Finite Group Databasebeta

← #14458512

Subgroups — #14458512

Order 1920.

924 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5543C211yesmaximal, characteristic, complemented
2C3×D4×D20C211yesmaximal, characteristic
3D4×C3⋊#168C211yesmaximal, characteristic, complemented
4#11748545C212yesmaximal, complemented
5#11748527C212yesmaximal, complemented
6#11748372C212yesmaximal, complemented
7C2×#90964C212yesmaximal, complemented
8#11747068C211yesmaximal, characteristic, complemented
9C4×#6655C211yesmaximal, characteristic, complemented
10#11746895C211yesmaximal, characteristic, complemented
11#11746880C211yesmaximal, characteristic, complemented
12#10615365—31nomaximal, Hall
13C2×C3×D4×D5—21no—
14C22×C3×D20C2212yes—
15C3×C4×C5×D4C2211yescharacteristic
16C22×C3⋊#168C2212yescomplemented
17C5×D4×Dic3—21no—
18C2×C5×C4⋊Dic3C2212yescomplemented
19C5×#924C2212yescomplemented
20C5×#890—21no—
21C5×#878—22no—
22C5×#869C2211yescharacteristic, complemented
23C4×C5×Dic6C2211yescharacteristic, complemented
24C3×#4076C2212yes—
25C3×#4041—21no—
26C3×#4028—22no—
27C3×#4020C2211yescharacteristic
28C3×C4×D20C2211yescharacteristic
29C2×#6655C2211yescharacteristic
30C2×#6655C2212yescomplemented
31#91114C2212yescomplemented
32C4⋊(C3⋊#168)C2211yescharacteristic, complemented
33C4×C3⋊#168C2211yescharacteristic, complemented
34#90986C2211yescharacteristic, complemented
35#90986C2212yescomplemented
36#90972C2211yescharacteristic, complemented
37#90964C2211yescharacteristic
38#90964C2212yes—
39#90964C2212yescomplemented
40#90964C2212yescomplemented
41#90964C2212yescomplemented
42#83483—51nomaximal, Hall
43C5×#554—31no—
44D4×D20D311yescharacteristic
45#65377—31no—
46#65373—31no—
47#65337—32no—
48#65335—32no—
49#65316—31no—
50C4×#688—31no—
51D4×#168—31no—
52#65257—32no—
53C2×#3941—32no—
54C23×C3×D5—22no—
55C2×C3×C5×D4D411yescharacteristic
56C22×C3×C4×C5C2312yes—
57C22×C5×Dic3—22no—
58C2×C5×Dic6—21no—
59C2×C5×Dic6C2311yescharacteristic
60C2×C3×C22⋊D5—22no—
61C3×D4×D5—44no—
62C2×C3×D20—24no—
63C2×C3×D20—22no—
64C2×C3×D20—21no—
65C2×C3×D20—21no—
66C2×C3×D20—42no—
67C2×C3×D20C2312yes—
68C2×C3×D20C2311yescharacteristic
69C2×C3×C4×D5—21no—
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5C2311yescharacteristic
73C2×C3⋊#168—42no—
74C2×C3⋊#168C2311yescharacteristic
75C2×C3⋊#168C2312yes—
76C2×C3⋊#168C2311yescharacteristic
77C5×#217—22no—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3—22no—
80C5×C4⋊Dic3C2311yescharacteristic
81C5×C4⋊Dic3C2312yes—
82C5×Dic3⋊C4—21no—
83C5×Dic3⋊C4—22no—
84C4×C5×Dic3—21no—
85C3×#685—21no—
86C3×#685—22no—
87C3×C4⋊Dic5C2311yescharacteristic
88#6655—22no—
89#6655—42no—
90C3×D42—51no—
91#5543D511yescharacteristic, complemented
92#5218—52no—
93C2×#832—52no—
94#5075—51no—
95#5070—51no—
96#5041—52no—
97#5034—52no—
98C4×#214—51no—
99D4×C3⋊C8—51no—
100#5008—51no—
101C2×D4×D5—21no—
102C22×D20C2×D312yes—
103C5×C4⋊Q8—31no—
104C5×#125—61no—
105C5×C22⋊Q8—32no—
106C5×C22⋊Q8—62no—
107C4×C5×Q8—31no—
108C4×C5×D4—61no—
109C4×C5×D4C2×D311yescharacteristic
110C2×C5×C4⋊C4—32no—
111C2×#688—31no—
112C2×#688—32no—
113#4076C2×D312yes—
114C22×#168—32no—
115#4041—21no—
116#4028—22no—
117#4020C2×D311yescharacteristic
118C4×D20C2×D311yescharacteristic
119#3967—31no—
120C22⋊#168—32no—
121#3941—31no—
122#3941—32no—
123#3941—32no—
124#3941—32no—
125#3941—32no—
126#3940—31no—
127#3940—32no—
128C4⋊#168—31no—
129C4×#168—31no—
130#3295—151noSylow
131C23×C3×C5C2×D412yes—
132C22×C3×D5—21no—
133C22×C3×D5—22no—
134C22×C3×D5—44no—
135C22×C3×D5—44no—
136C22×C3×D5—82no—
137C3×C5×D4SD1614yescomplemented
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2411yescharacteristic
144C2×C5×Dic3—21no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—82no—
148C5×Dic6—21no—
149C5×Dic6—21no—
150C5×Dic6—81no—
151C3×C22⋊D5—44no—
152C2×C3×Dic5—21no—
153C3×D20—22no—
154C3×D20—21no—
155C3×D20—22no—
156C3×D20—44no—
157C3×D20—42no—
158C3×D20—81no—
159C3×D20#4112yes—
160C3×C4×D5—42no—
161C3⋊#168—21no—
162C3⋊#168—21no—
163C3⋊#168—22no—
164C22×C3×D4—52no—
165C22×C3×D4—101no—
166C3×#127—51no—
167C3×#121—52no—
168C3×#121—101no—
169C3×#120—102no—
170C3×C4×D4—51no—
171C3×C4×D4C2×D511yescharacteristic
172D4×Dic3—21no—
173C2×#214—51no—
174C2×#214—52no—
175C2×C4⋊Dic3C2×D512yescomplemented
176#924C2×D512yescomplemented
177C22×C3⋊C8—52no—
178#890—21no—
179#878—22no—
180#869C2×D511yescharacteristic, complemented
181C4×Dic6C2×D511yescharacteristic, complemented
182#832—51no—
183#832—52no—
184#832—52no—
185#832—52no—
186#832—52no—
187#830—52no—
188#810—51no—
189#808—52no—
190#808—51no—
191C4⋊(C3⋊C8)—51no—
192C4×C3⋊C8—51no—
193C23×D5—22no—
194C2×C5×Q8—31no—
195C2×C5×Q8—61no—
196C2×C5×D4#6711yescharacteristic
197C22×C4×C5—62no—
198C22×C4×C5C22×D312yes—
199C2×C22⋊D5—22no—
200D4×D5—44no—
201C2×D20—24no—
202C2×D20—22no—
203C2×D20—21no—
204C2×D20—21no—
205C2×D20—42no—
206C2×D20C22×D312yes—
207C2×D20C22×D311yescharacteristic
208C2×C4×D5—21no—
209C5×C4⋊C4—31no—
210C5×C4⋊C4—32no—
211C5×C4⋊C4—62no—
212C5×C4⋊C4—61no—
213C5×C4⋊C4—61no—
214C5×C4⋊C4—62no—
215C5×C4⋊C4#6711yescharacteristic
216C5×C22⋊C4—62no—
217C5×C22⋊C4#6712yes—
218C42×C5—61no—
219C42×C5C22×D311yescharacteristic
220#688—62no—
221#688—122no—
222#685—21no—
223#685—22no—
224C4⋊Dic5C22×D311yescharacteristic
225C2×#168—31no—
226C2×#168—31no—
227C2×#168—32no—
228C2×#168—122no—
229#554—31no—
230D42—51no—
231#486—151no—
232#481—152no—
233#476—151no—
234#465—152no—
235#460—151no—
236#450—152no—
237C4×SD16—151no—
238C8×D4—151no—
239C2×#102—152no—
240C22×C3×C5—22no—
241C22×C3×C5—22no—
242C22×C3×C5C2×SD1614yes—
243C22×C3×C5C22×D411yescharacteristic
244C2×C3×D5—22no—
245C2×C3×D5—41no—
246C2×C3×D5—41no—
247C2×C3×D5—42no—
248C2×C3×D5—42no—
249C2×C3×D5—42no—
250C2×C3×D5—88no—
251C3×C4×C5—22no—
252C3×C4×C5—21no—
253C3×C4×C5C2×SD1612yes—
254C3×C4×C5C2×#4111yescharacteristic
255C3×C4×C5H(5,𝔽2)11yescharacteristic
256C3×Dic5—41no—
257C5×Dic3—41no—
258C5×Dic3—42no—
259C5×Dic3—41no—
260C24×C3—102no—
261C2×C3×D4—51no—
262C2×C3×D4—52no—
263C2×C3×D4—104no—
264C2×C3×D4—102no—
265C2×C3×D4—102no—
266C2×C3×D4—101no—
267C2×C3×D4—101no—
268C2×C3×D4—202no—
269C2×C3×D4—204no—
270C2×C3×D4C22⋊D511yescharacteristic
271C22×C3×C4—101no—
272C22×C3×C4C22×D512yes—
273C22×Dic3—22no—
274C2×Dic6—21no—
275C2×Dic6C22×D511yescharacteristic
276C3×C4⋊C4—51no—
277C3×C4⋊C4C22⋊D511yescharacteristic
278C3×C22⋊C4—101no—
279C3×C22⋊C4—102no—
280C3×C22⋊C4C22⋊D512yes—
281C3×C42C22×D511yescharacteristic
282#217—22no—
283#214—102no—
284#214—202no—
285C4⋊Dic3—21no—
286C4⋊Dic3—22no—
287C4⋊Dic3C22×D511yescharacteristic
288C4⋊Dic3C22×D512yes—
289Dic3⋊C4—21no—
290Dic3⋊C4—22no—
291C4×Dic3—21no—
292C2×C3⋊C8—51no—
293C2×C3⋊C8—51no—
294C2×C3⋊C8—52no—
295C2×C3⋊C8—202no—
296C23×C5C2×#6712yes—
297C22×D5—21no—
298C22×D5—22no—
299C22×D5—44no—
300C22×D5—44no—
301C22×D5—82no—
302C5×Q8—61no—
303C5×Q8—61no—
304C5×Q8—241no—
305C5×D4#21414yescomplemented
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—61no—
309C2×C4×C5—61no—
310C2×C4×C5—62no—
311C2×C4×C5—242no—
312C2×C4×C5C2×#6711yescharacteristic
313C2×C4×C5C2×#6712yes—
314C2×C4×C5C2×#6711yescharacteristic
315C2×C4×C5C23×D311yescharacteristic
316C22⋊D5—44no—
317C2×Dic5—21no—
318D20—22no—
319D20—21no—
320D20—22no—
321D20—44no—
322D20—42no—
323D20—81no—
324D20#23712yes—
325C4×D5—42no—
326#168—61no—
327#168—61no—
328#168—62no—
329C22×D4—52no—
330C22×D4—101no—
331C2×SD16—151no—
332C2×SD16—152no—
333C22×C8—152no—
334C4⋊Q8—31no—
335#127—51no—
336#125—61no—
337C22⋊Q8—32no—
338C22⋊Q8—62no—
339#121—52no—
340#121—101no—
341#120—102no—
342C4×Q8—31no—
343C4×D4—51no—
344C4×D4—61no—
345C4×D4D3×D511yescharacteristic
346C2×C4⋊C4—32no—
347#106—151no—
348#106—152no—
349C4⋊C8—151no—
350#103—151no—
351#102—151no—
352#102—152no—
353#102—152no—
354#102—152no—
355#102—152no—
356C22⋊C8—152no—
357C4×C8—151no—
358C2×C3×C5—24no—
359C2×C3×C5#54611yescharacteristic
360C2×C3×C5C22×SD1611yescharacteristic
361C2×C3×C5#57611yescharacteristic
362C3×D5—42no—
363C3×D5—82no—
364C23×C3—101no—
365C23×C3—102no—
366C23×C3—204no—
367C23×C3—204no—
368C23×C3—402no—
369C23×C3C2×C22⋊D512yes—
370C3×D4—52no—
371C3×D4—102no—
372C3×D4—101no—
373C3×D4—102no—
374C3×D4—204no—
375C3×D4—204no—
376C3×D4—202no—
377C3×D4—401no—
378C3×D4#68814yescomplemented
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—101no—
382C2×C3×C4—202no—
383C2×C3×C4C2×C22⋊D512yes—
384C2×C3×C4C2×C22⋊D511yescharacteristic
385C2×C3×C4C2×C22⋊D511yescharacteristic
386C2×C3×C4C23×D511yescharacteristic
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—82no—
391Dic6—21no—
392Dic6—21no—
393Dic6—81no—
394C3⋊C8—101no—
395C3⋊C8—101no—
396C3⋊C8—102no—
397C22×C5—22no—
398C22×C5—22no—
399C22×C5C2×#21414yes—
400C22×C5C22×#6711yescharacteristic
401C2×D5—22no—
402C2×D5—41no—
403C2×D5—41no—
404C2×D5—42no—
405C2×D5—42no—
406C2×D5—42no—
407C2×D5—88no—
408C4×C5—22no—
409C4×C5—21no—
410C4×C5—121no—
411C4×C5—121no—
412C4×C5—122no—
413C4×C5C2×#21412yes—
414C4×C5C2×#23711yescharacteristic
415C4×C5#100411yescharacteristic
416Dic5—41no—
417C24—102no—
418C2×Q8—31no—
419C2×Q8—61no—
420C2×D4—51no—
421C2×D4—52no—
422C2×D4—104no—
423C2×D4—102no—
424C2×D4—102no—
425C2×D4—101no—
426C2×D4—101no—
427C2×D4—202no—
428C2×D4—204no—
429C2×D4#120111yescharacteristic
430C22×C4—62no—
431C22×C4—101no—
432C22×C4C2×D3×D512yes—
433SD16—302no—
434SD16—602no—
435C2×C8—151no—
436C2×C8—151no—
437C2×C8—152no—
438C2×C8—602no—
439C4⋊C4—31no—
440C4⋊C4—32no—
441C4⋊C4—51no—
442C4⋊C4—62no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—62no—
446C4⋊C4#120111yescharacteristic
447C22⋊C4—62no—
448C22⋊C4—101no—
449C22⋊C4—102no—
450C22⋊C4#120112yes—
451C42—61no—
452C42C2×D3×D511yescharacteristic
453C3×C5#329511yescharacteristic, complemented, Hall
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—102no—
457C22×C3—201no—
458C22×C3—201no—
459C22×C3—202no—
460C22×C3—202no—
461C22×C3—202no—
462C22×C3—408no—
463C22×C3C2×#68814yes—
464C22×C3C22×C22⋊D511yescharacteristic
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—201no—
468C3×C4C2×#68812yes—
469C3×C4D5×#4111yescharacteristic
470C3×C4#414911yescharacteristic
471Dic3—41no—
472Dic3—42no—
473Dic3—41no—
474C2×C5—24no—
475C2×C5C22×#21411yescharacteristic
476C2×C5#580611yescharacteristic
477C2×C5#583511yescharacteristic
478D5—42no—
479D5—82no—
480C23—101no—
481C23—102no—
482C23—204no—
483C23—204no—
484C23—402no—
485C23C2×#120112yes—
486Q8—61no—
487Q8—61no—
488Q8—241no—
489D4—52no—
490D4—102no—
491D4—101no—
492D4—102no—
493D4—204no—
494D4—204no—
495D4—202no—
496D4—401no—
497D4#665514yescomplemented
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—62no—
503C2×C4—101no—
504C2×C4—202no—
505C2×C4—242no—
506C2×C4C2×#120111yescharacteristic
507C2×C4C2×#120112yes—
508C2×C4C2×#120111yescharacteristic
509C2×C4C22×D3×D511yescharacteristic
510C8—301no—
511C8—301no—
512C8—302no—
513C2×C3—24no—
514C2×C3—202no—
515C2×C3—402no—
516C2×C3C22×#68811yescharacteristic
517C2×C3#6616311yescharacteristic
518C2×C3#6616911yescharacteristic
519C5#8348311yescharacteristic, complemented, Sylow
520C22—22no—
521C22—22no—
522C22—102no—
523C22—201no—
524C22—201no—
525C22—202no—
526C22—202no—
527C22—202no—
528C22—408no—
529C22C2×#665514yes—
530C22C22×#120111yescharacteristic
531C4—21no—
532C4—22no—
533C4—121no—
534C4—122no—
535C4—121no—
536C4—201no—
537C4C2×#665512yes—
538C4D5×#23711yescharacteristic
539C4#9202611yescharacteristic
540C3#1061536511yescharacteristic, complemented, Sylow
541C2—24no—
542C2—202no—
543C2—402no—
544C2C22×#665511yescharacteristic
545C2#1175496111yescharacteristic
546C2#1175530111yescharacteristic
subgroupG/H
centerC22C22×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC3×D4×D20C2
hypercenterC4×D4D3×D5
hypocenterC3×C5#3295
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C5×#5543C2semidirect
3. D4×C3⋊#168C2semidirect
4. #11748545C2semidirect
5. #11748527C2semidirect
6. #11748372C2semidirect
7. C2×#90964C2semidirect
8. #11747068C2semidirect
9. C4×#6655C2semidirect
10. #11746895C2semidirect
11. #11746880C2semidirect
16. C22×C3⋊#168C22semidirect
18. C2×C5×C4⋊Dic3C22semidirect
19. C5×#924C22semidirect
22. C5×#869C22semidirect
23. C4×C5×Dic6C22semidirect
30. C2×#6655C22semidirect
31. #91114C22semidirect
32. C4⋊(C3⋊#168)C22semidirect
33. C4×C3⋊#168C22semidirect
34. #90986C22semidirect
35. #90986C22semidirect
36. #90972C22semidirect
39. #90964C22semidirect
40. #90964C22semidirect
41. #90964C22semidirect
91. #5543D5semidirect
137. C3×C5×D4SD16semidirect
175. C2×C4⋊Dic3C2×D5semidirect
176. #924C2×D5semidirect
180. #869C2×D5semidirect
181. C4×Dic6C2×D5semidirect
305. C5×D4#214semidirect
378. C3×D4#688semidirect
453. C3×C5#3295semidirect
497. D4#6655semidirect
519. C5#83483semidirect
540. C3#10615365semidirect

Maximal subgroups

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

subgroupindex
C5×#55432
C3×D4×D202
D4×C3⋊#1682
#117485452
#117485272
#117483722
C2×#909642
#117470682
C4×#66552
#117468952
#117468802
#106153653
#834835