Finite Group Databasebeta

← #12021995

Subgroups — #12021995

Order 1280.

1862 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
1C22×#4023C211yesmaximal, characteristic
2C2×#65221C211yesmaximal, characteristic, complemented
3C2×#64939C211yesmaximal, characteristic, complemented
4C5×#1437C211yesmaximal, characteristic
5#10602340C212yesmaximal
6#10602338C211yesmaximal, characteristic
7#10602337C212yesmaximal
8#10602329C211yesmaximal, characteristic, complemented
9#10600202C212yesmaximal, complemented
10#10600198C211yesmaximal, characteristic, complemented
11#10600196C212yesmaximal, complemented
12C24×Dic5C2211yescharacteristic
13C22×C5×C22⋊C4C2212yes—
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C22×C22⋊Dic5C2212yes—
17C22×C22⋊Dic5C2211yescharacteristic
18C22×#683C2211yescharacteristic
19C22×#683C2211yescharacteristic
20C22×#683C2212yes—
21C22×C4×Dic5C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C22×C4×Dic5C2211yescharacteristic
24C2×#4023—24no—
25C2×#4023—24no—
26C2×#4023C418yescomplemented
27C2×C5×#95C2211yescharacteristic
28C2×C5×#95C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2211yescharacteristic
37C2×#3963C2212yes—
38#65221—24no—
39#64939—24no—
40#13957—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2312yes—
45C23×C4×C5C2311yescharacteristic
46C23×Dic5—24no—
47C23×Dic5—22no—
48C23×Dic5—22no—
49C23×Dic5—24no—
50C23×Dic5C2×C414yes—
51C23×Dic5C2312yes—
52C23×Dic5C2311yescharacteristic
53C23×Dic5C2312yes—
54C23×Dic5C2311yescharacteristic
55C23×Dic5C2311yescharacteristic
56C23×Dic5C2311yescharacteristic
57C2×C5×C22⋊C4—22no—
58C2×C5×C22⋊C4—22no—
59C2×C5×C22⋊C4—24no—
60C2×C5×C22⋊C4—24no—
61C2×C5×C22⋊C4—42no—
62C2×C5×C22⋊C4C2×C414yes—
63C2×C42×C5—22no—
64C2×C42×C5—24no—
65C2×C22⋊Dic5—22no—
66C2×C22⋊Dic5—22no—
67C2×C22⋊Dic5—24no—
68C2×C22⋊Dic5—24no—
69C2×C22⋊Dic5—42no—
70C2×C22⋊Dic5C2×C414yes—
71C2×#683—22no—
72C2×#683—22no—
73C2×#683—22no—
74C2×#683—22no—
75C2×#683—24no—
76C2×#683—42no—
77C2×#683—42no—
78C2×#683C2×C414yes—
79C2×#683C2×C414yes—
80C2×C4×Dic5—22no—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—22no—
83C2×C4×Dic5—24no—
84C2×C4×Dic5C2×C414yes—
85C2×C4×Dic5C2×C414yes—
86C2×C4×Dic5D414yes—
87C2×C4×Dic5D414yes—
88#4023—232no—
89C5×#95—22no—
90C5×#95—42no—
91#3963—22no—
92#3963—24no—
93#3963—24no—
94#3963—24no—
95#3963—24no—
96#3963—24no—
97#3963—24no—
98#3963—24no—
99#3963—42no—
100C22×#117—51no—
101C2×#382—51no—
102C2×#376—51no—
103#1441—52no—
104#1439—52no—
105#1437—51no—
106#1437D511yescharacteristic
107#1436—51no—
108#1436—51no—
109#1434—52no—
110C22⋊#95—52no—
111C24×C5—21no—
112C24×C5—21no—
113C24×C5—22no—
114C24×C5—22no—
115C24×C5—22no—
116C24×C5—22no—
117C24×C5—22no—
118C24×C5C22×C412yes—
119C24×C5#4114yes—
120C24×C5C2411yescharacteristic
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C22×C4×C5—21no—
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×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5C22×C412yes—
141C22×C4×C5C22×C412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D412yes—
144C22×C4×C5#4114yes—
145C22×Dic5—216no—
146C22×Dic5—216no—
147C22×Dic5—28no—
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—28no—
156C22×Dic5—28no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5C22×C412yes—
175C22×Dic5C22×C412yes—
176C22×Dic5C22×C412yes—
177C22×Dic5C22×C412yes—
178C22×Dic5C2×D412yes—
179C22×Dic5C2×D412yes—
180C22×Dic5C2×D412yes—
181C22×Dic5C2×D412yes—
182C22×Dic5#4114yes—
183C22×Dic5#4114yes—
184C5×C22⋊C4—28no—
185C5×C22⋊C4—48no—
186C42×C5—28no—
187C22⋊Dic5—28no—
188C22⋊Dic5—48no—
189#683—28no—
190#683—28no—
191#683—48no—
192C4×Dic5—28no—
193C4×Dic5—28no—
194C4×Dic5C22⋊C4116yescomplemented
195C24×C4—51no—
196C2×#117—58no—
197C2×#117—104no—
198C2×#117—104no—
199C22×C4⋊C4—51no—
200C22×C4⋊C4—51no—
201C22×C4⋊C4—52no—
202C22×C22⋊C4—51no—
203C22×C22⋊C4—52no—
204C22×C22⋊C4C2×D512yes—
205C22×C22⋊C4C2×D511yescharacteristic
206C22×C42—51no—
207C22×C42—51no—
208C22×C42—51no—
209C22×C42C2×D511yescharacteristic
210#382—104no—
211#376—104no—
212C2×#95—52no—
213C2×#95—52no—
214C2×#95—52no—
215C2×#95—52no—
216C2×#95—52no—
217C2×#95—52no—
218C2×#95—51no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95C2×D511yescharacteristic
222C2×#95C2×D512yes—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—24no—
226C23×C5—24no—
227C23×C5—24no—
228C23×C5—24no—
229C23×C5—24no—
230C23×C5—24no—
231C23×C5—24no—
232C23×C5—24no—
233C23×C5—41no—
234C23×C5—41no—
235C23×C5—42no—
236C23×C5—41no—
237C23×C5—41no—
238C23×C5—42no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5—42no—
243C23×C5#11714yes—
244C23×C5C23×C411yescharacteristic
245C23×C5C22×D411yescharacteristic
246C23×C5C22×D411yescharacteristic
247C23×C5C2×#4112yes—
248C23×C5C2×#4112yes—
249C23×C5C2×#4112yes—
250C23×C5C2×#4112yes—
251C23×C5H(5,𝔽2)12yes—
252C23×C52−1+412yes—
253C2×C4×C5—24no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—24no—
257C2×C4×C5—22no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—24no—
262C2×C4×C5—42no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5C2×C22⋊C414yes—
277C2×Dic5—232no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—24no—
285C2×Dic5—24no—
286C2×Dic5—24no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—24no—
290C2×Dic5—22no—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—24no—
295C2×Dic5—42no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5C2×C22⋊C414yes—
310C2×Dic5C2×C22⋊C414yes—
311C2×Dic5#11714yes—
312C25C22×D511yescharacteristic
313C23×C4—51no—
314C23×C4—51no—
315C23×C4—51no—
316C23×C4—54no—
317C23×C4—51no—
318C23×C4—52no—
319C23×C4—52no—
320C23×C4—102no—
321C23×C4—102no—
322C23×C4—104no—
323C23×C4—104no—
324C23×C4C22×D512yes—
325C23×C4C22×D511yescharacteristic
326C23×C4C22×D512yes—
327C23×C4C22×D511yescharacteristic
328#117—1032no—
329C2×C4⋊C4—54no—
330C2×C4⋊C4—54no—
331C2×C4⋊C4—102no—
332C2×C4⋊C4—102no—
333C2×C4⋊C4—102no—
334C2×C4⋊C4—102no—
335C2×C4⋊C4—104no—
336C2×C4⋊C4—202no—
337C2×C4⋊C4—202no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—54no—
344C2×C22⋊C4—102no—
345C2×C22⋊C4—102no—
346C2×C22⋊C4—104no—
347C2×C22⋊C4—104no—
348C2×C22⋊C4—202no—
349C2×C22⋊C4C4×D514yes—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—54no—
353C2×C42—54no—
354C2×C42—54no—
355C2×C42—54no—
356C2×C42—102no—
357C2×C42—102no—
358C2×C42—104no—
359C2×C42—102no—
360#95—22no—
361#95—42no—
362#95—104no—
363#95—104no—
364#95—104no—
365#95—104no—
366#95—104no—
367#95—104no—
368#95—102no—
369#95—104no—
370#95—202no—
371C22×C5—28no—
372C22×C5—24no—
373C22×C5—42no—
374C22×C5—44no—
375C22×C5—44no—
376C22×C5—44no—
377C22×C5—44no—
378C22×C5—44no—
379C22×C5—42no—
380C22×C5C22×C22⋊C411yescharacteristic
381C22×C5C2×#11711yescharacteristic
382C22×C5C2×#11711yescharacteristic
383C22×C5C4×#4112yes—
384C22×C5#52012yes—
385C22×C5#52512yes—
386C22×C5#52512yes—
387C22×C5#52511yescharacteristic
388C22×C5#52511yescharacteristic
389C22×C5#53212yes—
390C22×C5#53511yescharacteristic
391C22×C5#53611yescharacteristic
392C22×C5#53712yes—
393C22×C5#55112yes—
394C22×C5#55112yes—
395C22×C5#55212yes—
396C22×C5#55212yes—
397C22×C5#55212yes—
398C22×C5#55212yes—
399C22×C5#55512yes—
400C22×C5#55512yes—
401C4×C5—24no—
402C4×C5—42no—
403C4×C5—44no—
404C4×C5—82no—
405Dic5—24no—
406Dic5—24no—
407Dic5—24no—
408Dic5—44no—
409Dic5—42no—
410Dic5—82no—
411C24—21no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24C2×C4×D512yes—
419C24#70914yes—
420C24C23×D511yescharacteristic
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
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—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—52no—
441C22×C4—52no—
442C22×C4—52no—
443C22×C4—52no—
444C22×C4—52no—
445C22×C4—52no—
446C22×C4—52no—
447C22×C4—54no—
448C22×C4—52no—
449C22×C4—54no—
450C22×C4—108no—
451C22×C4—108no—
452C22×C4—1016no—
453C22×C4—1016no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—108no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—101no—
465C22×C4—101no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4—102no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4C2×C4×D512yes—
480C22×C4C2×C4×D512yes—
481C22×C4D4×D512yes—
482C22×C4D4×D512yes—
483C22×C4#71114yes—
484C4⋊C4—108no—
485C4⋊C4—108no—
486C4⋊C4—208no—
487C22⋊C4—28no—
488C22⋊C4—48no—
489C22⋊C4—108no—
490C22⋊C4—208no—
491C42—28no—
492C42—516no—
493C42—108no—
494C42—108no—
495C2×C5—44no—
496C2×C5#228011yescharacteristic
497C2×C5#229711yescharacteristic
498C2×C5#231111yescharacteristic
499C2×C5#234312yes—
500C2×C5#234612yes—
501C2×C5#260912yes—
502C2×C5#260912yes—
503C2×C5#261012yes—
504C2×C5#261112yes—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23#401814yes—
526C23C22×C4×D511yescharacteristic
527C23C2×#70912yes—
528C23C2×D4×D511yescharacteristic
529C23C2×D4×D511yescharacteristic
530C23C2×#71112yes—
531C23#414412yes—
532C23D5×#4112yes—
533C23D5×#4112yes—
534C23#415012yes—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—54no—
559C2×C4—54no—
560C2×C4—54no—
561C2×C4—1032no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—102no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—104no—
570C2×C4—104no—
571C2×C4—104no—
572C2×C4—104no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—104no—
576C2×C4—104no—
577C2×C4—104no—
578C2×C4—104no—
579C2×C4—202no—
580C2×C4—202no—
581C2×C4—201no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4—201no—
585C2×C4—202no—
586C2×C4—202no—
587C2×C4—202no—
588C2×C4—202no—
589C2×C4—202no—
590C2×C4—202no—
591C2×C4—202no—
592C2×C4—202no—
593C2×C4D5×C22⋊C414yes—
594C5#1395711yescharacteristic, complemented, Sylow
595C22—28no—
596C22—24no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22C2×#401811yescharacteristic
605C22C2×D5×C22⋊C411yescharacteristic
606C22#6582912yes—
607C22D5×#11711yescharacteristic
608C22#6586512yes—
609C22C4×#71112yes—
610C22#6587512yes—
611C22#6588012yes—
612C22#6588112yes—
613C22#6588112yes—
614C22#6589012yes—
615C22#6592011yescharacteristic
616C22#6593012yes—
617C22#6594212yes—
618C22#6595012yes—
619C22#6598412yes—
620C22#6599011yescharacteristic
621C22#6600412yes—
622C22#6600911yescharacteristic
623C22#6604212yes—
624C22#6604911yescharacteristic
625C4—24no—
626C4—42no—
627C4—44no—
628C4—82no—
629C4—104no—
630C4—104no—
631C4—104no—
632C4—204no—
633C4—202no—
634C4—402no—
635C2—44no—
636C2#1060518612yes—
637C2#1060529211yescharacteristic
638C2#1060638911yescharacteristic
639C2#1060651012yes—
640C2#1060720412yes—
641C2#1060738212yes—
642C2#1060763012yes—
643C2#1060802511yescharacteristic
644C2#1060810312yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1437C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenter#1437D5
hypocenterC5#13957
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#65221C2semidirect
3. C2×#64939C2semidirect
8. #10602329C2semidirect
9. #10600202C2semidirect
10. #10600198C2semidirect
11. #10600196C2semidirect
26. C2×#4023C4semidirect
194. C4×Dic5C22⋊C4semidirect
594. C5#13957semidirect

Maximal subgroups

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

subgroupindex
C22×#40232
C2×#652212
C2×#649392
C5×#14372
#106023402
#106023382
#106023372
#106023292
#106002022
#106001982
#106001962
#139575