Finite Group Databasebeta

← #13088941

Subgroups — #13088941

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
1#10624226C211yesmaximal, characteristic, complemented
2D3×D60C211yesmaximal, characteristic, complemented
3#10624201C211yesmaximal, characteristic, complemented
4C5×#3799C211yesmaximal, characteristic, complemented
5D5×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
6C3×#6773C211yesmaximal, characteristic, complemented
7C3×#6771C211yesmaximal, characteristic, complemented
8#10624054C212yesmaximal, complemented
9#10624033C212yesmaximal, complemented
10#10624031C212yesmaximal, complemented
11#10624030C212yesmaximal, complemented
12#92035—31nomaximal
13#92019—31nomaximal
14C2×(C3⋊D3)⋊D5C2212yescomplemented
15C2×D3×D15C2212yescomplemented
16C2×D5×C3⋊D3C2212yescomplemented
17C4⋊(C3⋊D15)C2211yescharacteristic, complemented
18C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C3×D60C2211yescharacteristic, complemented
20C3×C4×D15C2211yescharacteristic, complemented
21C3×C4×C5×D3C2211yescharacteristic, complemented
22C3×C5×Dic6C2211yescharacteristic, complemented
23C32×C4×D5C2211yescharacteristic, complemented
24#67156C2212yescomplemented
25#67151C2212yescomplemented
26#67151C2212yescomplemented
27#67149C2212yescomplemented
28C5×#626C2212yescomplemented
29C5×#624C2212yescomplemented
30#67140C2212yescomplemented
31C3×#1204C2211yescharacteristic, complemented
32C3×#1203C2211yescharacteristic, complemented
33C3×#1202C2212yescomplemented
34C3×#1202C2211yescharacteristic, complemented
35C3×#1201C2211yescharacteristic, complemented
36C3×#1200C2212yescomplemented
37#64308—51nomaximal, Hall
38#6821—31no—
39C2×D60—31no—
40C5×#239—31no—
41C5×#235—31no—
42C3×#713—31no—
43C3×#709—31no—
44#6790—32no—
45#6790—32no—
46#6777—31no—
47#6777—32no—
48D3×D20—31no—
49D3×D20—32no—
50D3×D20—32no—
51D5×D12—31no—
52D5×D12—31no—
53D5×D12—31no—
54D5×D12—32no—
55D5×D12—61no—
56#6773—31no—
57#6773D311yescharacteristic, complemented
58#6771—32no—
59#6771D311yescharacteristic, complemented
60#6769—32no—
61C2×C3⋊D15C2312yes—
62C2×C5×C3⋊D3C2312yes—
63C2×C3×D15C2312yes—
64C2×C3×D15C2311yescharacteristic
65C2×C3×C5×D3C2311yescharacteristic
66C2×C32×D5C2311yescharacteristic
67(C3⋊D3)⋊D5—42no—
68D3×D15—42no—
69D5×C3⋊D3—42no—
70C32×C4×C5C2311yescharacteristic
71C3×Dic15C2311yescharacteristic
72C3×C5×Dic3C2312yes—
73C3×C5×Dic3C2311yescharacteristic
74C32×Dic5C2311yescharacteristic
75#4149—91noHall
76C2×C4⋊(C3⋊D3)—51no—
77C3×#235—51no—
78C3×#235—51no—
79#3811—52no—
80#3811—52no—
81D3×D12—51no—
82D3×D12—51no—
83D3×D12—52no—
84#3799—51no—
85#3799—52no—
86#3799D511yescharacteristic, complemented
87C22×D15—32no—
88C2×D3×D5—32no—
89C2×D3×D5—32no—
90C2×D3×D5—32no—
91C2×D3×D5—32no—
92C2×D3×D5—32no—
93C2×D3×D5—62no—
94C3×C5×Q8—31no—
95C2×C3×C4×C5—31no—
96D60—32no—
97D60—32no—
98D60—31no—
99D60—31no—
100D60—61no—
101D60C2×D311yescharacteristic, complemented
102C4×D15—32no—
103C4×D15C2×D311yescharacteristic, complemented
104C5×#67—32no—
105C5×D12—32no—
106C5×D12—31no—
107C5×D12—31no—
108C5×D12—61no—
109C4×C5×D3—32no—
110C4×C5×D3—32no—
111C4×C5×D3C2×D311yescharacteristic, complemented
112C5×Dic6C2×D311yescharacteristic, complemented
113C3×C22⋊D5—31no—
114C3×C22⋊D5—31no—
115C3×D20—31no—
116C3×D20—32no—
117C3×D20—31no—
118C3×C4×D5—21no—
119C3×C4×D5—31no—
120C3×C4×D5—32no—
121C3×C4×D5C2×D311yescharacteristic, complemented
122C3×C4×D5C2×D311yescharacteristic, complemented
123C3×Dic10—31no—
124#1204C2×D311yescharacteristic, complemented
125#1203—32no—
126#1203—32no—
127#1203—32no—
128#1203—62no—
129#1203C2×D311yescharacteristic, complemented
130#1202—32no—
131#1202—32no—
132#1202C2×D311yescharacteristic, complemented
133#1202C2×D312yescomplemented
134#1201—32no—
135#1201C2×D311yescharacteristic, complemented
136#1200—32no—
137#1200C2×D312yescomplemented
138#1009—151no—
139#1009—151no—
140C2×C32×C5C2411yescharacteristic
141C3⋊D15—22no—
142C5×C3⋊D3—22no—
143C3×D15—22no—
144C3×D15—21no—
145C3×C5×D3—21no—
146C32×D5—21no—
147C5×#41—91no—
148#713—31no—
149#713—91no—
150D4×D5—91no—
151D4×D5—92no—
152D4×D5—92no—
153D4×D5—91no—
154#709—31no—
155#709—92no—
156C2×D20—92no—
157C2×D20—91no—
158C22×C3⋊D3—52no—
159C2×D32—52no—
160C2×D32—52no—
161C2×C32×C4—51no—
162C4⋊(C3⋊D3)—52no—
163C4⋊(C3⋊D3)—51no—
164C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
165C3×#67—51no—
166C3×#67—52no—
167C3×#67—51no—
168C3×D12—51no—
169C3×D12—51no—
170C3×C4×D3—51no—
171C3×C4×D3—52no—
172C3×C4×D3C2×D511yescharacteristic, complemented
173C3×Dic6—51no—
174C3×Dic6C2×D511yescharacteristic, complemented
175#626—52no—
176#626—52no—
177#626—52no—
178#626C2×D512yescomplemented
179#624—52no—
180#624C2×D512yescomplemented
181C22×C3×C5—31no—
182C2×D15—32no—
183C2×D15—32no—
184C2×D15—62no—
185C2×D15—122no—
186C2×D15C22×D312yes—
187C2×D15C22×D311yescharacteristic
188C2×C5×D3—32no—
189C2×C5×D3—32no—
190C2×C5×D3—62no—
191C2×C5×D3C22×D311yescharacteristic
192C2×C3×D5—21no—
193C2×C3×D5—32no—
194C2×C3×D5—31no—
195C2×C3×D5C22×D311yescharacteristic
196C2×C3×D5C22×D311yescharacteristic
197D3×D5—122no—
198D3×D5—122no—
199D3×D5—122no—
200D3×D5—122no—
201D3×D5—122no—
202D3×D5—124no—
203C3×C4×C5—21no—
204C3×C4×C5—31no—
205C3×C4×C5—32no—
206C3×C4×C5C22×D311yescharacteristic
207C3×C4×C5C22×D311yescharacteristic
208Dic15C22×D311yescharacteristic
209C3×Dic5—21no—
210C3×Dic5—31no—
211C3×Dic5C22×D311yescharacteristic
212C3×Dic5C22×D311yescharacteristic
213C5×Dic3C22×D312yes—
214C5×Dic3C22×D311yescharacteristic
215C3×#41—151no—
216C3×#41—151no—
217#239—31no—
218#239—152no—
219#239—151no—
220D3×D4—152no—
221D3×D4—152no—
222D3×D4—151no—
223D3×D4—152no—
224D3×D4—152no—
225D3×D4—152no—
226D3×D4—151no—
227#235—31no—
228#235—51no—
229#235—51no—
230#235—152no—
231#235—151no—
232C2×D12—152no—
233C2×D12—151no—
234C2×D12—151no—
235C2×D12—151no—
236C2×D12—151no—
237C2×D12—301no—
238C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
239C22×D5—92no—
240C22×D5—92no—
241C22×D5—92no—
242C5×Q8—31no—
243C5×D4—92no—
244C5×D4—91no—
245C2×C4×C5—31no—
246C2×C4×C5—92no—
247C22⋊D5—31no—
248C22⋊D5—31no—
249C22⋊D5—92no—
250C22⋊D5—92no—
251D20—31no—
252D20—32no—
253D20—31no—
254D20—92no—
255D20—92no—
256D20—91no—
257C4×D5—31no—
258C4×D5—32no—
259C4×D5—92no—
260C4×D5D3211yescharacteristic, complemented
261Dic10—31no—
262C22×C32—51no—
263C2×C3⋊D3—52no—
264C2×C3⋊D3—202no—
265C2×C3⋊D3C22×D512yes—
266C2×C3×D3—52no—
267C2×C3×D3—51no—
268C2×C3×D3C22×D511yescharacteristic
269D32—202no—
270D32—202no—
271C32×C4—51no—
272C32×C4C22×D511yescharacteristic
273C3×Dic3—51no—
274C3×Dic3C22×D512yes—
275C3×Dic3C22×D511yescharacteristic
276H(5,𝔽2)—451noSylow
277C2×C3×C5—21no—
278C2×C3×C5—61no—
279C2×C3×C5C23×D311yescharacteristic
280C2×C3×C5C23×D311yescharacteristic
281D15—21no—
282D15—22no—
283D15—62no—
284D15—62no—
285D15—122no—
286C3×D5—22no—
287C3×D5—21no—
288C3×D5—21no—
289C3×D5—62no—
290C3×D5—61no—
291C5×D3—21no—
292C5×D3—62no—
293C5×D3—62no—
294C5×D3—122no—
295C22×D3—152no—
296C22×D3—152no—
297C22×D3—152no—
298C22×D3—152no—
299C22×D3—152no—
300C22×D3—152no—
301C22×D3—302no—
302C3×Q8—31no—
303C3×Q8—151no—
304C3×D4—151no—
305C3×D4—152no—
306C3×D4—151no—
307C3×D4—151no—
308C3×D4—151no—
309C2×C3×C4—31no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—101no—
313C2×C3×C4—152no—
314C2×C3×C4—151no—
315#67—32no—
316#67—51no—
317#67—51no—
318#67—52no—
319#67—152no—
320#67—152no—
321#67—152no—
322D12—32no—
323D12—31no—
324D12—31no—
325D12—51no—
326D12—51no—
327D12—61no—
328D12—152no—
329D12—152no—
330D12—152no—
331D12—152no—
332D12—152no—
333D12—151no—
334D12—151no—
335D12—301no—
336D12—302no—
337C4×D3—32no—
338C4×D3—32no—
339C4×D3—51no—
340C4×D3—52no—
341C4×D3—152no—
342C4×D3—152no—
343C4×D3D3×D511yescharacteristic, complemented
344Dic6—51no—
345Dic6D3×D511yescharacteristic, complemented
346C22×C5—31no—
347C22×C5—92no—
348C2×D5—32no—
349C2×D5—31no—
350C2×D5—92no—
351C2×D5—362no—
352C2×D5—362no—
353C2×D5—362no—
354C2×D5C2×D3211yescharacteristic
355C4×C5—31no—
356C4×C5—32no—
357C4×C5C2×D3211yescharacteristic
358Dic5—31no—
359Dic5C2×D3211yescharacteristic
360C2×C32—101no—
361C2×C32C23×D511yescharacteristic
362C3⋊D3—22no—
363C3⋊D3—102no—
364C3×D3—21no—
365C3×D3—101no—
366C3×D3—102no—
367#41—91no—
368#41—151no—
369#41—151no—
370#41—452no—
371#41—451no—
372C2×D4—451no—
373C2×D4—451no—
374C2×D4—452no—
375C2×D4—452no—
376C2×D4—451no—
377C2×D4—452no—
378C3×C5—21no—
379C3×C5#100911yescharacteristic, complemented
380C3×C5#100911yescharacteristic, complemented
381C22×C3—31no—
382C22×C3—51no—
383C22×C3—51no—
384C22×C3—101no—
385C22×C3—152no—
386C22×C3—151no—
387C2×D3—32no—
388C2×D3—32no—
389C2×D3—51no—
390C2×D3—52no—
391C2×D3—62no—
392C2×D3—152no—
393C2×D3—152no—
394C2×D3—302no—
395C2×D3—602no—
396C2×D3—602no—
397C2×D3—602no—
398C2×D3—602no—
399C2×D3—602no—
400C2×D3—602no—
401C2×D3—604no—
402C2×D3C2×D3×D511yescharacteristic
403C3×C4—21no—
404C3×C4—31no—
405C3×C4—32no—
406C3×C4—51no—
407C3×C4—51no—
408C3×C4—101no—
409C3×C4—151no—
410C3×C4C2×D3×D511yescharacteristic
411C3×C4C2×D3×D511yescharacteristic
412Dic3—51no—
413Dic3C2×D3×D511yescharacteristic
414Dic3C2×D3×D512yes—
415C2×C5—61no—
416C2×C5—182no—
417C2×C5C22×D3211yescharacteristic
418D5—21no—
419D5—62no—
420D5—61no—
421D5—182no—
422C32#414911yescharacteristic, complemented, Sylow
423C23—452no—
424C23—452no—
425C23—452no—
426Q8—31no—
427Q8—151no—
428D4—92no—
429D4—91no—
430D4—151no—
431D4—152no—
432D4—151no—
433D4—151no—
434D4—151no—
435D4—452no—
436D4—452no—
437D4—452no—
438D4—452no—
439D4—451no—
440C2×C4—31no—
441C2×C4—51no—
442C2×C4—92no—
443C2×C4—152no—
444C2×C4—151no—
445C2×C4—452no—
446C2×C3—21no—
447C2×C3—61no—
448C2×C3—102no—
449C2×C3—101no—
450C2×C3—101no—
451C2×C3—302no—
452C2×C3—301no—
453C2×C3C22×D3×D511yescharacteristic
454C2×C3C22×D3×D511yescharacteristic
455D3—21no—
456D3—62no—
457D3—62no—
458D3—101no—
459D3—102no—
460D3—122no—
461D3—302no—
462D3—302no—
463D3—602no—
464C5#6430811yescharacteristic, complemented, Sylow
465C22—31no—
466C22—51no—
467C22—92no—
468C22—152no—
469C22—151no—
470C22—452no—
471C22—1802no—
472C22—1802no—
473C22—1802no—
474C4—31no—
475C4—32no—
476C4—51no—
477C4—151no—
478C4D32×D511yescharacteristic
479C3—21no—
480C3#9201911yescharacteristic, complemented
481C3#9203511yescharacteristic, complemented
482C2—61no—
483C2—101no—
484C2—182no—
485C2—302no—
486C2—301no—
487C2—902no—
488C2C2×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. D3×D60C2semidirect
3. #10624201C2semidirect
4. C5×#3799C2semidirect
5. D5×C4⋊(C3⋊D3)C2semidirect
6. C3×#6773C2semidirect
7. C3×#6771C2semidirect
8. #10624054C2semidirect
9. #10624033C2semidirect
10. #10624031C2semidirect
11. #10624030C2semidirect
14. C2×(C3⋊D3)⋊D5C22semidirect
15. C2×D3×D15C22semidirect
16. C2×D5×C3⋊D3C22semidirect
17. C4⋊(C3⋊D15)C22semidirect
18. C5×C4⋊(C3⋊D3)C22semidirect
19. C3×D60C22semidirect
20. C3×C4×D15C22semidirect
21. C3×C4×C5×D3C22semidirect
22. C3×C5×Dic6C22semidirect
23. C32×C4×D5C22semidirect
24. #67156C22semidirect
25. #67151C22semidirect
26. #67151C22semidirect
27. #67149C22semidirect
28. C5×#626C22semidirect
29. C5×#624C22semidirect
30. #67140C22semidirect
31. C3×#1204C22semidirect
32. C3×#1203C22semidirect
33. C3×#1202C22semidirect
34. C3×#1202C22semidirect
35. C3×#1201C22semidirect
36. C3×#1200C22semidirect
57. #6773D3semidirect
59. #6771D3semidirect
86. #3799D5semidirect
101. D60C2×D3semidirect
103. C4×D15C2×D3semidirect
111. C4×C5×D3C2×D3semidirect
112. C5×Dic6C2×D3semidirect
121. C3×C4×D5C2×D3semidirect
122. C3×C4×D5C2×D3semidirect
124. #1204C2×D3semidirect
129. #1203C2×D3semidirect
132. #1202C2×D3semidirect
133. #1202C2×D3semidirect
135. #1201C2×D3semidirect
137. #1200C2×D3semidirect
164. C4⋊(C3⋊D3)C2×D5semidirect
172. C3×C4×D3C2×D5semidirect
174. C3×Dic6C2×D5semidirect
178. #626C2×D5semidirect
180. #624C2×D5semidirect
238. C32×C5H(5,𝔽2)semidirect
260. C4×D5D32semidirect
343. C4×D3D3×D5semidirect
345. Dic6D3×D5semidirect
379. C3×C5#1009semidirect
380. C3×C5#1009semidirect
422. C32#4149semidirect
464. C5#64308semidirect
480. C3#92019semidirect
481. C3#92035semidirect

Maximal subgroups

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

subgroupindex
#106242262
D3×D602
#106242012
C5×#37992
D5×C4⋊(C3⋊D3)2
C3×#67732
C3×#67712
#106240542
#106240332
#106240312
#106240302
#920353
#920193
#643085