Finite Group Databasebeta

← #14412955

Subgroups — #14412955

Order 1920.

1476 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11757059C211yesmaximal, characteristic
2#11756993C211yesmaximal, characteristic, complemented
3C3×#41×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4#11751720C213yesmaximal, complemented
5#11751719C213yesmaximal, complemented
6D5⋊#805C213yesmaximal, complemented
7C2×#91156C213yesmaximal, complemented
8#10617560—31nomaximal, Hall
9#92101C411yescharacteristic, complemented
10C5×#1009C411yescharacteristic, complemented
11C3×D5×#41C2211yescharacteristic
12#92034—21no—
13#92030—21no—
14#92025—23no—
15D3×#711—23no—
16#92018C413yescomplemented
17D5×#235C2213yes—
18C2×#6772C2213yes—
19#92001C413yescomplemented
20#91992—21no—
21#91990—23no—
22D5⋊#208C2213yescomplemented
23C2×#6758C2213yescomplemented
24C3×Q8×AGL(1,𝔽5)—21no—
25C3×D4×AGL(1,𝔽5)—23no—
26C3×#4130C2213yescomplemented
27C2×C3×C4×AGL(1,𝔽5)C2213yescomplemented
28#91158C2211yescharacteristic, complemented
29#91158C2213yescomplemented
30#91156C2213yescomplemented
31#91156C2216yescomplemented
32#91156C2213yescomplemented
33#82536—51nomaximal, Hall
34#66285—31no—
35#41×AGL(1,𝔽5)D311yescharacteristic, complemented
36#66264—31no—
37#65795—33no—
38D5⋊#104—33no—
39#65769—33no—
40C2×#4008—33no—
41C22×C3×AGL(1,𝔽5)—23no—
42C3×C5×#41C2×C411yescharacteristic
43Q8×D15—21no—
44#6819—23no—
45#6817C2×C413yescomplemented
46C2×Dic30C2×C413yescomplemented
47C5×#239—21no—
48C5×D3×D4—23no—
49C5×#235C2×C413yescomplemented
50C2×C5×D12C2×C413yescomplemented
51C3×#713D411yescharacteristic
52C3×Q8×D5D411yescharacteristic
53C3×#711D413yes—
54C3×D4×D5D413yes—
55C3×#709C2×C413yes—
56C2×C3×C4×D5C2313yes—
57D3×C22⋊D5—26no—
58D5×#67—23no—
59C2×#1201—23no—
60#6782—23no—
61C2×D3×Dic5—23no—
62#6780—26no—
63#6777—23no—
64D5×D12C2×C413yescomplemented
65C4×D3×D5—23no—
66#6773C2311yescharacteristic
67#6772—23no—
68#6772—23no—
69#6772C2×C416yescomplemented
70#6772C2313yes—
71#6771—23no—
72#6768—23no—
73D3×Dic10—23no—
74#6765—23no—
75D5×Dic6C2×C413yescomplemented
76#6762—23no—
77C2×C3⋊#170—23no—
78#6758—23no—
79#6758C2313yes—
80#6758C2311yescharacteristic
81D5⋊(C3⋊C8)—23no—
82C3×C22⋊AGL(1,𝔽5)—23no—
83C3×C4⋊AGL(1,𝔽5)—23no—
84C3×C4×AGL(1,𝔽5)—23no—
85C3×C4×AGL(1,𝔽5)C2311yescharacteristic
86C3×C4×AGL(1,𝔽5)C2313yes—
87#5964—51no—
88C3×C4×#41—51no—
89#5819—51no—
90#5005—53no—
91#5005—53no—
92C2×#805—53no—
93C2×#805—53no—
94C5×H(5,𝔽2)—31no—
95#4150—31no—
96#4150—33no—
97#4150—61no—
98D5×#41—33no—
99D5×#41—61no—
100D5×#41C2×D311yescharacteristic
101#4144—33no—
102#4144—63no—
103C2×#711—33no—
104C2×#711—63no—
105Q8×AGL(1,𝔽5)—21no—
106#4133—61no—
107D4×AGL(1,𝔽5)—23no—
108#4131—63no—
109#4130C2×D313yescomplemented
110C2×C4×AGL(1,𝔽5)C2×D313yescomplemented
111D5⋊M16—33no—
112C2×#700—33no—
113#4010—31no—
114#4010—33no—
115#4008—33no—
116#4008—33no—
117#4008—36no—
118#2901—151noSylow
119C22×C5×D3—23no—
120C22×C3×D5C2×D413yes—
121C2×D3×D5—23no—
122C2×C3×AGL(1,𝔽5)—21no—
123C2×C3×AGL(1,𝔽5)—23no—
124C2×C3×AGL(1,𝔽5)—46no—
125C3×C5×Q8C22⋊C411yescharacteristic
126C3×C5×D4C22⋊C413yes—
127C2×C3×C4×C5C22×C413yes—
128#1220—23no—
129C2×Dic15—23no—
130D60C22×C411yescharacteristic
131C4×D15—23no—
132Dic30—23no—
133Dic30C22×C413yes—
134C5×#67—23no—
135C5×D12—23no—
136C5×D12C22×C413yes—
137C4×C5×D3—23no—
138C5×Dic6C22×C411yescharacteristic
139C3×C22⋊D5C22⋊C416yes—
140C2×C3×Dic5C2×D413yes—
141C3×D20C22⋊C413yes—
142C3×C4×D5C22×C413yes—
143C3×C4×D5C2×D413yes—
144C3×C4×D5C2×D413yes—
145C3×C4×D5C2411yescharacteristic
146C3×Dic10C22⋊C413yes—
147#1204—23no—
148#1203—23no—
149#1202—21no—
150#1201—26no—
151#1201—23no—
152#1200—21no—
153D3×Dic5—26no—
154D3×Dic5—23no—
155D5×Dic3—23no—
156C3⋊#170—21no—
157C3⋊#170—23no—
158C2×C3×#41—51no—
159#1010—51no—
160#1010—53no—
161#1009—53no—
162#1009AGL(1,𝔽5)11yescharacteristic, complemented
163D3×#41—103no—
164D3×#41—101no—
165#1007—101no—
166#1004—103no—
167C2×#235—53no—
168C2×#235—53no—
169C3×C4×Q8—101no—
170C3×C4×D4—103no—
171C3×#117—53no—
172C2×C3×C42—53no—
173#948—103no—
174#948—101no—
175C2×#208—53no—
176C2×#208—53no—
177#805—53no—
178#805—56no—
179#805—53no—
180#805—51no—
181#805—53no—
182C22×AGL(1,𝔽5)—23no—
183C5×#41—33no—
184C5×#41—61no—
185C5×#41C4×D311yescharacteristic
186C2×C5×D4—33no—
187C2×C5×D4—63no—
188C2×C22⋊D5—63no—
189C2×C22⋊D5—66no—
190C22×Dic5—63no—
191#713—31no—
192#713D1211yescharacteristic
193Q8×D5—33no—
194Q8×D5—61no—
195Q8×D5#6711yescharacteristic
196#711—36no—
197#711—33no—
198#711—63no—
199#711—66no—
200#711—63no—
201#711—63no—
202#711—63no—
203#711—63no—
204#711D1213yes—
205D4×D5—33no—
206D4×D5—63no—
207D4×D5—63no—
208D4×D5#6713yes—
209#709—33no—
210#709—63no—
211#709—63no—
212#709C4×D313yes—
213C2×C4×D5—63no—
214C2×C4×D5C22×D313yes—
215C2×Dic10—33no—
216C2×Dic10—63no—
217C22⋊AGL(1,𝔽5)—23no—
218#704—63no—
219C2×#170—63no—
220C4⋊AGL(1,𝔽5)—23no—
221C4×AGL(1,𝔽5)—23no—
222C4×AGL(1,𝔽5)C22×D311yescharacteristic
223C4×AGL(1,𝔽5)C22×D313yes—
224#700—33no—
225#700—31no—
226#700—63no—
227D5⋊C8—63no—
228#585—151no—
229#568—151no—
230C4×#41—51no—
231#421—153no—
232#421—153no—
233C2×#104—153no—
234C2×#104—153no—
235C22×C3×C5C2×C22⋊C413yes—
236C2×D15—21no—
237C2×C5×D3—23no—
238C2×C5×D3—83no—
239C2×C3×D5—43no—
240C2×C3×D5C2×C22⋊C413yes—
241C2×C3×D5C22×D411yescharacteristic
242D3×D5—83no—
243C3×AGL(1,𝔽5)—22no—
244C3×AGL(1,𝔽5)—43no—
245C3×C4×C5C2×C22⋊C413yes—
246C3×C4×C5C23×C411yescharacteristic
247Dic15—23no—
248C3×Dic5C2×C22⋊C413yes—
249C3×Dic5C22×D411yescharacteristic
250C5×Dic3—21no—
251C3×#41—51no—
252C3×#41—53no—
253C3×#41—53no—
254C3×#41C2×AGL(1,𝔽5)11yescharacteristic
255C2×C3×Q8—51no—
256C2×C3×D4—53no—
257C22×C3×C4—53no—
258C22×C3×C4—103no—
259C2×#67—103no—
260C2×#67—103no—
261#239—21no—
262#239—103no—
263D3×Q8—103no—
264D3×Q8—101no—
265#237—103no—
266#237—106no—
267#237—103no—
268D3×D4—23no—
269D3×D4—103no—
270D3×D4—106no—
271#235—53no—
272#235—53no—
273#235—56no—
274#235—51no—
275#235—103no—
276#235—103no—
277#235—103no—
278#235—103no—
279#235C2×AGL(1,𝔽5)13yescomplemented
280C2×D12—53no—
281C2×D12C2×AGL(1,𝔽5)13yescomplemented
282C2×C4×D3—103no—
283C2×C4×D3—103no—
284C2×Dic6—53no—
285C2×Dic6—53no—
286C3×C4⋊C4—103no—
287C3×C22⋊C4—103no—
288C3×C42—51no—
289C3×C42—53no—
290C3×C42—103no—
291#208—53no—
292#208—51no—
293#208—103no—
294#208—103no—
295C2×C3⋊C8—103no—
296C2×C3⋊C8—103no—
297C23×C5—63no—
298C22×D5—63no—
299C22×D5C2×#6713yes—
300C2×AGL(1,𝔽5)—21no—
301C2×AGL(1,𝔽5)—23no—
302C2×AGL(1,𝔽5)—46no—
303C5×Q8—31no—
304C5×Q8(C2×D3)⋊C411yescharacteristic
305C5×D4—33no—
306C5×D4—63no—
307C5×D4—63no—
308C5×D4(C2×D3)⋊C413yes—
309C2×C4×C5—63no—
310C2×C4×C5C2×C4×D313yes—
311C22⋊D5—63no—
312C22⋊D5—66no—
313C22⋊D5—63no—
314C22⋊D5—63no—
315C22⋊D5(C2×D3)⋊C416yes—
316C2×Dic5—63no—
317C2×Dic5—66no—
318C2×Dic5—63no—
319C2×Dic5C2×D1213yes—
320D20—31no—
321D20—61no—
322D20(C2×D3)⋊C413yes—
323C4×D5—63no—
324C4×D5—63no—
325C4×D5—61no—
326C4×D5C2×C4×D313yes—
327C4×D5C2×D1213yes—
328C4×D5C2×#6713yes—
329C4×D5C23×D311yescharacteristic
330Dic10—33no—
331Dic10—63no—
332Dic10—63no—
333Dic10(C2×D3)⋊C413yes—
334#170—61no—
335#170—63no—
3362−1+4—153no—
3372−1+4—151no—
3382−1+4—301no—
339H(5,𝔽2)—31no—
340H(5,𝔽2)—153no—
341H(5,𝔽2)—303no—
342C2×#41—51no—
343C2×#41—153no—
344C2×#41—153no—
345C2×#41—303no—
346C2×#41—301no—
347#131—303no—
348#131—301no—
349C2×M16—153no—
350C2×M16—153no—
351C4×Q8—101no—
352C4×D4—103no—
353#117—53no—
354C2×C42—53no—
355#104—153no—
356#104—151no—
357#104—153no—
358#104—156no—
359#104—153no—
360C2×C3×C5—23no—
361C2×C3×C5C22×C22⋊C411yescharacteristic
362D15—41no—
363C3×D5—21no—
364C3×D5—23no—
365C5×D3—43no—
366C23×C3—53no—
367C22×D3—23no—
368C22×D3—103no—
369C3×Q8—53no—
370C3×Q8C22⋊AGL(1,𝔽5)11yescharacteristic
371C3×D4—56no—
372C3×D4—53no—
373C3×D4C22⋊AGL(1,𝔽5)13yes—
374C2×C3×C4—53no—
375C2×C3×C4—53no—
376C2×C3×C4—53no—
377C2×C3×C4—53no—
378C2×C3×C4—51no—
379C2×C3×C4—101no—
380C2×C3×C4—103no—
381C2×C3×C4—206no—
382C2×C3×C4C22×AGL(1,𝔽5)13yes—
383#67—23no—
384#67—101no—
385#67—103no—
386#67—106no—
387#67—103no—
388C2×Dic3—103no—
389C2×Dic3—103no—
390D12—23no—
391D12—51no—
392D12—103no—
393D12C22×AGL(1,𝔽5)13yes—
394C4×D3—23no—
395C4×D3—101no—
396C4×D3—103no—
397C4×D3—106no—
398C4×D3—103no—
399Dic6—53no—
400Dic6—103no—
401Dic6—103no—
402Dic6C22×AGL(1,𝔽5)11yescharacteristic
403C3⋊C8—101no—
404C3⋊C8—103no—
405C22×C5—63no—
406C22×C5—243no—
407C22×C5C2×(C2×D3)⋊C413yes—
408C2×D5—43no—
409C2×D5—61no—
410C2×D5—243no—
411C2×D5C2×(C2×D3)⋊C413yes—
412C2×D5C22×#6711yescharacteristic
413AGL(1,𝔽5)—22no—
414AGL(1,𝔽5)—43no—
415C4×C5—61no—
416C4×C5C2×(C2×D3)⋊C413yes—
417C4×C5C22×C4×D311yescharacteristic
418Dic5—63no—
419Dic5C2×(C2×D3)⋊C413yes—
420Dic5C22×D1211yescharacteristic
421#41—33no—
422#41—51no—
423#41—53no—
424#41—53no—
425#41—61no—
426#41—153no—
427#41—156no—
428#41—151no—
429#41—153no—
430#41—303no—
431#41—303no—
432#41—303no—
433#41—303no—
434#41—303no—
435#41—306no—
436#41—303no—
437#41—303no—
438#41D3×AGL(1,𝔽5)11yescharacteristic
439C2×Q8—51no—
440C2×Q8—153no—
441C2×Q8—153no—
442C2×Q8—303no—
443C2×Q8—301no—
444C2×D4—33no—
445C2×D4—53no—
446C2×D4—63no—
447C2×D4—153no—
448C2×D4—303no—
449C2×D4—306no—
450C2×D4—303no—
451C2×D4—303no—
452C22×C4—53no—
453C22×C4—103no—
454C22×C4—303no—
455C22×C4—303no—
456M16—153no—
457M16—151no—
458M16—303no—
459M16—303no—
460C2×C8—303no—
461C2×C8—303no—
462C4⋊C4—103no—
463C22⋊C4—103no—
464C42—51no—
465C42—53no—
466C42—103no—
467C3×C5#290111yescharacteristic, complemented, Hall
468C22×C3—53no—
469C22×C3—51no—
470C22×C3—203no—
471C22×C3C2×C22⋊AGL(1,𝔽5)13yes—
472C2×D3—23no—
473C2×D3—83no—
474C2×D3—101no—
475C2×D3—403no—
476C3×C4—53no—
477C3×C4—51no—
478C3×C4—102no—
479C3×C4—203no—
480C3×C4C2×C22⋊AGL(1,𝔽5)13yes—
481C3×C4C23×AGL(1,𝔽5)11yescharacteristic
482Dic3—21no—
483Dic3—103no—
484C2×C5—23no—
485C2×C5—123no—
486C2×C5C22×(C2×D3)⋊C411yescharacteristic
487D5—21no—
488D5—23no—
489D5—121no—
490C23—53no—
491C23—63no—
492C23—303no—
493Q8—31no—
494Q8—53no—
495Q8—153no—
496Q8—303no—
497Q8—303no—
498Q8(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
499D4—33no—
500D4—56no—
501D4—53no—
502D4—63no—
503D4—63no—
504D4—151no—
505D4—303no—
506D4—301no—
507D4—303no—
508D4—306no—
509D4—303no—
510D4(C2×D3)⋊AGL(1,𝔽5)13yes—
511C2×C4—53no—
512C2×C4—53no—
513C2×C4—53no—
514C2×C4—51no—
515C2×C4—53no—
516C2×C4—63no—
517C2×C4—101no—
518C2×C4—103no—
519C2×C4—206no—
520C2×C4—301no—
521C2×C4—303no—
522C2×C4—306no—
523C2×C4—303no—
524C2×C4—303no—
525C2×C4—303no—
526C2×C4C2×D3×AGL(1,𝔽5)13yes—
527C8—301no—
528C8—303no—
529C2×C3—23no—
530C2×C3—101no—
531C2×C3—103no—
532C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
533D3—43no—
534D3—201no—
535C5#8253611yescharacteristic, complemented, Sylow
536C22—51no—
537C22—53no—
538C22—63no—
539C22—203no—
540C22—243no—
541C22—301no—
542C22—1203no—
543C22C2×(C2×D3)⋊AGL(1,𝔽5)13yes—
544C4—51no—
545C4—53no—
546C4—61no—
547C4—102no—
548C4—203no—
549C4—303no—
550C4C2×(C2×D3)⋊AGL(1,𝔽5)13yes—
551C4C22×D3×AGL(1,𝔽5)11yescharacteristic
552C3#1061756011yescharacteristic, complemented, Sylow
553C2—23no—
554C2—101no—
555C2—103no—
556C2—123no—
557C2—601no—
558C2C22×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2C22×(C2×D3)⋊AGL(1,𝔽5)
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC3×C5×#41C2×C4
socleC2×C3×C5C22×C22⋊C4
prime radical#11757059C2
cosocleC3×C4×D5C24
hypercenter#41D3×AGL(1,𝔽5)
hypocenterC3×C5#2901
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11756993C2semidirect
3. C3×#41×AGL(1,𝔽5)C2semidirect
4. #11751720C2semidirect
5. #11751719C2semidirect
6. D5⋊#805C2semidirect
7. C2×#91156C2semidirect
9. #92101C4semidirect
10. C5×#1009C4semidirect
16. #92018C4semidirect
19. #92001C4semidirect
22. D5⋊#208C22semidirect
23. C2×#6758C22semidirect
26. C3×#4130C22semidirect
27. C2×C3×C4×AGL(1,𝔽5)C22semidirect
28. #91158C22semidirect
29. #91158C22semidirect
30. #91156C22semidirect
31. #91156C22semidirect
32. #91156C22semidirect
35. #41×AGL(1,𝔽5)D3semidirect
45. #6817C2×C4semidirect
46. C2×Dic30C2×C4semidirect
49. C5×#235C2×C4semidirect
50. C2×C5×D12C2×C4semidirect
64. D5×D12C2×C4semidirect
69. #6772C2×C4semidirect
75. D5×Dic6C2×C4semidirect
109. #4130C2×D3semidirect
110. C2×C4×AGL(1,𝔽5)C2×D3semidirect
162. #1009AGL(1,𝔽5)semidirect
279. #235C2×AGL(1,𝔽5)semidirect
281. C2×D12C2×AGL(1,𝔽5)semidirect
467. C3×C5#2901semidirect
535. C5#82536semidirect
552. C3#10617560semidirect

Maximal subgroups

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

subgroupindex
#117570592
#117569932
C3×#41×AGL(1,𝔽5)2
#117517202
#117517192
D5⋊#8052
C2×#911562
#106175603
#825365