Finite Group Databasebeta

← #14424323

Subgroups — #14424323

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×#92095C211yesmaximal, characteristic
2C22×#6716C211yesmaximal, characteristic, complemented
3C2×#91798C211yesmaximal, characteristic, complemented
4#11756285C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6#11751230C211yesmaximal, characteristic, complemented
7#11751103C211yesmaximal, characteristic, complemented
8#11750884C214yesmaximal, complemented
9#11750877C214yesmaximal, complemented
10#10610660—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×#1220—22no—
13#92095—24no—
14#92095—24no—
15C2×#6819—21no—
16C2×#6819C2212yes—
17C2×D4×D15—21no—
18C2×D4×D15C2212yes—
19C2×#6817C2211yescharacteristic
20C2×#6817C2211yescharacteristic
21C22×Dic30C2211yescharacteristic, complemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C3×C5×#117C2211yescharacteristic, complemented
24C2×#6716—24no—
25C2×#6716C2214yescomplemented
26C22×C3⋊#168C2211yescharacteristic, complemented
27#91798—24no—
28C2×Dic60C2212yescomplemented
29C2×#6706C2212yescomplemented
30#91786C2214yescomplemented
31#91775—22no—
32#91764—21no—
33#91758—21no—
34C3×C5×#102C2214yescomplemented
35#91104C2214yescomplemented
36#91094C2212yescomplemented
37#91093C2212yescomplemented
38#78929—51nomaximal, Hall
39C2×#4144—31no—
40C22×#687—31no—
41C2×#4055—31no—
42#65860—31no—
43C5×#418D311yescharacteristic, complemented
44#65418—31no—
45#65291—31no—
46#65072—34no—
47#65065—34no—
48C24×C3×C5—21no—
49C23×D15—22no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4D414yes—
53C2×C3×C5×D4C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×#1220—216no—
56C2×#1220—21no—
57C2×#1220—22no—
58C2×#1220—22no—
59C2×#1220D412yes—
60C22×Dic15—22no—
61C22×Dic15—21no—
62#6819—28no—
63#6819—24no—
64#6819—24no—
65D4×D15—28no—
66D4×D15—24no—
67D4×D15—24no—
68#6817—22no—
69#6817—22no—
70#6817—22no—
71#6817D414yes—
72C2×D60D411yescharacteristic
73C2×D60C2311yescharacteristic
74C2×C4×D15—21no—
75C2×C4×D15C2312yes—
76C2×Dic30—22no—
77C2×Dic30—22no—
78C2×Dic30—21no—
79C2×Dic30D411yescharacteristic
80C2×Dic30C2312yes—
81C2×Dic30C2311yescharacteristic
82C3×C5×M16—22no—
83C2×C3×C5×C8C2312yes—
84C3×C5×C4⋊C4C2312yes—
85C3×C5×C22⋊C4—21no—
86C3×C42×C5—21no—
87#6716—28no—
88#6714—24no—
89C4⋊Dic15—24no—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2312yes—
92Dic60—24no—
93#6706—24no—
94C2×#1004—51no—
95C22×#214—51no—
96C2×#909—51no—
97#5526—51no—
98C3×#418D511yescharacteristic, complemented
99#5129—51no—
100#5002—51no—
101#4783—54no—
102#4776—54no—
103C22×C5×D4C2×D311yescharacteristic
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×#709—31no—
113C22×Dic10—31no—
114C2×C5×M16C2×D311yescharacteristic, complemented
115C5×#117C2×D311yescharacteristic, complemented
116C2×#687—34no—
117C2×#687—64no—
118C22×#168—31no—
119#4055—64no—
120C2×Dic20—32no—
121C2×#677—32no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#102C2×D314yescomplemented
127#3952—34no—
128#3942—32no—
129#3941—32no—
130#3009—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—21no—
136C22×D15—28no—
137C22×D15—84no—
138C22×D15C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—22no—
141C3×C5×D4C2×D414yes—
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—28no—
148#1220—28no—
149#1220—24no—
150#1220—24no—
151C2×Dic15—28no—
152C2×Dic15—21no—
153C2×Dic15—22no—
154C2×Dic15—42no—
155C2×Dic15C2×D412yes—
156D60—22no—
157D60—21no—
158D60C2×D412yes—
159C4×D15—24no—
160C4×D15—22no—
161C4×D15—22no—
162Dic30—22no—
163Dic30—21no—
164Dic30—22no—
165Dic30—22no—
166Dic30—42no—
167Dic30C2×D412yes—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C22×C3×D4C2×D511yescharacteristic
171C22×#67—102no—
172#1004—104no—
173#1004—104no—
174C2×#237—52no—
175C2×#237—101no—
176C2×D3×D4—52no—
177C2×D3×D4—101no—
178C2×#235—51no—
179C2×#235—51no—
180C22×Dic6—51no—
181C2×C3×M16C2×D511yescharacteristic, complemented
182C3×#117C2×D511yescharacteristic, complemented
183C2×#214—54no—
184C2×#214—104no—
185C22×C3⋊C8—51no—
186#909—104no—
187C2×Dic12—52no—
188C2×#204—52no—
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#102C2×D514yescomplemented
194(C2×D3)⋊C8—54no—
195#810—52no—
196#809—52no—
197C24×C5—21no—
198C23×D5—62no—
199C2×C5×D4—21no—
200C2×C5×D4—22no—
201C2×C5×D4D1214yes—
202C2×C5×D4C22×D312yes—
203C22×C4×C5C22×D311yescharacteristic
204C2×C22⋊D5—32no—
205C2×C22⋊D5—62no—
206C2×C22⋊D5—62no—
207C2×C22⋊D5—616no—
208C2×C22⋊D5—61no—
209C22×Dic5—62no—
210C22×Dic5—61no—
211#711—64no—
212#711—68no—
213#711—64no—
214D4×D5—64no—
215D4×D5—68no—
216D4×D5—64no—
217#709—34no—
218#709—62no—
219#709—62no—
220#709—62no—
221C2×D20—31no—
222C2×D20—31no—
223C2×C4×D5—32no—
224C2×C4×D5—61no—
225C2×Dic10—31no—
226C2×Dic10—31no—
227C2×Dic10—32no—
228C2×Dic10—62no—
229C2×Dic10—62no—
230C2×Dic10—61no—
231C5×M16—22no—
232C2×C5×C8C22×D312yes—
233C5×C4⋊C4C22×D312yes—
234C5×C22⋊C4—21no—
235C42×C5—21no—
236#687—68no—
237#685—64no—
238C4⋊Dic5—64no—
239C2×#168—32no—
240C2×#168—62no—
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#418D1511yescharacteristic, complemented
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—24no—
254C22×C3×C5—44no—
255C22×C3×C5—84no—
256C22×C3×C5#12012yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×D15—22no—
259C2×D15—22no—
260C2×D15—24no—
261C2×D15—42no—
262C2×D15—88no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268Dic15—22no—
269Dic15—42no—
270C24×C3—21no—
271C23×D3—102no—
272C2×C3×D4—21no—
273C2×C3×D4—22no—
274C2×C3×D4D2014yes—
275C2×C3×D4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—52no—
278C2×#67—102no—
279C2×#67—102no—
280C2×#67—1016no—
281C2×#67—101no—
282C22×Dic3—102no—
283C22×Dic3—101no—
284#237—104no—
285#237—108no—
286#237—104no—
287D3×D4—104no—
288D3×D4—108no—
289D3×D4—104no—
290#235—54no—
291#235—102no—
292#235—102no—
293#235—102no—
294C2×D12—51no—
295C2×D12—51no—
296C2×C4×D3—52no—
297C2×C4×D3—101no—
298C2×Dic6—51no—
299C2×Dic6—51no—
300C2×Dic6—52no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304C3×M16—22no—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4C22×D512yes—
307C3×C22⋊C4—21no—
308C3×C42—21no—
309#214—108no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—104no—
312C2×C3⋊C8—52no—
313C2×C3⋊C8—102no—
314Dic12—104no—
315#204—104no—
316C23×C5—22no—
317C23×C5—24no—
318C23×C5—82no—
319C23×C5D3×D411yescharacteristic
320C22×D5—32no—
321C22×D5—61no—
322C22×D5—68no—
323C22×D5—244no—
324C5×D4—24no—
325C5×D4—22no—
326C5×D4C2×D1214yes—
327C2×C4×C5—22no—
328C2×C4×C5C2×D1212yes—
329C2×C4×C5D3×D411yescharacteristic
330C2×C4×C5D3×D412yes—
331C2×C4×C5C23×D311yescharacteristic
332C22⋊D5—68no—
333C22⋊D5—68no—
334C22⋊D5—64no—
335C22⋊D5—64no—
336C2×Dic5—32no—
337C2×Dic5—68no—
338C2×Dic5—61no—
339C2×Dic5—62no—
340C2×Dic5—122no—
341D20—32no—
342D20—62no—
343D20—61no—
344C4×D5—64no—
345C4×D5—62no—
346C4×D5—62no—
347Dic10—32no—
348Dic10—62no—
349Dic10—61no—
350Dic10—62no—
351Dic10—62no—
352Dic10—122no—
353C5×C8—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—151no—
360C2×#41—301no—
361C22×Q8—151no—
362C22×D4—152no—
363C22×D4—301no—
364C22×D4—302no—
365C22×D4C2×D1511yescharacteristic
366#137—304no—
367C2×Q16—152no—
368C2×SD16—154no—
369C2×SD16—152no—
370C2×SD16—304no—
371C2×M16C2×D1511yescharacteristic, complemented
372C22×C8—151no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117C2×D1511yescharacteristic, complemented
378#103—152no—
379#102—152no—
380#102C2×D1514yescomplemented
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—44no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—42no—
387D15—42no—
388C23×C3—22no—
389C23×C3—24no—
390C23×C3—82no—
391C23×C3D4×D511yescharacteristic
392C22×D3—52no—
393C22×D3—101no—
394C22×D3—108no—
395C22×D3—404no—
396C3×D4—22no—
397C3×D4—24no—
398C3×D4C2×D2014yes—
399C2×C3×C4—22no—
400C2×C3×C4C2×D2012yes—
401C2×C3×C4D4×D512yes—
402C2×C3×C4D4×D511yescharacteristic
403C2×C3×C4C23×D511yescharacteristic
404#67—108no—
405#67—108no—
406#67—104no—
407#67—104no—
408C2×Dic3—52no—
409C2×Dic3—108no—
410C2×Dic3—101no—
411C2×Dic3—102no—
412C2×Dic3—202no—
413D12—52no—
414D12—101no—
415D12—102no—
416C4×D3—102no—
417C4×D3—102no—
418C4×D3—104no—
419Dic6—52no—
420Dic6—101no—
421Dic6—102no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3×C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—44no—
430C22×C5—84no—
431C22×C5#88212yes—
432C22×C5C2×D3×D411yescharacteristic
433C2×D5—62no—
434C2×D5—62no—
435C2×D5—64no—
436C2×D5—122no—
437C2×D5—248no—
438C4×C5—42no—
439C4×C5#88212yes—
440C4×C5C22×D1211yescharacteristic
441C4×C5C2×D3×D411yescharacteristic
442Dic5—62no—
443Dic5—62no—
444Dic5—122no—
445C24—21no—
446C24—302no—
447#41—154no—
448#41—304no—
449#41—308no—
450#41—304no—
451#41—302no—
452#41—302no—
453#41—302no—
454C2×Q8—151no—
455C2×Q8—151no—
456C2×Q8—152no—
457C2×Q8—301no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—152no—
463C2×D4—151no—
464C2×D4—151no—
465C2×D4—302no—
466C2×D4—304no—
467C2×D4—308no—
468C2×D4—302no—
469C2×D4—3016no—
470C2×D4—301no—
471C2×D4—304no—
472C2×D4D6014yes—
473C2×D4C22×D1512yes—
474C22×C4—152no—
475C22×C4—302no—
476C22×C4—301no—
477C22×C4—301no—
478C22×C4C22×D1511yescharacteristic
479Q16—304no—
480SD16—304no—
481SD16—308no—
482M16—22no—
483C2×C8—152no—
484C2×C8—302no—
485C2×C8C22×D1512yes—
486C4⋊C4—304no—
487C4⋊C4C22×D1512yes—
488C22⋊C4—21no—
489C22⋊C4—304no—
490C42—21no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—24no—
494C22×C3—44no—
495C22×C3—84no—
496C22×C3#402812yes—
497C22×C3C2×D4×D511yescharacteristic
498C2×D3—102no—
499C2×D3—102no—
500C2×D3—104no—
501C2×D3—202no—
502C2×D3—408no—
503C3×C4—42no—
504C3×C4#402812yes—
505C3×C4C22×D2011yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507Dic3—102no—
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—44no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#577512yes—
514D5—122no—
515D5—122no—
516C23—22no—
517C23—24no—
518C23—82no—
519C23—152no—
520C23—308no—
521C23—301no—
522C23—1204no—
523C23D4×D1511yescharacteristic
524Q8—152no—
525Q8—302no—
526Q8—301no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—24no—
531D4—22no—
532D4—152no—
533D4—308no—
534D4—308no—
535D4—304no—
536D4—304no—
537D4—302no—
538D4—301no—
539D4C2×D6014yes—
540C2×C4—22no—
541C2×C4—152no—
542C2×C4—308no—
543C2×C4—301no—
544C2×C4—304no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—602no—
549C2×C4C2×D6012yes—
550C2×C4D4×D1511yescharacteristic
551C2×C4D4×D1512yes—
552C2×C4C23×D1511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—22no—
556C2×C3—44no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6610912yes—
559D3—202no—
560D3—202no—
561C5#7892911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—44no—
565C22—84no—
566C22—302no—
567C22—302no—
568C22—304no—
569C22—602no—
570C22—1208no—
571C22#9177112yes—
572C22C2×D4×D1511yescharacteristic
573C4—42no—
574C4—302no—
575C4—302no—
576C4—602no—
577C4#9177112yes—
578C4C22×D6011yescharacteristic
579C4C2×D4×D1511yescharacteristic
580C3#1061066011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—44no—
583C2—602no—
584C2—602no—
585C2C2×#9177111yescharacteristic
586C2#1175653412yes—
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
prime radicalC2×#92095C2
hypercenter#418D15
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C22×#6716C2semidirect
3. C2×#91798C2semidirect
4. #11756285C2semidirect
5. C3×C5×#418C2semidirect
6. #11751230C2semidirect
7. #11751103C2semidirect
8. #11750884C2semidirect
9. #11750877C2semidirect
21. C22×Dic30C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C3×C5×#117C22semidirect
25. C2×#6716C22semidirect
26. C22×C3⋊#168C22semidirect
28. C2×Dic60C22semidirect
29. C2×#6706C22semidirect
30. #91786C22semidirect
34. C3×C5×#102C22semidirect
35. #91104C22semidirect
36. #91094C22semidirect
37. #91093C22semidirect
43. C5×#418D3semidirect
98. C3×#418D5semidirect
114. C2×C5×M16C2×D3semidirect
115. C5×#117C2×D3semidirect
126. C5×#102C2×D3semidirect
181. C2×C3×M16C2×D5semidirect
182. C3×#117C2×D5semidirect
193. C3×#102C2×D5semidirect
249. #418D15semidirect
371. C2×M16C2×D15semidirect
377. #117C2×D15semidirect
380. #102C2×D15semidirect
491. C3×C5#3009semidirect
561. C5#78929semidirect
580. C3#10610660semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C22×#67162
C2×#917982
#117562852
C3×C5×#4182
#117512302
#117511032
#117508842
#117508772
#106106603
#789295