Finite Group Databasebeta

← #12102421

Subgroups — #12102421

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2276C211yesmaximal, characteristic, complemented
2C2×#65315C211yesmaximal, characteristic
3#10605146C211yesmaximal, characteristic
4#10602426C211yesmaximal, characteristic
5#10602426C211yesmaximal, characteristic
6C4×#3967C211yesmaximal, characteristic, complemented
7#10602422C211yesmaximal, characteristic, complemented
8#10602421C211yesmaximal, characteristic, complemented
9#10600274C211yesmaximal, characteristic, complemented
10C4×#3941C211yesmaximal, characteristic
11#10600256C211yesmaximal, characteristic
12#10600255C211yesmaximal, characteristic
13#10600254C211yesmaximal, characteristic, complemented
14#10600251C211yesmaximal, characteristic, complemented
15#10600228C211yesmaximal, characteristic, complemented
16C2×C4×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C22×#688C2211yescharacteristic
20C2×C4×D20C2211yescharacteristic
21C4×C5×C4⋊C4—21no—
22C4×C5×C4⋊C4—21no—
23C4×C5×C4⋊C4—21no—
24C4×C5×C4⋊C4C2211yescharacteristic
25C4×C5×C4⋊C4C2211yescharacteristic
26C4×C5×C4⋊C4C2211yescharacteristic
27C5×#376—21no—
28C5×#376C2211yescharacteristic
29C2×#3967C2211yescharacteristic
30C2×#3967C2211yescharacteristic
31C2×#3967C2211yescharacteristic
32#65315C418yescomplemented
33C2×#3941C2211yescharacteristic
34C2×#3941C2211yescharacteristic
35C2×#3941C2211yescharacteristic
36C2×#3940C2211yescharacteristic
37C2×#3939C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39#65228C2211yescharacteristic
40#65228C2211yescharacteristic
41C4×C4⋊Dic5C2211yescharacteristic
42#65217C2211yescharacteristic
43#65217C2211yescharacteristic
44C2×#3935C2211yescharacteristic
45C2×C4×#168C2211yescharacteristic
46#64958—21no—
47#64958—21no—
48#64956—21no—
49#64953—21no—
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64744C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#64744C2211yescharacteristic
60#21128—51nomaximal, Sylow
61C22×C5×Q8C2311yescharacteristic
62C22×D20C2311yescharacteristic
63C22×C4×D5—21no—
64C4×C5×Q8—22no—
65C4×C5×Q8—22no—
66C4×C5×Q8—22no—
67C4×C5×Q8—22no—
68C4×C5×Q8—22no—
69C4×C5×Q8C2×C414yes—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C42×C5—21no—
77C2×C42×C5—21no—
78C2×C42×C5—21no—
79C2×C42×C5—21no—
80C2×C42×C5—21no—
81C2×C42×C5—21no—
82C2×C42×C5C2311yescharacteristic
83C2×C42×C5C2311yescharacteristic
84C2×C42×C5C2311yescharacteristic
85C2×#688—22no—
86C2×#688—22no—
87C2×#688C2×C414yes—
88C2×#685—21no—
89C2×#685—21no—
90C2×#685—21no—
91C2×#685—21no—
92C2×#685—21no—
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4⋊Dic5C2311yescharacteristic
95C2×C4⋊Dic5C2311yescharacteristic
96C2×#683—21no—
97C2×#683—21no—
98C2×C4×Dic5—21no—
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2311yescharacteristic
102C22×#168C2311yescharacteristic
103C4×D20—22no—
104C4×D20C2×C414yes—
105C5×#95—21no—
106C5×#95—21no—
107C5×#95—21no—
108C5×#95D411yescharacteristic
109C5×#95D411yescharacteristic
110C5×#95D411yescharacteristic
111C5×#95D411yescharacteristic
112#3967—22no—
113#3967—22no—
114#3967C2×C414yes—
115#3963—21no—
116#3963—21no—
117#3963—21no—
118#3941—22no—
119#3941—22no—
120#3941C2×C414yes—
121#3940—22no—
122#3939—22no—
123#3939C2×C414yes—
124#3935C2×C414yes—
125C4×#168—22no—
126C2×#440—51no—
127#2319—51no—
128#2276D511yescharacteristic, complemented
129#1926—51no—
130#1925—51no—
131#1921—51no—
132#1918—51no—
133#1880—51no—
134#1880—51no—
135#1865—51no—
136#1863—51no—
137#1776—51no—
138C4×#103—51no—
139C4×#102—51no—
140#1746—51no—
141C23×D5—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8C22×C412yes—
146C2×C5×Q8#4112yes—
147C2×C5×Q8#4112yes—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2411yescharacteristic
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C2×D20—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20—21no—
166C2×D20—21no—
167C2×D20C22×C412yes—
168C2×C4×D5—24no—
169C2×C4×D5—81no—
170C2×C4×D5—81no—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4C22×C412yes—
178C5×C4⋊C4#4112yes—
179C5×C4⋊C4#4112yes—
180C42×C5—24no—
181C42×C5—24no—
182C42×C5—24no—
183C42×C5—21no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—81no—
188C42×C5—81no—
189C42×C5—81no—
190C42×C5C22×C412yes—
191C42×C5#4112yes—
192#688—28no—
193#685—24no—
194#685—81no—
195#685—81no—
196#685—81no—
197#685—81no—
198C4⋊Dic5—21no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5—21no—
203C4⋊Dic5C22×C412yes—
204#683—81no—
205#683—81no—
206C4×Dic5—81no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168C22×C412yes—
218C2×#168C22×C412yes—
219C22×SD16—51no—
220C2×C4×Q8C2×D511yescharacteristic
221C2×C4×Q8C2×D511yescharacteristic
222C2×C4×Q8C2×D511yescharacteristic
223C2×C4×D4—51no—
224#440—58no—
225C2×#107—51no—
226C2×#107—51no—
227C2×#106—51no—
228C2×#103—51no—
229C2×#103—51no—
230C2×#103—51no—
231C2×#102—51no—
232C2×#102—51no—
233C2×#102—51no—
234C2×#97—51no—
235C2×C4×C8—51no—
236#390—51no—
237#390—51no—
238#389—51no—
239#389—51no—
240#388—101no—
241#388—101no—
242#381—101no—
243#380—101no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—51no—
248C4×C4⋊C4C2×D511yescharacteristic
249C4×C4⋊C4C2×D511yescharacteristic
250C4×C4⋊C4C2×D511yescharacteristic
251#376—21no—
252#376C2×D511yescharacteristic
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#336—51no—
260#336—51no—
261#336—51no—
262#336—51no—
263C23×C5C22×D411yescharacteristic
264C22×D5—22no—
265C22×D5—41no—
266C22×D5—41no—
267C22×D5—41no—
268C22×D5—41no—
269C22×D5—41no—
270C22×D5—41no—
271C5×Q8—22no—
272C5×Q8—22no—
273C5×Q8—22no—
274C5×Q8#11714yes—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—22no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5C4×D412yes—
304C2×C4×C5C4×D412yes—
305C2×C4×C5C23×C411yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×C4×C5C2×#4111yescharacteristic
309C2×C4×C5C2×#4111yescharacteristic
310C2×C4×C5C2×#4111yescharacteristic
311C2×Dic5—22no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—41no—
320D20—22no—
321D20—22no—
322D20—41no—
323D20—41no—
324C4×D5—84no—
325#168—22no—
326#168—22no—
327#168—41no—
328#168—41no—
329C22×Q8C22×D511yescharacteristic
330C22×D4—51no—
331C23×C4—101no—
332C2×SD16—54no—
333C2×SD16—102no—
334C2×SD16—102no—
335C22×C8—51no—
336C22×C8—51no—
337C22×C8—51no—
338C22×C8—51no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8C4×D514yes—
345C4×D4—54no—
346C4×D4—102no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—51no—
351C2×C4⋊C4—51no—
352C2×C4⋊C4—51no—
353C2×C4⋊C4—101no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D511yescharacteristic
357C2×C4⋊C4C22×D511yescharacteristic
358C2×C22⋊C4—101no—
359C2×C22⋊C4—101no—
360C2×C22⋊C4—101no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—101no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—101no—
370C2×C42C22×D511yescharacteristic
371C2×C42C22×D511yescharacteristic
372C2×C42C22×D511yescharacteristic
373#107—54no—
374#107—102no—
375#106—102no—
376#103—54no—
377#103—102no—
378#103—102no—
379#102—54no—
380#102—102no—
381#102—102no—
382#97—54no—
383C4×C8—102no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—101no—
388#95—101no—
389#95—101no—
390#95C22⋊D511yescharacteristic
391#95C22⋊D511yescharacteristic
392#95C22⋊D511yescharacteristic
393#95C22⋊D511yescharacteristic
394C22×C5C2×C4×D411yescharacteristic
395C22×C5#52511yescharacteristic
396C22×C5#52511yescharacteristic
397C22×C5#57611yescharacteristic
398C22×C5#57711yescharacteristic
399C22×C5#57711yescharacteristic
400C22×C5#57811yescharacteristic
401C2×D5—42no—
402C2×D5—42no—
403C2×D5—42no—
404C2×D5—81no—
405C2×D5—81no—
406C2×D5—81no—
407C2×D5—81no—
408C4×C5—22no—
409C4×C5—22no—
410C4×C5—24no—
411C4×C5—41no—
412C4×C5—42no—
413C4×C5—41no—
414C4×C5—41no—
415C4×C5—42no—
416C4×C5—41no—
417C4×C5—42no—
418C4×C5C2×#11711yescharacteristic
419C4×C5C4×#4111yescharacteristic
420C4×C5#52911yescharacteristic
421C4×C5#52911yescharacteristic
422Dic5—42no—
423Dic5—81no—
424Dic5—81no—
425C24—101no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8C2×C4×D512yes—
430C2×Q8#70912yes—
431C2×Q8#70912yes—
432C2×D4—52no—
433C2×D4—101no—
434C2×D4—101no—
435C2×D4—101no—
436C2×D4—101no—
437C2×D4—101no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—101no—
443C22×C4—104no—
444C22×C4—101no—
445C22×C4—101no—
446C22×C4—401no—
447C22×C4—401no—
448C22×C4C2×C22⋊D511yescharacteristic
449C22×C4C2×C22⋊D511yescharacteristic
450C22×C4C2×C22⋊D511yescharacteristic
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C2×C22⋊D511yescharacteristic
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C23×D511yescharacteristic
455SD16—108no—
456C2×C8—52no—
457C2×C8—52no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8—101no—
463C2×C8—101no—
464C2×C8—101no—
465C2×C8—101no—
466C2×C8—101no—
467C2×C8—101no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—52no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—101no—
478C4⋊C4—101no—
479C4⋊C4—101no—
480C4⋊C4—401no—
481C4⋊C4—401no—
482C4⋊C4C2×C4×D512yes—
483C4⋊C4#70912yes—
484C4⋊C4#70912yes—
485C22⋊C4—104no—
486C22⋊C4—401no—
487C22⋊C4—401no—
488C22⋊C4—401no—
489C22⋊C4—401no—
490C42—24no—
491C42—24no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—24no—
497C42—81no—
498C42—81no—
499C42—81no—
500C42—401no—
501C42C2×C4×D512yes—
502C42#70912yes—
503C2×C5#229011yescharacteristic
504C2×C5#294611yescharacteristic
505C2×C5#294711yescharacteristic
506C2×C5#311911yescharacteristic
507C2×C5#312111yescharacteristic
508C2×C5#312411yescharacteristic
509C2×C5#312611yescharacteristic
510D5—82no—
511C23—102no—
512C23—201no—
513C23—201no—
514C23—201no—
515C23—201no—
516C23—201no—
517C23—201no—
518C23C22×C22⋊D511yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8#401814yes—
523D4—102no—
524D4—102no—
525D4—201no—
526D4—201no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—102no—
556C2×C4—201no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4—201no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—404no—
565C2×C4C4×C22⋊D512yes—
566C2×C4C4×C22⋊D512yes—
567C2×C4C22×C4×D511yescharacteristic
568C2×C4C2×#70911yescharacteristic
569C2×C4C2×#70911yescharacteristic
570C2×C4C2×#70911yescharacteristic
571C2×C4C2×#70911yescharacteristic
572C2×C4C2×#70911yescharacteristic
573C8—102no—
574C8—102no—
575C8—201no—
576C8—201no—
577C5#2112811yescharacteristic, complemented, Sylow
578C22—202no—
579C22—202no—
580C22—202no—
581C22—401no—
582C22—401no—
583C22—401no—
584C22—401no—
585C22C2×C4×C22⋊D511yescharacteristic
586C22#6612511yescharacteristic
587C22#6612511yescharacteristic
588C22#6616911yescharacteristic
589C22#6617111yescharacteristic
590C22#6617111yescharacteristic
591C22#6617211yescharacteristic
592C4—22no—
593C4—22no—
594C4—24no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—202no—
603C4—401no—
604C4—401no—
605C4C2×#401811yescharacteristic
606C4C4×#70911yescharacteristic
607C4#6581311yescharacteristic
608C4#6581311yescharacteristic
609C2—402no—
610C2#1061379611yescharacteristic
611C2#1061427811yescharacteristic
612C2#1061428311yescharacteristic
613C2#1061428411yescharacteristic
614C2#1061429511yescharacteristic
615C2#1061429911yescharacteristic
616C2#1061430011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2276C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#2276D5
hypocenterC5#21128
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
1. C5×#2276C2semidirect
6. C4×#3967C2semidirect
7. #10602422C2semidirect
8. #10602421C2semidirect
9. #10600274C2semidirect
13. #10600254C2semidirect
14. #10600251C2semidirect
15. #10600228C2semidirect
32. #65315C4semidirect
128. #2276D5semidirect
577. C5#21128semidirect

Maximal subgroups

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

subgroupindex
C5×#22762
C2×#653152
#106051462
#106024262
C4×#39672
#106024222
#106024212
#106002742
C4×#39412
#106002562
#106002552
#106002542
#106002512
#106002282
#211285