Finite Group Databasebeta

← #12065234

Subgroups — #12065234

Order 1280.

1174 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2#10611045C212yesmaximal, complemented
3C5×#1464C211yesmaximal, characteristic, complemented
4#10602365C211yesmaximal, characteristic, complemented
5#10601620C212yesmaximal, complemented
6#10598919C212yesmaximal, complemented
7#10598917C212yesmaximal, complemented
8#10598896C212yesmaximal, complemented
9#10598894C212yesmaximal, complemented
10C22×#709C411yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C22×C4⋊Dic5C411yescharacteristic
13#65857—24no—
14#65857—24no—
15C2×#4038C2211yescharacteristic
16C2×#4038C2211yescharacteristic
17C2×D5×C4⋊C4C411yescharacteristic
18C2×D5×C4⋊C4C411yescharacteristic
19C2×#4027C411yescharacteristic
20C2×#4027C411yescharacteristic
21C2×#4027C2211yescharacteristic
22C2×#4027C2211yescharacteristic
23C2×C4×D20C411yescharacteristic
24C2×C4×D20C2211yescharacteristic
25C2×C4×Dic10C411yescharacteristic
26C2×C4×Dic10C2211yescharacteristic
27C5×#432C2212yes—
28#65215C2212yes—
29#65117—24no—
30#65008C2214yes—
31#65007C2214yes—
32#64990—24no—
33C5×#324C2212yes—
34C5×#323C2212yes—
35#64748C2212yes—
36#64745C2212yes—
37#64688C2214yes—
38#64688C2214yes—
39#17372—51nomaximal, Sylow
40C23×C4×C5C2×C411yescharacteristic
41C22×C22⋊D5C2×C411yescharacteristic
42C22×C22⋊D5C2×C411yescharacteristic
43C23×Dic5C2×C411yescharacteristic
44C23×Dic5C2×C411yescharacteristic
45C2×#709—24no—
46C2×#709—24no—
47C2×#709—22no—
48C2×#709—22no—
49C22×D20C2×C411yescharacteristic
50C22×C4×D5C2×C411yescharacteristic
51C22×C4×D5C2×C411yescharacteristic
52C22×C4×D5C2×C411yescharacteristic
53C22×C4×D5C2×C411yescharacteristic
54C22×C4×D5C2×C411yescharacteristic
55C22×C4×D5C2×C411yescharacteristic
56C22×Dic10C2×C411yescharacteristic
57C5×#117—22no—
58C5×#117—22no—
59C2×C5×C4⋊C4C2×C411yescharacteristic
60C2×C5×C4⋊C4C2311yescharacteristic
61C2×C5×C22⋊C4C2×C411yescharacteristic
62C2×C5×C22⋊C4C2311yescharacteristic
63C2×C42×C5C2×C411yescharacteristic
64C2×C42×C5C2311yescharacteristic
65C2×#685C2×C411yescharacteristic
66C2×#685C2×C411yescharacteristic
67C2×#685C2311yescharacteristic
68C2×#685C2311yescharacteristic
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5C2×C411yescharacteristic
72C2×C4⋊Dic5C2×C411yescharacteristic
73C2×C4⋊Dic5C2×C411yescharacteristic
74C2×C4⋊Dic5C2×C411yescharacteristic
75C2×#683C2×C411yescharacteristic
76C2×#683C2×C411yescharacteristic
77C2×#683C2×C411yescharacteristic
78C2×#683C2×C411yescharacteristic
79C2×C4×Dic5C2×C411yescharacteristic
80C2×C4×Dic5C2×C411yescharacteristic
81C2×C4×Dic5C2311yescharacteristic
82C2×C4×Dic5C2311yescharacteristic
83#4038—24no—
84#4038—24no—
85D5×C4⋊C4—24no—
86D5×C4⋊C4—24no—
87#4027—24no—
88#4027—24no—
89#4027—24no—
90#4027—24no—
91C4×D20—22no—
92C4×D20—22no—
93C4×D20—22no—
94C4×D20—22no—
95C4×Dic10—22no—
96C4×Dic10—22no—
97C4×Dic10D414yes—
98C4×Dic10D414yes—
99C5×C4⋊C8—22no—
100C5×C22⋊C8C2314yes—
101C5×#97—22no—
102C22⋊#168C2314yes—
103#3947—24no—
104#3946—24no—
105#3946—24no—
106C8×Dic5—24no—
107C4⋊#168—22no—
108#3935—22no—
109C2×#520—51no—
110#2983—52no—
111#1466—52no—
112#1466—52no—
113#1464—52no—
114#1464—51no—
115#1464D511yescharacteristic, complemented
116#1460—52no—
117#1458—52no—
118C24×C5C22×C411yescharacteristic
119C23×D5C22×C411yescharacteristic
120C23×D5C22×C411yescharacteristic
121C22×C4×C5—22no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5C22×C411yescharacteristic
126C22×C4×C5C22×C411yescharacteristic
127C22×C4×C5C22×C411yescharacteristic
128C22×C4×C5C22×C411yescharacteristic
129C22×C4×C5C22×C411yescharacteristic
130C22×C4×C5C2411yescharacteristic
131C2×C22⋊D5—22no—
132C2×C22⋊D5—22no—
133C2×C22⋊D5—22no—
134C2×C22⋊D5—22no—
135C2×C22⋊D5—22no—
136C2×C22⋊D5M1614yes—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—81no—
140C22×Dic5—81no—
141C22×Dic5C22×C411yescharacteristic
142C22×Dic5C22×C411yescharacteristic
143C22×Dic5C22×C411yescharacteristic
144C22×Dic5C22×C411yescharacteristic
145C22×Dic5C22×C411yescharacteristic
146C22×Dic5C22×C411yescharacteristic
147#709—216no—
148#709—216no—
149C2×D20—22no—
150C2×D20—22no—
151C2×D20—21no—
152C2×D20—21no—
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—22no—
160C2×C4×D5—22no—
161C2×C4×D5—22no—
162C2×C4×D5—81no—
163C2×C4×D5—81no—
164C2×C4×D5—81no—
165C2×C4×D5—81no—
166C2×C4×D5M1614yes—
167C2×Dic10—22no—
168C2×Dic10—22no—
169C2×Dic10C22⋊C412yes—
170C2×Dic10C22⋊C412yes—
171C2×C5×C8—24no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—22no—
174C5×C22⋊C4—22no—
175C5×C22⋊C4—22no—
176C42×C5—21no—
177C42×C5—21no—
178C42×C5C2×D412yes—
179C42×C5C2×D412yes—
180#685—22no—
181#685—22no—
182#685—22no—
183#685—22no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—22no—
187C4⋊Dic5—22no—
188C4⋊Dic5C22⋊C412yes—
189C4⋊Dic5C22⋊C412yes—
190#683—22no—
191#683—22no—
192#683C22⋊C414yes—
193#683C22⋊C414yes—
194C4×Dic5—22no—
195C4×Dic5—22no—
196C4×Dic5C2×D414yes—
197C4×Dic5C2×D414yes—
198C2×#168—24no—
199C22×#41—51no—
200#520—104no—
201#520—104no—
202C2×C4×Q8—51no—
203C2×C4×Q8—51no—
204C2×C4×D4—51no—
205C2×C4×D4—51no—
206C2×C4×D4—51no—
207C2×C4×D4—51no—
208C2×C4×D4—51no—
209C2×C4×D4—51no—
210C2×#117—51no—
211C2×#117—51no—
212C2×#117C2×D511yescharacteristic
213C22×C4⋊C4—51no—
214C22×C4⋊C4—51no—
215C22×C4⋊C4—51no—
216#446—104no—
217#446—104no—
218#432—52no—
219#432—54no—
220#432C2×D512yes—
221#431—54no—
222#324—52no—
223#324C2×D512yes—
224#323—52no—
225#323—54no—
226#323—54no—
227#323C2×D512yes—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—41no—
231C23×C5—41no—
232C23×C5C2×M1612yes—
233C23×C5C23×C411yescharacteristic
234C22×D5—21no—
235C22×D5—21no—
236C22×D5—21no—
237C22×D5—21no—
238C22×D5—41no—
239C22×D5—41no—
240C22×D5—41no—
241C22×D5—41no—
242C22×D5C2×M1612yes—
243C22×D5C2×M1612yes—
244C2×C4×C5—24no—
245C2×C4×C5—24no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5C2×C22⋊C412yes—
257C2×C4×C5C2×C22⋊C411yescharacteristic
258C2×C4×C5C2×C22⋊C411yescharacteristic
259C2×C4×C5C2×M1612yes—
260C2×C4×C5C22×D412yes—
261C22⋊D5—28no—
262C22⋊D5—28no—
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5—21no—
267C2×Dic5—41no—
268C2×Dic5—41no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—82no—
272C2×Dic5—82no—
273C2×Dic5C2×C22⋊C412yes—
274C2×Dic5C2×C22⋊C412yes—
275C2×Dic5C2×C22⋊C412yes—
276C2×Dic5C2×C22⋊C412yes—
277C2×Dic5C2×M1612yes—
278C2×Dic5C2×M1612yes—
279D20—24no—
280D20—24no—
281C4×D5—28no—
282C4×D5—28no—
283C4×D5—82no—
284C4×D5—82no—
285C4×D5—82no—
286C4×D5—82no—
287Dic10—24no—
288Dic10—24no—
289C5×C8—44no—
290#168—44no—
291C2×#41—104no—
292C2×#41—104no—
293C2×#41—102no—
294C2×#41—102no—
295C22×Q8—51no—
296C22×D4—51no—
297C22×D4—51no—
298C22×D4—51no—
299C23×C4—51no—
300C23×C4—51no—
301C23×C4—51no—
302C23×C4—51no—
303C23×C4—51no—
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4C4×D511yescharacteristic
308C4×Q8—54no—
309C4×Q8—54no—
310C4×Q8—102no—
311C4×Q8—102no—
312C4×D4—102no—
313C4×D4—102no—
314C4×D4—104no—
315C4×D4—104no—
316C4×D4—104no—
317C4×D4—104no—
318C4×D4—102no—
319C4×D4—102no—
320#117—22no—
321#117—22no—
322#117—104no—
323#117—104no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4—51no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—104no—
333C2×C4⋊C4—104no—
334C2×C4⋊C4—102no—
335C2×C4⋊C4—102no—
336C2×C4⋊C4C4×D511yescharacteristic
337C2×C4⋊C4C22×D511yescharacteristic
338C2×C22⋊C4—51no—
339C2×C22⋊C4—51no—
340C2×C22⋊C4—51no—
341C2×C22⋊C4—51no—
342C2×C22⋊C4C4×D511yescharacteristic
343C2×C22⋊C4C22×D511yescharacteristic
344C2×C42—51no—
345C2×C42—51no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42C4×D511yescharacteristic
349C2×C42C22×D511yescharacteristic
350C4⋊C8—22no—
351C4⋊C8—104no—
352C4⋊C8—104no—
353C4⋊C8—102no—
354C22⋊C8—54no—
355C22⋊C8C22×D514yes—
356#97—22no—
357#97—104no—
358#97—102no—
359C4×C8—104no—
360C22×C5—21no—
361C22×C5—21no—
362C22×C5—22no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5C22×C22⋊C411yescharacteristic
368C22×C5C22×M1611yescharacteristic
369C22×C5C22×M1611yescharacteristic
370C2×D5—22no—
371C2×D5—22no—
372C2×D5—42no—
373C2×D5—42no—
374C2×D5—41no—
375C2×D5—41no—
376C2×D5—41no—
377C2×D5—41no—
378C2×D5—42no—
379C2×D5—42no—
380C4×C5—21no—
381C4×C5—21no—
382C4×C5—22no—
383C4×C5—22no—
384C4×C5—22no—
385C4×C5—41no—
386C4×C5—41no—
387Dic5—22no—
388Dic5—22no—
389Dic5—22no—
390Dic5—41no—
391Dic5—41no—
392Dic5#40714yes—
393C24—51no—
394C24—51no—
395C24C2×C4×D511yescharacteristic
396#41—1016no—
397#41—1016no—
398C2×Q8—52no—
399C2×Q8—52no—
400C2×Q8—102no—
401C2×Q8—102no—
402C2×D4—54no—
403C2×D4—102no—
404C2×D4—102no—
405C2×D4—102no—
406C2×D4—102no—
407C2×D4—102no—
408C2×D4—102no—
409C2×D4—102no—
410C2×D4—101no—
411C2×D4—101no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—51no—
417C22×C4—51no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—54no—
421C22×C4—51no—
422C22×C4—51no—
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—401no—
435C22×C4—401no—
436C22×C4—401no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4—401no—
440C22×C4C2×C4×D511yescharacteristic
441C22×C4C2×C4×D511yescharacteristic
442C22×C4C2×C4×D511yescharacteristic
443C22×C4C2×C4×D511yescharacteristic
444C22×C4C2×C4×D511yescharacteristic
445C22×C4C23×D511yescharacteristic
446C2×C8—24no—
447C2×C8—104no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—54no—
451C4⋊C4—54no—
452C4⋊C4—52no—
453C4⋊C4—52no—
454C4⋊C4—102no—
455C4⋊C4—102no—
456C4⋊C4—102no—
457C4⋊C4—102no—
458C4⋊C4—101no—
459C4⋊C4—101no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—102no—
463C22⋊C4—102no—
464C22⋊C4—102no—
465C22⋊C4—102no—
466C42—21no—
467C42—21no—
468C42—54no—
469C42—54no—
470C42—102no—
471C42—102no—
472C42D4×D512yes—
473C42D4×D512yes—
474C2×C5—21no—
475C2×C5—21no—
476C2×C5—42no—
477C2×C5C2×#40711yescharacteristic
478C2×C5#288411yescharacteristic
479C2×C5#288811yescharacteristic
480D5—42no—
481D5—42no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—52no—
487C23—52no—
488C23—101no—
489C23—101no—
490C23—101no—
491C23—101no—
492C23—201no—
493C23—201no—
494C23—201no—
495C23—201no—
496C23D5×M1612yes—
497C23C22×C4×D511yescharacteristic
498Q8—104no—
499Q8—104no—
500D4—104no—
501D4—104no—
502D4—108no—
503D4—108no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—52no—
519C2×C4—52no—
520C2×C4—52no—
521C2×C4—52no—
522C2×C4—108no—
523C2×C4—108no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—101no—
527C2×C4—101no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—402no—
533C2×C4—402no—
534C2×C4—402no—
535C2×C4—402no—
536C2×C4—402no—
537C2×C4—402no—
538C2×C4D5×C22⋊C411yescharacteristic
539C2×C4D5×C22⋊C411yescharacteristic
540C2×C4D5×C22⋊C412yes—
541C2×C4D5×M1612yes—
542C2×C4C2×D4×D512yes—
543C8—44no—
544C8—204no—
545C5#1737211yescharacteristic, complemented, Sylow
546C22—21no—
547C22—21no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—102no—
554C22—102no—
555C22—202no—
556C22—202no—
557C22—202no—
558C22—202no—
559C22—201no—
560C22—201no—
561C22—201no—
562C22—201no—
563C22C2×D5×C22⋊C411yescharacteristic
564C22C2×D5×M1611yescharacteristic
565C22C2×D5×M1611yescharacteristic
566C4—22no—
567C4—22no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—54no—
574C4—102no—
575C4—102no—
576C4—102no—
577C4—201no—
578C4—201no—
579C2—21no—
580C2—21no—
581C2—42no—
582C2—202no—
583C2—202no—
584C2D5×#40711yescharacteristic
585C2#1061042011yescharacteristic
586C2#1061044511yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1464C2
socleC22×C5C22×C22⋊C4
prime radicalC22×#709C4
cosocleC22×C4×C5C24
hypercenter#1464D5
hypocenterC5#17372
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #10611045C2semidirect
3. C5×#1464C2semidirect
4. #10602365C2semidirect
5. #10601620C2semidirect
6. #10598919C2semidirect
7. #10598917C2semidirect
8. #10598896C2semidirect
9. #10598894C2semidirect
115. #1464D5semidirect
545. C5#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#658572
#106110452
C5×#14642
#106023652
#106016202
#105989192
#105989172
#105988962
#105988942
#173725