Finite Group Databasebeta

← #12121146

Subgroups — #12121146

Order 1280.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614065C211yesmaximal, characteristic
2C2×#65052C211yesmaximal, characteristic
3#10605391C211yesmaximal, characteristic, complemented
4C4×C5×#102C211yesmaximal, characteristic
5#10602399C211yesmaximal, characteristic
6#10600963C211yesmaximal, characteristic
7Dic5×#102C211yesmaximal, characteristic
8Dic5×#102C211yesmaximal, characteristic
9C8×#683C211yesmaximal, characteristic, complemented
10#10600562C211yesmaximal, characteristic, complemented
11#10600254C211yesmaximal, characteristic, complemented
12#10599652C211yesmaximal, characteristic, complemented
13#10599611C211yesmaximal, characteristic, complemented
14#10599597C211yesmaximal, characteristic, complemented
15#10599590C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C2×D4×Dic5C2211yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C2×#4036C2211yescharacteristic
20C2×C5×#102C2211yescharacteristic
21C2×C5×#102C2211yescharacteristic
22C2×C4×C5×C8C2211yescharacteristic
23C4×C5×C4⋊C4C2211yescharacteristic
24C2×#3964C2211yescharacteristic
25C2×#3964C2211yescharacteristic
26#65499—21no—
27C2×#3963—21no—
28C2×#3949C2211yescharacteristic
29C2×#3949C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×C8×Dic5C2211yescharacteristic
32C2×C8×Dic5C2211yescharacteristic
33#65274C2211yescharacteristic
34#65266C2211yescharacteristic
35C4⋊C4×Dic5C2211yescharacteristic
36C4⋊C4×Dic5C2211yescharacteristic
37#65262C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40#65235—21no—
41#65231—21no—
42C4×#683—21no—
43C4×#683C2211yescharacteristic
44#65217—21no—
45#65217C2211yescharacteristic
46#65217C2211yescharacteristic
47#65052C418yescomplemented
48#64944—21no—
49#64943—21no—
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52#64771C2211yescharacteristic
53#64771C2211yescharacteristic
54#64770C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#22966—51nomaximal, Sylow
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×Dic5—21no—
64C23×Dic5—21no—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C4×C5×D4—22no—
68C4×C5×D4—41no—
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C42×C5—21no—
74C2×C42×C5C2311yescharacteristic
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—21no—
77D4×Dic5—22no—
78D4×Dic5—22no—
79D4×Dic5—41no—
80D4×Dic5C2×C414yes—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×C4⋊Dic5C2311yescharacteristic
87C2×#683—21no—
88C2×#683—21no—
89C2×#683—21no—
90C2×#683—21no—
91C2×#683D411yescharacteristic
92C2×#683D411yescharacteristic
93C2×C4×Dic5—21no—
94C2×C4×Dic5—21no—
95C2×C4×Dic5C2311yescharacteristic
96C2×C4×Dic5C2311yescharacteristic
97C22×#168C2311yescharacteristic
98C22×#168C2311yescharacteristic
99#4036—22no—
100#4036C2×C414yescomplemented
101C5×#102C2×C414yes—
102C5×#102Q814yes—
103C4×C5×C8—41no—
104C5×#95—21no—
105#3964—22no—
106#3964C2×C414yes—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963D411yescharacteristic
116#3963D411yescharacteristic
117#3949—41no—
118#3949C2×C414yescomplemented
119#3946C2×C414yescomplemented
120C8×Dic5—22no—
121C8×Dic5—41no—
122#3939—41no—
123#3939C2×C414yescomplemented
124C2×#478—51no—
125#2370—51no—
126#2349—51no—
127#1991—51no—
128#1949—51no—
129#1942—51no—
130#1941—51no—
131#1921—51no—
132#1920—51no—
133#1864—51no—
134#1862—51no—
135C8×C4⋊C4—51no—
136C4×#102—51no—
137C4×#102—51no—
138C4×#102D511yescharacteristic
139C24×C5—21no—
140C2×C5×D4—21no—
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4C22×C412yes—
144C2×C5×D4C2×Q812yes—
145C2×C5×D4#4112yes—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5—81no—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2411yescharacteristic
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—24no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—42no—
161C22×Dic5—42no—
162C22×Dic5—81no—
163C22×Dic5—81no—
164C22×Dic5—81no—
165C22×Dic5C2×D411yescharacteristic
166C22×Dic5C2×D411yescharacteristic
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8C22×C412yes—
174C2×C5×C8C2×Q812yes—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4C22×C412yes—
180C5×C4⋊C4C2×Q812yes—
181C5×C22⋊C4—42no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—81no—
185C22⋊Dic5—24no—
186C22⋊Dic5—42no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—24no—
190C4⋊Dic5—21no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—81no—
195C4⋊Dic5C22×C412yes—
196C4⋊Dic5C22×C412yes—
197#683—22no—
198#683—24no—
199#683—22no—
200#683—42no—
201#683—81no—
202#683—81no—
203C4×Dic5—21no—
204C4×Dic5—21no—
205C4×Dic5—42no—
206C4×Dic5—81no—
207C4×Dic5C22×C412yes—
208C4×Dic5#4112yes—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168C22×C412yes—
215C2×#125—51no—
216C2×C4×D4—51no—
217C2×C4×D4—51no—
218C2×C4×D4C2×D511yescharacteristic
219#478—58no—
220C2×#107—51no—
221C2×#107—51no—
222C2×#106—51no—
223C2×#106—51no—
224C2×C4⋊C8—51no—
225C2×#102—51no—
226C2×#102—51no—
227C2×#102C2×D511yescharacteristic
228C2×#102C2×D511yescharacteristic
229C2×C4×C8—51no—
230C2×C4×C8—51no—
231C2×C4×C8C2×D511yescharacteristic
232#389—51no—
233#389—51no—
234#389—51no—
235#389—51no—
236#389—101no—
237#387—101no—
238#387—101no—
239#385—101no—
240#385—101no—
241#382—51no—
242#380—101no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—51no—
245C4×C4⋊C4—51no—
246C4×C4⋊C4—101no—
247C4×C4⋊C4C2×D511yescharacteristic
248C2×#95—101no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340C2×D511yescharacteristic
256#340C2×D511yescharacteristic
257#336—51no—
258#336—51no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—41no—
263C23×C5—41no—
264C23×C5—41no—
265C23×C5—41no—
266C23×C5C22×D411yescharacteristic
267C5×D4—22no—
268C5×D4—22no—
269C5×D4—22no—
270C5×D4C4×Q814yes—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—22no—
274C2×C4×C5—21no—
275C2×C4×C5—22no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
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—84no—
286C2×C4×C5#13512yes—
287C2×C4×C5C23×C411yescharacteristic
288C2×C4×C5C22×Q811yescharacteristic
289C2×C4×C5C2×#4111yescharacteristic
290C2×C4×C5C2×#4111yescharacteristic
291C2×C4×C5H(5,𝔽2)11yescharacteristic
292C2×C4×C52−1+411yescharacteristic
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—21no—
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—84no—
315C2×Dic5—84no—
316C2×Dic5C4×D412yes—
317C2×Dic5C4×D412yes—
318C2×Dic5#13512yes—
319C5×C8—22no—
320C5×C8—22no—
321#168—22no—
322#168—41no—
323C22×D4C22×D511yescharacteristic
324C23×C4—21no—
325C23×C4—101no—
326C23×C4—101no—
327C22×C8—51no—
328C22×C8—51no—
329C22×C8C22×D511yescharacteristic
330C22×C8C22×D511yescharacteristic
331#125—54no—
332#125—102no—
333C4×D4—22no—
334C4×D4—41no—
335C4×D4—54no—
336C4×D4—102no—
337C4×D4—102no—
338C4×D4—201no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—51no—
342C2×C4⋊C4—51no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—101no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4—101no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4C22×D511yescharacteristic
352C2×C4⋊C4C22×D511yescharacteristic
353C2×C4⋊C4C22×D511yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—101no—
356C2×C22⋊C4—101no—
357C2×C42—21no—
358C2×C42—51no—
359C2×C42—51no—
360C2×C42—101no—
361C2×C42—101no—
362C2×C42C22×D511yescharacteristic
363#107—54no—
364#107—201no—
365#106—54no—
366#106—201no—
367C4⋊C8—54no—
368#102—54no—
369#102—102no—
370#102Dic1014yes—
371#102C4×D514yes—
372C4×C8—41no—
373C4×C8—102no—
374C4×C8—201no—
375#95—21no—
376#95—51no—
377#95—51no—
378#95—101no—
379#95—101no—
380#95—101no—
381#95—101no—
382#95—101no—
383#95—101no—
384#95—101no—
385#95—101no—
386C22×C5—24no—
387C22×C5—42no—
388C22×C5—42no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—42no—
393C22×C5C2×C4×D411yescharacteristic
394C22×C5#54611yescharacteristic
395C22×C5D4×Q811yescharacteristic
396C22×C5C2×#13511yescharacteristic
397C22×C5C2×#13511yescharacteristic
398C22×C5#57611yescharacteristic
399C22×C5#57711yescharacteristic
400C4×C5—22no—
401C4×C5—42no—
402C4×C5—42no—
403C4×C5—41no—
404C4×C5—81no—
405C4×C5C2×C4×Q811yescharacteristic
406C4×C5#52011yescharacteristic
407C4×C5#55211yescharacteristic
408C4×C5#55411yescharacteristic
409Dic5—22no—
410Dic5—22no—
411Dic5—22no—
412Dic5—42no—
413Dic5—42no—
414Dic5—41no—
415Dic5—81no—
416Dic5—81no—
417C24—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×Dic1012yes—
422C2×D4C2×C4×D512yes—
423C2×D4#70912yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—51no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—81no—
434C22×C4—104no—
435C22×C4—101no—
436C22×C4—101no—
437C22×C4—101no—
438C22×C4—101no—
439C22×C4—202no—
440C22×C4—202no—
441C22×C4—401no—
442C22×C4—401no—
443C22×C4—401no—
444C22×C4D4×D511yescharacteristic
445C22×C4D4×D511yescharacteristic
446C22×C4C23×D511yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—52no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8C2×Dic1012yes—
458C2×C8C2×C4×D512yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—52no—
464C4⋊C4—52no—
465C4⋊C4—102no—
466C4⋊C4—104no—
467C4⋊C4—104no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—102no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—202no—
476C4⋊C4—401no—
477C4⋊C4—401no—
478C4⋊C4—401no—
479C4⋊C4C2×Dic1012yes—
480C4⋊C4C2×C4×D512yes—
481C22⋊C4—42no—
482C22⋊C4—104no—
483C22⋊C4—202no—
484C42—21no—
485C42—21no—
486C42—52no—
487C42—52no—
488C42—81no—
489C42—101no—
490C42—101no—
491C42—202no—
492C42—401no—
493C2×C5—44no—
494C2×C5#236011yescharacteristic
495C2×C5C4×#13511yescharacteristic
496C2×C5#294611yescharacteristic
497C2×C5#329611yescharacteristic
498C2×C5#329711yescharacteristic
499C2×C5#338011yescharacteristic
500C2×C5#338111yescharacteristic
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C2×D4×D511yescharacteristic
509D4—22no—
510D4—22no—
511D4—22no—
512D4C4×Dic1014yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—52no—
528C2×C4—52no—
529C2×C4—52no—
530C2×C4—84no—
531C2×C4—101no—
532C2×C4—101no—
533C2×C4—101no—
534C2×C4—101no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—101no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—404no—
553C2×C4—404no—
554C2×C4#406212yes—
555C2×C4C22×Dic1011yescharacteristic
556C2×C4C22×C4×D511yescharacteristic
557C2×C4C2×#70911yescharacteristic
558C2×C4#414411yescharacteristic
559C2×C4D5×#4111yescharacteristic
560C2×C4#415011yescharacteristic
561C8—22no—
562C8—22no—
563C8—102no—
564C8—201no—
565C5#2296611yescharacteristic, complemented, Sylow
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22#6582411yescharacteristic
574C22D4×Dic1011yescharacteristic
575C22C4×D4×D511yescharacteristic
576C22#6609111yescharacteristic
577C22C2×#406211yescharacteristic
578C22D5×#13511yescharacteristic
579C22#6610411yescharacteristic
580C4—22no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—81no—
585C4—102no—
586C4—102no—
587C4—102no—
588C4—201no—
589C4—202no—
590C4—202no—
591C4—401no—
592C4—401no—
593C4C2×C4×Dic1011yescharacteristic
594C4#6586511yescharacteristic
595C4#6587311yescharacteristic
596C4#6588011yescharacteristic
597C2—44no—
598C2#1060519711yescharacteristic
599C2#1061049111yescharacteristic
600C2#1061057611yescharacteristic
601C2#1061130611yescharacteristic
602C2#1061131811yescharacteristic
603C2#1061136311yescharacteristic
604C2C4×#406211yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×#102C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC4×#102D5
hypocenterC5#22966
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. #10605391C2semidirect
9. C8×#683C2semidirect
10. #10600562C2semidirect
11. #10600254C2semidirect
12. #10599652C2semidirect
13. #10599611C2semidirect
14. #10599597C2semidirect
15. #10599590C2semidirect
47. #65052C4semidirect
100. #4036C2×C4semidirect
118. #3949C2×C4semidirect
119. #3946C2×C4semidirect
123. #3939C2×C4semidirect
565. C5#22966semidirect

Maximal subgroups

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

subgroupindex
#106140652
C2×#650522
#106053912
C4×C5×#1022
#106023992
#106009632
Dic5×#1022
C8×#6832
#106005622
#106002542
#105996522
#105996112
#105995972
#105995902
#229665