Finite Group Databasebeta

← #14424394

Subgroups — #14424394

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×#92100C211yesmaximal, characteristic
2C2×C3×#4097C211yesmaximal, characteristic, complemented
3#11754278C211yesmaximal, characteristic, complemented
4C22×#6658C211yesmaximal, characteristic, complemented
5C5×#5233C211yesmaximal, characteristic, complemented
6#11749309C214yesmaximal, complemented
7#11749064C214yesmaximal, complemented
8#11748396C211yesmaximal, characteristic, complemented
9#11748350C211yesmaximal, characteristic, complemented
10#10610566—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12#92100—24no—
13#92100—24no—
14C2×#6821C2212yes—
15C2×D4×D15—22no—
16C2×D4×D15C2212yes—
17C2×#6817—21no—
18C2×#6817C2211yescharacteristic
19C22×D60—21no—
20C22×D60C2211yescharacteristic
21C22×C3×Dic10C2211yescharacteristic, complemented
22C5×#926C2211yescharacteristic, complemented
23C22×C5×C3⋊C8C2211yescharacteristic, complemented
24C3×#4097—24no—
25C2×C3×#689C2212yescomplemented
26C2×C3×#687C2212yescomplemented
27C2×C3×#681C2211yescharacteristic, complemented
28#91546—22no—
29#91390C2214yescomplemented
30#91389—21no—
31#91374—21no—
32C2×#6658—24no—
33C2×#6658C2214yescomplemented
34C5×#835C2212yescomplemented
35C5×#832C2212yescomplemented
36#90969C2214yescomplemented
37#90957C2214yescomplemented
38#83982—51nomaximal, Hall
39C2×#4149—31no—
40C2×#4097D311yescharacteristic, complemented
41C22×#677—31no—
42#65860—31no—
43C5×#416—31no—
44#65399—31no—
45#65290—31no—
46#65099—34no—
47#65065—34no—
48C23×D15—21no—
49C23×D15—22no—
50C3×C5×#41—22no—
51C3×C5×#41D414yes—
52C2×C3×C5×Q8C2311yescharacteristic
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—22no—
58C2×#1220D412yes—
59#6821—24no—
60#6821—24no—
61D4×D15—216no—
62D4×D15—24no—
63D4×D15—24no—
64#6817—28no—
65#6817—22no—
66#6817—22no—
67C2×D60—28no—
68C2×D60—22no—
69C2×D60—21no—
70C2×D60—22no—
71C2×D60D411yescharacteristic
72C2×D60D414yes—
73C2×D60C2312yes—
74C2×C4×D15—22no—
75C2×C4×D15C2312yes—
76C2×Dic30D411yescharacteristic
77C22×C3×Dic5—21no—
78C2×C3×Dic10—22no—
79C2×C3×Dic10—22no—
80C2×C3×Dic10—21no—
81C2×C3×Dic10C2312yes—
82C5×#217—21no—
83C5×C4⋊Dic3C2312yes—
84C4×C5×Dic3—21no—
85C2×C5×C3⋊C8—22no—
86C2×C5×C3⋊C8C2312yes—
87C3×#689—24no—
88C3×#687—24no—
89C3×#681—22no—
90C2×C3×#168C2312yes—
91Dic3⋊Dic5—24no—
92#6667—24no—
93#6658—28no—
94C2×#1009—51no—
95C2×C3×#137—51no—
96#5816—51no—
97C22×#215—51no—
98#5233D511yescharacteristic, complemented
99#5187—54no—
100#5169—54no—
101#5131—51no—
102#5129—51no—
103C2×C5×#41C2×D311yescharacteristic
104#4149—64no—
105#4149—64no—
106C2×#713—32no—
107C2×D4×D5—32no—
108C2×D4×D5—62no—
109C2×#709—31no—
110C2×#709—61no—
111C22×D20—31no—
112C22×D20—61no—
113C22×Dic10C2×D311yescharacteristic, complemented
114C22×C5×C8—31no—
115C5×#117—31no—
116#4097—24no—
117C2×#689C2×D312yescomplemented
118C2×#687C2×D312yescomplemented
119C2×#681C2×D311yescharacteristic, complemented
120C2×#677—34no—
121C2×#677—64no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#103—32no—
127C5×#102—32no—
128#3952—34no—
129#3941—34no—
130#3009—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×D15—22no—
134C22×D15—28no—
135C22×D15—24no—
136C22×D15—84no—
137C22×D15—82no—
138C22×D15C2×D412yes—
139C3×C5×Q8—21no—
140C3×C5×Q8C2×D412yes—
141C3×C5×D4—24no—
142C3×C5×D4—21no—
143C3×C5×D4C2×D412yes—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—24no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2411yescharacteristic
150#1220—28no—
151#1220—24no—
152C2×Dic15C2×D412yes—
153D60—28no—
154D60—22no—
155D60—24no—
156D60—22no—
157D60C2×D414yes—
158C4×D15—28no—
159C4×D15—22no—
160C4×D15—22no—
161Dic30—22no—
162C2×C5×Dic3—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5—42no—
165C3×Dic10—22no—
166C3×Dic10—22no—
167C3×Dic10—42no—
168C3×#168—22no—
169C5×C3⋊C8—22no—
170C2×C3×#41C2×D511yescharacteristic
171C22×C3×Q8—51no—
172#1009—104no—
173#1009—104no—
174C2×#239—52no—
175C2×D3×D4—52no—
176C2×D3×D4—102no—
177C2×#235—51no—
178C2×#235—101no—
179C22×D12—51no—
180C22×D12—101no—
181C3×#137—104no—
182C2×C3×Q16—52no—
183C2×C3×SD16—52no—
184C2×C3×M16—51no—
185#947—101no—
186#946—54no—
187#944—101no—
188C2×#215—54no—
189C2×#215—104no—
190#929—102no—
191#926C2×D511yescharacteristic, complemented
192C22×C3⋊C8C2×D511yescharacteristic, complemented
193#835C2×D512yescomplemented
194#832C2×D512yescomplemented
195#821—54no—
196(C2×D3)⋊C8—54no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—22no—
200C5×#41#6714yes—
201C2×C5×Q8C22×D311yescharacteristic
202C2×C5×D4—21no—
203C2×C5×D4C22×D311yescharacteristic
204C22×C4×C5—21no—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—32no—
207C2×C22⋊D5—62no—
208C22×Dic5—21no—
209#713—64no—
210#713—64no—
211D4×D5—64no—
212D4×D5—616no—
213D4×D5—64no—
214#709—68no—
215#709—62no—
216#709—62no—
217C2×D20—34no—
218C2×D20—32no—
219C2×D20—31no—
220C2×D20—62no—
221C2×D20—68no—
222C2×D20—61no—
223C2×D20—62no—
224C2×C4×D5—32no—
225C2×C4×D5—62no—
226C2×Dic10—22no—
227C2×Dic10—22no—
228C2×Dic10—21no—
229C2×Dic10—31no—
230C2×Dic10C22×D312yes—
231C2×C5×C8—32no—
232C2×C5×C8—62no—
233C5×C4⋊C4—32no—
234C5×C22⋊C4—61no—
235C42×C5—61no—
236#689—24no—
237#687—24no—
238#685—64no—
239C4⋊Dic5—64no—
240#681—22no—
241C2×#168C22×D312yes—
242#677—68no—
243C2×H(5,𝔽2)—151no—
244C2×#137—51no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—31no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—42no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—24no—
259C2×D15—24no—
260C2×D15—44no—
261C2×D15—84no—
262C2×D15—88no—
263C3×C4×C5—22no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268C3×Dic5—42no—
269C5×Dic3—42no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—22no—
273C3×#41D2014yes—
274C2×C3×Q8—52no—
275C2×C3×Q8—101no—
276C2×C3×Q8—102no—
277C2×C3×Q8—102no—
278C2×C3×Q8C22×D511yescharacteristic
279C2×C3×D4—21no—
280C2×C3×D4C22×D511yescharacteristic
281C22×C3×C4—21no—
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—54no—
295C2×D12—52no—
296C2×D12—51no—
297C2×D12—102no—
298C2×D12—108no—
299C2×D12—102no—
300C2×D12—101no—
301C2×C4×D3—52no—
302C2×C4×D3—102no—
303C2×Dic6—51no—
304C3×Q16—104no—
305C3×SD16—104no—
306C3×M16—102no—
307C2×C3×C8—52no—
308#217—21no—
309#215—108no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3C22×D512yes—
312Dic3⋊C4—104no—
313C4×Dic3—21no—
314C2×C3⋊C8—22no—
315C2×C3⋊C8C22×D512yes—
316C23×C5—21no—
317C23×C5D3×D411yescharacteristic
318C22×D5—32no—
319C22×D5—62no—
320C22×D5—68no—
321C22×D5—64no—
322C22×D5—244no—
323C22×D5—242no—
324C5×Q8—21no—
325C5×Q8C2×#6712yes—
326C5×D4—24no—
327C5×D4—21no—
328C5×D4C2×#6712yes—
329C2×C4×C5—21no—
330C2×C4×C5—24no—
331C2×C4×C5—62no—
332C2×C4×C5D3×D411yescharacteristic
333C2×C4×C5D3×D412yes—
334C2×C4×C5C2×#6712yes—
335C2×C4×C5C23×D311yescharacteristic
336C22⋊D5—68no—
337C22⋊D5—64no—
338C2×Dic5—22no—
339C2×Dic5—32no—
340C2×Dic5—42no—
341D20—34no—
342D20—68no—
343D20—64no—
344D20—62no—
345D20—62no—
346C4×D5—68no—
347C4×D5—62no—
348C4×D5—62no—
349Dic10—22no—
350Dic10—22no—
351Dic10—42no—
352Dic10—62no—
353C5×C8—62no—
354#168—22no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—301no—
360C2×#41D3×D511yescharacteristic
361C22×Q8—51no—
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—31no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—31no—
378#103—32no—
379#102—32no—
380#102—154no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—42no—
387D15—44no—
388C23×C3—21no—
389C23×C3D4×D511yescharacteristic
390C22×D3—52no—
391C22×D3—102no—
392C22×D3—108no—
393C22×D3—104no—
394C22×D3—404no—
395C22×D3—402no—
396C3×Q8—21no—
397C3×Q8—102no—
398C3×Q8—102no—
399C3×Q8—202no—
400C3×Q8C2×D2012yes—
401C3×D4—24no—
402C3×D4—21no—
403C3×D4C2×D2012yes—
404C2×C3×C4—21no—
405C2×C3×C4—24no—
406C2×C3×C4—102no—
407C2×C3×C4—202no—
408C2×C3×C4C2×D2012yes—
409C2×C3×C4D4×D512yes—
410C2×C3×C4D4×D511yescharacteristic
411C2×C3×C4C23×D511yescharacteristic
412#67—108no—
413#67—104no—
414C2×Dic3—22no—
415C2×Dic3—52no—
416D12—54no—
417D12—108no—
418D12—102no—
419D12—104no—
420D12—102no—
421C4×D3—108no—
422C4×D3—102no—
423C4×D3—102no—
424Dic6—102no—
425C3×C8—102no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—22no—
429C22×C5—42no—
430C22×C5#93712yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—64no—
433C2×D5—64no—
434C2×D5—62no—
435C2×D5—124no—
436C2×D5—244no—
437C2×D5—248no—
438C4×C5—22no—
439C4×C5—122no—
440C4×C5#93712yes—
441C4×C5C2×D3×D411yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—42no—
444Dic5—62no—
445C24—302no—
446C24—301no—
447#41—22no—
448#41—308no—
449#41—304no—
450#41—304no—
451#41—302no—
452#41—302no—
453#41#120214yes—
454C2×Q8—52no—
455C2×Q8—101no—
456C2×Q8—102no—
457C2×Q8—102no—
458C2×Q8—151no—
459C2×Q8C2×D3×D511yescharacteristic
460C2×D4—21no—
461C2×D4—152no—
462C2×D4—154no—
463C2×D4—151no—
464C2×D4—152no—
465C2×D4—302no—
466C2×D4—308no—
467C2×D4—302no—
468C2×D4—304no—
469C2×D4—3016no—
470C2×D4—304no—
471C2×D4—302no—
472C2×D4—301no—
473C2×D4C2×D3×D511yescharacteristic
474C22×C4—21no—
475C22×C4—101no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—104no—
480SD16—104no—
481SD16—308no—
482M16—102no—
483C2×C8—32no—
484C2×C8—52no—
485C2×C8—62no—
486C4⋊C4—32no—
487C4⋊C4—304no—
488C22⋊C4—61no—
489C22⋊C4—304no—
490C42—61no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—42no—
495C22×C3#402812yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—104no—
498C2×D3—102no—
499C2×D3—104no—
500C2×D3—204no—
501C2×D3—404no—
502C2×D3—408no—
503C3×C4—22no—
504C3×C4—202no—
505C3×C4#402812yes—
506C3×C4C22×D2011yescharacteristic
507C3×C4C2×D4×D511yescharacteristic
508Dic3—42no—
509Dic3—102no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5#577812yes—
513C2×C5C2×#93711yescharacteristic
514D5—122no—
515D5—124no—
516C23—21no—
517C23—152no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23#679011yescharacteristic
524Q8—21no—
525Q8—102no—
526Q8—102no—
527Q8—202no—
528Q8—302no—
529Q8C2×#120212yes—
530D4—24no—
531D4—21no—
532D4—154no—
533D4—308no—
534D4—308no—
535D4—304no—
536D4—302no—
537D4—304no—
538D4—302no—
539D4C2×#120212yes—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—152no—
545C2×C4—202no—
546C2×C4—308no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4C2×#120212yes—
550C2×C4#679012yes—
551C2×C4#679011yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—102no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6610412yes—
559D3—202no—
560D3—204no—
561C5#8398211yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—42no—
565C22—304no—
566C22—304no—
567C22—302no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9157212yes—
572C22C2×#679011yescharacteristic
573C4—22no—
574C4—122no—
575C4—202no—
576C4—302no—
577C4#9157212yes—
578C4C22×#120211yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061056611yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—602no—
584C2—604no—
585C2#1175535412yes—
586C2C2×#9157211yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×#92100C2
hypercenterC2×#41D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4097C2semidirect
3. #11754278C2semidirect
4. C22×#6658C2semidirect
5. C5×#5233C2semidirect
6. #11749309C2semidirect
7. #11749064C2semidirect
8. #11748396C2semidirect
9. #11748350C2semidirect
21. C22×C3×Dic10C22semidirect
22. C5×#926C22semidirect
23. C22×C5×C3⋊C8C22semidirect
25. C2×C3×#689C22semidirect
26. C2×C3×#687C22semidirect
27. C2×C3×#681C22semidirect
29. #91390C22semidirect
33. C2×#6658C22semidirect
34. C5×#835C22semidirect
35. C5×#832C22semidirect
36. #90969C22semidirect
37. #90957C22semidirect
40. C2×#4097D3semidirect
98. #5233D5semidirect
113. C22×Dic10C2×D3semidirect
117. C2×#689C2×D3semidirect
118. C2×#687C2×D3semidirect
119. C2×#681C2×D3semidirect
191. #926C2×D5semidirect
192. C22×C3⋊C8C2×D5semidirect
193. #835C2×D5semidirect
194. #832C2×D5semidirect
491. C3×C5#3009semidirect
561. C5#83982semidirect
580. C3#10610566semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C2×C3×#40972
#117542782
C22×#66582
C5×#52332
#117493092
#117490642
#117483962
#117483502
#106105663
#839825