Finite Group Databasebeta

← #14439602

Subgroups — #14439602

Order 1920.

1192 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756284C211yesmaximal, characteristic, complemented
2C3×#65859C211yesmaximal, characteristic, complemented
3C2×#91312C211yesmaximal, characteristic, complemented
4C2×#91311C211yesmaximal, characteristic, complemented
5C2×#91301C211yesmaximal, characteristic, complemented
6C22×#6653C211yesmaximal, characteristic, complemented
7C5×#4997C211yesmaximal, characteristic, complemented
8#11747073C212yesmaximal, complemented
9#11747072C212yesmaximal, complemented
10#11747036C212yesmaximal, complemented
11#11747035C212yesmaximal, complemented
12#10610296—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C22×C3×D20C2211yescharacteristic
17C3×C5×#117C2211yescharacteristic
18#91784—22no—
19#91771—22no—
20#91764C2212yescomplemented
21#91763C2212yescomplemented
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C3×#4041—22no—
25C3×#4028—22no—
26C3×#4021C2212yescomplemented
27C3×#4020C2212yescomplemented
28C2×#6662C2211yescharacteristic, complemented
29C2×#6658C2212yescomplemented
30C2×#6658C2211yescharacteristic, complemented
31#91312—24no—
32#91311—24no—
33C2×#6657C2212yescomplemented
34C2×#6657C2211yescharacteristic, complemented
35#91301—24no—
36C2×#6653—24no—
37C2×#6653C2212yes—
38C2×#6653C2212yes—
39C2×#6653C2212yescomplemented
40C2×#6653C2211yescharacteristic, complemented
41C2×#6653C2212yescomplemented
42C5×#803C2212yescomplemented
43C4×C5×C3⋊C8C2212yescomplemented
44#83382—51nomaximal, Hall
45C2×#4054—31no—
46C2×#4054—31no—
47C2×#4050—31no—
48C22×D40—31no—
49#65859—31no—
50#65859D311yescharacteristic, complemented
51C5×#405—31no—
52#65001—32no—
53#65001—32no—
54#64985—32no—
55#64984—32no—
56C23×D15—21no—
57C23×C3×D5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—21no—
60#6817—22no—
61#6817—22no—
62C2×D60—22no—
63C2×D60—22no—
64C2×D60—21no—
65C2×D60—24no—
66C2×D60C2311yescharacteristic
67C2×D60C2312yes—
68C2×C4×D15—21no—
69C2×Dic30C2311yescharacteristic
70C2×C3×C22⋊D5—21no—
71C3×#709—22no—
72C3×#709—22no—
73C2×C3×D20—22no—
74C2×C3×D20—22no—
75C2×C3×D20—21no—
76C2×C3×D20—24no—
77C2×C3×D20C2312yes—
78C2×C3×D20C2311yescharacteristic
79C2×C3×C4×D5—21no—
80C2×C3×Dic10C2311yescharacteristic
81C3×C5×C4⋊C4D412yes—
82C3×C5×C22⋊C4D412yes—
83C3×C42×C5C2312yes—
84#6714—24no—
85C5×#208D414yescomplemented
86C2×C5×C3⋊C8D414yescomplemented
87C2×C5×C3⋊C8C2311yescharacteristic
88C2×C5×C3⋊C8C2311yescharacteristic
89C2×C5×C3⋊C8C2312yes—
90C3×#685—24no—
91#6662—22no—
92#6658—24no—
93#6658—22no—
94#6657—22no—
95#6657—24no—
96#6653—24no—
97#6653—24no—
98#6653—24no—
99#6653—22no—
100#6653—24no—
101C3×#535—51no—
102C2×#950—51no—
103C2×#949—51no—
104C2×#932—51no—
105C22×#213—51no—
106#5525—51no—
107#5074—52no—
108#5073—52no—
109#5059—52no—
110#5058—52no—
111#4997D511yescharacteristic, complemented
112C2×#709—31no—
113C2×#709C2×D311yescharacteristic
114C22×D20—31no—
115C22×D20C2×D311yescharacteristic
116C2×C5×M16—31no—
117C22×C5×C8—31no—
118C5×#117C2×D311yescharacteristic
119#4054—64no—
120#4054—64no—
121C2×Dic20—31no—
122#4050—64no—
123C2×D40—32no—
124C2×D40—32no—
125C2×D40—31no—
126C2×D40—32no—
127C2×D40—32no—
128C2×D40—64no—
129C2×#677—31no—
130C2×#677—32no—
131C2×#677—31no—
132C2×#677—32no—
133#4041—22no—
134#4041—62no—
135#4028—22no—
136#4028—62no—
137#4021—32no—
138#4021C2×D312yescomplemented
139#4020—32no—
140#4020C2×D312yescomplemented
141C5×#97—32no—
142C4×C5×C8—32no—
143#3155—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×D15—22no—
146C22×D15—21no—
147C22×D15—42no—
148C22×D15—82no—
149C22×C3×D5—22no—
150C22×C3×D5—21no—
151C22×C3×D5—42no—
152C22×C3×D5—82no—
153C2×C3×C4×C5C2×D414yes—
154C2×C3×C4×C5C2×D411yescharacteristic
155C2×C3×C4×C5C2×D412yes—
156C2×C3×C4×C5C2×D412yes—
157C2×C3×C4×C5C2411yescharacteristic
158#1220—42no—
159C2×Dic15—21no—
160D60—22no—
161D60—21no—
162D60—22no—
163D60—21no—
164D60—42no—
165D60—84no—
166C4×D15—42no—
167Dic30—21no—
168Dic30—21no—
169C3×C22⋊D5—42no—
170C2×C3×Dic5—21no—
171C3×D20—22no—
172C3×D20—21no—
173C3×D20—22no—
174C3×D20—21no—
175C3×D20—42no—
176C3×D20—84no—
177C3×C4×D5—42no—
178C3×Dic10—21no—
179C3×Dic10—21no—
180C5×C3⋊C8C2×D412yes—
181C5×C3⋊C8C2×D414yes—
182C5×C3⋊C8C2×D412yes—
183C2×C3×#41—51no—
184C22×C3×D4—51no—
185C2×#235—51no—
186C22×D12—51no—
187C3×#127—52no—
188C3×#124—52no—
189C3×#121—102no—
190C3×#120—102no—
191C3×#117C2×D511yescharacteristic
192#950—104no—
193#949—104no—
194C2×#216—51no—
195C2×#215—52no—
196C2×#215—51no—
197C2×#214—52no—
198C2×#214—51no—
199#932—104no—
200C2×#213—52no—
201C2×#213—52no—
202C2×#213—52no—
203C2×#213—52no—
204C2×#213—51no—
205C2×#213—104no—
206C2×#208C2×D511yescharacteristic
207C22×C3⋊C8C2×D511yescharacteristic
208#895—102no—
209#882—102no—
210#875—52no—
211#874—52no—
212#803C2×D512yescomplemented
213C4×C3⋊C8C2×D512yescomplemented
214C23×D5—21no—
215C23×D5—61no—
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—21no—
218C2×C22⋊D5—61no—
219#709—22no—
220#709—22no—
221#709—62no—
222#709—62no—
223C2×D20—22no—
224C2×D20—22no—
225C2×D20—21no—
226C2×D20—24no—
227C2×D20—32no—
228C2×D20—31no—
229C2×D20—62no—
230C2×D20—62no—
231C2×D20—61no—
232C2×D20—64no—
233C2×D20C22×D312yes—
234C2×D20C22×D311yescharacteristic
235C2×C4×D5—21no—
236C2×C4×D5—61no—
237C2×Dic10—31no—
238C2×Dic10C22×D311yescharacteristic
239C5×M16—34no—
240C2×C5×C8—34no—
241C2×C5×C8—31no—
242C2×C5×C8—31no—
243C2×C5×C8—32no—
244C5×C4⋊C4#6712yes—
245C5×C22⋊C4#6712yes—
246C42×C5C22×D312yes—
247#685—24no—
248#685—64no—
249Dic20—62no—
250D40—64no—
251D40—64no—
252D40—62no—
253D40—64no—
254D40—64no—
255#677—62no—
256#677—64no—
257#677—62no—
258#677—64no—
259C2×AGL(1,ℤ/8)—151no—
260C2×AGL(1,ℤ/8)—151no—
261C2×#135—151no—
262C22×D8—151no—
263#535—51no—
264#535—151no—
265#496—152no—
266#496—152no—
267#495—152no—
268#493—152no—
269#405—31no—
270C22×C3×C5—22no—
271C22×C3×C5#12712yes—
272C22×C3×C5C22×D411yescharacteristic
273C2×D15—41no—
274C2×D15—42no—
275C2×D15—42no—
276C2×D15—44no—
277C2×D15—84no—
278C2×C3×D5—41no—
279C2×C3×D5—42no—
280C2×C3×D5—42no—
281C2×C3×D5—44no—
282C2×C3×D5—84no—
283C3×C4×C5—24no—
284C3×C4×C5#12712yes—
285C3×C4×C5C22×D411yescharacteristic
286C3×C4×C5C22×D411yescharacteristic
287Dic15—41no—
288C3×Dic5—41no—
289C24×C3—101no—
290C23×D3—101no—
291C3×#41—102no—
292C3×#41—102no—
293C2×C3×Q8—51no—
294C2×C3×D4—52no—
295C2×C3×D4—51no—
296C2×C3×D4—101no—
297C2×C3×D4—102no—
298C2×C3×D4—102no—
299C2×C3×D4—101no—
300C2×C3×D4—104no—
301C22×C3×C4—101no—
302C22×C3×C4C22×D511yescharacteristic
303C2×#67—101no—
304#235—102no—
305#235—102no—
306C2×D12—51no—
307C2×D12—52no—
308C2×D12—102no—
309C2×D12—102no—
310C2×D12—101no—
311C2×D12—104no—
312C2×C4×D3—101no—
313C2×Dic6—51no—
314C3×C4⋊C4D2012yes—
315C3×C22⋊C4—104no—
316C3×C22⋊C4D2012yes—
317C3×C42C22×D512yes—
318#216—102no—
319#215—104no—
320#215—102no—
321#214—104no—
322#214—102no—
323#213—104no—
324#213—104no—
325#213—104no—
326#213—104no—
327#213—102no—
328(C2×D3)⋊C4—104no—
329#208D2014yescomplemented
330C2×C3⋊C8D2014yescomplemented
331C2×C3⋊C8C22×D511yescharacteristic
332C2×C3⋊C8C22×D512yes—
333C2×C3⋊C8C22×D511yescharacteristic
334C23×C5C2×#6711yescharacteristic
335C22×D5—22no—
336C22×D5—21no—
337C22×D5—42no—
338C22×D5—62no—
339C22×D5—61no—
340C22×D5—82no—
341C22×D5—122no—
342C22×D5—242no—
343C2×C4×C5D3×D412yes—
344C2×C4×C5D3×D412yes—
345C2×C4×C5C2×#6714yes—
346C2×C4×C5C2×#6711yescharacteristic
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—42no—
349C22⋊D5—122no—
350C2×Dic5—21no—
351C2×Dic5—61no—
352D20—22no—
353D20—21no—
354D20—22no—
355D20—21no—
356D20—42no—
357D20—62no—
358D20—61no—
359D20—62no—
360D20—61no—
361D20—84no—
362D20—122no—
363D20—244no—
364C4×D5—42no—
365C4×D5—122no—
366Dic10—21no—
367Dic10—21no—
368Dic10—61no—
369Dic10—61no—
370C5×C8—32no—
371C5×C8—34no—
372C5×C8—32no—
373C2×#41—51no—
374C2×#41—151no—
375C22×D4—51no—
376C22×D4—151no—
377AGL(1,ℤ/8)—304no—
378AGL(1,ℤ/8)—304no—
379#135—304no—
380C2×Q16—151no—
381C2×SD16—151no—
382C2×SD16—152no—
383C2×SD16—151no—
384C2×SD16—152no—
385C2×D8—152no—
386C2×D8—152no—
387C2×D8—151no—
388C2×D8—152no—
389C2×D8—152no—
390C2×D8—304no—
391C2×M16—31no—
392C22×C8—31no—
393#127—52no—
394#127—152no—
395#124—52no—
396#124—152no—
397#121—102no—
398#121—302no—
399#120—102no—
400#120—302no—
401#117D3×D511yescharacteristic
402#97—32no—
403C4×C8—32no—
404C2×C3×C5—22no—
405C2×C3×C5C2×#12711yescharacteristic
406C2×C3×C5#57612yes—
407D15—81no—
408D15—82no—
409C3×D5—81no—
410C3×D5—82no—
411C23×C3—102no—
412C23×C3—101no—
413C23×C3—202no—
414C23×C3—402no—
415C23×C3C2×D2011yescharacteristic
416C22×D3—102no—
417C22×D3—101no—
418C22×D3—202no—
419C22×D3—402no—
420C3×Q8—101no—
421C3×Q8—101no—
422C3×D4—102no—
423C3×D4—101no—
424C3×D4—102no—
425C3×D4—101no—
426C3×D4—202no—
427C3×D4—202no—
428C3×D4—404no—
429C2×C3×C4—101no—
430C2×C3×C4—202no—
431C2×C3×C4C2×D2012yes—
432C2×C3×C4C2×D2012yes—
433C2×C3×C4C2×D2014yes—
434C2×C3×C4C2×D2011yescharacteristic
435C2×C3×C4C23×D511yescharacteristic
436#67—202no—
437C2×Dic3—101no—
438D12—102no—
439D12—101no—
440D12—102no—
441D12—101no—
442D12—202no—
443D12—404no—
444C4×D3—202no—
445Dic6—101no—
446Dic6—101no—
447C3⋊C8C2×D2012yes—
448C3⋊C8C2×D2012yes—
449C3⋊C8C2×D2014yes—
450C22×C5—22no—
451C22×C5#94012yes—
452C22×C5C22×#6711yescharacteristic
453C2×D5—41no—
454C2×D5—42no—
455C2×D5—42no—
456C2×D5—44no—
457C2×D5—84no—
458C2×D5—121no—
459C2×D5—122no—
460C2×D5—122no—
461C2×D5—124no—
462C2×D5—244no—
463C4×C5—24no—
464C4×C5#94012yes—
465C4×C5C2×D3×D411yescharacteristic
466C4×C5C2×D3×D411yescharacteristic
467Dic5—41no—
468Dic5—121no—
469C24—101no—
470C24—301no—
471#41—102no—
472#41—102no—
473#41—302no—
474#41—302no—
475C2×Q8—51no—
476C2×Q8—151no—
477C2×D4—52no—
478C2×D4—51no—
479C2×D4—101no—
480C2×D4—102no—
481C2×D4—102no—
482C2×D4—101no—
483C2×D4—104no—
484C2×D4—152no—
485C2×D4—151no—
486C2×D4—301no—
487C2×D4—302no—
488C2×D4—302no—
489C2×D4—301no—
490C2×D4—304no—
491C22×C4—101no—
492C22×C4—301no—
493C22×C4C2×D3×D511yescharacteristic
494Q16—302no—
495SD16—302no—
496SD16—304no—
497SD16—302no—
498SD16—304no—
499D8—304no—
500D8—304no—
501D8—302no—
502D8—304no—
503D8—304no—
504M16—34no—
505C2×C8—34no—
506C2×C8—31no—
507C2×C8—32no—
508C2×C8—31no—
509C4⋊C4#120212yes—
510C22⋊C4—104no—
511C22⋊C4—304no—
512C22⋊C4#120212yes—
513C42C2×D3×D512yes—
514C3×C5#315511yescharacteristic, complemented, Hall
515C22×C3—22no—
516C22×C3—201no—
517C22×C3—202no—
518C22×C3—202no—
519C22×C3—204no—
520C22×C3—404no—
521C22×C3#402012yes—
522C22×C3C22×D2011yescharacteristic
523C2×D3—201no—
524C2×D3—202no—
525C2×D3—202no—
526C2×D3—204no—
527C2×D3—404no—
528C3×C4—24no—
529C3×C4—201no—
530C3×C4#402012yes—
531C3×C4C22×D2011yescharacteristic
532C3×C4C22×D2011yescharacteristic
533Dic3—201no—
534C2×C5—22no—
535C2×C5C2×#94011yescharacteristic
536C2×C5#583512yes—
537D5—82no—
538D5—81no—
539D5—242no—
540D5—241no—
541C23—102no—
542C23—101no—
543C23—202no—
544C23—302no—
545C23—301no—
546C23—402no—
547C23—602no—
548C23—1202no—
549C23C2×#120211yescharacteristic
550Q8—101no—
551Q8—101no—
552Q8—301no—
553Q8—301no—
554D4—102no—
555D4—101no—
556D4—102no—
557D4—101no—
558D4—202no—
559D4—202no—
560D4—302no—
561D4—301no—
562D4—302no—
563D4—301no—
564D4—404no—
565D4—602no—
566D4—602no—
567D4—1204no—
568C2×C4—101no—
569C2×C4—202no—
570C2×C4—301no—
571C2×C4—602no—
572C2×C4D3×D2012yes—
573C2×C4D3×D2012yes—
574C2×C4C2×#120214yes—
575C2×C4C2×#120211yescharacteristic
576C2×C4C22×D3×D511yescharacteristic
577C8—32no—
578C8—32no—
579C8—34no—
580C2×C3—22no—
581C2×C3—402no—
582C2×C3—401no—
583C2×C3C2×#402011yescharacteristic
584C2×C3#6608612yes—
585D3—402no—
586D3—401no—
587C5#8338211yescharacteristic, complemented, Sylow
588C22—22no—
589C22—202no—
590C22—204no—
591C22—201no—
592C22—202no—
593C22—404no—
594C22—602no—
595C22—604no—
596C22—601no—
597C22—602no—
598C22—1204no—
599C22#9145812yes—
600C22C22×#120211yescharacteristic
601C4—24no—
602C4—201no—
603C4—601no—
604C4#9145812yes—
605C4C2×D3×D2011yescharacteristic
606C4C2×D3×D2011yescharacteristic
607C3#1061029611yescharacteristic, complemented, Sylow
608C2—22no—
609C2—402no—
610C2—401no—
611C2—1202no—
612C2—1201no—
613C2C2×#9145811yescharacteristic
614C2#1175531112yes—
subgroupG/H
centerC22C22×#1202
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
hypercenter#117D3×D5
hypocenterC3×C5#3155
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #11756284C2semidirect
2. C3×#65859C2semidirect
3. C2×#91312C2semidirect
4. C2×#91311C2semidirect
5. C2×#91301C2semidirect
6. C22×#6653C2semidirect
7. C5×#4997C2semidirect
8. #11747073C2semidirect
9. #11747072C2semidirect
10. #11747036C2semidirect
11. #11747035C2semidirect
20. #91764C22semidirect
21. #91763C22semidirect
26. C3×#4021C22semidirect
27. C3×#4020C22semidirect
28. C2×#6662C22semidirect
29. C2×#6658C22semidirect
30. C2×#6658C22semidirect
33. C2×#6657C22semidirect
34. C2×#6657C22semidirect
39. C2×#6653C22semidirect
40. C2×#6653C22semidirect
41. C2×#6653C22semidirect
42. C5×#803C22semidirect
43. C4×C5×C3⋊C8C22semidirect
50. #65859D3semidirect
85. C5×#208D4semidirect
86. C2×C5×C3⋊C8D4semidirect
111. #4997D5semidirect
138. #4021C2×D3semidirect
140. #4020C2×D3semidirect
212. #803C2×D5semidirect
213. C4×C3⋊C8C2×D5semidirect
329. #208D20semidirect
330. C2×C3⋊C8D20semidirect
514. C3×C5#3155semidirect
587. C5#83382semidirect
607. C3#10610296semidirect

Maximal subgroups

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

subgroupindex
#117562842
C3×#658592
C2×#913122
C2×#913112
C2×#913012
C22×#66532
C5×#49972
#117470732
#117470722
#117470362
#117470352
#106102963
#833825