Finite Group Databasebeta

← #12984028

Subgroups — #12984028

Order 1280.

2090 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618181C211yesmaximal, characteristic
2C2×#66061C211yesmaximal, characteristic
3C5×#3209C211yesmaximal, characteristic, complemented
4#10615099C211yesmaximal, characteristic, complemented
5#10614953C211yesmaximal, characteristic, complemented
6#10613538C214yesmaximal, complemented
7#10613344C214yesmaximal, complemented
8#10612222C214yesmaximal, complemented
9#10612221C214yesmaximal, complemented
10#10611918C214yesmaximal, complemented
11D5×#451C214yesmaximal, complemented
12#10611041C211yesmaximal, characteristic, complemented
13#10611017C211yesmaximal, characteristic
14C22×Q8×D5C2212yes—
15C5×#533C2211yescharacteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×#4092C2212yes—
18C2×D5×Q16C2218yescomplemented
19#66064C2212yes—
20#66061—22no—
21#66061—22no—
22#66061C2214yes—
23#66061C2214yes—
24#66061C2212yes—
25#66024—22no—
26#65968—22no—
27D5×C22⋊Q8C2214yes—
28#65913—22no—
29#65913C2212yes—
30Q8×D20C2212yes—
31#65855—21no—
32#65853—21no—
33C2×#4042—22no—
34C2×#4039—21no—
35C2×#4034—21no—
36C2×C4×D20C2211yescharacteristic
37#65804C2211yescharacteristic
38C2×C4×Dic10C2211yescharacteristic
39C5×#494C2212yescomplemented
40C5×#487C2212yescomplemented
41C5×#462C2214yescomplemented
42C5×#451C2214yes—
43C5×#431C2211yescharacteristic
44#65517C2214yes—
45#65477C2218yescomplemented
46#65475C2218yescomplemented
47#65475C2218yescomplemented
48#65381C2212yescomplemented
49#65380C2212yescomplemented
50#65379C2212yescomplemented
51#65379C2214yescomplemented
52#65370C2212yescomplemented
53#65339C2212yes—
54#65333C2212yes—
55C4×#689C2214yescomplemented
56#65214C2211yescharacteristic
57#65139C2212yescomplemented
58#65138C2212yescomplemented
59#65123C2214yes—
60#65102C2218yescomplemented
61#65098C2218yescomplemented
62#65097C2218yescomplemented
63D5×#103C2218yescomplemented
64#65028C2212yes—
65D5×C22⋊C8C2214yes—
66#65011C2212yes—
67C4×Dic20C2214yescomplemented
68C8×D20C2214yes—
69#61563—51nomaximal, Sylow
70C22×C5×Q8C2312yes—
71C2×Q8×D5—24no—
72C2×Q8×D5—48no—
73C2×Q8×D5C2318yes—
74C2×Q8×D5C2314yes—
75C22×D20D411yescharacteristic
76C22×C4×D5—22no—
77C22×C4×D5C2312yes—
78C22×Dic10—22no—
79C22×Dic10C2311yescharacteristic
80C2×C5×Q16C2318yes—
81C5×C4⋊Q8—21no—
82C5×C4⋊Q8—21no—
83C5×C4⋊Q8C2312yes—
84C5×C4⋊Q8C2312yes—
85C5×C4⋊Q8C2311yescharacteristic
86C5×C4⋊Q8C2311yescharacteristic
87C5×#125—21no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8C2312yes—
90C4×C5×Q8—21no—
91C4×C5×Q8C2312yes—
92C5×#117—21no—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4—21no—
95C2×C42×C5C2311yescharacteristic
96#4092—24no—
97#4092—24no—
98#4092C2318yes—
99#4092C2314yes—
100C2×#689C2314yes—
101C2×#689C2318yes—
102C2×#685—22no—
103C2×#685D412yes—
104#4072—21no—
105C2×C4⋊Dic5D411yescharacteristic
106C2×C4⋊Dic5C2311yescharacteristic
107#4070—21no—
108#4070C2312yes—
109C2×#683—21no—
110C2×#683—22no—
111C2×C4×Dic5—21no—
112D5×Q16—416no—
113C2×Dic20C2314yes—
114C2×C8×D5C2318yes—
115#4043—24no—
116#4043C2314yes—
117#4042—24no—
118#4042—28no—
119#4042—28no—
120#4042—24no—
121#4042—24no—
122#4039—24no—
123#4039—24no—
124#4039—22no—
125#4039—22no—
126D5×C4⋊C4—24no—
127D5×C4⋊C4C2314yes—
128#4036—22no—
129#4034—24no—
130#4034—24no—
131#4034—22no—
132D5×C22⋊C4—24no—
133#4024—24no—
134C4×D20—22no—
135C4×D20—22no—
136C4×D20D412yes—
137C4×D20D412yes—
138C4×D20C2312yes—
139C4×D20C2312yes—
140#4016—21no—
141#4016—22no—
142#4015—22no—
143#4015C2311yescharacteristic
144#4015C2311yescharacteristic
145C4×Dic10—21no—
146C4×Dic10—21no—
147C4×Dic10—21no—
148C4×Dic10C2312yes—
149C4×Dic10C2312yes—
150C4×Dic10C2312yes—
151C5×C4⋊C8C2312yes—
152C5×#103C2318yes—
153C5×C22⋊C8C2312yes—
154C4×C5×C8C2312yes—
155#3967C2314yes—
156#3967C2318yes—
157C22⋊#168C2312yes—
158#3952C2318yes—
159#3950C2314yes—
160#3948C2314yes—
161#3948C2314yes—
162#3942C2314yes—
163#3939C2314yes—
164C4⋊#168C2312yes—
165C4×#168C2312yes—
166#3501—51no—
167C2×D4×Q8—51no—
168#3299—54no—
169#3299—54no—
170D4×Q16—54no—
171D4×Q16—54no—
172#3209—51no—
173#3209D511yescharacteristic, complemented
174#3189—54no—
175#3104—51no—
176#3104—51no—
177C2×#451—54no—
178#2972—51no—
179C23×D5C2×D412yes—
180C2×C5×Q8—22no—
181C2×C5×Q8—21no—
182C2×C5×Q8—44no—
183C2×C5×Q8C2412yes—
184C2×C5×Q8C2414yes—
185C22×C4×C5—22no—
186C22×C4×C5C2×D411yescharacteristic
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2411yescharacteristic
189C22×Dic5—21no—
190C22×Dic5—21no—
191C22×Dic5C2×D412yes—
192Q8×D5—216no—
193Q8×D5—48no—
194Q8×D5—416no—
195Q8×D5—48no—
196C2×D20Q1618yescomplemented
197C2×D20C2×D412yes—
198C2×D20C2×D412yes—
199C2×C4×D5—28no—
200C2×C4×D5—24no—
201C2×C4×D5—24no—
202C2×C4×D5—44no—
203C2×C4×D5—48no—
204C2×C4×D5C2×D414yes—
205C2×C4×D5C2414yes—
206C2×Dic10—21no—
207C2×Dic10—24no—
208C2×Dic10—28no—
209C2×Dic10—21no—
210C2×Dic10—24no—
211C2×Dic10—48no—
212C2×Dic10C2412yes—
213C2×Dic10C2412yes—
214C5×Q16—48no—
215C2×C5×C8C2414yes—
216C5×C4⋊C4—22no—
217C5×C4⋊C4—22no—
218C5×C4⋊C4—22no—
219C5×C4⋊C4—22no—
220C5×C4⋊C4—21no—
221C5×C4⋊C4—21no—
222C5×C4⋊C4—21no—
223C5×C4⋊C4—22no—
224C5×C4⋊C4—22no—
225C5×C4⋊C4C2412yes—
226C5×C22⋊C4—22no—
227C42×C5—22no—
228C42×C5C2×D411yescharacteristic
229C42×C5C2×D411yescharacteristic
230C42×C5C2411yescharacteristic
231C42×C5C2411yescharacteristic
232C22⋊Dic5—22no—
233#689—28no—
234#689—48no—
235#685—24no—
236#685—24no—
237#685—28no—
238#685C2×D414yes—
239#685C2×D414yes—
240C4⋊Dic5—21no—
241C4⋊Dic5—21no—
242C4⋊Dic5—21no—
243C4⋊Dic5—21no—
244C4⋊Dic5—24no—
245C4⋊Dic5C2×D412yes—
246C4⋊Dic5C2×D412yes—
247C4⋊Dic5C2412yes—
248C4⋊Dic5C2412yes—
249#683—22no—
250#683—22no—
251#683—22no—
252#683—22no—
253#683—24no—
254#683—24no—
255#683—28no—
256C4×Dic5—22no—
257C4×Dic5—22no—
258C4×Dic5—22no—
259C2×#168C2414yes—
260Dic20—28no—
261C8×D5—48no—
262C23×Q8—52no—
263C22×Q16—58no—
264#554—52no—
265#554—52no—
266#554—102no—
267#554—102no—
268D4×Q8—54no—
269D4×Q8—54no—
270D4×Q8—52no—
271D4×Q8—52no—
272D4×Q8—102no—
273D4×Q8—102no—
274#544—101no—
275#543—102no—
276#533—51no—
277#533C2×D511yescharacteristic
278C2×C4⋊Q8—101no—
279C2×C4⋊Q8C2×D511yescharacteristic
280C2×C22⋊Q8—52no—
281C2×C22⋊Q8—54no—
282C2×C22⋊Q8—102no—
283#518—101no—
284C2×C4×Q8—51no—
285C2×C4×D4—51no—
286C2×C4×D4—101no—
287#494—52no—
288#494C2×D512yescomplemented
289#487—52no—
290#487C2×D512yescomplemented
291#484—52no—
292#484—58no—
293#484—52no—
294#477—52no—
295#477—52no—
296#467—58no—
297#467—58no—
298#462—52no—
299#462—54no—
300#462—52no—
301#462C2×D514yescomplemented
302#451—54no—
303#451—58no—
304#451—58no—
305#451—52no—
306#451—58no—
307#451—52no—
308#451C2×D514yes—
309C4×Q16—54no—
310C4×Q16—54no—
311C8×D4—54no—
312C8×D4—54no—
313#431—51no—
314#431C2×D511yescharacteristic
315C2×#103—58no—
316C2×C22⋊C8—54no—
317C23×C5C22×D411yescharacteristic
318C22×D5—42no—
319C22×D5—42no—
320C22×D5C2×Q1618yescomplemented
321C22×D5C22×D414yes—
322C5×Q8—24no—
323C5×Q8—28no—
324C5×Q8—44no—
325C5×Q8—81no—
326C2×C4×C5—22no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—24no—
330C2×C4×C5—41no—
331C2×C4×C5—44no—
332C2×C4×C5C2×Q1614yescomplemented
333C2×C4×C5C22×D411yescharacteristic
334C2×C4×C5C22×D411yescharacteristic
335C2×C4×C5C22×D411yescharacteristic
336C2×C4×C5C22×D411yescharacteristic
337C2×C4×C5C22×D411yescharacteristic
338C2×C4×C5C2511yescharacteristic
339C2×Dic5—24no—
340C2×Dic5—22no—
341C2×Dic5—22no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—42no—
345C2×Dic5—42no—
346C2×Dic5C22×D414yes—
347D20C2×Q1618yescomplemented
348D20C2×Q1618yescomplemented
349C4×D5—28no—
350C4×D5—44no—
351C4×D5—416no—
352C4×D5—84no—
353C4×D5—84no—
354Dic10—24no—
355Dic10—22no—
356Dic10—24no—
357Dic10—416no—
358Dic10—84no—
359Dic10—81no—
360Dic10H(5,𝔽2)14yes—
361C5×C8—24no—
362#168—24no—
363C22×Q8—58no—
364C22×Q8—54no—
365C22×Q8—51no—
366C22×Q8—102no—
367C22×Q8—104no—
368C22×Q8—208no—
369C22×Q8C22×D512yes—
370C22×D4—51no—
371C23×C4—52no—
372C23×C4—102no—
373C2×Q16—54no—
374C2×Q16—58no—
375C2×Q16—54no—
376C2×Q16—2016no—
377C2×Q16C22×D518yes—
378C22×C8—58no—
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—51no—
382C4⋊Q8—51no—
383C4⋊Q8—102no—
384C4⋊Q8—104no—
385C4⋊Q8—104no—
386C4⋊Q8—102no—
387C4⋊Q8C22×D512yes—
388C4⋊Q8C22×D512yes—
389C4⋊Q8C22×D511yescharacteristic
390C4⋊Q8C22×D511yescharacteristic
391#125—21no—
392#125—102no—
393#125—102no—
394#125—101no—
395C22⋊Q8—21no—
396C22⋊Q8—58no—
397C22⋊Q8—54no—
398C22⋊Q8—52no—
399C22⋊Q8—54no—
400C22⋊Q8—104no—
401C22⋊Q8—108no—
402C22⋊Q8—108no—
403C22⋊Q8—104no—
404C22⋊Q8—104no—
405C22⋊Q8—104no—
406C22⋊Q8—104no—
407C22⋊Q8—104no—
408C22⋊Q8—104no—
409C22⋊Q8—101no—
410C22⋊Q8C22×D512yes—
411C4×Q8—21no—
412C4×Q8—52no—
413C4×Q8—52no—
414C4×Q8—52no—
415C4×Q8—101no—
416C4×Q8—101no—
417C4×Q8—101no—
418C4×Q8C22×D512yes—
419C4×D4—52no—
420C4×D4—52no—
421C4×D4—52no—
422C4×D4—52no—
423C4×D4—102no—
424C4×D4—102no—
425C4×D4—104no—
426C4×D4—104no—
427C4×D4—102no—
428C4×D4—102no—
429#117—21no—
430#117—101no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—51no—
434C2×C4⋊C4—51no—
435C2×C4⋊C4—54no—
436C2×C4⋊C4—104no—
437C2×C4⋊C4—102no—
438C2×C4⋊C4—101no—
439C2×C22⋊C4—52no—
440C2×C22⋊C4—104no—
441C2×C22⋊C4—102no—
442C2×C42—101no—
443C2×C42C22×D511yescharacteristic
444#107—54no—
445#107—54no—
446C4⋊C8—52no—
447C4⋊C8C22×D512yes—
448#103—58no—
449#103—54no—
450#103—54no—
451#103—54no—
452#103—54no—
453#103C22×D518yes—
454C22⋊C8—52no—
455C22⋊C8—58no—
456C22⋊C8C22×D512yes—
457C4×C8—52no—
458C4×C8C22×D512yes—
459C22×C5—21no—
460C22×C5—21no—
461C22×C5C22×Q1612yes—
462C22×C5C23×D411yescharacteristic
463C2×D5—24no—
464C2×D5—24no—
465C2×D5—48no—
466C2×D5C22×Q1618yes—
467C4×C5—21no—
468C4×C5—21no—
469C4×C5—21no—
470C4×C5—42no—
471C4×C5—44no—
472C4×C5—42no—
473C4×C5C22×Q1612yes—
474C4×C5C22×Q1612yes—
475C4×C5C2×H(5,𝔽2)12yes—
476Dic5—24no—
477Dic5—24no—
478Dic5—42no—
479Dic5—42no—
480Dic5—42no—
481C24—52no—
482C2×Q8—22no—
483C2×Q8—21no—
484C2×Q8—44no—
485C2×Q8—52no—
486C2×Q8—52no—
487C2×Q8—108no—
488C2×Q8—104no—
489C2×Q8—104no—
490C2×Q8—101no—
491C2×Q8—101no—
492C2×Q8—1016no—
493C2×Q8—208no—
494C2×Q8—208no—
495C2×Q8—208no—
496C2×Q8—2016no—
497C2×Q8C23×D514yes—
498C2×Q8C23×D512yes—
499C2×D4—52no—
500C2×D4—52no—
501C2×D4—58no—
502C22×C4—22no—
503C22×C4—52no—
504C22×C4—54no—
505C22×C4—54no—
506C22×C4—108no—
507C22×C4—104no—
508C22×C4—104no—
509C22×C4—101no—
510C22×C4—101no—
511C22×C4—204no—
512C22×C4—208no—
513C22×C4D4×D511yescharacteristic
514C22×C4D4×D511yescharacteristic
515C22×C4C23×D511yescharacteristic
516Q16—48no—
517Q16—108no—
518Q16—108no—
519Q16—208no—
520C2×C8—54no—
521C2×C8—208no—
522C2×C8C23×D514yes—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—52no—
533C4⋊C4—52no—
534C4⋊C4—52no—
535C4⋊C4—52no—
536C4⋊C4—104no—
537C4⋊C4—104no—
538C4⋊C4—104no—
539C4⋊C4—108no—
540C4⋊C4—101no—
541C4⋊C4—102no—
542C4⋊C4—102no—
543C4⋊C4—102no—
544C4⋊C4—101no—
545C4⋊C4—101no—
546C4⋊C4—102no—
547C4⋊C4—101no—
548C4⋊C4C23×D512yes—
549C22⋊C4—22no—
550C22⋊C4—54no—
551C22⋊C4—54no—
552C22⋊C4—104no—
553C22⋊C4—104no—
554C22⋊C4—108no—
555C22⋊C4—102no—
556C42—22no—
557C42—102no—
558C42—102no—
559C42—102no—
560C42D4×D511yescharacteristic
561C42D4×D511yescharacteristic
562C42C23×D511yescharacteristic
563C42C23×D511yescharacteristic
564C2×C5—22no—
565C2×C5#348411yescharacteristic
566C2×C5C23×Q1611yescharacteristic
567C2×C5#358611yescharacteristic
568D5—28no—
569C23—54no—
570C23—58no—
571C23—202no—
572C23—202no—
573C23C2×D4×D511yescharacteristic
574Q8—24no—
575Q8—28no—
576Q8—44no—
577Q8—54no—
578Q8—81no—
579Q8—104no—
580Q8—102no—
581Q8—104no—
582Q8—2016no—
583Q8—401no—
584Q8—404no—
585D4—58no—
586D4—58no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—54no—
594C2×C4—104no—
595C2×C4—102no—
596C2×C4—108no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—102no—
600C2×C4—204no—
601C2×C4—2016no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—404no—
605C2×C4—404no—
606C2×C4D5×Q1614yescomplemented
607C2×C4C2×D4×D511yescharacteristic
608C2×C4C2×D4×D511yescharacteristic
609C2×C4C2×D4×D511yescharacteristic
610C2×C4C2×D4×D511yescharacteristic
611C2×C4C2×D4×D511yescharacteristic
612C2×C4C24×D511yescharacteristic
613C8—24no—
614C8—104no—
615C5#6156311yescharacteristic, complemented, Sylow
616C22—21no—
617C22—21no—
618C22—58no—
619C22—104no—
620C22—104no—
621C22—208no—
622C22C2×D5×Q1612yes—
623C22C22×D4×D511yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—104no—
631C4—104no—
632C4—202no—
633C4—202no—
634C4—202no—
635C4C2×D5×Q1612yes—
636C4C2×D5×Q1612yes—
637C4D5×H(5,𝔽2)12yes—
638C2—22no—
639C2—108no—
640C2#1061823111yescharacteristic
641C2C22×D5×Q1611yescharacteristic
642C2#1061909811yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3209C2
socleC22×C5C23×D4
prime radicalC22×D20D4
hypercenter#3209D5
hypocenterC5#61563
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. C5×#3209C2semidirect
4. #10615099C2semidirect
5. #10614953C2semidirect
6. #10613538C2semidirect
7. #10613344C2semidirect
8. #10612222C2semidirect
9. #10612221C2semidirect
10. #10611918C2semidirect
11. D5×#451C2semidirect
12. #10611041C2semidirect
18. C2×D5×Q16C22semidirect
39. C5×#494C22semidirect
40. C5×#487C22semidirect
41. C5×#462C22semidirect
45. #65477C22semidirect
46. #65475C22semidirect
47. #65475C22semidirect
48. #65381C22semidirect
49. #65380C22semidirect
50. #65379C22semidirect
51. #65379C22semidirect
52. #65370C22semidirect
55. C4×#689C22semidirect
57. #65139C22semidirect
58. #65138C22semidirect
60. #65102C22semidirect
61. #65098C22semidirect
62. #65097C22semidirect
63. D5×#103C22semidirect
67. C4×Dic20C22semidirect
173. #3209D5semidirect
196. C2×D20Q16semidirect
288. #494C2×D5semidirect
290. #487C2×D5semidirect
301. #462C2×D5semidirect
320. C22×D5C2×Q16semidirect
332. C2×C4×C5C2×Q16semidirect
347. D20C2×Q16semidirect
348. D20C2×Q16semidirect
606. C2×C4D5×Q16semidirect
615. C5#61563semidirect

Maximal subgroups

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

subgroupindex
#106181812
C2×#660612
C5×#32092
#106150992
#106149532
#106135382
#106133442
#106122222
#106122212
#106119182
D5×#4512
#106110412
#106110172
#615635