Finite Group Databasebeta

← #14439253

Subgroups — #14439253

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#909C211yesmaximal, characteristic, complemented
2C2×C3×#4055C211yesmaximal, characteristic, complemented
3#11754586C211yesmaximal, characteristic
4C2×#91385C211yesmaximal, characteristic, complemented
5C2×#91309C211yesmaximal, characteristic, complemented
6C2×#91307C211yesmaximal, characteristic, complemented
7M16×Dic15C211yesmaximal, characteristic, complemented
8#11748041C212yesmaximal, complemented
9#11748010C212yesmaximal, complemented
10#11747983C212yesmaximal, complemented
11#11747981C212yesmaximal, complemented
12#10615703—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C2×C3×C5×M16C2211yescharacteristic
18C2×C4×Dic15C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20C5×#909—24no—
21C2×C5×Dic12C2212yes—
22C2×C5×#204C2212yes—
23C3×#4055—24no—
24C2×C3×Dic20C2212yes—
25C2×C3×#677C2212yes—
26#91571—22no—
27C2×#6669—22no—
28#91463C2211yescharacteristic
29C4×#1201—22no—
30#91385—22no—
31#91385C2212yes—
32#91385C2212yes—
33#91385C2211yescharacteristic
34#91370C2211yescharacteristic
35#91368C2211yescharacteristic
36C2×#6661C2212yes—
37C2×#6661C2212yes—
38#91309—24no—
39C2×#6656C2212yes—
40#91307—24no—
41C2×#6655C2212yes—
42#91099C2212yes—
43C8×Dic15C2212yes—
44#83972—51nomaximal, Hall
45C2×C5×#137—31no—
46#66166—31no—
47C2×#4097—31no—
48C2×#4090—31no—
49C2×#4088—31no—
50C2×#4055D311yescharacteristic, complemented
51#65479—32no—
52#65470—32no—
53#65464—32no—
54#65456—32no—
55M16×Dic5—31no—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15D412yes—
58C2×C5×#67—21no—
59C22×C5×Dic3—21no—
60C5×#235—22no—
61C5×#235—22no—
62C2×C5×D12C2311yescharacteristic
63C2×C4×C5×D3—21no—
64C2×C5×Dic6—22no—
65C2×C5×Dic6—22no—
66C2×C5×Dic6—21no—
67C2×C5×Dic6C2312yes—
68C2×C5×Dic6C2311yescharacteristic
69C2×C3×C22⋊D5—21no—
70C22×C3×Dic5—21no—
71C3×#709—22no—
72C3×#709—22no—
73C2×C3×D20C2311yescharacteristic
74C2×C3×C4×D5—21no—
75C2×C3×Dic10—22no—
76C2×C3×Dic10—22no—
77C2×C3×Dic10—21no—
78C2×C3×Dic10C2312yes—
79C2×C3×Dic10C2311yescharacteristic
80C2×#1201—22no—
81C3×C5×M16D414yescomplemented
82C2×C3×C5×C8C2312yes—
83C4×Dic15D412yes—
84C4×Dic15C2311yescharacteristic
85C4×Dic15C2311yescharacteristic
86#6710D414yescomplemented
87C2×C3⋊#168C2312yes—
88C5×Dic12—24no—
89C5×#204—24no—
90C3×Dic20—24no—
91C3×#677—24no—
92#6669—22no—
93#6669—24no—
94#6669—24no—
95(C2×D3)⋊Dic5—22no—
96#6665—22no—
97#6661—24no—
98#6661—24no—
99#6656—24no—
100#6655—24no—
101C2×C3×#137—51no—
102#5832—51no—
103C2×#951—51no—
104C2×#944—51no—
105C2×#942—51no—
106C2×#909D511yescharacteristic, complemented
107#5190—52no—
108#5183—52no—
109#5175—52no—
110#5167—52no—
111Dic3×M16—51no—
112C2×C5×#41—31no—
113C22×C5×Q8—31no—
114C2×#709C2×D311yescharacteristic
115C22×Dic10C2×D311yescharacteristic
116C5×#137—64no—
117C2×C5×Q16—32no—
118C2×C5×SD16—32no—
119C2×C5×M16C2×D311yescharacteristic
120#4097—64no—
121#4093—31no—
122#4090—32no—
123#4090—32no—
124#4090—31no—
125#4090—62no—
126C2×#689—32no—
127C2×#689—32no—
128#4088—64no—
129C2×#688—32no—
130#4086—31no—
131#4085—62no—
132#4082—31no—
133C2×#687—32no—
134C4×C22⋊D5—62no—
135C2×#683—62no—
136C2×C4×Dic5—31no—
137C2×#681—31no—
138#4055—24no—
139C2×Dic20C2×D312yes—
140C2×#677C2×D312yes—
141#3947—32no—
142C8×Dic5—32no—
143#3152—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C5×D3—21no—
146C22×C3×D5—21no—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2411yescharacteristic
151C2×Dic15—24no—
152C2×Dic15C2×D412yes—
153C2×Dic15C2×D412yes—
154C5×#67—42no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—42no—
158C5×D12—21no—
159C5×D12—21no—
160C4×C5×D3—42no—
161C5×Dic6—21no—
162C5×Dic6—22no—
163C5×Dic6—22no—
164C5×Dic6—21no—
165C5×Dic6—42no—
166C3×C22⋊D5—42no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—42no—
170C3×D20—21no—
171C3×D20—21no—
172C3×C4×D5—42no—
173C3×Dic10—22no—
174C3×Dic10—21no—
175C3×Dic10—22no—
176C3×Dic10—21no—
177C3×Dic10—42no—
178#1201—44no—
179C3×C5×C8C2×D414yes—
180C3⋊#168C2×D414yes—
181C2×C3×#41—51no—
182C22×C3×Q8—51no—
183C2×#235C2×D511yescharacteristic
184C22×Dic6C2×D511yescharacteristic
185C3×#137—104no—
186C2×C3×Q16—52no—
187C2×C3×SD16—52no—
188C2×C3×M16C2×D511yescharacteristic
189#951—104no—
190#947—51no—
191#944—52no—
192#944—52no—
193#944—51no—
194#944—102no—
195C2×#216—52no—
196C2×#216—52no—
197#942—104no—
198C2×#215—52no—
199#940—51no—
200#939—102no—
201#936—51no—
202C2×#214—52no—
203C4×#67—102no—
204C2×Dic3⋊C4—102no—
205C2×C4×Dic3—51no—
206C2×#208—51no—
207#909—24no—
208C2×Dic12C2×D512yes—
209C2×#204C2×D512yes—
210#815—52no—
211C8×Dic3—52no—
212C5×#41—62no—
213C5×#41—62no—
214C2×C5×Q8—32no—
215C2×C5×Q8—31no—
216C2×C5×Q8—61no—
217C2×C5×Q8—62no—
218C2×C5×Q8—62no—
219C2×C5×D4—31no—
220C2×C5×D4—61no—
221C22×C4×C5—61no—
222C22×C4×C5—61no—
223C22×C4×C5C22×D311yescharacteristic
224C2×C22⋊D5—21no—
225C2×C22⋊D5—62no—
226C22×Dic5—21no—
227C22×Dic5—32no—
228#709—22no—
229#709—22no—
230C2×D20C22×D311yescharacteristic
231C2×C4×D5—21no—
232C2×Dic10—22no—
233C2×Dic10—22no—
234C2×Dic10—21no—
235C2×Dic10C22×D312yes—
236C2×Dic10C22×D311yescharacteristic
237C5×Q16—64no—
238C5×SD16—64no—
239C5×M16#6714yescomplemented
240C2×C5×C8C22×D312yes—
241C22⋊Dic5—62no—
242#689—64no—
243#689—64no—
244#688—64no—
245#687—64no—
246#685—62no—
247#683—64no—
248#683—64no—
249#683—62no—
250C4×Dic5—32no—
251C4×Dic5—31no—
252C4×Dic5—31no—
253#681—34no—
254C2×#168—32no—
255Dic20—24no—
256#677—24no—
257C2×#137—31no—
258C2×#137—51no—
259C2×#137—151no—
260C2×#137—151no—
261#532—151no—
262C2×C4⋊Q8—151no—
263#497—152no—
264#497—152no—
265#494—152no—
266#492—152no—
267C4×M16—151no—
268C22×C3×C5—22no—
269C22×C3×C5#12712yes—
270C22×C3×C5C22×D411yescharacteristic
271C2×C5×D3—41no—
272C2×C5×D3—42no—
273C2×C3×D5—41no—
274C2×C3×D5—42no—
275C3×C4×C5#12712yes—
276C3×C4×C5C22×D411yescharacteristic
277C3×C4×C5C22×D411yescharacteristic
278Dic15—22no—
279Dic15#13714yescomplemented
280C3×Dic5—42no—
281C3×Dic5—41no—
282C5×Dic3—42no—
283C5×Dic3—41no—
284C3×#41—102no—
285C3×#41—102no—
286C2×C3×Q8—52no—
287C2×C3×Q8—51no—
288C2×C3×Q8—101no—
289C2×C3×Q8—102no—
290C2×C3×Q8—102no—
291C2×C3×D4—51no—
292C2×C3×D4—101no—
293C22×C3×C4—101no—
294C22×C3×C4—101no—
295C22×C3×C4C22×D511yescharacteristic
296C2×#67—21no—
297C2×#67—102no—
298C22×Dic3—21no—
299C22×Dic3—52no—
300#235—22no—
301#235—22no—
302C2×D12C22×D511yescharacteristic
303C2×C4×D3—21no—
304C2×Dic6—22no—
305C2×Dic6—22no—
306C2×Dic6—21no—
307C2×Dic6C22×D512yes—
308C2×Dic6C22×D511yescharacteristic
309C3×Q16—104no—
310C3×SD16—104no—
311C3×M16C22⋊D514yescomplemented
312C2×C3×C8C22×D512yes—
313#217—102no—
314#216—104no—
315#216—104no—
316#215—104no—
317#214—104no—
318(C2×D3)⋊C4—102no—
319Dic3⋊C4—104no—
320Dic3⋊C4—104no—
321Dic3⋊C4—102no—
322C4×Dic3—52no—
323C4×Dic3—51no—
324C4×Dic3—51no—
325#208—54no—
326C2×C3⋊C8—52no—
327Dic12—24no—
328#204—24no—
329C23×C5—61no—
330C23×C5D3×D411yescharacteristic
331C22×D5—21no—
332C5×Q8—62no—
333C5×Q8—61no—
334C5×Q8—62no—
335C5×Q8—61no—
336C5×Q8—122no—
337C5×D4—61no—
338C5×D4—61no—
339C5×D4—122no—
340C2×C4×C5—62no—
341C2×C4×C5—61no—
342C2×C4×C5—122no—
343C2×C4×C5—122no—
344C2×C4×C5D3×D411yescharacteristic
345C2×C4×C5D3×D412yes—
346C2×C4×C5C2×#6712yes—
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—42no—
349C22⋊D5—124no—
350C2×Dic5—22no—
351C2×Dic5—21no—
352C2×Dic5—32no—
353C2×Dic5—32no—
354C2×Dic5—42no—
355C2×Dic5—64no—
356D20—21no—
357D20—21no—
358C4×D5—42no—
359Dic10—22no—
360Dic10—21no—
361Dic10—22no—
362Dic10—21no—
363Dic10—42no—
364C5×C8C2×#6714yes—
365#168—34no—
366C2×#41—31no—
367C2×#41—51no—
368C22×Q8—31no—
369C22×Q8—51no—
370#137—64no—
371#137—104no—
372#137—304no—
373#137—304no—
374C2×Q16—32no—
375C2×Q16—52no—
376C2×Q16—152no—
377C2×Q16—152no—
378C2×SD16—32no—
379C2×SD16—52no—
380C2×SD16—152no—
381C2×SD16—152no—
382C2×M16—151no—
383C2×M16D3×D511yescharacteristic
384C4⋊Q8—152no—
385C4⋊Q8—152no—
386C4⋊Q8—151no—
387C4⋊Q8—302no—
388#127—151no—
389#124—151no—
390#124—151no—
391#121—302no—
392C4×D4—302no—
393C2×C4⋊C4—302no—
394C2×C42—151no—
395#97—152no—
396C4×C8—152no—
397C2×C3×C5—22no—
398C2×C3×C5C2×#12711yescharacteristic
399C2×C3×C5C2×#13712yes—
400C3×D5—81no—
401C5×D3—81no—
402C23×C3—101no—
403C23×C3D4×D511yescharacteristic
404C22×D3—21no—
405C3×Q8—102no—
406C3×Q8—101no—
407C3×Q8—102no—
408C3×Q8—101no—
409C3×Q8—202no—
410C3×D4—101no—
411C3×D4—101no—
412C3×D4—202no—
413C2×C3×C4—102no—
414C2×C3×C4—101no—
415C2×C3×C4—202no—
416C2×C3×C4—202no—
417C2×C3×C4D4×D512yes—
418C2×C3×C4D4×D511yescharacteristic
419C2×C3×C4C2×C22⋊D512yes—
420C2×C3×C4C23×D511yescharacteristic
421#67—42no—
422#67—204no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—42no—
426C2×Dic3—52no—
427C2×Dic3—52no—
428C2×Dic3—104no—
429D12—21no—
430D12—21no—
431C4×D3—42no—
432Dic6—22no—
433Dic6—21no—
434Dic6—22no—
435Dic6—21no—
436Dic6—42no—
437C3×C8C2×C22⋊D514yes—
438C3⋊C8—54no—
439C22×C5—22no—
440C22×C5—122no—
441C22×C5—121no—
442C22×C5#94012yes—
443C22×C5C2×D3×D411yescharacteristic
444C2×D5—42no—
445C2×D5—41no—
446C4×C5—121no—
447C4×C5—122no—
448C4×C5#94012yes—
449C4×C5C2×D3×D411yescharacteristic
450C4×C5C22×#6711yescharacteristic
451Dic5—34no—
452Dic5—42no—
453Dic5—41no—
454Dic5—62no—
455#41—62no—
456#41—62no—
457#41—102no—
458#41—102no—
459C2×Q8—31no—
460C2×Q8—32no—
461C2×Q8—51no—
462C2×Q8—52no—
463C2×Q8—61no—
464C2×Q8—62no—
465C2×Q8—62no—
466C2×Q8—101no—
467C2×Q8—102no—
468C2×Q8—102no—
469C2×D4—31no—
470C2×D4—51no—
471C2×D4—61no—
472C2×D4—101no—
473C2×D4—302no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4—152no—
479C22×C4C2×D3×D511yescharacteristic
480Q16—64no—
481Q16—104no—
482Q16—304no—
483Q16—304no—
484SD16—64no—
485SD16—104no—
486SD16—304no—
487SD16—304no—
488M16—154no—
489M16#120114yescomplemented
490C2×C8—152no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—304no—
493C4⋊C4—304no—
494C4⋊C4—302no—
495C22⋊C4—302no—
496C22⋊C4—302no—
497C42—152no—
498C42—151no—
499C42—151no—
500C3×C5#315211yescharacteristic, complemented, Hall
501C22×C3—22no—
502C22×C3—202no—
503C22×C3—201no—
504C22×C3#408612yes—
505C22×C3C2×D4×D511yescharacteristic
506C2×D3—42no—
507C2×D3—41no—
508C3×C4—201no—
509C3×C4—202no—
510C3×C4#408612yes—
511C3×C4C2×D4×D511yescharacteristic
512C3×C4C22×C22⋊D511yescharacteristic
513Dic3—42no—
514Dic3—41no—
515Dic3—54no—
516Dic3—102no—
517C2×C5—22no—
518C2×C5—241no—
519C2×C5D3×#13712yes—
520C2×C5C2×#94011yescharacteristic
521D5—81no—
522C23—61no—
523C23—101no—
524C23#677711yescharacteristic
525Q8—62no—
526Q8—61no—
527Q8—62no—
528Q8—61no—
529Q8—102no—
530Q8—101no—
531Q8—102no—
532Q8—101no—
533Q8—122no—
534Q8—202no—
535D4—61no—
536D4—61no—
537D4—101no—
538D4—101no—
539D4—122no—
540D4—202no—
541D4—604no—
542C2×C4—61no—
543C2×C4—62no—
544C2×C4—101no—
545C2×C4—102no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—202no—
551C2×C4—202no—
552C2×C4—304no—
553C2×C4#677711yescharacteristic
554C2×C4#677712yes—
555C2×C4C2×#120112yes—
556C2×C4C22×D3×D511yescharacteristic
557C8—154no—
558C8C2×#120114yes—
559C2×C3—22no—
560C2×C3—401no—
561C2×C3D5×#13712yes—
562C2×C3C2×#408611yescharacteristic
563D3—81no—
564C5#8397211yescharacteristic, complemented, Sylow
565C22—22no—
566C22—122no—
567C22—121no—
568C22—202no—
569C22—201no—
570C22#9146312yes—
571C22C2×#677711yescharacteristic
572C4—121no—
573C4—122no—
574C4—154no—
575C4—201no—
576C4—202no—
577C4—302no—
578C4#9146312yes—
579C4C2×#677711yescharacteristic
580C4C22×#120111yescharacteristic
581C3#1061570311yescharacteristic, complemented, Sylow
582C2—22no—
583C2—241no—
584C2—401no—
585C2#1175403412yes—
586C2C2×#9146311yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radical#11754586C2
hypercenterC2×M16D3×D5
hypocenterC3×C5#3152
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C5×#909C2semidirect
2. C2×C3×#4055C2semidirect
4. C2×#91385C2semidirect
5. C2×#91309C2semidirect
6. C2×#91307C2semidirect
7. M16×Dic15C2semidirect
8. #11748041C2semidirect
9. #11748010C2semidirect
10. #11747983C2semidirect
11. #11747981C2semidirect
50. C2×#4055D3semidirect
81. C3×C5×M16D4semidirect
86. #6710D4semidirect
106. C2×#909D5semidirect
239. C5×M16#67semidirect
279. Dic15#137semidirect
311. C3×M16C22⋊D5semidirect
489. M16#1201semidirect
500. C3×C5#3152semidirect
564. C5#83972semidirect
581. C3#10615703semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9092
C2×C3×#40552
#117545862
C2×#913852
C2×#913092
C2×#913072
M16×Dic152
#117480412
#117480102
#117479832
#117479812
#106157033
#839725