Finite Group Databasebeta

← #13087478

Subgroups — #13087478

Order 1440.

322 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×C3×C4×C5×D3C211yesmaximal, characteristic
2C2×#67153C211yesmaximal, characteristic, complemented
3#10623906C211yesmaximal, characteristic, complemented
4C3×#6712C211yesmaximal, characteristic, complemented
5Dic3⋊Dic15C211yesmaximal, characteristic, complemented
6(C2×D3)⋊Dic15C211yesmaximal, characteristic
7(C2×D3)⋊Dic15C211yesmaximal, characteristic
8#91813—31nomaximal
9#91352—31nomaximal
10C22×C3×C5×D3C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C3⋊Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×C5×Dic3C2211yescharacteristic
17C3×C4×C5×D3—22no—
18#67153—22no—
19#63851—51nomaximal, Hall
20C22×C3×C4×C5—31no—
21C2×Dic30—31no—
22C2×C4×C5×D3D311yescharacteristic
23C2×#1204—31no—
24#6719—31no—
25#6719—31no—
26C4⋊Dic15—31no—
27#6712—31no—
28#6712—31no—
29#6712—61no—
30#6712D311yescharacteristic, complemented
31C3×#683—31no—
32Dic3⋊Dic5—31no—
33(C2×D3)⋊Dic5—31no—
34(C2×D3)⋊Dic5—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×C5×D3—21no—
37C2×C3×C5×D3—21no—
38C2×C3×C5×D3Q812yescomplemented
39C32×C4×C5—21no—
40C3⋊Dic15—21no—
41C3⋊Dic15—21no—
42C3×Dic15—21no—
43C3×Dic15—21no—
44C3×C5×Dic3D412yes—
45#4070—91noHall
46C2×C3×C4×D3D511yescharacteristic
47C2×#627—51no—
48#3750—51no—
49C3×Dic3⋊C4—51no—
50Dic3⋊Dic3—51no—
51(C2×D3)⋊Dic3—51no—
52(C2×D3)⋊Dic3—51no—
53C23×C3×C5—31no—
54C22×C5×D3C2×D311yescharacteristic
55C2×C3×C4×C5—21no—
56C2×C3×C4×C5—31no—
57C2×C3×C4×C5—62no—
58C2×C3×C4×C5C2×D311yescharacteristic
59C2×C3×C4×C5C2×D311yescharacteristic
60C2×Dic15—31no—
61C2×Dic15—31no—
62C2×Dic15—31no—
63C2×Dic15—31no—
64C2×Dic15—61no—
65C2×Dic15—61no—
66C2×Dic15C2×D311yescharacteristic
67C2×Dic15C2×D311yescharacteristic
68Dic30—62no—
69C2×C5×Dic3C2×D311yescharacteristic
70C4×C5×D3—22no—
71C2×C3×Dic5—31no—
72C2×C3×Dic5—31no—
73#1204—62no—
74#924—151no—
75#896—151no—
76C2×C32×C5C2×D411yescharacteristic
77C2×C32×C5C2×Q811yescharacteristic
78C2×C32×C5#4111yescharacteristic
79C3×C5×D3—22no—
80C22×C4×C5—31no—
81C2×Dic10—91no—
82C22⋊Dic5—91no—
83C22⋊Dic5—91no—
84C4⋊Dic5—91no—
85#683—31no—
86#683—91no—
87C22×C3×D3C2×D511yescharacteristic
88C2×C32×C4C2×D511yescharacteristic
89C2×C3⋊Dic3—51no—
90C2×C3⋊Dic3—51no—
91C2×C3×Dic3—51no—
92C2×C3×Dic3—51no—
93C2×C3×Dic3C2×D511yescharacteristic
94C3×C4×D3—22no—
95#627—102no—
96C22×C3×C5—21no—
97C22×C3×C5—32no—
98C22×C3×C5—61no—
99C22×C3×C5—61no—
100C22×C3×C5C22×D311yescharacteristic
101C22×C3×C5C22×D311yescharacteristic
102C2×C5×D3—21no—
103C2×C5×D3—21no—
104C2×C5×D3Dic612yescomplemented
105C3×C4×C5—21no—
106C3×C4×C5—21no—
107C3×C4×C5—32no—
108C3×C4×C5—41no—
109Dic15—21no—
110Dic15—21no—
111Dic15—62no—
112Dic15—61no—
113Dic15—61no—
114Dic15—61no—
115Dic15—61no—
116Dic15—121no—
117C3×Dic5—61no—
118C3×Dic5—61no—
119C5×Dic3#6712yes—
120C22×C3×C4—31no—
121C2×C4×D3D1511yescharacteristic
122C2×Dic6—151no—
123C2×Dic6—151no—
124C3×C4⋊C4—151no—
125#217—151no—
126#217—151no—
127(C2×D3)⋊C4—151no—
128(C2×D3)⋊C4—151no—
129C4⋊Dic3—151no—
130Dic3⋊C4—51no—
131Dic3⋊C4—151no—
132Dic3⋊C4—151no—
133Dic3⋊C4—151no—
134Dic3⋊C4—301no—
135C32×C5C22⋊Q811yescharacteristic, complemented, Hall
136C23×C5—31no—
137C2×C4×C5—31no—
138C2×C4×C5—62no—
139C2×C4×C5D3211yescharacteristic
140C2×Dic5—31no—
141C2×Dic5—31no—
142C2×Dic5—91no—
143C2×Dic5—91no—
144Dic10—182no—
145C22×C32C22×D511yescharacteristic
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3Dic1012yescomplemented
149C32×C4—21no—
150C3⋊Dic3—101no—
151C3⋊Dic3—101no—
152C3×Dic3—101no—
153C3×Dic3—101no—
154C3×Dic3C22⋊D512yes—
155C22⋊Q8—451noSylow
156C2×C3×C5—21no—
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5—62no—
160C2×C3×C5C2×Dic611yescharacteristic
161C2×C3×C5#23511yescharacteristic
162C2×C3×C5#23511yescharacteristic
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5D3×Q811yescharacteristic
165C2×C3×C5C2×#6711yescharacteristic
166C5×D3—22no—
167C23×C3—31no—
168C22×D3C2×D1511yescharacteristic
169C2×C3×C4—21no—
170C2×C3×C4—31no—
171C2×C3×C4—62no—
172C2×C3×C4—151no—
173C2×C3×C4—151no—
174C2×C3×C4D3×D511yescharacteristic
175C2×C3×C4C2×D1511yescharacteristic
176C2×Dic3—51no—
177C2×Dic3—51no—
178C2×Dic3—151no—
179C2×Dic3—151no—
180C2×Dic3—151no—
181C2×Dic3—151no—
182C2×Dic3—301no—
183C2×Dic3—301no—
184C2×Dic3C2×D1511yescharacteristic
185C4×D3—22no—
186Dic6—302no—
187Dic6—302no—
188C22×C5—32no—
189C22×C5—61no—
190C22×C5—61no—
191C22×C5C2×D3211yescharacteristic
192C4×C5—21no—
193C4×C5—32no—
194Dic5—61no—
195Dic5—61no—
196Dic5—181no—
197Dic5—181no—
198C2×C32C2×Dic1011yescharacteristic
199C2×C32#70911yescharacteristic
200C2×C32C2×C22⋊D511yescharacteristic
201C3×D3—22no—
202C2×Q8—451no—
203C22×C4—31no—
204C4⋊C4—151no—
205C4⋊C4—451no—
206C4⋊C4—451no—
207C22⋊C4—451no—
208C22⋊C4—451no—
209C3×C5—21no—
210C3×C5#89611yescharacteristic, complemented
211C3×C5#92411yescharacteristic, complemented
212C22×C3—21no—
213C22×C3—32no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3C2×D3×D511yescharacteristic
217C22×C3C22×D1511yescharacteristic
218C2×D3—21no—
219C2×D3—21no—
220C2×D3Dic3012yescomplemented
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—32no—
224C3×C4—41no—
225C3×C4—301no—
226C3×C4—301no—
227Dic3—101no—
228Dic3—101no—
229Dic3—302no—
230Dic3—301no—
231Dic3—301no—
232Dic3—301no—
233Dic3—301no—
234Dic3—601no—
235Dic3#122012yes—
236C2×C5—62no—
237C2×C5D3×Dic611yescharacteristic
238C2×C5#379811yescharacteristic
239C2×C5D3×#6711yescharacteristic
240C32#407011yescharacteristic, complemented, Sylow
241C23—31no—
242Q8—902no—
243C2×C4—31no—
244C2×C4—62no—
245C2×C4—151no—
246C2×C4—151no—
247C2×C4—451no—
248C2×C4—451no—
249C2×C4D3×D1511yescharacteristic
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—62no—
254C2×C3D3×Dic1011yescharacteristic
255C2×C3#677111yescharacteristic
256C2×C3D3×C22⋊D511yescharacteristic
257C2×C3C2×Dic3011yescharacteristic
258C2×C3#681711yescharacteristic
259C2×C3C2×#122011yescharacteristic
260D3—22no—
261C5#6385111yescharacteristic, complemented, Sylow
262C22—32no—
263C22—61no—
264C22—61no—
265C22C2×D3×D1511yescharacteristic
266C4—21no—
267C4—32no—
268C4—301no—
269C4—301no—
270C4—901no—
271C4—901no—
272C3—21no—
273C3#9135211yescharacteristic, complemented
274C3#9181311yescharacteristic, complemented
275C2—62no—
276C2D3×Dic3011yescharacteristic
277C2#1062419911yescharacteristic
278C2D3×#122011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×D15
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC22×C3×C5×D3C22
hypercenterC2×C4D3×D15
hypocenterC32×C5C22⋊Q8
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#67153C2semidirect
3. #10623906C2semidirect
4. C3×#6712C2semidirect
5. Dic3⋊Dic15C2semidirect
30. #6712D3semidirect
38. C2×C3×C5×D3Q8semidirect
104. C2×C5×D3Dic6semidirect
135. C32×C5C22⋊Q8semidirect
148. C2×C3×D3Dic10semidirect
210. C3×C5#896semidirect
211. C3×C5#924semidirect
220. C2×D3Dic30semidirect
240. C32#4070semidirect
261. C5#63851semidirect
273. C3#91352semidirect
274. C3#91813semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D32
C2×#671532
#106239062
C3×#67122
Dic3⋊Dic152
(C2×D3)⋊Dic152
#918133
#913523
#638515