Finite Group Databasebeta

← #13087532

Subgroups — #13087532

Order 1440.

268 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×C5×#239C211yesmaximal, characteristic, complemented
2#10624197C211yesmaximal, characteristic, complemented
3#10623911C211yesmaximal, characteristic, complemented
4C3×#6718C211yesmaximal, characteristic, complemented
5#10623802C211yesmaximal, characteristic, complemented
6#10623800C211yesmaximal, characteristic, complemented
7D3×C3⋊#168C211yesmaximal, characteristic, complemented
8#91839—31nomaximal
9#91519—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11C4⋊(C3⋊D15)C2211yescharacteristic
12C3×Dic30C2211yescharacteristic
13C3×C5×D12—21no—
14C3×C5×D12C2211yescharacteristic
15C3×C4×C5×D3—21no—
16C3×C4×C5×D3C2211yescharacteristic
17#67152—21no—
18#67149—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic
20C3×C3⋊#168C2211yescharacteristic
21#63949—51nomaximal, Hall
22C3×C5×#41—31no—
23#6817—31no—
24C5×#239D311yescharacteristic, complemented
25#6769—31no—
26#6718D311yescharacteristic, complemented
27#6717—31no—
28#6717—31no—
29#6717—61no—
30#6716—31no—
31#6715—31no—
32C2×C3⋊#168—31no—
33C3×#689—31no—
34#6656—31no—
35#6654—31no—
36D3×#168—31no—
37C2×C3⋊D15—21no—
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3D411yescharacteristic
40C32×C4×C5—21no—
41C32×C4×C5C2311yescharacteristic
42C3×Dic15—21no—
43C3×C5×Dic3D411yescharacteristic
44#4096—91noHall
45C3×#239D511yescharacteristic, complemented
46#3799—51no—
47#3755—51no—
48C3×#216—51no—
49#3715—51no—
50#3714—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—21no—
53C3×C5×Q8C2×D311yescharacteristic
54C3×C5×Q8C2×D311yescharacteristic
55C3×C5×D4—31no—
56C3×C5×D4—61no—
57C2×C3×C4×C5—31no—
58C2×C3×C4×C5—61no—
59#1220—61no—
60D60—31no—
61D60—31no—
62D60—61no—
63C4×D15—61no—
64Dic30C2×D311yescharacteristic
65C5×D12—21no—
66C5×D12C2×D311yescharacteristic
67C4×C5×D3—21no—
68C4×C5×D3C2×D311yescharacteristic
69C3×Dic10—31no—
70#1203—61no—
71#1200—61no—
72C3⋊#168—31no—
73C3⋊#168—31no—
74C3⋊#168—61no—
75C3⋊#168C2×D311yescharacteristic
76C3×#168—31no—
77#950—151no—
78#919—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3⋊D15—41no—
81C3×C5×D3—21no—
82C3×C5×D3—41no—
83C5×#41—31no—
84#709—91no—
85#689—31no—
86#688—91no—
87#687—91no—
88#686—91no—
89C2×#168—91no—
90C32×Q8C2×D511yescharacteristic
91C4⋊(C3⋊D3)—51no—
92C3×D12—21no—
93C3×D12C2×D511yescharacteristic
94C3×C4×D3—21no—
95C3×C4×D3C2×D511yescharacteristic
96C3×Dic6—51no—
97#626—101no—
98#624—101no—
99C3⋊(C3⋊C8)—51no—
100C3×C3⋊C8—51no—
101C22×C3×C5—31no—
102C22×C3×C5—61no—
103C2×D15—61no—
104C2×D15—61no—
105C2×D15—121no—
106C2×C5×D3—21no—
107C2×C5×D3#6711yescharacteristic
108C3×C4×C5—21no—
109C3×C4×C5—21no—
110C3×C4×C5—21no—
111C3×C4×C5—31no—
112C3×C4×C5—41no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15—21no—
116C3×Dic5—61no—
117C5×Dic3#6711yescharacteristic
118C3×#41—31no—
119#239—151no—
120#239D1511yescharacteristic, complemented
121#235—151no—
122C3×Q16—151no—
123#216—51no—
124#215—151no—
125#215—151no—
126#215—151no—
127#215—301no—
128#214—151no—
129#213—151no—
130C2×C3⋊C8—151no—
131D24—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8D3211yescharacteristic
135C5×D4—31no—
136C5×D4—61no—
137C2×C4×C5—31no—
138C2×C4×C5—61no—
139C22⋊D5—181no—
140D20—91no—
141C4×D5—181no—
142Dic10—31no—
143#168—31no—
144#168—91no—
145C2×C3⋊D3—101no—
146C2×C3×D3—21no—
147C2×C3×D3C22⋊D511yescharacteristic
148C32×C4—21no—
149C32×C4C22×D511yescharacteristic
150C3×Dic3—101no—
151C3×Dic3C22⋊D511yescharacteristic
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—61no—
155C2×C3×C5—121no—
156C2×C3×C5D3×D411yescharacteristic
157C2×C3×C5C2×#6711yescharacteristic
158D15—121no—
159D15—121no—
160D15—241no—
161C5×D3—21no—
162C5×D3—41no—
163C3×Q8—21no—
164C3×Q8—151no—
165C3×Q8D3×D511yescharacteristic
166C3×Q8C2×D1511yescharacteristic
167C3×D4—31no—
168C3×D4—61no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171#67—301no—
172D12—21no—
173D12—151no—
174D12—151no—
175D12—301no—
176D12—301no—
177D12C2×D1511yescharacteristic
178C4×D3—21no—
179C4×D3—301no—
180C4×D3—301no—
181C4×D3C2×D1511yescharacteristic
182Dic6—51no—
183C3×C8—151no—
184C3⋊C8—51no—
185C3⋊C8—151no—
186C3⋊C8—151no—
187C3⋊C8—301no—
188C22×C5—31no—
189C22×C5—61no—
190C2×D5—181no—
191C4×C5—21no—
192C4×C5—31no—
193C4×C5C2×D3211yescharacteristic
194Dic5—61no—
195C2×C32C2×C22⋊D511yescharacteristic
196C3⋊D3—201no—
197C3×D3—21no—
198C3×D3—41no—
199#41—31no—
200#41—451no—
201Q16—151no—
202SD16—451no—
203SD16—451no—
204D8—451no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91911yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—31no—
210C22×C3—61no—
211C2×D3—21no—
212C2×D3—301no—
213C2×D3—301no—
214C2×D3—601no—
215C2×D3#122011yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—21no—
219C3×C4—31no—
220C3×C4—41no—
221C3×C4—301no—
222C3×C4C2×D3×D511yescharacteristic
223C3×C4C22×D1511yescharacteristic
224Dic3—101no—
225Dic3#122011yescharacteristic
226C2×C5—61no—
227C2×C5—121no—
228C2×C5D3×#6711yescharacteristic
229D5—361no—
230C32#409611yescharacteristic, complemented, Sylow
231Q8—151no—
232Q8D3×D1511yescharacteristic
233D4—31no—
234D4—61no—
235D4—451no—
236D4—901no—
237C2×C4—31no—
238C2×C4—61no—
239C2×C4—901no—
240C8—151no—
241C8—451no—
242C2×C3—21no—
243C2×C3—61no—
244C2×C3—121no—
245C2×C3D3×C22⋊D511yescharacteristic
246C2×C3C2×#122011yescharacteristic
247D3—21no—
248D3—41no—
249D3—601no—
250D3—601no—
251D3—1201no—
252C5#6394911yescharacteristic, complemented, Sylow
253C22—31no—
254C22—61no—
255C22—901no—
256C4—21no—
257C4—31no—
258C4—301no—
259C4C2×D3×D1511yescharacteristic
260C3—21no—
261C3#9151911yescharacteristic, complemented
262C3#9183911yescharacteristic, complemented
263C2—61no—
264C2—121no—
265C2—1801no—
266C2D3×#122011yescharacteristic
subgroupG/H
centerC2D3×#1220
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
hypercenterQ8D3×D15
hypocenterC32×C5#135
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×C5×#239C2semidirect
2. #10624197C2semidirect
3. #10623911C2semidirect
4. C3×#6718C2semidirect
5. #10623802C2semidirect
6. #10623800C2semidirect
7. D3×C3⋊#168C2semidirect
24. C5×#239D3semidirect
26. #6718D3semidirect
45. C3×#239D5semidirect
120. #239D15semidirect
133. C32×C5#135semidirect
207. C3×C5#919semidirect
208. C3×C5#950semidirect
230. C32#4096semidirect
252. C5#63949semidirect
261. C3#91519semidirect
262. C3#91839semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#2392
#106241972
#106239112
C3×#67182
#106238022
#106238002
D3×C3⋊#1682
#918393
#915193
#639495