Finite Group Databasebeta

← #12022201

Subgroups — #12022201

Order 1280.

2358 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D20C211yesmaximal, characteristic
2C2×C5×#389C211yesmaximal, characteristic, complemented
3C2×#65277C211yesmaximal, characteristic
4C2×#65277C211yesmaximal, characteristic
5C2×#65266C211yesmaximal, characteristic, complemented
6#10602363C214yesmaximal, complemented
7#10602356C211yesmaximal, characteristic, complemented
8C4×#3963C211yesmaximal, characteristic, complemented
9#10600218C214yesmaximal, complemented
10C23×D20C2211yescharacteristic
11C23×C4×D5C2212yes—
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2212yes—
15C22×C42×C5C2211yescharacteristic
16C22×#685C2212yes—
17C22×#685C2214yes—
18C22×#685C2214yes—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C2×C4×D20—24no—
23C2×C4×D20—24no—
24C2×C4×D20—24no—
25C5×#389—24no—
26C2×C5×#95C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2214yes—
29C2×#3963C2212yes—
30C2×#3963C2214yes—
31#65277—24no—
32#65277C418yescomplemented
33#65266—24no—
34#13979—51nomaximal, Sylow
35C24×D5C2312yes—
36C23×C4×C5C2312yes—
37C23×C4×C5C2312yes—
38C23×C4×C5C2311yescharacteristic
39C23×C4×C5C2311yescharacteristic
40C23×C4×C5C2311yescharacteristic
41C23×Dic5C2312yes—
42C23×Dic5C2312yes—
43C23×Dic5C2312yes—
44C22×D20—22no—
45C22×D20—22no—
46C22×D20—22no—
47C22×D20—22no—
48C22×D20C2×C414yes—
49C22×D20D414yes—
50C22×D20Q814yes—
51C22×C4×D5—24no—
52C22×C4×D5—24no—
53C22×C4×D5—24no—
54C22×C4×D5—28no—
55C22×C4×D5—44no—
56C2×C5×C4⋊C4—22no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—24no—
59C2×C5×C4⋊C4—22no—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4—22no—
62C2×C5×C4⋊C4—44no—
63C2×C5×C4⋊C4C2×C414yescomplemented
64C2×C42×C5—22no—
65C2×C42×C5—22no—
66C2×C42×C5D414yes—
67C2×#685—24no—
68C2×#685—24no—
69C2×#685—24no—
70C2×#685—28no—
71C2×#685—28no—
72C2×#685—44no—
73C2×#685—44no—
74C2×#685—44no—
75C2×#685C2×C4116yescomplemented
76C2×C4⋊Dic5—22no—
77C2×C4⋊Dic5—22no—
78C2×C4⋊Dic5D414yes—
79C2×C4×Dic5—22no—
80C2×C4×Dic5—22no—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—22no—
83C2×C4×Dic5—22no—
84C2×C4×Dic5C2×C414yescomplemented
85C4×D20—232no—
86C5×#95—24no—
87#3963—24no—
88#3963—28no—
89#3963—24no—
90#3963—44no—
91C22×C4×D4—51no—
92C2×#390—51no—
93C2×#390—51no—
94C2×#389D511yescharacteristic, complemented
95C2×#382—51no—
96#1439—54no—
97C4⋊#95—51no—
98C22⋊#95—54no—
99C4×#95—51no—
100C24×C5C2411yescharacteristic
101C23×D5—22no—
102C23×D5—22no—
103C23×D5—24no—
104C23×D5—22no—
105C23×D5—22no—
106C23×D5—24no—
107C23×D5—24no—
108C23×D5—24no—
109C23×D5C22×C414yes—
110C23×D5C2×D414yes—
111C23×D5C2×Q814yes—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
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—24no—
122C22×C4×C5—24no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—24no—
127C22×C4×C5—24no—
128C22×C4×C5C22×C414yes—
129C22×C4×C5C22×C412yes—
130C22×C4×C5C2×D412yes—
131C22×C4×C5C2×D412yes—
132C22×C4×C5C2×D412yes—
133C22×C4×C5C2×D412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×Q812yes—
136C22×C4×C5#4112yes—
137C22×C4×C5#4114yes—
138C22×C4×C5#4112yes—
139C22×Dic5—22no—
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—24no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—24no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5C22×C414yes—
156C22×Dic5C2×D414yes—
157C22×Dic5#4114yes—
158C2×D20—28no—
159C2×D20—28no—
160C2×D20—28no—
161C2×D20—28no—
162C2×D20—28no—
163C2×D20—28no—
164C2×D20C4⋊C4116yescomplemented
165C2×C4×D5—216no—
166C2×C4×D5—416no—
167C2×C4×D5—416no—
168C2×C4×D5—48no—
169C2×C4×D5—48no—
170C5×C4⋊C4—28no—
171C5×C4⋊C4—48no—
172C5×C4⋊C4—44no—
173C42×C5—28no—
174#685—216no—
175#685—416no—
176#685—416no—
177C4⋊Dic5—28no—
178C4×Dic5—28no—
179C4×Dic5—44no—
180C23×D4—51no—
181C24×C4—52no—
182C2×C4×D4—104no—
183C2×C4×D4—104no—
184C2×C4×D4—104no—
185C22×C4⋊C4—51no—
186C22×C4⋊C4C2×D511yescharacteristic
187C22×C4⋊C4C2×D511yescharacteristic
188C22×C4⋊C4C2×D512yes—
189C22×C22⋊C4—54no—
190C22×C22⋊C4—52no—
191C22×C22⋊C4—54no—
192C22×C42—51no—
193C22×C42—51no—
194C22×C42C2×D511yescharacteristic
195#390—58no—
196#390—104no—
197#389—24no—
198#382—104no—
199C2×#95—52no—
200C2×#95—52no—
201C2×#95—54no—
202C2×#95—54no—
203C2×#95C2×D512yes—
204C23×C5C23×C411yescharacteristic
205C23×C5C22×D411yescharacteristic
206C23×C5C22×D411yescharacteristic
207C23×C5C22×D411yescharacteristic
208C23×C5C22×Q811yescharacteristic
209C23×C5C2×#4111yescharacteristic
210C23×C5C2×#4111yescharacteristic
211C23×C5C2×#4112yes—
212C23×C5C2×#4112yes—
213C23×C5H(5,𝔽2)12yes—
214C23×C5H(5,𝔽2)12yes—
215C22×D5—24no—
216C22×D5—24no—
217C22×D5—28no—
218C22×D5—28no—
219C22×D5—24no—
220C22×D5—24no—
221C22×D5—24no—
222C22×D5—24no—
223C22×D5—28no—
224C22×D5—28no—
225C22×D5—28no—
226C22×D5—28no—
227C22×D5—42no—
228C22×D5—44no—
229C22×D5—42no—
230C22×D5—42no—
231C22×D5—42no—
232C22×D5—44no—
233C22×D5—44no—
234C22×D5—44no—
235C22×D5—44no—
236C22×D5—44no—
237C22×D5C2×C4⋊C418yes—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—24no—
243C2×C4×C5—28no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5—44no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—44no—
256C2×C4×C5—44no—
257C2×C4×C5C2×C4⋊C414yes—
258C2×C4×C5C4×D414yes—
259C2×C4×C5C4×D414yes—
260C2×C4×C5#12014yes—
261C2×C4×C5C22⋊Q814yes—
262C2×C4×C5C22⋊Q814yes—
263C2×C4×C5#12314yes—
264C2×C4×C5#12414yes—
265C2×Dic5—24no—
266C2×Dic5—24no—
267C2×Dic5—24no—
268C2×Dic5—24no—
269C2×Dic5—24no—
270C2×Dic5—24no—
271C2×Dic5—28no—
272C2×Dic5—24no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—44no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284D20—232no—
285C4×D5—432no—
286C25—52no—
287C22×D4—54no—
288C22×D4—54no—
289C22×D4—54no—
290C22×D4—102no—
291C22×D4—102no—
292C22×D4—102no—
293C22×D4—102no—
294C23×C4—52no—
295C23×C4—52no—
296C23×C4—52no—
297C23×C4—104no—
298C23×C4—104no—
299C23×C4—104no—
300C23×C4—108no—
301C23×C4—204no—
302C23×C4C22×D512yes—
303C23×C4C22×D512yes—
304C23×C4C22×D511yescharacteristic
305C23×C4C22×D511yescharacteristic
306C23×C4C22×D511yescharacteristic
307C4×D4—1032no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—54no—
316C2×C4⋊C4—102no—
317C2×C4⋊C4—102no—
318C2×C4⋊C4C4×D514yescomplemented
319C2×C22⋊C4—516no—
320C2×C22⋊C4—104no—
321C2×C22⋊C4—104no—
322C2×C22⋊C4—104no—
323C2×C22⋊C4—108no—
324C2×C22⋊C4—108no—
325C2×C22⋊C4—204no—
326C2×C22⋊C4—204no—
327C2×C22⋊C4—204no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—54no—
331C2×C42—102no—
332C2×C42—102no—
333C2×C42—102no—
334C2×C42—102no—
335C2×C42—102no—
336C2×C42C22⋊D514yes—
337#95—24no—
338#95—104no—
339#95—108no—
340#95—104no—
341#95—204no—
342C22×C5C22×C4⋊C411yescharacteristic
343C22×C5C2×C4×D411yescharacteristic
344C22×C5C2×C4×D411yescharacteristic
345C22×C5C4×#4112yes—
346C22×C5#51812yes—
347C22×C5C2×#12011yescharacteristic
348C22×C5C2×C22⋊Q811yescharacteristic
349C22×C5C2×C22⋊Q811yescharacteristic
350C22×C5C2×#12311yescharacteristic
351C22×C5C2×#12411yescharacteristic
352C22×C5#52912yes—
353C22×C5#53211yescharacteristic
354C22×C5#53511yescharacteristic
355C22×C5#53511yescharacteristic
356C22×C5#54012yes—
357C22×C5#54612yes—
358C22×C5#54612yes—
359C22×C5#54612yes—
360C22×C5#55012yes—
361C22×C5#55312yes—
362C22×C5#55412yes—
363C22×C5#55412yes—
364C22×C5#55512yes—
365C2×D5—216no—
366C2×D5—28no—
367C2×D5—44no—
368C2×D5—44no—
369C2×D5—44no—
370C2×D5—48no—
371C2×D5—48no—
372C2×D5—44no—
373C2×D5—44no—
374C2×D5—48no—
375C2×D5—44no—
376C4×C5—24no—
377C4×C5—44no—
378C4×C5—42no—
379C4×C5—82no—
380C4×C5#38518yes—
381Dic5—44no—
382Dic5—44no—
383Dic5—44no—
384C24—54no—
385C24—54no—
386C24—54no—
387C24—102no—
388C24—102no—
389C24—104no—
390C24—104no—
391C24—102no—
392C24—104no—
393C24—104no—
394C24—102no—
395C24C23×D511yescharacteristic
396C2×D4—516no—
397C2×D4—108no—
398C2×D4—108no—
399C2×D4—108no—
400C2×D4—108no—
401C2×D4—108no—
402C2×D4—108no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—54no—
420C22×C4—54no—
421C22×C4—54no—
422C22×C4—1016no—
423C22×C4—102no—
424C22×C4—102no—
425C22×C4—102no—
426C22×C4—102no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—102no—
430C22×C4—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—104no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—104no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—2016no—
440C22×C4—2016no—
441C22×C4—208no—
442C22×C4—208no—
443C22×C4C2×C4×D514yes—
444C22×C4C2×C4×D512yes—
445C22×C4#70912yes—
446C22×C4D4×D512yes—
447C22×C4D4×D512yes—
448C22×C4Q8×D512yes—
449C22×C4#71314yes—
450C22×C4#71312yes—
451C22×C4C2×C22⋊D512yes—
452C22×C4C2×C22⋊D512yes—
453C22×C4C2×C22⋊D512yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—108no—
458C22⋊C4—1016no—
459C22⋊C4—2016no—
460C22⋊C4—2016no—
461C42—28no—
462C42—108no—
463C42—204no—
464C2×C5C2×#38511yescharacteristic
465C2×C5C4×#12411yescharacteristic
466C2×C5#232811yescharacteristic
467C2×C5#234912yes—
468C2×C5#236312yes—
469C2×C5#242012yes—
470C2×C5#243111yescharacteristic
471C2×C5#243511yescharacteristic
472C2×C5#245412yes—
473C2×C5#256311yescharacteristic
474C2×C5#256611yescharacteristic
475D5—48no—
476C23—58no—
477C23—108no—
478C23—104no—
479C23—104no—
480C23—104no—
481C23—104no—
482C23—104no—
483C23—108no—
484C23—104no—
485C23—108no—
486C23—108no—
487C23—108no—
488C23—108no—
489C23—204no—
490C23—202no—
491C23—202no—
492C23—204no—
493C23—202no—
494C23—202no—
495C23—204no—
496C23—204no—
497C23—204no—
498C23—204no—
499C23C22×C4×D511yescharacteristic
500C23C2×#70911yescharacteristic
501C23C2×D4×D511yescharacteristic
502C23C2×D4×D511yescharacteristic
503C23#414412yes—
504C23C2×Q8×D511yescharacteristic
505C23C2×#71312yes—
506C23C2×#71311yescharacteristic
507C23D5×#4112yes—
508C23#414912yes—
509C23C22×C22⋊D511yescharacteristic
510D4—1032no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—104no—
531C2×C4—104no—
532C2×C4—104no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—104no—
536C2×C4—108no—
537C2×C4—104no—
538C2×C4—2032no—
539C2×C4—202no—
540C2×C4—202no—
541C2×C4—202no—
542C2×C4—202no—
543C2×C4—202no—
544C2×C4—202no—
545C2×C4—202no—
546C2×C4—204no—
547C2×C4—202no—
548C2×C4—202no—
549C2×C4—202no—
550C2×C4D5×C4⋊C414yes—
551C2×C4#403914yes—
552C2×C4#404014yes—
553C2×C4#404214yes—
554C2×C4C4×C22⋊D514yes—
555C2×C4#408314yes—
556C2×C4#409214yes—
557C2×C4#409314yes—
558C5#1397911yescharacteristic, complemented, Sylow
559C22—1016no—
560C22—108no—
561C22—204no—
562C22—208no—
563C22—208no—
564C22—204no—
565C22—208no—
566C22—204no—
567C22—204no—
568C22—204no—
569C22—204no—
570C22C2×D5×C4⋊C411yescharacteristic
571C22C2×#403911yescharacteristic
572C22C2×#404011yescharacteristic
573C22C2×#404211yescharacteristic
574C22#6587912yes—
575C22#6589612yes—
576C22C4×#71312yes—
577C22#6591412yes—
578C22#6593312yes—
579C22#6594612yes—
580C22#6596412yes—
581C22#6596412yes—
582C22#6602412yes—
583C22#6602912yes—
584C22#6606412yes—
585C22#6606812yes—
586C22C2×C4×C22⋊D511yescharacteristic
587C22C2×#408311yescharacteristic
588C22#6613911yescharacteristic
589C22C2×#409211yescharacteristic
590C22C2×#409311yescharacteristic
591C22#6616411yescharacteristic
592C22#6616611yescharacteristic
593C4—24no—
594C4—44no—
595C4—42no—
596C4—82no—
597C4—204no—
598C4—204no—
599C4—204no—
600C4#6527818yes—
601C2—208no—
602C2#1060771612yes—
603C2#1060774612yes—
604C2#1060793412yes—
605C2#1060794412yes—
606C2C2×#6527811yescharacteristic
607C2#1061406011yescharacteristic
608C2C4×#409311yescharacteristic
609C2#1061438311yescharacteristic
610C2#1061449211yescharacteristic
611C2#1061477111yescharacteristic
612C2#1061499111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#389C2
socle, cosocleC24×C5C24
prime radicalC23×D20C22
hypercenterC2×#389D5
hypocenterC5#13979
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C5×#389C2semidirect
5. C2×#65266C2semidirect
6. #10602363C2semidirect
7. #10602356C2semidirect
8. C4×#3963C2semidirect
9. #10600218C2semidirect
32. #65277C4semidirect
63. C2×C5×C4⋊C4C2×C4semidirect
75. C2×#685C2×C4semidirect
84. C2×C4×Dic5C2×C4semidirect
94. C2×#389D5semidirect
164. C2×D20C4⋊C4semidirect
318. C2×C4⋊C4C4×D5semidirect
558. C5#13979semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D202
C2×C5×#3892
C2×#652772
C2×#652662
#106023632
#106023562
C4×#39632
#106002182
#139795