Finite Group Databasebeta

← #14424273

Subgroups — #14424273

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92018C211yesmaximal, characteristic, complemented
2C3×#65859C211yesmaximal, characteristic, complemented
3C2×#91312C211yesmaximal, characteristic, complemented
4C22×#6652C211yesmaximal, characteristic, complemented
5#11751103C211yesmaximal, characteristic, complemented
6C5×#5001C211yesmaximal, characteristic, complemented
7#11748343C211yesmaximal, characteristic, complemented
8#11747510C214yesmaximal, complemented
9#11747480C214yesmaximal, complemented
10#10610659—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic
13C2×C3×#709C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic, complemented
15C2×D3×C22⋊D5—22no—
16#92018—24no—
17#92018—24no—
18C2×#6777—21no—
19C2×D5×D12C2212yescomplemented
20C2×#6772C2212yescomplemented
21C2×#6768—21no—
22C3×C5×#117C2211yescharacteristic, complemented
23C22×C3⋊#168C2211yescharacteristic, complemented
24C2×C5×#208C2211yescharacteristic, complemented
25C3×#4041C2214yescomplemented
26C3×#4028—22no—
27C3×#4021—21no—
28C3×#4020—21no—
29#91312—24no—
30C2×#6657C2212yescomplemented
31C2×#6653C2212yescomplemented
32C2×#6652—24no—
33C2×#6652C2214yescomplemented
34#91094C2212yescomplemented
35#91093C2212yescomplemented
36C5×#809C2214yescomplemented
37#90955C2214yescomplemented
38#83383—51nomaximal, Hall
39C2×#4144—31no—
40C22×#686—31no—
41C2×#4054—31no—
42#65859D311yescharacteristic, complemented
43C5×#418—31no—
44#65418—31no—
45#65291—31no—
46#65070—34no—
47#65062—34no—
48C23×C5×D3—21no—
49C23×C3×D5—21no—
50C22×D3×D5—22no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×#1220—21no—
53#6817—22no—
54#6817D414yescomplemented
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×Dic30C2311yescharacteristic
58C2×C5×D12—21no—
59C2×C5×D12—22no—
60C2×C5×D12D414yescomplemented
61C2×C5×D12C2312yes—
62C2×C3×C22⋊D5D412yes—
63C3×#709—22no—
64C3×#709—22no—
65C2×C3×D20—22no—
66C2×C3×D20—22no—
67C2×C3×D20—21no—
68C2×C3×D20—24no—
69C2×C3×D20D411yescharacteristic
70C2×C3×D20C2312yes—
71C2×C3×C4×D5C2312yes—
72C2×C3×Dic10D411yescharacteristic
73D3×C22⋊D5—216no—
74C2×#1203—22no—
75C2×#1201—22no—
76C2×D3×Dic5—22no—
77#6777—28no—
78D5×D12—24no—
79D5×D12—24no—
80#6772—24no—
81#6772—24no—
82#6768—28no—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C42×C5—21no—
86C2×C3⋊#168—22no—
87C2×C3⋊#168C2312yes—
88C5×#208—22no—
89C2×C5×C3⋊C8C2312yes—
90C3×#685—24no—
91#6657—24no—
92#6653—24no—
93#6652—28no—
94C2×#1009—51no—
95C3×#535—51no—
96C2×#932—51no—
97C22×#213—51no—
98#5234—51no—
99#5037—54no—
100#5036—54no—
101#5002—51no—
102#5001D511yescharacteristic, complemented
103C22×C5×D4—31no—
104C22×C22⋊D5—62no—
105#4144—64no—
106#4144—64no—
107C2×#711—32no—
108C2×#711—61no—
109C2×D4×D5—32no—
110C2×D4×D5—61no—
111C2×#709—31no—
112C2×#709C2×D311yescharacteristic
113C22×D20C2×D311yescharacteristic, complemented
114C2×C5×M16—31no—
115C5×#117C2×D311yescharacteristic, complemented
116C2×#686—34no—
117C2×#686—64no—
118C22×#168—31no—
119#4054—64no—
120C2×D40—32no—
121C2×#677—32no—
122#4041C2×D314yescomplemented
123#4028—22no—
124#4021—21no—
125#4020—21no—
126C5×#102—34no—
127#3952—34no—
128#3942—32no—
129#3941—32no—
130#3008—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—21no—
133C22×C5×D3—22no—
134C22×C5×D3—24no—
135C22×C5×D3—82no—
136C22×C3×D5—22no—
137C22×C3×D5—42no—
138C22×C3×D5—82no—
139C22×C3×D5C2×D412yes—
140C2×D3×D5—28no—
141C2×D3×D5—84no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2411yescharacteristic
147#1220—24no—
148C2×Dic15—21no—
149D60—21no—
150D60C2×D412yes—
151C4×D15—24no—
152Dic30—21no—
153Dic30C2×D412yes—
154C5×D12—24no—
155C5×D12—22no—
156C5×D12C2×D414yes—
157C3×C22⋊D5—24no—
158C2×C3×Dic5C2×D412yes—
159C3×D20—22no—
160C3×D20—22no—
161C3×D20—22no—
162C3×D20—42no—
163C3×D20—84no—
164C3×C4×D5—22no—
165C3×C4×D5—22no—
166C3×Dic10—22no—
167#1203—28no—
168#1201—28no—
169D3×Dic5—28no—
170C3⋊#168—22no—
171C5×C3⋊C8—22no—
172C2×C3×#41—51no—
173C22×C3×D4—51no—
174#1009—104no—
175#1009—104no—
176C2×#239—101no—
177C2×D3×D4—101no—
178C2×D3×D4—102no—
179C2×#235—51no—
180C2×#235—52no—
181C22×D12—52no—
182C22×D12C2×D511yescharacteristic
183C3×#127—101no—
184C3×#124—101no—
185C3×#121—54no—
186C3×#120—102no—
187C3×#117C2×D511yescharacteristic, complemented
188C2×#214—52no—
189#932—104no—
190C2×#213—54no—
191C2×#213—52no—
192C2×#213—104no—
193C2×#208C2×D511yescharacteristic, complemented
194C22×C3⋊C8—51no—
195#830—54no—
196#810—52no—
197#809—52no—
198#809C2×D514yescomplemented
199C24×C5—61no—
200C23×D5—21no—
201C23×D5—62no—
202C2×C5×D4—34no—
203C2×C5×D4—32no—
204C2×C5×D4—62no—
205C2×C5×D4—61no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—616no—
208C2×C22⋊D5—61no—
209C2×C22⋊D5—62no—
210C2×C22⋊D5—62no—
211C2×C22⋊D5#6712yes—
212C22×Dic5—62no—
213#711—68no—
214#711—64no—
215#711—64no—
216D4×D5—64no—
217D4×D5—64no—
218D4×D5—68no—
219#709—22no—
220#709—22no—
221#709—34no—
222#709—62no—
223C2×D20—22no—
224C2×D20—22no—
225C2×D20—21no—
226C2×D20—24no—
227C2×D20—31no—
228C2×D20#6711yescharacteristic
229C2×D20C22×D312yes—
230C2×C4×D5—61no—
231C2×C4×D5C22×D312yes—
232C2×Dic10—31no—
233C2×Dic10#6711yescharacteristic
234C5×M16—62no—
235C2×C5×C8—32no—
236C5×C4⋊C4C22×D312yes—
237C5×C22⋊C4—21no—
238C42×C5—21no—
239#686—68no—
240#685—24no—
241C2×#168—32no—
242C2×#168—62no—
243D40—64no—
244#677—64no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—51no—
249#449—154no—
250#447—154no—
251#418—31no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—42no—
259C2×C5×D3—24no—
260C2×C5×D3—44no—
261C2×C5×D3—84no—
262C2×C3×D5—22no—
263C2×C3×D5—24no—
264C2×C3×D5—42no—
265C2×C3×D5—44no—
266C2×C3×D5—84no—
267D3×D5—88no—
268C3×C4×C5—42no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272Dic15—22no—
273C3×Dic5—22no—
274C24×C3—101no—
275C23×D3—21no—
276C23×D3—102no—
277C3×#41—102no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×D4—52no—
281C2×C3×D4—51no—
282C2×C3×D4—52no—
283C2×C3×D4—104no—
284C2×C3×D4—102no—
285C2×C3×D4—102no—
286C2×C3×D4—101no—
287C22×C3×C4—52no—
288C22×C3×C4C22×D511yescharacteristic
289C2×#67—102no—
290C2×#67—101no—
291#239—108no—
292D3×D4—108no—
293D3×D4—1016no—
294#235—54no—
295#235—104no—
296#235—102no—
297#235—104no—
298C2×D12—21no—
299C2×D12—22no—
300C2×D12—51no—
301C2×D12—104no—
302C2×D12—102no—
303C2×D12—104no—
304C2×D12D2014yescomplemented
305C2×D12C22×D512yes—
306C2×C4×D3—102no—
307C2×C4×D3—101no—
308C2×Dic6—51no—
309C3×C4⋊C4C22×D512yes—
310C3×C22⋊C4—21no—
311C3×C22⋊C4—104no—
312C3×C42—21no—
313#214—104no—
314#213—104no—
315#213—108no—
316#208—22no—
317C2×C3⋊C8—52no—
318C2×C3⋊C8—102no—
319C2×C3⋊C8C22×D512yes—
320C23×C5—62no—
321C23×C5—64no—
322C23×C5—242no—
323C23×C5C2×#6711yescharacteristic
324C22×D5—22no—
325C22×D5—42no—
326C22×D5—68no—
327C22×D5—61no—
328C22×D5—82no—
329C22×D5—244no—
330C22×D5C2×#6712yes—
331C5×D4—34no—
332C5×D4—64no—
333C5×D4—62no—
334C2×C4×C5—22no—
335C2×C4×C5D3×D412yes—
336C2×C4×C5D3×D412yes—
337C2×C4×C5C2×#6711yescharacteristic
338C2×C4×C5C23×D311yescharacteristic
339C22⋊D5—24no—
340C22⋊D5—68no—
341C22⋊D5—68no—
342C22⋊D5—64no—
343C2×Dic5—68no—
344C2×Dic5—61no—
345C2×Dic5C2×#6712yes—
346D20—22no—
347D20—22no—
348D20—22no—
349D20—32no—
350D20—42no—
351D20—61no—
352D20—84no—
353C4×D5—22no—
354C4×D5—22no—
355C4×D5—64no—
356Dic10—22no—
357Dic10—32no—
358Dic10—61no—
359C5×C8—62no—
360#168—62no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—51no—
364C2×#41—152no—
365C2×#41—151no—
366C2×#41—301no—
367C22×D4—31no—
368C22×D4—51no—
369C22×D4—152no—
370C22×D4—301no—
371C22×D4—302no—
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—152no—
375C2×D8—154no—
376C2×D8—304no—
377C2×M16—31no—
378C22×C8—151no—
379#127—101no—
380#124—101no—
381#121—54no—
382#120—102no—
383#117D3×D511yescharacteristic, complemented
384#103—152no—
385#102—34no—
386#102—152no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5C2×#12011yescharacteristic
390C2×C3×C5#57612yes—
391D15—42no—
392C3×D5—42no—
393C3×D5—82no—
394C5×D3—44no—
395C23×C3—52no—
396C23×C3—102no—
397C23×C3—202no—
398C23×C3—402no—
399C23×C3D4×D511yescharacteristic
400C22×D3—22no—
401C22×D3—24no—
402C22×D3—82no—
403C22×D3—108no—
404C22×D3—101no—
405C22×D3—404no—
406C3×Q8—102no—
407C3×D4—104no—
408C3×D4—102no—
409C3×D4—102no—
410C3×D4—102no—
411C3×D4—202no—
412C3×D4—404no—
413C2×C3×C4—22no—
414C2×C3×C4—52no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4C2×D2012yes—
418C2×C3×C4D4×D512yes—
419C2×C3×C4D4×D511yescharacteristic
420C2×C3×C4C23×D511yescharacteristic
421#67—108no—
422#67—104no—
423C2×Dic3—101no—
424D12—22no—
425D12—24no—
426D12—52no—
427D12—108no—
428D12—101no—
429D12C2×D2014yes—
430C4×D3—108no—
431C4×D3—104no—
432Dic6—52no—
433Dic6—101no—
434C3⋊C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—64no—
438C22×C5—124no—
439C22×C5—244no—
440C22×C5#93712yes—
441C22×C5C22×#6711yescharacteristic
442C2×D5—22no—
443C2×D5—24no—
444C2×D5—42no—
445C2×D5—44no—
446C2×D5—62no—
447C2×D5—84no—
448C2×D5—122no—
449C2×D5—248no—
450C4×C5—42no—
451C4×C5#93712yes—
452C4×C5C2×D3×D411yescharacteristic
453C4×C5C2×D3×D411yescharacteristic
454Dic5—22no—
455Dic5—62no—
456C24—61no—
457C24—101no—
458C24—302no—
459#41—102no—
460#41—102no—
461#41—154no—
462#41—304no—
463#41—304no—
464#41—308no—
465#41—302no—
466C2×Q8—51no—
467C2×Q8—151no—
468C2×D4—34no—
469C2×D4—32no—
470C2×D4—52no—
471C2×D4—51no—
472C2×D4—52no—
473C2×D4—62no—
474C2×D4—61no—
475C2×D4—104no—
476C2×D4—102no—
477C2×D4—102no—
478C2×D4—101no—
479C2×D4—151no—
480C2×D4—302no—
481C2×D4—304no—
482C2×D4—304no—
483C2×D4—308no—
484C2×D4—3016no—
485C2×D4—302no—
486C2×D4—301no—
487C22×C4—52no—
488C22×C4—302no—
489C22×C4—301no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—304no—
492D8—304no—
493D8—308no—
494M16—62no—
495C2×C8—32no—
496C2×C8—152no—
497C2×C8—302no—
498C4⋊C4C2×D3×D512yes—
499C22⋊C4—21no—
500C22⋊C4—104no—
501C42—21no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—102no—
505C22×C3—104no—
506C22×C3—202no—
507C22×C3—204no—
508C22×C3—404no—
509C22×C3#402812yes—
510C22×C3C2×D4×D511yescharacteristic
511C2×D3—24no—
512C2×D3—44no—
513C2×D3—84no—
514C2×D3—102no—
515C2×D3—202no—
516C2×D3—408no—
517C3×C4—42no—
518C3×C4—102no—
519C3×C4#402812yes—
520C3×C4C22×D2011yescharacteristic
521C3×C4C2×D4×D511yescharacteristic
522Dic3—102no—
523C2×C5—22no—
524C2×C5—124no—
525C2×C5C2×#93711yescharacteristic
526C2×C5#583512yes—
527D5—42no—
528D5—82no—
529D5—122no—
530C23—52no—
531C23—62no—
532C23—64no—
533C23—102no—
534C23—202no—
535C23—242no—
536C23—308no—
537C23—301no—
538C23—402no—
539C23—1204no—
540C23D5×#6711yescharacteristic
541Q8—102no—
542Q8—152no—
543Q8—301no—
544D4—34no—
545D4—64no—
546D4—62no—
547D4—104no—
548D4—102no—
549D4—102no—
550D4—102no—
551D4—152no—
552D4—202no—
553D4—308no—
554D4—308no—
555D4—304no—
556D4—301no—
557D4—404no—
558C2×C4—22no—
559C2×C4—52no—
560C2×C4—102no—
561C2×C4—102no—
562C2×C4—308no—
563C2×C4—301no—
564C2×C4—304no—
565C2×C4D3×D2012yes—
566C2×C4#677712yes—
567C2×C4D5×#6711yescharacteristic
568C2×C4C22×D3×D511yescharacteristic
569C8—62no—
570C8—302no—
571C2×C3—22no—
572C2×C3—202no—
573C2×C3—402no—
574C2×C3C2×#402811yescharacteristic
575C2×C3#6610812yes—
576D3—44no—
577D3—202no—
578C5#8338311yescharacteristic, complemented, Sylow
579C22—22no—
580C22—64no—
581C22—104no—
582C22—102no—
583C22—124no—
584C22—204no—
585C22—202no—
586C22—244no—
587C22—302no—
588C22—404no—
589C22—602no—
590C22—1208no—
591C22#9147412yes—
592C22C2×D5×#6711yescharacteristic
593C4—42no—
594C4—102no—
595C4—302no—
596C4#9147412yes—
597C4C2×D3×D2011yescharacteristic
598C4C2×#677711yescharacteristic
599C3#1061065911yescharacteristic, complemented, Sylow
600C2—22no—
601C2—124no—
602C2—202no—
603C2—402no—
604C2—602no—
605C2C2×#9147411yescharacteristic
606C2#1175531612yes—
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
hypercenter#117D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#92018C2semidirect
2. C3×#65859C2semidirect
3. C2×#91312C2semidirect
4. C22×#6652C2semidirect
5. #11751103C2semidirect
6. C5×#5001C2semidirect
7. #11748343C2semidirect
8. #11747510C2semidirect
9. #11747480C2semidirect
14. C22×C3×D20C22semidirect
19. C2×D5×D12C22semidirect
20. C2×#6772C22semidirect
22. C3×C5×#117C22semidirect
23. C22×C3⋊#168C22semidirect
24. C2×C5×#208C22semidirect
25. C3×#4041C22semidirect
30. C2×#6657C22semidirect
31. C2×#6653C22semidirect
33. C2×#6652C22semidirect
34. #91094C22semidirect
35. #91093C22semidirect
36. C5×#809C22semidirect
37. #90955C22semidirect
42. #65859D3semidirect
54. #6817D4semidirect
60. C2×C5×D12D4semidirect
102. #5001D5semidirect
113. C22×D20C2×D3semidirect
115. C5×#117C2×D3semidirect
122. #4041C2×D3semidirect
187. C3×#117C2×D5semidirect
193. C2×#208C2×D5semidirect
198. #809C2×D5semidirect
304. C2×D12D20semidirect
383. #117D3×D5semidirect
502. C3×C5#3008semidirect
578. C5#83383semidirect
599. C3#10610659semidirect

Maximal subgroups

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

subgroupindex
C2×#920182
C3×#658592
C2×#913122
C22×#66522
#117511032
C5×#50012
#117483432
#117475102
#117474802
#106106593
#833835