Finite Group Databasebeta

← #12029670

Subgroups — #12029670

Order 1280.

1506 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65236C211yesmaximal, characteristic
2C5×#1439C211yesmaximal, characteristic
3#10602357C212yesmaximal, complemented
4#10602338C212yesmaximal
5#10602337C212yesmaximal
6#10602337C211yesmaximal, characteristic
7#10600217C212yesmaximal, complemented
8#10600211C212yesmaximal, complemented
9#10600195C212yesmaximal, complemented
10C24×Dic5C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C22⋊C4C2212yes—
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic5C2212yes—
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C4×Dic5C2212yes—
18C2×C5×#95C2212yes—
19C2×C5×#95C2211yescharacteristic
20C2×#3963C2212yes—
21C2×#3963C2212yes—
22C2×#3963C2212yes—
23C2×#3963C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2211yescharacteristic
30C2×#3963C2212yes—
31#65236C418yescomplemented
32#14596—51nomaximal, Sylow
33C25×C5C2311yescharacteristic
34C23×C4×C5C2312yes—
35C23×C4×C5C2311yescharacteristic
36C23×C4×C5C2311yescharacteristic
37C23×C4×C5C2312yes—
38C23×Dic5—22no—
39C23×Dic5—24no—
40C23×Dic5—41no—
41C23×Dic5—42no—
42C23×Dic5—41no—
43C23×Dic5C2312yes—
44C23×Dic5C2312yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C22×Dic10—42no—
48C22×Dic10—42no—
49C22×Dic10—41no—
50C22×Dic10—41no—
51C22×Dic10C2×C414yescomplemented
52C2×C5×C4⋊C4—24no—
53C2×C5×C4⋊C4—41no—
54C2×C5×C22⋊C4—24no—
55C2×C5×C22⋊C4—24no—
56C2×C5×C22⋊C4—41no—
57C2×C5×C22⋊C4—41no—
58C2×C5×C22⋊C4—42no—
59C2×C5×C22⋊C4C2×C414yes—
60C2×C22⋊Dic5—42no—
61C2×C22⋊Dic5—42no—
62C2×C22⋊Dic5—41no—
63C2×C22⋊Dic5—41no—
64C2×C22⋊Dic5C2×C418yes—
65C2×C22⋊Dic5Q814yescomplemented
66C2×C4×Dic5—24no—
67C2×C4×Dic5—42no—
68C2×C4×Dic5—42no—
69C5×#95—41no—
70C5×#95—42no—
71#3963—24no—
72#3963—24no—
73#3963—42no—
74#3963—42no—
75#3963—42no—
76#3963—42no—
77#3963—42no—
78#3963—42no—
79#3963—42no—
80#3963C2×C418yescomplemented
81#3963C2×C414yescomplemented
82C2×#393—51no—
83#1446—52no—
84#1443—52no—
85#1439D511yescharacteristic
86#1437—52no—
87#1436—52no—
88#1434—52no—
89C22⋊#95—52no—
90C22⋊#95—51no—
91C24×C5—22no—
92C24×C5—22no—
93C24×C5—22no—
94C24×C5—21no—
95C24×C5—22no—
96C24×C5—21no—
97C24×C5—21no—
98C24×C5—21no—
99C24×C5C22×C412yes—
100C24×C5C2×Q812yes—
101C24×C5#4112yes—
102C24×C5C2411yescharacteristic
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—22no—
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×C412yes—
125C22×C4×C5C22×C412yes—
126C22×C4×C5#4114yes—
127C22×Dic5—22no—
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—44no—
152C22×Dic5—48no—
153C22×Dic5—48no—
154C22×Dic5—44no—
155C22×Dic5—44no—
156C22×Dic5C22×C414yes—
157C22×Dic5C22×C414yes—
158C22×Dic5C2×Q814yes—
159C22×Dic5#4114yes—
160C2×Dic10—48no—
161C2×Dic10—44no—
162C2×Dic10—44no—
163C2×Dic10—48no—
164C2×Dic10—44no—
165C5×C4⋊C4—44no—
166C5×C22⋊C4—44no—
167C5×C22⋊C4—48no—
168C22⋊Dic5—48no—
169C22⋊Dic5—44no—
170C4×Dic5—48no—
171C23×Q8—51no—
172C24×C4—51no—
173C22×C4⋊C4C2×D511yescharacteristic
174C22×C22⋊C4—52no—
175C22×C22⋊C4—51no—
176C22×C22⋊C4C2×D512yes—
177C22×C22⋊C4C2×D511yescharacteristic
178C22×C42—52no—
179#393—58no—
180C2×#95—51no—
181C2×#95—52no—
182C2×#95—52no—
183C2×#95—52no—
184C2×#95—52no—
185C2×#95—52no—
186C2×#95—52no—
187C2×#95—52no—
188C2×#95—52no—
189C2×#95—52no—
190C2×#95—52no—
191C2×#95C2×D512yes—
192C2×#95C2×D511yescharacteristic
193C23×C5—24no—
194C23×C5—24no—
195C23×C5—22no—
196C23×C5—22no—
197C23×C5—22no—
198C23×C5—22no—
199C23×C5—22no—
200C23×C5—22no—
201C23×C5—24no—
202C23×C5—24no—
203C23×C5—24no—
204C23×C5—24no—
205C23×C5—42no—
206C23×C5—42no—
207C23×C5—42no—
208C23×C5—42no—
209C23×C5—41no—
210C23×C5—41no—
211C23×C5—42no—
212C23×C5—41no—
213C23×C5—42no—
214C23×C5—41no—
215C23×C5C4×Q814yes—
216C23×C5C23×C411yescharacteristic
217C23×C5C22×Q811yescharacteristic
218C23×C5C2×#4111yescharacteristic
219C23×C5C2×#4112yes—
220C23×C5C2×#4112yes—
221C23×C5H(5,𝔽2)12yes—
222C23×C5H(5,𝔽2)11yescharacteristic
223C23×C5H(5,𝔽2)11yescharacteristic
224C23×C5H(5,𝔽2)12yes—
225C23×C5H(5,𝔽2)11yescharacteristic
226C23×C52−1+411yescharacteristic
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—42no—
235C2×C4×C5—42no—
236C2×C4×C5—41no—
237C2×C4×C5—42no—
238C2×C4×C5—41no—
239C2×C4×C5—42no—
240C2×C4×C5—41no—
241C2×C4×C5—42no—
242C2×C4×C5—42no—
243C2×C4×C5—41no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×Dic5—24no—
252C2×Dic5—24no—
253C2×Dic5—24no—
254C2×Dic5—24no—
255C2×Dic5—24no—
256C2×Dic5—24no—
257C2×Dic5—24no—
258C2×Dic5—24no—
259C2×Dic5—24no—
260C2×Dic5—24no—
261C2×Dic5—24no—
262C2×Dic5—24no—
263C2×Dic5—416no—
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—
279C2×Dic5—42no—
280Dic10—416no—
281C25C22×D511yescharacteristic
282C22×Q8—54no—
283C22×Q8—202no—
284C22×Q8—201no—
285C22×Q8—201no—
286C22×Q8—202no—
287C23×C4—52no—
288C23×C4—52no—
289C23×C4—52no—
290C23×C4—52no—
291C23×C4—104no—
292C23×C4—102no—
293C23×C4—201no—
294C23×C4—202no—
295C23×C4—201no—
296C23×C4C22×D512yes—
297C23×C4C22×D511yescharacteristic
298C23×C4C22×D511yescharacteristic
299C23×C4C22×D512yes—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—41no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—58no—
308C2×C22⋊C4—54no—
309C2×C22⋊C4—201no—
310C2×C22⋊C4—202no—
311C2×C22⋊C4—202no—
312C2×C22⋊C4—201no—
313C2×C22⋊C4C4×D514yes—
314C2×C42—104no—
315C2×C42—202no—
316C2×C42—202no—
317#95—42no—
318#95—41no—
319#95—54no—
320#95—58no—
321#95—104no—
322#95—104no—
323#95—202no—
324#95—202no—
325#95—202no—
326#95—202no—
327#95—202no—
328#95—202no—
329#95—202no—
330C22×C5—24no—
331C22×C5—28no—
332C22×C5—44no—
333C22×C5—42no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—44no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—42no—
340C22×C5—44no—
341C22×C5C2×C4×Q811yescharacteristic
342C22×C5C4×#4112yes—
343C22×C5#51811yescharacteristic
344C22×C5#51812yes—
345C22×C5#52011yescharacteristic
346C22×C5#53811yescharacteristic
347C22×C5#53812yes—
348C22×C5#53812yes—
349C22×C5#53911yescharacteristic
350C22×C5#54212yes—
351C22×C5#54212yes—
352C22×C5#54311yescharacteristic
353C22×C5#54411yescharacteristic
354C22×C5#55112yes—
355C22×C5#55111yescharacteristic
356C22×C5#55112yes—
357C22×C5#55111yescharacteristic
358C22×C5#55412yes—
359C22×C5#55412yes—
360C22×C5#55512yes—
361C22×C5#56011yescharacteristic
362C22×C5#56012yes—
363C22×C5H(3,𝔽4)11yescharacteristic
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—44no—
367C4×C5—82no—
368Dic5—44no—
369Dic5—44no—
370Dic5—44no—
371Dic5—44no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24—21no—
380C24C2×C4×D512yes—
381C24#71112yes—
382C24Q8×D512yes—
383C24C23×D511yescharacteristic
384C2×Q8—204no—
385C2×Q8—208no—
386C2×Q8—204no—
387C2×Q8—208no—
388C2×Q8—204no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—54no—
411C22×C4—54no—
412C22×C4—54no—
413C22×C4—54no—
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—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—204no—
439C22×C4—208no—
440C22×C4—208no—
441C22×C4—204no—
442C22×C4—204no—
443C22×C4C2×C4×D512yes—
444C22×C4C2×C4×D512yes—
445C22×C4#71114yes—
446C4⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—208no—
450C22⋊C4—204no—
451C42—208no—
452C2×C5—44no—
453C2×C5#232911yescharacteristic
454C2×C5#233712yes—
455C2×C5#236812yes—
456C2×C5#237211yescharacteristic
457C2×C5#237511yescharacteristic
458C2×C5#273611yescharacteristic
459C2×C5#273712yes—
460C2×C5#274211yescharacteristic
461C2×C5#276011yescharacteristic
462C2×C5#276311yescharacteristic
463C2×C5#278412yes—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23#403314yes—
487C23C22×C4×D511yescharacteristic
488C23C2×#71111yescharacteristic
489C23C2×#71112yes—
490C23#414411yescharacteristic
491C23#414412yes—
492C23#414411yescharacteristic
493C23C2×Q8×D511yescharacteristic
494C23D5×#4112yes—
495C23#414911yescharacteristic
496C23#414912yes—
497C23#415011yescharacteristic
498Q8—2016no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—104no—
527C2×C4—104no—
528C2×C4—104no—
529C2×C4—104no—
530C2×C4—104no—
531C2×C4—104no—
532C2×C4—104no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—2016no—
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—
546C2×C4—202no—
547C2×C4—202no—
548C2×C4—202no—
549C2×C4—202no—
550C2×C4—202no—
551C2×C4—202no—
552C5#1459611yescharacteristic, complemented, Sylow
553C22—24no—
554C22—28no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22#6582011yescharacteristic
565C22C2×#403311yescharacteristic
566C22#6585711yescharacteristic
567C22C4×#71112yes—
568C22#6587912yes—
569C22#6592112yes—
570C22#6592111yescharacteristic
571C22#6593611yescharacteristic
572C22#6595012yes—
573C22#6595112yes—
574C22#6595111yescharacteristic
575C22#6595412yes—
576C22#6595711yescharacteristic
577C22#6596811yescharacteristic
578C22#6596912yes—
579C22#6597012yes—
580C22#6599212yes—
581C22#6599311yescharacteristic
582C22#6599612yes—
583C22#6601612yes—
584C22#6601711yescharacteristic
585C22#6603812yes—
586C22#6605411yescharacteristic
587C4—42no—
588C4—42no—
589C4—44no—
590C4—82no—
591C4—204no—
592C4—204no—
593C4—204no—
594C4—204no—
595C2—44no—
596C2#1060560511yescharacteristic
597C2#1060613012yes—
598C2#1060641111yescharacteristic
599C2#1060720312yes—
600C2#1060816511yescharacteristic
601C2#1060846811yescharacteristic
602C2#1060857411yescharacteristic
603C2#1060877912yes—
604C2#1060917712yes—
605C2#1060943711yescharacteristic
606C2#1060980711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1439C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenter#1439D5
hypocenterC5#14596
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #10602357C2semidirect
7. #10600217C2semidirect
8. #10600211C2semidirect
9. #10600195C2semidirect
31. #65236C4semidirect
51. C22×Dic10C2×C4semidirect
65. C2×C22⋊Dic5Q8semidirect
80. #3963C2×C4semidirect
81. #3963C2×C4semidirect
552. C5#14596semidirect

Maximal subgroups

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

subgroupindex
C2×#652362
C5×#14392
#106023572
#106023382
#106023372
#106002172
#106002112
#106001952
#145965