Finite Group Databasebeta

← #13089372

Subgroups — #13089372

Order 1440.

704 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624285C211yesmaximal, characteristic, complemented
2C3×D4×D15C211yesmaximal, characteristic, complemented
3C3×C5×D3×D4C211yesmaximal, characteristic, complemented
4D3×#1220C212yesmaximal, complemented
5#67×D15C212yesmaximal, complemented
6C2×#67150C212yesmaximal, complemented
7#10624208C212yesmaximal, complemented
8#10624205C211yesmaximal, characteristic, complemented
9#10624200C211yesmaximal, characteristic, complemented
10#10624199C211yesmaximal, characteristic, complemented
11#10624193C211yesmaximal, characteristic, complemented
12#92095—31nomaximal
13#92025—31nomaximal
14C22×C3×D15C2212yescomplemented
15C22×C3×C5×D3C2212yescomplemented
16C2×D3×D15C2212yescomplemented
17C32×C5×D4C2211yescharacteristic, complemented
18#67184C2212yescomplemented
19C2×C3⋊Dic15C2212yescomplemented
20C4×C3⋊D15C2211yescharacteristic, complemented
21#67180C2211yescharacteristic, complemented
22C3×#1220C2212yescomplemented
23C3×D60C2211yescharacteristic, complemented
24C3×C4×D15C2211yescharacteristic, complemented
25C3×C5×#67C2212yescomplemented
26C3×C5×D12C2211yescharacteristic, complemented
27C3×C4×C5×D3C2211yescharacteristic, complemented
28#67153C2211yescharacteristic, complemented
29#67152C2211yescharacteristic, complemented
30#67151C2211yescharacteristic, complemented
31#67150C2212yescomplemented
32#67150C2211yescharacteristic, complemented
33#67150C2212yescomplemented
34#67150C2212yescomplemented
35D3×Dic15C2212yescomplemented
36Dic3×D15C2212yescomplemented
37#64313—51nomaximal, Hall
38C2×C3×C5×D4—31no—
39C2×#1220—32no—
40C2×#1220—32no—
41#6819—31no—
42#6819—31no—
43#6819—32no—
44#6819—31no—
45#6819—61no—
46D4×D15—32no—
47D4×D15—31no—
48D4×D15D311yescharacteristic, complemented
49#6817—31no—
50#6817—31no—
51C5×D3×D4D311yescharacteristic, complemented
52C3×D4×D5—31no—
53D3×C22⋊D5—32no—
54D5×#67—32no—
55C2×#1201—32no—
56#6780—32no—
57#6777—31no—
58#6772—31no—
59#6771—31no—
60#6765—31no—
61C22×C32×C5C2312yes—
62C2×C3⋊D15C2311yescharacteristic
63C2×C3×D15—42no—
64C2×C3×D15C2312yes—
65C2×C3×D15C2311yescharacteristic
66C2×C3×C5×D3—42no—
67C2×C3×C5×D3C2312yes—
68C2×C3×C5×D3C2311yescharacteristic
69D3×D15—42no—
70C32×C4×C5C2311yescharacteristic
71C3⋊Dic15C2312yes—
72C3⋊Dic15C2311yescharacteristic
73C3×Dic15C2311yescharacteristic
74C3×C5×Dic3C2311yescharacteristic
75#4144—91noHall
76#3830—51no—
77C3×D3×D4—51no—
78C3×D3×D4D511yescharacteristic, complemented
79D3×#67—52no—
80D3×#67—52no—
81C2×#625—52no—
82#3805—52no—
83#3802—51no—
84#3798—51no—
85#3795—51no—
86#3795—51no—
87C23×C3×C5—32no—
88C22×D15—32no—
89C22×D15C2×D312yescomplemented
90C22×C5×D3C2×D312yescomplemented
91C22×C3×D5—32no—
92C2×D3×D5—32no—
93C3×C5×D4—21no—
94C3×C5×D4—32no—
95C3×C5×D4—31no—
96C3×C5×D4C2×D311yescharacteristic, complemented
97C3×C5×D4C2×D311yescharacteristic, complemented
98C2×C3×C4×C5—31no—
99#1220—32no—
100#1220—32no—
101#1220—32no—
102#1220—31no—
103#1220—32no—
104#1220—32no—
105#1220—31no—
106#1220—62no—
107#1220C2×D312yescomplemented
108C2×Dic15—32no—
109C2×Dic15—32no—
110C2×Dic15—32no—
111C2×Dic15—62no—
112D60—31no—
113D60C2×D311yescharacteristic, complemented
114C4×D15—32no—
115C4×D15—31no—
116C4×D15—31no—
117C4×D15—61no—
118C4×D15C2×D311yescharacteristic, complemented
119Dic30—31no—
120Dic30—31no—
121Dic30—31no—
122Dic30—61no—
123C5×#67C2×D312yescomplemented
124C5×D12C2×D311yescharacteristic, complemented
125C4×C5×D3C2×D311yescharacteristic, complemented
126C3×C22⋊D5—32no—
127C3×D20—31no—
128C3×C4×D5—31no—
129#1204—31no—
130#1203—31no—
131#1202—31no—
132#1201—32no—
133#1201—31no—
134#1201—32no—
135#1201—32no—
136D3×Dic5—32no—
137D5×Dic3—32no—
138#1004—151no—
139#1004—151no—
140C2×C32×C5—22no—
141C2×C32×C5C2411yescharacteristic
142C3⋊D15—21no—
143C3×D15—22no—
144C3×D15—21no—
145C3×C5×D3—22no—
146C3×C5×D3—21no—
147C2×C5×D4—31no—
148C2×C22⋊D5—92no—
149C2×C22⋊D5—92no—
150#711—91no—
151#711—92no—
152#711—91no—
153D4×D5—31no—
154D4×D5—91no—
155D4×D5—92no—
156#709—91no—
157#709—91no—
158C22×C3×D3—52no—
159C22×C3×D3C2×D512yescomplemented
160C2×D32—52no—
161C32×D4C2×D511yescharacteristic, complemented
162#638—52no—
163C2×C3⋊Dic3—52no—
164C4×C3⋊D3—51no—
165#634—51no—
166C3×#67—52no—
167C3×#67C2×D512yescomplemented
168C3×D12—51no—
169C3×D12C2×D511yescharacteristic, complemented
170C3×C4×D3—51no—
171C3×C4×D3C2×D511yescharacteristic, complemented
172#627—51no—
173#626—51no—
174#626—51no—
175#625—52no—
176#625—52no—
177#625—51no—
178#625—52no—
179D3×Dic3—52no—
180D3×Dic3—52no—
181C22×C3×C5—22no—
182C22×C3×C5—32no—
183C22×C3×C5—31no—
184C22×C3×C5—122no—
185C22×C3×C5C22×D312yes—
186C22×C3×C5C22×D312yes—
187C2×D15—31no—
188C2×D15—31no—
189C2×D15—42no—
190C2×D15—61no—
191C2×D15—122no—
192C2×D15C22×D312yes—
193C2×D15C22×D311yescharacteristic
194C2×C5×D3—42no—
195C2×C5×D3C22×D312yes—
196C2×C5×D3C22×D311yescharacteristic
197C2×C3×D5—32no—
198C2×C3×D5—31no—
199C2×C3×D5—122no—
200D3×D5—122no—
201C3×C4×C5—21no—
202C3×C4×C5—31no—
203C3×C4×C5C22×D311yescharacteristic
204C3×C4×C5C22×D311yescharacteristic
205Dic15—32no—
206Dic15—31no—
207Dic15—32no—
208Dic15—31no—
209Dic15—62no—
210Dic15—61no—
211Dic15C22×D311yescharacteristic
212C3×Dic5—31no—
213C5×Dic3C22×D311yescharacteristic
214C2×C3×D4—31no—
215C2×C3×D4—151no—
216C2×#67—152no—
217C2×#67—152no—
218C2×#67—152no—
219C2×#67—152no—
220#237—151no—
221#237—152no—
222#237—151no—
223#237—151no—
224#237—152no—
225#237—151no—
226#237—301no—
227D3×D4—51no—
228D3×D4—151no—
229D3×D4—152no—
230D3×D4—152no—
231D3×D4—151no—
232D3×D4D1511yescharacteristic, complemented
233#235—151no—
234#235—151no—
235#235—151no—
236#235—151no—
237C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
238C23×C5—32no—
239C22×D5—32no—
240C22×D5—92no—
241C5×D4—32no—
242C5×D4—31no—
243C5×D4D3211yescharacteristic, complemented
244C2×C4×C5—31no—
245C22⋊D5—32no—
246C22⋊D5—91no—
247C22⋊D5—91no—
248C22⋊D5—92no—
249C22⋊D5—92no—
250C22⋊D5—92no—
251C22⋊D5—92no—
252C2×Dic5—92no—
253C2×Dic5—92no—
254D20—31no—
255D20—91no—
256C4×D5—31no—
257C4×D5—92no—
258C4×D5—91no—
259Dic10—91no—
260Dic10—91no—
261C22×C32C22×D512yes—
262C2×C3⋊D3—51no—
263C2×C3×D3—42no—
264C2×C3×D3—52no—
265C2×C3×D3—51no—
266C2×C3×D3—202no—
267C2×C3×D3C22×D512yes—
268C2×C3×D3C22×D511yescharacteristic
269D32—202no—
270C32×C4C22×D511yescharacteristic
271C3⋊Dic3—52no—
272C3⋊Dic3—51no—
273C3×Dic3—51no—
274C3×Dic3C22×D511yescharacteristic
275H(5,𝔽2)—451noSylow
276C2×C3×C5—22no—
277C2×C3×C5—22no—
278C2×C3×C5—21no—
279C2×C3×C5—42no—
280C2×C3×C5—62no—
281C2×C3×C5—61no—
282C2×C3×C5C23×D311yescharacteristic
283C2×C3×C5C23×D311yescharacteristic
284D15—21no—
285D15—22no—
286D15—61no—
287D15—61no—
288D15—121no—
289C3×D5—62no—
290C3×D5—61no—
291C5×D3—22no—
292C5×D3—21no—
293C23×C3—32no—
294C23×C3—152no—
295C22×D3—52no—
296C22×D3—152no—
297C22×D3—152no—
298C22×D3C2×D1512yescomplemented
299C3×D4—21no—
300C3×D4—32no—
301C3×D4—31no—
302C3×D4—152no—
303C3×D4—151no—
304C3×D4D3×D511yescharacteristic, complemented
305C3×D4C2×D1511yescharacteristic, complemented
306C2×C3×C4—31no—
307C2×C3×C4—151no—
308#67—52no—
309#67—151no—
310#67—151no—
311#67—152no—
312#67—152no—
313#67—152no—
314#67—152no—
315#67—152no—
316#67—152no—
317#67—151no—
318#67—152no—
319#67—152no—
320#67—151no—
321#67—302no—
322#67C2×D1512yescomplemented
323C2×Dic3—152no—
324C2×Dic3—152no—
325C2×Dic3—152no—
326C2×Dic3—152no—
327C2×Dic3—302no—
328D12—51no—
329D12—151no—
330D12—151no—
331D12C2×D1511yescharacteristic, complemented
332C4×D3—51no—
333C4×D3—152no—
334C4×D3—152no—
335C4×D3—151no—
336C4×D3—151no—
337C4×D3—301no—
338C4×D3C2×D1511yescharacteristic, complemented
339Dic6—151no—
340Dic6—151no—
341Dic6—151no—
342Dic6—151no—
343Dic6—301no—
344C22×C5—32no—
345C22×C5—31no—
346C22×C5—122no—
347C22×C5C2×D3212yes—
348C2×D5—31no—
349C2×D5—32no—
350C2×D5—91no—
351C2×D5—122no—
352C2×D5—362no—
353C4×C5—31no—
354C4×C5C2×D3211yescharacteristic
355Dic5—31no—
356Dic5—92no—
357Dic5—91no—
358C2×C32—22no—
359C2×C32C23×D511yescharacteristic
360C3⋊D3—101no—
361C3×D3—22no—
362C3×D3—21no—
363C3×D3—102no—
364C3×D3—101no—
365#41—451no—
366#41—451no—
367#41—451no—
368#41—452no—
369#41—451no—
370C2×D4—31no—
371C2×D4—151no—
372C2×D4—451no—
373C2×D4—452no—
374C2×D4—452no—
375C2×D4—452no—
376C3×C5—21no—
377C3×C5#100411yescharacteristic, complemented
378C3×C5#100411yescharacteristic, complemented
379C22×C3—22no—
380C22×C3—32no—
381C22×C3—31no—
382C22×C3—122no—
383C22×C3—152no—
384C22×C3—151no—
385C22×C3—602no—
386C22×C3C2×D3×D512yes—
387C22×C3C22×D1512yes—
388C2×D3—42no—
389C2×D3—51no—
390C2×D3—52no—
391C2×D3—151no—
392C2×D3—151no—
393C2×D3—202no—
394C2×D3—301no—
395C2×D3—602no—
396C2×D3—602no—
397C2×D3C22×D1512yes—
398C2×D3C22×D1511yescharacteristic
399C3×C4—21no—
400C3×C4—31no—
401C3×C4—151no—
402C3×C4C2×D3×D511yescharacteristic
403C3×C4C22×D1511yescharacteristic
404Dic3—51no—
405Dic3—151no—
406Dic3—152no—
407Dic3—152no—
408Dic3—151no—
409Dic3—302no—
410Dic3—301no—
411Dic3C22×D1511yescharacteristic
412C2×C5—22no—
413C2×C5—62no—
414C2×C5—61no—
415C2×C5C22×D3211yescharacteristic
416D5—61no—
417D5—62no—
418D5—181no—
419C32#414411yescharacteristic, complemented, Sylow
420C23—32no—
421C23—152no—
422C23—452no—
423Q8—451no—
424Q8—451no—
425D4—32no—
426D4—31no—
427D4—152no—
428D4—151no—
429D4—451no—
430D4—451no—
431D4—451no—
432D4—452no—
433D4—452no—
434D4—452no—
435D4—452no—
436D4D3×D1511yescharacteristic, complemented
437C2×C4—31no—
438C2×C4—151no—
439C2×C4—452no—
440C2×C4—452no—
441C2×C4—452no—
442C2×C4—451no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—42no—
447C2×C3—62no—
448C2×C3—61no—
449C2×C3—302no—
450C2×C3—301no—
451C2×C3C22×D3×D511yescharacteristic
452C2×C3C23×D1511yescharacteristic
453D3—21no—
454D3—22no—
455D3—101no—
456D3—102no—
457D3—301no—
458D3—301no—
459D3—601no—
460C5#6431311yescharacteristic, complemented, Sylow
461C22—31no—
462C22—32no—
463C22—122no—
464C22—151no—
465C22—152no—
466C22—451no—
467C22—602no—
468C22—1802no—
469C22C2×D3×D1512yes—
470C4—31no—
471C4—151no—
472C4—452no—
473C4—451no—
474C4C2×D3×D1511yescharacteristic
475C3—21no—
476C3#9202511yescharacteristic, complemented
477C3#9209511yescharacteristic, complemented
478C2—22no—
479C2—61no—
480C2—62no—
481C2—301no—
482C2—302no—
483C2—901no—
484C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4D3×D15
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #10624285C2semidirect
2. C3×D4×D15C2semidirect
3. C3×C5×D3×D4C2semidirect
4. D3×#1220C2semidirect
5. #67×D15C2semidirect
6. C2×#67150C2semidirect
7. #10624208C2semidirect
8. #10624205C2semidirect
9. #10624200C2semidirect
10. #10624199C2semidirect
11. #10624193C2semidirect
14. C22×C3×D15C22semidirect
15. C22×C3×C5×D3C22semidirect
16. C2×D3×D15C22semidirect
17. C32×C5×D4C22semidirect
18. #67184C22semidirect
19. C2×C3⋊Dic15C22semidirect
20. C4×C3⋊D15C22semidirect
21. #67180C22semidirect
22. C3×#1220C22semidirect
23. C3×D60C22semidirect
24. C3×C4×D15C22semidirect
25. C3×C5×#67C22semidirect
26. C3×C5×D12C22semidirect
27. C3×C4×C5×D3C22semidirect
28. #67153C22semidirect
29. #67152C22semidirect
30. #67151C22semidirect
31. #67150C22semidirect
32. #67150C22semidirect
33. #67150C22semidirect
34. #67150C22semidirect
35. D3×Dic15C22semidirect
36. Dic3×D15C22semidirect
48. D4×D15D3semidirect
51. C5×D3×D4D3semidirect
78. C3×D3×D4D5semidirect
89. C22×D15C2×D3semidirect
90. C22×C5×D3C2×D3semidirect
96. C3×C5×D4C2×D3semidirect
97. C3×C5×D4C2×D3semidirect
107. #1220C2×D3semidirect
113. D60C2×D3semidirect
118. C4×D15C2×D3semidirect
123. C5×#67C2×D3semidirect
124. C5×D12C2×D3semidirect
125. C4×C5×D3C2×D3semidirect
159. C22×C3×D3C2×D5semidirect
161. C32×D4C2×D5semidirect
167. C3×#67C2×D5semidirect
169. C3×D12C2×D5semidirect
171. C3×C4×D3C2×D5semidirect
232. D3×D4D15semidirect
237. C32×C5H(5,𝔽2)semidirect
243. C5×D4D32semidirect
298. C22×D3C2×D15semidirect
304. C3×D4D3×D5semidirect
305. C3×D4C2×D15semidirect
322. #67C2×D15semidirect
331. D12C2×D15semidirect
338. C4×D3C2×D15semidirect
377. C3×C5#1004semidirect
378. C3×C5#1004semidirect
419. C32#4144semidirect
436. D4D3×D15semidirect
460. C5#64313semidirect
476. C3#92025semidirect
477. C3#92095semidirect

Maximal subgroups

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

subgroupindex
#106242852
C3×D4×D152
C3×C5×D3×D42
D3×#12202
#67×D152
C2×#671502
#106242082
#106242052
#106242002
#106241992
#106241932
#920953
#920253
#643135