Finite Group Databasebeta

← #12030007

Subgroups — #12030007

Order 1280.

1434 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#397C211yesmaximal, characteristic
2C2×#65239C212yesmaximal
3#10602358C212yesmaximal, complemented
4#10602341C212yesmaximal
5#10602339C211yesmaximal, characteristic
6#10602337C211yesmaximal, characteristic
7#10600212C212yesmaximal, complemented
8#10600209C212yesmaximal, complemented
9#10600205C212yesmaximal, complemented
10C24×Dic5C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12C22×C5×C22⋊C4C2212yes—
13C22×C5×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C4⋊Dic5C2212yes—
17C22×C4⋊Dic5C2212yes—
18C22×C4⋊Dic5C2211yescharacteristic
19C5×#397C418yescomplemented
20C2×C5×#95C2211yescharacteristic
21C2×C5×#95C2212yes—
22C2×#3963C2212yes—
23C2×#3963C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32#65239—28no—
33#14627—51nomaximal, Sylow
34C25×C5C2311yescharacteristic
35C23×C4×C5C2312yes—
36C23×C4×C5C2311yescharacteristic
37C23×C4×C5C2312yes—
38C23×C4×C5C2311yescharacteristic
39C23×Dic5—42no—
40C23×Dic5—42no—
41C23×Dic5—41no—
42C23×Dic5—41no—
43C23×Dic5D414yes—
44C23×Dic5C2312yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C2×C5×C4⋊C4—41no—
49C2×C5×C4⋊C4—41no—
50C2×C5×C4⋊C4C2×C414yescomplemented
51C2×C5×C22⋊C4—24no—
52C2×C5×C22⋊C4—24no—
53C2×C5×C22⋊C4—41no—
54C2×C5×C22⋊C4—41no—
55C2×C5×C22⋊C4C2×C418yes—
56C2×C5×C22⋊C4C2×C414yes—
57C2×C22⋊Dic5—24no—
58C2×C22⋊Dic5—22no—
59C2×C22⋊Dic5—41no—
60C2×C22⋊Dic5—42no—
61C2×C22⋊Dic5—41no—
62C2×C22⋊Dic5—42no—
63C2×C4⋊Dic5—24no—
64C2×C4⋊Dic5—24no—
65C2×C4⋊Dic5—42no—
66C2×C4⋊Dic5—42no—
67C2×C4⋊Dic5—42no—
68C2×C4⋊Dic5—42no—
69C2×C4⋊Dic5—42no—
70C2×C4⋊Dic5—41no—
71C5×#95C2×C414yescomplemented
72C5×#95C2×C418yescomplemented
73#3963—24no—
74#3963—24no—
75#3963—42no—
76#3963—42no—
77#3963—42no—
78#3963—42no—
79#3963—42no—
80#3963—42no—
81#3963—42no—
82#3963—41no—
83C2×#397D511yescharacteristic
84C2×#385—52no—
85#1444—52no—
86#1443—52no—
87#1442—52no—
88#1441—52no—
89#1440—52no—
90#1439—51no—
91C22⋊#95—51no—
92C24×C5—21no—
93C24×C5—22no—
94C24×C5—22no—
95C24×C5—21no—
96C24×C5—21no—
97C24×C5—22no—
98C24×C5—22no—
99C24×C5—21no—
100C24×C5C22×C412yes—
101C24×C5C2×D412yes—
102C24×C5#4112yes—
103C24×C5C2411yescharacteristic
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5C22×C414yes—
125C22×C4×C5C22×C412yes—
126C22×C4×C5C22×C414yes—
127C22×C4×C5C22×C412yes—
128C22×Dic5—22no—
129C22×Dic5—22no—
130C22×Dic5—22no—
131C22×Dic5—22no—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5—22no—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—22no—
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—44no—
156C22×Dic5—48no—
157C22×Dic5—44no—
158C22×Dic5—48no—
159C22×Dic5—44no—
160C22×Dic5C2×D414yes—
161C5×C4⋊C4—44no—
162C5×C22⋊C4—216no—
163C5×C22⋊C4—44no—
164C22⋊Dic5—48no—
165C22⋊Dic5—44no—
166C4⋊Dic5—48no—
167C4⋊Dic5—48no—
168C4⋊Dic5—44no—
169C24×C4—51no—
170C22×C4⋊C4—52no—
171C22×C4⋊C4—52no—
172C22×C4⋊C4—51no—
173C22×C4⋊C4C2×D511yescharacteristic
174C22×C22⋊C4—51no—
175C22×C22⋊C4—52no—
176C22×C22⋊C4C2×D512yes—
177C22×C22⋊C4C2×D511yescharacteristic
178#397Dic518yescomplemented
179#385—108no—
180C2×#95—52no—
181C2×#95—52no—
182C2×#95—52no—
183C2×#95—52no—
184C2×#95—52no—
185C2×#95—51no—
186C2×#95—52no—
187C2×#95—52no—
188C2×#95—52no—
189C2×#95—52no—
190C2×#95C2×D511yescharacteristic
191C2×#95C2×D512yes—
192C23×C5—22no—
193C23×C5—24no—
194C23×C5—24no—
195C23×C5—22no—
196C23×C5—24no—
197C23×C5—22no—
198C23×C5—24no—
199C23×C5—22no—
200C23×C5—22no—
201C23×C5—22no—
202C23×C5—24no—
203C23×C5—24no—
204C23×C5—41no—
205C23×C5—42no—
206C23×C5—41no—
207C23×C5—42no—
208C23×C5—42no—
209C23×C5—41no—
210C23×C5—41no—
211C23×C5—42no—
212C23×C5—42no—
213C23×C5—42no—
214C23×C5C4×D414yes—
215C23×C5C23×C411yescharacteristic
216C23×C5C22×D411yescharacteristic
217C23×C5C2×#4111yescharacteristic
218C23×C5H(5,𝔽2)11yescharacteristic
219C23×C5H(5,𝔽2)11yescharacteristic
220C23×C5H(5,𝔽2)11yescharacteristic
221C23×C5H(5,𝔽2)12yes—
222C23×C5H(5,𝔽2)12yes—
223C23×C52−1+412yes—
224C23×C52−1+412yes—
225C23×C52−1+411yescharacteristic
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—22no—
229C2×C4×C5—22no—
230C2×C4×C5—24no—
231C2×C4×C5—24no—
232C2×C4×C5—24no—
233C2×C4×C5—24no—
234C2×C4×C5—24no—
235C2×C4×C5—24no—
236C2×C4×C5—42no—
237C2×C4×C5—42no—
238C2×C4×C5—42no—
239C2×C4×C5—42no—
240C2×C4×C5—42no—
241C2×C4×C5—42no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—41no—
249C2×C4×C5—41no—
250C2×Dic5—24no—
251C2×Dic5—24no—
252C2×Dic5—24no—
253C2×Dic5—416no—
254C2×Dic5—42no—
255C2×Dic5—42no—
256C2×Dic5—42no—
257C2×Dic5—42no—
258C2×Dic5—42no—
259C2×Dic5—42no—
260C2×Dic5—42no—
261C2×Dic5—42no—
262C2×Dic5—42no—
263C2×Dic5—42no—
264C2×Dic5—42no—
265C2×Dic5—42no—
266C2×Dic5—42no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—42no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C25C22×D511yescharacteristic
280C23×C4—52no—
281C23×C4—54no—
282C23×C4—52no—
283C23×C4—52no—
284C23×C4—52no—
285C23×C4—201no—
286C23×C4—202no—
287C23×C4—202no—
288C23×C4—201no—
289C23×C4C22×D512yes—
290C23×C4C22×D512yes—
291C23×C4C22×D511yescharacteristic
292C23×C4C22×D511yescharacteristic
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—104no—
296C2×C4⋊C4—104no—
297C2×C4⋊C4—202no—
298C2×C4⋊C4—202no—
299C2×C4⋊C4—202no—
300C2×C4⋊C4—202no—
301C2×C4⋊C4—201no—
302C2×C4⋊C4—202no—
303C2×C4⋊C4C2×Dic514yescomplemented
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—104no—
309C2×C22⋊C4—102no—
310C2×C22⋊C4—201no—
311C2×C22⋊C4—201no—
312C2×C22⋊C4—202no—
313C2×C22⋊C4—202no—
314C2×C22⋊C4C2×Dic518yes—
315C2×C22⋊C4C2×Dic514yes—
316#95—104no—
317#95—104no—
318#95—201no—
319#95—202no—
320#95—202no—
321#95—202no—
322#95—202no—
323#95—202no—
324#95—202no—
325#95—202no—
326#95C2×Dic514yescomplemented
327#95C2×Dic518yescomplemented
328C22×C5—28no—
329C22×C5—24no—
330C22×C5—42no—
331C22×C5—44no—
332C22×C5—42no—
333C22×C5—42no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—44no—
337C22×C5—42no—
338C22×C5—44no—
339C22×C5C2×C4×D411yescharacteristic
340C22×C5#51811yescharacteristic
341C22×C5#52012yes—
342C22×C5#52012yes—
343C22×C5#52011yescharacteristic
344C22×C5#53411yescharacteristic
345C22×C5#53412yes—
346C22×C5#53612yes—
347C22×C5#53711yescharacteristic
348C22×C5#53811yescharacteristic
349C22×C5#53911yescharacteristic
350C22×C5#53912yes—
351C22×C5#53912yes—
352C22×C5#56011yescharacteristic
353C22×C5#56012yes—
354C22×C5H(3,𝔽4)11yescharacteristic
355C22×C5#56212yes—
356C22×C5#56212yes—
357C22×C5#56311yescharacteristic
358C22×C5#56311yescharacteristic
359C22×C5#56312yes—
360C22×C5#56312yes—
361C22×C5#56312yes—
362C4×C5—44no—
363C4×C5—44no—
364C4×C5—42no—
365C4×C5—42no—
366Dic5—44no—
367Dic5—82no—
368Dic5—82no—
369Dic5—82no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24D4×D512yes—
379C24#71112yes—
380C24C22×Dic512yes—
381C24C23×D511yescharacteristic
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—54no—
403C22×C4—102no—
404C22×C4—102no—
405C22×C4—102no—
406C22×C4—102no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—102no—
417C22×C4—102no—
418C22×C4—102no—
419C22×C4—102no—
420C22×C4—102no—
421C22×C4—102no—
422C22×C4—102no—
423C22×C4—102no—
424C22×C4—102no—
425C22×C4—102no—
426C22×C4—102no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—102no—
430C22×C4—204no—
431C22×C4—208no—
432C22×C4—204no—
433C22×C4—204no—
434C22×C4—208no—
435C22×C4C22×Dic514yes—
436C22×C4C22×Dic512yes—
437C22×C4C22×Dic514yes—
438C22×C4C22×Dic512yes—
439C4⋊C4—44no—
440C4⋊C4—208no—
441C4⋊C4—208no—
442C4⋊C4—204no—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—204no—
446C22⋊C4—208no—
447C2×C5—44no—
448C2×C5#232212yes—
449C2×C5#232511yescharacteristic
450C2×C5#237611yescharacteristic
451C2×C5#237911yescharacteristic
452C2×C5#238312yes—
453C2×C5#281211yescharacteristic
454C2×C5#281412yes—
455C2×C5#281711yescharacteristic
456C2×C5#282712yes—
457C2×C5#283411yescharacteristic
458C2×C5#283511yescharacteristic
459C23—22no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23D4×Dic514yes—
482C23C2×D4×D511yescharacteristic
483C23C2×#71111yescharacteristic
484C23#414411yescharacteristic
485C23#414411yescharacteristic
486C23#414911yescharacteristic
487C23#414912yes—
488C23#414912yes—
489C23#415011yescharacteristic
490C23#415012yes—
491C23#415012yes—
492C23C23×Dic511yescharacteristic
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—104no—
518C2×C4—104no—
519C2×C4—104no—
520C2×C4—2016no—
521C2×C4—202no—
522C2×C4—202no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—202no—
528C2×C4—202no—
529C2×C4—202no—
530C2×C4—202no—
531C2×C4—202no—
532C2×C4—202no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—202no—
539C2×C4—202no—
540C2×C4—202no—
541C2×C4—202no—
542C2×C4—202no—
543C2×C4—202no—
544C2×C4—202no—
545C2×C4—202no—
546C5#1462711yescharacteristic, complemented, Sylow
547C22—28no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22#6593111yescharacteristic
559C22#6593611yescharacteristic
560C22#6593911yescharacteristic
561C22#6595111yescharacteristic
562C22#6598212yes—
563C22#6598511yescharacteristic
564C22#6598712yes—
565C22#6599912yes—
566C22#6599912yes—
567C22#6600011yescharacteristic
568C22#6600112yes—
569C22#6600611yescharacteristic
570C22#6601711yescharacteristic
571C22#6601812yes—
572C22#6604512yes—
573C22#6604612yes—
574C22#6604612yes—
575C22#6604612yes—
576C22C2×D4×Dic511yescharacteristic
577C22#6613211yescharacteristic
578C22#6616111yescharacteristic
579C22#6616112yes—
580C22#6616112yes—
581C4—44no—
582C4—42no—
583C4—44no—
584C4—42no—
585C4—204no—
586C4—402no—
587C4—402no—
588C4—402no—
589C2—44no—
590C2#1060937711yescharacteristic
591C2#1060938911yescharacteristic
592C2#1060939011yescharacteristic
593C2#1060945711yescharacteristic
594C2#1060945812yes—
595C2#1060946412yes—
596C2#1061436511yescharacteristic
597C2#1061456712yes—
598C2#1061457111yescharacteristic
599C2#1061468811yescharacteristic
600C2#1061480712yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#397C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#397D5
hypocenterC5#14627
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. #10602358C2semidirect
7. #10600212C2semidirect
8. #10600209C2semidirect
9. #10600205C2semidirect
19. C5×#397C4semidirect
50. C2×C5×C4⋊C4C2×C4semidirect
71. C5×#95C2×C4semidirect
72. C5×#95C2×C4semidirect
178. #397Dic5semidirect
303. C2×C4⋊C4C2×Dic5semidirect
326. #95C2×Dic5semidirect
327. #95C2×Dic5semidirect
546. C5#14627semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3972
C2×#652392
#106023582
#106023412
#106023392
#106023372
#106002122
#106002092
#106002052
#146275