Finite Group Databasebeta

← #12109711

Subgroups — #12109711

Order 1280.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65408C211yesmaximal, characteristic
2#10605599C211yesmaximal, characteristic
3#10605599C211yesmaximal, characteristic
4C5×#1809C211yesmaximal, characteristic
5#10602781C211yesmaximal, characteristic
6#10601409C211yesmaximal, characteristic, complemented
7#10601408C211yesmaximal, characteristic, complemented
8#10601357C211yesmaximal, characteristic, complemented
9#10601356C211yesmaximal, characteristic, complemented
10#10600305C211yesmaximal, characteristic
11#10600304C211yesmaximal, characteristic
12#10599591C211yesmaximal, characteristic, complemented
13#10599590C211yesmaximal, characteristic, complemented
14#10599589C211yesmaximal, characteristic, complemented
15#10599588C211yesmaximal, characteristic, complemented
16C22×C5×M16C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C23×#168C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#106C2211yescharacteristic
21C5×#380C2211yescharacteristic
22C5×#380C2211yescharacteristic
23#65501C2211yescharacteristic
24#65501C2211yescharacteristic
25#65408C418yescomplemented
26C2×#3950C2211yescharacteristic
27C2×#3949C2211yescharacteristic
28C2×#3946C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32#65266C2211yescharacteristic
33#65266C2211yescharacteristic
34#65265C2211yescharacteristic
35#65265C2211yescharacteristic
36C4⋊C4×Dic5C2211yescharacteristic
37C4⋊C4×Dic5C2211yescharacteristic
38C4⋊C4×Dic5C2211yescharacteristic
39C4⋊C4×Dic5C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3940C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65234—21no—
45#65234—21no—
46#65233—21no—
47#65233—21no—
48#64943—21no—
49#64943—21no—
50#64942—21no—
51#64942—21no—
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54#64771C2211yescharacteristic
55#64771C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#21966—51nomaximal, Sylow
61C23×C4×C5C2311yescharacteristic
62C2×C5×M16—41no—
63C2×C5×M16—41no—
64C2×C5×M16C2×C414yes—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C22⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C22⋊Dic5—21no—
74C2×C22⋊Dic5—21no—
75C2×C22⋊Dic5D411yescharacteristic
76C2×C22⋊Dic5D411yescharacteristic
77#4072C2×C414yes—
78#4072Q814yes—
79C2×C4⋊Dic5D411yescharacteristic
80C2×C4⋊Dic5D411yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5C2311yescharacteristic
94C2×C4×Dic5C2311yescharacteristic
95C22×#168—41no—
96C22×#168—41no—
97C22×#168C2×C414yes—
98C22×#168C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C5×#107—41no—
101C5×#106—41no—
102C5×#95—21no—
103C5×#95—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3950—41no—
115#3949—41no—
116#3946C2×C414yes—
117#3946C2×C414yes—
118#3946C2×C414yes—
119#3946C2×C414yes—
120#3940—41no—
121#3940—41no—
122#3939—41no—
123#3939—41no—
124C2×#424—51no—
125#2314—51no—
126#2314—51no—
127#1949—51no—
128#1949—51no—
129#1949—51no—
130#1949—51no—
131#1948—51no—
132#1948—51no—
133#1947—51no—
134#1947—51no—
135#1809—51no—
136#1809D511yescharacteristic
137#1808—51no—
138#1807—51no—
139C24×C5C2×D411yescharacteristic
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×Q812yes—
151C22×C4×C5C2411yescharacteristic
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5C2×D411yescharacteristic
157C22×Dic5C2×D411yescharacteristic
158C22×Dic5C2×D411yescharacteristic
159C22×Dic5C2×D411yescharacteristic
160C5×M16—44no—
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8C22×C412yes—
166C2×C5×C8C22×C412yes—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C22⋊C4—81no—
176C5×C22⋊C4—81no—
177C22⋊Dic5—81no—
178C22⋊Dic5—81no—
179C22⋊Dic5C4⋊C414yes—
180C22⋊Dic5C4⋊C414yes—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5C4⋊C414yes—
190C4⋊Dic5C4⋊C414yes—
191#683—81no—
192#683—81no—
193#683—81no—
194#683—81no—
195C4×Dic5—81no—
196C4×Dic5—81no—
197C4×Dic5—81no—
198C4×Dic5—81no—
199C4×Dic5C22×C412yes—
200C4×Dic5C22×C412yes—
201C4×Dic5C2×Q812yes—
202C4×Dic5C2×Q812yes—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—44no—
208C2×#168C22×C412yes—
209C2×#168C22×C412yes—
210C22×M16C2×D511yescharacteristic
211C23×C8—51no—
212C2×#117—51no—
213C2×#107—51no—
214C2×#107—51no—
215C2×#107—51no—
216C2×#107C2×D511yescharacteristic
217C2×#106—51no—
218C2×#106—51no—
219C2×#106—51no—
220C2×#106C2×D511yescharacteristic
221#424—58no—
222C2×C4⋊C8—51no—
223C2×C4⋊C8—51no—
224C2×C4⋊C8—51no—
225C2×C4⋊C8—51no—
226#388—101no—
227#388—101no—
228#388—101no—
229#388—101no—
230#387—101no—
231#387—101no—
232#387—101no—
233#387—101no—
234#384—51no—
235#384—51no—
236#382—51no—
237#382—51no—
238#380—51no—
239#380—51no—
240#380C2×D511yescharacteristic
241#380C2×D511yescharacteristic
242C4×C4⋊C4—51no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—51no—
245C4×C4⋊C4—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340C2×D511yescharacteristic
253#340C2×D511yescharacteristic
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5C2×C4⋊C412yes—
261C23×C5C22×D411yescharacteristic
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5C2×C4⋊C412yes—
281C2×C4×C5C23×C411yescharacteristic
282C2×C4×C5C22×Q811yescharacteristic
283C2×C4×C5H(5,𝔽2)11yescharacteristic
284C2×C4×C5H(5,𝔽2)11yescharacteristic
285C2×C4×C52−1+411yescharacteristic
286C2×C4×C52−1+411yescharacteristic
287C2×Dic5—21no—
288C2×Dic5—21no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5C2×C4⋊C412yes—
308C2×Dic5C2×C4⋊C412yes—
309C2×Dic5C2×C4⋊C412yes—
310C2×Dic5C2×C4⋊C412yes—
311C5×C8—22no—
312C5×C8—22no—
313#168—22no—
314#168—22no—
315C23×C4C22×D511yescharacteristic
316C2×M16—41no—
317C2×M16—41no—
318C2×M16C4×D514yes—
319C22×C8—51no—
320C22×C8—51no—
321C22×C8—54no—
322C22×C8—201no—
323C22×C8—201no—
324C22×C8C22×D511yescharacteristic
325C22×C8C22×D511yescharacteristic
326#117—54no—
327#117—54no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4—101no—
336C2×C4⋊C4—101no—
337C2×C4⋊C4—101no—
338C2×C4⋊C4C22×D511yescharacteristic
339C2×C4⋊C4C22×D511yescharacteristic
340C2×C4⋊C4C22×D511yescharacteristic
341C2×C4⋊C4C22×D511yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—51no—
345C2×C22⋊C4—51no—
346C2×C22⋊C4—101no—
347C2×C22⋊C4—101no—
348C2×C42—51no—
349C2×C42—51no—
350C2×C42—101no—
351C2×C42—101no—
352C2×C42—101no—
353C2×C42—101no—
354#107—41no—
355#107—201no—
356#107—201no—
357#107—201no—
358#106—41no—
359#106—201no—
360#106—201no—
361#106—201no—
362C4⋊C8—54no—
363C4⋊C8—54no—
364C4⋊C8—54no—
365C4⋊C8—54no—
366#95—21no—
367#95—21no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95—101no—
372#95—101no—
373#95—101no—
374#95—101no—
375#95—101no—
376#95—101no—
377#95—101no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5—41no—
385C22×C5C22×C4⋊C411yescharacteristic
386C22×C5#53711yescharacteristic
387C22×C5#53711yescharacteristic
388C22×C5#57611yescharacteristic
389C22×C5#57711yescharacteristic
390C22×C5#57711yescharacteristic
391C22×C5#57811yescharacteristic
392C4×C5—22no—
393C4×C5—81no—
394C4×C5—81no—
395C4×C5—81no—
396C4×C5—81no—
397C4×C5#51811yescharacteristic
398C4×C5#51911yescharacteristic
399C4×C5#54411yescharacteristic
400C4×C5#54411yescharacteristic
401Dic5—22no—
402Dic5—22no—
403Dic5—22no—
404Dic5—22no—
405Dic5—81no—
406Dic5—81no—
407Dic5—81no—
408Dic5—81no—
409C24D4×D511yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—101no—
423C22×C4—101no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4C2×C4×D512yes—
427C22×C4D4×D511yescharacteristic
428C22×C4Q8×D512yes—
429C22×C4C23×D511yescharacteristic
430M16—44no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—52no—
436C2×C8—52no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8—204no—
442C2×C8C2×C4×D512yes—
443C2×C8C2×C4×D512yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—54no—
453C4⋊C4—54no—
454C4⋊C4—101no—
455C4⋊C4—101no—
456C4⋊C4—101no—
457C4⋊C4—101no—
458C4⋊C4—101no—
459C4⋊C4—101no—
460C4⋊C4—101no—
461C4⋊C4—101no—
462C4⋊C4—401no—
463C4⋊C4—401no—
464C4⋊C4—401no—
465C4⋊C4—401no—
466C22⋊C4—54no—
467C22⋊C4—54no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—401no—
471C22⋊C4—401no—
472C42—52no—
473C42—52no—
474C42—52no—
475C42—52no—
476C42—401no—
477C42—401no—
478C42—401no—
479C42—401no—
480C2×C5—42no—
481C2×C5#231611yescharacteristic
482C2×C5#291211yescharacteristic
483C2×C5#291311yescharacteristic
484C2×C5#325711yescharacteristic
485C2×C5#325711yescharacteristic
486C2×C5#325811yescharacteristic
487C2×C5#325811yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23D5×C4⋊C412yes—
495C23C2×D4×D511yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—52no—
515C2×C4—52no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4D5×C4⋊C412yes—
539C2×C4C22×C4×D511yescharacteristic
540C2×C4#414411yescharacteristic
541C2×C4#414411yescharacteristic
542C2×C4C2×Q8×D511yescharacteristic
543C2×C4#415011yescharacteristic
544C2×C4#415011yescharacteristic
545C8—22no—
546C8—22no—
547C8—102no—
548C8—102no—
549C5#2196611yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×D5×C4⋊C411yescharacteristic
558C22#6593111yescharacteristic
559C22#6593111yescharacteristic
560C22#6610811yescharacteristic
561C22#6610911yescharacteristic
562C22#6611211yescharacteristic
563C22#6611311yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—102no—
570C4—102no—
571C4—102no—
572C4—102no—
573C4—401no—
574C4—401no—
575C4—401no—
576C4—401no—
577C4#6582011yescharacteristic
578C4#6585011yescharacteristic
579C4#6595711yescharacteristic
580C4#6595711yescharacteristic
581C2—42no—
582C2#1060560011yescharacteristic
583C2#1061094011yescharacteristic
584C2#1061094111yescharacteristic
585C2#1061252711yescharacteristic
586C2#1061252811yescharacteristic
587C2#1061258811yescharacteristic
588C2#1061262811yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1809C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenter#1809D5
hypocenterC5#21966
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
6. #10601409C2semidirect
7. #10601408C2semidirect
8. #10601357C2semidirect
9. #10601356C2semidirect
12. #10599591C2semidirect
13. #10599590C2semidirect
14. #10599589C2semidirect
15. #10599588C2semidirect
25. #65408C4semidirect
549. C5#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#654082
#106055992
C5×#18092
#106027812
#106014092
#106014082
#106013572
#106013562
#106003052
#106003042
#105995912
#105995902
#105995892
#105995882
#219665