Finite Group Databasebeta

← #12099158

Subgroups — #12099158

Order 1280.

1276 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617632C211yesmaximal, characteristic
2C2×#65295C211yesmaximal, characteristic
3C2×#65295C211yesmaximal, characteristic
4C5×#1740C211yesmaximal, characteristic, complemented
5#10602386C211yesmaximal, characteristic, complemented
6#10602386C211yesmaximal, characteristic, complemented
7#10602385C211yesmaximal, characteristic, complemented
8#10599753C218yesmaximal, complemented
9C22×#709C2211yescharacteristic
10C2×C5×#117C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C2×#4072C2211yescharacteristic
15C2×#4072C2211yescharacteristic
16#65857—22no—
17#65856—22no—
18#65855C412yescomplemented
19D5×#117C414yescomplemented
20#65850C412yescomplemented
21#65838—22no—
22#65820C2214yes—
23C4×#709—22no—
24#65295—22no—
25#65295—22no—
26#65295—22no—
27#65295—22no—
28C2×#3938C2214yes—
29C2×#3938C2214yes—
30C5×#342C2214yes—
31#64747C2214yes—
32#64747C2214yes—
33#64746C2214yes—
34#20710—51nomaximal, Sylow
35C23×C4×C5C2311yescharacteristic
36C22×C22⋊D5C2312yes—
37C2×#709—21no—
38C2×#709—21no—
39C2×#709—21no—
40C2×#709—21no—
41C2×#709—21no—
42C2×#709—21no—
43C2×#709—21no—
44C2×#709—21no—
45C2×#709D412yes—
46C2×#709D412yes—
47C2×#709D412yes—
48C2×#709D412yes—
49C22×D20C2×C411yescharacteristic
50C22×C4×D5C2×C412yes—
51C22×Dic10C2×C411yescharacteristic
52C5×#117—21no—
53C5×#117—21no—
54C5×#117—21no—
55C5×#117—21no—
56C5×#117—21no—
57C5×#117—21no—
58C5×#117—21no—
59C5×#117—21no—
60C5×#117—21no—
61C5×#117—21no—
62C5×#117C2×C412yes—
63C5×#117C2×C412yes—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C22⋊C4C2312yes—
68C2×C5×C22⋊C4C2312yes—
69C2×C5×C22⋊C4C2312yes—
70C2×C42×C5—21no—
71C2×C42×C5—21no—
72C2×C42×C5—21no—
73C2×C22⋊Dic5C2312yes—
74C2×C22⋊Dic5C2312yes—
75C2×C22⋊Dic5C2312yes—
76C4×C22⋊D5—28no—
77#4072—21no—
78#4072—21no—
79#4072—21no—
80#4072—21no—
81#4072—21no—
82#4072—21no—
83#4072—21no—
84#4072—21no—
85#4072—21no—
86#4072—21no—
87#4072C2×C412yes—
88#4072C2×C412yes—
89C2×C4⋊Dic5—21no—
90C2×C4⋊Dic5—21no—
91C2×C4⋊Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5—21no—
94C2×C4×Dic5—21no—
95#4039—24no—
96#4038—24no—
97D5×C4⋊C4—24no—
98#4033—24no—
99#4027—28no—
100D5×C22⋊C4C2×C418yes—
101#4023C2×C418yes—
102C4×D20—24no—
103#4018—24no—
104C42×D5—24no—
105C4×Dic10—24no—
106#3938—28no—
107#3938—28no—
108#3428—51no—
109C2×#410—51no—
110C2×#410—51no—
111#1828—58no—
112#1740—51no—
113#1740—51no—
114#1740—51no—
115#1740D511yescharacteristic, complemented
116C24×C5C2411yescharacteristic
117C23×D5C22×C412yes—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5C22×C411yescharacteristic
122C22×C4×C5C22×C411yescharacteristic
123C22×C4×C5C2×D411yescharacteristic
124C22×C4×C5C2×D411yescharacteristic
125C22×C4×C5C2×D411yescharacteristic
126C22×C4×C5C2×D411yescharacteristic
127C22×C4×C5#4111yescharacteristic
128C22×C4×C5#4111yescharacteristic
129C22×C4×C5#4111yescharacteristic
130C22×C4×C5#4111yescharacteristic
131C22×C4×C5#4111yescharacteristic
132C22×C4×C5#4111yescharacteristic
133C22×C4×C5#4111yescharacteristic
134C22×C4×C5#4111yescharacteristic
135C2×C22⋊D5—22no—
136C2×C22⋊D5—22no—
137C2×C22⋊D5—22no—
138C2×C22⋊D5—22no—
139C2×C22⋊D5C2×D414yes—
140C2×C22⋊D5C2×D414yes—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5C22×C412yes—
145#709—22no—
146#709—22no—
147#709—22no—
148#709—22no—
149#709—22no—
150#709—22no—
151#709—22no—
152#709—22no—
153#709—22no—
154#709—22no—
155#709—22no—
156#709—22no—
157#709—22no—
158#709—22no—
159#709—22no—
160#709—22no—
161C2×D20—21no—
162C2×D20—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20C22⋊C412yes—
166C2×D20C22⋊C412yes—
167C2×C4×D5—22no—
168C2×C4×D5—22no—
169C2×C4×D5—22no—
170C2×C4×D5—22no—
171C2×C4×D5—28no—
172C2×C4×D5C22⋊C414yes—
173C2×C4×D5C22⋊C414yes—
174C2×Dic10—21no—
175C2×Dic10—21no—
176C2×Dic10—21no—
177C2×Dic10—21no—
178C2×Dic10C22⋊C412yes—
179C2×Dic10C22⋊C412yes—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—41no—
183C5×C4⋊C4—41no—
184C5×C4⋊C4—41no—
185C5×C4⋊C4—41no—
186C5×C22⋊C4—22no—
187C5×C22⋊C4—22no—
188C5×C22⋊C4—22no—
189C5×C22⋊C4—22no—
190C5×C22⋊C4—22no—
191C5×C22⋊C4C22×C414yes—
192C42×C5—22no—
193C42×C5—22no—
194C42×C5—41no—
195C42×C5—41no—
196C42×C5—41no—
197C42×C5—41no—
198C22⋊Dic5—22no—
199C22⋊Dic5—22no—
200C22⋊Dic5—22no—
201C22⋊Dic5—22no—
202C22⋊Dic5—22no—
203C22⋊Dic5C22×C414yes—
204#685—28no—
205C4⋊Dic5—22no—
206C4⋊Dic5—22no—
207C4⋊Dic5—41no—
208C4⋊Dic5—41no—
209C4⋊Dic5—41no—
210C4⋊Dic5—41no—
211#683—28no—
212C4×Dic5—22no—
213C4×Dic5—22no—
214C4×Dic5—28no—
215C4×Dic5—41no—
216C4×Dic5—41no—
217C4×Dic5—41no—
218C4×Dic5—41no—
219C22×#41—51no—
220#520—102no—
221#520—102no—
222#519—52no—
223#518—52no—
224#518—54no—
225C4×#41—102no—
226C4×#41—102no—
227C2×#117—51no—
228C2×#117—54no—
229C2×#117—51no—
230C2×#117—51no—
231C2×#117C2×D511yescharacteristic
232C2×#117C2×D511yescharacteristic
233C2×#117C2×D511yescharacteristic
234#410—102no—
235#410—102no—
236#410—102no—
237#410—102no—
238C2×#99—54no—
239C2×#99—54no—
240#342—54no—
241#342—54no—
242#342—54no—
243#342C2×D514yes—
244C23×C5—81no—
245C23×C5C23×C411yescharacteristic
246C23×C5C22×D411yescharacteristic
247C23×C5C22×D411yescharacteristic
248C23×C5C2×#4111yescharacteristic
249C23×C5C2×#4111yescharacteristic
250C23×C5C2×#4111yescharacteristic
251C23×C5C2×#4111yescharacteristic
252C22×D5—22no—
253C22×D5—22no—
254C22×D5—82no—
255C22×D5C2×C22⋊C414yes—
256C2×C4×C5—24no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—82no—
260C2×C4×C5—82no—
261C2×C4×C5—82no—
262C2×C4×C5C2×C22⋊C411yescharacteristic
263C2×C4×C5C2×C22⋊C411yescharacteristic
264C2×C4×C5C2×C22⋊C411yescharacteristic
265C2×C4×C5C2×C22⋊C411yescharacteristic
266C2×C4×C5#11711yescharacteristic
267C2×C4×C5#11711yescharacteristic
268C2×C4×C5#11711yescharacteristic
269C2×C4×C5#11711yescharacteristic
270C2×C4×C5#11711yescharacteristic
271C2×C4×C5#11711yescharacteristic
272C2×C4×C5#11711yescharacteristic
273C2×C4×C5#11711yescharacteristic
274C2×C4×C5#12311yescharacteristic
275C2×C4×C5#12311yescharacteristic
276C2×C4×C5#12311yescharacteristic
277C2×C4×C5#12311yescharacteristic
278C2×C4×C5#12311yescharacteristic
279C2×C4×C5#12311yescharacteristic
280C2×C4×C5#12311yescharacteristic
281C2×C4×C5#12311yescharacteristic
282C2×C4×C5#12311yescharacteristic
283C2×C4×C5#12311yescharacteristic
284C2×C4×C5#12311yescharacteristic
285C2×C4×C5#12311yescharacteristic
286C2×C4×C5#12311yescharacteristic
287C2×C4×C5#12311yescharacteristic
288C2×C4×C5#12311yescharacteristic
289C2×C4×C5#12311yescharacteristic
290C22⋊D5—24no—
291C22⋊D5—24no—
292C22⋊D5—24no—
293C22⋊D5—24no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—24no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—82no—
300C2×Dic5—82no—
301C2×Dic5—82no—
302C2×Dic5C2×C22⋊C414yes—
303D20—22no—
304D20—22no—
305D20—22no—
306D20—22no—
307C4×D5—24no—
308C4×D5—24no—
309C4×D5—24no—
310C4×D5—24no—
311C4×D5—88no—
312Dic10—22no—
313Dic10—22no—
314Dic10—22no—
315Dic10—22no—
316C2×#41—52no—
317C2×#41—52no—
318C2×#41—52no—
319C2×#41—52no—
320C2×#41—101no—
321C2×#41—101no—
322C2×#41—101no—
323C2×#41—101no—
324C2×#41—101no—
325C2×#41—101no—
326C2×#41—101no—
327C2×#41—101no—
328C22×Q8—51no—
329C22×D4—52no—
330C22×D4—51no—
331C23×C4—52no—
332C23×C4C22×D511yescharacteristic
333C4×Q8—104no—
334C4×Q8—104no—
335C4×D4—104no—
336C4×D4—108no—
337C4×D4—104no—
338C4×D4—108no—
339#117—21no—
340#117—21no—
341#117—21no—
342#117—21no—
343#117—21no—
344#117—21no—
345#117—21no—
346#117—21no—
347#117—21no—
348#117—21no—
349#117—52no—
350#117—52no—
351#117—58no—
352#117—104no—
353#117—104no—
354#117—101no—
355#117—101no—
356#117—101no—
357#117—101no—
358#117—101no—
359#117—101no—
360#117—101no—
361#117—101no—
362#117—101no—
363#117—101no—
364#117C4×D512yes—
365#117C4×D512yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—104no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—101no—
373C2×C22⋊C4—52no—
374C2×C22⋊C4—58no—
375C2×C22⋊C4—52no—
376C2×C22⋊C4—52no—
377C2×C22⋊C4C22×D512yes—
378C2×C22⋊C4C22×D512yes—
379C2×C22⋊C4C22×D512yes—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—104no—
384C2×C42—101no—
385C2×C42—101no—
386C2×C42—101no—
387#99—108no—
388#99—108no—
389C22×C5—41no—
390C22×C5—41no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5—41no—
394C22×C5—41no—
395C22×C5—41no—
396C22×C5C22×C22⋊C411yescharacteristic
397C22×C5C2×#11711yescharacteristic
398C22×C5C2×#11711yescharacteristic
399C22×C5C2×#12311yescharacteristic
400C22×C5C2×#12311yescharacteristic
401C22×C5C2×#12311yescharacteristic
402C22×C5C2×#12311yescharacteristic
403C2×D5—24no—
404C2×D5—44no—
405C2×D5—82no—
406C2×D5—82no—
407C4×C5—44no—
408C4×C5—82no—
409C4×C5—82no—
410C4×C5#38111yescharacteristic
411C4×C5#38111yescharacteristic
412C4×C5#38111yescharacteristic
413C4×C5#38111yescharacteristic
414C4×C5#38111yescharacteristic
415C4×C5#38111yescharacteristic
416C4×C5#38111yescharacteristic
417C4×C5#38111yescharacteristic
418Dic5—24no—
419Dic5—44no—
420Dic5—82no—
421Dic5—82no—
422C24—52no—
423C24C23×D511yescharacteristic
424#41—102no—
425#41—102no—
426#41—102no—
427#41—102no—
428#41—102no—
429#41—102no—
430#41—102no—
431#41—102no—
432#41—102no—
433#41—102no—
434#41—102no—
435#41—102no—
436#41—102no—
437#41—102no—
438#41—102no—
439#41—102no—
440C2×Q8—52no—
441C2×Q8—52no—
442C2×Q8—101no—
443C2×Q8—101no—
444C2×Q8—101no—
445C2×Q8—101no—
446C2×D4—54no—
447C2×D4—54no—
448C2×D4—52no—
449C2×D4—52no—
450C2×D4—102no—
451C2×D4—102no—
452C2×D4—102no—
453C2×D4—102no—
454C2×D4—101no—
455C2×D4—101no—
456C2×D4—101no—
457C2×D4—101no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—52no—
462C22×C4—54no—
463C22×C4—54no—
464C22×C4—108no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4C2×C4×D511yescharacteristic
473C22×C4C2×C4×D511yescharacteristic
474C22×C4#70911yescharacteristic
475C22×C4#70911yescharacteristic
476C22×C4#70911yescharacteristic
477C22×C4#70911yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4D4×D511yescharacteristic
480C22×C4D4×D511yescharacteristic
481C22×C4D4×D511yescharacteristic
482C22×C4#71111yescharacteristic
483C22×C4#71111yescharacteristic
484C22×C4#71311yescharacteristic
485C22×C4#71311yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—108no—
493C4⋊C4—102no—
494C4⋊C4—102no—
495C4⋊C4—201no—
496C4⋊C4—201no—
497C4⋊C4—201no—
498C4⋊C4—201no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—54no—
505C22⋊C4—108no—
506C22⋊C4—102no—
507C22⋊C4—102no—
508C22⋊C4—102no—
509C22⋊C4—102no—
510C22⋊C4—102no—
511C22⋊C4C2×C4×D514yes—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—108no—
519C42—102no—
520C42—102no—
521C42—201no—
522C42—201no—
523C42—201no—
524C42—201no—
525C2×C5—21no—
526C2×C5—21no—
527C2×C5—21no—
528C2×C5—21no—
529C2×C5—21no—
530C2×C5—21no—
531C2×C5—21no—
532C2×C5C2×#38111yescharacteristic
533D5—44no—
534C23—54no—
535C23—81no—
536C23—102no—
537C23—102no—
538C23—402no—
539C23C22×C4×D511yescharacteristic
540C23C2×#70911yescharacteristic
541C23C2×#70911yescharacteristic
542C23C2×D4×D511yescharacteristic
543C23C2×D4×D511yescharacteristic
544C23C2×#71111yescharacteristic
545C23C2×#71311yescharacteristic
546Q8—102no—
547Q8—102no—
548Q8—102no—
549Q8—102no—
550D4—104no—
551D4—104no—
552D4—104no—
553D4—104no—
554D4—102no—
555D4—102no—
556D4—102no—
557D4—102no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—54no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—104no—
570C2×C4—104no—
571C2×C4—104no—
572C2×C4—202no—
573C2×C4—202no—
574C2×C4—408no—
575C2×C4—402no—
576C2×C4—402no—
577C2×C4—402no—
578C2×C4#401811yescharacteristic
579C2×C4#401811yescharacteristic
580C2×C4#401811yescharacteristic
581C2×C4#401811yescharacteristic
582C2×C4D5×C22⋊C411yescharacteristic
583C2×C4D5×C22⋊C411yescharacteristic
584C2×C4D5×C22⋊C411yescharacteristic
585C2×C4D5×C22⋊C411yescharacteristic
586C2×C4#402911yescharacteristic
587C2×C4#402911yescharacteristic
588C2×C4#402911yescharacteristic
589C2×C4#402911yescharacteristic
590C2×C4#402911yescharacteristic
591C2×C4#402911yescharacteristic
592C2×C4#402911yescharacteristic
593C2×C4#402911yescharacteristic
594C2×C4#403811yescharacteristic
595C2×C4#403811yescharacteristic
596C2×C4#403811yescharacteristic
597C2×C4#403811yescharacteristic
598C2×C4#404011yescharacteristic
599C2×C4#404011yescharacteristic
600C2×C4#404011yescharacteristic
601C2×C4#404011yescharacteristic
602C2×C4#404011yescharacteristic
603C2×C4#404011yescharacteristic
604C2×C4#404011yescharacteristic
605C2×C4#404011yescharacteristic
606C5#2071011yescharacteristic, complemented, Sylow
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—104no—
615C22—204no—
616C22—402no—
617C22—402no—
618C22C2×#401811yescharacteristic
619C22C2×D5×C22⋊C411yescharacteristic
620C22C2×#402911yescharacteristic
621C22C2×#402911yescharacteristic
622C22C2×#403811yescharacteristic
623C22C2×#404011yescharacteristic
624C22C2×#404011yescharacteristic
625C4—44no—
626C4—82no—
627C4—82no—
628C4—104no—
629C4—204no—
630C4—402no—
631C4—402no—
632C4#6495311yescharacteristic
633C4#6495311yescharacteristic
634C4#6495311yescharacteristic
635C4#6495311yescharacteristic
636C4#6495311yescharacteristic
637C4#6495311yescharacteristic
638C4#6495311yescharacteristic
639C4#6495311yescharacteristic
640C2—21no—
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—204no—
648C2C2×#6495311yescharacteristic
subgroupG/H
centerC4#64953
commutatorC23×C5C23×C4
FrattiniC24C23×D5
FittingC5×#1740C2
socleC2×C5C2×#381
prime radicalC22×#709C22
cosocleC24×C5C24
hypercenter#1740D5
hypocenterC5#20710
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C5×#1740C2semidirect
5. #10602386C2semidirect
6. #10602386C2semidirect
7. #10602385C2semidirect
8. #10599753C2semidirect
18. #65855C4semidirect
19. D5×#117C4semidirect
20. #65850C4semidirect
115. #1740D5semidirect
606. C5#20710semidirect

Maximal subgroups

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

subgroupindex
#106176322
C2×#652952
C5×#17402
#106023862
#106023852
#105997532
#207105