Finite Group Databasebeta

← #13089416

Subgroups — #13089416

Order 1440.

718 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×#6819C211yesmaximal, characteristic, complemented
3C3×D4×D15C211yesmaximal, characteristic, complemented
4#10624236C212yesmaximal, complemented
5#10624235C212yesmaximal, complemented
6C2×#67156C212yesmaximal, complemented
7#10624230C212yesmaximal, complemented
8#10624226C211yesmaximal, characteristic, complemented
9#10624224C211yesmaximal, characteristic, complemented
10#10624223C211yesmaximal, characteristic, complemented
11#10624220C211yesmaximal, characteristic, complemented
12#92027—31nomaximal
13#92025—31nomaximal
14C22×C5×C3⋊D3C2212yescomplemented
15C22×C3×D15C2212yescomplemented
16C2×(C3⋊D3)⋊D5C2212yescomplemented
17C32×C5×D4C2211yescharacteristic, complemented
18C5×#638C2212yescomplemented
19C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C4×C5×C3⋊D3C2211yescharacteristic, complemented
21C3×#1220C2212yescomplemented
22C3×#1220C2212yescomplemented
23C2×C3×Dic15C2212yescomplemented
24C3×D60C2211yescharacteristic, complemented
25C3×C4×D15C2211yescharacteristic, complemented
26C3×C4×D15C2211yescharacteristic, complemented
27C3×Dic30C2211yescharacteristic, complemented
28#67158C2211yescharacteristic, complemented
29#67157C2211yescharacteristic, complemented
30#67156C2212yescomplemented
31#67156C2211yescharacteristic, complemented
32#67156C2211yescharacteristic, complemented
33#67156C2212yescomplemented
34#67156C2212yescomplemented
35#67155C2212yescomplemented
36#67154C2212yescomplemented
37#64314—51nomaximal, Hall
38#6819D311yescharacteristic, complemented
39D4×D15D311yescharacteristic, complemented
40C5×D3×D4—31no—
41C5×D3×D4—31no—
42C5×D3×D4—61no—
43C3×#711—31no—
44C3×D4×D5—31no—
45#6790—32no—
46D3×C22⋊D5—32no—
47D3×C22⋊D5—32no—
48D5×#67—32no—
49C2×#1203—32no—
50C2×#1201—32no—
51#6782—32no—
52#6780—32no—
53#6777—31no—
54D5×D12—31no—
55#6772—31no—
56#6771—31no—
57#6771—31no—
58#6769—31no—
59#6768—31no—
60#6765—31no—
61C22×C32×C5C2312yes—
62C2×C5×C3⋊D3—42no—
63C2×C5×C3⋊D3C2312yes—
64C2×C5×C3⋊D3C2311yescharacteristic
65C2×C3×D15—42no—
66C2×C3×D15C2312yes—
67C2×C3×D15C2311yescharacteristic
68C2×C3×D15C2311yescharacteristic
69(C3⋊D3)⋊D5—42no—
70C32×C4×C5C2311yescharacteristic
71C5×C3⋊Dic3C2311yescharacteristic
72C3×Dic15C2312yes—
73C3×Dic15C2311yescharacteristic
74C3×Dic15C2311yescharacteristic
75#4144—91noHall
76D4×C3⋊D3D511yescharacteristic, complemented
77C3×#237—51no—
78C3×D3×D4—51no—
79#3811—52no—
80D3×#67—52no—
81C2×#626—52no—
82#3804—52no—
83D3×D12—51no—
84#3799—51no—
85#3798—51no—
86#3796—51no—
87C22×D15C2×D312yescomplemented
88C22×C5×D3—32no—
89C22×C5×D3—32no—
90C22×C5×D3—62no—
91C22×C3×D5—32no—
92C2×D3×D5—32no—
93C2×D3×D5—32no—
94C3×C5×D4—21no—
95C3×C5×D4C2×D311yescharacteristic, complemented
96C3×C5×D4C2×D311yescharacteristic, complemented
97#1220C2×D312yescomplemented
98#1220C2×D312yescomplemented
99C2×Dic15C2×D312yescomplemented
100D60C2×D311yescharacteristic, complemented
101C4×D15C2×D311yescharacteristic, complemented
102C4×D15C2×D311yescharacteristic, complemented
103Dic30C2×D311yescharacteristic, complemented
104C5×#67—32no—
105C5×#67—32no—
106C5×#67—62no—
107C5×D12—31no—
108C5×D12—31no—
109C5×D12—61no—
110C4×C5×D3—31no—
111C4×C5×D3—31no—
112C4×C5×D3—61no—
113C3×C22⋊D5—32no—
114C3×C22⋊D5—32no—
115C2×C3×Dic5—32no—
116C3×D20—31no—
117C3×C4×D5—31no—
118C3×C4×D5—31no—
119C3×Dic10—31no—
120#1204—31no—
121#1204—31no—
122#1203—32no—
123#1203—32no—
124#1203—31no—
125#1203—31no—
126#1203—32no—
127#1202—31no—
128#1202—31no—
129#1201—32no—
130#1201—31no—
131#1201—31no—
132#1201—32no—
133#1201—32no—
134#1200—32no—
135D3×Dic5—32no—
136D3×Dic5—32no—
137D5×Dic3—32no—
138#1009—151no—
139#1004—151no—
140C2×C32×C5—22no—
141C2×C32×C5C2411yescharacteristic
142C5×C3⋊D3—21no—
143C5×C3⋊D3—22no—
144C3×D15—21no—
145C3×D15—22no—
146C3×D15—21no—
147C2×C5×D4—91no—
148C2×C22⋊D5—92no—
149C2×C22⋊D5—92no—
150#711—31no—
151#711—91no—
152#711—92no—
153D4×D5—31no—
154D4×D5—91no—
155D4×D5—92no—
156#709—91no—
157#709—91no—
158C22×C3⋊D3C2×D512yescomplemented
159C22×C3×D3—52no—
160C2×D32—52no—
161C32×D4C2×D511yescharacteristic, complemented
162#638C2×D512yescomplemented
163C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
164C4×C3⋊D3C2×D511yescharacteristic, complemented
165C3×#67—52no—
166C3×#67—52no—
167C2×C3×Dic3—52no—
168C3×D12—51no—
169C3×C4×D3—51no—
170C3×C4×D3—51no—
171C3×Dic6—51no—
172#627—51no—
173#626—52no—
174#626—51no—
175#626—51no—
176#626—52no—
177#626—52no—
178#625—51no—
179#624—52no—
180D3×Dic3—52no—
181C22×C3×C5—22no—
182C22×C3×C5C22×D312yes—
183C22×C3×C5C22×D312yes—
184C2×D15—42no—
185C2×D15C22×D312yes—
186C2×D15C22×D311yescharacteristic
187C2×D15C22×D311yescharacteristic
188C2×C5×D3—32no—
189C2×C5×D3—32no—
190C2×C5×D3—31no—
191C2×C5×D3—31no—
192C2×C5×D3—61no—
193C2×C5×D3—62no—
194C2×C5×D3—122no—
195C2×C5×D3—122no—
196C2×C5×D3—124no—
197C2×C3×D5—32no—
198C2×C3×D5—31no—
199C2×C3×D5—31no—
200C2×C3×D5—122no—
201D3×D5—122no—
202D3×D5—122no—
203C3×C4×C5—21no—
204C3×C4×C5C22×D311yescharacteristic
205C3×C4×C5C22×D311yescharacteristic
206Dic15C22×D312yes—
207Dic15C22×D311yescharacteristic
208Dic15C22×D311yescharacteristic
209C3×Dic5—31no—
210C3×Dic5—32no—
211C3×Dic5—31no—
212C5×Dic3—31no—
213C5×Dic3—31no—
214C5×Dic3—61no—
215C3×#41—151no—
216C2×C3×D4—151no—
217C2×#67—152no—
218C2×#67—152no—
219#239—151no—
220#239—151no—
221#237—51no—
222#237—151no—
223#237—152no—
224D3×D4—31no—
225D3×D4—31no—
226D3×D4—51no—
227D3×D4—61no—
228D3×D4—151no—
229D3×D4—152no—
230D3×D4—152no—
231D3×D4—152no—
232#235—151no—
233#235—151no—
234#235—151no—
235#235—152no—
236C2×D12—151no—
237C2×D12—152no—
238C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
239C23×C5—92no—
240C22×D5—32no—
241C22×D5—92no—
242C5×D4—92no—
243C5×D4—91no—
244C5×D4D3211yescharacteristic, complemented
245C2×C4×C5—91no—
246C22⋊D5—32no—
247C22⋊D5—32no—
248C22⋊D5—92no—
249C22⋊D5—91no—
250C22⋊D5—91no—
251C22⋊D5—92no—
252C22⋊D5—92no—
253C2×Dic5—32no—
254C2×Dic5—92no—
255D20—31no—
256D20—91no—
257C4×D5—31no—
258C4×D5—31no—
259C4×D5—92no—
260Dic10—31no—
261Dic10—91no—
262C22×C32C22×D512yes—
263C2×C3⋊D3—42no—
264C2×C3⋊D3C22×D512yes—
265C2×C3⋊D3C22×D511yescharacteristic
266C2×C3×D3—52no—
267C2×C3×D3—51no—
268C2×C3×D3—51no—
269C2×C3×D3—202no—
270D32—202no—
271C32×C4C22×D511yescharacteristic
272C3⋊Dic3C22×D511yescharacteristic
273C3×Dic3—52no—
274C3×Dic3—51no—
275C3×Dic3—51no—
276H(5,𝔽2)—451noSylow
277C2×C3×C5—22no—
278C2×C3×C5—22no—
279C2×C3×C5—21no—
280C2×C3×C5—42no—
281C2×C3×C5C23×D311yescharacteristic
282C2×C3×C5C23×D311yescharacteristic
283D15—22no—
284D15—21no—
285D15—21no—
286C3×D5—61no—
287C3×D5—62no—
288C3×D5—61no—
289C5×D3—62no—
290C5×D3—62no—
291C5×D3—61no—
292C5×D3—62no—
293C5×D3—61no—
294C5×D3—122no—
295C23×C3—152no—
296C22×D3—32no—
297C22×D3—32no—
298C22×D3—52no—
299C22×D3—62no—
300C22×D3—152no—
301C22×D3—152no—
302C3×Q8—151no—
303C3×D4—21no—
304C3×D4—152no—
305C3×D4—152no—
306C3×D4—151no—
307C3×D4D3×D511yescharacteristic, complemented
308C3×D4D3×D511yescharacteristic, complemented
309C2×C3×C4—152no—
310C2×C3×C4—151no—
311C2×C3×C4—151no—
312#67—32no—
313#67—32no—
314#67—52no—
315#67—52no—
316#67—62no—
317#67—151no—
318#67—151no—
319#67—151no—
320#67—152no—
321#67—151no—
322#67—152no—
323#67—152no—
324C2×Dic3—52no—
325C2×Dic3—152no—
326D12—31no—
327D12—31no—
328D12—51no—
329D12—61no—
330D12—151no—
331D12—152no—
332D12—152no—
333D12—152no—
334D12—151no—
335C4×D3—31no—
336C4×D3—31no—
337C4×D3—51no—
338C4×D3—51no—
339C4×D3—61no—
340C4×D3—152no—
341C4×D3—152no—
342C4×D3—152no—
343Dic6—51no—
344Dic6—151no—
345Dic6—151no—
346C22×C5—92no—
347C22×C5—91no—
348C22×C5—362no—
349C22×C5C2×D3212yes—
350C2×D5—31no—
351C2×D5—31no—
352C2×D5—32no—
353C2×D5—122no—
354C2×D5—362no—
355C4×C5—91no—
356C4×C5C2×D3211yescharacteristic
357Dic5—32no—
358Dic5—31no—
359Dic5—31no—
360C2×C32—22no—
361C2×C32C23×D511yescharacteristic
362C3⋊D3—21no—
363C3⋊D3—22no—
364C3×D3—101no—
365C3×D3—102no—
366C3×D3—101no—
367#41—151no—
368#41—451no—
369#41—451no—
370#41—452no—
371#41—451no—
372C2×D4—91no—
373C2×D4—151no—
374C2×D4—451no—
375C2×D4—452no—
376C2×D4—452no—
377C2×D4—452no—
378C3×C5—21no—
379C3×C5#100411yescharacteristic, complemented
380C3×C5#100911yescharacteristic, complemented
381C22×C3—22no—
382C22×C3—152no—
383C22×C3—151no—
384C22×C3—151no—
385C22×C3—602no—
386C22×C3C2×D3×D512yes—
387C22×C3C2×D3×D512yes—
388C2×D3—31no—
389C2×D3—31no—
390C2×D3—32no—
391C2×D3—32no—
392C2×D3—51no—
393C2×D3—51no—
394C2×D3—52no—
395C2×D3—61no—
396C2×D3—62no—
397C2×D3—122no—
398C2×D3—122no—
399C2×D3—124no—
400C2×D3—202no—
401C2×D3—602no—
402C2×D3—602no—
403C3×C4—21no—
404C3×C4—151no—
405C3×C4—151no—
406C3×C4—152no—
407C3×C4C2×D3×D511yescharacteristic
408C3×C4C2×D3×D511yescharacteristic
409Dic3—31no—
410Dic3—31no—
411Dic3—52no—
412Dic3—51no—
413Dic3—51no—
414Dic3—61no—
415C2×C5—22no—
416C2×C5—182no—
417C2×C5—181no—
418C2×C5C22×D3211yescharacteristic
419D5—61no—
420D5—62no—
421D5—61no—
422C32#414411yescharacteristic, complemented, Sylow
423C23—92no—
424C23—152no—
425C23—452no—
426Q8—151no—
427Q8—451no—
428D4—92no—
429D4—91no—
430D4—152no—
431D4—152no—
432D4—151no—
433D4—451no—
434D4—451no—
435D4—452no—
436D4—451no—
437D4—452no—
438D4—452no—
439D4(C3⋊D3)⋊D511yescharacteristic, complemented
440C2×C4—91no—
441C2×C4—152no—
442C2×C4—151no—
443C2×C4—151no—
444C2×C4—452no—
445C2×C4—452no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3—301no—
451C2×C3—302no—
452C2×C3—301no—
453C2×C3C22×D3×D511yescharacteristic
454C2×C3C22×D3×D511yescharacteristic
455D3—62no—
456D3—61no—
457D3—62no—
458D3—61no—
459D3—62no—
460D3—101no—
461D3—102no—
462D3—101no—
463D3—122no—
464C5#6431411yescharacteristic, complemented, Sylow
465C22—91no—
466C22—92no—
467C22—151no—
468C22—151no—
469C22—152no—
470C22—362no—
471C22—602no—
472C22—1802no—
473C22C2×(C3⋊D3)⋊D512yes—
474C4—91no—
475C4—151no—
476C4—151no—
477C4—152no—
478C4C2×(C3⋊D3)⋊D511yescharacteristic
479C3—21no—
480C3#9202511yescharacteristic, complemented
481C3#9202711yescharacteristic, complemented
482C2—22no—
483C2—181no—
484C2—182no—
485C2—301no—
486C2—302no—
487C2—301no—
488C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4(C3⋊D3)⋊D5
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. C5×D4×C3⋊D3C2semidirect
2. C3×#6819C2semidirect
3. C3×D4×D15C2semidirect
4. #10624236C2semidirect
5. #10624235C2semidirect
6. C2×#67156C2semidirect
7. #10624230C2semidirect
8. #10624226C2semidirect
9. #10624224C2semidirect
10. #10624223C2semidirect
11. #10624220C2semidirect
14. C22×C5×C3⋊D3C22semidirect
15. C22×C3×D15C22semidirect
16. C2×(C3⋊D3)⋊D5C22semidirect
17. C32×C5×D4C22semidirect
18. C5×#638C22semidirect
19. C5×C4⋊(C3⋊D3)C22semidirect
20. C4×C5×C3⋊D3C22semidirect
21. C3×#1220C22semidirect
22. C3×#1220C22semidirect
23. C2×C3×Dic15C22semidirect
24. C3×D60C22semidirect
25. C3×C4×D15C22semidirect
26. C3×C4×D15C22semidirect
27. C3×Dic30C22semidirect
28. #67158C22semidirect
29. #67157C22semidirect
30. #67156C22semidirect
31. #67156C22semidirect
32. #67156C22semidirect
33. #67156C22semidirect
34. #67156C22semidirect
35. #67155C22semidirect
36. #67154C22semidirect
38. #6819D3semidirect
39. D4×D15D3semidirect
76. D4×C3⋊D3D5semidirect
87. C22×D15C2×D3semidirect
95. C3×C5×D4C2×D3semidirect
96. C3×C5×D4C2×D3semidirect
97. #1220C2×D3semidirect
98. #1220C2×D3semidirect
99. C2×Dic15C2×D3semidirect
100. D60C2×D3semidirect
101. C4×D15C2×D3semidirect
102. C4×D15C2×D3semidirect
103. Dic30C2×D3semidirect
158. C22×C3⋊D3C2×D5semidirect
161. C32×D4C2×D5semidirect
162. #638C2×D5semidirect
163. C4⋊(C3⋊D3)C2×D5semidirect
164. C4×C3⋊D3C2×D5semidirect
238. C32×C5H(5,𝔽2)semidirect
244. C5×D4D32semidirect
307. C3×D4D3×D5semidirect
308. C3×D4D3×D5semidirect
379. C3×C5#1004semidirect
380. C3×C5#1009semidirect
422. C32#4144semidirect
439. D4(C3⋊D3)⋊D5semidirect
464. C5#64314semidirect
480. C3#92025semidirect
481. C3#92027semidirect

Maximal subgroups

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

subgroupindex
C5×D4×C3⋊D32
C3×#68192
C3×D4×D152
#106242362
#106242352
C2×#671562
#106242302
#106242262
#106242242
#106242232
#106242202
#920273
#920253
#643145