Finite Group Databasebeta

← #13088952

Subgroups — #13088952

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624220C211yesmaximal, characteristic, complemented
2#10624205C211yesmaximal, characteristic, complemented
3#10624198C211yesmaximal, characteristic, complemented
4C5×#3796C211yesmaximal, characteristic, complemented
5C4×D5×C3⋊D3C211yesmaximal, characteristic, complemented
6C3×#6773C211yesmaximal, characteristic, complemented
7C3×#6772C211yesmaximal, characteristic, complemented
8#10624049C212yesmaximal, complemented
9#10624045C212yesmaximal, complemented
10#10624030C212yesmaximal, complemented
11D3×#1200C212yesmaximal, complemented
12#92034—31nomaximal
13#92024—31nomaximal
14C2×D3×D15C2212yes—
15C2×D5×C3⋊D3C2211yescharacteristic
16C4×C3⋊D15C2211yescharacteristic
17C4×C5×C3⋊D3C2211yescharacteristic, complemented
18C3×D60C2211yescharacteristic, complemented
19C3×Dic30C2211yescharacteristic, complemented
20C3×C5×D12C2211yescharacteristic, complemented
21C3×C5×Dic6C2211yescharacteristic, complemented
22C32×C4×D5C2211yescharacteristic, complemented
23#67156C2212yescomplemented
24#67154C2212yescomplemented
25#67153C2212yescomplemented
26#67150C2212yescomplemented
27#67149C2212yes—
28C5×#626C2212yescomplemented
29C5×D3×Dic3C2212yescomplemented
30#67137C2211yescharacteristic
31C3⋊D3×Dic5C2211yescharacteristic, complemented
32D5×C3⋊Dic3C2211yescharacteristic, complemented
33C3×#1202C2212yescomplemented
34C3×#1201C2212yescomplemented
35C3×#1200C2212yescomplemented
36C3×D3×Dic5C2212yescomplemented
37#64306—51nomaximal, Hall
38Q8×D15—31no—
39D4×D15—31no—
40C5×#239—31no—
41C5×#237—31no—
42C3×#713—31no—
43C3×#711—31no—
44#6790—32no—
45C2×#1200—32no—
46#6782—32no—
47#6779—32no—
48#6777—31no—
49#6777—32no—
50C4×D3×D5—31no—
51C4×D3×D5—31no—
52C4×D3×D5—32no—
53C4×D3×D5—61no—
54#6773D311yescharacteristic, complemented
55#6772—32no—
56#6772D311yescharacteristic, complemented
57D15⋊Q8—31no—
58#6769—31no—
59#6765—31no—
60#6765—32no—
61C2×C3⋊D15C2311yescharacteristic
62C2×C5×C3⋊D3C2311yescharacteristic
63C2×C3×D15C2312yes—
64C2×C3×C5×D3C2312yes—
65C2×C32×D5C2311yescharacteristic
66D3×D15—24no—
67D5×C3⋊D3—22no—
68C32×C4×C5C2311yescharacteristic
69C3⋊Dic15C2311yescharacteristic
70C5×C3⋊Dic3C2311yescharacteristic
71C3×Dic15C2312yes—
72C3×C5×Dic3C2312yes—
73C32×Dic5C2311yescharacteristic
74D5×#41—91noHall
75C2×C4×C3⋊D3—51no—
76C3×#235—51no—
77C3×#235—51no—
78#3811—52no—
79#3805—52no—
80#3802—51no—
81C4×D32—52no—
82#3798—52no—
83#3797—51no—
84#3796—51no—
85#3796D511yescharacteristic, complemented
86C22×D15—32no—
87C2×D3×D5—32no—
88C2×D3×D5—31no—
89C2×D3×D5—31no—
90C2×D3×D5—61no—
91C3×C5×Q8—31no—
92C3×C5×D4—31no—
93#1220—32no—
94D60C2×D311yescharacteristic, complemented
95C4×D15—31no—
96C4×D15—31no—
97C4×D15—32no—
98C4×D15—61no—
99Dic30—32no—
100Dic30C2×D311yescharacteristic, complemented
101C5×#67—32no—
102C2×C5×Dic3—32no—
103C5×D12—32no—
104C5×D12C2×D311yescharacteristic, complemented
105C4×C5×D3—32no—
106C4×C5×D3—31no—
107C4×C5×D3—31no—
108C4×C5×D3—61no—
109C5×Dic6C2×D311yescharacteristic, complemented
110C3×C22⋊D5—32no—
111C2×C3×Dic5—32no—
112C3×D20—32no—
113C3×D20—31no—
114C3×C4×D5—21no—
115C3×C4×D5—32no—
116C3×C4×D5C2×D311yescharacteristic, complemented
117C3×C4×D5C2×D311yescharacteristic, complemented
118C3×Dic10—31no—
119#1204—32no—
120#1203—32no—
121#1202C2×D312yescomplemented
122#1201—32no—
123#1201—32no—
124#1201C2×D312yescomplemented
125#1200—32no—
126#1200—31no—
127#1200—31no—
128#1200—32no—
129#1200—61no—
130#1200C2×D312yescomplemented
131D3×Dic5—31no—
132D3×Dic5—31no—
133D3×Dic5—61no—
134D3×Dic5C2×D312yescomplemented
135D5×Dic3—31no—
136D5×Dic3—31no—
137D5×Dic3—32no—
138D5×Dic3—61no—
139D3×#41—151no—
140D3×#41—151no—
141C2×C32×C5C2411yescharacteristic
142C3⋊D15#4112yescomplemented
143C5×C3⋊D3—21no—
144C3×D15—22no—
145C3×C5×D3—22no—
146C32×D5—21no—
147C5×#41—91no—
148#713—31no—
149Q8×D5—91no—
150#711—31no—
151#711—92no—
152D4×D5—92no—
153D4×D5—91no—
154#709—91no—
155#709—92no—
156C2×C4×D5—92no—
157C2×C4×D5—91no—
158C22×C3⋊D3—51no—
159C2×D32—52no—
160C2×C32×C4—51no—
161C2×C3⋊Dic3—51no—
162C4×C3⋊D3—51no—
163C4×C3⋊D3—51no—
164C4×C3⋊D3—51no—
165C4×C3⋊D3C2×D511yescharacteristic, complemented
166C3×#67—52no—
167C3×#67—52no—
168C3×D12—51no—
169C3×D12C2×D511yescharacteristic, complemented
170C3×C4×D3—52no—
171C3×C4×D3—52no—
172C3×Dic6—51no—
173C3×Dic6C2×D511yescharacteristic, complemented
174#627—52no—
175#626—52no—
176#626C2×D512yescomplemented
177#625—52no—
178#624—52no—
179D3×Dic3—52no—
180D3×Dic3C2×D512yescomplemented
181C22×C3×C5—32no—
182C2×D15—31no—
183C2×D15—31no—
184C2×D15—61no—
185C2×D15—64no—
186C2×D15C22×D312yes—
187C2×C5×D3—31no—
188C2×C5×D3—31no—
189C2×C5×D3—61no—
190C2×C5×D3C22×D312yes—
191C2×C3×D5—21no—
192C2×C3×D5—32no—
193C2×C3×D5C22×D311yescharacteristic
194C2×C3×D5C22×D311yescharacteristic
195D3×D5—64no—
196D3×D5—62no—
197D3×D5—62no—
198D3×D5—64no—
199C3×C4×C5—21no—
200C3×C4×C5—32no—
201C3×C4×C5C22×D311yescharacteristic
202C3×C4×C5C22×D311yescharacteristic
203Dic15—31no—
204Dic15—31no—
205Dic15—61no—
206Dic15C22×D312yes—
207C3×Dic5—21no—
208C3×Dic5—32no—
209C3×Dic5C22×D311yescharacteristic
210C3×Dic5C22×D311yescharacteristic
211C5×Dic3—31no—
212C5×Dic3—31no—
213C5×Dic3—61no—
214C5×Dic3C22×D312yes—
215C3×#41—151no—
216C3×#41—151no—
217#239—31no—
218#239—151no—
219D3×Q8—151no—
220D3×Q8—151no—
221#237—31no—
222#237—151no—
223#237—152no—
224#237—152no—
225D3×D4—152no—
226D3×D4—151no—
227D3×D4—152no—
228D3×D4—151no—
229#235—51no—
230#235—51no—
231#235—152no—
232#235—152no—
233C2×C4×D3—152no—
234C2×C4×D3—151no—
235C2×C4×D3—151no—
236C2×C4×D3—152no—
237C2×C4×D3—301no—
238C32×C5C2×#4111yescharacteristic, complemented, Hall
239C22×D5—92no—
240C22×D5—91no—
241C5×Q8—31no—
242C5×D4—31no—
243C5×D4—92no—
244C2×C4×C5—92no—
245C2×C4×C5—91no—
246C22⋊D5—32no—
247C22⋊D5—92no—
248C22⋊D5—92no—
249C2×Dic5—32no—
250C2×Dic5—91no—
251D20—32no—
252D20—31no—
253C4×D5—32no—
254C4×D5—92no—
255C4×D5—92no—
256C4×D5—91no—
257C4×D5—91no—
258C4×D5—91no—
259C4×D5D3211yescharacteristic, complemented
260Dic10—31no—
261Dic10—92no—
262C22×C32—51no—
263C2×C3⋊D3—51no—
264C2×C3⋊D3—102no—
265C2×C3⋊D3C22×D511yescharacteristic
266C2×C3×D3—52no—
267C2×C3×D3C22×D512yes—
268D32—104no—
269C32×C4—51no—
270C32×C4C22×D511yescharacteristic
271C3⋊Dic3—51no—
272C3⋊Dic3C22×D511yescharacteristic
273C3×Dic3—52no—
274C3×Dic3C22×D512yes—
275C2×#41—451noSylow
276C2×C3×C5—21no—
277C2×C3×C5—62no—
278C2×C3×C5C23×D311yescharacteristic
279C2×C3×C5C23×D311yescharacteristic
280D15—22no—
281D15—32no—
282D15—32no—
283D15—62no—
284C3×D5—22no—
285C3×D5—21no—
286C3×D5—21no—
287C3×D5—62no—
288C5×D3—22no—
289C5×D3—62no—
290C5×D3—61no—
291C5×D3—61no—
292C22×D3—151no—
293C22×D3—151no—
294C22×D3—152no—
295C22×D3—152no—
296C22×D3—301no—
297C3×Q8—31no—
298C3×Q8—151no—
299C3×D4—31no—
300C3×D4—152no—
301C3×D4—152no—
302C3×D4—151no—
303C2×C3×C4—51no—
304C2×C3×C4—51no—
305C2×C3×C4—101no—
306C2×C3×C4—152no—
307C2×C3×C4—152no—
308#67—32no—
309#67—52no—
310#67—52no—
311#67—152no—
312#67—152no—
313#67—152no—
314C2×Dic3—32no—
315C2×Dic3—152no—
316C2×Dic3—151no—
317C2×Dic3—151no—
318C2×Dic3—301no—
319D12—32no—
320D12—51no—
321D12—152no—
322D12D3×D511yescharacteristic, complemented
323C4×D3—32no—
324C4×D3—31no—
325C4×D3—31no—
326C4×D3—52no—
327C4×D3—52no—
328C4×D3—61no—
329C4×D3—151no—
330C4×D3—151no—
331C4×D3—152no—
332C4×D3—152no—
333C4×D3—151no—
334C4×D3—151no—
335C4×D3—151no—
336C4×D3—151no—
337C4×D3—152no—
338C4×D3—301no—
339C4×D3—301no—
340C4×D3—301no—
341Dic6—51no—
342Dic6—152no—
343Dic6—152no—
344Dic6D3×D511yescharacteristic, complemented
345C22×C5—32no—
346C22×C5—91no—
347C2×D5—32no—
348C2×D5—91no—
349C2×D5—182no—
350C2×D5—184no—
351C2×D5C2×D3211yescharacteristic
352C4×C5—32no—
353C4×C5—91no—
354C4×C5C2×D3211yescharacteristic
355Dic5—32no—
356Dic5—91no—
357Dic5C2×D3211yescharacteristic
358C2×C32—101no—
359C2×C32C23×D511yescharacteristic
360C3⋊D3—21no—
361C3⋊D3—52no—
362C3×D3—22no—
363C3×D3—102no—
364#41—91no—
365#41—151no—
366#41—151no—
367#41—452no—
368#41—451no—
369#41—452no—
370C2×Q8—451no—
371C2×D4—451no—
372C2×D4—452no—
373C22×C4—451no—
374C22×C4—452no—
375C3×C5—21no—
376C3×C5D3×#4111yescharacteristic, complemented
377C3×C5D3×#4111yescharacteristic, complemented
378C22×C3—32no—
379C22×C3—51no—
380C22×C3—51no—
381C22×C3—101no—
382C22×C3—152no—
383C2×D3—31no—
384C2×D3—31no—
385C2×D3—52no—
386C2×D3—61no—
387C2×D3—151no—
388C2×D3—151no—
389C2×D3—301no—
390C2×D3—304no—
391C2×D3—302no—
392C2×D3—302no—
393C2×D3—304no—
394C2×D3—304no—
395C2×D3C2×D3×D512yes—
396C3×C4—21no—
397C3×C4—32no—
398C3×C4—51no—
399C3×C4—51no—
400C3×C4—101no—
401C3×C4—152no—
402C3×C4C2×D3×D511yescharacteristic
403C3×C4C2×D3×D511yescharacteristic
404Dic3—31no—
405Dic3—31no—
406Dic3—52no—
407Dic3—61no—
408Dic3—151no—
409Dic3—151no—
410Dic3—301no—
411Dic3C2×D3×D512yes—
412C2×C5—62no—
413C2×C5—181no—
414C2×C5C22×D3211yescharacteristic
415D5—21no—
416D5—62no—
417D5—92no—
418C32D5×#4111yescharacteristic, complemented, Sylow
419C23—452no—
420C23—451no—
421Q8—31no—
422Q8—151no—
423Q8—452no—
424D4—31no—
425D4—92no—
426D4—152no—
427D4—152no—
428D4—151no—
429D4—452no—
430D4—452no—
431C2×C4—51no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—152no—
435C2×C4—152no—
436C2×C4—452no—
437C2×C4—451no—
438C2×C4—451no—
439C2×C4—452no—
440C2×C4—451no—
441C2×C4—451no—
442C2×C3—21no—
443C2×C3—62no—
444C2×C3—102no—
445C2×C3—101no—
446C2×C3—101no—
447C2×C3—302no—
448C2×C3C22×D3×D511yescharacteristic
449C2×C3C22×D3×D511yescharacteristic
450D3—22no—
451D3—62no—
452D3—61no—
453D3—61no—
454D3—102no—
455D3—152no—
456D3—152no—
457D3—302no—
458C5#6430611yescharacteristic, complemented, Sylow
459C22—32no—
460C22—51no—
461C22—91no—
462C22—152no—
463C22—451no—
464C22—904no—
465C22—902no—
466C4—32no—
467C4—51no—
468C4—91no—
469C4—152no—
470C4—451no—
471C4D32×D511yescharacteristic
472C3—21no—
473C3#9202411yescharacteristic, complemented
474C3#9203411yescharacteristic, complemented
475C2—62no—
476C2—101no—
477C2—181no—
478C2—302no—
479C2—452no—
480C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624220C2semidirect
2. #10624205C2semidirect
3. #10624198C2semidirect
4. C5×#3796C2semidirect
5. C4×D5×C3⋊D3C2semidirect
6. C3×#6773C2semidirect
7. C3×#6772C2semidirect
8. #10624049C2semidirect
9. #10624045C2semidirect
10. #10624030C2semidirect
11. D3×#1200C2semidirect
17. C4×C5×C3⋊D3C22semidirect
18. C3×D60C22semidirect
19. C3×Dic30C22semidirect
20. C3×C5×D12C22semidirect
21. C3×C5×Dic6C22semidirect
22. C32×C4×D5C22semidirect
23. #67156C22semidirect
24. #67154C22semidirect
25. #67153C22semidirect
26. #67150C22semidirect
28. C5×#626C22semidirect
29. C5×D3×Dic3C22semidirect
31. C3⋊D3×Dic5C22semidirect
32. D5×C3⋊Dic3C22semidirect
33. C3×#1202C22semidirect
34. C3×#1201C22semidirect
35. C3×#1200C22semidirect
36. C3×D3×Dic5C22semidirect
54. #6773D3semidirect
56. #6772D3semidirect
85. #3796D5semidirect
94. D60C2×D3semidirect
100. Dic30C2×D3semidirect
104. C5×D12C2×D3semidirect
109. C5×Dic6C2×D3semidirect
116. C3×C4×D5C2×D3semidirect
117. C3×C4×D5C2×D3semidirect
121. #1202C2×D3semidirect
124. #1201C2×D3semidirect
130. #1200C2×D3semidirect
134. D3×Dic5C2×D3semidirect
142. C3⋊D15#41semidirect
165. C4×C3⋊D3C2×D5semidirect
169. C3×D12C2×D5semidirect
173. C3×Dic6C2×D5semidirect
176. #626C2×D5semidirect
180. D3×Dic3C2×D5semidirect
238. C32×C5C2×#41semidirect
259. C4×D5D32semidirect
322. D12D3×D5semidirect
344. Dic6D3×D5semidirect
376. C3×C5D3×#41semidirect
377. C3×C5D3×#41semidirect
418. C32D5×#41semidirect
458. C5#64306semidirect
473. C3#92024semidirect
474. C3#92034semidirect

Maximal subgroups

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

subgroupindex
#106242202
#106242052
#106241982
C5×#37962
C4×D5×C3⋊D32
C3×#67732
C3×#67722
#106240492
#106240452
#106240302
D3×#12002
#920343
#920243
#643065