Finite Group Databasebeta

← #12157349

Subgroups — #12157349

Order 1280.

704 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
1C5×#2441C211yesmaximal, characteristic
2#10605188C211yesmaximal, characteristic
3#10602532C211yesmaximal, characteristic, complemented
4#10602531C211yesmaximal, characteristic, complemented
5#10602374C211yesmaximal, characteristic
6#10602369C211yesmaximal, characteristic
7#10600417C211yesmaximal, characteristic, complemented
8#10600415C211yesmaximal, characteristic, complemented
9#10600395C211yesmaximal, characteristic, complemented
10#10600394C211yesmaximal, characteristic, complemented
11#10600365C211yesmaximal, characteristic, complemented
12#10600360C211yesmaximal, characteristic, complemented
13#10600306C211yesmaximal, characteristic
14#10600294C211yesmaximal, characteristic
15#10600243C211yesmaximal, characteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×#117C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#399—21no—
21C5×#396—21no—
22C5×#389—21no—
23C5×#389C2211yescharacteristic
24C5×#389C2211yescharacteristic
25C5×#387—21no—
26C5×#384C2211yescharacteristic
27C5×#380C2211yescharacteristic
28C4×C5×C22⋊C4C2211yescharacteristic
29#65501C2211yescharacteristic
30C2×C22⋊#168C2211yescharacteristic
31C2×#3940C2211yescharacteristic
32C2×#3940C2211yescharacteristic
33C2×#3939C2211yescharacteristic
34C2×#3939C2211yescharacteristic
35#65231—21no—
36C22⋊C4×Dic5—21no—
37C4×C4⋊Dic5C2211yescharacteristic
38C4×C4⋊Dic5C2211yescharacteristic
39#65217C2211yescharacteristic
40#65217C2211yescharacteristic
41C2×C4⋊#168C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43#64943—21no—
44#64943—21no—
45#64749C2211yescharacteristic
46#64749C2211yescharacteristic
47#64749C2211yescharacteristic
48#64749C2211yescharacteristic
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#26018—51nomaximal, Sylow
60C23×C4×C5C2311yescharacteristic
61C23×Dic5—21no—
62C5×#125—22no—
63C5×#125—41no—
64C5×#117—22no—
65C5×#117—41no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C5×C22⋊C4—21no—
78C2×C5×C22⋊C4D411yescharacteristic
79C2×C5×C22⋊C4D411yescharacteristic
80C2×C42×C5—21no—
81C2×C42×C5C2311yescharacteristic
82C2×C42×C5C2311yescharacteristic
83C2×C22⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5Q814yes—
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4⋊Dic5C2311yescharacteristic
90C2×C4⋊Dic5C2311yescharacteristic
91C2×#683—21no—
92C2×#683—21no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5—21no—
95C2×#681—22no—
96C2×#681—22no—
97C2×#681—41no—
98C22×#168C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2311yescharacteristic
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95D411yescharacteristic
106C5×#95D411yescharacteristic
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112C22⋊#168—22no—
113#3940—22no—
114#3940—22no—
115#3939—41no—
116#3939—41no—
117C4⋊#168—22no—
118C4⋊#168—41no—
119#2441D511yescharacteristic
120#2345—51no—
121#2100—51no—
122#2088—51no—
123#2086—51no—
124#2059—51no—
125#1989—51no—
126#1947—51no—
127#1920—51no—
128#1919—51no—
129#1854—51no—
130#1851—51no—
131#1823—51no—
132#1809—51no—
133#1736—51no—
134C24×C5C2×D411yescharacteristic
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×Q812yes—
149C22×C4×C5#4112yes—
150C22×C4×C5C2411yescharacteristic
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5—24no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—81no—
157C22×Dic5—81no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—42no—
166C5×C4⋊C4—42no—
167C5×C4⋊C4—42no—
168C5×C4⋊C4—42no—
169C5×C4⋊C4—81no—
170C5×C4⋊C4—81no—
171C5×C4⋊C4#4112yes—
172C5×C22⋊C4—22no—
173C5×C22⋊C4—41no—
174C5×C22⋊C4—42no—
175C5×C22⋊C4—81no—
176C42×C5—21no—
177C42×C5—21no—
178C42×C5—21no—
179C42×C5—21no—
180C42×C5—42no—
181C22⋊Dic5—42no—
182C4⋊Dic5—24no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5C2×Q812yes—
190C4⋊Dic5C2×Q812yes—
191#683—81no—
192#683—81no—
193C4×Dic5—81no—
194C4×Dic5—81no—
195#681—44no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168#4112yes—
208C22×M16—51no—
209C2×#125C2×D511yescharacteristic
210C2×#117C2×D511yescharacteristic
211C22×C4⋊C4—51no—
212C2×#107—51no—
213C2×#107—51no—
214C2×#106—51no—
215C2×#106—51no—
216C2×C4⋊C8—51no—
217C2×C4⋊C8—51no—
218C2×C22⋊C8—51no—
219#399—21no—
220#396—21no—
221#389—21no—
222#389—51no—
223#389—51no—
224#389C2×D511yescharacteristic
225#389C2×D511yescharacteristic
226#387—21no—
227#387—101no—
228#387—101no—
229#384C2×D511yescharacteristic
230#380—51no—
231#380—101no—
232#380C2×D511yescharacteristic
233C4×C4⋊C4—51no—
234C4×C4⋊C4—51no—
235C4×C22⋊C4—101no—
236C4×C22⋊C4C2×D511yescharacteristic
237#340—51no—
238#340—51no—
239#340—51no—
240#340—51no—
241#340—51no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#336—51no—
250#336—51no—
251C23×C5—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5C22⋊Q812yes—
258C23×C5C22×D411yescharacteristic
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—22no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5C22⋊Q812yes—
289C2×C4×C5#13712yes—
290C2×C4×C5C22×Q811yescharacteristic
291C2×C4×C5C2×#4111yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5H(5,𝔽2)11yescharacteristic
295C2×C4×C52−1+411yescharacteristic
296C2×Dic5—22no—
297C2×Dic5—22no—
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×Dic5—41no—
308C2×Dic5—84no—
309#168—22no—
310#168—41no—
311#168—41no—
312#168—41no—
313C23×C4—101no—
314C23×C4C22×D511yescharacteristic
315C2×M16—102no—
316C2×M16—102no—
317C2×M16—201no—
318C22×C8—51no—
319C22×C8—51no—
320C22×C8—51no—
321C22×C8—51no—
322#125—22no—
323#125—41no—
324#117—22no—
325#117—41no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—54no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—102no—
338C2×C4⋊C4—101no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4C22×D511yescharacteristic
342C2×C4⋊C4C22×D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—101no—
348C2×C22⋊C4C22⋊D511yescharacteristic
349C2×C22⋊C4C22⋊D511yescharacteristic
350C2×C42—21no—
351C2×C42—101no—
352C2×C42—101no—
353C2×C42C22×D511yescharacteristic
354C2×C42C22×D511yescharacteristic
355#107—201no—
356#107—201no—
357#106—102no—
358#106—102no—
359C4⋊C8—102no—
360C4⋊C8—201no—
361C22⋊C8—102no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—101no—
366#95—101no—
367#95—101no—
368#95—101no—
369#95—101no—
370#95C22⋊D511yescharacteristic
371#95C22⋊D511yescharacteristic
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—41no—
376C22×C5—41no—
377C22×C5—41no—
378C22×C5—41no—
379C22×C5C2×C22⋊Q811yescharacteristic
380C22×C5#54611yescharacteristic
381C22×C5#54711yescharacteristic
382C22×C5C2×#13711yescharacteristic
383C22×C5#57511yescharacteristic
384C22×C5#57611yescharacteristic
385C22×C5#57711yescharacteristic
386C4×C5—22no—
387C4×C5—22no—
388C4×C5—41no—
389C4×C5—42no—
390C4×C5—41no—
391C4×C5—41no—
392C4×C5—81no—
393C4×C5—81no—
394C4×C5—81no—
395C4×C5#53311yescharacteristic
396C4×C5#54411yescharacteristic
397C4×C5#55111yescharacteristic
398C4×C5#55211yescharacteristic
399Dic5—42no—
400Dic5—42no—
401Dic5—81no—
402Dic5—81no—
403C24C2×C22⋊D511yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4—104no—
415C22×C4—101no—
416C22×C4—101no—
417C22×C4—401no—
418C22×C4—401no—
419C22×C4Q8×D512yes—
420C22×C4#71312yes—
421C22×C4C2×C22⋊D511yescharacteristic
422C22×C4C2×C22⋊D511yescharacteristic
423C22×C4C2×C22⋊D511yescharacteristic
424C22×C4C2×C22⋊D511yescharacteristic
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C23×D511yescharacteristic
427M16—204no—
428C2×C8—52no—
429C2×C8—101no—
430C2×C8—101no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—101no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—52no—
452C4⋊C4—52no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—101no—
456C4⋊C4—101no—
457C4⋊C4—104no—
458C4⋊C4—101no—
459C4⋊C4—101no—
460C4⋊C4—101no—
461C4⋊C4—101no—
462C4⋊C4—401no—
463C4⋊C4—401no—
464C4⋊C4#71112yes—
465C22⋊C4—22no—
466C22⋊C4—41no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—202no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—401no—
476C42—401no—
477C2×C5—42no—
478C2×C5#243811yescharacteristic
479C2×C5#307811yescharacteristic
480C2×C5#308111yescharacteristic
481C2×C5#330911yescharacteristic
482C2×C5#333111yescharacteristic
483C2×C5#334611yescharacteristic
484C2×C5#335111yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#409212yes—
492C23C22×C22⋊D511yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—201no—
525C2×C4—201no—
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—404no—
535C2×C4#409212yes—
536C2×C4#409712yes—
537C2×C4C2×#71111yescharacteristic
538C2×C4#414411yescharacteristic
539C2×C4C2×Q8×D511yescharacteristic
540C2×C4C2×#71311yescharacteristic
541C2×C4D5×#4111yescharacteristic
542C2×C4#415011yescharacteristic
543C8—102no—
544C8—201no—
545C8—201no—
546C8—201no—
547C5#2601811yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22#6614011yescharacteristic
556C22C2×#409211yescharacteristic
557C22#6615611yescharacteristic
558C22C2×#409711yescharacteristic
559C22#6615811yescharacteristic
560C22#6616911yescharacteristic
561C22#6617111yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—202no—
572C4—202no—
573C4—401no—
574C4—401no—
575C4#6595611yescharacteristic
576C4#6595711yescharacteristic
577C4#6597111yescharacteristic
578C4#6597611yescharacteristic
579C2—42no—
580C2#1061449311yescharacteristic
581C2#1061509811yescharacteristic
582C2#1061520511yescharacteristic
583C2#1061524611yescharacteristic
584C2#1061530311yescharacteristic
585C2#1061532711yescharacteristic
586C2#1061534011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2441C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2441D5
hypocenterC5#26018
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
3. #10602532C2semidirect
4. #10602531C2semidirect
7. #10600417C2semidirect
8. #10600415C2semidirect
9. #10600395C2semidirect
10. #10600394C2semidirect
11. #10600365C2semidirect
12. #10600360C2semidirect
547. C5#26018semidirect

Maximal subgroups

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

subgroupindex
C5×#24412
#106051882
#106025322
#106025312
#106023742
#106023692
#106004172
#106004152
#106003952
#106003942
#106003652
#106003602
#106003062
#106002942
#106002432
#260185