Finite Group Databasebeta

← #12109833

Subgroups — #12109833

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10613955C211yesmaximal, characteristic
2C2×#65007C211yesmaximal, characteristic
3#10605595C211yesmaximal, characteristic
4C5×#1854C211yesmaximal, characteristic
5#10602369C211yesmaximal, characteristic
6#10601409C211yesmaximal, characteristic, complemented
7#10601399C211yesmaximal, characteristic, complemented
8#10601357C211yesmaximal, characteristic, complemented
9#10601347C211yesmaximal, characteristic, complemented
10#10600722C211yesmaximal, characteristic
11#10600299C211yesmaximal, characteristic
12#10599640C211yesmaximal, characteristic, complemented
13#10599640C211yesmaximal, characteristic, complemented
14#10599591C211yesmaximal, characteristic, complemented
15#10599590C211yesmaximal, characteristic, complemented
16C2×C5×#117C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19C2×#4023—21no—
20C2×C5×#107C2211yescharacteristic
21C2×C5×#106C2211yescharacteristic
22C2×C5×C22⋊C8C2211yescharacteristic
23C5×#380C2211yescharacteristic
24#65501C2211yescharacteristic
25C2×C22⋊#168C2211yescharacteristic
26C2×#3950C2211yescharacteristic
27C2×#3949C2211yescharacteristic
28C2×#3946C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×C8×Dic5C2211yescharacteristic
31C2×C8×Dic5C2211yescharacteristic
32#65266C2211yescharacteristic
33#65266C2211yescharacteristic
34#65265C2211yescharacteristic
35#65265C2211yescharacteristic
36#65265C2211yescharacteristic
37#65265C2211yescharacteristic
38C4⋊C4×Dic5C2211yescharacteristic
39C4⋊C4×Dic5C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3940C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65231—21no—
45#65231—21no—
46#65231—21no—
47C4×#683—21no—
48C4×#683—21no—
49#65007C418yescomplemented
50#64939—21no—
51#64938—21no—
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54#64771C2211yescharacteristic
55#64771C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#21984—51nomaximal, Sylow
61C23×C4×C5C2311yescharacteristic
62C23×Dic5D411yescharacteristic
63C23×Dic5D411yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×#117—41no—
67C5×#117—41no—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—21no—
77#4072—41no—
78#4072—41no—
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×#683—21no—
84C2×#683—21no—
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683—21no—
89C2×#683—21no—
90C2×#683—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5C2×C414yes—
95C2×C4×Dic5D411yescharacteristic
96C2×C4×Dic5D411yescharacteristic
97C2×C4×Dic5Q814yes—
98C2×C4×Dic5C2311yescharacteristic
99C2×C4×Dic5C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2311yescharacteristic
102#4023—44no—
103C5×#107—41no—
104C5×#106—41no—
105C5×C22⋊C8C2×C414yes—
106C5×#95—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114C22⋊#168C2×C414yes—
115#3950—41no—
116#3949—41no—
117#3946C2×C414yes—
118#3946C2×C414yes—
119C8×Dic5C2×C414yes—
120C8×Dic5C2×C414yes—
121#3940—41no—
122#3940—41no—
123#3939—41no—
124#3939—41no—
125C2×#431—51no—
126#2285—51no—
127#2285—51no—
128#1949—51no—
129#1949—51no—
130#1949—51no—
131#1949—51no—
132#1854—51no—
133#1854—51no—
134#1854—51no—
135#1854D511yescharacteristic
136#1845—51no—
137#1844—51no—
138#1834—51no—
139#1834—51no—
140C24×C5C2×D411yescharacteristic
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×C5—21no—
149C22×C4×C5C22×C412yes—
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×Q812yes—
152C22×C4×C5C2411yescharacteristic
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5C4⋊C414yes—
162C22×Dic5C4⋊C414yes—
163C22×Dic5C2×D411yescharacteristic
164C22×Dic5C2×D411yescharacteristic
165C22×Dic5C2×D411yescharacteristic
166C22×Dic5C2×D411yescharacteristic
167C2×C5×C8—21no—
168C2×C5×C8—21no—
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8C22×C412yes—
172C2×C5×C8C22×C412yes—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4—42no—
183C42×C5—42no—
184C22⋊Dic5—42no—
185C22⋊Dic5—42no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194#683—42no—
195#683—42no—
196#683—42no—
197#683—42no—
198#683—42no—
199#683—42no—
200#683—42no—
201#683—42no—
202C4×Dic5—42no—
203C4×Dic5—42no—
204C4×Dic5—42no—
205C4×Dic5C4⋊C414yes—
206C4×Dic5C4⋊C414yes—
207C4×Dic5C22×C412yes—
208C4×Dic5C22×C412yes—
209C4×Dic5C2×Q812yes—
210C4×Dic5C2×Q812yes—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168C22×C412yes—
216C2×#168C22×C412yes—
217C2×#117—51no—
218C2×#117—101no—
219C2×#117C2×D511yescharacteristic
220C22×C42—51no—
221#431—58no—
222C2×#107—51no—
223C2×#107—51no—
224C2×#107—51no—
225C2×#107C2×D511yescharacteristic
226C2×#106—51no—
227C2×#106—51no—
228C2×#106—51no—
229C2×#106C2×D511yescharacteristic
230C2×C4⋊C8—51no—
231C2×C4⋊C8—51no—
232C2×C22⋊C8—51no—
233C2×C22⋊C8C2×D511yescharacteristic
234C2×C4×C8—51no—
235C2×C4×C8—51no—
236#384—51no—
237#384—51no—
238#384—51no—
239#384—51no—
240#382—51no—
241#382—51no—
242#382—101no—
243#380—51no—
244#380—101no—
245#380—101no—
246#380—101no—
247#380C2×D511yescharacteristic
248C4×C4⋊C4—51no—
249C4×C4⋊C4—51no—
250C4×C4⋊C4—101no—
251C4×C4⋊C4—101no—
252C4×C4⋊C4—101no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340C2×D511yescharacteristic
260#340C2×D511yescharacteristic
261C23×C5—21no—
262C23×C5—21no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5—21no—
267C23×C5C2×C4⋊C412yes—
268C23×C5C22×D411yescharacteristic
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
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×C5C2×C4⋊C412yes—
288C2×C4×C5C23×C411yescharacteristic
289C2×C4×C5C22×Q811yescharacteristic
290C2×C4×C5H(5,𝔽2)11yescharacteristic
291C2×C4×C5H(5,𝔽2)11yescharacteristic
292C2×C4×C52−1+411yescharacteristic
293C2×C4×C52−1+411yescharacteristic
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—28no—
299C2×Dic5—28no—
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—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5C2×C4⋊C412yes—
313C2×Dic5C2×C4⋊C412yes—
314C2×Dic5C2×C4⋊C412yes—
315C2×Dic5C2×C4⋊C412yes—
316C2×Dic5#13512yes—
317C2×Dic5#13512yes—
318C2×Dic5AGL(1,ℤ/8)12yes—
319C2×Dic5#13712yes—
320C5×C8—22no—
321C5×C8—22no—
322#168—22no—
323#168—22no—
324C23×C4—51no—
325C23×C4—51no—
326C23×C4C22×D511yescharacteristic
327C22×C8—51no—
328C22×C8—51no—
329C22×C8C22×D511yescharacteristic
330C22×C8C22×D511yescharacteristic
331#117—41no—
332#117—41no—
333#117—204no—
334#117—201no—
335#117—201no—
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⋊C4—101no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4—101no—
345C2×C4⋊C4—101no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D511yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—101no—
355C2×C22⋊C4—101no—
356C2×C42—21no—
357C2×C42—54no—
358C2×C42—51no—
359C2×C42—51no—
360C2×C42—54no—
361C2×C42—51no—
362C2×C42—51no—
363C2×C42—101no—
364C2×C42—101no—
365C2×C42—101no—
366#107—41no—
367#107—201no—
368#107—201no—
369#107—201no—
370#106—41no—
371#106—201no—
372#106—201no—
373#106—201no—
374C4⋊C8—54no—
375C4⋊C8—54no—
376C22⋊C8—54no—
377C22⋊C8C4×D514yes—
378C4×C8—54no—
379C4×C8—54no—
380#95—21no—
381#95—101no—
382#95—101no—
383#95—101no—
384#95—101no—
385#95—101no—
386#95—101no—
387#95—101no—
388C22×C5—22no—
389C22×C5—22no—
390C22×C5—22no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5—41no—
394C22×C5—41no—
395C22×C5C22×C4⋊C411yescharacteristic
396C22×C5#53511yescharacteristic
397C22×C5#53611yescharacteristic
398C22×C5C2×#13511yescharacteristic
399C22×C5C2×#13511yescharacteristic
400C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C5C2×#13711yescharacteristic
402C4×C5—22no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5—81no—
407C4×C5#52011yescharacteristic
408C4×C5#52011yescharacteristic
409C4×C5#54411yescharacteristic
410C4×C5#54411yescharacteristic
411Dic5—22no—
412Dic5—22no—
413Dic5—81no—
414Dic5—81no—
415Dic5—81no—
416Dic5—81no—
417Dic5#42714yes—
418Dic5#42814yes—
419C24D4×D511yescharacteristic
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—51no—
429C22×C4—51no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—54no—
433C22×C4—54no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4C2×C4×D512yes—
443C22×C4D4×D511yescharacteristic
444C22×C4Q8×D512yes—
445C22×C4C23×D511yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—52no—
451C2×C8—52no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8C2×C4×D512yes—
457C2×C8C2×C4×D512yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—101no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—101no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—202no—
475C4⋊C4—202no—
476C4⋊C4—202no—
477C4⋊C4—202no—
478C4⋊C4—202no—
479C4⋊C4—202no—
480C4⋊C4—202no—
481C4⋊C4—202no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—202no—
485C22⋊C4—202no—
486C42—42no—
487C42—52no—
488C42—52no—
489C42—52no—
490C42—52no—
491C42—54no—
492C42—54no—
493C42—202no—
494C42—202no—
495C42—202no—
496C2×C5—42no—
497C2×C5#231711yescharacteristic
498C2×C5C2×#42711yescharacteristic
499C2×C5C2×#42811yescharacteristic
500C2×C5#320711yescharacteristic
501C2×C5#321011yescharacteristic
502C2×C5#322811yescharacteristic
503C2×C5#323111yescharacteristic
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23D5×C4⋊C412yes—
511C23C2×D4×D511yescharacteristic
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
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—52no—
531C2×C4—52no—
532C2×C4—52no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—52no—
537C2×C4—52no—
538C2×C4—108no—
539C2×C4—108no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
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—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4D5×C4⋊C412yes—
557C2×C4C22×C4×D511yescharacteristic
558C2×C4#414411yescharacteristic
559C2×C4#414411yescharacteristic
560C2×C4C2×Q8×D511yescharacteristic
561C2×C4#415011yescharacteristic
562C2×C4#415011yescharacteristic
563C8—22no—
564C8—22no—
565C8—102no—
566C8—102no—
567C5#2198411yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×D5×C4⋊C411yescharacteristic
576C22#6592611yescharacteristic
577C22#6598211yescharacteristic
578C22D5×#13511yescharacteristic
579C22D5×#13511yescharacteristic
580C22D5×AGL(1,ℤ/8)11yescharacteristic
581C22D5×#13711yescharacteristic
582C4—22no—
583C4—54no—
584C4—54no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—102no—
590C4—102no—
591C4—401no—
592C4—401no—
593C4—401no—
594C4—401no—
595C4#6588011yescharacteristic
596C4#6588011yescharacteristic
597C4#6595711yescharacteristic
598C4#6595711yescharacteristic
599C2—42no—
600C2#1060611311yescharacteristic
601C2D5×#42711yescharacteristic
602C2D5×#42811yescharacteristic
603C2#1061180711yescharacteristic
604C2#1061195111yescharacteristic
605C2#1061308711yescharacteristic
606C2#1061323111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1854C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenter#1854D5
hypocenterC5#21984
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
6. #10601409C2semidirect
7. #10601399C2semidirect
8. #10601357C2semidirect
9. #10601347C2semidirect
12. #10599640C2semidirect
13. #10599640C2semidirect
14. #10599591C2semidirect
15. #10599590C2semidirect
49. #65007C4semidirect
567. C5#21984semidirect

Maximal subgroups

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

subgroupindex
#106139552
C2×#650072
#106055952
C5×#18542
#106023692
#106014092
#106013992
#106013572
#106013472
#106007222
#106002992
#105996402
#105995912
#105995902
#219845