Finite Group Databasebeta

← #13087510

Subgroups — #13087510

Order 1440.

288 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 60 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#10624285C211yesmaximal, characteristic, complemented
2#10624194C211yesmaximal, characteristic, complemented
3C3×#6715C211yesmaximal, characteristic, complemented
4C3×C5×#214C211yesmaximal, characteristic, complemented
5#10623809C211yesmaximal, characteristic, complemented
6#10623803C211yesmaximal, characteristic, complemented
7#10623794C211yesmaximal, characteristic, complemented
8#91805—31nomaximal
9#91497—31nomaximal
10C32×C5×D4C2211yescharacteristic
11#67184—21no—
12C2×C3⋊Dic15—21no—
13C4×C3⋊D15C2211yescharacteristic
14#67180C2211yescharacteristic
15C3×D60C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic
17#67151—21no—
18Dic3×D15—21no—
19C3×C3⋊#168C2211yescharacteristic
20C3×C5×C3⋊C8C2211yescharacteristic
21#63936—51nomaximal, Hall
22#6821—31no—
23#6819—31no—
24#6819—31no—
25#6819—61no—
26#6766—31no—
27C3×C5×SD16—31no—
28#6718—31no—
29#6715D311yescharacteristic, complemented
30#6706—31no—
31C8×D15—31no—
32C5×#214D311yescharacteristic, complemented
33C3×#686—31no—
34#6663—31no—
35#6657—31no—
36#6648—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15D411yescharacteristic
39C2×C3×D15—21no—
40C32×C4×C5C2311yescharacteristic
41C3⋊Dic15—21no—
42C3⋊Dic15D411yescharacteristic
43C3×C5×Dic3—21no—
44#4062—91noHall
45#3830—51no—
46#3796—51no—
47C3×#214D511yescharacteristic, complemented
48C3×#213—51no—
49#3719—51no—
50#3716—51no—
51#3710—51no—
52C3×C5×Q8—31no—
53C3×C5×D4—21no—
54C3×C5×D4C2×D311yescharacteristic
55C3×C5×D4C2×D311yescharacteristic
56#1220—61no—
57#1220—61no—
58#1220—62no—
59C2×Dic15—61no—
60C2×Dic15—61no—
61C2×Dic15—62no—
62D60—61no—
63D60C2×D311yescharacteristic
64C4×D15—31no—
65C4×D15—31no—
66C4×D15—61no—
67C4×D15—61no—
68Dic30—31no—
69Dic30—31no—
70Dic30—61no—
71C5×Dic6C2×D311yescharacteristic
72C3×D20—31no—
73#1202—61no—
74D5×Dic3—61no—
75C3×C5×C8—31no—
76C3⋊#168C2×D311yescharacteristic
77C3×#168—31no—
78C5×C3⋊C8C2×D311yescharacteristic
79#916—151no—
80#912—151no—
81C2×C32×C5—41no—
82C2×C32×C5C2×D411yescharacteristic
83C3⋊D15—21no—
84C3×D15—41no—
85#713—91no—
86#711—91no—
87C5×SD16—31no—
88#689—91no—
89#686—31no—
90#677—91no—
91C8×D5—91no—
92C32×D4C2×D511yescharacteristic
93#638—101no—
94C2×C3⋊Dic3—101no—
95C4×C3⋊D3—51no—
96#634—51no—
97C3×D12—51no—
98C3×Dic6C2×D511yescharacteristic
99#626—101no—
100D3×Dic3—101no—
101C3×C3⋊C8—51no—
102C3×C3⋊C8C2×D511yescharacteristic
103C22×C3×C5—22no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C2×D15—21no—
107C2×D15—31no—
108C2×D15—31no—
109C2×D15—61no—
110C2×C3×D5—61no—
111C3×C4×C5—21no—
112C3×C4×C5—61no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15—31no—
116Dic15—31no—
117Dic15—61no—
118Dic15—62no—
119Dic15—61no—
120Dic15—61no—
121C5×Dic3—21no—
122#239—151no—
123#237—151no—
124#237—151no—
125#237—151no—
126#237—301no—
127C3×SD16—31no—
128C3×D8—151no—
129#216—151no—
130#214—151no—
131#214D1511yescharacteristic, complemented
132#213—51no—
133Dic12—151no—
134#204—151no—
135D3×C8—151no—
136D3×C8—151no—
137C32×C5#13511yescharacteristic, complemented, Hall
138C5×Q8—31no—
139C5×D4D3211yescharacteristic
140C22⋊D5—181no—
141C2×Dic5—181no—
142D20—31no—
143D20—181no—
144C4×D5—91no—
145C4×D5—181no—
146Dic10—91no—
147C5×C8—31no—
148#168—31no—
149C22×C32—21no—
150C2×C3⋊D3—51no—
151C2×C3×D3—101no—
152C32×C4C22×D511yescharacteristic
153C3⋊Dic3—51no—
154C3⋊Dic3—101no—
155C3×Dic3—21no—
156#135—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—42no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5D3×D411yescharacteristic
162C2×C3×C5D3×D411yescharacteristic
163D15—41no—
164D15—61no—
165D15—61no—
166D15—121no—
167C3×D5—121no—
168C3×Q8—31no—
169C3×D4—21no—
170C3×D4—151no—
171C3×D4D3×D511yescharacteristic
172C3×D4C2×D1511yescharacteristic
173#67—301no—
174#67—301no—
175#67—301no—
176#67—302no—
177C2×Dic3—301no—
178C2×Dic3—301no—
179C2×Dic3—301no—
180C2×Dic3—302no—
181D12—51no—
182D12—301no—
183C4×D3—151no—
184C4×D3—151no—
185C4×D3—301no—
186C4×D3—301no—
187Dic6—151no—
188Dic6—151no—
189Dic6—301no—
190Dic6C2×D1511yescharacteristic
191C3×C8—31no—
192C3×C8—151no—
193C3⋊C8—51no—
194C3⋊C8C2×D1511yescharacteristic
195C22×C5—21no—
196C2×D5—61no—
197C2×D5—91no—
198C4×C5—61no—
199C4×C5C2×D3211yescharacteristic
200Dic5—91no—
201Dic5—181no—
202C2×C32—41no—
203C2×C32D4×D511yescharacteristic
204C3⋊D3—101no—
205C3×D3—201no—
206#41—451no—
207#41—451no—
208Q16—451no—
209SD16—31no—
210SD16—451no—
211D8—151no—
212C2×C8—451no—
213C3×C5—21no—
214C3×C5#91211yescharacteristic, complemented
215C3×C5#91611yescharacteristic, complemented
216C22×C3—22no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—301no—
220C2×D3—101no—
221C2×D3—151no—
222C2×D3—151no—
223C2×D3—301no—
224C3×C4—21no—
225C3×C4—61no—
226C3×C4C2×D3×D511yescharacteristic
227C3×C4C22×D1511yescharacteristic
228Dic3—21no—
229Dic3—151no—
230Dic3—151no—
231Dic3—301no—
232Dic3—302no—
233Dic3—301no—
234Dic3—301no—
235C2×C5—41no—
236C2×C5#381111yescharacteristic
237D5—121no—
238D5—181no—
239C32#406211yescharacteristic, complemented, Sylow
240Q8—31no—
241Q8—451no—
242D4—151no—
243D4—901no—
244D4—901no—
245D4D3×D1511yescharacteristic
246C2×C4—451no—
247C2×C4—901no—
248C2×C4—901no—
249C8—31no—
250C8—151no—
251C2×C3—21no—
252C2×C3—42no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—601no—
256C2×C3#679011yescharacteristic
257C2×C3D4×D1511yescharacteristic
258D3—201no—
259D3—301no—
260D3—301no—
261D3—601no—
262C5#6393611yescharacteristic, complemented, Sylow
263C22—21no—
264C22—301no—
265C22—451no—
266C4—61no—
267C4—451no—
268C4—901no—
269C4C2×D3×D1511yescharacteristic
270C3—21no—
271C3#9149711yescharacteristic, complemented
272C3#9180511yescharacteristic, complemented
273C2—41no—
274C2—601no—
275C2—901no—
276C2#1062421811yescharacteristic
subgroupG/H
centerC2#10624218
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5#135
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #10624285C2semidirect
2. #10624194C2semidirect
3. C3×#6715C2semidirect
4. C3×C5×#214C2semidirect
5. #10623809C2semidirect
6. #10623803C2semidirect
7. #10623794C2semidirect
29. #6715D3semidirect
32. C5×#214D3semidirect
47. C3×#214D5semidirect
131. #214D15semidirect
137. C32×C5#135semidirect
214. C3×C5#912semidirect
215. C3×C5#916semidirect
239. C32#4062semidirect
262. C5#63936semidirect
271. C3#91497semidirect
272. C3#91805semidirect

Maximal subgroups

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

subgroupindex
#106242852
#106241942
C3×#67152
C3×C5×#2142
#106238092
#106238032
#106237942
#918053
#914973
#639365