Finite Group Databasebeta

← #12022127

Subgroups — #12022127

Order 1280.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#95C211yesmaximal, characteristic
2C2×#65499C211yesmaximal, characteristic
3C2×#65500C211yesmaximal, characteristic
4C2×#65500C211yesmaximal, characteristic
5C2×C4×C22⋊Dic5C211yesmaximal, characteristic
6#10602920C211yesmaximal, characteristic
7#10602920C211yesmaximal, characteristic
8#10600211C212yesmaximal, complemented
9#10600207C212yesmaximal, complemented
10#10600198C212yesmaximal, complemented
11#10600196C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×C4×C5C2211yescharacteristic
14C24×C4×C5C2211yescharacteristic
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2211yescharacteristic
19C22×#683C2211yescharacteristic
20C22×#683C2211yescharacteristic
21C22×C4×Dic5C2212yes—
22C2×C5×#95—24no—
23C2×C5×#95C2212yes—
24C2×C5×#95C2212yes—
25#65500—24no—
26#65500C418yescomplemented
27#65499—24no—
28C4×C22⋊Dic5—24no—
29C2×#3963C2211yescharacteristic
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2211yescharacteristic
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38C2×#3963C2212yes—
39C2×#3963C2212yes—
40#13972—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—22no—
45C23×C4×C5—22no—
46C23×C4×C5—22no—
47C23×C4×C5—22no—
48C23×C4×C5—22no—
49C23×C4×C5—22no—
50C23×C4×C5—22no—
51C23×C4×C5—24no—
52C23×C4×C5—24no—
53C23×C4×C5—22no—
54C23×C4×C5—22no—
55C23×C4×C5—42no—
56C23×C4×C5—42no—
57C23×C4×C5C2×C414yes—
58C23×C4×C5D414yes—
59C23×C4×C5C2312yes—
60C23×C4×C5C2311yescharacteristic
61C23×C4×C5C2312yes—
62C23×C4×C5C2311yescharacteristic
63C23×Dic5C2311yescharacteristic
64C23×Dic5C2311yescharacteristic
65C23×Dic5C2312yes—
66C23×Dic5C2311yescharacteristic
67C23×Dic5C2311yescharacteristic
68C23×Dic5C2312yes—
69C2×C22⋊Dic5—22no—
70C2×C22⋊Dic5—22no—
71C2×C22⋊Dic5—22no—
72C2×C22⋊Dic5—22no—
73C2×C22⋊Dic5—22no—
74C2×C22⋊Dic5—22no—
75C2×C22⋊Dic5—22no—
76C2×C22⋊Dic5—22no—
77C2×C22⋊Dic5—22no—
78C2×C22⋊Dic5C2×C414yes—
79C2×C22⋊Dic5C2×C414yes—
80C2×C22⋊Dic5D414yes—
81C2×#683—22no—
82C2×#683—22no—
83C2×#683—24no—
84C2×#683—42no—
85C2×C4×Dic5—24no—
86C2×C4×Dic5—24no—
87C2×C4×Dic5—42no—
88C5×#95—24no—
89C5×#95—24no—
90#3963—22no—
91#3963—22no—
92#3963—24no—
93#3963—24no—
94#3963—24no—
95#3963—24no—
96#3963—24no—
97#3963—42no—
98#3963—42no—
99#3963C2×C418yescomplemented
100#3963C2×C418yescomplemented
101C2×#385—51no—
102C2×#381—51no—
103C2×#381—51no—
104C2×C4×C22⋊C4—51no—
105C22×#95D511yescharacteristic
106#1443—52no—
107#1442—52no—
108#1436—52no—
109#1434—52no—
110C22⋊#95—51no—
111C22⋊#95—51no—
112C24×C5—22no—
113C24×C5—22no—
114C24×C5—22no—
115C24×C5—22no—
116C24×C5C22×C412yes—
117C24×C5C2×D412yes—
118C24×C5C2×D412yes—
119C24×C5#4112yes—
120C24×C5#4112yes—
121C24×C5#4112yes—
122C24×C5#4112yes—
123C24×C5C2411yescharacteristic
124C22×C4×C5—21no—
125C22×C4×C5—21no—
126C22×C4×C5—22no—
127C22×C4×C5—28no—
128C22×C4×C5—28no—
129C22×C4×C5—28no—
130C22×C4×C5—28no—
131C22×C4×C5—28no—
132C22×C4×C5—22no—
133C22×C4×C5—21no—
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×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—21no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—28no—
149C22×C4×C5—44no—
150C22×C4×C5—48no—
151C22×C4×C5—48no—
152C22×C4×C5—48no—
153C22×C4×C5—48no—
154C22×C4×C5—44no—
155C22×C4×C5—44no—
156C22×C4×C5—44no—
157C22×C4×C5—44no—
158C22×C4×C5—44no—
159C22×C4×C5—44no—
160C22×C4×C5C22×C414yes—
161C22×C4×C5C2×D412yes—
162C22×C4×C5C2×D412yes—
163C22×C4×C5#4114yes—
164C22×C4×C5#4114yes—
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—21no—
169C22×Dic5—21no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5—22no—
185C22×Dic5—21no—
186C22×Dic5—21no—
187C22×Dic5—21no—
188C22×Dic5—21no—
189C22×Dic5—21no—
190C22×Dic5—21no—
191C22×Dic5—22no—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5C22×C412yes—
196C22×Dic5C22×C414yes—
197C22×Dic5C22×C412yes—
198C22×Dic5C2×D414yes—
199C22⋊Dic5—28no—
200C22⋊Dic5—28no—
201C22⋊Dic5—28no—
202C22⋊Dic5—44no—
203#683—44no—
204#683—44no—
205C4×Dic5—48no—
206C24×C4C2×D511yescharacteristic
207C24×C4C2×D511yescharacteristic
208C24×C4C2×D511yescharacteristic
209C22×C4⋊C4—51no—
210C22×C4⋊C4—51no—
211C22×C22⋊C4—51no—
212C22×C22⋊C4—51no—
213C22×C22⋊C4—51no—
214C22×C22⋊C4—51no—
215C22×C42—52no—
216#385—104no—
217#381—58no—
218#381—104no—
219C4×C22⋊C4—104no—
220C2×#95—24no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—52no—
224C2×#95—52no—
225C2×#95—51no—
226C2×#95—52no—
227C2×#95—52no—
228C2×#95—52no—
229C2×#95—51no—
230C2×#95—52no—
231C2×#95—52no—
232C2×#95C2×D512yes—
233C2×#95C2×D512yes—
234C23×C5—24no—
235C23×C5—24no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—24no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—24no—
242C23×C5—24no—
243C23×C5—24no—
244C23×C5—24no—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5—24no—
248C23×C5—24no—
249C23×C5—24no—
250C23×C5#11714yes—
251C23×C5C4×D414yes—
252C23×C5C4×D414yes—
253C23×C5#12114yes—
254C23×C5#12314yes—
255C23×C5#12414yes—
256C23×C5#12614yes—
257C23×C5C23×C411yescharacteristic
258C23×C5C22×D411yescharacteristic
259C23×C5C22×D411yescharacteristic
260C23×C5C2×#4111yescharacteristic
261C23×C5C2×#4112yes—
262C23×C5C2×#4112yes—
263C23×C5C2×#4111yescharacteristic
264C23×C5C2×#4111yescharacteristic
265C23×C5C2×#4111yescharacteristic
266C23×C5H(5,𝔽2)12yes—
267C23×C52−1+412yes—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—24no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
278C2×C4×C5—22no—
279C2×C4×C5—24no—
280C2×C4×C5—24no—
281C2×C4×C5—416no—
282C2×C4×C5—416no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×C4×C5—42no—
286C2×C4×C5—42no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—416no—
294C2×C4×C5#12314yes—
295C2×Dic5—24no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—24no—
299C2×Dic5—24no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5—24no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321C2×Dic5—41no—
322C2×Dic5—41no—
323C2×Dic5—41no—
324C2×Dic5—41no—
325C2×Dic5—42no—
326C2×Dic5—42no—
327C2×Dic5—42no—
328C2×Dic5—42no—
329C25C22×D511yescharacteristic
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—24no—
340C23×C4—24no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—42no—
344C23×C4—42no—
345C23×C4—51no—
346C23×C4—51no—
347C23×C4—52no—
348C23×C4—52no—
349C23×C4—51no—
350C23×C4—51no—
351C23×C4C4×D514yes—
352C23×C4C22⋊D514yes—
353C23×C4C22×D512yes—
354C23×C4C22×D512yes—
355C23×C4C22×D511yescharacteristic
356C23×C4C22×D511yescharacteristic
357C2×C4⋊C4—104no—
358C2×C4⋊C4—102no—
359C2×C4⋊C4—102no—
360C2×C4⋊C4—202no—
361C2×C22⋊C4—54no—
362C2×C22⋊C4—54no—
363C2×C22⋊C4—54no—
364C2×C22⋊C4—102no—
365C2×C22⋊C4—102no—
366C2×C22⋊C4—102no—
367C2×C22⋊C4—102no—
368C2×C22⋊C4—102no—
369C2×C22⋊C4—102no—
370C2×C22⋊C4—102no—
371C2×C22⋊C4—102no—
372C2×C22⋊C4—102no—
373C2×C42—104no—
374C2×C42—104no—
375C2×C42—202no—
376#95—24no—
377#95—24no—
378#95—58no—
379#95—58no—
380#95—104no—
381#95—104no—
382#95—104no—
383#95—104no—
384#95—104no—
385#95—102no—
386#95—102no—
387#95—202no—
388#95—202no—
389C22×C5—24no—
390C22×C5—24no—
391C22×C5—28no—
392C22×C5—28no—
393C22×C5—24no—
394C22×C5—24no—
395C22×C5—28no—
396C22×C5—28no—
397C22×C5—24no—
398C22×C5—24no—
399C22×C5#38818yes—
400C22×C5C2×#11711yescharacteristic
401C22×C5C2×C4×D411yescharacteristic
402C22×C5C2×C4×D411yescharacteristic
403C22×C5C4×#4112yes—
404C22×C5#52012yes—
405C22×C5C2×#12111yescharacteristic
406C22×C5C2×#12311yescharacteristic
407C22×C5C2×#12311yescharacteristic
408C22×C5#52511yescharacteristic
409C22×C5C2×#12411yescharacteristic
410C22×C5C2×#12611yescharacteristic
411C22×C5#52912yes—
412C22×C5#53411yescharacteristic
413C22×C5#53611yescharacteristic
414C22×C5#53912yes—
415C22×C5#54612yes—
416C22×C5#54712yes—
417C22×C5#55112yes—
418C22×C5#55112yes—
419C22×C5#55112yes—
420C22×C5#55212yes—
421C22×C5#55212yes—
422C22×C5#55212yes—
423C4×C5—24no—
424C4×C5—44no—
425C4×C5—42no—
426C4×C5—44no—
427Dic5—42no—
428Dic5—44no—
429Dic5—44no—
430Dic5—42no—
431Dic5—81no—
432Dic5—81no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24C2×C4×D512yes—
438C24#70912yes—
439C24D4×D512yes—
440C24#71112yes—
441C24#71112yes—
442C24#71112yes—
443C24C2×C22⋊D512yes—
444C24C23×D511yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—52no—
482C22×C4—54no—
483C22×C4—52no—
484C22×C4—54no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—101no—
492C22×C4—101no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—102no—
498C22×C4—102no—
499C22×C4—102no—
500C22×C4—102no—
501C22×C4—102no—
502C22×C4—102no—
503C22×C4—101no—
504C22×C4—101no—
505C22×C4—102no—
506C22×C4—102no—
507C22×C4—101no—
508C22×C4—101no—
509C22×C4—101no—
510C22×C4—101no—
511C22×C4—102no—
512C22×C4—102no—
513C22×C4—101no—
514C22×C4—101no—
515C22×C4C2×C4×D514yes—
516C22×C4#70914yes—
517C22×C4#70914yes—
518C22×C4C2×C22⋊D512yes—
519C22×C4C2×C22⋊D512yes—
520C4⋊C4—204no—
521C4⋊C4—204no—
522C22⋊C4—108no—
523C22⋊C4—108no—
524C22⋊C4—108no—
525C22⋊C4—204no—
526C42—208no—
527C2×C5—28no—
528C2×C5C2×#38811yescharacteristic
529C2×C5C4×#12311yescharacteristic
530C2×C5#232211yescharacteristic
531C2×C5#234312yes—
532C2×C5#235912yes—
533C2×C5#241211yescharacteristic
534C2×C5#241311yescharacteristic
535C2×C5#241912yes—
536C2×C5#249312yes—
537C2×C5#250311yescharacteristic
538C2×C5#256111yescharacteristic
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23#402314yes—
556C23#402514yes—
557C23#402714yes—
558C23#402914yes—
559C23C4×C22⋊D514yes—
560C23#408214yes—
561C23#408514yes—
562C23C22×C4×D511yescharacteristic
563C23C2×#70912yes—
564C23C2×#70912yes—
565C23C2×#70911yescharacteristic
566C23C2×D4×D511yescharacteristic
567C23C2×#71111yescharacteristic
568C23C2×#71111yescharacteristic
569C23C2×#71111yescharacteristic
570C23#414412yes—
571C23#414712yes—
572C23C22×C22⋊D511yescharacteristic
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—416no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—416no—
599C2×C4—104no—
600C2×C4—104no—
601C2×C4—104no—
602C2×C4—104no—
603C2×C4—104no—
604C2×C4—104no—
605C2×C4—104no—
606C2×C4—104no—
607C2×C4—102no—
608C2×C4—102no—
609C2×C4—202no—
610C2×C4—202no—
611C2×C4—202no—
612C2×C4—202no—
613C2×C4—202no—
614C2×C4—202no—
615C2×C4—202no—
616C2×C4—202no—
617C2×C4—202no—
618C2×C4—202no—
619C2×C4—201no—
620C2×C4—201no—
621C2×C4—201no—
622C2×C4—201no—
623C2×C4—202no—
624C2×C4—202no—
625C2×C4—202no—
626C2×C4—202no—
627C2×C4—201no—
628C2×C4—201no—
629C2×C4—201no—
630C2×C4—201no—
631C2×C4—201no—
632C2×C4—201no—
633C2×C4#407514yes—
634C5#1397211yescharacteristic, complemented, Sylow
635C22—28no—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—28no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22#6523418yes—
646C22C4×#70912yes—
647C22#6581312yes—
648C22C2×#402311yescharacteristic
649C22C2×#402511yescharacteristic
650C22C2×#402711yescharacteristic
651C22C2×#402911yescharacteristic
652C22#6582412yes—
653C22#6582912yes—
654C22#6582912yes—
655C22#6582912yes—
656C22#6583812yes—
657C22#6584112yes—
658C22#6584712yes—
659C22#6584712yes—
660C22#6584712yes—
661C22#6584912yes—
662C22C2×C4×C22⋊D511yescharacteristic
663C22C2×#407511yescharacteristic
664C22#6612511yescharacteristic
665C22C2×#408211yescharacteristic
666C22C2×#408511yescharacteristic
667C22#6613811yescharacteristic
668C22#6615011yescharacteristic
669C4—24no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—204no—
674C4—202no—
675C4—204no—
676C4—202no—
677C4—401no—
678C4—401no—
679C2—28no—
680C2#1060506612yes—
681C2#1060506812yes—
682C2#1060508412yes—
683C2#1060510612yes—
684C2C4×#407511yescharacteristic
685C2C2×#6523411yescharacteristic
686C2#1061383811yescharacteristic
687C2#1061384711yescharacteristic
688C2#1061388911yescharacteristic
689C2#1061390111yescharacteristic
690C2#1061391011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC22×C5×#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22×#95D5
hypocenterC5#13972
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
8. #10600211C2semidirect
9. #10600207C2semidirect
10. #10600198C2semidirect
11. #10600196C2semidirect
26. #65500C4semidirect
99. #3963C2×C4semidirect
100. #3963C2×C4semidirect
634. C5#13972semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#952
C2×#654992
C2×#655002
C2×C4×C22⋊Dic52
#106029202
#106002112
#106002072
#106001982
#106001962
#139725