Finite Group Databasebeta

← #12026323

Subgroups — #12026323

Order 1280.

1454 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#393C211yesmaximal, characteristic
2C2×#65519C211yesmaximal, characteristic, complemented
3C2×#65240C211yesmaximal, characteristic
4#10602927C211yesmaximal, characteristic, complemented
5#10602341C212yesmaximal
6#10602339C211yesmaximal, characteristic
7#10602338C212yesmaximal
8#10600209C212yesmaximal, complemented
9#10600205C211yesmaximal, characteristic, complemented
10#10600200C212yesmaximal, complemented
11Dic5×#95C211yesmaximal, characteristic, complemented
12C23×C5×Q8C2211yescharacteristic
13C22×C5×C22⋊C4C2211yescharacteristic
14C22×C5×C22⋊C4C2212yes—
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C22⋊Dic5C2212yes—
18C22×C4⋊Dic5C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4×Dic5C2212yes—
21C22×C4×Dic5C2211yescharacteristic
22C5×#393C418yescomplemented
23C2×C5×#95C2211yescharacteristic
24C2×C5×#95C2212yes—
25#65519—24no—
26C2×#3963C2212yes—
27C2×#3963C2211yescharacteristic
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2211yescharacteristic
31C2×#3963C2211yescharacteristic
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37#65240—24no—
38#14323—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C22×C5×Q8—22no—
41C22×C5×Q8—22no—
42C22×C5×Q8—42no—
43C22×C5×Q8—42no—
44C22×C5×Q8C2×C414yes—
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2312yes—
48C23×C4×C5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2312yes—
51C23×Dic5C2311yescharacteristic
52C23×Dic5C2311yescharacteristic
53C23×Dic5C2312yes—
54C23×Dic5C2311yescharacteristic
55C2×C5×C22⋊C4—22no—
56C2×C5×C22⋊C4—22no—
57C2×C5×C22⋊C4—42no—
58C2×C5×C22⋊C4—42no—
59C2×C5×C22⋊C4C2×C414yes—
60C2×C5×C22⋊C4C2×C418yes—
61C2×C22⋊Dic5—22no—
62C2×C22⋊Dic5—22no—
63C2×C22⋊Dic5—24no—
64C2×C22⋊Dic5—24no—
65C2×C22⋊Dic5—42no—
66C2×C22⋊Dic5—42no—
67C2×C22⋊Dic5—42no—
68C2×C4⋊Dic5—22no—
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—24no—
71C2×C4⋊Dic5—42no—
72C2×C4×Dic5—24no—
73C2×C4×Dic5—42no—
74C2×C4×Dic5—42no—
75C2×C4×Dic5—42no—
76C2×C4×Dic5D414yes—
77C5×#95C2×C418yes—
78C5×#95C2×C414yes—
79#3963—22no—
80#3963—22no—
81#3963—22no—
82#3963—24no—
83#3963—24no—
84#3963—24no—
85#3963—24no—
86#3963—42no—
87#3963—42no—
88#3963—42no—
89#3963—42no—
90C2×#393D511yescharacteristic
91C2×#391—51no—
92C2×#388—51no—
93#1446—51no—
94#1443—52no—
95#1442—51no—
96#1440—52no—
97#1439—51no—
98#1437—52no—
99#1436—52no—
100C4×#95—51no—
101C24×C5—21no—
102C24×C5—22no—
103C24×C5—22no—
104C24×C5—21no—
105C24×C5—21no—
106C24×C5—22no—
107C24×C5—22no—
108C24×C5—21no—
109C24×C5—21no—
110C24×C5—21no—
111C24×C5C22×C412yes—
112C24×C5C2411yescharacteristic
113C2×C5×Q8—28no—
114C2×C5×Q8—48no—
115C2×C5×Q8—44no—
116C2×C5×Q8—44no—
117C2×C5×Q8—48no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—21no—
121C22×C4×C5—22no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5C22×C414yes—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C22×C414yes—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C2×D412yes—
141C22×C4×C5#4112yes—
142C22×Dic5—22no—
143C22×Dic5—21no—
144C22×Dic5—21no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—21no—
153C22×Dic5—22no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5—22no—
168C22×Dic5—21no—
169C22×Dic5—21no—
170C22×Dic5—22no—
171C22×Dic5—21no—
172C22×Dic5C2×D412yes—
173C22×Dic5C2×D412yes—
174C22×Dic5#4114yes—
175C22×Dic5#4114yes—
176C5×C22⋊C4—28no—
177C5×C22⋊C4—48no—
178C22⋊Dic5—44no—
179C22⋊Dic5—48no—
180C22⋊Dic5—44no—
181C4⋊Dic5—44no—
182C4⋊Dic5—44no—
183C4×Dic5—44no—
184C4×Dic5—48no—
185C23×Q8C2×D511yescharacteristic
186C22×C4⋊C4—51no—
187C22×C4⋊C4—51no—
188C22×C22⋊C4—51no—
189C22×C22⋊C4—52no—
190C22×C22⋊C4—51no—
191C22×C22⋊C4C2×D511yescharacteristic
192C22×C22⋊C4C2×D512yes—
193C22×C42—51no—
194C22×C42—52no—
195#393Dic518yescomplemented
196#391—104no—
197#388—104no—
198C2×#95—51no—
199C2×#95—52no—
200C2×#95—52no—
201C2×#95—52no—
202C2×#95—52no—
203C2×#95—51no—
204C2×#95—51no—
205C2×#95—52no—
206C2×#95—52no—
207C2×#95—52no—
208C2×#95—52no—
209C2×#95C2×D511yescharacteristic
210C2×#95C2×D512yes—
211C23×C5—22no—
212C23×C5—24no—
213C23×C5—24no—
214C23×C5—22no—
215C23×C5—22no—
216C23×C5—41no—
217C23×C5—42no—
218C23×C5—41no—
219C23×C5—41no—
220C23×C5—41no—
221C23×C5—41no—
222C23×C5—41no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—41no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5—41no—
232C23×C5—42no—
233C23×C5—42no—
234C23×C5C23×C411yescharacteristic
235C23×C5C22×D411yescharacteristic
236C23×C5C2×#4111yescharacteristic
237C23×C5C2×#4112yes—
238C23×C5C2×#4112yes—
239C23×C5H(5,𝔽2)12yes—
240C23×C5H(5,𝔽2)12yes—
241C23×C5H(5,𝔽2)11yescharacteristic
242C23×C52−1+411yescharacteristic
243C23×C52−1+411yescharacteristic
244C23×C52−1+411yescharacteristic
245C5×Q8—416no—
246C2×C4×C5—24no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—24no—
250C2×C4×C5—22no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—24no—
257C2×C4×C5—24no—
258C2×C4×C5—41no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5C4×D414yes—
270C2×Dic5—22no—
271C2×Dic5—22no—
272C2×Dic5—24no—
273C2×Dic5—24no—
274C2×Dic5—24no—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—24no—
279C2×Dic5—24no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—41no—
287C2×Dic5—41no—
288C2×Dic5—42no—
289C2×Dic5—41no—
290C2×Dic5—42no—
291C2×Dic5—41no—
292C2×Dic5—41no—
293C2×Dic5—41no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5#12414yes—
304C25C22×D511yescharacteristic
305C22×Q8—22no—
306C22×Q8—22no—
307C22×Q8—42no—
308C22×Q8—42no—
309C22×Q8C2×Dic514yes—
310C23×C4—51no—
311C23×C4—51no—
312C23×C4—52no—
313C23×C4—52no—
314C23×C4—51no—
315C23×C4—51no—
316C23×C4C22×D511yescharacteristic
317C23×C4C22×D512yes—
318C23×C4C22×D512yes—
319C23×C4C22×D511yescharacteristic
320C2×C4⋊C4—102no—
321C2×C4⋊C4—104no—
322C2×C4⋊C4—102no—
323C2×C4⋊C4—202no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—104no—
329C2×C22⋊C4—104no—
330C2×C22⋊C4—102no—
331C2×C22⋊C4—102no—
332C2×C22⋊C4—202no—
333C2×C22⋊C4—202no—
334C2×C22⋊C4—202no—
335C2×C22⋊C4C2×Dic514yes—
336C2×C22⋊C4C2×Dic518yes—
337C2×C42—54no—
338C2×C42—104no—
339C2×C42—202no—
340C2×C42—202no—
341C2×C42—202no—
342#95—104no—
343#95—104no—
344#95—102no—
345#95—104no—
346#95—104no—
347#95—102no—
348#95—102no—
349#95—202no—
350#95—202no—
351#95—202no—
352#95—202no—
353#95C2×Dic518yes—
354#95C2×Dic514yes—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—44no—
358C22×C5—44no—
359C22×C5—42no—
360C22×C5—81no—
361C22×C5—81no—
362C22×C5—82no—
363C22×C5—82no—
364C22×C5—81no—
365C22×C5—81no—
366C22×C5C2×C4×D411yescharacteristic
367C22×C5C4×#4112yes—
368C22×C5#51812yes—
369C22×C5#51911yescharacteristic
370C22×C5#52011yescharacteristic
371C22×C5C2×#12411yescharacteristic
372C22×C5#52912yes—
373C22×C5#53211yescharacteristic
374C22×C5#53411yescharacteristic
375C22×C5#53511yescharacteristic
376C22×C5#53611yescharacteristic
377C22×C5#53711yescharacteristic
378C22×C5#53812yes—
379C22×C5#53812yes—
380C22×C5#53812yes—
381C22×C5#53912yes—
382C22×C5#53912yes—
383C22×C5#54111yescharacteristic
384C22×C5#54212yes—
385C22×C5#54212yes—
386C22×C5#54211yescharacteristic
387C22×C5#54212yes—
388C22×C5#54212yes—
389C4×C5—24no—
390C4×C5—44no—
391C4×C5—44no—
392C4×C5—42no—
393Dic5—24no—
394Dic5—44no—
395Dic5—42no—
396Dic5—81no—
397Dic5—81no—
398Dic5—82no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—22no—
407C24—21no—
408C24—21no—
409C24C22×Dic512yes—
410C24C23×D511yescharacteristic
411C2×Q8—28no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—52no—
435C22×C4—54no—
436C22×C4—52no—
437C22×C4—54no—
438C22×C4—102no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—102no—
444C22×C4—102no—
445C22×C4—101no—
446C22×C4—102no—
447C22×C4—102no—
448C22×C4—102no—
449C22×C4—102no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—102no—
453C22×C4—101no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—101no—
466C22×C4—102no—
467C22×C4—101no—
468C22×C4D4×D512yes—
469C22×C4#71112yes—
470C22×C4C22×Dic512yes—
471C22×C4C22×Dic514yes—
472C22×C4C22×Dic514yes—
473C22×C4C22×Dic512yes—
474C4⋊C4—204no—
475C4⋊C4—204no—
476C22⋊C4—28no—
477C22⋊C4—48no—
478C22⋊C4—204no—
479C22⋊C4—204no—
480C22⋊C4—208no—
481C42—204no—
482C42—208no—
483C2×C5—82no—
484C2×C5C4×#12411yescharacteristic
485C2×C5#232111yescharacteristic
486C2×C5#232311yescharacteristic
487C2×C5#233212yes—
488C2×C5#233712yes—
489C2×C5#263911yescharacteristic
490C2×C5#264011yescharacteristic
491C2×C5#265012yes—
492C2×C5#265312yes—
493C2×C5#266211yescharacteristic
494C2×C5#266311yescharacteristic
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×D4×D511yescharacteristic
519C23C2×#71111yescharacteristic
520C23#414412yes—
521C23#414412yes—
522C23#414711yescharacteristic
523C23#414711yescharacteristic
524C23D5×#4112yes—
525C23D5×#4112yes—
526C23#414911yescharacteristic
527C23#415011yescharacteristic
528C23C23×Dic511yescharacteristic
529Q8—416no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—54no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—104no—
557C2×C4—104no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—104no—
561C2×C4—102no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—202no—
565C2×C4—202no—
566C2×C4—202no—
567C2×C4—202no—
568C2×C4—202no—
569C2×C4—202no—
570C2×C4—201no—
571C2×C4—202no—
572C2×C4—201no—
573C2×C4—201no—
574C2×C4—202no—
575C2×C4—201no—
576C2×C4—202no—
577C2×C4—202no—
578C2×C4—202no—
579C2×C4—202no—
580C2×C4—202no—
581C2×C4—202no—
582C2×C4—201no—
583C2×C4—201no—
584C2×C4—202no—
585C2×C4—202no—
586C2×C4—202no—
587C2×C4D4×Dic514yes—
588C5#1432311yescharacteristic, complemented, Sylow
589C22—42no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—81no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22#6592311yescharacteristic
601C22#6592611yescharacteristic
602C22#6592712yes—
603C22#6595911yescharacteristic
604C22#6596211yescharacteristic
605C22#6596311yescharacteristic
606C22#6597212yes—
607C22#6597212yes—
608C22#6597412yes—
609C22#6597412yes—
610C22#6597711yescharacteristic
611C22#6600211yescharacteristic
612C22#6600312yes—
613C22#6600312yes—
614C22D5×#12411yescharacteristic
615C22#6601212yes—
616C22#6601212yes—
617C22#6601412yes—
618C22C2×D4×Dic511yescharacteristic
619C22#6613212yes—
620C22#6614611yescharacteristic
621C22#41×Dic512yes—
622C22#6616111yescharacteristic
623C4—24no—
624C4—44no—
625C4—42no—
626C4—44no—
627C4—104no—
628C4—204no—
629C4—202no—
630C4—401no—
631C4—402no—
632C4—401no—
633C2—82no—
634C2#1060854611yescharacteristic
635C2#1060854712yes—
636C2#1060854812yes—
637C2#1060855011yescharacteristic
638C2#1060855111yescharacteristic
639C2#1060860311yescharacteristic
640C2#1061430911yescharacteristic
641C2#1061447511yescharacteristic
642C2#1061447712yes—
643C2Dic5×#12411yescharacteristic
644C2#1061468512yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#393C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#393D5
hypocenterC5#14323
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
2. C2×#65519C2semidirect
4. #10602927C2semidirect
8. #10600209C2semidirect
9. #10600205C2semidirect
10. #10600200C2semidirect
11. Dic5×#95C2semidirect
22. C5×#393C4semidirect
195. #393Dic5semidirect
588. C5#14323semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3932
C2×#655192
C2×#652402
#106029272
#106023412
#106023392
#106023382
#106002092
#106002052
#106002002
Dic5×#952
#143235