Finite Group Databasebeta

← #14424363

Subgroups — #14424363

Order 1920.

1504 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×#92019C211yesmaximal, characteristic
2C22×#6717C211yesmaximal, characteristic, complemented
3C5×#5816C211yesmaximal, characteristic, complemented
4C2×C3×#4088C211yesmaximal, characteristic, complemented
5#11749408C214yesmaximal, complemented
6#11749398C214yesmaximal, complemented
7#11748396C211yesmaximal, characteristic, complemented
8#11748364C211yesmaximal, characteristic, complemented
9#11748347C211yesmaximal, characteristic, complemented
10#10615135—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic, complemented
12C22×D60C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×#6790—22no—
16#92019—24no—
17#92019—24no—
18C2×D3×D20C2212yes—
19C2×D5×D12—21no—
20C2×#6773—21no—
21C2×#6769C2212yes—
22C2×#6717—24no—
23C2×#6717C2214yescomplemented
24C22×C3⋊#168C2211yescharacteristic, complemented
25C5×#947—21no—
26C5×#946C2214yescomplemented
27C5×#944—21no—
28C5×#929—22no—
29C5×#926C2211yescharacteristic, complemented
30C2×C3×#689C2212yescomplemented
31C3×#4088—24no—
32C2×C3×#688C2212yescomplemented
33C2×C3×#681C2211yescharacteristic, complemented
34#90970C2212yescomplemented
35#90969C2214yescomplemented
36#90964C2212yescomplemented
37(C2×D3)⋊#168C2214yescomplemented
38#83982—51nomaximal, Hall
39C2×#4149—31no—
40C5×#536—31no—
41C2×#4088D311yescharacteristic, complemented
42C22×#688—31no—
43#65523—31no—
44#65336—34no—
45#65335—34no—
46#65291—31no—
47#65290—31no—
48C23×D15—21no—
49C22×D3×D5—22no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8C2312yes—
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×D60—21no—
57C2×D60—22no—
58C2×D60D414yes—
59C2×D60C2312yes—
60C2×C5×#67D412yes—
61C5×#235—22no—
62C5×#235—22no—
63C2×C5×D12D411yescharacteristic
64C2×C4×C5×D3C2312yes—
65C2×C5×Dic6D411yescharacteristic
66C2×C3×C22⋊D5—21no—
67C3×#709—22no—
68C3×#709D414yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72#6790—216no—
73C2×#1203—22no—
74C2×#1202—22no—
75C2×#1200—22no—
76D3×D20—24no—
77D3×D20—24no—
78D5×D12—28no—
79#6773—28no—
80#6769—24no—
81#6769—24no—
82#6717—28no—
83C2×C3⋊#168—22no—
84C2×C3⋊#168C2312yes—
85C5×#217—21no—
86C5×(C2×D3)⋊C4—24no—
87C5×C4⋊Dic3C2312yes—
88C5×Dic3⋊C4—24no—
89C4×C5×Dic3—21no—
90C3×#689—24no—
91C3×#688—24no—
92C3×#681—22no—
93C2×C3×#168C2312yes—
94C2×#1009—51no—
95C2×C3×#137—51no—
96#5816D511yescharacteristic, complemented
97C22×#215—51no—
98#5233—51no—
99#5187—54no—
100#5169—54no—
101#5131—51no—
102#5129—51no—
103C2×C5×#41—31no—
104C22×C5×Q8C2×D311yescharacteristic, complemented
105#4149—64no—
106#4149—64no—
107C2×#713—61no—
108C2×D4×D5—61no—
109C2×D4×D5—62no—
110C2×#709—32no—
111C2×#709C2×D311yescharacteristic
112C22×D20—31no—
113C22×D20—32no—
114C5×C4⋊Q8—61no—
115C5×#124—61no—
116C5×#123—62no—
117C5×C22⋊Q8—34no—
118C5×#117—31no—
119C2×#689C2×D312yescomplemented
120#4088—24no—
121C2×#688—34no—
122C2×#688—64no—
123C2×#688C2×D312yescomplemented
124C2×#681C2×D311yescharacteristic, complemented
125C22×#168—31no—
126C22⋊#168—34no—
127#3942—32no—
128#3941—32no—
129#3941—34no—
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—22no—
133C22×D15—24no—
134C22×D15—82no—
135C22×C5×D3C2×D412yes—
136C22×C3×D5—21no—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C3×C5×Q8—22no—
140C3×C5×Q8—22no—
141C3×C5×Q8—42no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—42no—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2411yescharacteristic
148D60—24no—
149D60—22no—
150D60C2×D414yes—
151C5×#67—24no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3C2×D412yes—
154C5×D12—22no—
155C4×C5×D3—22no—
156C4×C5×D3—22no—
157C5×Dic6—22no—
158C3×C22⋊D5—24no—
159C2×C3×Dic5—21no—
160C3×D20—21no—
161C3×D20C2×D412yes—
162C3×C4×D5—24no—
163C3×Dic10—21no—
164C3×Dic10C2×D412yes—
165#1203—28no—
166#1202—28no—
167#1200—28no—
168C3⋊#168—22no—
169C3×#168—22no—
170C2×C3×#41—51no—
171C22×C3×Q8C2×D511yescharacteristic, complemented
172#1009—104no—
173#1009—104no—
174C2×#239—52no—
175C2×D3×D4—52no—
176C2×D3×D4—102no—
177C2×#235—101no—
178C2×#235C2×D511yescharacteristic
179C22×D12—51no—
180C22×D12—101no—
181C3×#137—104no—
182C2×C3×Q16—52no—
183C2×C3×SD16—52no—
184C2×C3×M16—51no—
185#947—21no—
186#946C2×D514yescomplemented
187#944—21no—
188C2×#215—54no—
189C2×#215—104no—
190#929—22no—
191#926C2×D511yescharacteristic, complemented
192C22×C3⋊C8—51no—
193#835—52no—
194#832—52no—
195#821—54no—
196(C2×D3)⋊C8—54no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—62no—
200C5×#41—62no—
201C2×C5×Q8—21no—
202C2×C5×Q8—22no—
203C2×C5×Q8—22no—
204C2×C5×Q8—31no—
205C2×C5×Q8C22×D312yes—
206C2×C5×D4—32no—
207C2×C5×D4—31no—
208C22×C4×C5—21no—
209C22×C4×C5—32no—
210C22×C4×C5C22×D311yescharacteristic
211C2×C22⋊D5—21no—
212C2×C22⋊D5—62no—
213#713—68no—
214D4×D5—68no—
215D4×D5—616no—
216#709—22no—
217#709—64no—
218#709—64no—
219#709#6714yes—
220C2×D20—34no—
221C2×D20—32no—
222C2×D20—64no—
223C2×D20—62no—
224C2×D20—61no—
225C2×D20—62no—
226C2×D20—64no—
227C2×D20C22×D311yescharacteristic
228C2×C4×D5—21no—
229C2×C4×D5—62no—
230C2×Dic10C22×D311yescharacteristic
231C5×C4⋊C4—32no—
232C5×C4⋊C4—64no—
233C5×C22⋊C4—64no—
234C5×C22⋊C4—61no—
235C42×C5—61no—
236#689—24no—
237#688—24no—
238#688—68no—
239#681—22no—
240C2×#168—32no—
241C2×#168—62no—
242C2×#168C22×D312yes—
243C2×H(5,𝔽2)—151no—
244C2×#137—51no—
245C22×SD16—151no—
246#536—31no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—24no—
256C2×D15—44no—
257C2×D15—84no—
258C2×C5×D3—22no—
259C2×C5×D3—24no—
260C2×C3×D5—22no—
261C2×C3×D5—42no—
262D3×D5—88no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267C3×Dic5—22no—
268C5×Dic3—22no—
269C5×Dic3—42no—
270C23×D3—101no—
271C23×D3—102no—
272C3×#41—54no—
273C3×#41—102no—
274C2×C3×Q8—21no—
275C2×C3×Q8—22no—
276C2×C3×Q8—22no—
277C2×C3×Q8—51no—
278C2×C3×Q8C22×D512yes—
279C2×C3×D4—51no—
280C2×C3×D4—101no—
281C22×C3×C4—21no—
282C22×C3×C4—101no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—102no—
285C2×#67C22⋊D512yes—
286#239—104no—
287#239—104no—
288D3×D4—104no—
289D3×D4—104no—
290D3×D4—1016no—
291#235—22no—
292#235—22no—
293#235—108no—
294C2×D12—54no—
295C2×D12—52no—
296C2×D12—108no—
297C2×D12—102no—
298C2×D12—102no—
299C2×D12—101no—
300C2×D12C22⋊D511yescharacteristic
301C2×C4×D3—102no—
302C2×C4×D3C22×D512yes—
303C2×Dic6C22⋊D511yescharacteristic
304C3×Q16—104no—
305C3×SD16—104no—
306C3×M16—102no—
307C2×C3×C8—52no—
308#217—21no—
309#215—108no—
310(C2×D3)⋊C4—24no—
311C4⋊Dic3C22×D512yes—
312Dic3⋊C4—24no—
313C4×Dic3—21no—
314C2×C3⋊C8—52no—
315C2×C3⋊C8—102no—
316C23×C5—32no—
317C23×C5D3×D411yescharacteristic
318C22×D5—21no—
319C22×D5—68no—
320C22×D5—62no—
321C22×D5—64no—
322C22×D5—244no—
323C22×D5—242no—
324C5×Q8—22no—
325C5×Q8—22no—
326C5×Q8—42no—
327C5×Q8—62no—
328C5×D4—64no—
329C5×D4—62no—
330C2×C4×C5—22no—
331C2×C4×C5—32no—
332C2×C4×C5—42no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5D3×D411yescharacteristic
337C2×C4×C5D3×D412yes—
338C2×C4×C5C2×#6712yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—24no—
341C22⋊D5—68no—
342C2×Dic5—21no—
343D20—21no—
344D20—34no—
345D20—68no—
346D20—64no—
347D20—62no—
348D20C2×#6712yes—
349C4×D5—24no—
350C4×D5—68no—
351Dic10—21no—
352Dic10C2×#6712yes—
353#168—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—31no—
358C2×#41—51no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8D3×D511yescharacteristic, complemented
362C22×D4—152no—
363C22×D4—151no—
364C22×D4—301no—
365C22×D4—302no—
366#137—104no—
367C2×Q16—52no—
368C2×SD16—52no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—51no—
372C22×C8—151no—
373C4⋊Q8—61no—
374#124—61no—
375#123—62no—
376C22⋊Q8—34no—
377#117—31no—
378#103—152no—
379#102—154no—
380#102—152no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—44no—
386C3×D5—42no—
387C5×D3—42no—
388C23×C3—101no—
389C23×C3C2×C22⋊D511yescharacteristic
390C22×D3—108no—
391C22×D3—102no—
392C22×D3—104no—
393C22×D3—404no—
394C22×D3—402no—
395C22×D3C2×C22⋊D512yes—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—42no—
399C3×Q8—52no—
400C3×Q8—101no—
401C3×D4—52no—
402C3×D4—104no—
403C3×D4—101no—
404C2×C3×C4—22no—
405C2×C3×C4—42no—
406C2×C3×C4—101no—
407C2×C3×C4—104no—
408C2×C3×C4D4×D512yes—
409C2×C3×C4D4×D512yes—
410C2×C3×C4C2×C22⋊D511yescharacteristic
411C2×C3×C4C23×D511yescharacteristic
412#67—24no—
413#67—108no—
414C2×Dic3—22no—
415C2×Dic3C2×C22⋊D512yes—
416D12—22no—
417D12—54no—
418D12—108no—
419D12—102no—
420D12—104no—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—108no—
424Dic6—22no—
425C3×C8—102no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—64no—
429C22×C5—62no—
430C22×C5#93712yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—22no—
433C2×D5—42no—
434C2×D5—64no—
435C2×D5—124no—
436C2×D5—248no—
437C2×D5—244no—
438C4×C5—42no—
439C4×C5—62no—
440C4×C5—122no—
441C4×C5#93712yes—
442C4×C5C2×D3×D411yescharacteristic
443C4×C5C22×#6711yescharacteristic
444Dic5—22no—
445C24—302no—
446C24—301no—
447#41—54no—
448#41—62no—
449#41—62no—
450#41—102no—
451#41—304no—
452#41—304no—
453#41—308no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—31no—
458C2×Q8—51no—
459C2×Q8C2×D3×D512yes—
460C2×D4—32no—
461C2×D4—31no—
462C2×D4—51no—
463C2×D4—101no—
464C2×D4—154no—
465C2×D4—152no—
466C2×D4—302no—
467C2×D4—304no—
468C2×D4—304no—
469C2×D4—308no—
470C2×D4—302no—
471C2×D4—3016no—
472C2×D4—302no—
473C2×D4—301no—
474C22×C4—21no—
475C22×C4—32no—
476C22×C4—101no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—104no—
480SD16—104no—
481SD16—308no—
482M16—102no—
483C2×C8—52no—
484C2×C8—152no—
485C2×C8—302no—
486C4⋊C4—32no—
487C4⋊C4—64no—
488C22⋊C4—64no—
489C22⋊C4—61no—
490C42—61no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—102no—
494C22×C3—202no—
495C22×C3#408312yes—
496C22×C3C22×C22⋊D511yescharacteristic
497C2×D3—22no—
498C2×D3—24no—
499C2×D3—104no—
500C2×D3—204no—
501C2×D3—408no—
502C2×D3—404no—
503C3×C4—42no—
504C3×C4—102no—
505C3×C4#408312yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C2×D4×D511yescharacteristic
508Dic3—22no—
509Dic3—42no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5#577812yes—
513C2×C5C2×#93711yescharacteristic
514D5—42no—
515D5—124no—
516C23—32no—
517C23—101no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23D3×C22⋊D511yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—42no—
527Q8—52no—
528Q8—62no—
529Q8—101no—
530D4—52no—
531D4—64no—
532D4—62no—
533D4—104no—
534D4—101no—
535D4—154no—
536D4—308no—
537D4—308no—
538D4—302no—
539D4—304no—
540C2×C4—22no—
541C2×C4—32no—
542C2×C4—42no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—101no—
547C2×C4—104no—
548C2×C4—308no—
549C2×C4D5×#6712yes—
550C2×C4D3×C22⋊D511yescharacteristic
551C2×C4#679012yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—102no—
554C8—302no—
555C2×C3—22no—
556C2×C3—202no—
557C2×C3C2×#408311yescharacteristic
558C2×C3#6617112yes—
559D3—42no—
560D3—204no—
561C5#8398211yescharacteristic, complemented, Sylow
562C22—22no—
563C22—64no—
564C22—62no—
565C22—102no—
566C22—202no—
567C22—304no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9157712yes—
572C22C2×D3×C22⋊D511yescharacteristic
573C4—42no—
574C4—62no—
575C4—102no—
576C4—122no—
577C4#9157712yes—
578C4C2×D5×#6711yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061513511yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—202no—
584C2—604no—
585C2#1175509212yes—
586C2C2×#9157711yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92019C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#6717C2semidirect
3. C5×#5816C2semidirect
4. C2×C3×#4088C2semidirect
5. #11749408C2semidirect
6. #11749398C2semidirect
7. #11748396C2semidirect
8. #11748364C2semidirect
9. #11748347C2semidirect
11. C22×C3×C5×Q8C22semidirect
23. C2×#6717C22semidirect
24. C22×C3⋊#168C22semidirect
26. C5×#946C22semidirect
29. C5×#926C22semidirect
30. C2×C3×#689C22semidirect
32. C2×C3×#688C22semidirect
33. C2×C3×#681C22semidirect
34. #90970C22semidirect
35. #90969C22semidirect
36. #90964C22semidirect
37. (C2×D3)⋊#168C22semidirect
41. C2×#4088D3semidirect
96. #5816D5semidirect
104. C22×C5×Q8C2×D3semidirect
119. C2×#689C2×D3semidirect
123. C2×#688C2×D3semidirect
124. C2×#681C2×D3semidirect
171. C22×C3×Q8C2×D5semidirect
186. #946C2×D5semidirect
191. #926C2×D5semidirect
361. C22×Q8D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#83982semidirect
580. C3#10615135semidirect

Maximal subgroups

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

subgroupindex
C2×#920192
C22×#67172
C5×#58162
C2×C3×#40882
#117494082
#117493982
#117483962
#117483642
#117483472
#106151353
#839825