Finite Group Databasebeta

← #14424385

Subgroups — #14424385

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×#92018C211yesmaximal, characteristic
2#11756285C211yesmaximal, characteristic, complemented
3C22×#6659C211yesmaximal, characteristic, complemented
4C2×#91313C211yesmaximal, characteristic, complemented
5C5×#5001C211yesmaximal, characteristic, complemented
6C3×#65291C211yesmaximal, characteristic, complemented
7#11748349C211yesmaximal, characteristic, complemented
8#11747521C214yesmaximal, complemented
9#11747484C214yesmaximal, complemented
10#10610660—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×Dic30C2211yescharacteristic, complemented
13C22×C5×D12C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×D3×C22⋊D5—22no—
16#92018—24no—
17#92018—24no—
18C2×#6777C2212yes—
19C2×D5×D12—21no—
20C2×#6772—21no—
21C2×#6768C2212yes—
22C3×C5×#117C2211yescharacteristic, complemented
23#91786C2214yescomplemented
24#91775—22no—
25#91764—21no—
26#91758—21no—
27C2×C5×#208C2211yescharacteristic, complemented
28C22×C3×#168C2211yescharacteristic, complemented
29C2×#6662C2212yescomplemented
30C2×#6659—24no—
31C2×#6659C2214yescomplemented
32#91313—24no—
33C2×#6657C2212yescomplemented
34C5×#809C2214yescomplemented
35C3×#3942C2212yescomplemented
36C3×#3941C2212yescomplemented
37#90957C2214yescomplemented
38#78835—51nomaximal, Hall
39C2×#4144—31no—
40C22×#687—31no—
41C2×#4055—31no—
42#65860—31no—
43C5×#418—31no—
44#65418—31no—
45#65291D311yescharacteristic, complemented
46#65072—34no—
47#65065—34no—
48C23×C5×D3—21no—
49C22×D3×D5—22no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220D412yes—
52C22×Dic15—21no—
53#6817—22no—
54#6817—22no—
55C2×D60D411yescharacteristic
56C2×C4×D15C2312yes—
57C2×Dic30—22no—
58C2×Dic30—22no—
59C2×Dic30—21no—
60C2×Dic30D411yescharacteristic
61C2×Dic30C2312yes—
62C2×C5×D12—21no—
63C2×C5×D12—22no—
64C2×C5×D12D414yes—
65C2×C5×D12C2312yes—
66C2×C3×C22⋊D5—21no—
67C3×#709—22no—
68C3×#709D414yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72D3×C22⋊D5—216no—
73C2×#1203—22no—
74C2×#1201—22no—
75C2×D3×Dic5—22no—
76#6777—24no—
77#6777—24no—
78D5×D12—28no—
79#6772—28no—
80#6768—24no—
81#6768—24no—
82C3×C5×C4⋊C4C2312yes—
83C3×C5×C22⋊C4—21no—
84C3×C42×C5—21no—
85#6714—24no—
86C4⋊Dic15—24no—
87C5×#208—22no—
88C2×C5×C3⋊C8C2312yes—
89C2×C3×#168—22no—
90C2×C3×#168C2312yes—
91#6662—24no—
92#6659—28no—
93#6657—24no—
94C2×#1009—51no—
95C2×#951—51no—
96C22×#204—51no—
97#5526—51no—
98C3×#416—51no—
99#5110—51no—
100#5001D511yescharacteristic, complemented
101#4808—54no—
102#4776—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×Dic10—31no—
114C2×C5×M16—31no—
115C5×#117C2×D311yescharacteristic, complemented
116C2×#687—34no—
117C2×#687—64no—
118C22×#168C2×D311yescharacteristic, complemented
119#4055—64no—
120C2×Dic20—32no—
121C2×#677—32no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#102—34no—
127#3952—34no—
128#3942C2×D312yescomplemented
129#3941C2×D312yescomplemented
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—22no—
134C22×C5×D3—24no—
135C22×C5×D3—82no—
136C22×C3×D5—21no—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2411yescharacteristic
144#1220—24no—
145C2×Dic15—22no—
146C2×Dic15—42no—
147C2×Dic15C2×D412yes—
148D60—22no—
149C4×D15—22no—
150C4×D15—22no—
151Dic30—22no—
152Dic30—22no—
153Dic30—22no—
154Dic30—42no—
155C5×D12—24no—
156C5×D12—22no—
157C5×D12C2×D414yes—
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#1201—28no—
167D3×Dic5—28no—
168C3×#168—22no—
169C5×C3⋊C8—22no—
170C2×C3×#41—51no—
171#1009—104no—
172#1009—104no—
173C2×#239—52no—
174C2×D3×D4—52no—
175C2×D3×D4—102no—
176C2×#235—51no—
177C2×#235—101no—
178C22×D12—101no—
179C22×D12C2×D511yescharacteristic
180C22×Dic6—51no—
181C22×C3×C8—51no—
182C3×#117C2×D511yescharacteristic, complemented
183#951—104no—
184C2×#216—52no—
185C2×#214—52no—
186C2×#208C2×D511yescharacteristic, complemented
187C2×#204—54no—
188C2×#204—104no—
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#103—52no—
194C3×#102—52no—
195(C2×D3)⋊C8—54no—
196#809C2×D514yescomplemented
197C24×C5—61no—
198C23×D5—62no—
199C2×C5×D4—34no—
200C2×C5×D4—32no—
201C2×C5×D4—62no—
202C2×C5×D4—61no—
203C22×C4×C5C22×D311yescharacteristic
204C2×C22⋊D5—21no—
205C2×C22⋊D5—32no—
206C2×C22⋊D5—62no—
207C2×C22⋊D5—62no—
208C2×C22⋊D5—616no—
209C22×Dic5—62no—
210C22×Dic5—61no—
211#711—64no—
212#711—68no—
213#711—64no—
214D4×D5—68no—
215D4×D5—64no—
216D4×D5—64no—
217#709—22no—
218#709—62no—
219#709—62no—
220#709D1214yes—
221C2×D20—31no—
222C2×D20C22×D311yescharacteristic
223C2×C4×D5—21no—
224C2×C4×D5—32no—
225C2×Dic10—31no—
226C2×Dic10—32no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—61no—
230C2×Dic10C22×D311yescharacteristic
231C5×M16—62no—
232C2×C5×C8—32no—
233C5×C4⋊C4C22×D312yes—
234C5×C22⋊C4—21no—
235C42×C5—21no—
236#687—68no—
237#685—64no—
238C4⋊Dic5—64no—
239C2×#168—22no—
240C2×#168C22×D312yes—
241Dic20—64no—
242#677—64no—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—31no—
250#416—51no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C2×C5×D3—24no—
258C2×C5×D3—44no—
259C2×C5×D3—84no—
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
267Dic15—22no—
268Dic15—42no—
269C3×Dic5—22no—
270C23×D3—21no—
271C23×D3—102no—
272C3×#41—54no—
273C3×#41—102no—
274C2×C3×Q8—51no—
275C2×C3×D4—51no—
276C2×C3×D4—101no—
277C22×C3×C4—101no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—52no—
280C2×#67—102no—
281C22×Dic3—101no—
282#239—104no—
283#239—104no—
284D3×D4—104no—
285D3×D4—1016no—
286D3×D4—104no—
287#235—108no—
288#235—102no—
289#235—102no—
290C2×D12—21no—
291C2×D12—22no—
292C2×D12—51no—
293C2×D12—108no—
294C2×D12—102no—
295C2×D12D2014yes—
296C2×D12C22×D512yes—
297C2×C4×D3—52no—
298C2×C4×D3—102no—
299C2×Dic6—51no—
300C2×Dic6—52no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304C2×C3×C8—52no—
305C2×C3×C8—102no—
306C3×C4⋊C4C22×D512yes—
307C3×C22⋊C4—21no—
308C3×C42—21no—
309#216—104no—
310#214—104no—
311(C2×D3)⋊C4—104no—
312C4⋊Dic3—104no—
313#208—22no—
314C2×C3⋊C8C22×D512yes—
315#204—108no—
316C23×C5—62no—
317C23×C5—64no—
318C23×C5—242no—
319C23×C5D3×D411yescharacteristic
320C22×D5—21no—
321C22×D5—32no—
322C22×D5—68no—
323C22×D5—244no—
324C5×D4—34no—
325C5×D4—64no—
326C5×D4—62no—
327C2×C4×C5—22no—
328C2×C4×C5C2×D1212yes—
329C2×C4×C5D3×D412yes—
330C2×C4×C5D3×D411yescharacteristic
331C2×C4×C5C23×D311yescharacteristic
332C22⋊D5—24no—
333C22⋊D5—68no—
334C22⋊D5—68no—
335C22⋊D5—64no—
336C2×Dic5—21no—
337C2×Dic5—32no—
338C2×Dic5—68no—
339C2×Dic5—62no—
340C2×Dic5—122no—
341D20—21no—
342D20—62no—
343D20C2×D1212yes—
344C4×D5—24no—
345C4×D5—62no—
346C4×D5—62no—
347Dic10—21no—
348Dic10—62no—
349Dic10—62no—
350Dic10—62no—
351Dic10—122no—
352Dic10C2×D1212yes—
353C5×C8—62no—
354#168—22no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—51no—
358C2×#41—151no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8—151no—
362C22×D4—31no—
363C22×D4—152no—
364C22×D4—301no—
365C22×D4—302no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—152no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—31no—
372C22×C8—51no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117D3×D511yescharacteristic, complemented
378#103—52no—
379#102—34no—
380#102—52no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—42no—
386C3×D5—42no—
387C5×D3—44no—
388C23×C3—101no—
389C23×C3D4×D511yescharacteristic
390C22×D3—22no—
391C22×D3—24no—
392C22×D3—52no—
393C22×D3—82no—
394C22×D3—108no—
395C22×D3—404no—
396C3×Q8—52no—
397C3×Q8—101no—
398C3×D4—52no—
399C3×D4—104no—
400C3×D4—101no—
401C2×C3×C4—22no—
402C2×C3×C4—101no—
403C2×C3×C4—104no—
404C2×C3×C4C2×D2012yes—
405C2×C3×C4D4×D512yes—
406C2×C3×C4D4×D511yescharacteristic
407C2×C3×C4C23×D511yescharacteristic
408#67—108no—
409#67—104no—
410C2×Dic3—52no—
411C2×Dic3—102no—
412C2×Dic3—202no—
413D12—22no—
414D12—24no—
415D12—108no—
416D12—102no—
417D12C2×D2014yes—
418C4×D3—108no—
419C4×D3—102no—
420C4×D3—102no—
421Dic6—102no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3×C8—102no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—64no—
429C22×C5—124no—
430C22×C5—244no—
431C22×C5#88212yes—
432C22×C5C2×D3×D411yescharacteristic
433C2×D5—22no—
434C2×D5—42no—
435C2×D5—62no—
436C2×D5—64no—
437C2×D5—248no—
438C4×C5—42no—
439C4×C5#88212yes—
440C4×C5C22×D1211yescharacteristic
441C4×C5C2×D3×D411yescharacteristic
442Dic5—22no—
443Dic5—62no—
444Dic5—122no—
445C24—61no—
446C24—302no—
447#41—54no—
448#41—102no—
449#41—308no—
450#41—304no—
451#41—304no—
452#41—302no—
453#41—302no—
454C2×Q8—51no—
455C2×Q8—151no—
456C2×Q8—152no—
457C2×Q8—301no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×D4—34no—
461C2×D4—32no—
462C2×D4—51no—
463C2×D4—62no—
464C2×D4—61no—
465C2×D4—101no—
466C2×D4—152no—
467C2×D4—151no—
468C2×D4—308no—
469C2×D4—302no—
470C2×D4—302no—
471C2×D4—304no—
472C2×D4—304no—
473C2×D4—3016no—
474C22×C4—101no—
475C22×C4—152no—
476C22×C4—302no—
477C22×C4—301no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—304no—
481SD16—308no—
482M16—62no—
483C2×C8—32no—
484C2×C8—52no—
485C2×C8—102no—
486C4⋊C4—304no—
487C4⋊C4C2×D3×D512yes—
488C22⋊C4—21no—
489C22⋊C4—304no—
490C42—21no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—102no—
494C22×C3—202no—
495C22×C3#402812yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—24no—
498C2×D3—44no—
499C2×D3—84no—
500C2×D3—102no—
501C2×D3—104no—
502C2×D3—408no—
503C3×C4—42no—
504C3×C4—102no—
505C3×C4#402812yes—
506C3×C4C22×D2011yescharacteristic
507C3×C4C2×D4×D511yescharacteristic
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—124no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#577012yes—
514D5—42no—
515D5—122no—
516C23—62no—
517C23—64no—
518C23—101no—
519C23—152no—
520C23—242no—
521C23—308no—
522C23—1204no—
523C23#679011yescharacteristic
524Q8—52no—
525Q8—101no—
526Q8—302no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—34no—
531D4—52no—
532D4—64no—
533D4—62no—
534D4—104no—
535D4—101no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—22no—
541C2×C4—101no—
542C2×C4—104no—
543C2×C4—152no—
544C2×C4—308no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—602no—
549C2×C4D5×D1212yes—
550C2×C4D3×D2012yes—
551C2×C4#679011yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—102no—
555C2×C3—22no—
556C2×C3—202no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6610912yes—
559D3—44no—
560D3—202no—
561C5#7883511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—64no—
564C22—102no—
565C22—124no—
566C22—202no—
567C22—244no—
568C22—304no—
569C22—302no—
570C22—1208no—
571C22#9147612yes—
572C22C2×#679011yescharacteristic
573C4—42no—
574C4—102no—
575C4—302no—
576C4—602no—
577C4#9147612yes—
578C4C2×D5×D1211yescharacteristic
579C4C2×D3×D2011yescharacteristic
580C3#1061066011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—124no—
583C2—202no—
584C2—602no—
585C2C2×#9147611yescharacteristic
586C2#1175533312yes—
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#92018C2
hypercenter#117D3×D5
hypocenterC3×C5#3009
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
2. #11756285C2semidirect
3. C22×#6659C2semidirect
4. C2×#91313C2semidirect
5. C5×#5001C2semidirect
6. C3×#65291C2semidirect
7. #11748349C2semidirect
8. #11747521C2semidirect
9. #11747484C2semidirect
12. C22×Dic30C22semidirect
22. C3×C5×#117C22semidirect
23. #91786C22semidirect
27. C2×C5×#208C22semidirect
28. C22×C3×#168C22semidirect
29. C2×#6662C22semidirect
31. C2×#6659C22semidirect
33. C2×#6657C22semidirect
34. C5×#809C22semidirect
35. C3×#3942C22semidirect
36. C3×#3941C22semidirect
37. #90957C22semidirect
45. #65291D3semidirect
100. #5001D5semidirect
115. C5×#117C2×D3semidirect
118. C22×#168C2×D3semidirect
128. #3942C2×D3semidirect
129. #3941C2×D3semidirect
182. C3×#117C2×D5semidirect
186. C2×#208C2×D5semidirect
196. #809C2×D5semidirect
377. #117D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#78835semidirect
580. C3#10610660semidirect

Maximal subgroups

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

subgroupindex
C2×#920182
#117562852
C22×#66592
C2×#913132
C5×#50012
C3×#652912
#117483492
#117475212
#117474842
#106106603
#788355