Finite Group Databasebeta

← #12272491

Subgroups — #12272491

Order 1280.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2851C211yesmaximal, characteristic, complemented
2#10614759C212yesmaximal, complemented
3#10613805C211yesmaximal, characteristic, complemented
4C2×C4×#685C211yesmaximal, characteristic
5#10607734C212yesmaximal, complemented
6#10607733C212yesmaximal, complemented
7#10607093C214yesmaximal, complemented
8#10607092C214yesmaximal, complemented
9#10606322C214yesmaximal
10#10606235C214yesmaximal, complemented
11#10605894C212yesmaximal, complemented
12D5×#380C212yesmaximal
13#10605729C212yesmaximal, complemented
14C23×C4×D5C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic, complemented
16C22×C42×C5C2211yescharacteristic
17C22×#685C2212yes—
18C22×C4×Dic5C2211yescharacteristic
19C2×D5×C4⋊C4C2212yes—
20C2×D5×C4⋊C4C2212yes—
21C2×#4036C2212yescomplemented
22C2×D5×C22⋊C4C2214yes—
23C2×#4016C2211yescharacteristic, complemented
24C5×#389C2212yescomplemented
25C5×#387C2214yescomplemented
26C5×#385C2214yes—
27C5×#382C2211yescharacteristic, complemented
28C5×#380C2212yes—
29#65501C2212yes—
30#65499C2214yes—
31C2×#3963C2212yes—
32#65278C2214yes—
33#65278C2214yes—
34#65276C2212yes—
35#65276C2212yes—
36#65276C2214yes—
37#65276C2214yes—
38#65274C2212yescomplemented
39#65273C2214yescomplemented
40#65272C2214yescomplemented
41#65271C2214yescomplemented
42#65269C2214yescomplemented
43#65266C2212yescomplemented
44#65262C2212yescomplemented
45#65239C2214yes—
46#65235C2214yes—
47#65231C2214yes—
48C4×#685C2212yes—
49C4×#685C2212yes—
50C4×#685C2212yes—
51C4×#685C2212yes—
52#65225C2211yescharacteristic, complemented
53#65222C2214yescomplemented
54#65217C2212yescomplemented
55#64957C2214yes—
56#64957C2214yes—
57#64956C2214yes—
58#64956C2214yes—
59#64956C2214yes—
60D5×#95C2214yes—
61#64948C2214yescomplemented
62#64947C2214yescomplemented
63#64947C2214yescomplemented
64#64946C2214yescomplemented
65#64944C2214yescomplemented
66#64943C2214yescomplemented
67#35020—51nomaximal, Sylow
68C24×D5C2311yescharacteristic
69C23×C4×C5C2311yescharacteristic
70C23×C4×C5C2312yes—
71C23×Dic5C2312yes—
72C23×Dic5C2311yescharacteristic
73C22×C4×D5—42no—
74C22×C4×D5—42no—
75C22×C4×D5Q818yes—
76C22×C4×D5C2312yes—
77C22×C4×D5C2314yes—
78C22×C4×D5C2314yes—
79C22×C4×D5C2312yes—
80C5×#125—42no—
81C2×C5×C4⋊C4C2312yes—
82C2×C5×C4⋊C4C2312yes—
83C2×C5×C4⋊C4C2314yes—
84C2×C5×C4⋊C4C2314yes—
85C2×C5×C22⋊C4C2314yes—
86C2×C42×C5—42no—
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2311yescharacteristic
89C2×C42×C5C2312yes—
90C2×C22⋊Dic5C2314yes—
91C2×#685—28no—
92C2×#685C2314yes—
93C2×#685C2314yes—
94C2×#685C2312yes—
95C2×#685C2312yes—
96C2×#685C2312yes—
97C2×#685C2312yes—
98C2×C4⋊Dic5C2312yes—
99C2×C4⋊Dic5C2312yes—
100C2×C4⋊Dic5C2314yes—
101C2×C4⋊Dic5C2314yes—
102C2×#683C2314yes—
103C2×#683C2314yes—
104C2×#683C2314yes—
105C2×#683C2314yes—
106C2×#683C2314yes—
107C2×#683C2314yes—
108C2×C4×Dic5—42no—
109C2×C4×Dic5C2312yes—
110C2×C4×Dic5C2311yescharacteristic
111C2×C4×Dic5C2311yescharacteristic
112D5×C4⋊C4—44no—
113D5×C4⋊C4—44no—
114#4036—44no—
115D5×C22⋊C4—216no—
116#4016—42no—
117C5×#95C2314yes—
118C5×#95C2314yes—
119C5×#95C2314yes—
120#3963C2314yes—
121#3963C2314yes—
122#3963C2314yes—
123#3963C2314yes—
124#3963C2314yes—
125#3963C2312yes—
126#3963C2314yes—
127#3963C2312yes—
128#3963C2312yes—
129#3963C2314yes—
130#3963C2312yes—
131#3963C2314yes—
132#3963C2314yes—
133#2851—51no—
134#2851—52no—
135#2851D511yescharacteristic, complemented
136#2646—54no—
137#2569—54no—
138#2569—54no—
139#2442—52no—
140#2442—52no—
141#2442—52no—
142#2442—52no—
143#2315—54no—
144C2×#380—52no—
145C2×C4×C22⋊C4—51no—
146C24×C5C2411yescharacteristic
147C23×D5—41no—
148C23×D5—41no—
149C23×D5—42no—
150C23×D5C2×Q814yes—
151C23×D5#4118yes—
152C23×D5C2412yes—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—24no—
156C22×C4×C5—42no—
157C22×C4×C5—42no—
158C22×C4×C5C2×Q814yes—
159C22×C4×C5C2412yes—
160C22×C4×C5C2412yes—
161C22×C4×C5C2414yes—
162C22×C4×C5C2414yes—
163C22×C4×C5C2411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—24no—
166C22×Dic5—41no—
167C22×Dic5—41no—
168C22×Dic5—42no—
169C22×Dic5—42no—
170C22×Dic5C2×Q814yes—
171C22×Dic5C2414yes—
172C22×Dic5C2414yes—
173C22×Dic5C2412yes—
174C22×Dic5C2412yes—
175C22×Dic5C2412yes—
176C2×C4×D5—24no—
177C2×C4×D5—24no—
178C2×C4×D5—28no—
179C2×C4×D5—28no—
180C2×C4×D5—416no—
181C2×C4×D5—42no—
182C2×C4×D5—42no—
183C2×C4×D5—44no—
184C2×C4×D5—44no—
185C2×C4×D5—44no—
186C2×C4×D5—44no—
187C2×C4×D5C2×Q818yes—
188C2×C4×D5C2×Q818yes—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—24no—
191C5×C4⋊C4—44no—
192C5×C4⋊C4—44no—
193C5×C22⋊C4—28no—
194C42×C5—22no—
195C42×C5—22no—
196C42×C5—42no—
197C22⋊Dic5—28no—
198#685—28no—
199#685—28no—
200#685—24no—
201#685—24no—
202#685—24no—
203#685—24no—
204C4⋊Dic5—24no—
205C4⋊Dic5—24no—
206C4⋊Dic5—44no—
207C4⋊Dic5—44no—
208#683—44no—
209#683—44no—
210#683—44no—
211#683—44no—
212#683—44no—
213#683—44no—
214C4×Dic5—24no—
215C4×Dic5—41no—
216C4×Dic5—41no—
217C24×C4—51no—
218C2×#125—52no—
219C2×#125—51no—
220C2×#125C2×D511yescharacteristic, complemented
221C22×C4⋊C4—52no—
222C22×C4⋊C4—52no—
223C22×C22⋊C4—52no—
224C22×C22⋊C4—54no—
225C22×C42—51no—
226C22×C42C2×D511yescharacteristic
227#400—54no—
228#400—54no—
229#400—54no—
230#400—54no—
231#398—54no—
232#398—54no—
233#398—54no—
234#398—54no—
235#389—52no—
236#389—52no—
237#389—52no—
238#389C2×D512yescomplemented
239#387—54no—
240#387—54no—
241#387—54no—
242#387C2×D514yescomplemented
243#385—54no—
244#385—54no—
245#385—54no—
246#385—54no—
247#385—54no—
248#385—54no—
249#385—54no—
250#385C2×D514yes—
251#382—52no—
252#382—51no—
253#382C2×D511yescharacteristic, complemented
254#380—52no—
255#380—54no—
256#380—54no—
257#380—52no—
258#380—52no—
259#380—54no—
260#380—54no—
261#380—54no—
262#380—54no—
263#380C2×D512yes—
264C4×C22⋊C4—52no—
265C4×C22⋊C4—52no—
266C4×C22⋊C4—52no—
267C4×C22⋊C4—52no—
268C2×#95—52no—
269C2×#95—54no—
270C23×C5—21no—
271C23×C5—21no—
272C23×C5—22no—
273C23×C5C22×Q812yes—
274C23×C5C2×#4114yes—
275C23×C5C2511yescharacteristic
276C22×D5—22no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—44no—
281C22×D5—44no—
282C22×D5—48no—
283C22×D5—48no—
284C22×D5#125116yescomplemented
285C22×D5C22×Q814yes—
286C22×D5C2×#4114yes—
287C22×D5C2×#4114yes—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—28no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—24no—
294C2×C4×C5—24no—
295C2×C4×C5—24no—
296C2×C4×C5—22no—
297C2×C4×C5—24no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—24no—
301C2×C4×C5—24no—
302C2×C4×C5—22no—
303C2×C4×C5—48no—
304C2×C4×C5C22×Q812yes—
305C2×C4×C5C22×Q812yes—
306C2×C4×C5H(5,𝔽2)12yes—
307C2×C4×C5H(5,𝔽2)12yes—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—24no—
313C2×Dic5—24no—
314C2×Dic5—22no—
315C2×Dic5—22no—
316C2×Dic5—24no—
317C2×Dic5—24no—
318C2×Dic5—24no—
319C2×Dic5—22no—
320C2×Dic5—24no—
321C2×Dic5—22no—
322C2×Dic5—44no—
323C2×Dic5—48no—
324C2×Dic5C22×Q814yes—
325C4×D5—216no—
326C4×D5—48no—
327C4×D5—88no—
328C4×D5—88no—
329C25—51no—
330C23×C4—51no—
331C23×C4—52no—
332C23×C4—58no—
333C23×C4—52no—
334C23×C4—52no—
335C23×C4—54no—
336C23×C4—54no—
337C23×C4—202no—
338C23×C4—202no—
339C23×C4C22×D511yescharacteristic
340C23×C4C22×D512yes—
341#125—42no—
342#125—202no—
343#125—204no—
344C2×C4⋊C4—52no—
345C2×C4⋊C4—52no—
346C2×C4⋊C4—54no—
347C2×C4⋊C4—54no—
348C2×C4⋊C4—54no—
349C2×C4⋊C4—54no—
350C2×C4⋊C4—54no—
351C2×C4⋊C4—54no—
352C2×C4⋊C4—54no—
353C2×C4⋊C4—54no—
354C2×C4⋊C4—204no—
355C2×C4⋊C4—204no—
356C2×C4⋊C4C22×D512yes—
357C2×C4⋊C4C22×D512yes—
358C2×C4⋊C4C22×D514yes—
359C2×C4⋊C4C22×D514yes—
360C2×C22⋊C4—54no—
361C2×C22⋊C4—52no—
362C2×C22⋊C4—52no—
363C2×C22⋊C4—52no—
364C2×C22⋊C4—52no—
365C2×C22⋊C4—54no—
366C2×C22⋊C4—54no—
367C2×C22⋊C4—1016no—
368C2×C22⋊C4—108no—
369C2×C22⋊C4C22×D514yes—
370C2×C42—42no—
371C2×C42—52no—
372C2×C42—51no—
373C2×C42—51no—
374C2×C42—202no—
375C2×C42C22×D511yescharacteristic
376C2×C42C22×D511yescharacteristic
377C2×C42C22×D512yes—
378#95—52no—
379#95—52no—
380#95—52no—
381#95—52no—
382#95—54no—
383#95—54no—
384#95—54no—
385#95—54no—
386#95—54no—
387#95—54no—
388#95—54no—
389#95—54no—
390#95—54no—
391#95C22×D514yes—
392#95C22×D514yes—
393#95C22×D514yes—
394C22×C5—22no—
395C22×C5—22no—
396C22×C5—24no—
397C22×C5—24no—
398C22×C5C2×#12514yes—
399C22×C5C23×Q811yescharacteristic
400C22×C5C22×#4112yes—
401C22×C5C2×H(5,𝔽2)11yescharacteristic
402C22×C5C2×H(5,𝔽2)11yescharacteristic
403C22×C5#58512yes—
404C2×D5—24no—
405C2×D5—24no—
406C2×D5—24no—
407C2×D5—24no—
408C2×D5—24no—
409C2×D5—24no—
410C2×D5—416no—
411C2×D5C2×#12518yes—
412C4×C5—22no—
413C4×C5—44no—
414C4×C5—44no—
415C4×C5—42no—
416C4×C5—42no—
417C4×C5#54314yes—
418Dic5—24no—
419Dic5—44no—
420Dic5—44no—
421Dic5—42no—
422Dic5—42no—
423C24—52no—
424C24—54no—
425C24—58no—
426C24—201no—
427C24—201no—
428C24—202no—
429C24C23×D511yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—52no—
436C22×C4—54no—
437C22×C4—58no—
438C22×C4—58no—
439C22×C4—52no—
440C22×C4—52no—
441C22×C4—54no—
442C22×C4—54no—
443C22×C4—108no—
444C22×C4—108no—
445C22×C4—104no—
446C22×C4—104no—
447C22×C4—104no—
448C22×C4—2016no—
449C22×C4—202no—
450C22×C4—202no—
451C22×C4—204no—
452C22×C4—204no—
453C22×C4—204no—
454C22×C4—204no—
455C22×C4—202no—
456C22×C4—201no—
457C22×C4—201no—
458C22×C4—202no—
459C22×C4Q8×D514yes—
460C22×C4C23×D512yes—
461C22×C4C23×D512yes—
462C22×C4C23×D514yes—
463C22×C4C23×D514yes—
464C22×C4C23×D511yescharacteristic
465C22×C4C23×D511yescharacteristic
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—104no—
471C4⋊C4—104no—
472C4⋊C4—204no—
473C4⋊C4—204no—
474C4⋊C4—204no—
475C4⋊C4—204no—
476C4⋊C4—204no—
477C4⋊C4—204no—
478C4⋊C4—204no—
479C4⋊C4—204no—
480C22⋊C4—28no—
481C22⋊C4—108no—
482C22⋊C4—108no—
483C22⋊C4—104no—
484C22⋊C4—104no—
485C22⋊C4—104no—
486C22⋊C4—104no—
487C22⋊C4—108no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—104no—
492C42—201no—
493C42—201no—
494C2×C5—24no—
495C2×C5C22×#12511yescharacteristic
496C2×C5C2×#54311yescharacteristic
497C2×C5#345811yescharacteristic
498C2×C5#350512yes—
499C2×C5#350512yes—
500D5—28no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—54no—
505C23—54no—
506C23—54no—
507C23—516no—
508C23—102no—
509C23—102no—
510C23—102no—
511C23—102no—
512C23—204no—
513C23—204no—
514C23—208no—
515C23—208no—
516C23C2×Q8×D512yes—
517C23D5×#4114yes—
518C23C24×D511yescharacteristic
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—48no—
535C2×C4—54no—
536C2×C4—102no—
537C2×C4—102no—
538C2×C4—1016no—
539C2×C4—104no—
540C2×C4—104no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—104no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—104no—
547C2×C4—104no—
548C2×C4—102no—
549C2×C4—102no—
550C2×C4—104no—
551C2×C4—208no—
552C2×C4—204no—
553C2×C4—208no—
554C2×C4—408no—
555C2×C4—408no—
556C2×C4#414412yes—
557C2×C4C2×Q8×D512yes—
558C2×C4C2×Q8×D512yes—
559C2×C4#414912yes—
560C5#3502011yescharacteristic, complemented, Sylow
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—58no—
566C22—104no—
567C22—104no—
568C22—104no—
569C22—104no—
570C22—104no—
571C22—104no—
572C22—2016no—
573C22D5×#12514yes—
574C22C2×#414411yescharacteristic
575C22C22×Q8×D511yescharacteristic
576C22C2×D5×#4112yes—
577C22C2×#414911yescharacteristic
578C22#6628312yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—104no—
585C4—204no—
586C4—202no—
587C4—202no—
588C4—204no—
589C4#6596814yes—
590C2—24no—
591C2—108no—
592C2C2×#6596811yescharacteristic
593C2#1061794512yes—
594C2C2×D5×#12511yescharacteristic
595C2#1061805912yes—
596C2#1061817511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutator, socle, cosocleC23×C5C25
FittingC5×#2851C2
prime radicalC24×D5C23
hypercenter#2851D5
hypocenterC5#35020
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. C5×#2851C2semidirect
2. #10614759C2semidirect
3. #10613805C2semidirect
5. #10607734C2semidirect
6. #10607733C2semidirect
7. #10607093C2semidirect
8. #10607092C2semidirect
10. #10606235C2semidirect
11. #10605894C2semidirect
13. #10605729C2semidirect
15. C2×C5×#125C22semidirect
21. C2×#4036C22semidirect
23. C2×#4016C22semidirect
24. C5×#389C22semidirect
25. C5×#387C22semidirect
27. C5×#382C22semidirect
38. #65274C22semidirect
39. #65273C22semidirect
40. #65272C22semidirect
41. #65271C22semidirect
42. #65269C22semidirect
43. #65266C22semidirect
44. #65262C22semidirect
52. #65225C22semidirect
53. #65222C22semidirect
54. #65217C22semidirect
61. #64948C22semidirect
62. #64947C22semidirect
63. #64947C22semidirect
64. #64946C22semidirect
65. #64944C22semidirect
66. #64943C22semidirect
135. #2851D5semidirect
220. C2×#125C2×D5semidirect
238. #389C2×D5semidirect
242. #387C2×D5semidirect
253. #382C2×D5semidirect
284. C22×D5#125semidirect
560. C5#35020semidirect

Maximal subgroups

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

subgroupindex
C5×#28512
#106147592
#106138052
C2×C4×#6852
#106077342
#106077332
#106070932
#106070922
#106063222
#106062352
#106058942
D5×#3802
#106057292
#350205