Finite Group Databasebeta

← #12118753

Subgroups — #12118753

Order 1280.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614243C211yesmaximal, characteristic, complemented
2C4×#4096C211yesmaximal, characteristic
3#10610771C211yesmaximal, characteristic, complemented
4C5×#1806C211yesmaximal, characteristic, complemented
5#10601309C211yesmaximal, characteristic, complemented
6#10601170C212yesmaximal
7#10601163C212yesmaximal, complemented
8#10600287C211yesmaximal, characteristic, complemented
9#10600270C211yesmaximal, characteristic, complemented
10#10599539C212yesmaximal, complemented
11#10599531C212yesmaximal, complemented
12C4×C5×#41C2211yescharacteristic
13C2×#4096C2211yescharacteristic
14C2×#4094C2212yes—
15C4×#709C2211yescharacteristic
16C5×#423C2211yescharacteristic
17C2×C5×#104C2212yes—
18C4×C5×C4⋊C4C2211yescharacteristic
19#65522C2211yescharacteristic
20#65418C2212yes—
21#65411C2212yes—
22C4×#689C411yescharacteristic, complemented
23C4×#689C411yescharacteristic, complemented
24C4×#688C411yescharacteristic, complemented
25C4×#688C411yescharacteristic, complemented
26C4×#687C411yescharacteristic, complemented
27C4×#687C411yescharacteristic, complemented
28C4×#686C411yescharacteristic, complemented
29C4×#686C411yescharacteristic, complemented
30#65291C2211yescharacteristic
31#65288—22no—
32#65288C2212yes—
33#65287C2211yescharacteristic
34C2×#3937C2212yes—
35#65213—21no—
36#65213C2211yescharacteristic
37C2×C4⋊#168—21no—
38C2×C4⋊#168C2211yescharacteristic
39C2×#3935C2212yes—
40C2×C4×#168C2211yescharacteristic
41#65129—22no—
42#65124—21no—
43#65122—21no—
44C5×#339C2212yes—
45#64778C2212yes—
46#64778C2212yes—
47#64744C2212yes—
48#64743C2212yes—
49#22782—51nomaximal, Sylow
50C2×C5×#41C2311yescharacteristic
51C2×#709C2311yescharacteristic
52C2×C5×M16C2312yes—
53C4×C5×Q8C2×C411yescharacteristic
54C4×C5×Q8C2×C411yescharacteristic
55C4×C5×D4—21no—
56C4×C5×D4—21no—
57C4×C5×D4C2×C411yescharacteristic
58C4×C5×D4C2×C411yescharacteristic
59C5×#117—21no—
60C5×#117C2311yescharacteristic
61C2×C5×C4⋊C4—21no—
62C2×C42×C5—21no—
63C2×C42×C5C2312yes—
64C2×C42×C5C2311yescharacteristic
65#4096—22no—
66#4096—22no—
67#4094—24no—
68#4094—42no—
69C2×#689C2×C411yescharacteristic
70C2×#688C2×C411yescharacteristic
71C2×#687C2×C411yescharacteristic
72C2×#686C2×C411yescharacteristic
73C4×C22⋊D5—21no—
74C4×C22⋊D5—21no—
75#4072C2311yescharacteristic
76C2×#681—22no—
77C2×#681C2312yes—
78C22×#168—22no—
79C22×#168C2311yescharacteristic
80C22×#168C2311yescharacteristic
81C22×#168C2312yes—
82C4×D20C2×C411yescharacteristic
83C4×D20C2×C411yescharacteristic
84#4018—21no—
85C42×D5—21no—
86C4×Dic10C2×C411yescharacteristic
87C4×Dic10C2×C411yescharacteristic
88C5×C4⋊C8—21no—
89C5×C4⋊C8—21no—
90C5×#104—24no—
91C5×#95—21no—
92#3967C2×C411yescharacteristic
93#3967C2×C411yescharacteristic
94#3964C2×C411yescharacteristic
95#3964C2×C411yescharacteristic
96#3952—22no—
97#3952—22no—
98#3947—22no—
99C8×Dic5—22no—
100#3942C2×C411yescharacteristic
101#3942C2×C411yescharacteristic
102#3941C2×C411yescharacteristic
103#3941C2×C411yescharacteristic
104#3940C2×C411yescharacteristic
105#3940C2×C411yescharacteristic
106#3939C2×C411yescharacteristic
107#3939C2×C411yescharacteristic
108#3937—42no—
109C4⋊#168—21no—
110C4⋊#168—21no—
111C4⋊#168—22no—
112C4⋊#168—22no—
113C4⋊#168D412yes—
114C4⋊#168D411yescharacteristic
115C4⋊#168D411yescharacteristic
116#3935—42no—
117C4×#168C2×C411yescharacteristic
118C4×#168C2×C411yescharacteristic
119C4×#168C2×C411yescharacteristic
120C4×#168C2×C411yescharacteristic
121C4×#135—51no—
122#2925—51no—
123#2906—51no—
124#1938—52no—
125#1922—52no—
126#1876—52no—
127#1837—51no—
128#1806—51no—
129#1806D511yescharacteristic, complemented
130#1770—51no—
131#1765—52no—
132C5×#41—21no—
133C5×#41—21no—
134C5×#41#4112yes—
135C5×#41#4112yes—
136C2×C5×Q8C22×C411yescharacteristic
137C2×C5×D4—21no—
138C2×C5×D4C22×C411yescharacteristic
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—22no—
143C22×C4×C5#4111yescharacteristic
144C22×C4×C5#4111yescharacteristic
145C22×C4×C5C2411yescharacteristic
146C2×C22⋊D5—21no—
147#709—21no—
148#709—21no—
149#709—21no—
150#709—21no—
151C2×D20C22×C411yescharacteristic
152C2×C4×D5—21no—
153C2×C4×D5—21no—
154C2×C4×D5—21no—
155C2×Dic10C22×C411yescharacteristic
156C5×M16—22no—
157C5×M16—22no—
158C2×C5×C8—22no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—42no—
162C5×C4⋊C4C22×C411yescharacteristic
163C5×C4⋊C4C22×C411yescharacteristic
164C5×C22⋊C4—21no—
165C5×C22⋊C4—21no—
166C5×C22⋊C4—21no—
167C42×C5—21no—
168C42×C5—21no—
169C42×C5—21no—
170C42×C5—22no—
171C42×C5—22no—
172C42×C5C22×C411yescharacteristic
173C42×C5C22×C411yescharacteristic
174C42×C5C2×D411yescharacteristic
175C42×C5C2×D411yescharacteristic
176C22⋊Dic5—21no—
177#689—22no—
178#688—22no—
179#687—22no—
180#686—22no—
181#685—21no—
182#685—21no—
183C4⋊Dic5C22×C411yescharacteristic
184C4⋊Dic5C22×C411yescharacteristic
185#683—21no—
186#683—21no—
187C4×Dic5—21no—
188C4×Dic5—21no—
189C4×Dic5—21no—
190#681—22no—
191#681—22no—
192#681—44no—
193C2×#168—21no—
194C2×#168—21no—
195C2×#168—22no—
196C2×#168—21no—
197C2×#168—22no—
198C2×#168—21no—
199C2×#168—22no—
200C2×#168—44no—
201C2×#168C22×C411yescharacteristic
202C2×#168C22×C411yescharacteristic
203C2×#168C22×C411yescharacteristic
204C2×#168C22×C411yescharacteristic
205C2×#168C2×D412yes—
206C2×#168C2×D412yes—
207C2×#135—51no—
208C2×#131—52no—
209C4×#41—51no—
210C4×#41C2×D511yescharacteristic
211C4×Q16—51no—
212C4×Q16—51no—
213C4×SD16—51no—
214C4×SD16—51no—
215C4×SD16—51no—
216C4×SD16—51no—
217C4×D8—51no—
218C4×D8—51no—
219#433—102no—
220#432—101no—
221#431—101no—
222#427—51no—
223#424—52no—
224#424—102no—
225#423—51no—
226#423—101no—
227#423C2×D511yescharacteristic
228C2×C4⋊C8—51no—
229C2×C4⋊C8—101no—
230C2×#104—52no—
231C2×#104C2×D512yes—
232#416—51no—
233#416—51no—
234#408—52no—
235#405—52no—
236C2×#97—52no—
237C2×C4×C8—51no—
238C4×C4⋊C4C2×D511yescharacteristic
239#341—52no—
240#341—52no—
241#339—52no—
242#339C2×D512yes—
243#336—52no—
244C23×C5—21no—
245C23×C5C2×#4111yescharacteristic
246C22×D5—21no—
247C5×Q8—21no—
248C5×Q8#11712yes—
249C5×D4—22no—
250C5×D4—22no—
251C5×D4—21no—
252C5×D4#11712yes—
253C2×C4×C5—21no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—41no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5—81no—
265C2×C4×C5C4×D411yescharacteristic
266C2×C4×C5C4×D411yescharacteristic
267C2×C4×C5C4×D411yescharacteristic
268C2×C4×C5C4×D411yescharacteristic
269C2×C4×C5C23×C411yescharacteristic
270C2×C4×C5C22×D411yescharacteristic
271C2×C4×C5C2×#4111yescharacteristic
272C2×C4×C5C2×#4111yescharacteristic
273C2×C4×C5H(5,𝔽2)11yescharacteristic
274C2×C4×C52−1+411yescharacteristic
275C22⋊D5—41no—
276C22⋊D5—41no—
277C2×Dic5—21no—
278C2×Dic5—21no—
279C2×Dic5—21no—
280D20—21no—
281D20—21no—
282C4×D5—41no—
283C4×D5—41no—
284C4×D5—42no—
285C4×D5—81no—
286Dic10—21no—
287Dic10—21no—
288C5×C8—42no—
289#168—21no—
290#168—21no—
291#168—21no—
292#168—21no—
293#168—22no—
294#168—22no—
295#168—42no—
296C2×#41—51no—
297C2×#41C22×D511yescharacteristic
298#135—102no—
299#135—102no—
300C2×Q16—51no—
301C2×SD16—51no—
302C2×SD16—51no—
303C2×D8—51no—
304#131—104no—
305#131—202no—
306C2×M16—52no—
307C2×M16—102no—
308C2×M16C22×D512yes—
309C22×C8—52no—
310C22×C8—51no—
311C22×C8—51no—
312C22×C8—102no—
313C4×Q8—51no—
314C4×Q8—51no—
315C4×Q8C4×D511yescharacteristic
316C4×Q8C4×D511yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—51no—
320C4×D4—51no—
321C4×D4—101no—
322C4×D4—101no—
323C4×D4C4×D511yescharacteristic
324C4×D4C4×D511yescharacteristic
325#117—21no—
326#117—51no—
327#117—101no—
328#117C22×D511yescharacteristic
329C2×C4⋊C4—21no—
330C2×C42—21no—
331C2×C42—101no—
332C2×C42C22×D512yes—
333C2×C42C22×D511yescharacteristic
334#107—51no—
335#107—51no—
336#106—51no—
337#106—51no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—51no—
341C4⋊C8—51no—
342C4⋊C8—52no—
343C4⋊C8—102no—
344C4⋊C8—102no—
345C4⋊C8—101no—
346C4⋊C8—101no—
347#104—24no—
348#104—202no—
349#103—51no—
350#103—51no—
351#103—51no—
352#103—51no—
353#102—51no—
354#102—51no—
355#102—51no—
356#102—51no—
357C22⋊C8—102no—
358C22⋊C8—102no—
359#97—102no—
360#97—202no—
361C4×C8—51no—
362C4×C8—51no—
363C4×C8—51no—
364C4×C8—51no—
365C4×C8—102no—
366#95—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—22no—
370C22×C5—41no—
371C22×C5—41no—
372C22×C5C2×C4×D411yescharacteristic
373C22×C5#55211yescharacteristic
374C22×C5#55311yescharacteristic
375C2×D5—41no—
376C2×D5—41no—
377C2×D5—41no—
378C4×C5—22no—
379C4×C5—21no—
380C4×C5—21no—
381C4×C5—21no—
382C4×C5—41no—
383C4×C5—41no—
384C4×C5—82no—
385C4×C5#44412yes—
386C4×C5C2×#11711yescharacteristic
387C4×C5#52011yescharacteristic
388C4×C5#54611yescharacteristic
389C4×C5#54811yescharacteristic
390Dic5—41no—
391Dic5—41no—
392Dic5—41no—
393#41—21no—
394#41—21no—
395#41—101no—
396#41—101no—
397#41—101no—
398#41—101no—
399#41#71112yes—
400#41#71312yes—
401C2×Q8—51no—
402C2×Q8C2×C4×D511yescharacteristic
403C2×D4—21no—
404C2×D4—51no—
405C2×D4—101no—
406C2×D4C2×C4×D511yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—101no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4#71111yescharacteristic
415C22×C4#71111yescharacteristic
416C22×C4C23×D511yescharacteristic
417Q16—102no—
418SD16—102no—
419SD16—102no—
420D8—102no—
421M16—22no—
422M16—22no—
423M16—102no—
424M16—102no—
425M16—204no—
426C2×C8—22no—
427C2×C8—51no—
428C2×C8—51no—
429C2×C8—51no—
430C2×C8—51no—
431C2×C8—52no—
432C2×C8—52no—
433C2×C8—101no—
434C2×C8—102no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—102no—
438C2×C8—101no—
439C2×C8—102no—
440C2×C8—204no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—51no—
445C4⋊C4—51no—
446C4⋊C4—101no—
447C4⋊C4—101no—
448C4⋊C4C2×C4×D511yescharacteristic
449C4⋊C4C2×C4×D511yescharacteristic
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—21no—
453C22⋊C4—101no—
454C22⋊C4—101no—
455C22⋊C4—101no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—101no—
462C42—101no—
463C42—101no—
464C42C2×C4×D511yescharacteristic
465C42C2×C4×D511yescharacteristic
466C42D4×D511yescharacteristic
467C42D4×D511yescharacteristic
468C2×C5—21no—
469C2×C5—21no—
470C2×C5—42no—
471C2×C5#235211yescharacteristic
472C2×C5C2×#44411yescharacteristic
473C2×C5#295711yescharacteristic
474D5—81no—
475C23—21no—
476C23—101no—
477C23C2×#71111yescharacteristic
478Q8—21no—
479Q8—101no—
480Q8—101no—
481Q8#403812yes—
482D4—22no—
483D4—22no—
484D4—21no—
485D4—101no—
486D4—101no—
487D4—201no—
488D4—201no—
489D4#403812yes—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—202no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—401no—
509C2×C4#402711yescharacteristic
510C2×C4#402711yescharacteristic
511C2×C4#402711yescharacteristic
512C2×C4#402711yescharacteristic
513C2×C4C22×C4×D511yescharacteristic
514C2×C4C2×D4×D511yescharacteristic
515C2×C4C2×#71111yescharacteristic
516C2×C4C2×#71311yescharacteristic
517C2×C4#414911yescharacteristic
518C2×C4#415011yescharacteristic
519C8—42no—
520C8—101no—
521C8—101no—
522C8—101no—
523C8—101no—
524C8—102no—
525C8—102no—
526C8—202no—
527C5#2278211yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—201no—
534C22—201no—
535C22—201no—
536C22C2×#402711yescharacteristic
537C22#6595211yescharacteristic
538C22#6597611yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—22no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—201no—
547C4—201no—
548C4—201no—
549C4#6511212yes—
550C4C2×#403811yescharacteristic
551C4#6585711yescharacteristic
552C4#6593211yescharacteristic
553C4#6596411yescharacteristic
554C2—21no—
555C2—21no—
556C2—42no—
557C2—401no—
558C2#1060561411yescharacteristic
559C2C2×#6511211yescharacteristic
560C2#1061072111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1806C2
socleC22×C5C2×C4×D4
prime radicalC2×#4096C22
cosocleC22×C4×C5C24
hypercenter#1806D5
hypocenterC5#22782
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
1. #10614243C2semidirect
3. #10610771C2semidirect
4. C5×#1806C2semidirect
5. #10601309C2semidirect
7. #10601163C2semidirect
8. #10600287C2semidirect
9. #10600270C2semidirect
10. #10599539C2semidirect
11. #10599531C2semidirect
22. C4×#689C4semidirect
23. C4×#689C4semidirect
24. C4×#688C4semidirect
25. C4×#688C4semidirect
26. C4×#687C4semidirect
27. C4×#687C4semidirect
28. C4×#686C4semidirect
29. C4×#686C4semidirect
129. #1806D5semidirect
527. C5#22782semidirect

Maximal subgroups

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

subgroupindex
#106142432
C4×#40962
#106107712
C5×#18062
#106013092
#106011702
#106011632
#106002872
#106002702
#105995392
#105995312
#227825