Finite Group Databasebeta

← #13088944

Subgroups — #13088944

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624219C211yesmaximal, characteristic, complemented
2#10624199C211yesmaximal, characteristic, complemented
3#10624198C211yesmaximal, characteristic, complemented
4C5×D3×Dic6C211yesmaximal, characteristic, complemented
5#10624149C211yesmaximal, characteristic, complemented
6C3×#6766C211yesmaximal, characteristic, complemented
7C3×#6765C211yesmaximal, characteristic, complemented
8#10624045C212yesmaximal, complemented
9#10624044C212yesmaximal, complemented
10#10624041C212yesmaximal, complemented
11D5×#627C212yesmaximal, complemented
12#92028—31nomaximal
13#92000—31nomaximal
14C4×C3⋊D15C2211yescharacteristic
15C5×#634C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C4×C5×D3C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×D20C2211yescharacteristic
21#67158C2212yes—
22#67154C2212yes—
23#67153C2211yescharacteristic
24#67153C2212yes—
25#67152C2211yescharacteristic
26#67151C2211yescharacteristic
27#67150C2211yescharacteristic
28#67149C2212yes—
29C5×#627C2212yes—
30C5×D3×Dic3C2212yes—
31#67139C2212yes—
32D5×C3⋊Dic3C2212yes—
33C3×#1202C2212yes—
34C3×#1201C2212yes—
35C3×D5×Dic3C2212yes—
36C3×D5×Dic3C2212yes—
37#64309—51nomaximal, Hall
38Q8×D15—31no—
39#6817—31no—
40C5×D3×Q8—31no—
41C2×C5×Dic6—31no—
42C3×#713—31no—
43C3×#709—31no—
44#6779—32no—
45#6779—32no—
46#6773—32no—
47#6771—31no—
48#6771—32no—
49D15⋊Q8—31no—
50D15⋊Q8—31no—
51#6766—31no—
52#6766—31no—
53#6766—32no—
54#6766—61no—
55#6766D311yescharacteristic, complemented
56#6765—32no—
57#6765D311yescharacteristic, complemented
58D5×Dic6—31no—
59D5×Dic6—32no—
60D5×Dic6—32no—
61C2×C3⋊D15C2311yescharacteristic
62C2×C3×D15C2311yescharacteristic
63C2×C3×C5×D3C2311yescharacteristic
64C2×C32×D5C2312yes—
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2311yescharacteristic
67C5×C3⋊Dic3C2312yes—
68C3×Dic15C2311yescharacteristic
69C3×Dic15C2312yes—
70C3×C5×Dic3C2311yescharacteristic
71C3×C5×Dic3C2312yes—
72#4147—91noHall
73#3830—51no—
74C3×#237—51no—
75C3×#237—51no—
76C2×#627—52no—
77#3805—52no—
78#3804—52no—
79#3804—52no—
80#3798—51no—
81#3797—51no—
82D3×Dic6—51no—
83D3×Dic6D511yescharacteristic, complemented
84C3×C5×Q8—31no—
85C2×C3×C4×C5—31no—
86#1220—31no—
87#1220—31no—
88D60—31no—
89C4×D15—31no—
90C4×D15—31no—
91C4×D15—32no—
92C4×D15—61no—
93C4×D15C2×D311yescharacteristic
94Dic30—31no—
95Dic30—32no—
96Dic30C2×D311yescharacteristic
97C2×C5×Dic3—32no—
98C4×C5×D3—32no—
99C4×C5×D3C2×D311yescharacteristic
100C5×Dic6—31no—
101C5×Dic6—31no—
102C5×Dic6—32no—
103C5×Dic6—32no—
104C5×Dic6—61no—
105C5×Dic6C2×D311yescharacteristic
106C3×C22⋊D5—32no—
107C3×D20—21no—
108C3×D20—32no—
109C3×D20C2×D311yescharacteristic
110C3×D20C2×D311yescharacteristic
111C3×C4×D5—32no—
112C3×C4×D5—32no—
113C3×C4×D5—31no—
114C3×Dic10—31no—
115#1204—32no—
116#1204—32no—
117#1204—31no—
118#1204—32no—
119#1203—31no—
120#1202—32no—
121#1202—32no—
122#1202—31no—
123#1202—62no—
124#1202C2×D312yes—
125#1201—31no—
126#1201C2×D312yes—
127#1200—32no—
128#1200—32no—
129D5×Dic3—32no—
130D5×Dic3—32no—
131D5×Dic3—32no—
132D5×Dic3—62no—
133D5×Dic3C2×D312yes—
134D5×Dic3C2×D312yes—
135#1010—151no—
136#1010—151no—
137C2×C32×C5C2411yescharacteristic
138C3⋊D15—21no—
139C3×D15—21no—
140C3×C5×D3—21no—
141C32×D5—22no—
142C2×C5×Q8—91no—
143#713—31no—
144#713—92no—
145#713—91no—
146Q8×D5—91no—
147Q8×D5—91no—
148Q8×D5—92no—
149#709—31no—
150#709—92no—
151#709—91no—
152#709—92no—
153C32×D4—51no—
154#638—52no—
155C2×C3⋊Dic3—52no—
156C4×C3⋊D3—51no—
157#634C2×D511yescharacteristic
158C3×#67—52no—
159C3×#67—52no—
160C2×C3×Dic3—52no—
161C2×C3×Dic3—52no—
162C3×C4×D3—51no—
163C3×C4×D3C2×D511yescharacteristic
164C3×Dic6—51no—
165C3×Dic6C2×D511yescharacteristic
166#627—52no—
167#627—52no—
168#627—51no—
169#627C2×D512yes—
170#626—51no—
171#626—51no—
172#625—51no—
173#624—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—31no—
177C2×D15—31no—
178C2×D15—31no—
179C2×D15—61no—
180C2×D15C22×D311yescharacteristic
181C2×C5×D3C22×D311yescharacteristic
182C2×C3×D5—22no—
183C2×C3×D5—31no—
184C2×C3×D5C22×D312yes—
185C2×C3×D5C22×D312yes—
186C3×C4×C5—21no—
187C3×C4×C5—31no—
188C3×C4×C5—32no—
189C3×C4×C5C22×D311yescharacteristic
190C3×C4×C5C22×D311yescharacteristic
191Dic15—31no—
192Dic15—31no—
193Dic15—61no—
194Dic15C22×D312yes—
195Dic15C22×D311yescharacteristic
196C3×Dic5—32no—
197C3×Dic5—31no—
198C5×Dic3—32no—
199C5×Dic3—32no—
200C5×Dic3—62no—
201C5×Dic3C22×D311yescharacteristic
202C5×Dic3C22×D312yes—
203C3×#41—151no—
204C3×#41—151no—
205D3×Q8—31no—
206D3×Q8—151no—
207D3×Q8—151no—
208D3×Q8—151no—
209#237—51no—
210#237—51no—
211#237—152no—
212#237—151no—
213#237—151no—
214#237—152no—
215#237—152no—
216#237—152no—
217#237—301no—
218#235—152no—
219#235—152no—
220#235—151no—
221#235—151no—
222C2×Dic6—31no—
223C2×Dic6—151no—
224C2×Dic6—152no—
225C2×Dic6—152no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—31no—
228C5×Q8—92no—
229C5×Q8—91no—
230C2×C4×C5—31no—
231C2×C4×C5—92no—
232C22⋊D5—32no—
233C22⋊D5—91no—
234C22⋊D5—91no—
235D20—32no—
236D20—91no—
237D20—92no—
238D20D3211yescharacteristic
239C4×D5—32no—
240C4×D5—32no—
241C4×D5—31no—
242C4×D5—92no—
243C4×D5—92no—
244C4×D5—92no—
245C4×D5—91no—
246Dic10—31no—
247Dic10—92no—
248Dic10—92no—
249Dic10—91no—
250C22×C32—52no—
251C2×C3⋊D3—51no—
252C2×C3×D3—51no—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4C22×D511yescharacteristic
255C3⋊Dic3—51no—
256C3⋊Dic3C22×D512yes—
257C3×Dic3—52no—
258C3×Dic3—51no—
259C3×Dic3C22×D512yes—
260C3×Dic3C22×D511yescharacteristic
2612−1+4—451noSylow
262C2×C3×C5—21no—
263C2×C3×C5—61no—
264C2×C3×C5C23×D311yescharacteristic
265C2×C3×C5C23×D311yescharacteristic
266D15—21no—
267D15—61no—
268D15—61no—
269D15—121no—
270C3×D5—22no—
271C3×D5—22no—
272C3×D5—42no—
273C3×D5—61no—
274C5×D3—21no—
275C3×Q8—31no—
276C3×Q8—151no—
277C3×D4—51no—
278C3×D4—51no—
279C3×D4—101no—
280C3×D4—152no—
281C3×D4—152no—
282C2×C3×C4—31no—
283C2×C3×C4—152no—
284C2×C3×C4—152no—
285C2×C3×C4—151no—
286#67—52no—
287#67—52no—
288#67—152no—
289#67—152no—
290#67—151no—
291#67—151no—
292#67—151no—
293#67—151no—
294#67—302no—
295C2×Dic3—32no—
296C2×Dic3—52no—
297C2×Dic3—52no—
298C2×Dic3—152no—
299C2×Dic3—152no—
300C2×Dic3—152no—
301C2×Dic3—302no—
302D12—151no—
303D12—151no—
304C4×D3—32no—
305C4×D3—51no—
306C4×D3—152no—
307C4×D3—152no—
308C4×D3—151no—
309C4×D3—151no—
310C4×D3—152no—
311C4×D3—301no—
312C4×D3D3×D511yescharacteristic
313Dic6—31no—
314Dic6—31no—
315Dic6—32no—
316Dic6—32no—
317Dic6—51no—
318Dic6—61no—
319Dic6—152no—
320Dic6—152no—
321Dic6—151no—
322Dic6—152no—
323Dic6—151no—
324Dic6—152no—
325Dic6D3×D511yescharacteristic
326C22×C5—31no—
327C2×D5—31no—
328C2×D5—91no—
329C2×D5C2×D3212yes—
330C4×C5—31no—
331C4×C5—32no—
332C4×C5—92no—
333C4×C5C2×D3211yescharacteristic
334Dic5—32no—
335Dic5—31no—
336Dic5—91no—
337C2×C32—102no—
338C2×C32C23×D511yescharacteristic
339C3⋊D3—101no—
340C3×D3—21no—
341C3×D3—101no—
342#41—151no—
343#41—151no—
344#41—451no—
345#41—452no—
346#41—452no—
347#41—452no—
348#41—451no—
349C2×Q8—91no—
350C2×Q8—452no—
351C2×Q8—451no—
352C2×Q8—451no—
353C3×C5—21no—
354C3×C5#101011yescharacteristic, complemented
355C3×C5#101011yescharacteristic, complemented
356C22×C3—31no—
357C22×C3—52no—
358C22×C3—52no—
359C22×C3—102no—
360C22×C3—151no—
361C2×D3—51no—
362C2×D3—151no—
363C2×D3—151no—
364C2×D3—301no—
365C2×D3C2×D3×D511yescharacteristic
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—32no—
369C3×C4—151no—
370C3×C4—152no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C2×D3×D511yescharacteristic
373Dic3—32no—
374Dic3—32no—
375Dic3—52no—
376Dic3—51no—
377Dic3—62no—
378Dic3—151no—
379Dic3—151no—
380Dic3—301no—
381Dic3C2×D3×D511yescharacteristic
382Dic3C2×D3×D512yes—
383C2×C5—61no—
384C2×C5C22×D3211yescharacteristic
385D5—22no—
386D5—61no—
387D5—181no—
388C32#414711yescharacteristic, complemented, Sylow
389Q8—31no—
390Q8—92no—
391Q8—91no—
392Q8—151no—
393Q8—451no—
394Q8—452no—
395Q8—452no—
396D4—51no—
397D4—152no—
398D4—152no—
399D4—451no—
400D4—452no—
401D4—451no—
402D4—451no—
403C2×C4—31no—
404C2×C4—92no—
405C2×C4—152no—
406C2×C4—152no—
407C2×C4—151no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C4—451no—
412C2×C3—21no—
413C2×C3—61no—
414C2×C3—102no—
415C2×C3—102no—
416C2×C3—202no—
417C2×C3—301no—
418C2×C3C22×D3×D511yescharacteristic
419C2×C3C22×D3×D511yescharacteristic
420D3—21no—
421D3—101no—
422D3—301no—
423D3—301no—
424D3—601no—
425C5#6430911yescharacteristic, complemented, Sylow
426C22—31no—
427C22—52no—
428C22—151no—
429C22—451no—
430C4—31no—
431C4—32no—
432C4—92no—
433C4—151no—
434C4—152no—
435C4—451no—
436C4D32×D511yescharacteristic
437C3—21no—
438C3#9200011yescharacteristic, complemented
439C3#9202811yescharacteristic, complemented
440C2—61no—
441C2—102no—
442C2—301no—
443C2—901no—
444C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624219C2semidirect
2. #10624199C2semidirect
3. #10624198C2semidirect
4. C5×D3×Dic6C2semidirect
5. #10624149C2semidirect
6. C3×#6766C2semidirect
7. C3×#6765C2semidirect
8. #10624045C2semidirect
9. #10624044C2semidirect
10. #10624041C2semidirect
11. D5×#627C2semidirect
55. #6766D3semidirect
57. #6765D3semidirect
83. D3×Dic6D5semidirect
226. C32×C52−1+4semidirect
354. C3×C5#1010semidirect
355. C3×C5#1010semidirect
388. C32#4147semidirect
425. C5#64309semidirect
438. C3#92000semidirect
439. C3#92028semidirect

Maximal subgroups

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

subgroupindex
#106242192
#106241992
#106241982
C5×D3×Dic62
#106241492
C3×#67662
C3×#67652
#106240452
#106240442
#106240412
D5×#6272
#920283
#920003
#643095