Finite Group Databasebeta

← #12989736

Subgroups — #12989736

Order 1280.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3550C211yesmaximal, characteristic
2#10618159C211yesmaximal, characteristic, complemented
3#10615450C212yesmaximal, complemented
4#10615448C212yesmaximal, complemented
5#10615440C212yesmaximal, complemented
6Q8×#689C212yesmaximal, complemented
7#10615409C214yesmaximal, complemented
8#10615289C214yesmaximal, complemented
9#10615198C214yesmaximal, complemented
10#10615185C214yesmaximal, complemented
11#10615099C211yesmaximal, characteristic, complemented
12#10615062C211yesmaximal, characteristic, complemented
13#10615058C211yesmaximal, characteristic, complemented
14#10615047C211yesmaximal, characteristic
15#10615003C211yesmaximal, characteristic
16C5×#563—22no—
17C5×#559C2212yes—
18C5×Q82C2212yes—
19C5×#556—21no—
20C5×#556C2212yes—
21C5×D4×Q8—21no—
22C5×D4×Q8C2212yes—
23C5×#542—22no—
24C5×#542C2214yes—
25C5×#536—22no—
26C5×#533C2211yescharacteristic
27C5×#533C2211yescharacteristic
28C5×#532C2211yescharacteristic
29#66056—21no—
30#66055—21no—
31#66019—22no—
32#65957—22no—
33#65956—22no—
34#65901C2214yes—
35#65901C2212yes—
36Q8×Dic10C2212yes—
37#65853C2212yes—
38#65850C2212yes—
39#65804C2211yescharacteristic
40C2×C4×Dic10C2211yescharacteristic
41#65383C2214yes—
42#65380C2212yes—
43#65380C2212yes—
44#65380C2214yes—
45#65379C2212yes—
46#65371C2212yes—
47#65369C2212yes—
48#65369C2214yes—
49#65368C2211yescharacteristic
50#65367C2211yescharacteristic
51#65367C2214yes—
52#65364C2212yes—
53#65359C2211yescharacteristic
54#65359C2211yescharacteristic
55#65345C2214yes—
56#65344C2214yes—
57#65343C2214yes—
58#65343C2214yes—
59#65339C2212yes—
60#65334C2214yes—
61#65333C2212yes—
62#65329C2212yes—
63#65324C2214yes—
64#65322C2212yes—
65#65319C2214yes—
66C4×#689C2212yes—
67C4×#689C2212yes—
68C4×#689C2214yes—
69#65313C2214yes—
70#65313C2212yes—
71Q8×#168C2212yes—
72#65311C2214yes—
73#65310C2212yes—
74#65310C2214yes—
75#65309C2212yes—
76#65307C2214yes—
77#65307C2214yes—
78C4×#687C2214yes—
79#65298C2214yes—
80#65298C2214yes—
81#65296C2214yes—
82#65289C2214yes—
83#65215C2211yescharacteristic
84#65214C2211yescharacteristic
85#61854—51nomaximal, Sylow
86C2×C5×#41—21no—
87C22×C5×Q8—22no—
88C22×Dic10C2311yescharacteristic
89C5×C4⋊Q8—24no—
90C5×C4⋊Q8—22no—
91C5×C4⋊Q8D411yescharacteristic
92C5×C4⋊Q8D411yescharacteristic
93C5×C4⋊Q8C2311yescharacteristic
94C5×C4⋊Q8C2311yescharacteristic
95C5×C4⋊Q8C2314yes—
96C5×C4⋊Q8C2311yescharacteristic
97C5×#127C2311yescharacteristic
98C5×#126—24no—
99C5×#126—24no—
100C5×#125—21no—
101C5×#125D412yes—
102C5×#124—22no—
103C5×#124—24no—
104C5×#124—21no—
105C5×#124C2314yes—
106C5×#123—24no—
107C5×#123—24no—
108C5×C22⋊Q8—22no—
109C5×C22⋊Q8—21no—
110C5×C22⋊Q8—22no—
111C5×C22⋊Q8—24no—
112C5×C22⋊Q8—24no—
113C5×C22⋊Q8D412yes—
114C5×C22⋊Q8D412yes—
115C5×C22⋊Q8C2312yes—
116C5×#121—21no—
117C5×#121C2312yes—
118C4×C5×Q8—22no—
119C4×C5×Q8—21no—
120C4×C5×Q8C2312yes—
121C4×C5×Q8C2314yes—
122C4×C5×Q8C2312yes—
123C4×C5×Q8C2312yes—
124C4×C5×D4—22no—
125C4×C5×D4—21no—
126C4×C5×D4C2314yes—
127C4×C5×D4C2312yes—
128C5×#117—21no—
129C5×#117—22no—
130C5×#117C2312yes—
131C2×C42×C5C2311yescharacteristic
132Q8×Dic5—22no—
133#4090—22no—
134#4090—24no—
135C2×#689C2312yes—
136C2×#689C2312yes—
137C2×#687C2314yes—
138#4072C2312yes—
139C2×C4⋊Dic5C2311yescharacteristic
140#4070—21no—
141#4070—21no—
142C2×#683—21no—
143C2×C4×Dic5—21no—
144#4036—22no—
145#4035—24no—
146#4035—24no—
147#4035—24no—
148#4034—22no—
149#4034—22no—
150#4034—24no—
151#4033—22no—
152#4033—22no—
153#4033—22no—
154#4033—24no—
155#4024—24no—
156#4023—24no—
157#4016C2312yes—
158#4016C2314yes—
159#4015C2311yescharacteristic
160#4015C2311yescharacteristic
161#4015C2314yes—
162C4×Dic10—21no—
163C4×Dic10—21no—
164C4×Dic10—21no—
165C4×Dic10—21no—
166C4×Dic10C2312yes—
167C4×Dic10C2312yes—
168C4×Dic10C2314yes—
169C4×Dic10C2314yes—
170#3967C2314yes—
171#3967C2312yes—
172#3967C2314yes—
173#3967C2312yes—
174#3964C2314yes—
175#3964C2314yes—
176#3964C2314yes—
177C22⋊#168C2312yes—
178C22⋊#168C2312yes—
179#3942C2312yes—
180#3942C2312yes—
181#3942C2314yes—
182#3940C2314yes—
183#3940C2314yes—
184#3939C2312yes—
185#3939C2314yes—
186#3939C2312yes—
187#3939C2314yes—
188#3939C2314yes—
189C4⋊#168C2314yes—
190C4⋊#168C2314yes—
191C4⋊#168C2312yes—
192C4⋊#168C2312yes—
193#3935C2314yes—
194#3935C2312yes—
195C4×#168C2314yes—
196C4×#168C2311yescharacteristic
197C4×#168C2311yescharacteristic
198#3550D511yescharacteristic
199#3502—51no—
200#3387—52no—
201#3386—54no—
202Q8×Q16—52no—
203#3377—54no—
204#3375—52no—
205#3365—52no—
206#3243—54no—
207#3237—51no—
208#3228—51no—
209#3121—54no—
210#3112—51no—
211#3104—51no—
212#2985—51no—
213C5×#41—82no—
214C2×C5×Q8—21no—
215C2×C5×Q8—24no—
216C2×C5×Q8—21no—
217C2×C5×Q8—84no—
218C2×C5×Q8C2×D412yes—
219C2×C5×Q8C2×D412yes—
220C2×C5×Q8C2412yes—
221C2×C5×D4—22no—
222C2×C5×D4—21no—
223C2×C5×D4—21no—
224C2×C5×D4C2412yes—
225C22×C4×C5—22no—
226C22×C4×C5—24no—
227C22×C4×C5C2×D411yescharacteristic
228C22×C4×C5C2×D411yescharacteristic
229C22×C4×C5C2411yescharacteristic
230C22×Dic5—21no—
231C22×Dic5—21no—
232C2×Dic10—21no—
233C2×Dic10—24no—
234C2×Dic10—21no—
235C2×Dic10—21no—
236C2×Dic10—24no—
237C2×Dic10C2412yes—
238C5×C4⋊C4—21no—
239C5×C4⋊C4—21no—
240C5×C4⋊C4—21no—
241C5×C4⋊C4—21no—
242C5×C4⋊C4—22no—
243C5×C4⋊C4—24no—
244C5×C4⋊C4—24no—
245C5×C4⋊C4—22no—
246C5×C4⋊C4C2×D412yes—
247C5×C4⋊C4C2×D412yes—
248C5×C4⋊C4C2×D412yes—
249C5×C4⋊C4C2×D412yes—
250C5×C4⋊C4C2×D414yes—
251C5×C4⋊C4C2×D414yes—
252C5×C4⋊C4C2412yes—
253C5×C4⋊C4C2412yes—
254C5×C22⋊C4—22no—
255C5×C22⋊C4—24no—
256C5×C22⋊C4—22no—
257C5×C22⋊C4—24no—
258C5×C22⋊C4—22no—
259C5×C22⋊C4—24no—
260C5×C22⋊C4C2×D414yes—
261C42×C5—22no—
262C42×C5—24no—
263C42×C5C2×D411yescharacteristic
264C42×C5C2×D411yescharacteristic
265C42×C5C2411yescharacteristic
266C42×C5C2411yescharacteristic
267C42×C5C2414yes—
268C22⋊Dic5—22no—
269#689—24no—
270#689—24no—
271#687—28no—
272C4⋊Dic5—21no—
273C4⋊Dic5—21no—
274C4⋊Dic5—21no—
275C4⋊Dic5—24no—
276C4⋊Dic5C2412yes—
277C4⋊Dic5C2412yes—
278C4⋊Dic5C2412yes—
279#683—22no—
280#683—22no—
281#683—22no—
282#683—22no—
283#683—24no—
284#683—24no—
285#683—24no—
286#683—24no—
287C4×Dic5—22no—
288C4×Dic5—22no—
289C4×Dic5—22no—
290C4×Dic5—24no—
291C4×Dic5—24no—
292C2×#168C2412yes—
293C2×#168C2414yes—
294C2×#168C2412yes—
295#563—22no—
296#559C2×D512yes—
297Q82—52no—
298Q82—101no—
299Q82C2×D512yes—
300#557—52no—
301#557—54no—
302#557—101no—
303#557—102no—
304#556—21no—
305#556C2×D512yes—
306D4×Q8—21no—
307D4×Q8C2×D512yes—
308#544—52no—
309#544—102no—
310#542—22no—
311#542C2×D514yes—
312#536—22no—
313#533—51no—
314#533—102no—
315#533C2×D511yescharacteristic
316#533C2×D511yescharacteristic
317#532C2×D511yescharacteristic
318#519—52no—
319C2×C4×Q8—51no—
320#501—52no—
321#501—54no—
322#500—51no—
323#500—54no—
324#498—51no—
325#491—52no—
326#489—54no—
327#488—54no—
328#486—51no—
329#486—51no—
330#485—54no—
331#484—52no—
332#482—54no—
333#480—52no—
334#479—54no—
335#478—54no—
336#478—54no—
337#477—52no—
338#477—52no—
339#477—54no—
340#477—52no—
341#477—54no—
342#475—54no—
343#475—52no—
344#474—54no—
345#464—54no—
346#464—54no—
347#462—52no—
348#460—52no—
349#451—52no—
350#448—52no—
351#446—52no—
352#446—54no—
353C8×Q8—52no—
354#441—54no—
355#440—54no—
356#440—54no—
357C4×Q16—52no—
358C4×Q16—52no—
359C4×Q16—54no—
360C4×SD16—54no—
361#433—54no—
362#432—51no—
363#431—51no—
364C23×C5—22no—
365C23×C5C22×D411yescharacteristic
366C5×Q8—24no—
367C5×Q8—22no—
368C5×Q8—22no—
369C5×Q8—42no—
370C5×Q8—42no—
371C5×Q8—84no—
372C5×Q8—81no—
373C5×D4—24no—
374C5×D4—42no—
375C5×D4—81no—
376C5×D4—82no—
377C2×C4×C5—22no—
378C2×C4×C5—22no—
379C2×C4×C5—22no—
380C2×C4×C5—22no—
381C2×C4×C5—22no—
382C2×C4×C5—41no—
383C2×C4×C5—44no—
384C2×C4×C5—84no—
385C2×C4×C5C22×D411yescharacteristic
386C2×C4×C5C22×D411yescharacteristic
387C2×C4×C5C22×D414yes—
388C2×C4×C5C22×D414yes—
389C2×C4×C5C22×D411yescharacteristic
390C2×C4×C5C22×D411yescharacteristic
391C2×C4×C5C22×D411yescharacteristic
392C2×C4×C5C2511yescharacteristic
393C2×Dic5—24no—
394C2×Dic5—22no—
395C2×Dic5—22no—
396C2×Dic5—22no—
397C2×Dic5—22no—
398C2×Dic5—81no—
399Dic10—24no—
400Dic10—22no—
401Dic10—42no—
402Dic10—81no—
403Dic10—84no—
404Dic102−1+414yes—
405#168—24no—
406#168—22no—
407#168—22no—
408C2×#41—21no—
409C22×Q8—22no—
410C22×Q8—51no—
411C2×Q16—52no—
412C2×Q16—52no—
413C2×SD16—54no—
414C4⋊Q8—24no—
415C4⋊Q8—22no—
416C4⋊Q8—51no—
417C4⋊Q8—51no—
418C4⋊Q8—54no—
419C4⋊Q8—102no—
420C4⋊Q8—104no—
421C4⋊Q8—102no—
422C4⋊Q8—102no—
423C4⋊Q8—104no—
424C4⋊Q8C22⋊D511yescharacteristic
425C4⋊Q8C22⋊D511yescharacteristic
426C4⋊Q8C22×D511yescharacteristic
427C4⋊Q8C22×D511yescharacteristic
428C4⋊Q8C22×D511yescharacteristic
429C4⋊Q8C22×D514yes—
430#127C22×D511yescharacteristic
431#126—24no—
432#126—24no—
433#125—21no—
434#125—52no—
435#125—54no—
436#125—104no—
437#125—102no—
438#125—104no—
439#125—104no—
440#125C22⋊D512yes—
441#124—22no—
442#124—24no—
443#124—21no—
444#124C22×D514yes—
445#123—24no—
446#123—24no—
447C22⋊Q8—22no—
448C22⋊Q8—21no—
449C22⋊Q8—22no—
450C22⋊Q8—24no—
451C22⋊Q8—24no—
452C22⋊Q8—101no—
453C22⋊Q8—101no—
454C22⋊Q8—104no—
455C22⋊Q8C22⋊D512yes—
456C22⋊Q8C22⋊D512yes—
457C22⋊Q8C22×D512yes—
458#121—21no—
459#121C22×D512yes—
460C4×Q8—22no—
461C4×Q8—21no—
462C4×Q8—52no—
463C4×Q8—52no—
464C4×Q8—54no—
465C4×Q8—54no—
466C4×Q8—102no—
467C4×Q8—102no—
468C4×Q8—102no—
469C4×Q8—104no—
470C4×Q8—102no—
471C4×Q8—101no—
472C4×Q8—101no—
473C4×Q8—101no—
474C4×Q8—101no—
475C4×Q8C22×D512yes—
476C4×Q8C22×D514yes—
477C4×Q8C22×D512yes—
478C4×Q8C22×D512yes—
479C4×D4—22no—
480C4×D4—21no—
481C4×D4C22×D512yes—
482C4×D4C22×D514yes—
483#117—21no—
484#117—22no—
485#117—52no—
486#117—104no—
487#117C22×D512yes—
488C2×C4⋊C4—51no—
489C2×C4⋊C4—101no—
490C2×C42—101no—
491C2×C42C22×D511yescharacteristic
492#107—52no—
493#107—54no—
494#107—52no—
495#107—54no—
496#107—54no—
497#106—54no—
498#106—54no—
499C4⋊C8—52no—
500C4⋊C8—52no—
501C4⋊C8—54no—
502C4⋊C8—54no—
503#103—52no—
504#103—52no—
505#103—54no—
506#103—54no—
507#103—52no—
508#103—52no—
509#103—54no—
510#102—54no—
511#102—54no—
512#102—54no—
513C22⋊C8—52no—
514C22⋊C8—52no—
515#97—52no—
516#97—54no—
517C4×C8—51no—
518C4×C8—51no—
519C4×C8—54no—
520C22×C5—21no—
521C22×C5—21no—
522C22×C5—21no—
523C22×C5—42no—
524C22×C5—42no—
525C22×C5—42no—
526C22×C5C23×D411yescharacteristic
527C4×C5—21no—
528C4×C5—24no—
529C4×C5—21no—
530C4×C5—21no—
531C4×C5—21no—
532C4×C5—24no—
533C4×C5—42no—
534C4×C5—42no—
535C4×C5—42no—
536C4×C5#57812yes—
537C4×C5C2×2−1+412yes—
538Dic5—24no—
539Dic5—42no—
540Dic5—42no—
541Dic5—42no—
542#41—82no—
543C2×Q8—21no—
544C2×Q8—24no—
545C2×Q8—21no—
546C2×Q8—52no—
547C2×Q8—84no—
548C2×Q8—104no—
549C2×Q8—101no—
550C2×Q8—101no—
551C2×Q8—101no—
552C2×Q8—104no—
553C2×Q8C2×C22⋊D512yes—
554C2×Q8C2×C22⋊D512yes—
555C2×Q8C23×D512yes—
556C2×D4—22no—
557C2×D4—21no—
558C2×D4—21no—
559C2×D4C23×D512yes—
560C22×C4—22no—
561C22×C4—24no—
562C22×C4—101no—
563C22×C4—101no—
564C22×C4C2×C22⋊D511yescharacteristic
565C22×C4C2×C22⋊D511yescharacteristic
566C22×C4C23×D511yescharacteristic
567Q16—104no—
568Q16—104no—
569SD16—108no—
570C2×C8—52no—
571C2×C8—52no—
572C2×C8—54no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—22no—
578C4⋊C4—24no—
579C4⋊C4—24no—
580C4⋊C4—22no—
581C4⋊C4—52no—
582C4⋊C4—52no—
583C4⋊C4—52no—
584C4⋊C4—101no—
585C4⋊C4—102no—
586C4⋊C4—102no—
587C4⋊C4—104no—
588C4⋊C4—104no—
589C4⋊C4—102no—
590C4⋊C4—101no—
591C4⋊C4—104no—
592C4⋊C4—102no—
593C4⋊C4—104no—
594C4⋊C4—101no—
595C4⋊C4—104no—
596C4⋊C4C2×C22⋊D512yes—
597C4⋊C4C2×C22⋊D512yes—
598C4⋊C4C2×C22⋊D512yes—
599C4⋊C4C2×C22⋊D512yes—
600C4⋊C4C2×C22⋊D514yes—
601C4⋊C4C2×C22⋊D514yes—
602C4⋊C4C23×D512yes—
603C4⋊C4C23×D512yes—
604C22⋊C4—22no—
605C22⋊C4—24no—
606C22⋊C4—22no—
607C22⋊C4—24no—
608C22⋊C4—22no—
609C22⋊C4—24no—
610C22⋊C4—102no—
611C22⋊C4C2×C22⋊D514yes—
612C42—22no—
613C42—24no—
614C42—102no—
615C42—102no—
616C42—102no—
617C42—104no—
618C42—104no—
619C42C2×C22⋊D511yescharacteristic
620C42C2×C22⋊D511yescharacteristic
621C42C23×D511yescharacteristic
622C42C23×D514yes—
623C42C23×D511yescharacteristic
624C2×C5—41no—
625C2×C5—82no—
626C2×C5#348611yescharacteristic
627C2×C5C2×#57811yescharacteristic
628C2×C5#358511yescharacteristic
629C23—22no—
630C23C22×C22⋊D511yescharacteristic
631Q8—24no—
632Q8—22no—
633Q8—22no—
634Q8—42no—
635Q8—42no—
636Q8—54no—
637Q8—84no—
638Q8—81no—
639Q8—104no—
640Q8—102no—
641Q8—202no—
642Q8—401no—
643Q8—404no—
644D4—24no—
645D4—42no—
646D4—81no—
647D4—82no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—41no—
654C2×C4—44no—
655C2×C4—84no—
656C2×C4—104no—
657C2×C4—102no—
658C2×C4—102no—
659C2×C4—102no—
660C2×C4—102no—
661C2×C4—401no—
662C2×C4C22×C22⋊D514yes—
663C2×C4C22×C22⋊D511yescharacteristic
664C2×C4C22×C22⋊D511yescharacteristic
665C2×C4C22×C22⋊D514yes—
666C2×C4C22×C22⋊D511yescharacteristic
667C2×C4C22×C22⋊D511yescharacteristic
668C2×C4C22×C22⋊D511yescharacteristic
669C2×C4C24×D511yescharacteristic
670C8—104no—
671C8—102no—
672C8—102no—
673C5#6185411yescharacteristic, complemented, Sylow
674C22—21no—
675C22—21no—
676C22—21no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22C23×C22⋊D511yescharacteristic
681C4—21no—
682C4—21no—
683C4—24no—
684C4—21no—
685C4—24no—
686C4—21no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—104no—
691C4—202no—
692C4—202no—
693C4—202no—
694C4#6617212yes—
695C4D5×2−1+412yes—
696C2—41no—
697C2—82no—
698C2#1061917811yescharacteristic
699C2C2×#6617211yescharacteristic
700C2#1061919511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3550C2
socleC22×C5C23×D4
prime radicalC5×#532C22
hypercenter#3550D5
hypocenterC5#61854
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #533 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #10618159C2semidirect
3. #10615450C2semidirect
4. #10615448C2semidirect
5. #10615440C2semidirect
6. Q8×#689C2semidirect
7. #10615409C2semidirect
8. #10615289C2semidirect
9. #10615198C2semidirect
10. #10615185C2semidirect
11. #10615099C2semidirect
12. #10615062C2semidirect
13. #10615058C2semidirect
673. C5#61854semidirect

Maximal subgroups

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

subgroupindex
C5×#35502
#106181592
#106154502
#106154482
#106154402
Q8×#6892
#106154092
#106152892
#106151982
#106151852
#106150992
#106150622
#106150582
#106150472
#106150032
#618545