Finite Group Databasebeta

← #14412954

Subgroups — #14412954

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#11757060C211yesmaximal, characteristic
2#41×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×#66264C211yesmaximal, characteristic, complemented
4#11751723C213yesmaximal, complemented
5C2×#91157C213yesmaximal, complemented
6#11751717C213yesmaximal, complemented
7D5⋊#825C213yesmaximal, complemented
8#10617562—31nomaximal, Hall
9#92100C411yescharacteristic, complemented
10C5×#1010C411yescharacteristic, complemented
11C3×D5×#41C2211yescharacteristic
12D3×#713—21no—
13#92029—21no—
14#92026—23no—
15#92024—23no—
16#92019C413yescomplemented
17D5×#235C2213yes—
18C2×#6773C2213yes—
19#92000C413yescomplemented
20Q8×ΓL(1,𝔽16)—21no—
21D4×ΓL(1,𝔽16)—23no—
22#91989C2213yescomplemented
23C2×C4×ΓL(1,𝔽16)C2213yescomplemented
24C3×#4133—21no—
25C3×#4131—23no—
26C3×D5⋊M16C2213yescomplemented
27C2×C3×#700C2213yescomplemented
28#91157C2213yescomplemented
29#91157C2213yescomplemented
30#91157C2216yescomplemented
31#91155C2211yescharacteristic, complemented
32#91155C2213yescomplemented
33#83817—51nomaximal, Hall
34#66287—31no—
35#41×AGL(1,𝔽5)—31no—
36#66264D311yescharacteristic, complemented
37#65795—33no—
38D5⋊#104—33no—
39#65784—33no—
40C2×#4010—33no—
41C22×ΓL(1,𝔽16)—23no—
42C3×C5×#41C2×C411yescharacteristic
43#6821—21no—
44D4×D15—23no—
45#6817C2×C413yescomplemented
46C2×D60C2×C413yescomplemented
47C5×D3×Q8—21no—
48C5×#237—23no—
49C5×#235C2×C413yescomplemented
50C2×C5×Dic6C2×C413yescomplemented
51C3×#713D411yescharacteristic
52C3×Q8×D5D411yescharacteristic
53C3×#711D413yes—
54C3×D4×D5D413yes—
55C3×#709C2×C413yes—
56C2×C3×C4×D5C2313yes—
57#6790—26no—
58D5×#67—23no—
59C2×#1202—23no—
60C2×#1200—23no—
61#6782—23no—
62#6779—26no—
63D3×D20—23no—
64D5×D12C2×C413yescomplemented
65C4×D3×D5—23no—
66#6773—23no—
67#6773—23no—
68#6773C2×C416yescomplemented
69#6773C2313yes—
70#6772C2311yescharacteristic
71#6771—23no—
72D15⋊Q8—23no—
73#6769—23no—
74#6766—23no—
75D5×Dic6C2×C413yescomplemented
76#6763—23no—
77C4⋊ΓL(1,𝔽16)—23no—
78C4×ΓL(1,𝔽16)—23no—
79C4×ΓL(1,𝔽16)C2311yescharacteristic
80C4×ΓL(1,𝔽16)C2313yes—
81C3×#704—23no—
82C2×C3×#170—23no—
83C3×#700—23no—
84C3×#700C2313yes—
85C3×#700C2311yescharacteristic
86C3×D5⋊C8—23no—
87#5964—51no—
88C3×#568—51no—
89Dic3×#41—51no—
90#5139—53no—
91#5139—53no—
92C2×#825—53no—
93C2×#825—53no—
94C5×2−1+4—31no—
95#4149—31no—
96#4149—33no—
97#4149—63no—
98D5×#41—33no—
99D5×#41—63no—
100D5×#41C2×D311yescharacteristic
101#4147—33no—
102#4147—61no—
103C2×#713—33no—
104C2×#713—61no—
105Q8×AGL(1,𝔽5)—61no—
106#4133—21no—
107D4×AGL(1,𝔽5)—63no—
108#4131—23no—
109#4130—33no—
110C2×C4×AGL(1,𝔽5)—33no—
111D5⋊M16C2×D313yescomplemented
112C2×#700C2×D313yescomplemented
113#4010—33no—
114#4010—33no—
115#4010—36no—
116#4008—31no—
117#4008—33no—
118#2901—151noSylow
119C22×D15—23no—
120C22×C3×D5C2×D413yes—
121C2×D3×D5—23no—
122C2×ΓL(1,𝔽16)—21no—
123C2×ΓL(1,𝔽16)—23no—
124C2×ΓL(1,𝔽16)—46no—
125C3×C5×Q8C22⋊C411yescharacteristic
126C3×C5×D4C22⋊C413yes—
127C2×C3×C4×C5C22×C413yes—
128#1220—23no—
129D60—23no—
130D60C22×C413yes—
131C4×D15—23no—
132Dic30C22×C411yescharacteristic
133C5×#67—23no—
134C2×C5×Dic3—23no—
135C5×D12C22×C411yescharacteristic
136C4×C5×D3—23no—
137C5×Dic6—23no—
138C5×Dic6C22×C413yes—
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—26no—
150#1202—23no—
151#1201—21no—
152#1200—26no—
153#1200—23no—
154D3×Dic5—21no—
155D5×Dic3—23no—
156C3×#170—21no—
157C3×#170—23no—
158C2×C3×#41—51no—
159#1010—53no—
160#1010AGL(1,𝔽5)11yescharacteristic, complemented
161#1009—51no—
162#1009—53no—
163D3×#41—103no—
164D3×#41—101no—
165#1007—101no—
166#1004—103no—
167C2×#235—53no—
168C2×#235—53no—
169C3×#131—103no—
170C3×#131—101no—
171C2×C3×M16—53no—
172C2×C3×M16—53no—
173Q8×Dic3—101no—
174D4×Dic3—103no—
175#926—53no—
176C2×C4×Dic3—53no—
177#825—51no—
178#825—53no—
179#825—53no—
180#825—53no—
181#825—56no—
182C22×AGL(1,𝔽5)—63no—
183C5×#41—33no—
184C5×#41—63no—
185C5×#41C4×D311yescharacteristic
186C2×C5×Q8—33no—
187C2×C5×Q8—61no—
188#713—36no—
189#713—33no—
190#713—61no—
191#713—63no—
192#713—63no—
193#713—63no—
194#713#6711yescharacteristic
195Q8×D5—33no—
196Q8×D5—63no—
197Q8×D5D1211yescharacteristic
198#711—31no—
199#711—63no—
200#711#6713yes—
201D4×D5—33no—
202D4×D5—63no—
203D4×D5—63no—
204D4×D5—66no—
205D4×D5D1213yes—
206#709—33no—
207#709—66no—
208#709—63no—
209#709—63no—
210#709C4×D313yes—
211C2×D20—33no—
212C2×D20—63no—
213C2×D20—63no—
214C2×C4×D5—63no—
215C2×C4×D5—63no—
216C2×C4×D5C22×D313yes—
217C22⋊AGL(1,𝔽5)—63no—
218#704—23no—
219C2×#170—23no—
220C4⋊AGL(1,𝔽5)—63no—
221C4×AGL(1,𝔽5)—31no—
222C4×AGL(1,𝔽5)—33no—
223C4×AGL(1,𝔽5)—63no—
224#700—23no—
225#700C22×D313yes—
226#700C22×D311yescharacteristic
227D5⋊C8—23no—
228#585—151no—
229#568—51no—
230C4×#41—151no—
231#421—153no—
232#421—153no—
233C2×#104—153no—
234C2×#104—153no—
235C22×C3×C5C2×C22⋊C413yes—
236C2×D15—23no—
237C2×D15—83no—
238C2×C5×D3—21no—
239C2×C3×D5—43no—
240C2×C3×D5C2×C22⋊C413yes—
241C2×C3×D5C22×D411yescharacteristic
242D3×D5—83no—
243ΓL(1,𝔽16)—22no—
244ΓL(1,𝔽16)—43no—
245C3×C4×C5C2×C22⋊C413yes—
246C3×C4×C5C23×C411yescharacteristic
247Dic15—21no—
248C3×Dic5C2×C22⋊C413yes—
249C3×Dic5C22×D411yescharacteristic
250C5×Dic3—23no—
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—
258C2×#67—103no—
259C2×#67—103no—
260C22×Dic3—103no—
261#239—103no—
262#239—101no—
263D3×Q8—21no—
264D3×Q8—103no—
265#237—23no—
266#237—103no—
267#237—106no—
268D3×D4—103no—
269D3×D4—106no—
270D3×D4—103no—
271#235—53no—
272#235—51no—
273#235—53no—
274#235—56no—
275#235—103no—
276#235—103no—
277#235—103no—
278#235—103no—
279#235C2×AGL(1,𝔽5)13yescomplemented
280C2×D12—53no—
281C2×D12—53no—
282C2×C4×D3—103no—
283C2×C4×D3—103no—
284C2×Dic6—53no—
285C2×Dic6C2×AGL(1,𝔽5)13yescomplemented
286C3×M16—51no—
287C3×M16—53no—
288C3×M16—103no—
289C3×M16—103no—
290C2×C3×C8—103no—
291C2×C3×C8—103no—
292#217—103no—
293C4⋊Dic3—103no—
294C4×Dic3—51no—
295C4×Dic3—53no—
296C4×Dic3—103no—
297C22×D5—63no—
298C22×D5—63no—
299C22×D5C2×D1213yes—
300C2×AGL(1,𝔽5)—61no—
301C2×AGL(1,𝔽5)—63no—
302C2×AGL(1,𝔽5)—126no—
303C5×Q8—33no—
304C5×Q8—63no—
305C5×Q8(C2×D3)⋊C411yescharacteristic
306C5×D4—31no—
307C5×D4—63no—
308C5×D4(C2×D3)⋊C413yes—
309C2×C4×C5—63no—
310C2×C4×C5—63no—
311C2×C4×C5C2×C4×D313yes—
312C22⋊D5—63no—
313C22⋊D5—61no—
314C22⋊D5—63no—
315C22⋊D5(C2×D3)⋊C416yes—
316C2×Dic5—61no—
317C2×Dic5C2×#6713yes—
318D20—33no—
319D20—66no—
320D20—63no—
321D20—63no—
322D20(C2×D3)⋊C413yes—
323C4×D5—63no—
324C4×D5—63no—
325C4×D5—66no—
326C4×D5—63no—
327C4×D5C2×C4×D313yes—
328C4×D5C2×D1213yes—
329C4×D5C2×#6713yes—
330C4×D5C23×D311yescharacteristic
331Dic10—31no—
332Dic10—63no—
333Dic10(C2×D3)⋊C413yes—
334#170—21no—
335#170—23no—
3362−1+4—31no—
3372−1+4—153no—
3382−1+4—301no—
339H(5,𝔽2)—153no—
340H(5,𝔽2)—151no—
341H(5,𝔽2)—303no—
342C2×#41—51no—
343C2×#41—153no—
344C2×#41—153no—
345C2×#41—301no—
346C2×#41—303no—
347#131—103no—
348#131—101no—
349C2×M16—53no—
350C2×M16—53no—
351C4×Q8—301no—
352C4×D4—303no—
353#117—153no—
354C2×C42—153no—
355#104—151no—
356#104—153no—
357#104—153no—
358#104—153no—
359#104—156no—
360C2×C3×C5—23no—
361C2×C3×C5C22×C22⋊C411yescharacteristic
362D15—43no—
363C3×D5—21no—
364C3×D5—23no—
365C5×D3—41no—
366C23×C3—53no—
367C22×D3—103no—
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×C4C22×AGL(1,𝔽5)13yes—
380#67—23no—
381#67—103no—
382#67—106no—
383#67—101no—
384#67—103no—
385C2×Dic3—23no—
386C2×Dic3—103no—
387C2×Dic3—101no—
388C2×Dic3—103no—
389C2×Dic3—206no—
390D12—53no—
391D12—103no—
392D12—103no—
393D12C22×AGL(1,𝔽5)11yescharacteristic
394C4×D3—23no—
395C4×D3—103no—
396C4×D3—106no—
397C4×D3—101no—
398C4×D3—103no—
399Dic6—23no—
400Dic6—51no—
401Dic6—103no—
402Dic6C22×AGL(1,𝔽5)13yes—
403C3×C8—101no—
404C3×C8—103no—
405C22×C5—61no—
406C22×C5C2×(C2×D3)⋊C413yes—
407C2×D5—43no—
408C2×D5—63no—
409C2×D5—243no—
410C2×D5—243no—
411C2×D5C2×(C2×D3)⋊C413yes—
412C2×D5C22×D1211yescharacteristic
413AGL(1,𝔽5)—62no—
414AGL(1,𝔽5)—123no—
415C4×C5—63no—
416C4×C5C2×(C2×D3)⋊C413yes—
417C4×C5C22×C4×D311yescharacteristic
418Dic5—61no—
419Dic5C2×(C2×D3)⋊C413yes—
420Dic5C22×#6711yescharacteristic
421#41—33no—
422#41—51no—
423#41—53no—
424#41—53no—
425#41—63no—
426#41—151no—
427#41—153no—
428#41—156no—
429#41—153no—
430#41—303no—
431#41—303no—
432#41—303no—
433#41—303no—
434#41—306no—
435#41—303no—
436#41—303no—
437#41—301no—
438#41D3×AGL(1,𝔽5)11yescharacteristic
439C2×Q8—33no—
440C2×Q8—51no—
441C2×Q8—61no—
442C2×Q8—153no—
443C2×Q8—303no—
444C2×D4—53no—
445C2×D4—153no—
446C2×D4—153no—
447C2×D4—303no—
448C2×D4—303no—
449C2×D4—306no—
450C2×D4—303no—
451C2×D4—303no—
452C22×C4—53no—
453C22×C4—303no—
454C22×C4—303no—
455C22×C4—303no—
456M16—51no—
457M16—53no—
458M16—103no—
459M16—103no—
460C2×C8—103no—
461C2×C8—103no—
462C4⋊C4—303no—
463C22⋊C4—303no—
464C42—151no—
465C42—153no—
466C42—303no—
467C3×C5#290111yescharacteristic, complemented, Hall
468C22×C3—53no—
469C22×C3—51no—
470C22×C3—203no—
471C22×C3C2×C22⋊AGL(1,𝔽5)13yes—
472C2×D3—21no—
473C2×D3—103no—
474C2×D3—403no—
475C2×D3—403no—
476C3×C4—53no—
477C3×C4—51no—
478C3×C4C2×C22⋊AGL(1,𝔽5)13yes—
479C3×C4C23×AGL(1,𝔽5)11yescharacteristic
480Dic3—23no—
481Dic3—102no—
482Dic3—101no—
483Dic3—203no—
484C2×C5—23no—
485C2×C5—121no—
486C2×C5C22×(C2×D3)⋊C411yescharacteristic
487D5—21no—
488D5—23no—
489D5—123no—
490C23—53no—
491C23—303no—
492C23—303no—
493Q8—33no—
494Q8—53no—
495Q8—63no—
496Q8—151no—
497Q8—303no—
498Q8(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
499D4—31no—
500D4—56no—
501D4—53no—
502D4—63no—
503D4—153no—
504D4—303no—
505D4—303no—
506D4—306no—
507D4—301no—
508D4—303no—
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—63no—
518C2×C4—303no—
519C2×C4—306no—
520C2×C4—301no—
521C2×C4—303no—
522C2×C4—301no—
523C2×C4—303no—
524C2×C4—303no—
525C2×C4—606no—
526C2×C4C2×D3×AGL(1,𝔽5)13yes—
527C8—101no—
528C8—103no—
529C2×C3—23no—
530C2×C3—101no—
531C2×C3—103no—
532C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
533D3—41no—
534D3—203no—
535C5#8381711yescharacteristic, complemented, Sylow
536C22—51no—
537C22—53no—
538C22—61no—
539C22—203no—
540C22—303no—
541C22—1203no—
542C22—1203no—
543C22C2×(C2×D3)⋊AGL(1,𝔽5)13yes—
544C4—51no—
545C4—53no—
546C4—63no—
547C4—302no—
548C4—301no—
549C4—603no—
550C4C2×(C2×D3)⋊AGL(1,𝔽5)13yes—
551C4C22×D3×AGL(1,𝔽5)11yescharacteristic
552C3#1061756211yescharacteristic, complemented, Sylow
553C2—23no—
554C2—101no—
555C2—103no—
556C2—121no—
557C2—603no—
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#11757060C2
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. #41×ΓL(1,𝔽16)C2semidirect
3. C3×#66264C2semidirect
4. #11751723C2semidirect
5. C2×#91157C2semidirect
6. #11751717C2semidirect
7. D5⋊#825C2semidirect
9. #92100C4semidirect
10. C5×#1010C4semidirect
16. #92019C4semidirect
19. #92000C4semidirect
22. #91989C22semidirect
23. C2×C4×ΓL(1,𝔽16)C22semidirect
26. C3×D5⋊M16C22semidirect
27. C2×C3×#700C22semidirect
28. #91157C22semidirect
29. #91157C22semidirect
30. #91157C22semidirect
31. #91155C22semidirect
32. #91155C22semidirect
36. #66264D3semidirect
45. #6817C2×C4semidirect
46. C2×D60C2×C4semidirect
49. C5×#235C2×C4semidirect
50. C2×C5×Dic6C2×C4semidirect
64. D5×D12C2×C4semidirect
68. #6773C2×C4semidirect
75. D5×Dic6C2×C4semidirect
111. D5⋊M16C2×D3semidirect
112. C2×#700C2×D3semidirect
160. #1010AGL(1,𝔽5)semidirect
279. #235C2×AGL(1,𝔽5)semidirect
285. C2×Dic6C2×AGL(1,𝔽5)semidirect
467. C3×C5#2901semidirect
535. C5#83817semidirect
552. C3#10617562semidirect

Maximal subgroups

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

subgroupindex
#117570602
#41×ΓL(1,𝔽16)2
C3×#662642
#117517232
C2×#911572
#117517172
D5⋊#8252
#106175623
#838175