Finite Group Databasebeta

← #13087507

Subgroups — #13087507

Order 1440.

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

normal subgroups only

This group has 292 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×D3×D4C211yesmaximal, characteristic, complemented
2#10624193C211yesmaximal, characteristic, complemented
3#10623910C211yesmaximal, characteristic, complemented
4C3×#6715C211yesmaximal, characteristic, complemented
5#10623805C211yesmaximal, characteristic, complemented
6#10623798C211yesmaximal, characteristic, complemented
7#10623796C211yesmaximal, characteristic, complemented
8#91821—31nomaximal
9#91494—31nomaximal
10C22×C3×C5×D3—21no—
11C32×C5×D4C2211yescharacteristic
12#67180C2211yescharacteristic, complemented
13C3×D60C2211yescharacteristic, complemented
14C3×C5×#67—21no—
15C3×C5×D12C2211yescharacteristic
16C3×C4×C5×D3C2211yescharacteristic
17#67150—21no—
18Dic3×D15—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21#63932—51nomaximal, Hall
22C2×C3×C5×D4—31no—
23#6817—31no—
24C5×D3×D4D311yescharacteristic, complemented
25#6765—31no—
26#6716—31no—
27#6716—31no—
28#6716—31no—
29#6716—61no—
30#6715—31no—
31#6715D311yescharacteristic, complemented
32#6710—31no—
33C3×#686—31no—
34#6659—31no—
35#6652—31no—
36#6650—31no—
37C22×C32×C5—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—42no—
41C2×C3×C5×D3D411yescharacteristic
42C32×C4×C5C2311yescharacteristic
43C3⋊Dic15—21no—
44C3×C5×Dic3D411yescharacteristic
45#4078—91noHall
46C3×D3×D4D511yescharacteristic, complemented
47#3795—51no—
48#3754—51no—
49C3×#213—51no—
50#3716—51no—
51#3713—51no—
52#3711—51no—
53C23×C3×C5—61no—
54C22×C5×D3—21no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4—61no—
58C3×C5×D4C2×D311yescharacteristic
59C3×C5×D4C2×D311yescharacteristic
60C2×C3×C4×C5—31no—
61#1220—61no—
62D60C2×D311yescharacteristic, complemented
63C4×D15—61no—
64Dic30—31no—
65Dic30—31no—
66Dic30—61no—
67C5×#67—21no—
68C5×D12C2×D311yescharacteristic
69C4×C5×D3C2×D311yescharacteristic
70C3×D20—31no—
71#1201—61no—
72D5×Dic3—61no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C3⋊#168C2×D311yescharacteristic, complemented
77C3×#168—31no—
78#932—151no—
79#911—151no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×D15—41no—
83C3×C5×D3—21no—
84C3×C5×D3—41no—
85C2×C5×D4—31no—
86#709—91no—
87#687—91no—
88#687—91no—
89#686—31no—
90#686—91no—
91#681—91no—
92C22×C3×D3—21no—
93C32×D4C2×D511yescharacteristic
94#634—51no—
95C3×#67—21no—
96C3×D12—51no—
97C3×D12C2×D511yescharacteristic
98C3×C4×D3C2×D511yescharacteristic
99#625—101no—
100D3×Dic3—101no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8—51no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—31no—
106C22×C3×C5—41no—
107C22×C3×C5—61no—
108C22×C3×C5—122no—
109C2×D15—21no—
110C2×C5×D3—21no—
111C2×C5×D3—42no—
112C2×C5×D3#6711yescharacteristic
113C2×C3×D5—61no—
114C3×C4×C5—21no—
115C3×C4×C5—31no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118Dic15—61no—
119Dic15—61no—
120Dic15—121no—
121C5×Dic3#6711yescharacteristic
122C2×C3×D4—31no—
123#237—151no—
124D3×D4D1511yescharacteristic, complemented
125#235—151no—
126C3×D8—151no—
127#214—151no—
128#214—151no—
129#214—151no—
130#214—301no—
131#213—51no—
132#213—151no—
133#213—151no—
134#208—151no—
135#204—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—61no—
139C5×D4—31no—
140C5×D4—61no—
141C5×D4D3211yescharacteristic
142C2×C4×C5—31no—
143C22⋊D5—181no—
144D20—31no—
145C4×D5—181no—
146Dic10—91no—
147#168—31no—
148#168—91no—
149C22×C32—21no—
150C2×C3×D3—21no—
151C2×C3×D3—42no—
152C2×C3×D3—101no—
153C2×C3×D3C22⋊D511yescharacteristic
154C32×C4C22×D511yescharacteristic
155C3⋊Dic3—101no—
156C3×Dic3C22⋊D511yescharacteristic
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—61no—
162C2×C3×C5—81no—
163C2×C3×C5—121no—
164C2×C3×C5D3×D411yescharacteristic
165C2×C3×C5C2×#6711yescharacteristic
166D15—41no—
167C3×D5—121no—
168C5×D3—21no—
169C5×D3—41no—
170C23×C3—61no—
171C22×D3—21no—
172C3×D4—21no—
173C3×D4—31no—
174C3×D4—61no—
175C3×D4—151no—
176C3×D4D3×D511yescharacteristic
177C3×D4C2×D1511yescharacteristic
178C2×C3×C4—31no—
179#67—21no—
180#67—301no—
181#67—301no—
182C2×Dic3—301no—
183D12—51no—
184D12C2×D1511yescharacteristic
185C4×D3—301no—
186C4×D3C2×D1511yescharacteristic
187Dic6—151no—
188Dic6—151no—
189Dic6—301no—
190C3×C8—151no—
191C3⋊C8—51no—
192C3⋊C8—151no—
193C3⋊C8—151no—
194C3⋊C8—301no—
195C22×C5—21no—
196C22×C5—31no—
197C22×C5—61no—
198C22×C5—122no—
199C2×D5—61no—
200C4×C5—31no—
201C4×C5C2×D3211yescharacteristic
202Dic5—181no—
203C2×C32—41no—
204C2×C32C2×C22⋊D511yescharacteristic
205C3×D3—21no—
206C3×D3—41no—
207C3×D3—201no—
208#41—451no—
209C2×D4—31no—
210SD16—451no—
211SD16—451no—
212D8—151no—
213D8—451no—
214M16—451no—
215C3×C5—21no—
216C3×C5#91111yescharacteristic, complemented
217C3×C5#93211yescharacteristic, complemented
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—31no—
221C22×C3—41no—
222C22×C3—61no—
223C22×C3—122no—
224C22×C3—301no—
225C2×D3—21no—
226C2×D3—42no—
227C2×D3—101no—
228C2×D3#122011yescharacteristic
229C3×C4—21no—
230C3×C4—31no—
231C3×C4C2×D3×D511yescharacteristic
232C3×C4C22×D1511yescharacteristic
233Dic3—301no—
234Dic3—301no—
235Dic3—601no—
236Dic3#122011yescharacteristic
237C2×C5—41no—
238C2×C5—61no—
239C2×C5—121no—
240C2×C5D3×#6711yescharacteristic
241D5—121no—
242C32#407811yescharacteristic, complemented, Sylow
243C23—61no—
244Q8—451no—
245D4—31no—
246D4—61no—
247D4—151no—
248D4—901no—
249D4D3×D1511yescharacteristic
250C2×C4—31no—
251C2×C4—901no—
252C8—151no—
253C8—451no—
254C2×C3—21no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—61no—
258C2×C3—81no—
259C2×C3—121no—
260C2×C3—601no—
261C2×C3D3×C22⋊D511yescharacteristic
262C2×C3C2×#122011yescharacteristic
263D3—21no—
264D3—41no—
265D3—201no—
266C5#6393211yescharacteristic, complemented, Sylow
267C22—21no—
268C22—31no—
269C22—61no—
270C22—122no—
271C22—301no—
272C4—31no—
273C4—901no—
274C4C2×D3×D1511yescharacteristic
275C3—21no—
276C3#9149411yescharacteristic, complemented
277C3#9182111yescharacteristic, complemented
278C2—41no—
279C2—61no—
280C2—121no—
281C2—601no—
282C2D3×#122011yescharacteristic
subgroupG/H
centerC2D3×#1220
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5AGL(1,ℤ/8)
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. C3×C5×D3×D4C2semidirect
2. #10624193C2semidirect
3. #10623910C2semidirect
4. C3×#6715C2semidirect
5. #10623805C2semidirect
6. #10623798C2semidirect
7. #10623796C2semidirect
12. #67180C22semidirect
13. C3×D60C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C3⋊#168C22semidirect
24. C5×D3×D4D3semidirect
31. #6715D3semidirect
46. C3×D3×D4D5semidirect
62. D60C2×D3semidirect
76. C3⋊#168C2×D3semidirect
124. D3×D4D15semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
216. C3×C5#911semidirect
217. C3×C5#932semidirect
242. C32#4078semidirect
266. C5#63932semidirect
276. C3#91494semidirect
277. C3#91821semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×D42
#106241932
#106239102
C3×#67152
#106238052
#106237982
#106237962
#918213
#914943
#639325