6308	—	2	4	no	—
38	#5518	—	9	13	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
40	C₂×#634	—	3	26	no	—
41	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
42	C₃²×C₂²⋊C₄	C₂×D₃	1	26	yes	—
43	C₃²×C₄²	C₂×D₃	1	13	yes	—
44	C₃²×C₄²	C₂×D₃	1	13	yes	—
45	#3757	—	3	52	no	—
46	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
47	C₄⋊(C₃⋊Dic₃)	—	3	52	no	—
48	#3750	—	3	52	no	—
49	#3750	—	3	52	no	—
50	#3750	—	3	52	no	—
51	C₄×C₃⋊Dic₃	—	3	52	no	—
52	C₂²×C₃³	—	2	1	no	—
53	C₂²×C₃³	—	2	2	no	—
54	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
55	C₃³×C₄	—	2	1	no	—
56	C₃³×C₄	—	2	2	no	—
57	C₃³×C₄	—	2	2	no	—
58	C₃³×C₄	#41	1	2	yes	—
59	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
60	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
61	C₃×#117	C₃⋊D₃	1	13	yes	—
62	#924	—	9	26	no	—
63	#889	—	9	52	no	—
64	#879	—	9	52	no	—
65	#870	—	9	13	no	—
66	#869	—	9	13	no	—
67	C₄×Dic₆	—	9	26	no	—
68	C₂³×C₃²	C₂²×D₃	1	13	yes	—
69	C₂×C₃²×C₄	—	2	26	no	—
70	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
71	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
72	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
74	C₂×C₃⋊Dic₃	—	3	52	no	—
75	C₂×C₃⋊Dic₃	—	3	52	no	—
76	#634	—	6	52	no	—
77	#541	—	27	1	no	Sylow
78	C₂×C₃³	—	4	1	no	—
79	C₂×C₃³	C₂×#41	1	1	yes	characteristic
80	C₂×C₃³	2₋¹⁺⁴	1	2	yes	—
81	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
82	C₂×Dic₆	—	9	26	no	—
83	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
84	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
85	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
86	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
87	#217	—	9	52	no	—
88	C₄⋊Dic₃	—	9	26	no	—
89	C₄⋊Dic₃	—	9	52	no	—
90	Dic₃⋊C₄	—	9	52	no	—
91	Dic₃⋊C₄	—	9	52	no	—
92	Dic₃⋊C₄	—	9	52	no	—
93	C₄×Dic₃	—	9	52	no	—
94	C₂²×C₃²	—	2	13	no	—
95	C₂²×C₃²	—	2	26	no	—
96	C₂²×C₃²	C₂³×D₃	1	13	yes	—
97	C₃²×C₄	—	2	13	no	—
98	C₃²×C₄	—	2	26	no	—
99	C₃²×C₄	—	2	26	no	—
100	C₃²×C₄	#235	1	26	yes	—
101	C₃⋊Dic₃	—	6	52	no	—
102	C₃⋊Dic₃	—	6	52	no	—
103	C₄⋊Q₈	—	27	1	no	—
104	#126	—	27	4	no	—
105	#125	—	27	4	no	—
106	#125	—	27	1	no	—
107	C₂²⋊Q₈	—	27	2	no	—
108	C₄×Q₈	—	27	2	no	—
109	#117	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
110	C₃³	#541	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
112	C₂×C₃×C₄	—	2	26	no	—
113	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
114	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
115	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
116	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
117	C₂×Dic₃	—	9	52	no	—
118	C₂×Dic₃	—	9	52	no	—
119	Dic₆	—	18	52	no	—
120	C₂×C₃²	—	4	13	no	—
121	C₂×C₃²	C₂×#235	1	13	yes	—
122	C₂×C₃²	#1010	1	26	yes	—
123	C₂×Q₈	—	27	2	no	—
124	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
125	C₄⋊C₄	—	27	2	no	—
126	C₄⋊C₄	—	27	4	no	—
127	C₄⋊C₄	—	27	4	no	—
128	C₄⋊C₄	—	27	4	no	—
129	C₄⋊C₄	—	27	4	no	—
130	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
131	C₂²⋊C₄	—	27	4	no	—
132	C₂²⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
133	C₄²	—	27	4	no	—
134	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
135	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
136	C₂²×C₃	—	2	13	no	—
137	C₂²×C₃	—	2	26	no	—
138	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
139	C₃×C₄	—	2	13	no	—
140	C₃×C₄	—	2	26	no	—
141	C₃×C₄	—	2	26	no	—
142	C₃×C₄	#3828	1	26	yes	—
143	Dic₃	—	18	52	no	—
144	Dic₃	—	18	52	no	—
145	C₃²	#5518	1	13	yes	complemented
146	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
147	Q₈	—	54	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	27	4	no	—
150	C₂×C₄	—	27	4	no	—
151	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
152	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
153	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
155	C₂×C₃	—	4	13	no	—
156	C₂×C₃	C₂×#3828	1	13	yes	—
157	C₂×C₃	#64367	1	26	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	54	4	no	—
165	C₄	—	54	4	no	—
166	C₄	#89076	1	2	yes	—
167	C₃	#10595989	1	13	yes	complemented
168	C₂	—	4	1	no	—
169	C₂	C₂×#89076	1	1	yes	characteristic
170	C₂	#11724685	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#117

C₂

yes

maximal, characteristic

#11723965

C₂

yes

maximal

#11723930

C₂

yes

maximal, complemented

#11723920

C₂

yes

maximal

#11723911

C₂

yes

maximal, characteristic, complemented

#11723910

C₂

yes

maximal, characteristic, complemented

C₄×#6308

C₂

yes

maximal, complemented

#10595989

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂×#6308

C₂²

yes

—

C₃³×C₄⋊C₄

C₂²

yes

—

C₃³×C₂²⋊C₄

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₃³×C₄²

C₂²

yes

characteristic

#88864

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

C₂²

yes

—

#88857

C₂²

yes

—

#88857

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₃²×#117

D₃

yes

—

—

—

—

—

—

—

C₂³×C₃³

C₂³

yes

characteristic

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

#6308

—

#5518

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂×#634

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₂²⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

#3757

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

#3750

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

C₂⁴

yes

characteristic

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

#41

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃×#117

C₃⋊D₃

yes

—

—

—

—

—

—

C₄×Dic₆

—

C₂³×C₃²

C₂²×D₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#541

—

Sylow

C₂×C₃³

—

C₂×C₃³

C₂×#41

yes

characteristic

C₂×C₃³

2₋¹⁺⁴

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₂²⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂³×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

100

C₃²×C₄

#235

yes

—

101

C₃⋊Dic₃

—

102

C₃⋊Dic₃

—

103

C₄⋊Q₈

—

104

#126

—

105

#125

—

106

#125

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

#117

C₃⋊(C₃⋊D₃)

yes

characteristic

110

C₃³

#541

yes

characteristic, complemented, Sylow

111

C₂³×C₃

C₂²×C₃⋊D₃

yes

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

114

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

115

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

116

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

Dic₆

—

120

C₂×C₃²

—

121

C₂×C₃²

C₂×#235

yes

—

122

C₂×C₃²

#1010

yes

—

123

C₂×Q₈

—

124

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

133

C₄²

—

134

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

135

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

C₂³×C₃⋊D₃

yes

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

#3828

yes

—

143

Dic₃

—

144

Dic₃

—

145

C₃²

#5518

yes

complemented

146

C₂³

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

152

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

153

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

154

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

155

C₂×C₃

—

156

C₂×C₃

C₂×#3828

yes

—

157

C₂×C₃

#64367

yes

—

158

C₂²

—

159

C₂²

—

160

C₂²

C₂³×C₃⋊(C₃⋊D₃)

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#89076

yes

—

167

C₃

#10595989

yes

complemented

168

C₂

—

169

C₂

C₂×#89076

yes

characteristic

170

C₂

#11724685

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₃³×#117	C₂
prime radical	C₂³×C₃³	C₂³
hypercenter	#117	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#541
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊(C₃⋊D₃)

commutator, socle, cosocle

C₂²×C₃³

C₂⁴

Fitting

C₃³×#117

C₂

prime radical

C₂³×C₃³

C₂³

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂²×C₃³ ▷ C₃³

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
3. #11723930	C₂	semidirect
5. #11723911	C₂	semidirect
6. #11723910	C₂	semidirect
7. C₄×#6308	C₂	semidirect
110. C₃³	#541	semidirect
145. C₃²	#5518	semidirect
167. C₃	#10595989	semidirect

H ≅ G/N

type

3. #11723930

C₂

semidirect

5. #11723911

C₂

semidirect

6. #11723910

C₂

semidirect

7. C₄×#6308

C₂

semidirect

110. C₃³

#541

semidirect

145. C₃²

#5518

semidirect

167. C₃

#10595989

semidirect

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

subgroup	index
C₃³×#117	2
#11723965	2
#11723930	2
#11723920	2
#11723911	2
#11723910	2
C₄×#6308	2
#10595989	3

subgroup

index

C₃³×#117

Subgroups — #13179646

Conjugacy classes of subgroups

Complements

Maximal subgroups