Finite Group Databasebeta

← #12028575

Subgroups — #12028575

Order 1280.

1550 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65234C212yesmaximal
2C2×#65235C211yesmaximal, characteristic
3C5×#1439C211yesmaximal, characteristic
4#10602359C212yesmaximal, complemented
5#10602339C211yesmaximal, characteristic
6#10602337C212yesmaximal
7#10600207C212yesmaximal, complemented
8#10600200C212yesmaximal, complemented
9#10600197C212yesmaximal, complemented
10C24×Dic5C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12C22×C5×C22⋊C4C2212yes—
13C22×C5×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#683C2212yes—
18C22×#683C2212yes—
19C22×C4×Dic5C2212yes—
20C2×C5×#95C2212yes—
21C2×C5×#95C2211yescharacteristic
22C2×#3963C2212yes—
23C2×#3963C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31#65235C418yescomplemented
32#65234—28no—
33#14514—51nomaximal, Sylow
34C25×C5C2311yescharacteristic
35C23×C4×C5C2312yes—
36C23×C4×C5C2311yescharacteristic
37C23×C4×C5C2311yescharacteristic
38C23×C4×C5C2312yes—
39C23×Dic5—41no—
40C23×Dic5—41no—
41C23×Dic5—42no—
42C23×Dic5—42no—
43C23×Dic5C2×C414yes—
44C23×Dic5C2312yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C2×C5×C4⋊C4—41no—
49C2×C5×C4⋊C4—42no—
50C2×C5×C22⋊C4—24no—
51C2×C5×C22⋊C4—24no—
52C2×C5×C22⋊C4—41no—
53C2×C5×C22⋊C4—41no—
54C2×C5×C22⋊C4—42no—
55C2×C5×C22⋊C4C2×C414yes—
56C2×C22⋊Dic5—24no—
57C2×C22⋊Dic5—24no—
58C2×C22⋊Dic5—42no—
59C2×C22⋊Dic5C2×C414yes—
60C2×C22⋊Dic5D414yes—
61C2×C22⋊Dic5Q814yes—
62C2×C4⋊Dic5—24no—
63C2×C4⋊Dic5—41no—
64C2×#683—24no—
65C2×#683—24no—
66C2×#683—42no—
67C2×#683—42no—
68C2×#683—42no—
69C2×#683C2×C418yes—
70C2×C4×Dic5—24no—
71C2×C4×Dic5—24no—
72C2×C4×Dic5—42no—
73C5×#95—24no—
74C5×#95—41no—
75#3963—24no—
76#3963—24no—
77#3963—42no—
78#3963—42no—
79#3963—42no—
80#3963—42no—
81#3963—42no—
82#3963—41no—
83#3963C2×C418yes—
84C2×#388—52no—
85C2×#385—51no—
86#1442—52no—
87#1441—52no—
88#1439—51no—
89#1439D511yescharacteristic
90#1436—52no—
91C4⋊#95—52no—
92C22⋊#95—52no—
93C24×C5—22no—
94C24×C5—22no—
95C24×C5—22no—
96C24×C5—21no—
97C24×C5—21no—
98C24×C5—22no—
99C24×C5—21no—
100C24×C5—21no—
101C24×C5C22×C412yes—
102C24×C5C2×D412yes—
103C24×C5C2×Q812yes—
104C24×C5C2411yescharacteristic
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
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×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5C22×C412yes—
127C22×C4×C5C22×C412yes—
128C22×C4×C5#4114yes—
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—48no—
153C22×Dic5—44no—
154C22×Dic5—48no—
155C22×Dic5—44no—
156C22×Dic5—44no—
157C22×Dic5C22×C414yes—
158C22×Dic5C22×C414yes—
159C22×Dic5C2×D414yes—
160C22×Dic5C2×Q814yes—
161C22×Dic5#4114yes—
162C5×C4⋊C4—44no—
163C5×C22⋊C4—44no—
164C5×C22⋊C4—48no—
165C22⋊Dic5—48no—
166C22⋊Dic5C4⋊C4116yescomplemented
167C4⋊Dic5—44no—
168#683—48no—
169#683—48no—
170C4×Dic5—48no—
171C24×C4—51no—
172C22×C4⋊C4—52no—
173C22×C4⋊C4—52no—
174C22×C4⋊C4—51no—
175C22×C4⋊C4C2×D511yescharacteristic
176C22×C22⋊C4—51no—
177C22×C22⋊C4—52no—
178C22×C22⋊C4C2×D512yes—
179C22×C22⋊C4C2×D511yescharacteristic
180C22×C42—52no—
181#388—108no—
182#385—58no—
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×#95—51no—
192C2×#95C2×D512yes—
193C2×#95C2×D511yescharacteristic
194C23×C5—24no—
195C23×C5—22no—
196C23×C5—24no—
197C23×C5—22no—
198C23×C5—22no—
199C23×C5—22no—
200C23×C5—22no—
201C23×C5—22no—
202C23×C5—24no—
203C23×C5—24no—
204C23×C5—24no—
205C23×C5—24no—
206C23×C5—42no—
207C23×C5—42no—
208C23×C5—42no—
209C23×C5—42no—
210C23×C5—41no—
211C23×C5—42no—
212C23×C5—41no—
213C23×C5—42no—
214C23×C5—41no—
215C23×C5—41no—
216C23×C5C2×C4⋊C414yes—
217C23×C5C23×C411yescharacteristic
218C23×C5C22×D411yescharacteristic
219C23×C5C22×Q811yescharacteristic
220C23×C5C2×#4112yes—
221C23×C5C2×#4112yes—
222C23×C5H(5,𝔽2)11yescharacteristic
223C23×C5H(5,𝔽2)11yescharacteristic
224C23×C5H(5,𝔽2)12yes—
225C23×C5H(5,𝔽2)12yes—
226C23×C52−1+411yescharacteristic
227C23×C52−1+411yescharacteristic
228C2×C4×C5—24no—
229C2×C4×C5—24no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—24no—
233C2×C4×C5—24no—
234C2×C4×C5—24no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
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—41no—
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×C4×C5—42no—
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—24no—
264C2×Dic5—416no—
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—
280C2×Dic5C2×C4⋊C418yes—
281C25C22×D511yescharacteristic
282C23×C4—52no—
283C23×C4—52no—
284C23×C4—52no—
285C23×C4—54no—
286C23×C4—52no—
287C23×C4—201no—
288C23×C4—201no—
289C23×C4—202no—
290C23×C4—202no—
291C23×C4C22×D512yes—
292C23×C4C22×D511yescharacteristic
293C23×C4C22×D511yescharacteristic
294C23×C4C22×D512yes—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—58no—
298C2×C4⋊C4—104no—
299C2×C4⋊C4—104no—
300C2×C4⋊C4—104no—
301C2×C4⋊C4—202no—
302C2×C4⋊C4—202no—
303C2×C4⋊C4—202no—
304C2×C4⋊C4—201no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—54no—
311C2×C22⋊C4—54no—
312C2×C22⋊C4—54no—
313C2×C22⋊C4—104no—
314C2×C22⋊C4—104no—
315C2×C22⋊C4—202no—
316C2×C22⋊C4C4×D514yes—
317C2×C42—104no—
318C2×C42—104no—
319C2×C42—202no—
320#95—24no—
321#95—41no—
322#95—58no—
323#95—104no—
324#95—104no—
325#95—202no—
326#95—201no—
327#95—202no—
328#95—202no—
329#95—202no—
330#95—202no—
331C22×C5—24no—
332C22×C5—28no—
333C22×C5—44no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—44no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—42no—
340C22×C5—44no—
341C22×C5—42no—
342C22×C5C22×C4⋊C411yescharacteristic
343C22×C5C4×#4112yes—
344C22×C5#51811yescharacteristic
345C22×C5#51812yes—
346C22×C5#51911yescharacteristic
347C22×C5#53711yescharacteristic
348C22×C5#53711yescharacteristic
349C22×C5#53912yes—
350C22×C5#53912yes—
351C22×C5#54212yes—
352C22×C5#54212yes—
353C22×C5#54411yescharacteristic
354C22×C5#54411yescharacteristic
355C22×C5#54612yes—
356C22×C5#54612yes—
357C22×C5#55111yescharacteristic
358C22×C5#55312yes—
359C22×C5#55412yes—
360C22×C5#55412yes—
361C22×C5#55911yescharacteristic
362C22×C5#56012yes—
363C22×C5#56011yescharacteristic
364C22×C5H(3,𝔽4)11yescharacteristic
365C4×C5—42no—
366C4×C5—42no—
367C4×C5—44no—
368C4×C5—82no—
369Dic5—28no—
370Dic5—44no—
371Dic5—44no—
372Dic5—82no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24C2×C4×D512yes—
382C24D4×D512yes—
383C24Q8×D512yes—
384C24C23×D511yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—54no—
407C22×C4—54no—
408C22×C4—54no—
409C22×C4—54no—
410C22×C4—54no—
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—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—204no—
435C22×C4—208no—
436C22×C4—204no—
437C22×C4—208no—
438C22×C4—204no—
439C22×C4C2×C4×D512yes—
440C22×C4C2×C4×D512yes—
441C22×C4#71114yes—
442C4⋊C4—44no—
443C4⋊C4—208no—
444C4⋊C4—208no—
445C4⋊C4—204no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—516no—
449C22⋊C4—208no—
450C42—208no—
451C2×C5—44no—
452C2×C5#231611yescharacteristic
453C2×C5#233212yes—
454C2×C5#234912yes—
455C2×C5#237411yescharacteristic
456C2×C5#237511yescharacteristic
457C2×C5#269112yes—
458C2×C5#269311yescharacteristic
459C2×C5#271211yescharacteristic
460C2×C5#276311yescharacteristic
461C2×C5#279012yes—
462C2×C5#280711yescharacteristic
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23D5×C4⋊C414yes—
486C23C22×C4×D511yescharacteristic
487C23C2×D4×D511yescharacteristic
488C23C2×#71112yes—
489C23#414411yescharacteristic
490C23#414411yescharacteristic
491C23#414412yes—
492C23C2×Q8×D511yescharacteristic
493C23D5×#4112yes—
494C23#414912yes—
495C23#415011yescharacteristic
496C23#415011yescharacteristic
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—58no—
522C2×C4—104no—
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—2016no—
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—
546C2×C4—202no—
547C2×C4—202no—
548C2×C4—202no—
549C2×C4—202no—
550C5#1451411yescharacteristic, complemented, Sylow
551C22—24no—
552C22—28no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22#6582011yescharacteristic
563C22C2×D5×C4⋊C411yescharacteristic
564C22#6585011yescharacteristic
565C22C4×#71112yes—
566C22#6587912yes—
567C22#6592512yes—
568C22#6592811yescharacteristic
569C22#6593111yescharacteristic
570C22#6593111yescharacteristic
571C22#6593512yes—
572C22#6593712yes—
573C22#6595412yes—
574C22#6595711yescharacteristic
575C22#6595711yescharacteristic
576C22#6596912yes—
577C22#6597012yes—
578C22#6598812yes—
579C22#6599912yes—
580C22#6600011yescharacteristic
581C22#6600312yes—
582C22#6601311yescharacteristic
583C22#6604112yes—
584C22#6604711yescharacteristic
585C4—42no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—108no—
590C4—204no—
591C4—204no—
592C4—402no—
593C2—44no—
594C2#1060560011yescharacteristic
595C2#1060612512yes—
596C2#1060661311yescharacteristic
597C2#1060719512yes—
598C2#1060803911yescharacteristic
599C2#1060845811yescharacteristic
600C2#1060857511yescharacteristic
601C2#1060874312yes—
602C2#1060912612yes—
603C2#1060930511yescharacteristic
604C2#1060981911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1439C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenter#1439D5
hypocenterC5#14514
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
4. #10602359C2semidirect
7. #10600207C2semidirect
8. #10600200C2semidirect
9. #10600197C2semidirect
31. #65235C4semidirect
166. C22⋊Dic5C4⋊C4semidirect
550. C5#14514semidirect

Maximal subgroups

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

subgroupindex
C2×#652342
C2×#652352
C5×#14392
#106023592
#106023392
#106023372
#106002072
#106002002
#106001972
#145145