Finite Group Databasebeta

← #12068515

Subgroups — #12068515

Order 1280.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2989C211yesmaximal, characteristic
2#10614252C211yesmaximal, characteristic
3#10614244C211yesmaximal, characteristic
4#10601590C211yesmaximal, characteristic, complemented
5#10601589C211yesmaximal, characteristic, complemented
6#10601588C211yesmaximal, characteristic, complemented
7#10601587C211yesmaximal, characteristic, complemented
8#10601541C211yesmaximal, characteristic, complemented
9#10601540C211yesmaximal, characteristic, complemented
10#10601288C211yesmaximal, characteristic, complemented
11#10601287C211yesmaximal, characteristic, complemented
12#10601285C211yesmaximal, characteristic, complemented
13#10601284C211yesmaximal, characteristic, complemented
14#10601216C211yesmaximal, characteristic, complemented
15#10601215C211yesmaximal, characteristic, complemented
16C4×C5×#41C2211yescharacteristic
17C5×#446—21no—
18C5×#436—21no—
19C5×#435—21no—
20C5×#435—21no—
21C5×#432C2211yescharacteristic
22C5×#432C2211yescharacteristic
23C5×#431C2211yescharacteristic
24C5×#423C2211yescharacteristic
25C5×#423C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27#65523C412yes—
28#65313—21no—
29#65301—21no—
30#65300—21no—
31#65300—21no—
32#65289C412yes—
33#65289C412yes—
34#65288C412yes—
35#65215C2211yescharacteristic
36#65215C2211yescharacteristic
37#65214C2211yescharacteristic
38#65214C2211yescharacteristic
39#65214C2211yescharacteristic
40#65214C2211yescharacteristic
41#65214C2211yescharacteristic
42#65213C2211yescharacteristic
43#65213C2211yescharacteristic
44C2×C4⋊#168C2211yescharacteristic
45C4×#681C2211yescharacteristic
46C2×C4×#168C2211yescharacteristic
47#64700C2211yescharacteristic
48#64700C2211yescharacteristic
49#64700C2211yescharacteristic
50#64700C2211yescharacteristic
51#64700C2211yescharacteristic
52#64700C2211yescharacteristic
53#64699C2211yescharacteristic
54#64699C2211yescharacteristic
55#64699C2211yescharacteristic
56#64699C2211yescharacteristic
57#64699C2211yescharacteristic
58#64699C2211yescharacteristic
59#64699C2211yescharacteristic
60#64699C2211yescharacteristic
61#64699C2211yescharacteristic
62#64699C2211yescharacteristic
63#17825—51nomaximal, Sylow
64C2×C5×#41C2×C411yescharacteristic
65C2×C5×M16—21no—
66C2×C5×M16—21no—
67C22×C5×C8—21no—
68C4×C5×Q8D411yescharacteristic
69C4×C5×Q8Q811yescharacteristic
70C4×C5×D4D411yescharacteristic
71C4×C5×D4D411yescharacteristic
72C4×C5×D4D411yescharacteristic
73C4×C5×D4Q811yescharacteristic
74C4×C5×D4Q811yescharacteristic
75C4×C5×D4Q811yescharacteristic
76C5×#117C2×C411yescharacteristic
77C5×#117C2×C411yescharacteristic
78C5×#117C2×C411yescharacteristic
79C2×C42×C5C2311yescharacteristic
80C2×C42×C5C2311yescharacteristic
81C2×C42×C5C2311yescharacteristic
82C2×#681—21no—
83C2×#681—21no—
84C2×#681C2×C412yes—
85C22×#168—21no—
86C22×#168C2×C412yes—
87C5×C4⋊C8—21no—
88C5×C4⋊C8—21no—
89C5×C4⋊C8—21no—
90C5×C4⋊C8C2311yescharacteristic
91C5×C4⋊C8C2311yescharacteristic
92C5×C4⋊C8C2311yescharacteristic
93C5×C4⋊C8C2311yescharacteristic
94C5×C22⋊C8—21no—
95C5×C22⋊C8—21no—
96C5×C22⋊C8—21no—
97C5×#97—21no—
98C5×#97—21no—
99C4×C5×C8—21no—
100C22⋊#168—21no—
101C22⋊#168—21no—
102C22⋊#168—21no—
103C22⋊#168C2×C412yes—
104C22⋊#168C2×C412yes—
105C22⋊#168C2×C412yes—
106C22⋊#168C2×C412yes—
107C4⋊#168—21no—
108C4⋊#168—21no—
109C4⋊#168—21no—
110C4⋊#168C2×C412yes—
111C4⋊#168C2×C412yes—
112C4⋊#168C2×C412yes—
113C4⋊#168C2×C412yes—
114C4⋊#168C2311yescharacteristic
115C4⋊#168C2311yescharacteristic
116C4⋊#168C2311yescharacteristic
117C4⋊#168C2311yescharacteristic
118#3935—21no—
119#3935—21no—
120#3935C2×C412yes—
121C4×#168—21no—
122C4×#168C2×C412yes—
123C4×#168C2311yescharacteristic
124C4×#168C2311yescharacteristic
125C4×#168C2311yescharacteristic
126C4×#168C2311yescharacteristic
127#2989—51no—
128#2989D511yescharacteristic
129#2922—51no—
130#1574—51no—
131#1574—51no—
132#1573—51no—
133#1573—51no—
134#1572—51no—
135#1572—51no—
136#1571—51no—
137#1571—51no—
138#1569—51no—
139#1569—51no—
140#1568—51no—
141#1567—51no—
142C5×#41—42no—
143C2×C5×Q8C4⋊C411yescharacteristic
144C2×C5×D4C4⋊C411yescharacteristic
145C2×C5×D4C4⋊C411yescharacteristic
146C2×C5×D4C4⋊C411yescharacteristic
147C22×C4×C5C22×C411yescharacteristic
148C22×C4×C5C22×C411yescharacteristic
149C22×C4×C5C22×C411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×Q811yescharacteristic
154C22×C4×C5C2×Q811yescharacteristic
155C22×C4×C5C2×Q811yescharacteristic
156C5×M16—81no—
157C5×M16—81no—
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C2×C5×C8—21no—
162C2×C5×C8—81no—
163C5×C4⋊C4C4⋊C411yescharacteristic
164C5×C4⋊C4C4⋊C411yescharacteristic
165C5×C4⋊C4C4⋊C411yescharacteristic
166C5×C4⋊C4C4⋊C411yescharacteristic
167C5×C4⋊C4C4⋊C411yescharacteristic
168C5×C4⋊C4C4⋊C411yescharacteristic
169C5×C22⋊C4C4⋊C411yescharacteristic
170C5×C22⋊C4C4⋊C411yescharacteristic
171C5×C22⋊C4C4⋊C411yescharacteristic
172C5×C22⋊C4C4⋊C411yescharacteristic
173C5×C22⋊C4C4⋊C411yescharacteristic
174C5×C22⋊C4C4⋊C411yescharacteristic
175C42×C5C22×C411yescharacteristic
176C42×C5C22×C411yescharacteristic
177C42×C5C22×C411yescharacteristic
178C42×C5C2×D411yescharacteristic
179C42×C5C2×D411yescharacteristic
180C42×C5C2×D411yescharacteristic
181C42×C5C2×Q811yescharacteristic
182C42×C5C2×Q811yescharacteristic
183C42×C5C2×Q811yescharacteristic
184C42×C5C2411yescharacteristic
185#681—42no—
186#681—81no—
187#681—81no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—21no—
191C2×#168—21no—
192C2×#168—42no—
193C2×#168—81no—
194C2×#168C22×C412yes—
195C2×#168C22×C412yes—
196C2×#168C22×C412yes—
197C2×#168C22×C412yes—
198C4×#41C2×D511yescharacteristic
199#446—21no—
200#446—101no—
201#436—21no—
202#436—101no—
203#435—21no—
204#435—21no—
205#435—101no—
206#435—101no—
207#433—52no—
208#433—52no—
209#432—51no—
210#432—51no—
211#432C2×D511yescharacteristic
212#432C2×D511yescharacteristic
213#431—51no—
214#431—51no—
215#431—51no—
216#431—51no—
217#431—51no—
218#431C2×D511yescharacteristic
219#424—52no—
220#423—51no—
221#423—51no—
222#423C2×D511yescharacteristic
223#423C2×D511yescharacteristic
224C2×C4⋊C8—51no—
225C2×C4⋊C8C2×D511yescharacteristic
226#408—52no—
227C4×M16—51no—
228C2×C4×C8—51no—
229#335—51no—
230#335—51no—
231#335—51no—
232#335—51no—
233#335—51no—
234#335—51no—
235#335—51no—
236#335—51no—
237#335—51no—
238#335—51no—
239#334—51no—
240#334—51no—
241#334—51no—
242#334—51no—
243#334—51no—
244#334—51no—
245C23×C5C2×C4⋊C411yescharacteristic
246C23×C5C2×C4⋊C411yescharacteristic
247C23×C5C2×C4⋊C411yescharacteristic
248C5×Q8—41no—
249C5×D4—41no—
250C5×D4—41no—
251C5×D4—41no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5C2×C4⋊C411yescharacteristic
265C2×C4×C5C2×C4⋊C411yescharacteristic
266C2×C4×C5C2×C4⋊C411yescharacteristic
267C2×C4×C5C2×C4⋊C411yescharacteristic
268C2×C4×C5C2×C4⋊C411yescharacteristic
269C2×C4×C5C2×C4⋊C411yescharacteristic
270C2×C4×C5C2×C4⋊C411yescharacteristic
271C2×C4×C5C2×C4⋊C411yescharacteristic
272C2×C4×C5C2×C4⋊C411yescharacteristic
273C2×C4×C5C23×C411yescharacteristic
274C2×C4×C5C22×D411yescharacteristic
275C2×C4×C5C22×Q811yescharacteristic
276C5×C8—41no—
277C5×C8—41no—
278C5×C8—41no—
279C5×C8—41no—
280#168—22no—
281#168—22no—
282#168—22no—
283#168—22no—
284#168—41no—
285#168—41no—
286#168—41no—
287#168—41no—
288C2×#41C4×D511yescharacteristic
289C2×M16—21no—
290C2×M16—21no—
291C2×M16—52no—
292C2×M16—101no—
293C2×M16—101no—
294C22×C8—21no—
295C22×C8—52no—
296C22×C8—101no—
297C4×Q8Dic1011yescharacteristic
298C4×Q8C22⋊D511yescharacteristic
299C4×D4Dic1011yescharacteristic
300C4×D4Dic1011yescharacteristic
301C4×D4Dic1011yescharacteristic
302C4×D4C22⋊D511yescharacteristic
303C4×D4C22⋊D511yescharacteristic
304C4×D4C22⋊D511yescharacteristic
305#117C4×D511yescharacteristic
306#117C4×D511yescharacteristic
307#117C4×D511yescharacteristic
308C2×C42C22×D511yescharacteristic
309C2×C42C22×D511yescharacteristic
310C2×C42C22×D511yescharacteristic
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—52no—
315C4⋊C8—52no—
316C4⋊C8—52no—
317C4⋊C8—52no—
318C4⋊C8—51no—
319C4⋊C8—51no—
320C4⋊C8—51no—
321C4⋊C8—51no—
322C4⋊C8—101no—
323C4⋊C8—101no—
324C4⋊C8—101no—
325C4⋊C8C22×D511yescharacteristic
326C4⋊C8C22×D511yescharacteristic
327C4⋊C8C22×D511yescharacteristic
328C4⋊C8C22×D511yescharacteristic
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8—52no—
333C22⋊C8—52no—
334C22⋊C8—52no—
335C22⋊C8—52no—
336C22⋊C8—101no—
337C22⋊C8—101no—
338C22⋊C8—101no—
339#97—21no—
340#97—21no—
341#97—52no—
342#97—101no—
343#97—101no—
344C4×C8—21no—
345C4×C8—52no—
346C4×C8—51no—
347C4×C8—51no—
348C4×C8—51no—
349C4×C8—51no—
350C4×C8—101no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—21no—
360C22×C5C22×C4⋊C411yescharacteristic
361C4×C5—21no—
362C4×C5—21no—
363C4×C5—21no—
364C4×C5—21no—
365C4×C5—21no—
366C4×C5—21no—
367C4×C5—21no—
368C4×C5—21no—
369C4×C5—21no—
370C4×C5—21no—
371C4×C5#56812yes—
372C4×C5#57611yescharacteristic
373C4×C5#57811yescharacteristic
374#41—42no—
375C2×Q8#68311yescharacteristic
376C2×D4#68311yescharacteristic
377C2×D4#68311yescharacteristic
378C2×D4#68311yescharacteristic
379C22×C4C2×Dic1011yescharacteristic
380C22×C4C2×Dic1011yescharacteristic
381C22×C4C2×Dic1011yescharacteristic
382C22×C4C2×C4×D511yescharacteristic
383C22×C4C2×C4×D511yescharacteristic
384C22×C4C2×C4×D511yescharacteristic
385C22×C4C2×C22⋊D511yescharacteristic
386C22×C4C2×C22⋊D511yescharacteristic
387C22×C4C2×C22⋊D511yescharacteristic
388M16—81no—
389M16—81no—
390M16—202no—
391M16—401no—
392M16—401no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—52no—
398C2×C8—52no—
399C2×C8—52no—
400C2×C8—52no—
401C2×C8—81no—
402C2×C8—101no—
403C2×C8—101no—
404C2×C8—101no—
405C2×C8—101no—
406C2×C8—202no—
407C2×C8—401no—
408C4⋊C4#68311yescharacteristic
409C4⋊C4#68311yescharacteristic
410C4⋊C4#68311yescharacteristic
411C4⋊C4#68311yescharacteristic
412C4⋊C4#68311yescharacteristic
413C4⋊C4#68311yescharacteristic
414C22⋊C4#68311yescharacteristic
415C22⋊C4#68311yescharacteristic
416C22⋊C4#68311yescharacteristic
417C22⋊C4#68311yescharacteristic
418C22⋊C4#68311yescharacteristic
419C22⋊C4#68311yescharacteristic
420C42C2×Dic1011yescharacteristic
421C42C2×Dic1011yescharacteristic
422C42C2×Dic1011yescharacteristic
423C42C2×C4×D511yescharacteristic
424C42C2×C4×D511yescharacteristic
425C42C2×C4×D511yescharacteristic
426C42C2×C22⋊D511yescharacteristic
427C42C2×C22⋊D511yescharacteristic
428C42C2×C22⋊D511yescharacteristic
429C42C23×D511yescharacteristic
430C2×C5—41no—
431C2×C5—41no—
432C2×C5—41no—
433C2×C5#290511yescharacteristic
434C2×C5#291311yescharacteristic
435C2×C5#291611yescharacteristic
436C23C2×#68311yescharacteristic
437C23C2×#68311yescharacteristic
438C23C2×#68311yescharacteristic
439Q8—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4C2×#68311yescharacteristic
456C2×C4C2×#68311yescharacteristic
457C2×C4C2×#68311yescharacteristic
458C2×C4C2×#68311yescharacteristic
459C2×C4C2×#68311yescharacteristic
460C2×C4C2×#68311yescharacteristic
461C2×C4C2×#68311yescharacteristic
462C2×C4C2×#68311yescharacteristic
463C2×C4C2×#68311yescharacteristic
464C2×C4C22×Dic1011yescharacteristic
465C2×C4C22×C4×D511yescharacteristic
466C2×C4C22×C22⋊D511yescharacteristic
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—102no—
472C8—102no—
473C8—102no—
474C8—102no—
475C8—201no—
476C8—201no—
477C8—201no—
478C8—201no—
479C5#1782511yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22C22×#68311yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4#6607912yes—
501C4#6616911yescharacteristic
502C4#6617211yescharacteristic
503C2—41no—
504C2—41no—
505C2—41no—
506C2#1061429311yescharacteristic
507C2#1061548111yescharacteristic
508C2#1061554111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2989C2
socleC22×C5C22×C4⋊C4
prime radicalC2×C5×#41C2×C4
cosocleC42×C5C24
hypercenter#2989D5
hypocenterC5#17825
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
4. #10601590C2semidirect
5. #10601589C2semidirect
6. #10601588C2semidirect
7. #10601587C2semidirect
8. #10601541C2semidirect
9. #10601540C2semidirect
10. #10601288C2semidirect
11. #10601287C2semidirect
12. #10601285C2semidirect
13. #10601284C2semidirect
14. #10601216C2semidirect
15. #10601215C2semidirect
479. C5#17825semidirect

Maximal subgroups

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

subgroupindex
C5×#29892
#106142522
#106142442
#106015902
#106015892
#106015882
#106015872
#106015412
#106015402
#106012882
#106012872
#106012852
#106012842
#106012162
#106012152
#178255