Finite Group Databasebeta

← #14484836

Subgroups — #14484836

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756335C211yesmaximal, characteristic, complemented
2C5×#5569C211yesmaximal, characteristic, complemented
3C3×C4×#713C211yesmaximal, characteristic, complemented
4#11755257C211yesmaximal, characteristic, complemented
5#11755251C211yesmaximal, characteristic, complemented
6#11755200C211yesmaximal, characteristic, complemented
7Q8×#1200C211yesmaximal, characteristic, complemented
8#11753799C213yesmaximal, complemented
9#11753768C213yesmaximal, complemented
10#11753682C213yesmaximal, complemented
11#11753521C213yesmaximal, complemented
12#11752705C213yesmaximal, complemented
13#11752700C213yesmaximal, complemented
14#11752685C213yesmaximal, complemented
15C4×#6773C213yesmaximal, complemented
16#10618917—31nomaximal, Hall
17C2×#6821C2211yescharacteristic
18C2×Q8×D15C2211yescharacteristic
19C2×C3×#713C2211yescharacteristic
20C2×#6773C2213yes—
21C3×C4×C5×Q8C2211yescharacteristic
22#91786C2213yes—
23#91784C2213yes—
24#91764C2213yes—
25C4×D60C2213yes—
26C5×Q8×Dic3C2211yescharacteristic
27C5×Q8×Dic3C2211yescharacteristic
28C5×#890C2213yes—
29C5×#888C2213yes—
30C5×#870C2213yes—
31C4×C5×Dic6C2213yes—
32C3×Q8×Dic5C2211yescharacteristic
33C3×#4039C2213yes—
34C3×#4038C2213yes—
35C3×C4×D20C2213yes—
36C3×C42×D5C2213yes—
37#91465C2213yes—
38#91465C2213yes—
39#91459C2216yes—
40#91458C2213yes—
41#91455C2216yes—
42#91450C2213yes—
43#91444C2213yes—
44#91444C2216yes—
45#91444C2213yes—
46C4×#1202C2213yes—
47C4×#1202C2213yes—
48#91437C2213yes—
49#91428C2213yes—
50#91418C2213yes—
51#91418C2213yes—
52#91402C2211yescharacteristic
53#91402C2213yes—
54C4×#1200C2213yes—
55C4×#1200C2213yes—
56#91387C2216yes—
57#91364C2213yes—
58#91362C2216yes—
59#91360C2213yes—
60#91358C2213yes—
61#91356C2216yes—
62#91348C2216yes—
63#91347C2213yes—
64#91340C2213yes—
65#91333C2213yes—
66Dic5×Dic6C2213yes—
67#85842—51nomaximal, Hall
68C5×#557—31no—
69#66066—33no—
70#66059—33no—
71#66029—33no—
72#66029—33no—
73#66025—33no—
74#66025—33no—
75#65919—33no—
76#65911—31no—
77#65910—31no—
78#65910—31no—
79C4×#713—31no—
80C4×#713—33no—
81C4×#713D311yescharacteristic, complemented
82C4×Q8×D5—31no—
83C2×C3×C5×Q8C2311yescharacteristic
84#6821—24no—
85Q8×D15—24no—
86C2×D60C2313yes—
87C2×D60C2313yes—
88C2×C4×D15C2313yes—
89C2×C4×D15C2313yes—
90C2×Dic30C2313yes—
91C2×C5×Dic6C2313yes—
92C3×#713—24no—
93C2×C3×D20C2313yes—
94C2×C3×C4×D5C2313yes—
95C2×C3×C4×D5C2313yes—
96C2×C3×C4×D5C2313yes—
97C2×#1202C2313yes—
98C2×#1202C2313yes—
99C2×#1202C2316yes—
100C2×#1200C2313yes—
101C2×#1200C2313yes—
102C2×#1200C2313yes—
103C2×#1200C2311yescharacteristic
104#6773—212no—
105C3×C5×C4⋊C4C2313yes—
106C3×C42×C5C2313yes—
107#6714C2311yescharacteristic
108#6714C2313yes—
109#6714C2316yes—
110C4⋊Dic15C2313yes—
111C5×C4⋊Dic3C2313yes—
112C5×C4⋊Dic3C2313yes—
113C5×C4⋊Dic3C2313yes—
114C5×Dic3⋊C4C2311yescharacteristic
115C5×Dic3⋊C4C2313yes—
116C5×Dic3⋊C4C2316yes—
117C4×C5×Dic3C2313yes—
118C4×C5×Dic3C2313yes—
119C3×#685C2316yes—
120C3×C4⋊Dic5C2313yes—
121C3×C4×Dic5C2313yes—
122C3×C4×Dic5C2311yescharacteristic
123C3×C4×Dic5C2313yes—
124Dic3⋊Dic5C2311yescharacteristic
125Dic3⋊Dic5C2313yes—
126#6669C2316yes—
127#6667C2311yescharacteristic
128#6667C2313yes—
129#6667C2316yes—
130#6665C2313yes—
131#6665C2313yes—
132#6665C2316yes—
133Dic3×Dic5C2311yescharacteristic
134Dic3×Dic5C2313yes—
135Dic3×Dic5C2313yes—
136Dic3×Dic5C2313yes—
137C3×C4×#41—51no—
138#5834—51no—
139#5831—51no—
140#5576—51no—
141C4×#239—51no—
142C4×D3×Q8—51no—
143#5569D511yescharacteristic, complemented
144#5559—53no—
145#5558—53no—
146#5557—53no—
147#5548—53no—
148#5533—53no—
149#5530—53no—
150#5479—53no—
151C4×#235—53no—
152C2×#713—31no—
153C2×#713—33no—
154C2×#713C2×D311yescharacteristic
155C2×Q8×D5—31no—
156C5×C4⋊Q8—33no—
157C5×#125—33no—
158C5×#125—33no—
159C4×C5×Q8—31no—
160C4×C5×Q8—31no—
161C4×C5×Q8—33no—
162C4×C5×Q8C2×D311yescharacteristic
163#4093—36no—
164Q8×Dic5—33no—
165Q8×Dic5C2×D311yescharacteristic
166#4044—36no—
167#4044—36no—
168#4043—33no—
169#4043—33no—
170#4041—33no—
171#4041—33no—
172#4041—33no—
173#4041—33no—
174#4041—36no—
175#4040—36no—
176#4040—36no—
177#4039—36no—
178#4039—33no—
179#4039—33no—
180#4039—33no—
181#4039—33no—
182#4039C2×D313yes—
183#4038—33no—
184#4038—33no—
185#4038—31no—
186#4038—33no—
187#4038C2×D313yes—
188D5×C4⋊C4—33no—
189#4033—33no—
190#4021—33no—
191#4021—33no—
192#4020—33no—
193C4×D20—33no—
194C4×D20—33no—
195C4×D20—33no—
196C4×D20—33no—
197C4×D20C2×D313yes—
198C42×D5—33no—
199C42×D5—33no—
200C42×D5C2×D313yes—
201C4×Dic10—33no—
202#3480—151noSylow
203C22×D15C2413yes—
204C22×D15C2411yescharacteristic
205C22×C3×D5C2413yes—
206C3×C5×Q8#4114yescomplemented
207C2×C3×C4×C5C2413yes—
208C2×C3×C4×C5C2411yescharacteristic
209C2×C3×C4×C5C2413yes—
210C2×Dic15C2411yescharacteristic
211C2×Dic15C2413yes—
212D60—26no—
213D60—26no—
214C4×D15—26no—
215C4×D15—26no—
216Dic30—26no—
217C2×C5×Dic3C2411yescharacteristic
218C2×C5×Dic3C2413yes—
219C2×C5×Dic3C2413yes—
220C2×C5×Dic3C2411yescharacteristic
221C5×Dic6—26no—
222C2×C3×Dic5C2413yes—
223C2×C3×Dic5C2411yescharacteristic
224C2×C3×Dic5C2411yescharacteristic
225C3×D20—26no—
226C3×C4×D5—26no—
227C3×C4×D5—26no—
228C3×C4×D5—43no—
229#1202—26no—
230#1202—26no—
231#1202—46no—
232#1200—26no—
233#1200—43no—
234#1200—43no—
235#1200#4114yescomplemented
236C2×C3×#41—51no—
237C2×#239—51no—
238C2×D3×Q8—51no—
239C2×#235—53no—
240C3×C4×Q8—51no—
241C3×C4×Q8C2×D511yescharacteristic
242C3×C4×D4—53no—
243C3×C4×D4—53no—
244C3×#117—53no—
245C2×C3×C42—53no—
246Q8×Dic3C2×D511yescharacteristic
247Q8×Dic3C2×D511yescharacteristic
248#940—53no—
249#938—53no—
250#938—53no—
251#936—53no—
252#930—53no—
253#929—56no—
254C4×#67—53no—
255C4×#67—53no—
256C4×#67—56no—
257C4×#67—53no—
258C4×#67—53no—
259#926—53no—
260#924—53no—
261#897—53no—
262#895—53no—
263#893—53no—
264#892—53no—
265D3×C4⋊C4—53no—
266#890C2×D513yes—
267#888C2×D513yes—
268#887—53no—
269#885—56no—
270#884—56no—
271#883—56no—
272#879—56no—
273#876—56no—
274#875—53no—
275C4×D12—53no—
276C4×D12—53no—
277C4×D12—53no—
278#872—51no—
279#872—53no—
280C42×D3—53no—
281C42×D3—53no—
282#870C2×D513yes—
283C4×Dic6—53no—
284C4×Dic6—53no—
285C4×Dic6C2×D513yes—
286C2×C5×Q8—33no—
287C2×C5×Q8C22×D311yescharacteristic
288#713—24no—
289#713—612no—
290#713—64no—
291Q8×D5—64no—
292C2×D20—33no—
293C2×D20—33no—
294C2×D20—33no—
295C2×D20—33no—
296C2×D20—36no—
297C2×D20C22×D313yes—
298C2×C4×D5—31no—
299C2×C4×D5—33no—
300C2×C4×D5—33no—
301C2×C4×D5—33no—
302C2×C4×D5—33no—
303C2×C4×D5—33no—
304C2×C4×D5C22×D313yes—
305C2×C4×D5C22×D313yes—
306C2×C4×D5C22×D313yes—
307C2×Dic10—33no—
308C5×C4⋊C4—31no—
309C5×C4⋊C4—33no—
310C5×C4⋊C4—33no—
311C5×C4⋊C4—33no—
312C5×C4⋊C4—33no—
313C5×C4⋊C4—36no—
314C5×C4⋊C4C22×D313yes—
315C42×C5—33no—
316C42×C5—33no—
317C42×C5C22×D313yes—
318#685—33no—
319#685—33no—
320#685—36no—
321#685—31no—
322#685—33no—
323#685—36no—
324#685—31no—
325#685—36no—
326#685—33no—
327#685C22×D316yes—
328C4⋊Dic5—33no—
329C4⋊Dic5—31no—
330C4⋊Dic5—33no—
331C4⋊Dic5C22×D313yes—
332#683—36no—
333C4×Dic5—33no—
334C4×Dic5—31no—
335C4×Dic5—33no—
336C4×Dic5—33no—
337C4×Dic5C22×D313yes—
338C4×Dic5C22×D311yescharacteristic
339C4×Dic5C22×D313yes—
340#559—153no—
341#557—31no—
342#556—153no—
343#553—153no—
344#553—153no—
345#550—151no—
346#548—151no—
347#548—151no—
348#532—153no—
349#529—153no—
350#529—153no—
351C4×#41—51no—
352C4×#41—153no—
353C4×#41—151no—
354C2×C4×Q8—151no—
355C22×C3×C5C2511yescharacteristic
356C2×D15—21no—
357C2×D15—21no—
358C2×D15—23no—
359C2×D15—23no—
360C2×D15—23no—
361C2×D15C2×#4112yes—
362C2×C3×D5—23no—
363C2×C3×D5—23no—
364C2×C3×D5—23no—
365C3×C4×C5—23no—
366C3×C4×C5—21no—
367C3×C4×C5C2×#4116yes—
368Dic15—23no—
369Dic15—21no—
370C3×Dic5—21no—
371C3×Dic5—23no—
372C3×Dic5C2×#4112yes—
373C5×Dic3—23no—
374C5×Dic3—21no—
375C5×Dic3—23no—
376C5×Dic3C2×#4112yes—
377C3×#41—104no—
378C2×C3×Q8C22×D511yescharacteristic
379C2×C3×D4—53no—
380C22×C3×C4—53no—
381C22×C3×C4—53no—
382C22×C3×C4—53no—
383C2×#67—53no—
384C2×#67—53no—
385C2×#67—56no—
386#239—104no—
387D3×Q8—104no—
388#235—1012no—
389C2×D12—53no—
390C2×D12—53no—
391C2×C4×D3—51no—
392C2×C4×D3—53no—
393C2×C4×D3—53no—
394C2×C4×D3—53no—
395C2×C4×D3—53no—
396C2×C4×D3—53no—
397C2×Dic6—53no—
398C2×Dic6C22×D513yes—
399C3×C4⋊C4—53no—
400C3×C4⋊C4C22×D513yes—
401C3×C22⋊C4—56no—
402C3×C42—53no—
403C3×C42—53no—
404C3×C42—51no—
405C3×C42C22×D513yes—
406#217—53no—
407#217—56no—
408#217—53no—
409(C2×D3)⋊C4—51no—
410(C2×D3)⋊C4—53no—
411(C2×D3)⋊C4—56no—
412(C2×D3)⋊C4—51no—
413(C2×D3)⋊C4—53no—
414(C2×D3)⋊C4—56no—
415C4⋊Dic3—53no—
416C4⋊Dic3C22×D513yes—
417C4⋊Dic3C22×D513yes—
418C4⋊Dic3C22×D513yes—
419Dic3⋊C4—56no—
420Dic3⋊C4—51no—
421Dic3⋊C4—53no—
422Dic3⋊C4C22×D511yescharacteristic
423Dic3⋊C4C22×D513yes—
424Dic3⋊C4C22×D516yes—
425C4×Dic3—53no—
426C4×Dic3—53no—
427C4×Dic3—53no—
428C4×Dic3—51no—
429C4×Dic3C22×D513yes—
430C4×Dic3C22×D513yes—
431C22×D5—31no—
432C22×D5—33no—
433C22×D5C23×D313yes—
434C5×Q8—66no—
435C5×Q8#23514yescomplemented
436C2×C4×C5—31no—
437C2×C4×C5—33no—
438C2×C4×C5—33no—
439C2×C4×C5—31no—
440C2×C4×C5C23×D313yes—
441C2×C4×C5C23×D311yescharacteristic
442C2×C4×C5C23×D313yes—
443C2×Dic5—31no—
444C2×Dic5—33no—
445C2×Dic5C23×D313yes—
446C2×Dic5C23×D311yescharacteristic
447C2×Dic5C23×D311yescharacteristic
448D20—26no—
449D20—66no—
450D20—66no—
451D20—66no—
452D20—66no—
453D20—126no—
454C4×D5—26no—
455C4×D5—26no—
456C4×D5—34no—
457C4×D5—43no—
458C4×D5—66no—
459C4×D5—66no—
460C4×D5—66no—
461C4×D5—123no—
462C4×D5—123no—
463Dic10—66no—
464C2×#41—51no—
465C2×#41—153no—
466C2×#41—151no—
467C22×Q8—151no—
468C4⋊Q8—33no—
469#127—153no—
470#126—156no—
471#126—156no—
472#125—33no—
473#125—33no—
474#124—153no—
475#124—156no—
476#124—153no—
477#123—156no—
478#123—156no—
479C22⋊Q8—153no—
480C22⋊Q8—153no—
481#121—156no—
482#121—153no—
483#121—153no—
484#121—153no—
485#121—153no—
486C4×Q8—31no—
487C4×Q8—31no—
488C4×Q8—33no—
489C4×Q8—51no—
490C4×Q8—153no—
491C4×Q8—153no—
492C4×Q8—153no—
493C4×Q8D3×D511yescharacteristic
494C4×D4—53no—
495C4×D4—53no—
496C4×D4—153no—
497C4×D4—153no—
498C4×D4—153no—
499C4×D4—153no—
500C4×D4—153no—
501C4×D4—153no—
502C4×D4—153no—
503C4×D4—156no—
504C4×D4—153no—
505#117—53no—
506#117—151no—
507#117—153no—
508#117—153no—
509#117—153no—
510C2×C4⋊C4—153no—
511C2×C42—53no—
512C2×C42—153no—
513C2×C42—153no—
514C2×C3×C5C22×#4111yescharacteristic
515C2×C3×C5C22×#4111yescharacteristic
516C2×C3×C5#58511yescharacteristic
517D15—22no—
518D15—43no—
519C3×D5—43no—
520C23×C3—53no—
521C22×D3—51no—
522C22×D3—53no—
523C3×Q8#71314yescomplemented
524C3×D4—106no—
525C2×C3×C4—51no—
526C2×C3×C4—53no—
527C2×C3×C4—51no—
528C2×C3×C4—106no—
529C2×C3×C4—106no—
530C2×C3×C4—203no—
531C2×C3×C4C23×D511yescharacteristic
532C2×C3×C4C23×D513yes—
533C2×C3×C4C23×D513yes—
534#67—106no—
535#67—106no—
536#67—206no—
537C2×Dic3—51no—
538C2×Dic3—53no—
539C2×Dic3C23×D511yescharacteristic
540C2×Dic3C23×D513yes—
541C2×Dic3C23×D513yes—
542C2×Dic3C23×D511yescharacteristic
543D12—106no—
544D12—106no—
545C4×D3—54no—
546C4×D3—106no—
547C4×D3—106no—
548C4×D3—106no—
549C4×D3—203no—
550C4×D3—203no—
551Dic6—26no—
552Dic6—106no—
553C22×C5C24×D311yescharacteristic
554C2×D5—23no—
555C2×D5—23no—
556C2×D5—23no—
557C2×D5—32no—
558C2×D5—61no—
559C2×D5—61no—
560C2×D5—63no—
561C2×D5—63no—
562C2×D5—63no—
563C4×C5—23no—
564C4×C5—21no—
565C4×C5—32no—
566C4×C5—63no—
567C4×C5—61no—
568C4×C5—63no—
569C4×C5C2×#23516yes—
570Dic5—21no—
571Dic5—23no—
572Dic5—63no—
573Dic5—61no—
574Dic5D3×#4112yes—
575#41—104no—
576#41—3012no—
577#41—304no—
578C2×Q8—33no—
579C2×Q8—153no—
580C2×Q8—304no—
581C2×Q8C2×D3×D511yescharacteristic
582C2×D4—53no—
583C2×D4—153no—
584C2×D4—153no—
585C2×D4—156no—
586C2×D4—153no—
587C2×D4—153no—
588C22×C4—53no—
589C22×C4—53no—
590C22×C4—53no—
591C22×C4—151no—
592C22×C4—153no—
593C22×C4—153no—
594C22×C4—153no—
595C22×C4—153no—
596C22×C4—153no—
597C4⋊C4—33no—
598C4⋊C4—31no—
599C4⋊C4—33no—
600C4⋊C4—33no—
601C4⋊C4—33no—
602C4⋊C4—36no—
603C4⋊C4—53no—
604C4⋊C4—156no—
605C4⋊C4—151no—
606C4⋊C4—153no—
607C4⋊C4—153no—
608C4⋊C4C2×D3×D513yes—
609C22⋊C4—56no—
610C22⋊C4—151no—
611C22⋊C4—153no—
612C22⋊C4—156no—
613C22⋊C4—153no—
614C22⋊C4—153no—
615C22⋊C4—156no—
616C22⋊C4—151no—
617C22⋊C4—153no—
618C22⋊C4—156no—
619C42—33no—
620C42—33no—
621C42—51no—
622C42—53no—
623C42—53no—
624C42—151no—
625C42—153no—
626C42—153no—
627C42—153no—
628C42C2×D3×D513yes—
629C3×C5#348011yescharacteristic, complemented, Hall
630C22×C3—103no—
631C22×C3—103no—
632C22×C3—103no—
633C22×C3C24×D511yescharacteristic
634C2×D3—52no—
635C2×D3—101no—
636C2×D3—101no—
637C2×D3—103no—
638C2×D3—103no—
639C2×D3—103no—
640C3×C4—23no—
641C3×C4—21no—
642C3×C4—52no—
643C3×C4—101no—
644C3×C4—103no—
645C3×C4C2×#71316yes—
646Dic3—23no—
647Dic3—21no—
648Dic3—23no—
649Dic3—103no—
650Dic3—101no—
651Dic3D5×#4112yes—
652C2×C5C22×#23511yescharacteristic
653C2×C5C2×D3×#4111yescharacteristic
654C2×C5#596411yescharacteristic
655D5—43no—
656D5—62no—
657D5—123no—
658C23—53no—
659C23—151no—
660C23—153no—
661Q8—66no—
662Q8—306no—
663Q8#677314yescomplemented
664D4—106no—
665D4—306no—
666D4—306no—
667D4—306no—
668D4—306no—
669D4—606no—
670C2×C4—31no—
671C2×C4—31no—
672C2×C4—33no—
673C2×C4—33no—
674C2×C4—51no—
675C2×C4—51no—
676C2×C4—53no—
677C2×C4—106no—
678C2×C4—106no—
679C2×C4—154no—
680C2×C4—151no—
681C2×C4—153no—
682C2×C4—203no—
683C2×C4—306no—
684C2×C4—306no—
685C2×C4—306no—
686C2×C4—603no—
687C2×C4—603no—
688C2×C4C22×D3×D511yescharacteristic
689C2×C4C22×D3×D513yes—
690C2×C4C22×D3×D513yes—
691C2×C3—203no—
692C2×C3C22×#71311yescharacteristic
693C2×C3C2×D5×#4111yescharacteristic
694C2×C3#6628711yescharacteristic
695D3—102no—
696D3—203no—
697C5#8584211yescharacteristic, complemented, Sylow
698C22—103no—
699C22—103no—
700C22—103no—
701C22—152no—
702C22—301no—
703C22—301no—
704C22—303no—
705C22—303no—
706C22—303no—
707C22C23×D3×D511yescharacteristic
708C4—21no—
709C4—23no—
710C4—32no—
711C4—52no—
712C4—61no—
713C4—63no—
714C4—63no—
715C4—101no—
716C4—103no—
717C4—301no—
718C4—303no—
719C4C2×#677316yes—
720C3#1061891711yescharacteristic, complemented, Sylow
721C2—203no—
722C2—302no—
723C2—603no—
724C2C22×#677311yescharacteristic
725C2#1175704811yescharacteristic
726C2D3×D5×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3480
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11756335C2semidirect
2. C5×#5569C2semidirect
3. C3×C4×#713C2semidirect
4. #11755257C2semidirect
5. #11755251C2semidirect
6. #11755200C2semidirect
7. Q8×#1200C2semidirect
8. #11753799C2semidirect
9. #11753768C2semidirect
10. #11753682C2semidirect
11. #11753521C2semidirect
12. #11752705C2semidirect
13. #11752700C2semidirect
14. #11752685C2semidirect
15. C4×#6773C2semidirect
81. C4×#713D3semidirect
143. #5569D5semidirect
206. C3×C5×Q8#41semidirect
235. #1200#41semidirect
435. C5×Q8#235semidirect
523. C3×Q8#713semidirect
629. C3×C5#3480semidirect
663. Q8#6773semidirect
697. C5#85842semidirect
720. C3#10618917semidirect

Maximal subgroups

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

subgroupindex
#117563352
C5×#55692
C3×C4×#7132
#117552572
#117552512
#117552002
Q8×#12002
#117537992
#117537682
#117536822
#117535212
#117527052
#117527002
#117526852
C4×#67732
#106189173
#858425