Group #13144838 — subgroups

Subgroups — #13144838

Order 1728.

910 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 277 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×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88534	C₂	1	3	yes	maximal
3	#11720572	C₂	1	3	yes	maximal
4	#10592929	—	3	1	no	maximal
5	#10588910	—	3	3	no	maximal
6	C₂×C₃×Q₈×C₉	—	2	6	no	—
7	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
8	C₂²×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₂²	1	3	yes	—
10	#88534	—	2	6	no	—
11	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
12	C₂×#3750	—	3	3	no	—
13	C₂²×Q₈×C₉	D₃	1	3	yes	—
14	#63658	—	3	3	no	—
15	C₂×Dic₉⋊C₄	—	3	9	no	—
16	C₉⋊#95	—	3	9	no	—
17	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
18	C₃×Q₈×C₉	—	2	8	no	—
19	C₂×C₃×C₄×C₉	—	2	6	no	—
20	C₂×C₃×C₄×C₉	D₄	1	6	yes	—
21	C₂×C₃⋊Dic₉	—	2	3	no	—
22	C₂×C₃⋊Dic₉	—	2	3	no	—
23	C₂×C₃⋊Dic₉	—	2	3	no	—
24	C₂×C₃⋊Dic₉	Q₈	1	6	yes	complemented
25	#5230	—	9	3	no	—
26	#5230	—	9	1	no	—
27	C₂×C₃²×Q₈	—	2	6	no	—
28	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
29	C₂²×C₃⋊Dic₃	—	3	3	no	—
30	C₂²×C₃⋊Dic₃	—	3	1	no	—
31	#3750	—	6	6	no	—
32	C₂×Q₈×C₉	—	2	18	no	—
33	C₂²×C₄×C₉	C₂×D₃	1	9	yes	—
34	C₂²×Dic₉	—	3	9	no	—
35	C₂²×Dic₉	—	3	3	no	—
36	Dic₉⋊C₄	—	6	18	no	—
37	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
38	C₂²×C₃×C₉	C₂×Q₈	1	3	yes	—
39	C₂²×C₃×C₉	#41	1	1	yes	characteristic
40	C₃×C₄×C₉	—	2	6	no	—
41	C₃⋊Dic₉	—	2	6	no	—
42	C₃⋊Dic₉	—	4	1	no	—
43	C₂²×C₃×Q₈	D₉	1	3	yes	—
44	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
45	C₂×Dic₃⋊C₄	—	9	9	no	—
46	C₂×Dic₃⋊C₄	—	9	3	no	—
47	C₃⋊#95	—	9	9	no	—
48	C₃⋊#95	—	9	3	no	—
49	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
50	C₃²×Q₈	—	2	8	no	—
51	C₂×C₃²×C₄	—	2	6	no	—
52	C₂×C₃²×C₄	#67	1	6	yes	—
53	C₂×C₃⋊Dic₃	—	3	6	no	—
54	C₂×C₃⋊Dic₃	—	6	3	no	—
55	C₂×C₃⋊Dic₃	—	6	3	no	—
56	C₂×C₃⋊Dic₃	—	6	3	no	—
57	C₂³×C₉	C₂²×D₃	1	3	yes	—
58	Q₈×C₉	—	2	24	no	—
59	C₂×C₄×C₉	—	2	18	no	—
60	C₂×C₄×C₉	#67	1	18	yes	—
61	C₂×Dic₉	—	3	18	no	—
62	C₂×Dic₉	—	6	9	no	—
63	C₂×Dic₉	—	6	9	no	—
64	C₂×Dic₉	—	6	9	no	—
65	#395	—	27	1	no	Sylow
66	C₂×C₃×C₉	#120	1	1	yes	characteristic
67	C₂×C₃×C₉	C₂²⋊Q₈	1	3	yes	—
68	C₂×C₃×C₉	C₄⋊Q₈	1	3	yes	—
69	C₂×C₃×Q₈	—	2	18	no	—
70	C₂×C₃×Q₈	—	2	6	no	—
71	C₂²×C₃×C₄	C₂×D₉	1	9	yes	—
72	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
73	C₂²×Dic₃	—	9	9	no	—
74	C₂²×Dic₃	—	9	3	no	—
75	C₂²×Dic₃	—	9	3	no	—
76	C₂²×Dic₃	—	9	1	no	—
77	Dic₃⋊C₄	—	18	18	no	—
78	Dic₃⋊C₄	—	18	6	no	—
79	C₂²×C₃²	D₃×Q₈	1	3	yes	—
80	C₂²×C₃²	#239	1	1	yes	characteristic
81	C₂²×C₃²	C₂×#67	1	3	yes	—
82	C₃²×C₄	—	2	6	no	—
83	C₃⋊Dic₃	—	6	6	no	—
84	C₃⋊Dic₃	—	12	1	no	—
85	C₂²×C₉	D₃×Q₈	1	9	yes	—
86	C₂²×C₉	#239	1	3	yes	—
87	C₂²×C₉	C₂×#67	1	9	yes	—
88	C₄×C₉	—	2	18	no	—
89	Dic₉	—	6	18	no	—
90	Dic₉	—	12	3	no	—
91	C₂²×Q₈	C₃⋊D₉	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	27	3	no	—
93	#95	—	27	3	no	—
94	C₃×C₉	#395	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₃	C₂²×D₉	1	3	yes	—
96	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
97	C₃×Q₈	—	2	8	no	—
98	C₃×Q₈	—	2	24	no	—
99	C₂×C₃×C₄	—	2	18	no	—
100	C₂×C₃×C₄	—	2	6	no	—
101	C₂×C₃×C₄	#611	1	18	yes	—
102	C₂×C₃×C₄	#638	1	6	yes	—
103	C₂×Dic₃	—	9	18	no	—
104	C₂×Dic₃	—	9	6	no	—
105	C₂×Dic₃	—	18	9	no	—
106	C₂×Dic₃	—	18	9	no	—
107	C₂×Dic₃	—	18	3	no	—
108	C₂×Dic₃	—	18	3	no	—
109	C₂×Dic₃	—	18	9	no	—
110	C₂×Dic₃	—	18	3	no	—
111	C₂×C₃²	#944	1	3	yes	—
112	C₂×C₃²	#946	1	3	yes	—
113	C₂×C₃²	#953	1	1	yes	characteristic
114	C₂×C₉	#944	1	9	yes	—
115	C₂×C₉	#946	1	9	yes	—
116	C₂×C₉	#953	1	3	yes	—
117	C₂×Q₈	—	2	6	no	—
118	C₂²×C₄	—	27	3	no	—
119	C₂²×C₄	—	27	1	no	—
120	C₂²×C₄	C₂×C₃⋊D₉	1	3	yes	—
121	C₄⋊C₄	—	54	6	no	—
122	C₂²×C₃	Q₈×D₉	1	9	yes	—
123	C₂²×C₃	#3701	1	3	yes	—
124	C₂²×C₃	C₂×#611	1	9	yes	—
125	C₂²×C₃	Q₈×C₃⋊D₃	1	3	yes	—
126	C₂²×C₃	#3832	1	1	yes	characteristic
127	C₂²×C₃	C₂×#638	1	3	yes	—
128	C₃×C₄	—	2	18	no	—
129	C₃×C₄	—	2	6	no	—
130	Dic₃	—	18	18	no	—
131	Dic₃	—	18	6	no	—
132	Dic₃	—	36	3	no	—
133	Dic₃	—	36	1	no	—
134	C₃²	#5230	1	1	yes	characteristic
135	C₉	#5230	1	3	yes	complemented
136	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
137	Q₈	—	2	8	no	—
138	C₂×C₄	—	2	6	no	—
139	C₂×C₄	—	27	6	no	—
140	C₂×C₄	—	54	3	no	—
141	C₂×C₄	—	54	3	no	—
142	C₂×C₄	—	54	3	no	—
143	C₂×C₄	#6233	1	6	yes	—
144	C₂×C₃	#63506	1	9	yes	—
145	C₂×C₃	#63508	1	9	yes	—
146	C₂×C₃	#63515	1	3	yes	—
147	C₂×C₃	#64153	1	3	yes	—
148	C₂×C₃	#64155	1	3	yes	—
149	C₂×C₃	#64162	1	1	yes	characteristic
150	C₂²	Q₈×C₃⋊D₉	1	3	yes	—
151	C₂²	#88746	1	1	yes	characteristic
152	C₂²	C₂×#6233	1	3	yes	—
153	C₄	—	2	6	no	—
154	C₄	—	54	6	no	—
155	C₄	—	108	1	no	—
156	C₃	#10588910	1	3	yes	complemented
157	C₃	#10592929	1	1	yes	characteristic
158	C₂	#11721854	1	3	yes	—
159	C₂	#11721856	1	3	yes	—
160	C₂	#11721874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×Q₈×C₉

C₂

yes

maximal, characteristic

C₂×#88534

C₂

yes

maximal

#11720572

C₂

yes

—

—

C₂×C₃×Q₈×C₉

—

C₂²×C₃×C₄×C₉

C₂²

yes

—

C₂²×C₃⋊Dic₉

C₂²

yes

characteristic

C₂²×C₃⋊Dic₉

C₂²

yes

—

#88534

—

C₂²×C₃²×Q₈

D₃

yes

characteristic

C₂×#3750

—

C₂²×Q₈×C₉

D₃

yes

—

#63658

—

C₂×Dic₉⋊C₄

—

C₉⋊#95

—

C₂³×C₃×C₉

C₂³

yes

characteristic

C₃×Q₈×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

Q₈

yes

complemented

#5230

—

#5230

—

C₂×C₃²×Q₈

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

C₂²×C₃⋊Dic₃

—

#3750

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

C₂×D₃

yes

—

C₂²×Dic₉

—

C₂²×Dic₉

—

Dic₉⋊C₄

—

C₂²×C₃×C₉

C₂×D₄

yes

—

C₂²×C₃×C₉

C₂×Q₈

yes

—

C₂²×C₃×C₉

#41

yes

characteristic

C₃×C₄×C₉

—

C₃⋊Dic₉

—

C₃⋊Dic₉

—

C₂²×C₃×Q₈

D₉

yes

—

C₂²×C₃×Q₈

C₃⋊D₃

yes

characteristic

C₂×Dic₃⋊C₄

—

C₂×Dic₃⋊C₄

—

C₃⋊#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂³×C₉

C₂²×D₃

yes

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

C₂²⋊Q₈

yes

—

C₂×C₃×C₉

C₄⋊Q₈

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×D₉

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂²×C₃²

D₃×Q₈

yes

—

C₂²×C₃²

#239

yes

characteristic

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂²×C₉

D₃×Q₈

yes

—

C₂²×C₉

#239

yes

—

C₂²×C₉

C₂×#67

yes

—

C₄×C₉

—

Dic₉

—

Dic₉

—

C₂²×Q₈

C₃⋊D₉

yes

characteristic

C₂×C₄⋊C₄

—

#95

—

C₃×C₉

#395

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂²×D₉

yes

—

C₂³×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

#611

yes

—

102

C₂×C₃×C₄

#638

yes

—

103

C₂×Dic₃

—

104

C₂×Dic₃

—

105

C₂×Dic₃

—

106

C₂×Dic₃

—

107

C₂×Dic₃

—

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

C₂×C₃²

#944

yes

—

112

C₂×C₃²

#946

yes

—

113

C₂×C₃²

#953

yes

characteristic

114

C₂×C₉

#944

yes

—

115

C₂×C₉

#946

yes

—

116

C₂×C₉

#953

yes

—

117

C₂×Q₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×C₃⋊D₉

yes

—

121

C₄⋊C₄

—

122

C₂²×C₃

Q₈×D₉

yes

—

123

C₂²×C₃

#3701

yes

—

124

C₂²×C₃

C₂×#611

yes

—

125

C₂²×C₃

Q₈×C₃⋊D₃

yes

—

126

C₂²×C₃

#3832

yes

characteristic

127

C₂²×C₃

C₂×#638

yes

—

128

C₃×C₄

—

129

C₃×C₄

—

130

Dic₃

—

131

Dic₃

—

132

Dic₃

—

133

Dic₃

—

134

C₃²

#5230

yes

characteristic

135

C₉

#5230

yes

complemented

136

C₂³

C₂²×C₃⋊D₉

yes

characteristic

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#6233

yes

—

144

C₂×C₃

#63506

yes

—

145

C₂×C₃

#63508

yes

—

146

C₂×C₃

#63515

yes

—

147

C₂×C₃

#64153

yes

—

148

C₂×C₃

#64155

yes

—

149

C₂×C₃

#64162

yes

characteristic

150

C₂²

Q₈×C₃⋊D₉

yes

—

151

C₂²

#88746

yes

characteristic

152

C₂²

C₂×#6233

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

108

—

156

C₃

#10588910

yes

complemented

157

C₃

#10592929

yes

characteristic

158

C₂

#11721854

yes

—

159

C₂

#11721856

yes

—

160

C₂

#11721874

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator, cosocle	C₂³×C₃×C₉	C₂³
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	C₂²×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#395
LCS	G ▷ C₂³×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×C₃⋊D₉

commutator, cosocle

C₂³×C₃×C₉

C₂³

Frattini

C₂³×C₃

C₂²×C₃⋊D₃

Fitting

C₂²×C₃×Q₈×C₉

C₂

socle, prime radical

C₂³×C₃²

C₂²×D₃

hypercenter

C₂²×Q₈

C₃⋊D₉

hypocenter

C₃×C₉

#395

LCS

G ▷ C₂³×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
24. C₂×C₃⋊Dic₉	Q₈	semidirect
94. C₃×C₉	#395	semidirect
135. C₉	#5230	semidirect
156. C₃	#10588910	semidirect

H ≅ G/N

type

24. C₂×C₃⋊Dic₉

Q₈

semidirect

94. C₃×C₉

#395

semidirect

135. C₉

#5230

semidirect

156. C₃

#10588910

semidirect

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂×#88534	2
#11720572	2
#10592929	3
#10588910	3

subgroup

index

C₂²×C₃×Q₈×C₉

Subgroups — #13144838

Conjugacy classes of subgroups

Complements

Maximal subgroups