Finite Group Databasebeta

← #14412918

Subgroups — #14412918

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
1C3×#66285C211yesmaximal, characteristic
2#41×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3#11756993C211yesmaximal, characteristic, complemented
4#11752454C213yesmaximal, complemented
5D5⋊#837C213yesmaximal, complemented
6#11752428C213yesmaximal, complemented
7C2×#91237C213yesmaximal, complemented
8#10617560—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C411yescharacteristic, complemented
10C3×#4150—21no—
11C3×#4150C413yescomplemented
12C3×#4150C411yescharacteristic, complemented
13C3×D5×#41—21no—
14C3×D5×#41C2213yes—
15C3×D5×#41C2211yescharacteristic
16C3×#4144—23no—
17C3×#4144C413yescomplemented
18C2×C3×#711—23no—
19C2×C3×#711C2213yes—
20Q8×ΓL(1,𝔽16)—21no—
21#91992—21no—
22D4×ΓL(1,𝔽16)—23no—
23#91990—23no—
24#91989C2213yescomplemented
25C2×C4×ΓL(1,𝔽16)C2213yescomplemented
26D5⋊#208C2213yescomplemented
27C2×#6758C2213yescomplemented
28#91239C2211yescharacteristic, complemented
29#91239C2213yescomplemented
30#91237C2213yescomplemented
31#91237C2213yescomplemented
32#91237C2216yescomplemented
33#84120—51nomaximal, Hall
34#66285D311yescharacteristic
35#41×AGL(1,𝔽5)—31no—
36#66264—31no—
37#65795—33no—
38D5⋊#104—33no—
39#65769—33no—
40C2×#4008—33no—
41C22×ΓL(1,𝔽16)—23no—
42C3×C5×#41—21no—
43C3×C5×#41C2×C411yescharacteristic
44C3×C5×#41C2×C413yescomplemented
45C2×C3×C5×D4—23no—
46C2×C3×C5×D4C2×C413yescomplemented
47C2×C3×C22⋊D5—23no—
48C2×C3×C22⋊D5—26no—
49C22×C3×Dic5—23no—
50C3×#713D411yescharacteristic
51C3×#713C2311yescharacteristic
52C3×Q8×D5—21no—
53C3×Q8×D5C2×C413yescomplemented
54C3×Q8×D5D411yescharacteristic
55C3×#711—23no—
56C3×#711—26no—
57C3×#711—23no—
58C3×#711—23no—
59C3×#711—23no—
60C3×#711—23no—
61C3×#711C2×C416yescomplemented
62C3×#711D413yes—
63C3×#711C2313yes—
64C3×D4×D5—23no—
65C3×D4×D5—23no—
66C3×D4×D5C2×C413yescomplemented
67C3×D4×D5D413yes—
68C3×#709—23no—
69C3×#709—23no—
70C3×#709C2×C413yes—
71C3×#709C2×C413yescomplemented
72C2×C3×C4×D5—23no—
73C2×C3×C4×D5C2313yes—
74C2×C3×Dic10—23no—
75C2×C3×Dic10C2×C413yescomplemented
76#6763—23no—
77#6762—23no—
78C2×C3⋊#170—23no—
79C4⋊ΓL(1,𝔽16)—23no—
80C4×ΓL(1,𝔽16)—23no—
81C4×ΓL(1,𝔽16)C2311yescharacteristic
82C4×ΓL(1,𝔽16)C2313yes—
83#6758—23no—
84#6758C2313yes—
85#6758C2311yescharacteristic
86D5⋊(C3⋊C8)—23no—
87C3×#585—51no—
88Dic3×#41—51no—
89#5819—51no—
90#5236—53no—
91#5236—53no—
92C2×#837—53no—
93C2×#837—53no—
94C5×H(5,𝔽2)Dic311yescharacteristic, complemented
95#4150—21no—
96#4150Dic313yescomplemented
97#4150Dic311yescharacteristic, complemented
98D5×#41—21no—
99D5×#41C2×D313yes—
100D5×#41C2×D311yescharacteristic
101#4144—23no—
102#4144Dic313yescomplemented
103C2×#711—23no—
104C2×#711C2×D313yes—
105Q8×AGL(1,𝔽5)—61no—
106#4133—61no—
107D4×AGL(1,𝔽5)—63no—
108#4131—63no—
109#4130—33no—
110C2×C4×AGL(1,𝔽5)—33no—
111D5⋊M16—33no—
112C2×#700—33no—
113#4010—31no—
114#4010—33no—
115#4008—33no—
116#4008—33no—
117#4008—36no—
118#2901—151noSylow
119C23×C3×C5—23no—
120C22×C3×D5—23no—
121C22×C3×D5C2×D413yes—
122C2×ΓL(1,𝔽16)—21no—
123C2×ΓL(1,𝔽16)—23no—
124C2×ΓL(1,𝔽16)—46no—
125C3×C5×Q8C22⋊C411yescharacteristic
126C3×C5×Q8C22×C411yescharacteristic
127C3×C5×D4—23no—
128C3×C5×D4—23no—
129C3×C5×D4C22⋊C413yes—
130C3×C5×D4C22×C413yes—
131C2×C3×C4×C5—23no—
132C2×C3×C4×C5C22×C413yes—
133C3×C22⋊D5—23no—
134C3×C22⋊D5—26no—
135C3×C22⋊D5—23no—
136C3×C22⋊D5—23no—
137C3×C22⋊D5C22⋊C416yes—
138C2×C3×Dic5—23no—
139C2×C3×Dic5—26no—
140C2×C3×Dic5—23no—
141C2×C3×Dic5C2×D413yes—
142C3×D20—21no—
143C3×D20C22⋊C413yes—
144C3×D20C22×C411yescharacteristic
145C3×C4×D5—23no—
146C3×C4×D5—23no—
147C3×C4×D5—21no—
148C3×C4×D5C22×C413yes—
149C3×C4×D5C2×D413yes—
150C3×C4×D5C2×D413yes—
151C3×C4×D5C2411yescharacteristic
152C3×Dic10—23no—
153C3×Dic10—23no—
154C3×Dic10C22⋊C413yes—
155C3×Dic10C22×C413yes—
156C3⋊#170—21no—
157C3⋊#170—23no—
158C3×2−1+4—51no—
159C3×2−1+4—53no—
160C3×2−1+4—101no—
161C3×H(5,𝔽2)—53no—
162C3×H(5,𝔽2)—103no—
163C3×H(5,𝔽2)AGL(1,𝔽5)11yescharacteristic, complemented
164C2×C3×#41—51no—
165C2×C3×#41—53no—
166C2×C3×#41—53no—
167C2×C3×#41—103no—
168C2×C3×#41—101no—
169#948—103no—
170#948—101no—
171Q8×Dic3—101no—
172D4×Dic3—103no—
173#926—53no—
174C2×C4×Dic3—53no—
175C2×#208—53no—
176C2×#208—53no—
177#837—53no—
178#837—53no—
179#837—56no—
180#837—51no—
181#837—53no—
182C22×AGL(1,𝔽5)—63no—
183C5×#41—21no—
184C5×#41C2×Dic311yescharacteristic
185C5×#41C2×Dic313yescomplemented
186C2×C5×D4—23no—
187C2×C5×D4C2×Dic313yescomplemented
188C2×C22⋊D5—23no—
189C2×C22⋊D5—26no—
190C22×Dic5—23no—
191#713#6711yescharacteristic
192#713C22×D311yescharacteristic
193Q8×D5—21no—
194Q8×D5C2×Dic313yescomplemented
195Q8×D5#6711yescharacteristic
196#711—23no—
197#711—26no—
198#711—23no—
199#711—23no—
200#711—23no—
201#711—23no—
202#711C2×Dic316yescomplemented
203#711#6713yes—
204#711C22×D313yes—
205D4×D5—23no—
206D4×D5—23no—
207D4×D5C2×Dic313yescomplemented
208D4×D5#6713yes—
209#709—23no—
210#709—23no—
211#709C2×Dic313yes—
212#709C2×Dic313yescomplemented
213C2×C4×D5—23no—
214C2×C4×D5C22×D313yes—
215C2×Dic10—23no—
216C2×Dic10C2×Dic313yescomplemented
217C22⋊AGL(1,𝔽5)—63no—
218#704—63no—
219C2×#170—63no—
220C4⋊AGL(1,𝔽5)—63no—
221C4×AGL(1,𝔽5)—31no—
222C4×AGL(1,𝔽5)—33no—
223C4×AGL(1,𝔽5)—63no—
224#700—33no—
225#700—31no—
226#700—63no—
227D5⋊C8—63no—
228#585—51no—
229#568—151no—
230C4×#41—151no—
231#421—153no—
232#421—153no—
233C2×#104—153no—
234C2×#104—153no—
235C22×C3×C5—23no—
236C22×C3×C5—83no—
237C22×C3×C5C2×C22⋊C413yes—
238C2×C3×D5—21no—
239C2×C3×D5—43no—
240C2×C3×D5—83no—
241C2×C3×D5C2×C22⋊C413yes—
242C2×C3×D5C22×D411yescharacteristic
243ΓL(1,𝔽16)—22no—
244ΓL(1,𝔽16)—43no—
245C3×C4×C5—21no—
246C3×C4×C5C2×C22⋊C413yes—
247C3×C4×C5C23×C411yescharacteristic
248C3×Dic5—23no—
249C3×Dic5C2×C22⋊C413yes—
250C3×Dic5C22×D411yescharacteristic
251C3×#41—21no—
252C3×#41—51no—
253C3×#41—53no—
254C3×#41—53no—
255C3×#41—51no—
256C3×#41—53no—
257C3×#41—56no—
258C3×#41—53no—
259C3×#41—103no—
260C3×#41—103no—
261C3×#41—103no—
262C3×#41—103no—
263C3×#41—103no—
264C3×#41—106no—
265C3×#41—103no—
266C3×#41—103no—
267C3×#41C2×AGL(1,𝔽5)11yescharacteristic
268C3×#41C2×AGL(1,𝔽5)13yescomplemented
269C2×C3×Q8—51no—
270C2×C3×Q8—53no—
271C2×C3×Q8—53no—
272C2×C3×Q8—103no—
273C2×C3×Q8—101no—
274C2×C3×D4—23no—
275C2×C3×D4—53no—
276C2×C3×D4—53no—
277C2×C3×D4—103no—
278C2×C3×D4—103no—
279C2×C3×D4—106no—
280C2×C3×D4—103no—
281C2×C3×D4C2×AGL(1,𝔽5)13yescomplemented
282C22×C3×C4—53no—
283C22×C3×C4—103no—
284C22×C3×C4—103no—
285C22×Dic3—103no—
286#217—103no—
287C4⋊Dic3—103no—
288C4×Dic3—51no—
289C4×Dic3—53no—
290C4×Dic3—103no—
291#208—53no—
292#208—51no—
293#208—103no—
294#208—103no—
295C2×C3⋊C8—103no—
296C2×C3⋊C8—103no—
297C23×C5—23no—
298C22×D5—23no—
299C22×D5C2×#6713yes—
300C2×AGL(1,𝔽5)—61no—
301C2×AGL(1,𝔽5)—63no—
302C2×AGL(1,𝔽5)—126no—
303C5×Q8#21711yescharacteristic
304C5×Q8C22×Dic311yescharacteristic
305C5×D4—23no—
306C5×D4—23no—
307C5×D4#21713yes—
308C5×D4C22×Dic313yes—
309C2×C4×C5—23no—
310C2×C4×C5C22×Dic313yes—
311C22⋊D5—23no—
312C22⋊D5—26no—
313C22⋊D5—23no—
314C22⋊D5—23no—
315C22⋊D5#21716yes—
316C2×Dic5—23no—
317C2×Dic5—26no—
318C2×Dic5—23no—
319C2×Dic5C2×#6713yes—
320D20—21no—
321D20#21713yes—
322D20C22×Dic311yescharacteristic
323C4×D5—23no—
324C4×D5—23no—
325C4×D5—21no—
326C4×D5C22×Dic313yes—
327C4×D5C2×#6713yes—
328C4×D5C2×#6713yes—
329C4×D5C23×D311yescharacteristic
330Dic10—23no—
331Dic10—23no—
332Dic10#21713yes—
333Dic10C22×Dic313yes—
334#170—61no—
335#170—63no—
3362−1+4—51no—
3372−1+4—53no—
3382−1+4—101no—
339H(5,𝔽2)—53no—
340H(5,𝔽2)—103no—
341H(5,𝔽2)ΓL(1,𝔽16)11yescharacteristic, complemented
342C2×#41—51no—
343C2×#41—53no—
344C2×#41—53no—
345C2×#41—103no—
346C2×#41—101no—
347#131—303no—
348#131—301no—
349C2×M16—153no—
350C2×M16—153no—
351C4×Q8—301no—
352C4×D4—303no—
353#117—153no—
354C2×C42—153no—
355#104—153no—
356#104—151no—
357#104—153no—
358#104—156no—
359#104—153no—
360C2×C3×C5—23no—
361C2×C3×C5—43no—
362C2×C3×C5C22×C22⋊C411yescharacteristic
363C3×D5—21no—
364C3×D5—23no—
365C3×D5—41no—
366C23×C3—23no—
367C23×C3—53no—
368C23×C3—103no—
369C3×Q8—53no—
370C3×Q8—53no—
371C3×Q8—103no—
372C3×Q8—103no—
373C3×Q8C22⋊AGL(1,𝔽5)11yescharacteristic
374C3×Q8C22×AGL(1,𝔽5)11yescharacteristic
375C3×D4—23no—
376C3×D4—23no—
377C3×D4—56no—
378C3×D4—53no—
379C3×D4—51no—
380C3×D4—101no—
381C3×D4—103no—
382C3×D4—106no—
383C3×D4—103no—
384C3×D4—103no—
385C3×D4C22⋊AGL(1,𝔽5)13yes—
386C3×D4C22×AGL(1,𝔽5)13yes—
387C2×C3×C4—23no—
388C2×C3×C4—53no—
389C2×C3×C4—53no—
390C2×C3×C4—53no—
391C2×C3×C4—53no—
392C2×C3×C4—51no—
393C2×C3×C4—101no—
394C2×C3×C4—103no—
395C2×C3×C4—103no—
396C2×C3×C4—106no—
397C2×C3×C4—103no—
398C2×C3×C4—103no—
399C2×C3×C4C22×AGL(1,𝔽5)13yes—
400C2×Dic3—101no—
401C2×Dic3—103no—
402C2×Dic3—206no—
403C3⋊C8—101no—
404C3⋊C8—103no—
405C22×C5—23no—
406C22×C5—83no—
407C22×C5C2×#21713yes—
408C2×D5—21no—
409C2×D5—43no—
410C2×D5—83no—
411C2×D5C2×#21713yes—
412C2×D5C22×#6711yescharacteristic
413AGL(1,𝔽5)—62no—
414AGL(1,𝔽5)—123no—
415C4×C5—21no—
416C4×C5C2×#21713yes—
417C4×C5C23×Dic311yescharacteristic
418Dic5—23no—
419Dic5C2×#21713yes—
420Dic5C22×#6711yescharacteristic
421#41—21no—
422#41—51no—
423#41—53no—
424#41—53no—
425#41—51no—
426#41—53no—
427#41—56no—
428#41—53no—
429#41—103no—
430#41—103no—
431#41—103no—
432#41—103no—
433#41—103no—
434#41—106no—
435#41—103no—
436#41—103no—
437#41C2×ΓL(1,𝔽16)11yescharacteristic
438#41C2×ΓL(1,𝔽16)13yescomplemented
439C2×Q8—51no—
440C2×Q8—53no—
441C2×Q8—53no—
442C2×Q8—103no—
443C2×Q8—101no—
444C2×D4—23no—
445C2×D4—53no—
446C2×D4—53no—
447C2×D4—103no—
448C2×D4—103no—
449C2×D4—106no—
450C2×D4—103no—
451C2×D4C2×ΓL(1,𝔽16)13yescomplemented
452C22×C4—53no—
453C22×C4—103no—
454C22×C4—103no—
455C22×C4—303no—
456M16—153no—
457M16—151no—
458M16—303no—
459M16—303no—
460C2×C8—303no—
461C2×C8—303no—
462C4⋊C4—303no—
463C22⋊C4—303no—
464C42—151no—
465C42—153no—
466C42—303no—
467C3×C5#290111yescharacteristic, complemented, Hall
468C22×C3—23no—
469C22×C3—53no—
470C22×C3—51no—
471C22×C3—83no—
472C22×C3—101no—
473C22×C3—203no—
474C22×C3—403no—
475C22×C3C2×C22⋊AGL(1,𝔽5)13yes—
476C3×C4—21no—
477C3×C4—53no—
478C3×C4—51no—
479C3×C4—103no—
480C3×C4C2×C22⋊AGL(1,𝔽5)13yes—
481C3×C4C23×AGL(1,𝔽5)11yescharacteristic
482Dic3—102no—
483Dic3—203no—
484C2×C5—23no—
485C2×C5—43no—
486C2×C5C22×#21711yescharacteristic
487D5—21no—
488D5—23no—
489D5—41no—
490C23—23no—
491C23—53no—
492C23—103no—
493Q8—53no—
494Q8—53no—
495Q8—103no—
496Q8—103no—
497Q8#676311yescharacteristic
498Q8C22×ΓL(1,𝔽16)11yescharacteristic
499D4—23no—
500D4—23no—
501D4—56no—
502D4—53no—
503D4—51no—
504D4—101no—
505D4—103no—
506D4—103no—
507D4—106no—
508D4—103no—
509D4#676313yes—
510D4C22×ΓL(1,𝔽16)13yes—
511C2×C4—23no—
512C2×C4—53no—
513C2×C4—53no—
514C2×C4—53no—
515C2×C4—51no—
516C2×C4—53no—
517C2×C4—101no—
518C2×C4—103no—
519C2×C4—103no—
520C2×C4—106no—
521C2×C4—103no—
522C2×C4—103no—
523C2×C4—301no—
524C2×C4—303no—
525C2×C4—606no—
526C2×C4C22×ΓL(1,𝔽16)13yes—
527C8—301no—
528C8—303no—
529C2×C3—23no—
530C2×C3—43no—
531C2×C3—101no—
532C2×C3—103no—
533C2×C3—201no—
534C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
535C5#8412011yescharacteristic, complemented, Sylow
536C22—23no—
537C22—51no—
538C22—53no—
539C22—83no—
540C22—101no—
541C22—203no—
542C22—403no—
543C22C2×#676313yes—
544C4—21no—
545C4—51no—
546C4—53no—
547C4—103no—
548C4—302no—
549C4—603no—
550C4C2×#676313yes—
551C4C23×ΓL(1,𝔽16)11yescharacteristic
552C3#1061756011yescharacteristic, complemented, Sylow
553C2—23no—
554C2—43no—
555C2—101no—
556C2—103no—
557C2—201no—
558C2C22×#676311yescharacteristic
subgroupG/H
centerC2C22×#6763
commutatorC3×C4×C5C23×C4
FrattiniC4C23×ΓL(1,𝔽16)
FittingC3×C5×H(5,𝔽2)C4
socleC2×C3×C5C22×C22⋊C4
prime radicalC3×#66285C2
cosocleC3×C4×D5C24
hypercenterH(5,𝔽2)ΓL(1,𝔽16)
hypocenterC3×C5#2901
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #41×ΓL(1,𝔽16)C2semidirect
3. #11756993C2semidirect
4. #11752454C2semidirect
5. D5⋊#837C2semidirect
6. #11752428C2semidirect
7. C2×#91237C2semidirect
9. C3×C5×H(5,𝔽2)C4semidirect
11. C3×#4150C4semidirect
12. C3×#4150C4semidirect
17. C3×#4144C4semidirect
24. #91989C22semidirect
25. C2×C4×ΓL(1,𝔽16)C22semidirect
26. D5⋊#208C22semidirect
27. C2×#6758C22semidirect
28. #91239C22semidirect
29. #91239C22semidirect
30. #91237C22semidirect
31. #91237C22semidirect
32. #91237C22semidirect
44. C3×C5×#41C2×C4semidirect
46. C2×C3×C5×D4C2×C4semidirect
53. C3×Q8×D5C2×C4semidirect
61. C3×#711C2×C4semidirect
66. C3×D4×D5C2×C4semidirect
71. C3×#709C2×C4semidirect
75. C2×C3×Dic10C2×C4semidirect
94. C5×H(5,𝔽2)Dic3semidirect
96. #4150Dic3semidirect
97. #4150Dic3semidirect
102. #4144Dic3semidirect
163. C3×H(5,𝔽2)AGL(1,𝔽5)semidirect
185. C5×#41C2×Dic3semidirect
187. C2×C5×D4C2×Dic3semidirect
194. Q8×D5C2×Dic3semidirect
202. #711C2×Dic3semidirect
207. D4×D5C2×Dic3semidirect
212. #709C2×Dic3semidirect
216. C2×Dic10C2×Dic3semidirect
268. C3×#41C2×AGL(1,𝔽5)semidirect
281. C2×C3×D4C2×AGL(1,𝔽5)semidirect
341. H(5,𝔽2)ΓL(1,𝔽16)semidirect
438. #41C2×ΓL(1,𝔽16)semidirect
451. C2×D4C2×ΓL(1,𝔽16)semidirect
467. C3×C5#2901semidirect
535. C5#84120semidirect
552. C3#10617560semidirect

Maximal subgroups

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

subgroupindex
C3×#662852
#41×ΓL(1,𝔽16)2
#117569932
#117524542
D5⋊#8372
#117524282
C2×#912372
#106175603
#841205