Finite Group Databasebeta

← #12172026

Subgroups — #12172026

Order 1280.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2360C211yesmaximal, characteristic
2#10605242C211yesmaximal, characteristic
3#10602488C211yesmaximal, characteristic
4#10602487C211yesmaximal, characteristic
5C4×#3967C211yesmaximal, characteristic, complemented
6#10602421C211yesmaximal, characteristic, complemented
7#10602348C211yesmaximal, characteristic
8#10602346C211yesmaximal, characteristic
9#10600476C211yesmaximal, characteristic, complemented
10#10600464C211yesmaximal, characteristic, complemented
11#10600412C211yesmaximal, characteristic, complemented
12#10600395C211yesmaximal, characteristic, complemented
13#10600261C211yesmaximal, characteristic, complemented
14#10600251C211yesmaximal, characteristic, complemented
15#10600244C211yesmaximal, characteristic
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C5×#391C2211yescharacteristic
20C5×#389—21no—
21C5×#389C2211yescharacteristic
22C5×#387—21no—
23C5×#385—21no—
24C4×C5×C4⋊C4C2211yescharacteristic
25C4×C5×C22⋊C4—21no—
26C4×C5×C22⋊C4C2211yescharacteristic
27C2×#3967C2211yescharacteristic
28C2×#3967C2211yescharacteristic
29C2×#3967C2211yescharacteristic
30C2×#3967C2211yescharacteristic
31#65501C2211yescharacteristic
32#65501C2211yescharacteristic
33C2×C22⋊#168C2211yescharacteristic
34C2×C22⋊#168C2211yescharacteristic
35C2×#3940C2211yescharacteristic
36C2×#3939C2211yescharacteristic
37C2×#3939C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39#65241—21no—
40#65233—21no—
41#65225C2211yescharacteristic
42C4×C4⋊Dic5C2211yescharacteristic
43#65217C2211yescharacteristic
44#65217C2211yescharacteristic
45C2×C4⋊#168C2211yescharacteristic
46C2×C4×#168C2211yescharacteristic
47#64947—21no—
48#64940—21no—
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#27035—51nomaximal, Sylow
60C22×C5×Q8C2311yescharacteristic
61C23×C4×C5—21no—
62C23×C4×C5C2311yescharacteristic
63C5×C22⋊Q8—22no—
64C5×C22⋊Q8—22no—
65C4×C5×Q8—22no—
66C4×C5×Q8—41no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—41no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C5×C22⋊C4D411yescharacteristic
78C2×C5×C22⋊C4D411yescharacteristic
79C2×C42×C5—21no—
80C2×C42×C5—21no—
81C2×C42×C5C2311yescharacteristic
82C2×C42×C5C2311yescharacteristic
83C2×C22⋊Dic5—21no—
84C2×C22⋊Dic5—21no—
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×#683—21no—
90C2×#683—21no—
91C2×C4×Dic5—21no—
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C5×#95—21no—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95D411yescharacteristic
101C5×#95D411yescharacteristic
102#3967—22no—
103#3967—22no—
104#3967—22no—
105#3967—41no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113C22⋊#168—22no—
114C22⋊#168—41no—
115#3940—22no—
116#3939—22no—
117#3939—22no—
118#3939—41no—
119C4⋊#168—22no—
120C4×#168—41no—
121#2495—51no—
122#2360D511yescharacteristic
123#2074—51no—
124#2068—51no—
125#2036—51no—
126#2035—51no—
127#1949—51no—
128#1942—51no—
129#1923—51no—
130#1920—51no—
131#1918—51no—
132#1863—51no—
133#1852—51no—
134#1823—51no—
135C4×#103—51no—
136C24×C5C2×D411yescharacteristic
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8#4112yes—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—42no—
154C22×C4×C5—42no—
155C22×C4×C5—81no—
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—42no—
172C5×C4⋊C4—42no—
173C5×C4⋊C4—81no—
174C5×C4⋊C4#4112yes—
175C5×C4⋊C4#4112yes—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—22no—
179C42×C5—21no—
180C42×C5—21no—
181C42×C5—21no—
182C42×C5—21no—
183C42×C5—42no—
184C42×C5—81no—
185C22⋊Dic5—42no—
186C22⋊Dic5—81no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195#683—81no—
196#683—81no—
197C4×Dic5—42no—
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—21no—
208C2×#168—21no—
209C2×#168#4112yes—
210C2×C22⋊Q8C2×D511yescharacteristic
211C2×C4×Q8C2×D511yescharacteristic
212C22×C4⋊C4C2×D511yescharacteristic
213C2×#107—51no—
214C2×#107—51no—
215C2×#107—51no—
216C2×#106—51no—
217C2×C4⋊C8—51no—
218C2×#103—51no—
219C2×#103—51no—
220C2×#103—51no—
221C2×#103—51no—
222C2×C22⋊C8—51no—
223C2×C22⋊C8—51no—
224C2×C4×C8—51no—
225#400—101no—
226#397—101no—
227#391C2×D511yescharacteristic
228#389—21no—
229#389—51no—
230#389—51no—
231#389C2×D511yescharacteristic
232#388—101no—
233#387—21no—
234#385—21no—
235#383—101no—
236#382—51no—
237#380—51no—
238#380—51no—
239C4×C4⋊C4—51no—
240C4×C4⋊C4C2×D511yescharacteristic
241C4×C22⋊C4—21no—
242C4×C22⋊C4C2×D511yescharacteristic
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340—51no—
251#336—51no—
252#336—51no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5#13512yes—
260C23×C5C22×D411yescharacteristic
261C5×Q8—22no—
262C5×Q8—22no—
263C5×Q8—41no—
264C5×Q8—41no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
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×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—84no—
296C2×C4×C5#13712yes—
297C2×C4×C5C2×#4111yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×C4×C5C2×#4111yescharacteristic
300C2×C4×C5C2×#4111yescharacteristic
301C2×C4×C5H(5,𝔽2)11yescharacteristic
302C2×C4×C52−1+411yescharacteristic
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315#168—22no—
316#168—41no—
317#168—41no—
318#168—41no—
319C22×Q8C22×D511yescharacteristic
320C23×C4—21no—
321C23×C4C22×D511yescharacteristic
322C22×C8—51no—
323C22×C8—51no—
324C22×C8—51no—
325C22×C8—51no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C4×Q8—22no—
329C4×Q8—41no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—41no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—51no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4—101no—
342C2×C4⋊C4C22×D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—101no—
347C2×C22⋊C4—101no—
348C2×C22⋊C4C22⋊D511yescharacteristic
349C2×C22⋊C4C22⋊D511yescharacteristic
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—101no—
353C2×C42C22×D511yescharacteristic
354C2×C42C22×D511yescharacteristic
355#107—102no—
356#107—102no—
357#107—201no—
358#106—102no—
359C4⋊C8—102no—
360#103—102no—
361#103—102no—
362#103—102no—
363#103—201no—
364C22⋊C8—102no—
365C22⋊C8—201no—
366C4×C8—201no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—101no—
372#95—101no—
373#95—101no—
374#95—101no—
375#95—101no—
376#95—101no—
377#95—101no—
378#95C22⋊D511yescharacteristic
379#95C22⋊D511yescharacteristic
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—41no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5#52511yescharacteristic
388C22×C5#54611yescharacteristic
389C22×C5#54711yescharacteristic
390C22×C5C2×#13511yescharacteristic
391C22×C5C2×#13711yescharacteristic
392C22×C5#57611yescharacteristic
393C22×C5#57811yescharacteristic
394C4×C5—22no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—42no—
398C4×C5—41no—
399C4×C5—42no—
400C4×C5—41no—
401C4×C5—42no—
402C4×C5—81no—
403C4×C5#52911yescharacteristic
404C4×C5#53911yescharacteristic
405C4×C5#55311yescharacteristic
406C4×C5#55611yescharacteristic
407Dic5—81no—
408Dic5—81no—
409Dic5—81no—
410Dic5—81no—
411C24C2×C22⋊D511yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8#70912yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—101no—
432C22×C4—101no—
433C22×C4—101no—
434C22×C4—101no—
435C22×C4C2×C22⋊D511yescharacteristic
436C22×C4C2×C22⋊D511yescharacteristic
437C22×C4C2×C22⋊D511yescharacteristic
438C22×C4C2×C22⋊D511yescharacteristic
439C22×C4C2×C22⋊D511yescharacteristic
440C22×C4C23×D511yescharacteristic
441C2×C8—52no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—101no—
445C2×C8—101no—
446C2×C8—101no—
447C2×C8—101no—
448C2×C8—101no—
449C2×C8—101no—
450C2×C8—101no—
451C2×C8—101no—
452C2×C8—101no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—101no—
462C4⋊C4—101no—
463C4⋊C4—101no—
464C4⋊C4—101no—
465C4⋊C4—101no—
466C4⋊C4—101no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—401no—
470C4⋊C4—401no—
471C4⋊C4#70912yes—
472C4⋊C4#71112yes—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—202no—
477C22⋊C4—401no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—81no—
484C42—202no—
485C2×C5—42no—
486C2×C5#246611yescharacteristic
487C2×C5#310811yescharacteristic
488C2×C5#312411yescharacteristic
489C2×C5#329711yescharacteristic
490C2×C5#331211yescharacteristic
491C2×C5#333611yescharacteristic
492C2×C5#335711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#409612yes—
500C23C22×C22⋊D511yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—201no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4#408812yes—
549C2×C4C2×#70911yescharacteristic
550C2×C4C2×#70911yescharacteristic
551C2×C4C2×#71111yescharacteristic
552C2×C4#414411yescharacteristic
553C2×C4D5×#4111yescharacteristic
554C2×C4#415011yescharacteristic
555C8—102no—
556C8—201no—
557C8—201no—
558C8—201no—
559C5#2703511yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#6612511yescharacteristic
568C22#6614011yescharacteristic
569C22C2×#408811yescharacteristic
570C22C2×#409611yescharacteristic
571C22#6615811yescharacteristic
572C22#6616911yescharacteristic
573C22#6617211yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—401no—
584C4—401no—
585C4—401no—
586C4—401no—
587C4#6587211yescharacteristic
588C4#6587611yescharacteristic
589C4#6589211yescharacteristic
590C4#6589811yescharacteristic
591C2—42no—
592C2#1061410011yescharacteristic
593C2#1061417411yescharacteristic
594C2#1061430011yescharacteristic
595C2#1061522611yescharacteristic
596C2#1061528311yescharacteristic
597C2#1061532111yescharacteristic
598C2#1061535811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2360C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2360D5
hypocenterC5#27035
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
5. C4×#3967C2semidirect
6. #10602421C2semidirect
9. #10600476C2semidirect
10. #10600464C2semidirect
11. #10600412C2semidirect
12. #10600395C2semidirect
13. #10600261C2semidirect
14. #10600251C2semidirect
559. C5#27035semidirect

Maximal subgroups

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

subgroupindex
C5×#23602
#106052422
#106024882
#106024872
C4×#39672
#106024212
#106023482
#106023462
#106004762
#106004642
#106004122
#106003952
#106002612
#106002512
#106002442
#270355