Finite Group Databasebeta

← #12984027

Subgroups — #12984027

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×Q8×D20C211yesmaximal, characteristic
3C5×#3104C211yesmaximal, characteristic, complemented
4#10615101C211yesmaximal, characteristic, complemented
5#10614178C211yesmaximal, characteristic, complemented
6#10613219C214yesmaximal, complemented
7#10612850C214yesmaximal, complemented
8#10612221C214yesmaximal, complemented
9D5×#451C214yesmaximal, complemented
10#10611466C214yesmaximal, complemented
11Q16×D20C214yesmaximal, complemented
12#10611044C211yesmaximal, characteristic, complemented
13#10611017C211yesmaximal, characteristic
14C22×Q8×D5C2212yes—
15C5×#533C2211yescharacteristic
16C2×C4×C5×Q8C2211yescharacteristic
17C2×D5×Q16C2218yescomplemented
18#66064C2212yes—
19#66061C2212yes—
20#66024—22no—
21#65968—22no—
22D5×C22⋊Q8C2214yes—
23#65913—22no—
24#65913C2212yes—
25Q8×D20—22no—
26Q8×D20—22no—
27Q8×D20C2212yes—
28Q8×D20C2214yes—
29Q8×D20C2214yes—
30#65855—21no—
31#65853—21no—
32C2×#4043—22no—
33C2×#4043C2212yes—
34C2×C4×D20—21no—
35C2×C4×D20C2211yescharacteristic
36#65804C2211yescharacteristic
37C2×#4015—21no—
38C2×#4015C2211yescharacteristic
39C5×#484C2212yes—
40C5×#477C2212yescomplemented
41C5×#462C2212yescomplemented
42C5×#451C2212yes—
43C4×C5×Q16C2214yescomplemented
44C5×#431C2211yescharacteristic
45#65517C2212yes—
46#65475C2218yescomplemented
47#65382C2212yescomplemented
48#65381C2212yescomplemented
49#65339C2214yes—
50#65320C2212yescomplemented
51#65320C2214yescomplemented
52C4×#689C2214yescomplemented
53#65310C2212yescomplemented
54#65259C2212yes—
55#65214C2211yescharacteristic
56#65176C2218yescomplemented
57#65173C2218yescomplemented
58#65138C2214yescomplemented
59#65123C2214yes—
60#65102C2218yescomplemented
61#65097C2218yescomplemented
62#65097C2218yescomplemented
63D5×#103C2218yescomplemented
64#65028C2214yes—
65D5×C22⋊C8C2214yes—
66#64988C2212yescomplemented
67C8×D20C2214yes—
68#64970C2212yescomplemented
69#61563—51nomaximal, Sylow
70C22×C5×Q8C2311yescharacteristic
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×Dic10C2312yes—
80C2×C5×Q16C2314yes—
81C5×C4⋊Q8C2311yescharacteristic
82C5×C4⋊Q8C2311yescharacteristic
83C5×#125—21no—
84C5×C22⋊Q8—21no—
85C5×C22⋊Q8C2312yes—
86C4×C5×Q8—21no—
87C4×C5×Q8—21no—
88C4×C5×Q8—21no—
89C4×C5×Q8C2312yes—
90C4×C5×Q8C2312yes—
91C4×C5×Q8C2312yes—
92C5×#117—21no—
93C2×C5×C4⋊C4—21no—
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C42×C5—21no—
96C2×C42×C5C2311yescharacteristic
97#4092—24no—
98#4092C2314yes—
99C2×#689C2314yes—
100C2×#689C2318yes—
101C2×#685—22no—
102C2×#685D412yes—
103#4072—21no—
104C2×C4⋊Dic5—21no—
105C2×C4⋊Dic5—21no—
106C2×C4⋊Dic5—22no—
107C2×C4⋊Dic5D411yescharacteristic
108#4070—21no—
109#4070C2312yes—
110D5×Q16—416no—
111C2×Dic20C2318yes—
112C2×C8×D5C2318yes—
113#4043—24no—
114#4043—28no—
115#4043—28no—
116#4043—24no—
117#4043C2314yes—
118#4043C2318yes—
119#4042—24no—
120#4042—24no—
121#4042—24no—
122#4039—22no—
123#4039—22no—
124D5×C4⋊C4—24no—
125D5×C4⋊C4C2314yes—
126#4036—22no—
127#4034—22no—
128D5×C22⋊C4—24no—
129#4024—24no—
130C4×D20—24no—
131C4×D20—24no—
132C4×D20—22no—
133C4×D20—22no—
134C4×D20D412yes—
135C4×D20D412yes—
136C4×D20C2312yes—
137C4×D20C2312yes—
138#4016—21no—
139#4016—22no—
140#4015—21no—
141#4015—21no—
142#4015—24no—
143#4015—24no—
144#4015—22no—
145#4015C2312yes—
146#4015C2312yes—
147#4015C2311yescharacteristic
148#4015C2311yescharacteristic
149C4×Dic10—21no—
150C4×Dic10C2312yes—
151C5×#107C2314yes—
152C5×C4⋊C8C2312yes—
153C5×#103C2314yes—
154C5×#103C2314yes—
155C5×C22⋊C8C2312yes—
156C4×C5×C8C2312yes—
157#3967C2314yes—
158C22⋊#168C2312yes—
159#3952C2318yes—
160#3948C2318yes—
161#3942C2314yes—
162#3942C2318yes—
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#3209—51no—
174#3189—54no—
175#3104—51no—
176#3104D511yescharacteristic, complemented
177C2×#451—54no—
178#2972—51no—
179C23×D5C2×D412yes—
180C2×C5×Q8—24no—
181C2×C5×Q8—21no—
182C2×C5×Q8—21no—
183C2×C5×Q8C2412yes—
184C2×C5×Q8C2412yes—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5C2×D411yescharacteristic
188C22×C4×C5C2×D411yescharacteristic
189C22×C4×C5C2411yescharacteristic
190C22×Dic5—22no—
191C22×Dic5C2×D412yes—
192Q8×D5—216no—
193Q8×D5—48no—
194Q8×D5—48no—
195Q8×D5—416no—
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—22no—
207C2×Dic10—28no—
208C2×Dic10—21no—
209C2×Dic10—24no—
210C2×Dic10—44no—
211C2×Dic10—48no—
212C2×Dic10C2414yes—
213C2×Dic10C2412yes—
214C5×Q16—28no—
215C2×C5×C8C2414yes—
216C5×C4⋊C4—22no—
217C5×C4⋊C4—22no—
218C5×C4⋊C4—21no—
219C5×C4⋊C4—21no—
220C5×C4⋊C4—21no—
221C5×C4⋊C4—21no—
222C5×C4⋊C4—22no—
223C5×C4⋊C4—22no—
224C5×C4⋊C4C2412yes—
225C5×C4⋊C4C2412yes—
226C5×C22⋊C4—22no—
227C42×C5—22no—
228C42×C5—22no—
229C42×C5—22no—
230C42×C5C2×D411yescharacteristic
231C42×C5C2×D411yescharacteristic
232C42×C5C2411yescharacteristic
233C42×C5C2411yescharacteristic
234C22⋊Dic5—22no—
235#689—28no—
236#689—48no—
237#685—28no—
238#685—24no—
239#685—24no—
240#685C2×D414yes—
241#685C2×D414yes—
242C4⋊Dic5—21no—
243C4⋊Dic5—22no—
244C4⋊Dic5—22no—
245C4⋊Dic5—21no—
246C4⋊Dic5—22no—
247C4⋊Dic5—22no—
248C4⋊Dic5—21no—
249C4⋊Dic5—28no—
250C4⋊Dic5—24no—
251C4⋊Dic5C2×D412yes—
252C4⋊Dic5C2×D412yes—
253C4⋊Dic5C2412yes—
254#683—22no—
255#683—22no—
256#683—24no—
257#683—24no—
258C4×Dic5—22no—
259C2×#168C2414yes—
260Dic20—48no—
261C8×D5—48no—
262C23×Q8—52no—
263C22×Q16—58no—
264#554—52no—
265#554—52no—
266#554—102no—
267#554—102no—
268D4×Q8—52no—
269D4×Q8—54no—
270D4×Q8—54no—
271D4×Q8—52no—
272D4×Q8—102no—
273D4×Q8—102no—
274#544—101no—
275#543—102no—
276#533—51no—
277#533C2×D511yescharacteristic
278C2×C4⋊Q8—51no—
279C2×C4⋊Q8—101no—
280C2×C22⋊Q8—54no—
281C2×C22⋊Q8—52no—
282C2×C22⋊Q8—102no—
283#518—101no—
284C2×C4×Q8C2×D511yescharacteristic
285C2×C4×D4—51no—
286C2×C4×D4—101no—
287#494—52no—
288#494—52no—
289#487—52no—
290#487—52no—
291#484—52no—
292#484—58no—
293#484C2×D512yes—
294#477—52no—
295#477C2×D512yescomplemented
296#467—58no—
297#467—58no—
298#462—52no—
299#462—54no—
300#462—54no—
301#462C2×D512yescomplemented
302#451—52no—
303#451—58no—
304#451—54no—
305#451—58no—
306#451—58no—
307#451—54no—
308#451C2×D512yes—
309C4×Q16—54no—
310C4×Q16C2×D514yescomplemented
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—22no—
324C5×Q8—24no—
325C5×Q8—81no—
326C5×Q8H(5,𝔽2)14yes—
327C2×C4×C5—24no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—41no—
334C2×C4×C5—42no—
335C2×C4×C5C2×Q1614yescomplemented
336C2×C4×C5C22×D411yescharacteristic
337C2×C4×C5C22×D411yescharacteristic
338C2×C4×C5C22×D411yescharacteristic
339C2×C4×C5C22×D411yescharacteristic
340C2×C4×C5C22×D411yescharacteristic
341C2×C4×C5C2511yescharacteristic
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—24no—
345C2×Dic5—42no—
346C2×Dic5—44no—
347C2×Dic5C22×D414yes—
348D20C2×Q1618yescomplemented
349D20C2×Q1618yescomplemented
350C4×D5—28no—
351C4×D5—44no—
352C4×D5—416no—
353C4×D5—84no—
354C4×D5—84no—
355Dic10—24no—
356Dic10—28no—
357Dic10—416no—
358Dic10—44no—
359Dic10—84no—
360Dic10—81no—
361C5×C8—24no—
362#168—24no—
363C22×Q8—58no—
364C22×Q8—54no—
365C22×Q8—52no—
366C22×Q8—102no—
367C22×Q8—104no—
368C22×Q8—208no—
369C22×Q8C22×D511yescharacteristic
370C22×D4—51no—
371C23×C4—52no—
372C23×C4—102no—
373C2×Q16—54no—
374C2×Q16—58no—
375C2×Q16—58no—
376C2×Q16—2016no—
377C2×Q16C22×D514yes—
378C22×C8—58no—
379C4⋊Q8—51no—
380C4⋊Q8—52no—
381C4⋊Q8—52no—
382C4⋊Q8—51no—
383C4⋊Q8—104no—
384C4⋊Q8—104no—
385C4⋊Q8—102no—
386C4⋊Q8—102no—
387C4⋊Q8—101no—
388C4⋊Q8—101no—
389C4⋊Q8C22×D511yescharacteristic
390C4⋊Q8C22×D511yescharacteristic
391#125—21no—
392#125—102no—
393#125—102no—
394#125—101no—
395C22⋊Q8—21no—
396C22⋊Q8—54no—
397C22⋊Q8—52no—
398C22⋊Q8—58no—
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—21no—
413C4×Q8—21no—
414C4×Q8—52no—
415C4×Q8—101no—
416C4×Q8C22×D512yes—
417C4×Q8C22×D512yes—
418C4×Q8C22×D512yes—
419C4×D4—52no—
420C4×D4—52no—
421C4×D4—52no—
422C4×D4—52no—
423C4×D4—104no—
424C4×D4—104no—
425C4×D4—102no—
426C4×D4—102no—
427C4×D4—102no—
428C4×D4—102no—
429#117—21no—
430#117—101no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—51no—
433C2×C4⋊C4—54no—
434C2×C4⋊C4—104no—
435C2×C4⋊C4—102no—
436C2×C4⋊C4—101no—
437C2×C4⋊C4—101no—
438C2×C4⋊C4C22×D511yescharacteristic
439C2×C22⋊C4—52no—
440C2×C22⋊C4—104no—
441C2×C22⋊C4—102no—
442C2×C42—21no—
443C2×C42C22×D511yescharacteristic
444#107—54no—
445#107C22×D514yes—
446C4⋊C8—52no—
447C4⋊C8C22×D512yes—
448#103—54no—
449#103—58no—
450#103—58no—
451#103—54no—
452#103C22×D514yes—
453#103C22×D514yes—
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—24no—
470C4×C5—21no—
471C4×C5—42no—
472C4×C5—42no—
473C4×C5—42no—
474C4×C5C22×Q1612yes—
475C4×C5C22×Q1612yes—
476C4×C5C2×H(5,𝔽2)12yes—
477Dic5—24no—
478Dic5—42no—
479Dic5—44no—
480Dic5—42no—
481C24—52no—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—54no—
486C2×Q8—52no—
487C2×Q8—108no—
488C2×Q8—104no—
489C2×Q8—102no—
490C2×Q8—101no—
491C2×Q8—1016no—
492C2×Q8—204no—
493C2×Q8—208no—
494C2×Q8—208no—
495C2×Q8—208no—
496C2×Q8—2016no—
497C2×Q8C23×D512yes—
498C2×Q8C23×D512yes—
499C2×D4—52no—
500C2×D4—52no—
501C2×D4—58no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—52no—
505C22×C4—54no—
506C22×C4—54no—
507C22×C4—108no—
508C22×C4—104no—
509C22×C4—104no—
510C22×C4—102no—
511C22×C4—204no—
512C22×C4—208no—
513C22×C4D4×D511yescharacteristic
514C22×C4D4×D511yescharacteristic
515C22×C4C23×D511yescharacteristic
516Q16—28no—
517Q16—108no—
518Q16—208no—
519Q16—208no—
520C2×C8—54no—
521C2×C8—208no—
522C2×C8C23×D514yes—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—52no—
532C4⋊C4—52no—
533C4⋊C4—52no—
534C4⋊C4—104no—
535C4⋊C4—104no—
536C4⋊C4—104no—
537C4⋊C4—108no—
538C4⋊C4—102no—
539C4⋊C4—102no—
540C4⋊C4—102no—
541C4⋊C4—102no—
542C4⋊C4—102no—
543C4⋊C4—101no—
544C4⋊C4—101no—
545C4⋊C4—102no—
546C4⋊C4—101no—
547C4⋊C4C23×D512yes—
548C4⋊C4C23×D512yes—
549C22⋊C4—22no—
550C22⋊C4—54no—
551C22⋊C4—54no—
552C22⋊C4—108no—
553C22⋊C4—104no—
554C22⋊C4—104no—
555C22⋊C4—102no—
556C42—22no—
557C42—22no—
558C42—22no—
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—22no—
576Q8—24no—
577Q8—81no—
578Q8—104no—
579Q8—108no—
580Q8—2016no—
581Q8—204no—
582Q8—401no—
583Q8—404no—
584Q8#414914yes—
585D4—58no—
586D4—58no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—54no—
596C2×C4—108no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—104no—
600C2×C4—204no—
601C2×C4—2016no—
602C2×C4—202no—
603C2×C4—204no—
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—24no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—104no—
632C4—204no—
633C4—202no—
634C4—202no—
635C4C2×D5×Q1612yes—
636C4C2×D5×Q1612yes—
637C4C2×#414912yes—
638C2—22no—
639C2—108no—
640C2#1061794011yescharacteristic
641C2C22×D5×Q1611yescharacteristic
642C2#1061910211yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3104C2
socleC22×C5C23×D4
prime radicalC22×D20D4
hypercenter#3104D5
hypocenterC5#61563
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. C5×#3104C2semidirect
4. #10615101C2semidirect
5. #10614178C2semidirect
6. #10613219C2semidirect
7. #10612850C2semidirect
8. #10612221C2semidirect
9. D5×#451C2semidirect
10. #10611466C2semidirect
11. Q16×D20C2semidirect
12. #10611044C2semidirect
17. C2×D5×Q16C22semidirect
40. C5×#477C22semidirect
41. C5×#462C22semidirect
43. C4×C5×Q16C22semidirect
46. #65475C22semidirect
47. #65382C22semidirect
48. #65381C22semidirect
50. #65320C22semidirect
51. #65320C22semidirect
52. C4×#689C22semidirect
53. #65310C22semidirect
56. #65176C22semidirect
57. #65173C22semidirect
58. #65138C22semidirect
60. #65102C22semidirect
61. #65097C22semidirect
62. #65097C22semidirect
63. D5×#103C22semidirect
66. #64988C22semidirect
68. #64970C22semidirect
176. #3104D5semidirect
196. C2×D20Q16semidirect
295. #477C2×D5semidirect
301. #462C2×D5semidirect
310. C4×Q16C2×D5semidirect
320. C22×D5C2×Q16semidirect
335. C2×C4×C5C2×Q16semidirect
348. D20C2×Q16semidirect
349. 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×Q8×D202
C5×#31042
#106151012
#106141782
#106132192
#106128502
#106122212
D5×#4512
#106114662
Q16×D202
#106110442
#106110172
#615635