Finite Group Databasebeta

← #12021312

Subgroups — #12021312

Order 1280.

1862 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#95C211yesmaximal, characteristic
2C2×#65501C211yesmaximal, characteristic
3C2×C4×C22⋊Dic5C211yesmaximal, characteristic
4#10602920C211yesmaximal, characteristic
5#10602920C211yesmaximal, characteristic
6#10602920C211yesmaximal, characteristic
7#10602920C211yesmaximal, characteristic
8#10600209C212yesmaximal, complemented
9#10600205C212yesmaximal, complemented
10#10600200C212yesmaximal, complemented
11#10600194C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×C4×C5C2211yescharacteristic
14C24×C4×C5C2211yescharacteristic
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2211yescharacteristic
19C22×C4⋊Dic5C2212yes—
20C22×C4×Dic5C2212yes—
21C2×C5×#95C418yescomplemented
22C2×C5×#95C2211yescharacteristic
23C2×C5×#95C2211yescharacteristic
24C2×C5×#95C2211yescharacteristic
25C2×C5×#95C2211yescharacteristic
26#65501—24no—
27C4×C22⋊Dic5—24no—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38#13883—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C23×C4×C5—22no—
41C23×C4×C5—22no—
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—22no—
45C23×C4×C5—22no—
46C23×C4×C5—24no—
47C23×C4×C5—24no—
48C23×C4×C5—42no—
49C23×C4×C5—42no—
50C23×C4×C5—42no—
51C23×C4×C5—42no—
52C23×C4×C5C2×C414yes—
53C23×C4×C5C2×C414yes—
54C23×C4×C5C2×C414yes—
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2311yescharacteristic
59C23×C4×C5C2311yescharacteristic
60C23×C4×C5C2311yescharacteristic
61C23×Dic5C2312yes—
62C23×Dic5C2312yes—
63C23×Dic5C2312yes—
64C23×Dic5C2312yes—
65C2×C22⋊Dic5—22no—
66C2×C22⋊Dic5—22no—
67C2×C22⋊Dic5—22no—
68C2×C22⋊Dic5—22no—
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⋊Dic5D414yes—
76C2×C22⋊Dic5Q814yes—
77C2×C4⋊Dic5—24no—
78C2×C4⋊Dic5—24no—
79C2×C4⋊Dic5—42no—
80C2×C4×Dic5—24no—
81C2×C4×Dic5—24no—
82C2×C4×Dic5—42no—
83C5×#95C2×C414yes—
84C5×#95C2×C414yes—
85C5×#95C2×C414yes—
86C5×#95C2×C414yes—
87#3963—24no—
88#3963—24no—
89#3963—24no—
90#3963—24no—
91#3963—42no—
92#3963—42no—
93#3963—42no—
94#3963—42no—
95#3963—42no—
96#3963—42no—
97C2×#380—51no—
98C2×C4×C22⋊C4—51no—
99C22×#95D511yescharacteristic
100#1443—52no—
101#1442—52no—
102#1436—52no—
103#1434—52no—
104C22⋊#95—51no—
105C22⋊#95—51no—
106C22⋊#95—51no—
107C22⋊#95—51no—
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—22no—
112C24×C5C22×C412yes—
113C24×C5C2×D412yes—
114C24×C5C2×Q812yes—
115C24×C5#4112yes—
116C24×C5#4112yes—
117C24×C5#4112yes—
118C24×C5#4112yes—
119C24×C5C2411yescharacteristic
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—28no—
127C22×C4×C5—28no—
128C22×C4×C5—28no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—28no—
146C22×C4×C5—28no—
147C22×C4×C5—44no—
148C22×C4×C5—44no—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5—48no—
152C22×C4×C5—48no—
153C22×C4×C5—48no—
154C22×C4×C5—48no—
155C22×C4×C5—48no—
156C22×C4×C5—48no—
157C22×C4×C5C22×C412yes—
158C22×C4×C5C22×C412yes—
159C22×C4×C5C22×C412yes—
160C22×C4×C5C22×C412yes—
161C22×C4×C5C22×C412yes—
162C22×C4×C5C22×C412yes—
163C22×C4×C5#4114yes—
164C22×C4×C5#4114yes—
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×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—22no—
186C22×Dic5—22no—
187C22×Dic5—22no—
188C22×Dic5—22no—
189C22×Dic5—22no—
190C22×Dic5—22no—
191C22×Dic5C2×D414yes—
192C22×Dic5C2×Q814yes—
193C22⋊Dic5—28no—
194C22⋊Dic5—28no—
195C22⋊Dic5—44no—
196C22⋊Dic5—44no—
197C4⋊Dic5—48no—
198C4×Dic5—48no—
199C24×C4C2×D511yescharacteristic
200C24×C4C2×D511yescharacteristic
201C24×C4C2×D511yescharacteristic
202C22×C4⋊C4—52no—
203C22×C22⋊C4—51no—
204C22×C22⋊C4—51no—
205C22×C22⋊C4—51no—
206C22×C22⋊C4—51no—
207C22×C42—52no—
208#380—104no—
209C4×C22⋊C4—104no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95—52no—
213C2×#95—52no—
214C2×#95—52no—
215C2×#95—52no—
216C2×#95—52no—
217C2×#95—52no—
218C2×#95—52no—
219C2×#95—52no—
220C2×#95Dic518yescomplemented
221C2×#95C2×D511yescharacteristic
222C2×#95C2×D511yescharacteristic
223C2×#95C2×D511yescharacteristic
224C2×#95C2×D511yescharacteristic
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—22no—
229C23×C5—24no—
230C23×C5—24no—
231C23×C5—24no—
232C23×C5—24no—
233C23×C5—24no—
234C23×C5—24no—
235C23×C5—24no—
236C23×C5—24no—
237C23×C5—24no—
238C23×C5—24no—
239C23×C5—24no—
240C23×C5—24no—
241C23×C5#11714yes—
242C23×C5C4×D414yes—
243C23×C5C4×Q814yes—
244C23×C5C22⋊Q814yes—
245C23×C5#12314yes—
246C23×C5#12514yes—
247C23×C5#12614yes—
248C23×C5C23×C411yescharacteristic
249C23×C5C22×D411yescharacteristic
250C23×C5C22×Q811yescharacteristic
251C23×C5C2×#4111yescharacteristic
252C23×C5C2×#4111yescharacteristic
253C23×C5C2×#4112yes—
254C23×C5C2×#4112yes—
255C23×C5C2×#4111yescharacteristic
256C23×C5C2×#4111yescharacteristic
257C23×C5H(5,𝔽2)12yes—
258C23×C5H(5,𝔽2)12yes—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—416no—
280C2×C4×C5—416no—
281C2×C4×C5—416no—
282C2×C4×C5—42no—
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#11714yes—
291C2×C4×C5#11714yes—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—24no—
295C2×Dic5—24no—
296C2×Dic5—24no—
297C2×Dic5—24no—
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×Dic5—42no—
310C2×Dic5—42no—
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—42no—
320C25C22×D511yescharacteristic
321C23×C4—22no—
322C23×C4—22no—
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—24no—
328C23×C4—24no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4—42no—
332C23×C4—42no—
333C23×C4—52no—
334C23×C4—52no—
335C23×C4—52no—
336C23×C4—52no—
337C23×C4C2×Dic514yes—
338C23×C4C2×Dic514yes—
339C23×C4C2×Dic514yes—
340C23×C4C22×D511yescharacteristic
341C23×C4C22×D511yescharacteristic
342C23×C4C22×D511yescharacteristic
343C23×C4C22×D511yescharacteristic
344C23×C4C22×D511yescharacteristic
345C23×C4C22×D511yescharacteristic
346C2×C4⋊C4—104no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4—202no—
349C2×C22⋊C4—54no—
350C2×C22⋊C4—54no—
351C2×C22⋊C4—102no—
352C2×C22⋊C4—102no—
353C2×C22⋊C4—102no—
354C2×C22⋊C4—102no—
355C2×C22⋊C4—102no—
356C2×C22⋊C4—102no—
357C2×C22⋊C4—102no—
358C2×C22⋊C4—102no—
359C2×C22⋊C4—102no—
360C2×C22⋊C4—102no—
361C2×C42—104no—
362C2×C42—104no—
363C2×C42—202no—
364#95—104no—
365#95—104no—
366#95—104no—
367#95—104no—
368#95—202no—
369#95—202no—
370#95—202no—
371#95—202no—
372#95—202no—
373#95—202no—
374#95C2×Dic514yes—
375#95C2×Dic514yes—
376#95C2×Dic514yes—
377#95C2×Dic514yes—
378C22×C5—24no—
379C22×C5—24no—
380C22×C5—24no—
381C22×C5—24no—
382C22×C5—24no—
383C22×C5—28no—
384C22×C5—28no—
385C22×C5—28no—
386C22×C5—28no—
387C22×C5—24no—
388C22×C5#38718yes—
389C22×C5C2×#11711yescharacteristic
390C22×C5C2×#11711yescharacteristic
391C22×C5C2×#11711yescharacteristic
392C22×C5C2×C4×D411yescharacteristic
393C22×C5C2×C4×Q811yescharacteristic
394C22×C5#51811yescharacteristic
395C22×C5#51811yescharacteristic
396C22×C5C2×C22⋊Q811yescharacteristic
397C22×C5C2×#12311yescharacteristic
398C22×C5C2×#12511yescharacteristic
399C22×C5C2×#12611yescharacteristic
400C22×C5#52912yes—
401C22×C5#52912yes—
402C22×C5#53812yes—
403C22×C5#53812yes—
404C22×C5#54612yes—
405C22×C5#54612yes—
406C22×C5#55112yes—
407C22×C5#55112yes—
408C22×C5#55312yes—
409C22×C5#55312yes—
410C22×C5#55412yes—
411C22×C5#55412yes—
412C4×C5—24no—
413C4×C5—24no—
414C4×C5—42no—
415C4×C5—42no—
416C4×C5—42no—
417C4×C5—42no—
418Dic5—44no—
419Dic5—44no—
420Dic5—82no—
421Dic5—82no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24#70912yes—
427C24#70912yes—
428C24D4×D512yes—
429C24#71112yes—
430C24Q8×D512yes—
431C24#71312yes—
432C24C22×Dic512yes—
433C24C23×D511yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—54no—
472C22×C4—54no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4—102no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—102no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—102no—
498C22×C4—102no—
499C22×C4#70914yes—
500C22×C4#70914yes—
501C22×C4C22×Dic512yes—
502C22×C4C22×Dic512yes—
503C22×C4C22×Dic512yes—
504C22×C4C22×Dic512yes—
505C22×C4C22×Dic512yes—
506C22×C4C22×Dic512yes—
507C4⋊C4—208no—
508C22⋊C4—108no—
509C22⋊C4—108no—
510C22⋊C4—204no—
511C22⋊C4—204no—
512C42—208no—
513C2×C5—28no—
514C2×C5#228311yescharacteristic
515C2×C5C2×#38711yescharacteristic
516C2×C5#231211yescharacteristic
517C2×C5#234211yescharacteristic
518C2×C5#234211yescharacteristic
519C2×C5#234511yescharacteristic
520C2×C5#234511yescharacteristic
521C2×C5#244412yes—
522C2×C5#245012yes—
523C2×C5#250512yes—
524C2×C5#251012yes—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23#403514yes—
542C23#404014yes—
543C23#404214yes—
544C23#404414yes—
545C23#407214yes—
546C23D4×Dic514yes—
547C23Q8×Dic514yes—
548C23C2×#70912yes—
549C23C2×#70912yes—
550C23C2×#70911yescharacteristic
551C23C2×#70911yescharacteristic
552C23C2×D4×D511yescharacteristic
553C23C2×#71111yescharacteristic
554C23#414412yes—
555C23#414412yes—
556C23C2×Q8×D511yescharacteristic
557C23C2×#71311yescharacteristic
558C23C23×Dic511yescharacteristic
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—104no—
591C2×C4—104no—
592C2×C4—104no—
593C2×C4—104no—
594C2×C4—104no—
595C2×C4—104no—
596C2×C4—202no—
597C2×C4—202no—
598C2×C4—202no—
599C2×C4—202no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—202no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4—202no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4—202no—
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#407214yes—
619C2×C4#407214yes—
620C5#1388311yescharacteristic, complemented, Sylow
621C22—28no—
622C22—28no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22#6527018yes—
632C22#6581312yes—
633C22#6581312yes—
634C22#6582412yes—
635C22#6582412yes—
636C22#6582812yes—
637C22#6582812yes—
638C22#6582912yes—
639C22#6582912yes—
640C22C2×#403511yescharacteristic
641C22C2×#404011yescharacteristic
642C22C2×#404211yescharacteristic
643C22#6584512yes—
644C22#6584512yes—
645C22C2×#404411yescharacteristic
646C22#6584812yes—
647C22#6584812yes—
648C22C2×#407211yescharacteristic
649C22C2×#407211yescharacteristic
650C22C2×#407211yescharacteristic
651C22C2×D4×Dic511yescharacteristic
652C22#6613211yescharacteristic
653C22#6613211yescharacteristic
654C22C2×Q8×Dic511yescharacteristic
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—204no—
662C4—204no—
663C4—402no—
664C4—402no—
665C2—28no—
666C2#1060508212yes—
667C2#1060508612yes—
668C2#1060508712yes—
669C2#1060510812yes—
670C2#1061379111yescharacteristic
671C2#1061382411yescharacteristic
672C2#1061382811yescharacteristic
673C2#1061382811yescharacteristic
674C2C2×#6527011yescharacteristic
675C2#1061389111yescharacteristic
676C2#1061389111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC22×C5×#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22×#95D5
hypocenterC5#13883
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
8. #10600209C2semidirect
9. #10600205C2semidirect
10. #10600200C2semidirect
11. #10600194C2semidirect
21. C2×C5×#95C4semidirect
220. C2×#95Dic5semidirect
620. C5#13883semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#952
C2×#655012
C2×C4×C22⋊Dic52
#106029202
#106002092
#106002052
#106002002
#106001942
#138835