Finite Group Databasebeta

← #12026326

Subgroups — #12026326

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×#65263C211yesmaximal, characteristic, complemented
2C2×#65236C211yesmaximal, characteristic
3C2×#65240C211yesmaximal, characteristic
4C5×#1439C211yesmaximal, characteristic
5#10602360C211yesmaximal, characteristic, complemented
6#10602342C212yesmaximal
7#10602338C212yesmaximal
8#10600217C211yesmaximal, characteristic, complemented
9#10600210C212yesmaximal, complemented
10#10600200C212yesmaximal, complemented
11Dic5×#95C211yesmaximal, characteristic, complemented
12C23×Dic10C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2212yes—
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2212yes—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4×Dic5C2212yes—
21C22×C4×Dic5C2211yescharacteristic
22C2×C5×#95C2211yescharacteristic
23C2×C5×#95C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2211yescharacteristic
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2211yescharacteristic
34C2×#3963C2212yes—
35#65263—24no—
36#65240—24no—
37#65236C418yescomplemented
38#14323—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C23×C4×C5C2312yes—
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×Dic5C2311yescharacteristic
45C23×Dic5C2311yescharacteristic
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C22×Dic10—22no—
51C22×Dic10—22no—
52C22×Dic10—42no—
53C22×Dic10—42no—
54C22×Dic10C2×C414yes—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4—42no—
57C2×C5×C22⋊C4—22no—
58C2×C5×C22⋊C4—22no—
59C2×C5×C22⋊C4—24no—
60C2×C5×C22⋊C4—42no—
61C2×C5×C22⋊C4—42no—
62C2×C5×C22⋊C4C2×C414yes—
63C2×C22⋊Dic5—22no—
64C2×C22⋊Dic5—22no—
65C2×C22⋊Dic5—24no—
66C2×C22⋊Dic5—42no—
67C2×C22⋊Dic5—42no—
68C2×C22⋊Dic5—42no—
69C2×C22⋊Dic5C2×C418yes—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—24no—
72C2×C4×Dic5—24no—
73C2×C4×Dic5—42no—
74C2×C4×Dic5—42no—
75C2×C4×Dic5—42no—
76C2×C4×Dic5D414yes—
77C5×#95—22no—
78C5×#95—24no—
79#3963—22no—
80#3963—24no—
81#3963—22no—
82#3963—24no—
83#3963—24no—
84#3963—42no—
85#3963—42no—
86#3963—42no—
87#3963—42no—
88#3963C2×C418yes—
89#3963C2×C414yes—
90C2×#393—51no—
91C2×#391—51no—
92C2×#388—51no—
93#1446—51no—
94#1443—52no—
95#1442—51no—
96#1440—52no—
97#1439D511yescharacteristic
98#1437—52no—
99#1436—52no—
100C4×#95—51no—
101C24×C5—21no—
102C24×C5—22no—
103C24×C5—21no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5—22no—
107C24×C5—21no—
108C24×C5—21no—
109C24×C5—21no—
110C24×C5—21no—
111C24×C5C22×C412yes—
112C24×C5C2411yescharacteristic
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
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×C5C22×C412yes—
133C22×C4×C5C22×C412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5#4114yes—
136C22×C4×C5#4112yes—
137C22×Dic5—21no—
138C22×Dic5—22no—
139C22×Dic5—21no—
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C22×Dic5C22×C414yes—
167C22×Dic5C22×C414yes—
168C22×Dic5C2×D412yes—
169C22×Dic5C2×D412yes—
170C22×Dic5#4114yes—
171C2×Dic10—28no—
172C2×Dic10—48no—
173C2×Dic10—44no—
174C2×Dic10—44no—
175C2×Dic10—48no—
176C5×C4⋊C4—44no—
177C5×C22⋊C4—28no—
178C5×C22⋊C4—48no—
179C22⋊Dic5—48no—
180C22⋊Dic5—44no—
181C22⋊Dic5—44no—
182C4⋊Dic5—44no—
183C4×Dic5—44no—
184C4×Dic5—48no—
185C23×Q8—51no—
186C22×C4⋊C4—51no—
187C22×C4⋊C4C2×D511yescharacteristic
188C22×C22⋊C4—52no—
189C22×C22⋊C4—51no—
190C22×C22⋊C4—51no—
191C22×C22⋊C4C2×D511yescharacteristic
192C22×C22⋊C4C2×D512yes—
193C22×C42—51no—
194C22×C42—52no—
195#393—58no—
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—41no—
230C23×C5—42no—
231C23×C5—42no—
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)11yescharacteristic
241C23×C5H(5,𝔽2)12yes—
242C23×C52−1+411yescharacteristic
243C23×C52−1+411yescharacteristic
244C23×C52−1+411yescharacteristic
245C2×C4×C5—24no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—22no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—41no—
255C2×C4×C5—42no—
256C2×C4×C5—41no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
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×C5C4×D414yes—
269C2×Dic5—22no—
270C2×Dic5—22no—
271C2×Dic5—24no—
272C2×Dic5—24no—
273C2×Dic5—24no—
274C2×Dic5—24no—
275C2×Dic5—24no—
276C2×Dic5—24no—
277C2×Dic5—24no—
278C2×Dic5—24no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—42no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—42no—
290C2×Dic5—42no—
291C2×Dic5—42no—
292C2×Dic5—41no—
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5#12414yes—
303Dic10—416no—
304C25C22×D511yescharacteristic
305C22×Q8—54no—
306C22×Q8—102no—
307C22×Q8—102no—
308C22×Q8—202no—
309C22×Q8—202no—
310C23×C4—51no—
311C23×C4—52no—
312C23×C4—52no—
313C23×C4—51no—
314C23×C4—51no—
315C23×C4—51no—
316C23×C4C22×D512yes—
317C23×C4C22×D511yescharacteristic
318C23×C4C22×D512yes—
319C23×C4C22×D511yescharacteristic
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—102no—
323C2×C4⋊C4—104no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—58no—
330C2×C22⋊C4—104no—
331C2×C22⋊C4—102no—
332C2×C22⋊C4—102no—
333C2×C22⋊C4—202no—
334C2×C22⋊C4—202no—
335C2×C22⋊C4—202no—
336C2×C22⋊C4C4×D514yes—
337C2×C42—54no—
338C2×C42—104no—
339C2×C42—202no—
340C2×C42—202no—
341C2×C42—202no—
342#95—22no—
343#95—24no—
344#95—54no—
345#95—58no—
346#95—102no—
347#95—104no—
348#95—104no—
349#95—102no—
350#95—104no—
351#95—202no—
352#95—202no—
353#95—202no—
354#95—202no—
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#54211yescharacteristic
385C22×C5#54212yes—
386C22×C5#54212yes—
387C22×C5#54212yes—
388C22×C5#54212yes—
389C4×C5—24no—
390C4×C5—44no—
391C4×C5—42no—
392C4×C5—82no—
393Dic5—24no—
394Dic5—44no—
395Dic5—44no—
396Dic5—42no—
397Dic5—81no—
398Dic5—81no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—22no—
406C24—22no—
407C24—21no—
408C24—21no—
409C24C2×C4×D512yes—
410C24C23×D511yescharacteristic
411C2×Q8—108no—
412C2×Q8—208no—
413C2×Q8—204no—
414C2×Q8—208no—
415C2×Q8—204no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
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—22no—
435C22×C4—52no—
436C22×C4—54no—
437C22×C4—54no—
438C22×C4—54no—
439C22×C4—52no—
440C22×C4—101no—
441C22×C4—102no—
442C22×C4—101no—
443C22×C4—102no—
444C22×C4—102no—
445C22×C4—102no—
446C22×C4—102no—
447C22×C4—102no—
448C22×C4—102no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—102no—
452C22×C4—101no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—101no—
459C22×C4—101no—
460C22×C4—101no—
461C22×C4—102no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—101no—
468C22×C4—101no—
469C22×C4C2×C4×D512yes—
470C22×C4C2×C4×D512yes—
471C22×C4D4×D512yes—
472C22×C4#71114yes—
473C22×C4#71112yes—
474C4⋊C4—44no—
475C4⋊C4—204no—
476C22⋊C4—28no—
477C22⋊C4—48no—
478C22⋊C4—204no—
479C22⋊C4—208no—
480C22⋊C4—204no—
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—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23C22×C4×D511yescharacteristic
519C23C2×D4×D511yescharacteristic
520C23C2×#71112yes—
521C23C2×#71111yescharacteristic
522C23#414411yescharacteristic
523C23#414412yes—
524C23#414711yescharacteristic
525C23D5×#4112yes—
526C23#414912yes—
527C23#415011yescharacteristic
528C23#415011yescharacteristic
529Q8—2016no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
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—104no—
560C2×C4—104no—
561C2×C4—104no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—104no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—202no—
568C2×C4—202no—
569C2×C4—202no—
570C2×C4—202no—
571C2×C4—202no—
572C2×C4—202no—
573C2×C4—202no—
574C2×C4—202no—
575C2×C4—202no—
576C2×C4—202no—
577C2×C4—201no—
578C2×C4—201no—
579C2×C4—201no—
580C2×C4—202no—
581C2×C4—201no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4—201no—
585C2×C4—201no—
586C2×C4—202no—
587C2×C4#402714yes—
588C5#1432311yescharacteristic, complemented, Sylow
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—81no—
595C22—81no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—82no—
600C22C2×#402711yescharacteristic
601C22#6583811yescharacteristic
602C22#6585011yescharacteristic
603C22C4×#71112yes—
604C22#6587912yes—
605C22#6592611yescharacteristic
606C22#6592712yes—
607C22#6593412yes—
608C22#6593711yescharacteristic
609C22#6593712yes—
610C22#6594211yescharacteristic
611C22#6595112yes—
612C22#6595911yescharacteristic
613C22#6596211yescharacteristic
614C22#6597412yes—
615C22#6597712yes—
616C22#6598711yescharacteristic
617C22#6599112yes—
618C22#6599912yes—
619C22#6600312yes—
620C22D5×#12411yescharacteristic
621C22#6604112yes—
622C22#6605011yescharacteristic
623C4—24no—
624C4—44no—
625C4—42no—
626C4—82no—
627C4—104no—
628C4—204no—
629C4—204no—
630C4—202no—
631C4—401no—
632C4—401no—
633C2—82no—
634C2#1060560211yescharacteristic
635C2#1060612212yes—
636C2#1060660611yescharacteristic
637C2#1060719512yes—
638C2#1060802311yescharacteristic
639C2#1060844011yescharacteristic
640C2#1060854611yescharacteristic
641C2#1060872312yes—
642C2#1060907112yes—
643C2#1060941811yescharacteristic
644C2#1060978911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1439C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenter#1439D5
hypocenterC5#14323
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#65263C2semidirect
5. #10602360C2semidirect
8. #10600217C2semidirect
9. #10600210C2semidirect
10. #10600200C2semidirect
11. Dic5×#95C2semidirect
37. #65236C4semidirect
588. C5#14323semidirect

Maximal subgroups

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

subgroupindex
C2×#652632
C2×#652362
C2×#652402
C5×#14392
#106023602
#106023422
#106023382
#106002172
#106002102
#106002002
Dic5×#952
#143235