Finite Group Databasebeta

← #14412545

Subgroups — #14412545

Order 1920.

1544 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6821C211yesmaximal, characteristic
2C2×M16×D15C211yesmaximal, characteristic, complemented
3D15×#117C211yesmaximal, characteristic, complemented
4C3×C5×#419C211yesmaximal, characteristic, complemented
5#11751326C211yesmaximal, characteristic, complemented
6#11751232C211yesmaximal, characteristic, complemented
7#11751102C211yesmaximal, characteristic, complemented
8#11750905C214yesmaximal, complemented
9#11750904C214yesmaximal, complemented
10#10610687—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×#6821—24no—
13C2×#6821—21no—
14C2×#6821—21no—
15C2×#6821C418yescomplemented
16C2×#6821C2212yes—
17C2×#6821C2212yes—
18C22×D60—21no—
19C22×D60C2211yescharacteristic
20C22×C4×D15—21no—
21C22×C4×D15C2211yescharacteristic
22C2×C3×C5×M16C2211yescharacteristic
23C3×C5×#117C2211yescharacteristic
24#91815C2211yescharacteristic
25C2×#6710C2211yescharacteristic
26M16×D15—24no—
27C2×#6705C2212yescomplemented
28C2×C8×D15C2212yescomplemented
29#91781C2212yescomplemented
30C4⋊C4×D15C2212yescomplemented
31C22⋊C4×D15—21no—
32#91766—21no—
33#91761—21no—
34C42×D15—21no—
35C3×C5×#103C2214yescomplemented
36#91119C2214yescomplemented
37#91105C2214yescomplemented
38#91093C2214yescomplemented
39#78956—51nomaximal, Hall
40C22×#713—31no—
41C2×D5×M16—31no—
42D5×#117—31no—
43C5×#419D311yescharacteristic, complemented
44#65514—31no—
45#65420—31no—
46#65290—31no—
47#65093—34no—
48#65092—34no—
49C23×D15—21no—
50C23×D15D411yescharacteristic
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8C2×C414yescomplemented
54C2×C3×C5×Q8C2312yes—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15D411yescharacteristic
58#6821—28no—
59#6821—28no—
60#6821—24no—
61#6821—24no—
62#6821C2×C418yes—
63#6821C2×C418yescomplemented
64C2×D60—28no—
65C2×D60—22no—
66C2×D60—22no—
67C2×D60—21no—
68C2×D60—22no—
69C2×D60C2×C414yescomplemented
70C2×D60C2312yes—
71C2×C4×D15—28no—
72C2×C4×D15—22no—
73C2×C4×D15—22no—
74C2×C4×D15—22no—
75C2×C4×D15C2×C414yes—
76C2×C4×D15D411yescharacteristic
77C2×C4×D15D411yescharacteristic
78C2×C4×D15D414yes—
79C2×C4×D15C2311yescharacteristic
80C2×C4×D15C2311yescharacteristic
81C3×C5×M16—22no—
82C2×C3×C5×C8C2312yes—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C42×C5—21no—
86#6719—21no—
87#6714—22no—
88C4⋊Dic15C2312yes—
89#6712—22no—
90C4×Dic15—21no—
91C4×Dic15—22no—
92#6710—22no—
93C2×C3⋊#168C2312yes—
94#6705—24no—
95C8×D15—24no—
96C22×#239—51no—
97C2×D3×M16—51no—
98D3×#117—51no—
99C3×#419D511yescharacteristic, complemented
100#5225—51no—
101#5131—51no—
102#5001—51no—
103#4804—54no—
104#4803—54no—
105C22×C5×Q8C2×D311yescharacteristic
106C2×#713—38no—
107C2×#713—32no—
108C2×#713—32no—
109C2×#713—61no—
110C2×#713—61no—
111C2×#713—64no—
112C22×D20—31no—
113C22×D20—61no—
114C22×C4×D5—31no—
115C22×C4×D5—61no—
116C2×C5×M16C2×D311yescharacteristic
117C5×#117C2×D311yescharacteristic
118#4072—31no—
119C2×#681—31no—
120D5×M16—64no—
121C2×#676—32no—
122C2×C8×D5—32no—
123#4038—32no—
124D5×C4⋊C4—32no—
125D5×C22⋊C4—61no—
126#4023—61no—
127#4018—61no—
128C42×D5—61no—
129C5×#103C2×D314yescomplemented
130#3967—34no—
131#3953—34no—
132#3941—34no—
133#2896—151noSylow
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—24no—
136C22×D15—22no—
137C22×D15—42no—
138C22×D15—44no—
139C22×D15C22⋊C414yes—
140C22×D15C2×D411yescharacteristic
141C22×D15C2×D411yescharacteristic
142C3×C5×Q8—24no—
143C3×C5×Q8—22no—
144C3×C5×Q8C22×C414yes—
145C2×C3×C4×C5—24no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5C22×C412yes—
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2411yescharacteristic
152C2×Dic15—22no—
153C2×Dic15C22⋊C414yes—
154C2×Dic15C2×D411yescharacteristic
155C2×Dic15C2×D411yescharacteristic
156D60—28no—
157D60—28no—
158D60—24no—
159D60—22no—
160D60C22×C414yes—
161C4×D15—28no—
162C4×D15—28no—
163C4×D15—44no—
164C4×D15C22⋊C414yes—
165C4×D15C22⋊C414yes—
166C4×D15C22×C412yes—
167C4×D15C22×C412yes—
168C4×D15C2×D412yes—
169C4×D15C2×D412yes—
170C3×C5×C8—22no—
171C3⋊#168—22no—
172C22×C3×Q8C2×D511yescharacteristic
173C2×#239—58no—
174C2×#239—52no—
175C2×#239—52no—
176C2×#239—101no—
177C2×#239—101no—
178C2×#239—104no—
179C22×D12—51no—
180C22×D12—101no—
181C22×C4×D3—51no—
182C22×C4×D3—101no—
183C2×C3×M16C2×D511yescharacteristic
184C3×#117C2×D511yescharacteristic
185#926—51no—
186C2×#208—51no—
187D3×M16—104no—
188C2×ΓL(1,𝔽25)—52no—
189C2×D3×C8—52no—
190#892—52no—
191D3×C4⋊C4—52no—
192D3×C22⋊C4—101no—
193#877—101no—
194#872—101no—
195C42×D3—101no—
196C3×#103C2×D514yescomplemented
197#835—54no—
198#821—54no—
199#809—54no—
200C23×D5—31no—
201C23×D5—61no—
202C2×C5×Q8—22no—
203C2×C5×Q8—21no—
204C2×C5×Q8C4×D314yescomplemented
205C2×C5×Q8C22×D312yes—
206C22×C4×C5—21no—
207C22×C4×C5C22×D311yescharacteristic
208C22×Dic5—31no—
209#713—38no—
210#713—38no—
211#713—68no—
212#713—68no—
213#713—64no—
214#713—64no—
215C2×D20—34no—
216C2×D20—32no—
217C2×D20—68no—
218C2×D20—62no—
219C2×D20—62no—
220C2×D20—61no—
221C2×D20—62no—
222C2×C4×D5—34no—
223C2×C4×D5—31no—
224C2×C4×D5—31no—
225C2×C4×D5—34no—
226C2×C4×D5—31no—
227C2×C4×D5—31no—
228C2×C4×D5—68no—
229C2×C4×D5—62no—
230C2×C4×D5—62no—
231C2×C4×D5—62no—
232C5×M16—22no—
233C2×C5×C8C22×D312yes—
234C5×C4⋊C4C22×D312yes—
235C5×C22⋊C4—21no—
236C42×C5—21no—
237C22⋊Dic5—61no—
238#685—62no—
239C4⋊Dic5—32no—
240#683—62no—
241C4×Dic5—62no—
242C4×Dic5—61no—
243#681—62no—
244C2×#168—32no—
245#676—64no—
246C8×D5—64no—
247C22×#41—151no—
248C22×M16—151no—
249C2×#117—151no—
250#419—151no—
251#419D1511yescharacteristic, complemented
252#418—151no—
253#418—151no—
254#417—154no—
255#416—154no—
256C22×C3×C5—22no—
257C22×C3×C5C2×C22⋊C412yes—
258C22×C3×C5C22×D411yescharacteristic
259C2×D15—22no—
260C2×D15—22no—
261C2×D15—24no—
262C2×D15—44no—
263C2×D15—48no—
264C2×D15—44no—
265C2×D15C2×C22⋊C412yes—
266C2×D15C2×C22⋊C412yes—
267C3×C4×C5—24no—
268C3×C4×C5—42no—
269C3×C4×C5C2×C22⋊C412yes—
270C3×C4×C5C23×C411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272Dic15—42no—
273Dic15C2×C22⋊C412yes—
274Dic15C2×C22⋊C412yes—
275C23×D3—51no—
276C23×D3—101no—
277C2×C3×Q8—21no—
278C2×C3×Q8—22no—
279C2×C3×Q8C4×D514yescomplemented
280C2×C3×Q8C22×D512yes—
281C22×C3×C4—21no—
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—51no—
284#239—58no—
285#239—58no—
286#239—104no—
287#239—104no—
288#239—108no—
289#239—108no—
290C2×D12—54no—
291C2×D12—52no—
292C2×D12—102no—
293C2×D12—102no—
294C2×D12—108no—
295C2×D12—102no—
296C2×D12—101no—
297C2×C4×D3—54no—
298C2×C4×D3—51no—
299C2×C4×D3—51no—
300C2×C4×D3—54no—
301C2×C4×D3—51no—
302C2×C4×D3—51no—
303C2×C4×D3—102no—
304C2×C4×D3—102no—
305C2×C4×D3—108no—
306C2×C4×D3—102no—
307C3×M16—22no—
308C2×C3×C8C22×D512yes—
309C3×C4⋊C4C22×D512yes—
310C3×C22⋊C4—21no—
311C3×C42—21no—
312#217—101no—
313(C2×D3)⋊C4—102no—
314C4⋊Dic3—52no—
315Dic3⋊C4—102no—
316C4×Dic3—102no—
317C4×Dic3—101no—
318#208—102no—
319C2×C3⋊C8—52no—
320ΓL(1,𝔽25)—104no—
321D3×C8—104no—
322C23×C5D3×D411yescharacteristic
323C22×D5—31no—
324C22×D5—31no—
325C22×D5—34no—
326C22×D5—64no—
327C22×D5—62no—
328C22×D5—122no—
329C22×D5—124no—
330C5×Q8—24no—
331C5×Q8—22no—
332C5×Q8C2×C4×D314yes—
333C2×C4×C5—24no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5C2×C4×D312yes—
337C2×C4×C5D3×D411yescharacteristic
338C2×C4×C5D3×D412yes—
339C2×C4×C5C23×D311yescharacteristic
340C2×Dic5—34no—
341C2×Dic5—31no—
342C2×Dic5—31no—
343C2×Dic5—62no—
344D20—34no—
345D20—68no—
346D20—68no—
347D20—64no—
348D20—62no—
349C4×D5—34no—
350C4×D5—34no—
351C4×D5—32no—
352C4×D5—32no—
353C4×D5—32no—
354C4×D5—32no—
355C4×D5—68no—
356C4×D5—68no—
357C4×D5—124no—
358C5×C8—22no—
359#168—62no—
360C2×#41—158no—
361C2×#41—152no—
362C2×#41—152no—
363C2×#41—301no—
364C2×#41—301no—
365C2×#41—304no—
366C22×Q8C2×D1511yescharacteristic
367C22×D4—151no—
368C22×D4—301no—
369C23×C4—151no—
370C23×C4—301no—
371C2×M16—151no—
372C2×M16—152no—
373C2×M16—304no—
374C2×M16C2×D1511yescharacteristic
375C22×C8—152no—
376#117—151no—
377#117—152no—
378#117—301no—
379#117—301no—
380#117C2×D1511yescharacteristic
381C2×C4⋊C4—152no—
382C2×C22⋊C4—301no—
383C2×C42—301no—
384#103—154no—
385#103C2×D1514yescomplemented
386#102—154no—
387#102—154no—
388C2×C3×C5—22no—
389C2×C3×C5C22×C22⋊C411yescharacteristic
390C2×C3×C5#57512yes—
391D15—22no—
392D15—22no—
393D15—44no—
394C23×C3D4×D511yescharacteristic
395C22×D3—51no—
396C22×D3—51no—
397C22×D3—54no—
398C22×D3—102no—
399C22×D3—104no—
400C22×D3—202no—
401C22×D3—204no—
402C3×Q8—22no—
403C3×Q8—24no—
404C3×Q8C2×C4×D514yes—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4C2×C4×D512yes—
409C2×C3×C4D4×D512yes—
410C2×C3×C4D4×D511yescharacteristic
411C2×C3×C4C23×D511yescharacteristic
412C2×Dic3—51no—
413C2×Dic3—51no—
414C2×Dic3—54no—
415C2×Dic3—102no—
416D12—54no—
417D12—108no—
418D12—108no—
419D12—102no—
420D12—104no—
421C4×D3—52no—
422C4×D3—52no—
423C4×D3—52no—
424C4×D3—52no—
425C4×D3—54no—
426C4×D3—54no—
427C4×D3—108no—
428C4×D3—108no—
429C4×D3—204no—
430C3×C8—22no—
431C3⋊C8—102no—
432C22×C5—22no—
433C22×C5D3×C22⋊C412yes—
434C22×C5C2×D3×D411yescharacteristic
435C2×D5—32no—
436C2×D5—32no—
437C2×D5—62no—
438C2×D5—62no—
439C2×D5—64no—
440C2×D5—124no—
441C2×D5—128no—
442C2×D5—124no—
443C4×C5—24no—
444C4×C5—42no—
445C4×C5D3×C22⋊C412yes—
446C4×C5C22×C4×D311yescharacteristic
447C4×C5C2×D3×D411yescharacteristic
448Dic5—32no—
449Dic5—32no—
450Dic5—122no—
451C24—151no—
452C24—301no—
453#41—158no—
454#41—158no—
455#41—308no—
456#41—308no—
457#41—304no—
458#41—304no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8C4×D1514yescomplemented
462C2×Q8C22×D1512yes—
463C2×D4—154no—
464C2×D4—152no—
465C2×D4—308no—
466C2×D4—302no—
467C2×D4—302no—
468C2×D4—301no—
469C2×D4—302no—
470C22×C4—21no—
471C22×C4—151no—
472C22×C4—154no—
473C22×C4—151no—
474C22×C4—151no—
475C22×C4—154no—
476C22×C4—151no—
477C22×C4—151no—
478C22×C4—308no—
479C22×C4—302no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4C22×D1511yescharacteristic
483M16—22no—
484M16—304no—
485M16—302no—
486C2×C8—152no—
487C2×C8—304no—
488C2×C8C22×D1512yes—
489C4⋊C4—152no—
490C4⋊C4—302no—
491C4⋊C4C22×D1512yes—
492C22⋊C4—21no—
493C22⋊C4—302no—
494C22⋊C4—301no—
495C42—21no—
496C42—302no—
497C42—301no—
498C3×C5#289611yescharacteristic, complemented, Hall
499C22×C3—22no—
500C22×C3D5×C22⋊C412yes—
501C22×C3C2×D4×D511yescharacteristic
502C2×D3—52no—
503C2×D3—52no—
504C2×D3—102no—
505C2×D3—102no—
506C2×D3—104no—
507C2×D3—204no—
508C2×D3—208no—
509C2×D3—204no—
510C3×C4—24no—
511C3×C4—42no—
512C3×C4D5×C22⋊C412yes—
513C3×C4C22×C4×D511yescharacteristic
514C3×C4C2×D4×D511yescharacteristic
515Dic3—52no—
516Dic3—52no—
517Dic3—202no—
518C2×C5—22no—
519C2×C5C2×D3×C22⋊C411yescharacteristic
520C2×C5#577712yes—
521D5—62no—
522D5—62no—
523D5—124no—
524C23—151no—
525C23—151no—
526C23—154no—
527C23—304no—
528C23—302no—
529C23—602no—
530C23—604no—
531C23D4×D1511yescharacteristic
532Q8—24no—
533Q8—22no—
534Q8C2×C4×D1514yes—
535D4—154no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—154no—
544C2×C4—151no—
545C2×C4—151no—
546C2×C4—154no—
547C2×C4—154no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—152no—
551C2×C4—152no—
552C2×C4—308no—
553C2×C4—308no—
554C2×C4—302no—
555C2×C4—604no—
556C2×C4C2×C4×D1512yes—
557C2×C4D4×D1511yescharacteristic
558C2×C4D4×D1512yes—
559C2×C4C23×D1511yescharacteristic
560C8—22no—
561C8—302no—
562C2×C3—22no—
563C2×C3C2×D5×C22⋊C411yescharacteristic
564C2×C3#6611112yes—
565D3—102no—
566D3—102no—
567D3—204no—
568C5#7895611yescharacteristic, complemented, Sylow
569C22—22no—
570C22—152no—
571C22—152no—
572C22—302no—
573C22—302no—
574C22—304no—
575C22—604no—
576C22—608no—
577C22—604no—
578C22C22⋊C4×D1512yes—
579C22C2×D4×D1511yescharacteristic
580C4—24no—
581C4—42no—
582C4—152no—
583C4—152no—
584C4—602no—
585C4C22⋊C4×D1512yes—
586C4C22×C4×D1511yescharacteristic
587C4C2×D4×D1511yescharacteristic
588C3#1061068711yescharacteristic, complemented, Sylow
589C2—22no—
590C2—302no—
591C2—302no—
592C2—604no—
593C2C2×C22⋊C4×D1511yescharacteristic
594C2#1175653612yes—
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#419C2
socleC22×C3×C5C22×D4
prime radicalC22×#6821C2
cosocleC2×C3×C4×C5C24
hypercenter#419D15
hypocenterC3×C5#2896
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C2×M16×D15C2semidirect
3. D15×#117C2semidirect
4. C3×C5×#419C2semidirect
5. #11751326C2semidirect
6. #11751232C2semidirect
7. #11751102C2semidirect
8. #11750905C2semidirect
9. #11750904C2semidirect
15. C2×#6821C4semidirect
27. C2×#6705C22semidirect
28. C2×C8×D15C22semidirect
29. #91781C22semidirect
30. C4⋊C4×D15C22semidirect
35. C3×C5×#103C22semidirect
36. #91119C22semidirect
37. #91105C22semidirect
38. #91093C22semidirect
43. C5×#419D3semidirect
53. C2×C3×C5×Q8C2×C4semidirect
63. #6821C2×C4semidirect
69. C2×D60C2×C4semidirect
99. C3×#419D5semidirect
129. C5×#103C2×D3semidirect
196. C3×#103C2×D5semidirect
204. C2×C5×Q8C4×D3semidirect
251. #419D15semidirect
279. C2×C3×Q8C4×D5semidirect
385. #103C2×D15semidirect
461. C2×Q8C4×D15semidirect
498. C3×C5#2896semidirect
568. C5#78956semidirect
588. C3#10610687semidirect

Maximal subgroups

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

subgroupindex
C22×#68212
C2×M16×D152
D15×#1172
C3×C5×#4192
#117513262
#117512322
#117511022
#117509052
#117509042
#106106873
#789565