Finite Group Databasebeta

← #13085735

Subgroups — #13085735

Order 1440.

268 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#10624201C211yesmaximal, characteristic, complemented
2C3×#6768C211yesmaximal, characteristic, complemented
3#10623817C211yesmaximal, characteristic, complemented
4#10623800C211yesmaximal, characteristic, complemented
5C5×#3715C211yesmaximal, characteristic, complemented
6#10623763C211yesmaximal, characteristic, complemented
7C3×#6661C211yesmaximal, characteristic, complemented
8#91523—31nomaximal
9#91492—31nomaximal
10C4⋊(C3⋊D15)C2211yescharacteristic
11C3×C4×D15C2211yescharacteristic
12C3×C5×D12C2211yescharacteristic
13C3×C5×Dic6C2211yescharacteristic
14C32×Dic10C2211yescharacteristic
15#67151—21no—
16#67149—21no—
17C3×#1203—21no—
18C3×D3×Dic5—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C3⋊#168C2211yescharacteristic
21#63949—51nomaximal, Hall
22#6821—31no—
23C3×#711—31no—
24#6773—31no—
25#6768D311yescharacteristic, complemented
26#6715—31no—
27C5×#215—31no—
28C5×#214—31no—
29C3×#689—31no—
30#6661D311yescharacteristic, complemented
31#6658—31no—
32#6658—31no—
33#6658—61no—
34#6654—31no—
35#6648—31no—
36D5×C3⋊C8—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40C32×C4×C5C2311yescharacteristic
41C3×Dic15D411yescharacteristic
42C3×C5×Dic3—21no—
43C32×Dic5—21no—
44#4062—91noHall
45C3×#239—51no—
46#3799—51no—
47#3755—51no—
48C3×#216—51no—
49#3715D511yescharacteristic, complemented
50#3714—51no—
51D3×C3⋊C8—51no—
52C3×C5×Q8—31no—
53C3×C5×D4—31no—
54D60—31no—
55D60—31no—
56D60—61no—
57D60—61no—
58C4×D15—61no—
59C4×D15C2×D311yescharacteristic
60C5×D12C2×D311yescharacteristic
61C5×Dic6C2×D311yescharacteristic
62C3×C22⋊D5—61no—
63C2×C3×Dic5—61no—
64C3×C4×D5—31no—
65C3×Dic10—21no—
66C3×Dic10C2×D311yescharacteristic
67C3×Dic10C2×D311yescharacteristic
68#1203—21no—
69#1202—61no—
70#1200—61no—
71D3×Dic5—21no—
72C3⋊#168C2×D311yescharacteristic
73C3×#168—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77#950—151no—
78#919—151no—
79C2×C32×C5C2×D411yescharacteristic
80C3⋊D15—41no—
81C3×D15—21no—
82C3×C5×D3—41no—
83#713—91no—
84#711—31no—
85C5×SD16—91no—
86#689—31no—
87#686—91no—
88#677—91no—
89C8×D5—91no—
90C32×Q8—51no—
91C4⋊(C3⋊D3)—51no—
92C3×D12—101no—
93C3×D12C2×D511yescharacteristic
94C3×C4×D3—51no—
95C3×C4×D3—101no—
96C3×Dic6C2×D511yescharacteristic
97#626—101no—
98#624—101no—
99C3⋊(C3⋊C8)C2×D511yescharacteristic
100C3×C3⋊C8—51no—
101C22×C3×C5—61no—
102C2×D15—61no—
103C2×D15—61no—
104C2×D15—121no—
105C2×D15#6711yescharacteristic
106C2×C5×D3—21no—
107C2×C3×D5—31no—
108C3×C4×C5—21no—
109C3×C4×C5—61no—
110C3×C4×C5C22×D311yescharacteristic
111C3×C4×C5C22×D311yescharacteristic
112Dic15#6711yescharacteristic
113C3×Dic5—21no—
114C3×Dic5—21no—
115C3×Dic5—31no—
116C3×Dic5—41no—
117C5×Dic3—21no—
118C3×#41—151no—
119#239—51no—
120#239—151no—
121#235—151no—
122C3×Q16—151no—
123#216—51no—
124#215—31no—
125#215—151no—
126#215—151no—
127#215—301no—
128#214—31no—
129#213—151no—
130C2×C3⋊C8—151no—
131D24—151no—
132D3×C8—151no—
133C32×C5#13511yescharacteristic, complemented, Hall
134C5×Q8—31no—
135C5×D4—31no—
136C22⋊D5—61no—
137C2×Dic5—61no—
138D20—91no—
139D20—181no—
140C4×D5—31no—
141C4×D5—181no—
142Dic10D3211yescharacteristic
143C5×C8—91no—
144#168—31no—
145C2×C3⋊D3—101no—
146C2×C3×D3—21no—
147C2×C3×D3—51no—
148C32×C4—101no—
149C32×C4C22×D511yescharacteristic
150C3×Dic3—21no—
151C3×Dic3—51no—
152#135—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—121no—
155C2×C3×C5D3×D411yescharacteristic
156C2×C3×C5C2×#6711yescharacteristic
157D15—21no—
158D15—121no—
159D15—121no—
160D15—241no—
161C3×D5—61no—
162C5×D3—41no—
163C3×Q8—31no—
164C3×Q8—51no—
165C3×Q8—51no—
166C3×Q8—101no—
167C3×D4—31no—
168C3×D4—301no—
169C2×C3×C4—151no—
170C2×C3×C4—301no—
171#67—301no—
172D12—101no—
173D12—151no—
174D12—151no—
175D12—301no—
176D12—301no—
177D12D3×D511yescharacteristic
178C4×D3—51no—
179C4×D3—101no—
180C4×D3—301no—
181C4×D3—301no—
182Dic6D3×D511yescharacteristic
183C3×C8—151no—
184C3⋊C8—31no—
185C3⋊C8—31no—
186C3⋊C8—51no—
187C3⋊C8—61no—
188C22×C5—61no—
189C2×D5—31no—
190C2×D5—181no—
191C4×C5—61no—
192C4×C5C2×D3211yescharacteristic
193Dic5—21no—
194Dic5—31no—
195C2×C32D4×D511yescharacteristic
196C3⋊D3—201no—
197C3×D3—41no—
198C3×D3—101no—
199#41—151no—
200#41—451no—
201Q16—151no—
202SD16—91no—
203SD16—451no—
204D8—451no—
205C2×C8—451no—
206C3×C5—21no—
207C3×C5#91911yescharacteristic, complemented
208C3×C5#95011yescharacteristic, complemented
209C22×C3—61no—
210C22×C3—151no—
211C2×D3—21no—
212C2×D3—51no—
213C2×D3—301no—
214C2×D3—301no—
215C2×D3—601no—
216C3×C4—21no—
217C3×C4—61no—
218C3×C4—101no—
219C3×C4—101no—
220C3×C4—151no—
221C3×C4—201no—
222C3×C4C2×D3×D511yescharacteristic
223C3×C4C2×D3×D511yescharacteristic
224Dic3—21no—
225Dic3—51no—
226C2×C5—121no—
227C2×C5D3×#6711yescharacteristic
228D5—61no—
229D5—361no—
230C32#406211yescharacteristic, complemented, Sylow
231Q8—31no—
232Q8—51no—
233D4—31no—
234D4—301no—
235D4—451no—
236D4—901no—
237C2×C4—151no—
238C2×C4—301no—
239C2×C4—901no—
240C8—91no—
241C8—151no—
242C2×C3—21no—
243C2×C3—121no—
244C2×C3—301no—
245C2×C3D5×#6711yescharacteristic
246C2×C3#679011yescharacteristic
247D3—41no—
248D3—101no—
249D3—601no—
250D3—601no—
251D3—1201no—
252C5#6394911yescharacteristic, complemented, Sylow
253C22—61no—
254C22—151no—
255C22—901no—
256C4—61no—
257C4—101no—
258C4—151no—
259C4D32×D511yescharacteristic
260C3—21no—
261C3#9149211yescharacteristic, complemented
262C3#9152311yescharacteristic, complemented
263C2—121no—
264C2—301no—
265C2—1801no—
266C2#1062402711yescharacteristic
subgroupG/H
centerC2#10624027
commutator, Fitting, cosocleC32×C4×C5C23
Frattini, hypercenterC4D32×D5
socleC2×C32×C5C2×D4
hypocenterC32×C5#135
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624201C2semidirect
2. C3×#6768C2semidirect
3. #10623817C2semidirect
4. #10623800C2semidirect
5. C5×#3715C2semidirect
6. #10623763C2semidirect
7. C3×#6661C2semidirect
25. #6768D3semidirect
30. #6661D3semidirect
49. #3715D5semidirect
133. C32×C5#135semidirect
207. C3×C5#919semidirect
208. C3×C5#950semidirect
230. C32#4062semidirect
252. C5#63949semidirect
261. C3#91492semidirect
262. C3#91523semidirect

Maximal subgroups

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

subgroupindex
#106242012
C3×#67682
#106238172
#106238002
C5×#37152
#106237632
C3×#66612
#915233
#914923
#639495