Finite Group Databasebeta

← #13088959

Subgroups — #13088959

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#10624226C211yesmaximal, characteristic, complemented
2#10624205C211yesmaximal, characteristic, complemented
3#10624199C211yesmaximal, characteristic, complemented
4C5×D3×D12C211yesmaximal, characteristic, complemented
5#10624151C211yesmaximal, characteristic, complemented
6C3×#6769C211yesmaximal, characteristic, complemented
7C3×#6768C211yesmaximal, characteristic, complemented
8#10624049C212yesmaximal, complemented
9#10624034C212yesmaximal, complemented
10#10624033C212yesmaximal, complemented
11D3×#1203C212yesmaximal, complemented
12#92027—31nomaximal
13#92018—31nomaximal
14C2×(C3⋊D3)⋊D5C2212yescomplemented
15C2×D3×D15C2212yescomplemented
16C2×C5×D32C2212yescomplemented
17C4×C3⋊D15C2211yescharacteristic, complemented
18C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C3×D60C2211yescharacteristic, complemented
20C3×C4×D15C2211yescharacteristic, complemented
21C3×C5×D12C2211yescharacteristic, complemented
22C3×C4×C5×D3C2211yescharacteristic, complemented
23C32×Dic10C2211yescharacteristic, complemented
24#67156C2212yescomplemented
25#67153C2211yescharacteristic, complemented
26#67152C2211yescharacteristic, complemented
27#67151C2211yescharacteristic, complemented
28#67150C2211yescharacteristic, complemented
29#67150C2212yescomplemented
30C5×#626C2212yescomplemented
31#67140C2212yescomplemented
32C3⋊D3×Dic5C2212yescomplemented
33C3×#1203C2212yescomplemented
34C3×#1203C2212yescomplemented
35C3×#1200C2212yescomplemented
36C3×D3×Dic5C2212yescomplemented
37#64320—51nomaximal, Hall
38D4×D15—31no—
39#6817—31no—
40C5×D3×D4—31no—
41C2×C5×D12—31no—
42C3×#711—31no—
43C3×#709—31no—
44#6790—32no—
45D3×C22⋊D5—32no—
46D3×C22⋊D5—32no—
47D3×C22⋊D5—32no—
48C2×#1203—32no—
49#6782—32no—
50#6777—31no—
51#6777—31no—
52D5×D12—31no—
53D5×D12—32no—
54#6772—32no—
55#6771—31no—
56#6769D311yescharacteristic, complemented
57#6768—31no—
58#6768—31no—
59#6768—61no—
60#6768D311yescharacteristic, complemented
61C2×C3⋊D15C2311yescharacteristic
62C2×C5×C3⋊D3C2312yes—
63C2×C3×D15C2312yes—
64C2×C3×D15C2311yescharacteristic
65C2×C3×C5×D3C2312yes—
66C2×C3×C5×D3C2311yescharacteristic
67(C3⋊D3)⋊D5—42no—
68D3×D15—42no—
69C5×D32—42no—
70C32×C4×C5C2311yescharacteristic
71C3⋊Dic15C2311yescharacteristic
72C3×Dic15C2311yescharacteristic
73C3×C5×Dic3C2311yescharacteristic
74C32×Dic5C2312yes—
75#4144—91noHall
76#3832—51no—
77C3×#239—51no—
78C3×#239—51no—
79#3802—52no—
80#3802—51no—
81D3×D12—52no—
82D3×D12—51no—
83D3×D12—52no—
84D3×D12D511yescharacteristic, complemented
85#3798—51no—
86#3798—52no—
87C22×D15—32no—
88C22×C5×D3—32no—
89C22×C5×D3—32no—
90C2×D3×D5—32no—
91C2×D3×D5—32no—
92C2×D3×D5—32no—
93C3×C5×D4—31no—
94C2×C3×C4×C5—31no—
95#1220—32no—
96#1220—31no—
97#1220—31no—
98D60—31no—
99D60C2×D311yescharacteristic, complemented
100C4×D15—31no—
101C4×D15—31no—
102C4×D15—61no—
103C4×D15C2×D311yescharacteristic, complemented
104Dic30—31no—
105C5×#67—32no—
106C5×D12—32no—
107C5×D12—31no—
108C5×D12—31no—
109C5×D12—61no—
110C5×D12C2×D311yescharacteristic, complemented
111C4×C5×D3C2×D311yescharacteristic, complemented
112C3×C22⋊D5—32no—
113C3×C22⋊D5—32no—
114C2×C3×Dic5—32no—
115C3×D20—31no—
116C3×C4×D5—31no—
117C3×C4×D5—32no—
118C3×Dic10—21no—
119C3×Dic10C2×D311yescharacteristic, complemented
120C3×Dic10C2×D311yescharacteristic, complemented
121#1204—31no—
122#1203—32no—
123#1203—31no—
124#1203—32no—
125#1203—32no—
126#1203—62no—
127#1203C2×D312yescomplemented
128#1203C2×D312yescomplemented
129#1202—31no—
130#1201—31no—
131#1201—32no—
132#1201—32no—
133#1200C2×D312yescomplemented
134D3×Dic5—32no—
135D3×Dic5—32no—
136D3×Dic5—62no—
137D3×Dic5C2×D312yescomplemented
138#1009—151no—
139#1009—151no—
140C2×C32×C5C2411yescharacteristic
141C3⋊D15—21no—
142C5×C3⋊D3—22no—
143C3×D15—22no—
144C3×D15—21no—
145C3×C5×D3—21no—
146C3×C5×D3—22no—
147C2×C5×D4—91no—
148C2×C22⋊D5—92no—
149C2×C22⋊D5—92no—
150#711—31no—
151#711—92no—
152#711—91no—
153D4×D5—91no—
154D4×D5—92no—
155D4×D5—91no—
156#709—31no—
157#709—91no—
158C2×D32—52no—
159C2×D32—52no—
160C2×D32C2×D512yescomplemented
161C32×Q8—51no—
162C4⋊(C3⋊D3)—52no—
163C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
164C4×C3⋊D3—52no—
165C4×C3⋊D3—51no—
166C3×D12—52no—
167C3×D12—52no—
168C3×D12—51no—
169C3×D12C2×D511yescharacteristic, complemented
170C3×C4×D3—52no—
171C3×C4×D3—51no—
172C3×C4×D3—52no—
173C3×C4×D3C2×D511yescharacteristic, complemented
174#627—51no—
175#626—52no—
176#626—51no—
177#626—51no—
178#626C2×D512yescomplemented
179#625—51no—
180#625—52no—
181C22×C3×C5—32no—
182C22×C3×C5—31no—
183C2×D15—31no—
184C2×D15—31no—
185C2×D15—61no—
186C2×D15—122no—
187C2×D15C22×D312yes—
188C2×D15C22×D311yescharacteristic
189C2×C5×D3—32no—
190C2×C5×D3—32no—
191C2×C5×D3—62no—
192C2×C5×D3—122no—
193C2×C5×D3—122no—
194C2×C5×D3C22×D312yes—
195C2×C5×D3C22×D311yescharacteristic
196C2×C3×D5—32no—
197C2×C3×D5—31no—
198D3×D5—122no—
199D3×D5—122no—
200D3×D5—122no—
201C3×C4×C5—21no—
202C3×C4×C5—31no—
203C3×C4×C5C22×D311yescharacteristic
204C3×C4×C5C22×D311yescharacteristic
205Dic15—31no—
206Dic15—31no—
207Dic15—61no—
208Dic15C22×D311yescharacteristic
209C3×Dic5—22no—
210C3×Dic5—31no—
211C3×Dic5C22×D312yes—
212C3×Dic5C22×D312yes—
213C5×Dic3C22×D311yescharacteristic
214C3×#41—151no—
215C3×#41—151no—
216#239—51no—
217#239—51no—
218#239—151no—
219#239—151no—
220#239—301no—
221D3×D4—31no—
222D3×D4—151no—
223D3×D4—152no—
224D3×D4—152no—
225D3×D4—151no—
226D3×D4—152no—
227D3×D4—152no—
228D3×D4—151no—
229#235—152no—
230#235—152no—
231#235—151no—
232#235—151no—
233C2×D12—31no—
234C2×D12—152no—
235C2×D12—151no—
236C2×D12—152no—
237C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
238C23×C5—92no—
239C22×D5—92no—
240C22×D5—92no—
241C5×D4—31no—
242C5×D4—92no—
243C5×D4—91no—
244C2×C4×C5—31no—
245C22⋊D5—32no—
246C22⋊D5—32no—
247C22⋊D5—92no—
248C22⋊D5—91no—
249C22⋊D5—91no—
250C22⋊D5—92no—
251C22⋊D5—92no—
252C2×Dic5—32no—
253C2×Dic5—92no—
254D20—31no—
255D20—91no—
256C4×D5—31no—
257C4×D5—32no—
258C4×D5—91no—
259Dic10—91no—
260Dic10D3211yescharacteristic, complemented
261C2×C3⋊D3—51no—
262C2×C3⋊D3C22×D512yes—
263C2×C3×D3—52no—
264C2×C3×D3—51no—
265C2×C3×D3C22×D511yescharacteristic
266C2×C3×D3C22×D512yes—
267D32—42no—
268D32—202no—
269D32—202no—
270C32×C4—52no—
271C32×C4C22×D511yescharacteristic
272C3⋊Dic3—51no—
273C3×Dic3—51no—
274C3×Dic3C22×D511yescharacteristic
275H(5,𝔽2)—451noSylow
276C2×C3×C5—21no—
277C2×C3×C5—61no—
278C2×C3×C5—62no—
279C2×C3×C5C23×D311yescharacteristic
280C2×C3×C5C23×D311yescharacteristic
281D15—21no—
282D15—22no—
283D15—61no—
284D15—61no—
285D15—121no—
286C3×D5—61no—
287C3×D5—62no—
288C5×D3—22no—
289C5×D3—21no—
290C5×D3—62no—
291C5×D3—62no—
292C5×D3—122no—
293C22×D3—32no—
294C22×D3—32no—
295C22×D3—152no—
296C22×D3—152no—
297C22×D3—152no—
298C22×D3—152no—
299C3×Q8—51no—
300C3×Q8—51no—
301C3×Q8—101no—
302C3×D4—31no—
303C3×D4—152no—
304C3×D4—152no—
305C3×D4—151no—
306C2×C3×C4—31no—
307C2×C3×C4—152no—
308C2×C3×C4—152no—
309C2×C3×C4—151no—
310#67—32no—
311#67—152no—
312#67—151no—
313#67—152no—
314#67—151no—
315#67—152no—
316#67—151no—
317#67—151no—
318D12—32no—
319D12—31no—
320D12—31no—
321D12—52no—
322D12—52no—
323D12—51no—
324D12—61no—
325D12—151no—
326D12—152no—
327D12—152no—
328D12—152no—
329D12—151no—
330D12—302no—
331D12D3×D511yescharacteristic, complemented
332C4×D3—52no—
333C4×D3—51no—
334C4×D3—52no—
335C4×D3—152no—
336C4×D3—152no—
337C4×D3—151no—
338C4×D3—151no—
339C4×D3—301no—
340C4×D3—302no—
341C4×D3D3×D511yescharacteristic, complemented
342Dic6—151no—
343Dic6—151no—
344C22×C5—32no—
345C22×C5—31no—
346C22×C5—92no—
347C22×C5—362no—
348C2×D5—32no—
349C2×D5—31no—
350C2×D5—91no—
351C2×D5—362no—
352C2×D5—362no—
353C4×C5—31no—
354C4×C5C2×D3211yescharacteristic
355Dic5—31no—
356Dic5—91no—
357Dic5C2×D3212yes—
358C2×C32C23×D511yescharacteristic
359C3⋊D3—22no—
360C3⋊D3—101no—
361C3×D3—21no—
362C3×D3—22no—
363C3×D3—102no—
364C3×D3—101no—
365#41—151no—
366#41—151no—
367#41—451no—
368#41—452no—
369#41—451no—
370C2×D4—91no—
371C2×D4—452no—
372C2×D4—451no—
373C2×D4—452no—
374C2×D4—451no—
375C2×D4—452no—
376C3×C5—21no—
377C3×C5#100911yescharacteristic, complemented
378C3×C5#100911yescharacteristic, complemented
379C22×C3—32no—
380C22×C3—31no—
381C22×C3—151no—
382C22×C3—152no—
383C2×D3—32no—
384C2×D3—32no—
385C2×D3—52no—
386C2×D3—51no—
387C2×D3—62no—
388C2×D3—122no—
389C2×D3—122no—
390C2×D3—151no—
391C2×D3—151no—
392C2×D3—301no—
393C2×D3—602no—
394C2×D3—602no—
395C2×D3—602no—
396C2×D3—602no—
397C2×D3C2×D3×D512yes—
398C2×D3C2×D3×D511yescharacteristic
399C3×C4—21no—
400C3×C4—31no—
401C3×C4—52no—
402C3×C4—52no—
403C3×C4—102no—
404C3×C4—151no—
405C3×C4C2×D3×D511yescharacteristic
406C3×C4C2×D3×D511yescharacteristic
407Dic3—51no—
408Dic3—151no—
409Dic3—151no—
410Dic3—301no—
411Dic3C2×D3×D511yescharacteristic
412C2×C5—62no—
413C2×C5—61no—
414C2×C5—182no—
415C2×C5C22×D3211yescharacteristic
416D5—62no—
417D5—61no—
418D5—181no—
419C32#414411yescharacteristic, complemented, Sylow
420C23—92no—
421C23—452no—
422C23—452no—
423Q8—51no—
424Q8—451no—
425D4—31no—
426D4—92no—
427D4—91no—
428D4—152no—
429D4—152no—
430D4—151no—
431D4—451no—
432D4—452no—
433D4—451no—
434D4—452no—
435D4—451no—
436D4—452no—
437C2×C4—31no—
438C2×C4—152no—
439C2×C4—152no—
440C2×C4—151no—
441C2×C4—452no—
442C2×C4—451no—
443C2×C3—21no—
444C2×C3—62no—
445C2×C3—61no—
446C2×C3—301no—
447C2×C3—302no—
448C2×C3C22×D3×D511yescharacteristic
449C2×C3C22×D3×D511yescharacteristic
450D3—21no—
451D3—22no—
452D3—62no—
453D3—62no—
454D3—102no—
455D3—101no—
456D3—122no—
457D3—301no—
458D3—301no—
459D3—601no—
460C5#6432011yescharacteristic, complemented, Sylow
461C22—31no—
462C22—32no—
463C22—92no—
464C22—152no—
465C22—151no—
466C22—362no—
467C22—451no—
468C22—1802no—
469C22—1802no—
470C4—31no—
471C4—52no—
472C4—151no—
473C4—451no—
474C4D32×D511yescharacteristic
475C3—21no—
476C3#9201811yescharacteristic, complemented
477C3#9202711yescharacteristic, complemented
478C2—61no—
479C2—62no—
480C2—182no—
481C2—302no—
482C2—301no—
483C2—901no—
484C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624226C2semidirect
2. #10624205C2semidirect
3. #10624199C2semidirect
4. C5×D3×D12C2semidirect
5. #10624151C2semidirect
6. C3×#6769C2semidirect
7. C3×#6768C2semidirect
8. #10624049C2semidirect
9. #10624034C2semidirect
10. #10624033C2semidirect
11. D3×#1203C2semidirect
14. C2×(C3⋊D3)⋊D5C22semidirect
15. C2×D3×D15C22semidirect
16. C2×C5×D32C22semidirect
17. C4×C3⋊D15C22semidirect
18. C5×C4⋊(C3⋊D3)C22semidirect
19. C3×D60C22semidirect
20. C3×C4×D15C22semidirect
21. C3×C5×D12C22semidirect
22. C3×C4×C5×D3C22semidirect
23. C32×Dic10C22semidirect
24. #67156C22semidirect
25. #67153C22semidirect
26. #67152C22semidirect
27. #67151C22semidirect
28. #67150C22semidirect
29. #67150C22semidirect
30. C5×#626C22semidirect
31. #67140C22semidirect
32. C3⋊D3×Dic5C22semidirect
33. C3×#1203C22semidirect
34. C3×#1203C22semidirect
35. C3×#1200C22semidirect
36. C3×D3×Dic5C22semidirect
56. #6769D3semidirect
60. #6768D3semidirect
84. D3×D12D5semidirect
99. D60C2×D3semidirect
103. C4×D15C2×D3semidirect
110. C5×D12C2×D3semidirect
111. C4×C5×D3C2×D3semidirect
119. C3×Dic10C2×D3semidirect
120. C3×Dic10C2×D3semidirect
127. #1203C2×D3semidirect
128. #1203C2×D3semidirect
133. #1200C2×D3semidirect
137. D3×Dic5C2×D3semidirect
160. C2×D32C2×D5semidirect
163. C4⋊(C3⋊D3)C2×D5semidirect
169. C3×D12C2×D5semidirect
173. C3×C4×D3C2×D5semidirect
178. #626C2×D5semidirect
237. C32×C5H(5,𝔽2)semidirect
260. Dic10D32semidirect
331. D12D3×D5semidirect
341. C4×D3D3×D5semidirect
377. C3×C5#1009semidirect
378. C3×C5#1009semidirect
419. C32#4144semidirect
460. C5#64320semidirect
476. C3#92018semidirect
477. C3#92027semidirect

Maximal subgroups

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

subgroupindex
#106242262
#106242052
#106241992
C5×D3×D122
#106241512
C3×#67692
C3×#67682
#106240492
#106240342
#106240332
D3×#12032
#920273
#920183
#643205