Finite Group Databasebeta

← #13086956

Subgroups — #13086956

Order 1440.

792 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊D15C211yesmaximal, characteristic, complemented
3C2×#67137C211yesmaximal, characteristic, complemented
4C5×#3752C211yesmaximal, characteristic, complemented
5C32×#685C211yesmaximal, characteristic, complemented
6#10623771C211yesmaximal, characteristic, complemented
7#10623770C211yesmaximal, characteristic, complemented
8#91473—34nomaximal
9C22×C3⋊D15C2211yescharacteristic
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C32×D5C2211yescharacteristic
12C2×D5×C3⋊D3—22no—
13C2×D5×C3⋊D3—22no—
14C2×D5×C3⋊D3C414yescomplemented
15C2×C32×C4×C5C2211yescharacteristic, complemented
16C2×C3⋊Dic15C2211yescharacteristic, complemented
17C4×C3⋊D15—22no—
18C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
19C2×C32×Dic5C2211yescharacteristic, complemented
20#67137—22no—
21C22⋊C4×C3⋊D3—51nomaximal, Hall
22C22×D3×D5—34no—
23C2×C4×D15—34no—
24C2×#1200—34no—
25C5×(C2×D3)⋊C4—34no—
26C3×#685D314yescomplemented
27(C2×D3)⋊Dic5—34no—
28#6665—34no—
29C22×C32×C5C2311yescharacteristic
30C2×C3⋊D15C2×C412yes—
31C2×C3⋊D15D412yes—
32C2×C3⋊D15D412yes—
33C2×C5×C3⋊D3—21no—
34C2×C5×C3⋊D3—21no—
35C2×C5×C3⋊D3C2×C412yescomplemented
36C2×C32×D5—21no—
37C2×C32×D5—21no—
38C2×C32×D5C2×C412yescomplemented
39D5×C3⋊D3—28no—
40C32×C4×C5—21no—
41C3⋊Dic15—21no—
42C5×C3⋊Dic3—21no—
43C32×Dic5—21no—
44D5×C22⋊C4—91noHall
45C23×C3⋊D3—51no—
46C2×C4×C3⋊D3—51no—
47C2×C4×C3⋊D3—51no—
48C32×C22⋊C4—51no—
49#3757—51no—
50#3752—51no—
51#3752D511yescharacteristic, complemented
52C22×D15—34no—
53C22×C5×D3—34no—
54C22×C3×D5C2×D314yes—
55C2×D3×D5—316no—
56C2×D3×D5—68no—
57C2×D3×D5—68no—
58C2×C3×C4×C5C2×D314yescomplemented
59C2×Dic15—34no—
60C4×D15—68no—
61C2×C5×Dic3—34no—
62C2×C3×Dic5C2×D314yescomplemented
63#1200—68no—
64D3×C22⋊C4—154no—
65C2×C32×C5C22×C411yescharacteristic
66C2×C32×C5C2×D411yescharacteristic
67C2×C32×C5C2×D411yescharacteristic
68C3⋊D15C22⋊C414yescomplemented
69C5×C3⋊D3—22no—
70C32×D5—22no—
71C23×D5—91no—
72C2×C4×D5—91no—
73C2×C4×D5—91no—
74C5×C22⋊C4—91no—
75C22⋊Dic5—91no—
76#685—91no—
77#685C3⋊D311yescharacteristic, complemented
78C23×C32—51no—
79C22×C3⋊D3—51no—
80C22×C3⋊D3—54no—
81C22×C3⋊D3—102no—
82C22×C3⋊D3—102no—
83C22×C3⋊D3C2×D511yescharacteristic
84C2×C32×C4—51no—
85C2×C32×C4C2×D511yescharacteristic, complemented
86C2×C3⋊Dic3—51no—
87C2×C3⋊Dic3C2×D511yescharacteristic, complemented
88C4×C3⋊D3—102no—
89C4×C3⋊D3—102no—
90C22×C3×C5C22×D314yes—
91C2×D15—38no—
92C2×D15—38no—
93C2×D15—38no—
94C2×C5×D3—38no—
95C2×C5×D3—64no—
96C2×C5×D3—64no—
97C2×C3×D5—24no—
98C2×C3×D5—24no—
99C2×C3×D5C4×D318yescomplemented
100D3×D5—632no—
101C3×C4×C5—24no—
102Dic15—64no—
103C3×Dic5—24no—
104C5×Dic3—64no—
105C23×D3—154no—
106C2×C4×D3—154no—
107C2×C4×D3—154no—
108C3×C22⋊C4—54no—
109#217—154no—
110(C2×D3)⋊C4—34no—
111(C2×D3)⋊C4—154no—
112C32×C5C2×C22⋊C411yescharacteristic, complemented, Hall
113C23×C5—91no—
114C22×D5—91no—
115C22×D5—94no—
116C22×D5—182no—
117C22×D5—182no—
118C22×D5C2×C3⋊D311yescharacteristic
119C2×C4×C5—91no—
120C2×C4×C5C2×C3⋊D311yescharacteristic, complemented
121C2×Dic5—91no—
122C2×Dic5C2×C3⋊D311yescharacteristic, complemented
123C4×D5—182no—
124C4×D5—182no—
125C22×C32—52no—
126C22×C32—101no—
127C22×C32—101no—
128C22×C32C22×D511yescharacteristic
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3—52no—
132C2×C3⋊D3—52no—
133C2×C3⋊D3—52no—
134C2×C3⋊D3—108no—
135C2×C3⋊D3C4×D512yescomplemented
136C32×C4—21no—
137C32×C4—101no—
138C3⋊Dic3—21no—
139C3⋊Dic3—101no—
140C2×C22⋊C4—451noSylow
141C2×C3×C5C2×C4×D314yes—
142C2×C3×C5D3×D414yes—
143C2×C3×C5D3×D414yes—
144D15—316no—
145C3×D5—28no—
146C5×D3—68no—
147C23×C3—54no—
148C22×D3—34no—
149C22×D3—1516no—
150C22×D3—154no—
151C22×D3—308no—
152C22×D3—308no—
153C2×C3×C4—54no—
154C2×C3×C4D3×D514yescomplemented
155C2×Dic3—34no—
156C2×Dic3—154no—
157C4×D3—308no—
158C4×D3—308no—
159C22×C5—92no—
160C22×C5—181no—
161C22×C5—181no—
162C22×C5C22×C3⋊D311yescharacteristic
163C2×D5—21no—
164C2×D5—21no—
165C2×D5—92no—
166C2×D5—92no—
167C2×D5—92no—
168C2×D5—188no—
169C2×D5C4×C3⋊D312yescomplemented
170C4×C5—21no—
171C4×C5—181no—
172Dic5—21no—
173Dic5—181no—
174C2×C32—102no—
175C2×C32C2×C4×D511yescharacteristic
176C2×C32D4×D511yescharacteristic
177C2×C32D4×D511yescharacteristic
178C3⋊D3—22no—
179C3⋊D3—54no—
180C24—451no—
181C22×C4—451no—
182C22×C4—451no—
183C22⋊C4—51no—
184C22⋊C4—91no—
185C22⋊C4—451no—
186C22⋊C4—451no—
187C3×C5D3×C22⋊C414yescomplemented
188C22×C3—58no—
189C22×C3—104no—
190C22×C3—104no—
191C22×C3C2×D3×D514yes—
192C2×D3—38no—
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—158no—
196C2×D3—158no—
197C2×D3—158no—
198C2×D3—3032no—
199C3×C4—24no—
200C3×C4—104no—
201Dic3—64no—
202Dic3—304no—
203C2×C5—182no—
204C2×C5C2×C4×C3⋊D311yescharacteristic
205C2×C5D4×C3⋊D311yescharacteristic
206C2×C5D4×C3⋊D311yescharacteristic
207D5—22no—
208D5—94no—
209C32D5×C22⋊C411yescharacteristic, complemented, Sylow
210C23—51no—
211C23—91no—
212C23—454no—
213C23—451no—
214C23—902no—
215C23—902no—
216C2×C4—51no—
217C2×C4—91no—
218C2×C4—451no—
219C2×C4—902no—
220C2×C4—902no—
221C2×C4D5×C3⋊D311yescharacteristic, complemented
222C2×C3—108no—
223C2×C3C4×D3×D514yes—
224C2×C3#677714yes—
225C2×C3#679014yes—
226D3—68no—
227D3—1516no—
228C5C22⋊C4×C3⋊D311yescharacteristic, complemented, Sylow
229C22—52no—
230C22—92no—
231C22—101no—
232C22—101no—
233C22—181no—
234C22—181no—
235C22—452no—
236C22—452no—
237C22—452no—
238C22—908no—
239C22C2×D5×C3⋊D311yescharacteristic
240C4—21no—
241C4—101no—
242C4—181no—
243C4—901no—
244C3#9147314yescomplemented
245C2—102no—
246C2—182no—
247C2—454no—
248C2C4×D5×C3⋊D311yescharacteristic
249C2#1062416011yescharacteristic
250C2#1062417311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×C3⋊D3
commutatorC2×C32×C5C22×C4
FittingC2×C32×C4×C5C22
socle, cosocleC22×C32×C5C23
prime radicalC22×D5×C3⋊D3C2
hypercenterC2×C4D5×C3⋊D3
hypocenterC32×C5C2×C22⋊C4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×C3⋊D15C2semidirect
3. C2×#67137C2semidirect
4. C5×#3752C2semidirect
5. C32×#685C2semidirect
6. #10623771C2semidirect
7. #10623770C2semidirect
14. C2×D5×C3⋊D3C4semidirect
15. C2×C32×C4×C5C22semidirect
16. C2×C3⋊Dic15C22semidirect
18. C2×C5×C3⋊Dic3C22semidirect
19. C2×C32×Dic5C22semidirect
26. C3×#685D3semidirect
35. C2×C5×C3⋊D3C2×C4semidirect
38. C2×C32×D5C2×C4semidirect
51. #3752D5semidirect
58. C2×C3×C4×C5C2×D3semidirect
62. C2×C3×Dic5C2×D3semidirect
68. C3⋊D15C22⋊C4semidirect
77. #685C3⋊D3semidirect
85. C2×C32×C4C2×D5semidirect
87. C2×C3⋊Dic3C2×D5semidirect
99. C2×C3×D5C4×D3semidirect
112. C32×C5C2×C22⋊C4semidirect
120. C2×C4×C5C2×C3⋊D3semidirect
122. C2×Dic5C2×C3⋊D3semidirect
135. C2×C3⋊D3C4×D5semidirect
154. C2×C3×C4D3×D5semidirect
169. C2×D5C4×C3⋊D3semidirect
187. C3×C5D3×C22⋊C4semidirect
209. C32D5×C22⋊C4semidirect
221. C2×C4D5×C3⋊D3semidirect
228. C5C22⋊C4×C3⋊D3semidirect
244. C3#91473semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C3⋊D32
C2×C4×C3⋊D152
C2×#671372
C5×#37522
C32×#6852
#106237712
#106237702
#914733
C22⋊C4×C3⋊D35