Finite Group Databasebeta

← #14427809

Subgroups — #14427809

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91808C211yesmaximal, characteristic, complemented
2C22×C8×D15C211yesmaximal, characteristic, complemented
3#11756386C212yesmaximal, complemented
4C3×C5×#467C211yesmaximal, characteristic, complemented
5#11751289C211yesmaximal, characteristic, complemented
6#11751216C211yesmaximal, characteristic, complemented
7#11751149C212yesmaximal, complemented
8#11750984C211yesmaximal, characteristic, complemented
9#11750981C211yesmaximal, characteristic, complemented
10#11750908C212yesmaximal, complemented
11#11750905C212yesmaximal, complemented
12#10612489—31nomaximal, Hall
13C2×#6821C2212yes—
14C22×C4×D15C2211yescharacteristic
15C2×C3×C5×Q16C2211yescharacteristic, complemented
16C22×C3×C5×C8C2211yescharacteristic
17C3×C5×C22⋊Q8C2212yes—
18#91835C2212yes—
19C2×#6717C2212yescomplemented
20#91819C2212yes—
21C22×C3⋊#168C2211yescharacteristic
22#91808—24no—
23C2×D120C2211yescharacteristic, complemented
24C2×C8×D15—24no—
25C2×C8×D15C2211yescharacteristic
26C2×C8×D15C2211yescharacteristic
27C2×C8×D15C2211yescharacteristic
28C2×C8×D15C2211yescharacteristic
29#91784C2212yes—
30#91783—22no—
31#91782—22no—
32#91781C2212yes—
33#91771—22no—
34#91770—22no—
35C3×C5×#107C2211yescharacteristic, complemented
36C3×C5×#103C2212yescomplemented
37#91119C2212yescomplemented
38#91105C2212yescomplemented
39#91102C2211yescharacteristic, complemented
40#91093C2212yescomplemented
41#91092C2212yescomplemented
42#80758—51nomaximal, Hall
43C2×#4065—31no—
44C22×C8×D5—31no—
45#65961—32no—
46C5×#467D311yescharacteristic, complemented
47#65477—31no—
48#65404—31no—
49#65337—32no—
50#65172—31no—
51#65169—31no—
52#65096—32no—
53#65093—32no—
54C23×D15D411yescharacteristic
55C2×C3×C5×Q8C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—22no—
58C22×Dic15D411yescharacteristic
59#6821—24no—
60#6821—24no—
61C2×D60—22no—
62C2×D60—22no—
63C2×D60C2312yes—
64C2×C4×D15—22no—
65C2×C4×D15—22no—
66C2×C4×D15—22no—
67C2×C4×D15D411yescharacteristic
68C2×C4×D15D411yescharacteristic
69C2×C4×D15C2311yescharacteristic
70C2×C4×D15C2311yescharacteristic
71C3×C5×Q16—22no—
72C2×C3×C5×C8—22no—
73C2×C3×C5×C8C2311yescharacteristic
74C2×C3×C5×C8C2311yescharacteristic
75C3×C5×C4⋊C4—22no—
76C3×C5×C4⋊C4C2312yes—
77C3×C5×C22⋊C4—22no—
78#6717—24no—
79#6714—22no—
80#6714—22no—
81#6714—22no—
82#6714—22no—
83C4⋊Dic15C2312yes—
84#6712—22no—
85C4×Dic15—22no—
86C2×C3⋊#168—22no—
87C2×C3⋊#168C2311yescharacteristic
88C2×C3⋊#168C2311yescharacteristic
89D120—22no—
90C8×D15—22no—
91C8×D15—22no—
92C8×D15—22no—
93C8×D15D414yescomplemented
94C2×#919—51no—
95C22×D3×C8—51no—
96#5627—52no—
97C3×#467D511yescharacteristic, complemented
98#5188—51no—
99#5115—51no—
100#5048—52no—
101#4883—51no—
102#4880—51no—
103#4807—52no—
104#4804—52no—
105C2×#713—32no—
106C22×C4×D5—31no—
107C2×C5×Q16C2×D311yescharacteristic, complemented
108C22×C5×C8C2×D311yescharacteristic
109C5×C22⋊Q8C2×D312yes—
110#4092—32no—
111C2×#688—32no—
112#4076—32no—
113C22×#168—31no—
114#4065—64no—
115C2×D40—31no—
116C2×C8×D5—31no—
117C2×C8×D5—31no—
118C2×C8×D5—31no—
119C2×C8×D5—31no—
120C2×C8×D5—64no—
121#4041—32no—
122#4040—62no—
123#4039—62no—
124#4038—32no—
125#4028—62no—
126#4027—62no—
127C5×#107C2×D311yescharacteristic, complemented
128C5×#103C2×D312yescomplemented
129#3967—32no—
130#3953—32no—
131#3950—31no—
132#3941—32no—
133#3940—32no—
134#3050—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15C2×D411yescharacteristic
141C22×D15C2×D411yescharacteristic
142C3×C5×Q8—22no—
143C3×C5×Q8—22no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2411yescharacteristic
150#1220—44no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15C2×D411yescharacteristic
154C2×Dic15C2×D411yescharacteristic
155D60—22no—
156D60—22no—
157D60—44no—
158D60—44no—
159C4×D15—22no—
160C4×D15—21no—
161C4×D15—21no—
162C4×D15—22no—
163C4×D15—44no—
164C4×D15C2×D412yes—
165C4×D15#4112yes—
166C3×C5×C8—21no—
167C3×C5×C8C2×D412yes—
168C3⋊#168—21no—
169C3⋊#168C2×D412yes—
170C2×#239—52no—
171C22×C4×D3—51no—
172C2×C3×Q16C2×D511yescharacteristic, complemented
173C22×C3×C8C2×D511yescharacteristic
174C3×C22⋊Q8C2×D512yes—
175#946—52no—
176C2×#215—52no—
177#930—52no—
178C22×C3⋊C8—51no—
179#919—104no—
180C2×D24—51no—
181C2×D3×C8—51no—
182C2×D3×C8—51no—
183C2×D3×C8—51no—
184C2×D3×C8—51no—
185C2×D3×C8—104no—
186#895—52no—
187#894—102no—
188#893—102no—
189#892—52no—
190#882—102no—
191#67⋊C4—102no—
192C3×#107C2×D511yescharacteristic, complemented
193C3×#103C2×D512yescomplemented
194#835—52no—
195#821—52no—
196#818—51no—
197#809—52no—
198#808—52no—
199C23×D5—31no—
200C2×C5×Q8C22×D312yes—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—62no—
203C22×Dic5—31no—
204#713—64no—
205#713—64no—
206C2×D20—32no—
207C2×D20—62no—
208C2×D20—62no—
209C2×C4×D5—31no—
210C2×C4×D5—31no—
211C2×C4×D5—31no—
212C2×C4×D5—31no—
213C2×C4×D5—62no—
214C2×C4×D5—62no—
215C2×C4×D5—62no—
216C5×Q16—22no—
217C2×C5×C8—22no—
218C2×C5×C8C22×D311yescharacteristic
219C2×C5×C8C22×D311yescharacteristic
220C5×C4⋊C4—22no—
221C5×C4⋊C4C22×D312yes—
222C5×C22⋊C4—22no—
223#688—64no—
224#685—62no—
225#685—62no—
226#685—62no—
227#685—62no—
228C4⋊Dic5—32no—
229#683—62no—
230C4×Dic5—62no—
231C2×#168—31no—
232C2×#168—31no—
233C2×#168—62no—
234D40—62no—
235C8×D5—34no—
236C8×D5—62no—
237C8×D5—62no—
238C8×D5—62no—
239C2×#135—151no—
240C23×C8—151no—
241#525—152no—
242#467—151no—
243#467D1511yescharacteristic, complemented
244#466—151no—
245#466—151no—
246#465—152no—
247#465—152no—
248#427—151no—
249#416—152no—
250C22×C3×C5—21no—
251C22×C3×C5—21no—
252C22×C3×C5#13512yescomplemented
253C22×C3×C5C22×D411yescharacteristic
254C2×D15—28no—
255C2×D15—21no—
256C2×D15—21no—
257C2×D15—21no—
258C2×D15—21no—
259C2×D15—42no—
260C2×D15—42no—
261C2×D15—42no—
262C2×D15#12112yes—
263C2×D15#13512yescomplemented
264C3×C4×C5—21no—
265C3×C4×C5—42no—
266C3×C4×C5—42no—
267C3×C4×C5C22×D411yescharacteristic
268C3×C4×C5C2×#4111yescharacteristic
269Dic15—21no—
270Dic15—42no—
271Dic15#12112yes—
272C23×D3—51no—
273C2×C3×Q8C22×D512yes—
274C22×C3×C4C22×D511yescharacteristic
275C2×#67—102no—
276C22×Dic3—51no—
277#239—104no—
278#239—104no—
279C2×D12—52no—
280C2×D12—102no—
281C2×D12—102no—
282C2×C4×D3—51no—
283C2×C4×D3—51no—
284C2×C4×D3—51no—
285C2×C4×D3—51no—
286C2×C4×D3—102no—
287C2×C4×D3—102no—
288C2×C4×D3—102no—
289C3×Q16—22no—
290C2×C3×C8—22no—
291C2×C3×C8C22×D511yescharacteristic
292C2×C3×C8C22×D511yescharacteristic
293C3×C4⋊C4—22no—
294C3×C4⋊C4C22×D512yes—
295C3×C22⋊C4—22no—
296#215—104no—
297(C2×D3)⋊C4—102no—
298(C2×D3)⋊C4—102no—
299(C2×D3)⋊C4—102no—
300(C2×D3)⋊C4—102no—
301C4⋊Dic3—52no—
302Dic3⋊C4—102no—
303C4×Dic3—102no—
304C2×C3⋊C8—51no—
305C2×C3⋊C8—51no—
306C2×C3⋊C8—102no—
307D24—102no—
308D3×C8—54no—
309D3×C8—102no—
310D3×C8—102no—
311D3×C8—102no—
312C23×C5D3×D411yescharacteristic
313C22×D5—31no—
314C22×D5—31no—
315C22×D5—62no—
316C22×D5—62no—
317C22×D5—62no—
318C22×D5—62no—
319C5×Q8—22no—
320C5×Q8—22no—
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—22no—
324C2×C4×C5—22no—
325C2×C4×C5D3×D411yescharacteristic
326C2×C4×C5C23×D311yescharacteristic
327C22⋊D5—124no—
328C2×Dic5—31no—
329C2×Dic5—31no—
330C2×Dic5—62no—
331C2×Dic5—62no—
332D20—62no—
333D20—62no—
334D20—124no—
335D20—124no—
336C4×D5—32no—
337C4×D5—32no—
338C4×D5—62no—
339C4×D5—62no—
340C4×D5—61no—
341C4×D5—61no—
342C4×D5—124no—
343C5×C8—21no—
344C5×C8D3×D412yes—
345#168—32no—
346#168—61no—
347C2×#41—152no—
348C23×C4—151no—
349#135—304no—
350C2×Q16C2×D1511yescharacteristic, complemented
351C2×SD16—152no—
352C2×D8—151no—
353C22×C8—151no—
354C22×C8—151no—
355C22×C8—151no—
356C22×C8—151no—
357C22×C8—151no—
358C22×C8—304no—
359C22×C8C2×D1511yescharacteristic
360#123—302no—
361C22⋊Q8—152no—
362C22⋊Q8C2×D1512yes—
363#121—152no—
364#121—152no—
365#120—302no—
366C4×D4—302no—
367C4×D4—302no—
368#117—152no—
369#107—151no—
370#107C2×D1511yescharacteristic, complemented
371#106—152no—
372#103—152no—
373#103C2×D1512yescomplemented
374#102—152no—
375#102—152no—
376C2×C3×C5—22no—
377C2×C3×C5C2×#12111yescharacteristic
378C2×C3×C5C2×#13511yescharacteristic
379C2×C3×C5C2×#13511yescharacteristic
380D15—22no—
381D15—22no—
382D15—82no—
383C23×C3D4×D511yescharacteristic
384C22×D3—51no—
385C22×D3—51no—
386C22×D3—102no—
387C22×D3—102no—
388C22×D3—102no—
389C22×D3—102no—
390C3×Q8—22no—
391C3×Q8—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4D4×D511yescharacteristic
397C2×C3×C4C23×D511yescharacteristic
398#67—204no—
399C2×Dic3—51no—
400C2×Dic3—51no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403D12—102no—
404D12—102no—
405D12—204no—
406D12—204no—
407C4×D3—52no—
408C4×D3—52no—
409C4×D3—102no—
410C4×D3—102no—
411C4×D3—101no—
412C4×D3—101no—
413C4×D3—204no—
414C3×C8—21no—
415C3×C8D4×D512yes—
416C3⋊C8—52no—
417C3⋊C8—101no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5#91912yescomplemented
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—32no—
423C2×D5—32no—
424C2×D5—68no—
425C2×D5—61no—
426C2×D5—61no—
427C2×D5—61no—
428C2×D5—61no—
429C2×D5—122no—
430C2×D5—122no—
431C2×D5—122no—
432C4×C5—21no—
433C4×C5—42no—
434C4×C5—42no—
435C4×C5C2×D3×D411yescharacteristic
436C4×C5D3×#4111yescharacteristic
437Dic5—32no—
438Dic5—61no—
439Dic5—122no—
440C24—151no—
441#41—304no—
442#41—304no—
443C2×Q8C22×D1512yes—
444C2×D4—152no—
445C2×D4—302no—
446C2×D4—302no—
447C2×D4—302no—
448C22×C4—151no—
449C22×C4—151no—
450C22×C4—151no—
451C22×C4—151no—
452C22×C4—151no—
453C22×C4—302no—
454C22×C4—302no—
455C22×C4—302no—
456C22×C4C22×D1511yescharacteristic
457Q16—22no—
458SD16—304no—
459D8—302no—
460C2×C8—22no—
461C2×C8—151no—
462C2×C8—154no—
463C2×C8—151no—
464C2×C8—302no—
465C2×C8—302no—
466C2×C8—302no—
467C2×C8—302no—
468C2×C8C22×D1511yescharacteristic
469C2×C8C22×D1511yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—152no—
472C4⋊C4—302no—
473C4⋊C4C22×D1512yes—
474C22⋊C4—22no—
475C22⋊C4—302no—
476C22⋊C4—302no—
477C22⋊C4—302no—
478C22⋊C4—302no—
479C42—302no—
480C3×C5#305011yescharacteristic, complemented, Hall
481C22×C3—21no—
482C22×C3—21no—
483C22×C3#406512yescomplemented
484C22×C3C2×D4×D511yescharacteristic
485C2×D3—52no—
486C2×D3—52no—
487C2×D3—108no—
488C2×D3—101no—
489C2×D3—101no—
490C2×D3—101no—
491C2×D3—101no—
492C2×D3—202no—
493C2×D3—202no—
494C2×D3—202no—
495C3×C4—21no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4C2×D4×D511yescharacteristic
499C3×C4D5×#4111yescharacteristic
500Dic3—52no—
501Dic3—101no—
502Dic3—202no—
503C2×C5—22no—
504C2×C5D3×#12111yescharacteristic
505C2×C5C2×#91911yescharacteristic
506C2×C5D3×#13511yescharacteristic
507D5—62no—
508D5—62no—
509D5—242no—
510C23—151no—
511C23—151no—
512C23—302no—
513C23—302no—
514C23—302no—
515C23—302no—
516C23D4×D1511yescharacteristic
517Q8—22no—
518Q8—22no—
519D4—302no—
520D4—302no—
521D4—604no—
522D4—604no—
523D4—604no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—151no—
529C2×C4—152no—
530C2×C4—152no—
531C2×C4—151no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—301no—
536C2×C4—301no—
537C2×C4—302no—
538C2×C4—604no—
539C2×C4D4×D1511yescharacteristic
540C2×C4C23×D1511yescharacteristic
541C8—21no—
542C8—152no—
543C8—301no—
544C8D4×D1512yes—
545C2×C3—22no—
546C2×C3D5×#12111yescharacteristic
547C2×C3C2×#406511yescharacteristic
548C2×C3D5×#13511yescharacteristic
549D3—102no—
550D3—102no—
551D3—402no—
552C5#8075811yescharacteristic, complemented, Sylow
553C22—21no—
554C22—21no—
555C22—152no—
556C22—152no—
557C22—308no—
558C22—301no—
559C22—301no—
560C22—301no—
561C22—301no—
562C22—602no—
563C22—602no—
564C22—602no—
565C22#9180812yescomplemented
566C22C2×D4×D1511yescharacteristic
567C4—21no—
568C4—42no—
569C4—42no—
570C4—152no—
571C4—301no—
572C4—602no—
573C4C2×D4×D1511yescharacteristic
574C4#41×D1511yescharacteristic
575C3#1061248911yescharacteristic, complemented, Sylow
576C2—22no—
577C2—302no—
578C2—302no—
579C2—1202no—
580C2D15×#12111yescharacteristic
581C2C2×#9180811yescharacteristic
582C2D15×#13511yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#467C2
socleC22×C3×C5C22×D4
hypercenter#467D15
hypocenterC3×C5#3050
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#91808C2semidirect
2. C22×C8×D15C2semidirect
3. #11756386C2semidirect
4. C3×C5×#467C2semidirect
5. #11751289C2semidirect
6. #11751216C2semidirect
7. #11751149C2semidirect
8. #11750984C2semidirect
9. #11750981C2semidirect
10. #11750908C2semidirect
11. #11750905C2semidirect
15. C2×C3×C5×Q16C22semidirect
19. C2×#6717C22semidirect
23. C2×D120C22semidirect
35. C3×C5×#107C22semidirect
36. C3×C5×#103C22semidirect
37. #91119C22semidirect
38. #91105C22semidirect
39. #91102C22semidirect
40. #91093C22semidirect
41. #91092C22semidirect
46. C5×#467D3semidirect
93. C8×D15D4semidirect
97. C3×#467D5semidirect
107. C2×C5×Q16C2×D3semidirect
127. C5×#107C2×D3semidirect
128. C5×#103C2×D3semidirect
172. C2×C3×Q16C2×D5semidirect
192. C3×#107C2×D5semidirect
193. C3×#103C2×D5semidirect
243. #467D15semidirect
252. C22×C3×C5#135semidirect
263. C2×D15#135semidirect
350. C2×Q16C2×D15semidirect
370. #107C2×D15semidirect
373. #103C2×D15semidirect
420. C22×C5#919semidirect
480. C3×C5#3050semidirect
483. C22×C3#4065semidirect
552. C5#80758semidirect
565. C22#91808semidirect
575. C3#10612489semidirect

Maximal subgroups

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

subgroupindex
C2×#918082
C22×C8×D152
#117563862
C3×C5×#4672
#117512892
#117512162
#117511492
#117509842
#117509812
#117509082
#117509052
#106124893
#807585