Finite Group Databasebeta

← #14424274

Subgroups — #14424274

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
2#11756284C211yesmaximal, characteristic, complemented
3C22×#6654C211yesmaximal, characteristic, complemented
4C2×#91301C211yesmaximal, characteristic, complemented
5C5×#5001C211yesmaximal, characteristic, complemented
6C3×#65291C211yesmaximal, characteristic, complemented
7#11748349C211yesmaximal, characteristic, complemented
8#11747520C214yesmaximal, complemented
9#11747485C214yesmaximal, complemented
10#10610659—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×D60C2211yescharacteristic, complemented
13C22×C5×D12C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×D3×C22⋊D5—22no—
16#92018—24no—
17#92018—24no—
18C2×#6777C2212yescomplemented
19C2×D5×D12—21no—
20C2×#6772—21no—
21C2×#6768C2212yescomplemented
22C3×C5×#117C2211yescharacteristic, complemented
23#91784C2214yescomplemented
24#91771—22no—
25#91764—21no—
26#91763—21no—
27C2×C5×#208C2211yescharacteristic, complemented
28C22×C3×#168C2211yescharacteristic, complemented
29C2×#6658C2212yescomplemented
30C2×#6654—24no—
31C2×#6654C2214yescomplemented
32#91301—24no—
33C2×#6653C2212yescomplemented
34C5×#809C2214yescomplemented
35C3×#3942C2212yescomplemented
36C3×#3941C2212yescomplemented
37#90957C2214yescomplemented
38#78834—51nomaximal, Hall
39C2×#4144—31no—
40C22×#686—31no—
41C2×#4054—31no—
42#65859—31no—
43C5×#418—31no—
44#65418—31no—
45#65291D311yescharacteristic, complemented
46#65070—34no—
47#65062—34no—
48C23×D15—21no—
49C23×C5×D3—21no—
50C22×D3×D5—22no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×#1220D412yes—
53#6817—22no—
54#6817—22no—
55C2×D60—22no—
56C2×D60—22no—
57C2×D60—21no—
58C2×D60—24no—
59C2×D60D411yescharacteristic
60C2×D60C2312yes—
61C2×C4×D15C2312yes—
62C2×Dic30D411yescharacteristic
63C2×C5×D12—21no—
64C2×C5×D12—22no—
65C2×C5×D12D414yescomplemented
66C2×C5×D12C2312yes—
67C2×C3×C22⋊D5—21no—
68C3×#709—22no—
69C3×#709D414yescomplemented
70C2×C3×D20C2311yescharacteristic
71C2×C3×C4×D5—21no—
72C2×C3×Dic10C2311yescharacteristic
73D3×C22⋊D5—216no—
74C2×#1203—22no—
75C2×#1201—22no—
76C2×D3×Dic5—22no—
77#6777—24no—
78#6777—24no—
79D5×D12—28no—
80#6772—28no—
81#6768—24no—
82#6768—24no—
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C42×C5—21no—
86#6714—24no—
87C5×#208—22no—
88C2×C5×C3⋊C8C2312yes—
89C2×C3×#168—22no—
90C2×C3×#168C2312yes—
91#6658—24no—
92#6654—28no—
93#6653—24no—
94C2×#1009—51no—
95C2×#949—51no—
96C22×D24—51no—
97#5525—51no—
98C3×#416—51no—
99#5110—51no—
100#5001D511yescharacteristic, complemented
101#4810—54no—
102#4773—54no—
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×D20—31no—
114C2×C5×M16—31no—
115C5×#117C2×D311yescharacteristic, complemented
116C2×#686—34no—
117C2×#686—64no—
118C22×#168C2×D311yescharacteristic, complemented
119#4054—64no—
120C2×D40—32no—
121C2×#677—32no—
122#4041—34no—
123#4028—62no—
124#4021—61no—
125#4020—61no—
126C5×#102—34no—
127#3952—34no—
128#3942C2×D312yescomplemented
129#3941C2×D312yescomplemented
130#3008—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—22no—
133C22×D15—42no—
134C22×D15—82no—
135C22×D15C2×D412yes—
136C22×C5×D3—22no—
137C22×C5×D3—24no—
138C22×C5×D3—82no—
139C22×C3×D5—21no—
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×Dic15C2×D412yes—
149D60—22no—
150D60—22no—
151D60—22no—
152D60—42no—
153D60—84no—
154C4×D15—22no—
155C4×D15—22no—
156Dic30—22no—
157C5×D12—24no—
158C5×D12—22no—
159C5×D12C2×D414yes—
160C3×C22⋊D5—24no—
161C2×C3×Dic5—21no—
162C3×D20—21no—
163C3×D20C2×D412yes—
164C3×C4×D5—24no—
165C3×Dic10—21no—
166C3×Dic10C2×D412yes—
167#1203—28no—
168#1201—28no—
169D3×Dic5—28no—
170C3×#168—22no—
171C5×C3⋊C8—22no—
172C2×C3×#41—51no—
173#1009—104no—
174#1009—104no—
175C2×#239—52no—
176C2×D3×D4—52no—
177C2×D3×D4—102no—
178C2×#235—51no—
179C2×#235—101no—
180C22×D12—51no—
181C22×D12—101no—
182C22×D12C2×D511yescharacteristic
183C22×C3×C8—51no—
184C3×#117C2×D511yescharacteristic, complemented
185#949—104no—
186C2×#215—52no—
187C2×#213—52no—
188C2×#208C2×D511yescharacteristic, complemented
189C2×D24—54no—
190C2×D24—104no—
191#895—54no—
192#882—102no—
193#875—101no—
194#874—101no—
195C3×#103—52no—
196C3×#102—52no—
197(C2×D3)⋊C8—54no—
198#809C2×D514yescomplemented
199C24×C5—61no—
200C23×D5—62no—
201C23×D5—61no—
202C2×C5×D4—34no—
203C2×C5×D4—32no—
204C2×C5×D4—62no—
205C2×C5×D4—61no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—21no—
208C2×C22⋊D5—32no—
209C2×C22⋊D5—62no—
210C2×C22⋊D5—62no—
211C2×C22⋊D5—616no—
212C22×Dic5—62no—
213#711—64no—
214#711—68no—
215#711—64no—
216D4×D5—68no—
217D4×D5—64no—
218D4×D5—64no—
219#709—22no—
220#709—62no—
221#709—62no—
222#709D1214yescomplemented
223C2×D20—31no—
224C2×D20—32no—
225C2×D20—62no—
226C2×D20—62no—
227C2×D20—61no—
228C2×D20—64no—
229C2×D20C22×D311yescharacteristic
230C2×C4×D5—21no—
231C2×C4×D5—32no—
232C2×Dic10—31no—
233C2×Dic10C22×D311yescharacteristic
234C5×M16—62no—
235C2×C5×C8—32no—
236C5×C4⋊C4C22×D312yes—
237C5×C22⋊C4—21no—
238C42×C5—21no—
239#686—68no—
240#685—64no—
241C2×#168—22no—
242C2×#168C22×D312yes—
243D40—64no—
244#677—64no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—31no—
252#416—51no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—24no—
259C2×D15—42no—
260C2×D15—44no—
261C2×D15—84no—
262C2×C5×D3—24no—
263C2×C5×D3—44no—
264C2×C5×D3—84no—
265C2×C3×D5—22no—
266C2×C3×D5—42no—
267D3×D5—88no—
268C3×C4×C5—42no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272Dic15—22no—
273C3×Dic5—22no—
274C23×D3—21no—
275C23×D3—102no—
276C23×D3—101no—
277C3×#41—54no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×D4—51no—
281C2×C3×D4—101no—
282C22×C3×C4—101no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—52no—
285C2×#67—102no—
286#239—104no—
287#239—104no—
288D3×D4—104no—
289D3×D4—1016no—
290D3×D4—104no—
291#235—108no—
292#235—102no—
293#235—102no—
294C2×D12—21no—
295C2×D12—22no—
296C2×D12—51no—
297C2×D12—52no—
298C2×D12—108no—
299C2×D12—102no—
300C2×D12—102no—
301C2×D12—102no—
302C2×D12—101no—
303C2×D12—104no—
304C2×D12D2014yescomplemented
305C2×D12C22×D512yes—
306C2×C4×D3—52no—
307C2×C4×D3—102no—
308C2×Dic6—51no—
309C2×C3×C8—52no—
310C2×C3×C8—102no—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—21no—
313C3×C42—21no—
314#215—104no—
315#213—104no—
316(C2×D3)⋊C4—104no—
317#208—22no—
318C2×C3⋊C8C22×D512yes—
319D24—108no—
320C23×C5—62no—
321C23×C5—64no—
322C23×C5—242no—
323C23×C5D3×D411yescharacteristic
324C22×D5—21no—
325C22×D5—32no—
326C22×D5—68no—
327C22×D5—62no—
328C22×D5—122no—
329C22×D5—244no—
330C22×D5—242no—
331C5×D4—34no—
332C5×D4—64no—
333C5×D4—62no—
334C2×C4×C5—22no—
335C2×C4×C5C2×D1212yes—
336C2×C4×C5D3×D412yes—
337C2×C4×C5D3×D411yescharacteristic
338C2×C4×C5C23×D311yescharacteristic
339C22⋊D5—24no—
340C22⋊D5—68no—
341C22⋊D5—68no—
342C22⋊D5—64no—
343C2×Dic5—21no—
344C2×Dic5—32no—
345C2×Dic5—68no—
346D20—21no—
347D20—62no—
348D20—62no—
349D20—62no—
350D20—122no—
351D20—244no—
352D20C2×D1212yes—
353C4×D5—24no—
354C4×D5—62no—
355C4×D5—62no—
356Dic10—21no—
357Dic10—62no—
358Dic10C2×D1212yes—
359C5×C8—62no—
360#168—22no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—51no—
364C2×#41—151no—
365C2×#41—152no—
366C2×#41—301no—
367C22×D4—31no—
368C22×D4—152no—
369C22×D4—151no—
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—51no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117D3×D511yescharacteristic, complemented
384#103—52no—
385#102—34no—
386#102—52no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5C2×#12011yescharacteristic
390C2×C3×C5#57612yes—
391D15—42no—
392D15—82no—
393C3×D5—42no—
394C5×D3—44no—
395C23×C3—101no—
396C23×C3D4×D511yescharacteristic
397C22×D3—22no—
398C22×D3—24no—
399C22×D3—52no—
400C22×D3—82no—
401C22×D3—108no—
402C22×D3—102no—
403C22×D3—202no—
404C22×D3—404no—
405C22×D3—402no—
406C3×Q8—52no—
407C3×Q8—101no—
408C3×D4—52no—
409C3×D4—104no—
410C3×D4—101no—
411C2×C3×C4—22no—
412C2×C3×C4—101no—
413C2×C3×C4—104no—
414C2×C3×C4C2×D2012yes—
415C2×C3×C4D4×D512yes—
416C2×C3×C4D4×D511yescharacteristic
417C2×C3×C4C23×D511yescharacteristic
418#67—108no—
419#67—104no—
420C2×Dic3—52no—
421D12—22no—
422D12—24no—
423D12—108no—
424D12—102no—
425D12—102no—
426D12—102no—
427D12—202no—
428D12—404no—
429D12C2×D2014yes—
430C4×D3—108no—
431C4×D3—102no—
432C4×D3—102no—
433Dic6—102no—
434C3×C8—102no—
435C3⋊C8—22no—
436C22×C5—22no—
437C22×C5—64no—
438C22×C5—124no—
439C22×C5—244no—
440C22×C5#88212yes—
441C22×C5C2×D3×D411yescharacteristic
442C2×D5—22no—
443C2×D5—42no—
444C2×D5—62no—
445C2×D5—64no—
446C2×D5—122no—
447C2×D5—124no—
448C2×D5—248no—
449C2×D5—244no—
450C4×C5—42no—
451C4×C5#88212yes—
452C4×C5C22×D1211yescharacteristic
453C4×C5C2×D3×D411yescharacteristic
454Dic5—22no—
455Dic5—62no—
456C24—61no—
457C24—302no—
458C24—301no—
459#41—54no—
460#41—102no—
461#41—308no—
462#41—304no—
463#41—304no—
464#41—302no—
465#41—302no—
466C2×Q8—51no—
467C2×Q8—151no—
468C2×D4—34no—
469C2×D4—32no—
470C2×D4—51no—
471C2×D4—62no—
472C2×D4—61no—
473C2×D4—101no—
474C2×D4—152no—
475C2×D4—151no—
476C2×D4—152no—
477C2×D4—304no—
478C2×D4—308no—
479C2×D4—302no—
480C2×D4—302no—
481C2×D4—304no—
482C2×D4—304no—
483C2×D4—3016no—
484C2×D4—302no—
485C2×D4—302no—
486C2×D4—301no—
487C22×C4—101no—
488C22×C4—152no—
489C22×C4—302no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—304no—
492D8—304no—
493D8—308no—
494M16—62no—
495C2×C8—32no—
496C2×C8—52no—
497C2×C8—102no—
498C4⋊C4C2×D3×D512yes—
499C22⋊C4—21no—
500C22⋊C4—304no—
501C42—21no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—102no—
505C22×C3—202no—
506C22×C3#402812yes—
507C22×C3C2×D4×D511yescharacteristic
508C2×D3—24no—
509C2×D3—44no—
510C2×D3—84no—
511C2×D3—102no—
512C2×D3—104no—
513C2×D3—202no—
514C2×D3—204no—
515C2×D3—408no—
516C2×D3—404no—
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×#88211yescharacteristic
526C2×C5#576912yes—
527D5—42no—
528D5—122no—
529D5—242no—
530C23—62no—
531C23—64no—
532C23—101no—
533C23—152no—
534C23—242no—
535C23—308no—
536C23—302no—
537C23—602no—
538C23—1202no—
539C23—1204no—
540C23#679011yescharacteristic
541Q8—52no—
542Q8—101no—
543Q8—302no—
544D4—34no—
545D4—52no—
546D4—64no—
547D4—62no—
548D4—104no—
549D4—101no—
550D4—308no—
551D4—308no—
552D4—304no—
553D4—302no—
554D4—302no—
555D4—302no—
556D4—602no—
557D4—1204no—
558C2×C4—22no—
559C2×C4—101no—
560C2×C4—104no—
561C2×C4—152no—
562C2×C4—308no—
563C2×C4—302no—
564C2×C4—302no—
565C2×C4D5×D1212yes—
566C2×C4D3×D2012yes—
567C2×C4#679011yescharacteristic
568C2×C4C22×D3×D511yescharacteristic
569C8—62no—
570C8—102no—
571C2×C3—22no—
572C2×C3—202no—
573C2×C3C2×#402811yescharacteristic
574C2×C3#6610812yes—
575D3—44no—
576D3—202no—
577D3—402no—
578C5#7883411yescharacteristic, complemented, Sylow
579C22—22no—
580C22—64no—
581C22—102no—
582C22—124no—
583C22—202no—
584C22—244no—
585C22—304no—
586C22—302no—
587C22—604no—
588C22—602no—
589C22—1204no—
590C22—1208no—
591C22#9147612yes—
592C22C2×#679011yescharacteristic
593C4—42no—
594C4—102no—
595C4—302no—
596C4#9147612yes—
597C4C2×D5×D1211yescharacteristic
598C4C2×D3×D2011yescharacteristic
599C3#1061065911yescharacteristic, complemented, Sylow
600C2—22no—
601C2—124no—
602C2—202no—
603C2—602no—
604C2—1202no—
605C2C2×#9147611yescharacteristic
606C2#1175533212yes—
subgroupG/H
centerC22C2×#6790
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. #11756284C2semidirect
3. C22×#6654C2semidirect
4. C2×#91301C2semidirect
5. C5×#5001C2semidirect
6. C3×#65291C2semidirect
7. #11748349C2semidirect
8. #11747520C2semidirect
9. #11747485C2semidirect
12. C22×D60C22semidirect
18. C2×#6777C22semidirect
21. C2×#6768C22semidirect
22. C3×C5×#117C22semidirect
23. #91784C22semidirect
27. C2×C5×#208C22semidirect
28. C22×C3×#168C22semidirect
29. C2×#6658C22semidirect
31. C2×#6654C22semidirect
33. C2×#6653C22semidirect
34. C5×#809C22semidirect
35. C3×#3942C22semidirect
36. C3×#3941C22semidirect
37. #90957C22semidirect
45. #65291D3semidirect
65. C2×C5×D12D4semidirect
69. C3×#709D4semidirect
100. #5001D5semidirect
115. C5×#117C2×D3semidirect
118. C22×#168C2×D3semidirect
128. #3942C2×D3semidirect
129. #3941C2×D3semidirect
184. C3×#117C2×D5semidirect
188. C2×#208C2×D5semidirect
198. #809C2×D5semidirect
222. #709D12semidirect
304. C2×D12D20semidirect
383. #117D3×D5semidirect
502. C3×C5#3008semidirect
578. C5#78834semidirect
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
#117562842
C22×#66542
C2×#913012
C5×#50012
C3×#652912
#117483492
#117475202
#117474852
#106106593
#788345