Finite Group Databasebeta

← #12984054

Subgroups — #12984054

Order 1280.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#547C211yesmaximal, characteristic, complemented
2#10617689C211yesmaximal, characteristic, complemented
3#10615811C214yesmaximal, complemented
4#10615261C214yesmaximal, complemented
5#10615255C214yesmaximal, complemented
6#10615242C214yesmaximal, complemented
7#10615241C214yesmaximal, complemented
8#10614953C211yesmaximal, characteristic, complemented
9#10614944C211yesmaximal, characteristic, complemented
10#10614468C214yesmaximal, complemented
11#10614045C211yesmaximal, characteristic, complemented
12#10614039C211yesmaximal, characteristic, complemented
13#10614032C211yesmaximal, characteristic, complemented
14C22×C5×#41C2212yes—
15C5×#547—22no—
16C5×#547—22no—
17C5×#547C2214yes—
18C5×#547C2214yes—
19C2×C5×C4⋊Q8C2211yescharacteristic
20C2×C5×#123—22no—
21C2×C5×C22⋊Q8C2212yes—
22C2×C5×#121—21no—
23C2×C4×C5×D4C2211yescharacteristic
24C2×C4×C5×D4C2211yescharacteristic
25C22×C5×C4⋊C4—21no—
26C2×#4096C2218yescomplemented
27#66139—21no—
28#66125C2214yes—
29#65897C2212yescomplemented
30#65893—22no—
31#65892—22no—
32#65887C2212yescomplemented
33#65885C2212yescomplemented
34#65876C2212yescomplemented
35#65838—21no—
36#65828—22no—
37#65811C2211yescharacteristic
38#65804C2211yescharacteristic
39#65528C2218yescomplemented
40#65527C2218yescomplemented
41#65522C2218yescomplemented
42#65517C2214yes—
43#65516C2214yes—
44#65507C2212yes—
45#65506C2212yes—
46C2×C22⋊#168C2214yes—
47#65381C2212yescomplemented
48#65379C2214yescomplemented
49#65374C2212yescomplemented
50#65372C2214yescomplemented
51#65370C2212yescomplemented
52#65370C2212yescomplemented
53#65340C2218yescomplemented
54#65337C2218yescomplemented
55#65336C2218yescomplemented
56#65334C2218yescomplemented
57#65308C2212yescomplemented
58C4×#687C2214yescomplemented
59#65304C2212yescomplemented
60C4×#686C2214yescomplemented
61D4×#168C2214yescomplemented
62D4×#168C2214yescomplemented
63#65297C2212yescomplemented
64#65296C2212yescomplemented
65#65260C2212yes—
66#65256C2212yes—
67#65214C2211yescharacteristic
68#65214C2211yescharacteristic
69#61565—51nomaximal, Sylow
70C2×C5×#41—24no—
71C2×C5×#41—48no—
72C2×C5×#41C2318yes—
73C22×C5×Q8C2312yes—
74C22×C5×D4—22no—
75C22×C5×D4D411yescharacteristic
76C22×C5×D4C2311yescharacteristic
77C23×C4×C5—21no—
78C23×C4×C5—22no—
79C23×C4×C5C2312yes—
80C2×#709—21no—
81C2×#709C2314yes—
82C5×C4⋊Q8—21no—
83C5×C4⋊Q8—21no—
84C5×C4⋊Q8C2312yes—
85C5×C4⋊Q8C2312yes—
86C5×#123—28no—
87C5×#123—28no—
88C5×C22⋊Q8—24no—
89C5×C22⋊Q8C2318yes—
90C5×#121—24no—
91C5×#121—24no—
92C4×C5×D4—21no—
93C4×C5×D4—21no—
94C4×C5×D4D412yes—
95C4×C5×D4D412yes—
96C4×C5×D4C2312yes—
97C4×C5×D4C2312yes—
98C4×C5×D4C2312yes—
99C4×C5×D4C2312yes—
100C2×C5×C4⋊C4—21no—
101C2×C5×C4⋊C4—22no—
102C2×C5×C4⋊C4—21no—
103C2×C5×C4⋊C4—24no—
104C2×C5×C4⋊C4—24no—
105C2×C5×C4⋊C4D411yescharacteristic
106C2×C5×C4⋊C4C2311yescharacteristic
107C2×C5×C22⋊C4—22no—
108C2×C5×C22⋊C4—21no—
109C2×C5×C22⋊C4D412yes—
110C2×C42×C5C2311yescharacteristic
111#4099—24no—
112#4096—416no—
113C2×#689C2318yes—
114C2×#688C2318yes—
115#4086—22no—
116#4084—22no—
117#4084—22no—
118#4082—22no—
119C2×#687C2314yes—
120C2×#686C2314yes—
121#4076—21no—
122#4076C2312yes—
123#4076C2314yes—
124#4075—24no—
125C4×C22⋊D5—24no—
126C4×C22⋊D5—24no—
127#4072—21no—
128#4072C2314yes—
129#4070—21no—
130#4070C2312yes—
131#4070C2314yes—
132C22×#168C2318yes—
133#4039—22no—
134#4038—22no—
135D5×C4⋊C4—22no—
136#4033—22no—
137#4031—24no—
138#4030—24no—
139#4029—24no—
140#4025—24no—
141#4021—21no—
142#4020C2311yescharacteristic
143C4×D20—21no—
144C4×D20C2312yes—
145#4016—21no—
146#4015C2311yescharacteristic
147#4015C2311yescharacteristic
148#4015C2311yescharacteristic
149C4×Dic10—21no—
150C4×Dic10C2312yes—
151#3967C2318yes—
152#3967C2318yes—
153#3964C2314yes—
154#3964C2314yes—
155C22⋊#168C2318yes—
156C22⋊#168C2312yes—
157C22⋊#168C2312yes—
158#3942C2314yes—
159#3941C2314yes—
160#3940C2314yes—
161#3939C2314yes—
162C4⋊#168C2312yes—
163C4⋊#168C2312yes—
164C4×#168C2312yes—
165C4×#168C2312yes—
166#3506—51no—
167C2×#547D511yescharacteristic, complemented
168#3303—54no—
169#3302—54no—
170#3284—54no—
171#3283—54no—
172#3209—51no—
173#3206—51no—
174#3190—54no—
175#3101—51no—
176#3100—51no—
177#3002—54no—
178#2972—51no—
179C24×C5—21no—
180C24×C5C2×D412yes—
181C5×#41—216no—
182C5×#41—48no—
183C5×#41—416no—
184C2×C5×Q8—22no—
185C2×C5×Q8—44no—
186C2×C5×Q8C2414yes—
187C2×C5×D4—24no—
188C2×C5×D4—28no—
189C2×C5×D4—24no—
190C2×C5×D4—21no—
191C2×C5×D4—48no—
192C2×C5×D4C2×D412yes—
193C2×C5×D4C2×D412yes—
194C2×C5×D4C2412yes—
195C22×C4×C5—21no—
196C22×C4×C5—22no—
197C22×C4×C5—22no—
198C22×C4×C5—22no—
199C22×C4×C5—28no—
200C22×C4×C5—44no—
201C22×C4×C5—48no—
202C22×C4×C5—44no—
203C22×C4×C5C2×D411yescharacteristic
204C22×C4×C5C2×D412yes—
205C22×C4×C5C2×D414yes—
206C22×C4×C5C2×D411yescharacteristic
207C22×C4×C5C2411yescharacteristic
208C22×C4×C5C2414yes—
209C2×C22⋊D5—22no—
210C2×C22⋊D5—24no—
211C2×C22⋊D5—24no—
212#709—48no—
213#709—44no—
214C2×D20—21no—
215C2×D20—21no—
216C2×D20C2412yes—
217C2×C4×D5—22no—
218C2×C4×D5—24no—
219C2×Dic10—21no—
220C2×Dic10—21no—
221C2×Dic10C2412yes—
222C5×C4⋊C4—22no—
223C5×C4⋊C4—22no—
224C5×C4⋊C4—28no—
225C5×C4⋊C4—22no—
226C5×C4⋊C4—22no—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C2×D412yes—
229C5×C4⋊C4C2×D412yes—
230C5×C4⋊C4C2412yes—
231C5×C22⋊C4—28no—
232C5×C22⋊C4—22no—
233C5×C22⋊C4—22no—
234C5×C22⋊C4C2×D414yes—
235C5×C22⋊C4C2×D414yes—
236C42×C5C2×D411yescharacteristic
237C42×C5C2×D411yescharacteristic
238C42×C5C2411yescharacteristic
239C42×C5C2411yescharacteristic
240C22⋊Dic5—22no—
241C22⋊Dic5—24no—
242C22⋊Dic5—24no—
243#689—48no—
244#688—48no—
245#687—28no—
246#686—28no—
247#685—22no—
248#685—22no—
249#685—24no—
250C4⋊Dic5—21no—
251C4⋊Dic5—21no—
252C4⋊Dic5—21no—
253C4⋊Dic5—21no—
254C4⋊Dic5C2412yes—
255C4⋊Dic5C2412yes—
256#683—22no—
257#683—22no—
258#683—24no—
259C4×Dic5—22no—
260C4×Dic5—24no—
261C2×#168—48no—
262C2×#168C2414yes—
263C2×#168C2414yes—
264C22×#41C2×D512yes—
265C2×#135—58no—
266#559—102no—
267#556—52no—
268#555—52no—
269#553—102no—
270#550—52no—
271#547—22no—
272#547—22no—
273#547—52no—
274#547C2×D514yes—
275#547C2×D514yes—
276#538—102no—
277#535—101no—
278#533—51no—
279#532—51no—
280C2×C4⋊Q8C2×D511yescharacteristic
281#525—54no—
282C2×#123—22no—
283C2×C22⋊Q8C2×D512yes—
284C2×#121—21no—
285#520—101no—
286C2×C4×D4C2×D511yescharacteristic
287C2×C4×D4C2×D511yescharacteristic
288C22×C4⋊C4—21no—
289#494—52no—
290#492—52no—
291#487—52no—
292#487—52no—
293#485—58no—
294#483—52no—
295#480—52no—
296#476—52no—
297#474—52no—
298#467—58no—
299#465—58no—
300#462—54no—
301#461—54no—
302#461—52no—
303#459—52no—
304#452—58no—
305#452—58no—
306#451—54no—
307#450—52no—
308#449—58no—
309#448—54no—
310#447—52no—
311C4×SD16—54no—
312C4×D8—54no—
313C8×D4—54no—
314C8×D4—54no—
315#431—51no—
316#431—51no—
317#416—58no—
318C2×C22⋊C8—54no—
319C23×C5—24no—
320C23×C5—22no—
321C23×C5—24no—
322C23×C5—42no—
323C23×C5—42no—
324C23×C5—42no—
325C23×C5—42no—
326C23×C5C22×D414yes—
327C23×C5C22×D411yescharacteristic
328C22×D5—22no—
329C5×Q8—28no—
330C5×Q8—44no—
331C5×D4—24no—
332C5×D4—24no—
333C5×D4—24no—
334C5×D4—22no—
335C5×D4—416no—
336C5×D4H(5,𝔽2)14yes—
337C2×C4×C5—28no—
338C2×C4×C5—22no—
339C2×C4×C5—22no—
340C2×C4×C5—24no—
341C2×C4×C5—42no—
342C2×C4×C5—44no—
343C2×C4×C5—41no—
344C2×C4×C5—416no—
345C2×C4×C5—44no—
346C2×C4×C5—42no—
347C2×C4×C5—44no—
348C2×C4×C5—44no—
349C2×C4×C5#13514yescomplemented
350C2×C4×C5C22×D414yes—
351C2×C4×C5C22×D411yescharacteristic
352C2×C4×C5C22×D411yescharacteristic
353C2×C4×C5C22×D411yescharacteristic
354C2×C4×C5C22×D411yescharacteristic
355C2×C4×C5C22×D411yescharacteristic
356C2×C4×C5C2511yescharacteristic
357C22⋊D5—44no—
358C22⋊D5—84no—
359C22⋊D5—84no—
360C2×Dic5—22no—
361C2×Dic5—22no—
362C2×Dic5—22no—
363D20—24no—
364D20—42no—
365D20—81no—
366C4×D5—44no—
367C4×D5—84no—
368Dic10—24no—
369Dic10—42no—
370Dic10—81no—
371#168—24no—
372#168—24no—
373C2×#41—24no—
374C2×#41—48no—
375C2×#41—54no—
376C2×#41—101no—
377C2×#41C22×D518yes—
378C22×Q8C22×D512yes—
379C22×D4—22no—
380C22×D4C22⋊D511yescharacteristic
381C22×D4C22×D511yescharacteristic
382C23×C4—21no—
383C23×C4—22no—
384C23×C4C22×D512yes—
385#135—2016no—
386C2×Q16—58no—
387C2×SD16—54no—
388C2×SD16—58no—
389C2×D8—54no—
390C22×C8—58no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8—51no—
394C4⋊Q8—51no—
395C4⋊Q8—51no—
396C4⋊Q8C22×D512yes—
397C4⋊Q8C22×D512yes—
398#127—51no—
399#127—102no—
400#126—104no—
401#125—101no—
402#124—102no—
403#124—104no—
404#124—101no—
405#123—28no—
406#123—28no—
407#123—104no—
408#123—104no—
409C22⋊Q8—24no—
410C22⋊Q8—52no—
411C22⋊Q8—54no—
412C22⋊Q8—101no—
413C22⋊Q8C22×D518yes—
414#121—24no—
415#121—24no—
416#121—52no—
417#121—54no—
418#121—102no—
419#121—102no—
420#121—101no—
421#121—104no—
422#120—104no—
423C4×Q8—52no—
424C4×Q8—102no—
425C4×Q8—101no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—52no—
429C4×D4—104no—
430C4×D4—104no—
431C4×D4—102no—
432C4×D4—101no—
433C4×D4C22⋊D512yes—
434C4×D4C22⋊D512yes—
435C4×D4C22×D512yes—
436C4×D4C22×D512yes—
437C4×D4C22×D512yes—
438C4×D4C22×D512yes—
439#117—54no—
440#117—101no—
441#117—102no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—24no—
446C2×C4⋊C4—24no—
447C2×C4⋊C4—102no—
448C2×C4⋊C4C22⋊D511yescharacteristic
449C2×C4⋊C4C22×D511yescharacteristic
450C2×C22⋊C4—22no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4C22⋊D512yes—
453C2×C42C22×D511yescharacteristic
454#107—54no—
455#106—54no—
456C4⋊C8—52no—
457C4⋊C8—52no—
458#103—58no—
459#103—58no—
460#103—54no—
461#102—54no—
462#102—54no—
463#102—54no—
464C22⋊C8—52no—
465C22⋊C8—52no—
466C22⋊C8—58no—
467C4×C8—52no—
468C4×C8—52no—
469C22×C5—21no—
470C22×C5—21no—
471C22×C5—24no—
472C22×C5—24no—
473C22×C5—24no—
474C22×C5—24no—
475C22×C5—48no—
476C22×C5—42no—
477C22×C5—42no—
478C22×C5—84no—
479C22×C5C2×#13512yes—
480C22×C5C23×D411yescharacteristic
481C2×D5—42no—
482C2×D5—42no—
483C2×D5—42no—
484C4×C5—21no—
485C4×C5—21no—
486C4×C5—24no—
487C4×C5—21no—
488C4×C5—44no—
489C4×C5—42no—
490C4×C5C2×#13512yes—
491C4×C5C2×#13512yes—
492C4×C5C2×H(5,𝔽2)12yes—
493Dic5—42no—
494Dic5—42no—
495Dic5—42no—
496C24—21no—
497C24C2×C22⋊D512yes—
498#41—216no—
499#41—48no—
500#41—416no—
501#41—208no—
502#41—204no—
503C2×Q8—22no—
504C2×Q8—44no—
505C2×Q8—52no—
506C2×Q8—101no—
507C2×Q8—101no—
508C2×Q8C23×D514yes—
509C2×D4—24no—
510C2×D4—28no—
511C2×D4—24no—
512C2×D4—21no—
513C2×D4—48no—
514C2×D4—52no—
515C2×D4—102no—
516C2×D4—104no—
517C2×D4—104no—
518C2×D4—101no—
519C2×D4—101no—
520C2×D4C2×C22⋊D512yes—
521C2×D4C2×C22⋊D512yes—
522C2×D4C23×D512yes—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—28no—
528C22×C4—44no—
529C22×C4—48no—
530C22×C4—44no—
531C22×C4—102no—
532C22×C4—104no—
533C22×C4C2×C22⋊D511yescharacteristic
534C22×C4C2×C22⋊D512yes—
535C22×C4C2×C22⋊D514yes—
536C22×C4C2×C22⋊D511yescharacteristic
537C22×C4C23×D511yescharacteristic
538C22×C4C23×D514yes—
539Q16—208no—
540SD16—108no—
541SD16—208no—
542D8—108no—
543C2×C8—54no—
544C2×C8—54no—
545C2×C8—208no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—28no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—52no—
553C4⋊C4—52no—
554C4⋊C4—101no—
555C4⋊C4—104no—
556C4⋊C4—101no—
557C4⋊C4—102no—
558C4⋊C4—101no—
559C4⋊C4—102no—
560C4⋊C4—101no—
561C4⋊C4C2×C22⋊D512yes—
562C4⋊C4C2×C22⋊D512yes—
563C4⋊C4C23×D512yes—
564C22⋊C4—28no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—104no—
568C22⋊C4—102no—
569C22⋊C4—104no—
570C22⋊C4—104no—
571C22⋊C4—102no—
572C22⋊C4—102no—
573C22⋊C4C2×C22⋊D514yes—
574C22⋊C4C2×C22⋊D514yes—
575C42—104no—
576C42—102no—
577C42C2×C22⋊D511yescharacteristic
578C42C2×C22⋊D511yescharacteristic
579C42C23×D511yescharacteristic
580C42C23×D511yescharacteristic
581C2×C5—22no—
582C2×C5—44no—
583C2×C5—44no—
584C2×C5#348411yescharacteristic
585C2×C5C22×#13511yescharacteristic
586C2×C5#358611yescharacteristic
587D5—82no—
588C23—24no—
589C23—22no—
590C23—24no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—102no—
596C23C22×C22⋊D514yes—
597C23C22×C22⋊D511yescharacteristic
598Q8—28no—
599Q8—44no—
600Q8—104no—
601Q8—202no—
602Q8—401no—
603D4—24no—
604D4—24no—
605D4—24no—
606D4—22no—
607D4—416no—
608D4—104no—
609D4—204no—
610D4—202no—
611D4—401no—
612D4—404no—
613D4—404no—
614D4#414414yes—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—102no—
628C2×C4—102no—
629C2×C4—102no—
630C2×C4—204no—
631C2×C4—404no—
632C2×C4#409614yescomplemented
633C2×C4C22×C22⋊D514yes—
634C2×C4C22×C22⋊D511yescharacteristic
635C2×C4C22×C22⋊D511yescharacteristic
636C2×C4C22×C22⋊D511yescharacteristic
637C2×C4C22×C22⋊D511yescharacteristic
638C2×C4C22×C22⋊D511yescharacteristic
639C2×C4C24×D511yescharacteristic
640C8—104no—
641C8—104no—
642C5#6156511yescharacteristic, complemented, Sylow
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—48no—
650C22—42no—
651C22—42no—
652C22—84no—
653C22—202no—
654C22—202no—
655C22—202no—
656C22C2×#409612yes—
657C22C23×C22⋊D511yescharacteristic
658C4—24no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—44no—
663C4—42no—
664C4—202no—
665C4—202no—
666C4—202no—
667C4C2×#409612yes—
668C4C2×#409612yes—
669C4C2×#414412yes—
670C2—22no—
671C2—44no—
672C2—44no—
673C2—402no—
674C2#1061914011yescharacteristic
675C2C22×#409611yescharacteristic
676C2#1061918811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#547C2
socleC22×C5C23×D4
hypercenterC2×#547D5
hypocenterC5#61565
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C5×#547C2semidirect
2. #10617689C2semidirect
3. #10615811C2semidirect
4. #10615261C2semidirect
5. #10615255C2semidirect
6. #10615242C2semidirect
7. #10615241C2semidirect
8. #10614953C2semidirect
9. #10614944C2semidirect
10. #10614468C2semidirect
11. #10614045C2semidirect
12. #10614039C2semidirect
13. #10614032C2semidirect
26. C2×#4096C22semidirect
29. #65897C22semidirect
32. #65887C22semidirect
33. #65885C22semidirect
34. #65876C22semidirect
39. #65528C22semidirect
40. #65527C22semidirect
41. #65522C22semidirect
47. #65381C22semidirect
48. #65379C22semidirect
49. #65374C22semidirect
50. #65372C22semidirect
51. #65370C22semidirect
52. #65370C22semidirect
53. #65340C22semidirect
54. #65337C22semidirect
55. #65336C22semidirect
56. #65334C22semidirect
57. #65308C22semidirect
58. C4×#687C22semidirect
59. #65304C22semidirect
60. C4×#686C22semidirect
61. D4×#168C22semidirect
62. D4×#168C22semidirect
63. #65297C22semidirect
64. #65296C22semidirect
167. C2×#547D5semidirect
349. C2×C4×C5#135semidirect
632. C2×C4#4096semidirect
642. C5#61565semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5472
#106176892
#106158112
#106152612
#106152552
#106152422
#106152412
#106149532
#106149442
#106144682
#106140452
#106140392
#106140322
#615655