Finite Group Databasebeta

← #12027807

Subgroups — #12027807

Order 1280.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4070C211yesmaximal, characteristic
2C2×#64935C212yesmaximal, complemented
3C5×C22⋊#95C211yesmaximal, characteristic
4#10602920C211yesmaximal, characteristic
5#10602340C212yesmaximal
6#10602338C212yesmaximal
7#10600215C212yesmaximal, complemented
8#10600202C212yesmaximal, complemented
9#10600199C212yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C22⋊C4C2212yes—
13C22×C22⋊Dic5C2212yes—
14C22×C22⋊Dic5C2211yescharacteristic
15C22×C22⋊Dic5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C2×#4070—28no—
18C2×#4070C418yescomplemented
19C22×#683C2212yes—
20C22×#683C2212yes—
21C22×C4×Dic5C2212yes—
22C2×C5×#95C2212yes—
23C2×C5×#95C2211yescharacteristic
24C2×C5×#95C2211yescharacteristic
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33#64935—28no—
34#14449—51nomaximal, Sylow
35C25×C5C2311yescharacteristic
36C23×C4×C5—24no—
37C23×C4×C5—42no—
38C23×C4×C5—41no—
39C23×C4×C5—41no—
40C23×C4×C5C2×C414yes—
41C23×C4×C5C2312yes—
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C23×Dic5C2312yes—
49C22×Dic10—41no—
50C22×Dic10—42no—
51C22×Dic10—41no—
52C22×Dic10C2×C414yes—
53C22×Dic10D418yes—
54C2×C5×C22⋊C4—24no—
55C2×C5×C22⋊C4—24no—
56C2×C5×C22⋊C4—42no—
57C2×C22⋊Dic5—24no—
58C2×C22⋊Dic5—24no—
59C2×C22⋊Dic5—22no—
60C2×C22⋊Dic5—41no—
61C2×C22⋊Dic5—41no—
62C2×C22⋊Dic5—41no—
63C2×C22⋊Dic5—41no—
64C2×C22⋊Dic5C2×C418yes—
65C2×C22⋊Dic5Q814yescomplemented
66C2×C4⋊Dic5—24no—
67C2×C4⋊Dic5C2×C414yes—
68#4070—232no—
69C2×#683—24no—
70C2×#683—24no—
71C2×#683—24no—
72C2×#683—42no—
73C2×#683—42no—
74C2×#683C2×C418yes—
75C2×C4×Dic5—24no—
76C2×C4×Dic5—42no—
77C2×C4×Dic5—42no—
78C5×#95—24no—
79C5×#95—41no—
80C5×#95—41no—
81#3963—24no—
82#3963—24no—
83#3963—24no—
84#3963—24no—
85#3963—24no—
86#3963—42no—
87#3963—42no—
88#3963—42no—
89C22×C22⋊Q8—51no—
90C2×#391—52no—
91#1445—52no—
92#1441—52no—
93#1439—52no—
94#1437—52no—
95#1436—52no—
96C22⋊#95—51no—
97C22⋊#95D511yescharacteristic
98C24×C5—22no—
99C24×C5—22no—
100C24×C5—21no—
101C24×C5—21no—
102C24×C5—21no—
103C24×C5—22no—
104C24×C5—22no—
105C24×C5—21no—
106C24×C5C22×C412yes—
107C24×C5C2×Q812yes—
108C24×C5#4112yes—
109C24×C5C2411yescharacteristic
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—21no—
119C22×C4×C5—21no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—28no—
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—44no—
132C22×C4×C5—44no—
133C22×C4×C5—48no—
134C22×C4×C5—48no—
135C22×C4×C5C22×C412yes—
136C22×C4×C5C22×C412yes—
137C22×C4×C5C2×D414yes—
138C22×C4×C5#4114yes—
139C22×Dic5—22no—
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—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5C22×C414yes—
163C22×Dic5C22×C414yes—
164C22×Dic5C2×D414yes—
165C22×Dic5C2×D414yes—
166C22×Dic5C2×Q814yes—
167C2×Dic10—44no—
168C2×Dic10—44no—
169C2×Dic10—48no—
170C2×Dic10—48no—
171C2×Dic10C22⋊C4116yescomplemented
172C5×C22⋊C4—48no—
173C22⋊Dic5—216no—
174C22⋊Dic5—44no—
175C22⋊Dic5—44no—
176C4⋊Dic5—28no—
177#683—216no—
178#683—48no—
179C4×Dic5—48no—
180C23×Q8—51no—
181C24×C4C2×D511yescharacteristic
182C2×C22⋊Q8—58no—
183C2×C22⋊Q8—108no—
184C22×C4⋊C4—52no—
185C22×C4⋊C4—51no—
186C22×C4⋊C4—52no—
187C22×C22⋊C4—52no—
188C22×C22⋊C4—51no—
189C22×C22⋊C4—51no—
190C22×C22⋊C4C2×D512yes—
191C22×C42—52no—
192#391—108no—
193C2×#95—52no—
194C2×#95—52no—
195C2×#95—52no—
196C2×#95—52no—
197C2×#95—52no—
198C2×#95—52no—
199C2×#95—52no—
200C2×#95—52no—
201C2×#95C2×D512yes—
202C2×#95C2×D511yescharacteristic
203C2×#95C2×D511yescharacteristic
204C23×C5—22no—
205C23×C5—22no—
206C23×C5—22no—
207C23×C5—22no—
208C23×C5—22no—
209C23×C5—22no—
210C23×C5—22no—
211C23×C5—22no—
212C23×C5—22no—
213C23×C5—22no—
214C23×C5—24no—
215C23×C5—24no—
216C23×C5—24no—
217C23×C5—24no—
218C23×C5—42no—
219C23×C5—42no—
220C23×C5—42no—
221C23×C5—42no—
222C23×C5—42no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5C4×Q814yes—
227C23×C5C23×C411yescharacteristic
228C23×C5C22×D412yes—
229C23×C5C22×Q811yescharacteristic
230C23×C5C2×#4112yes—
231C23×C5C2×#4111yescharacteristic
232C23×C5H(5,𝔽2)11yescharacteristic
233C23×C5H(5,𝔽2)11yescharacteristic
234C23×C5H(5,𝔽2)12yes—
235C23×C5H(5,𝔽2)11yescharacteristic
236C23×C52−1+412yes—
237C23×C52−1+411yescharacteristic
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—24no—
246C2×C4×C5—24no—
247C2×C4×C5—24no—
248C2×C4×C5—416no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5C2×C22⋊C414yes—
263C2×Dic5—24no—
264C2×Dic5—24no—
265C2×Dic5—24no—
266C2×Dic5—24no—
267C2×Dic5—24no—
268C2×Dic5—24no—
269C2×Dic5—24no—
270C2×Dic5—24no—
271C2×Dic5—24no—
272C2×Dic5—24no—
273C2×Dic5—24no—
274C2×Dic5—24no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
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×Dic5C2×C22⋊C418yes—
291Dic10—416no—
292C25C22×D511yescharacteristic
293C22×Q8—54no—
294C22×Q8—58no—
295C22×Q8—201no—
296C22×Q8—202no—
297C22×Q8—201no—
298C23×C4—24no—
299C23×C4—41no—
300C23×C4—41no—
301C23×C4—42no—
302C23×C4—52no—
303C23×C4—52no—
304C23×C4—52no—
305C23×C4—52no—
306C23×C4C4×D514yes—
307C23×C4C22×D512yes—
308C23×C4C22×D511yescharacteristic
309C23×C4C22×D512yes—
310C23×C4C22×D511yescharacteristic
311C22⋊Q8—1032no—
312C2×C4⋊C4—58no—
313C2×C4⋊C4—54no—
314C2×C4⋊C4—104no—
315C2×C4⋊C4—104no—
316C2×C4⋊C4—104no—
317C2×C4⋊C4—104no—
318C2×C4⋊C4—202no—
319C2×C4⋊C4—202no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—58no—
324C2×C22⋊C4—54no—
325C2×C22⋊C4—104no—
326C2×C22⋊C4—104no—
327C2×C22⋊C4—102no—
328C2×C22⋊C4—201no—
329C2×C22⋊C4—201no—
330C2×C22⋊C4—201no—
331C2×C22⋊C4—201no—
332C2×C42—104no—
333C2×C42—202no—
334C2×C42—202no—
335#95—24no—
336#95—41no—
337#95—41no—
338#95—104no—
339#95—104no—
340#95—104no—
341#95—104no—
342#95—104no—
343#95—202no—
344#95—202no—
345#95—202no—
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—24no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—44no—
354C22×C5—44no—
355C22×C5—44no—
356C22×C5—44no—
357C22×C5C22×C22⋊C411yescharacteristic
358C22×C5C2×C4×Q811yescharacteristic
359C22×C5C4×#4111yescharacteristic
360C22×C5#51811yescharacteristic
361C22×C5#51811yescharacteristic
362C22×C5#51911yescharacteristic
363C22×C5#52011yescharacteristic
364C22×C5#53412yes—
365C22×C5#53412yes—
366C22×C5#53811yescharacteristic
367C22×C5#53912yes—
368C22×C5#53911yescharacteristic
369C22×C5#54212yes—
370C22×C5#54311yescharacteristic
371C22×C5#54411yescharacteristic
372C22×C5#54812yes—
373C22×C5#54812yes—
374C22×C5D4×Q812yes—
375C22×C5#55012yes—
376C22×C5#55112yes—
377C22×C5#55412yes—
378C22×C5#56212yes—
379C22×C5#56312yes—
380C4×C5—24no—
381C4×C5—42no—
382C4×C5—44no—
383C4×C5—82no—
384Dic5—28no—
385Dic5—44no—
386Dic5—44no—
387Dic5—82no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—22no—
395C24—21no—
396C24C2×C4×D512yes—
397C24#71112yes—
398C24Q8×D512yes—
399C24C23×D511yescharacteristic
400C2×Q8—516no—
401C2×Q8—204no—
402C2×Q8—204no—
403C2×Q8—208no—
404C2×Q8—208no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—54no—
431C22×C4—54no—
432C22×C4—54no—
433C22×C4—54no—
434C22×C4—54no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4—102no—
441C22×C4—102no—
442C22×C4—102no—
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—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4C2×C4×D512yes—
459C22×C4C2×C4×D512yes—
460C22×C4#70914yes—
461C22×C4D4×D514yes—
462C4⋊C4—1016no—
463C4⋊C4—108no—
464C4⋊C4—208no—
465C22⋊C4—48no—
466C22⋊C4—1016no—
467C22⋊C4—204no—
468C22⋊C4—204no—
469C42—208no—
470C2×C5—44no—
471C2×C5#230811yescharacteristic
472C2×C5#232911yescharacteristic
473C2×C5#233211yescharacteristic
474C2×C5Q8×C22⋊C411yescharacteristic
475C2×C5#234111yescharacteristic
476C2×C5#236811yescharacteristic
477C2×C5#237811yescharacteristic
478C2×C5#272812yes—
479C2×C5#272912yes—
480C2×C5#273012yes—
481C2×C5#273212yes—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23#403314yes—
505C23C22×C4×D511yescharacteristic
506C23C2×#70912yes—
507C23C2×D4×D512yes—
508C23C2×#71111yescharacteristic
509C23#414412yes—
510C23#414411yescharacteristic
511C23#414411yescharacteristic
512C23C2×Q8×D511yescharacteristic
513C23#414911yescharacteristic
514C23#415012yes—
515C23#415011yescharacteristic
516Q8—2016no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—58no—
542C2×C4—104no—
543C2×C4—104no—
544C2×C4—104no—
545C2×C4—104no—
546C2×C4—104no—
547C2×C4—104no—
548C2×C4—104no—
549C2×C4—104no—
550C2×C4—104no—
551C2×C4—104no—
552C2×C4—104no—
553C2×C4—104no—
554C2×C4—202no—
555C2×C4—202no—
556C2×C4—202no—
557C2×C4—202no—
558C2×C4—202no—
559C2×C4—202no—
560C2×C4—202no—
561C2×C4—202no—
562C2×C4—202no—
563C2×C4—202no—
564C2×C4—202no—
565C2×C4—202no—
566C2×C4—202no—
567C2×C4—202no—
568C2×C4—202no—
569C2×C4D5×C22⋊C414yes—
570C5#1444911yescharacteristic, complemented, Sylow
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22C4×#70911yescharacteristic
583C22C2×D5×C22⋊C411yescharacteristic
584C22#6582011yescharacteristic
585C22#6582011yescharacteristic
586C22#6582912yes—
587C22C2×#403311yescharacteristic
588C22#6584512yes—
589C22#6585011yescharacteristic
590C22#6585711yescharacteristic
591C22#6588612yes—
592C22#6588712yes—
593C22#6589512yes—
594C22#6589812yes—
595C22#6592312yes—
596C22#6592312yes—
597C22#6593212yes—
598C22#6593611yescharacteristic
599C22#6595111yescharacteristic
600C22#6595711yescharacteristic
601C22#6596612yes—
602C22#6596811yescharacteristic
603C22#6598512yes—
604C22#6600112yes—
605C4—24no—
606C4—42no—
607C4—44no—
608C4—82no—
609C4—108no—
610C4—204no—
611C4—204no—
612C4—402no—
613C2—44no—
614C2#1060507011yescharacteristic
615C2#1060517911yescharacteristic
616C2#1060518511yescharacteristic
617C2#1060558011yescharacteristic
618C2#1060560511yescharacteristic
619C2#1060560711yescharacteristic
620C2#1060579311yescharacteristic
621C2#1060627012yes—
622C2#1060627212yes—
623C2#1060655812yes—
624C2#1060655912yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×C22⋊#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22⋊#95D5
hypocenterC5#14449
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#64935C2semidirect
7. #10600215C2semidirect
8. #10600202C2semidirect
9. #10600199C2semidirect
18. C2×#4070C4semidirect
65. C2×C22⋊Dic5Q8semidirect
171. C2×Dic10C22⋊C4semidirect
570. C5#14449semidirect

Maximal subgroups

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

subgroupindex
C22×#40702
C2×#649352
C5×C22⋊#952
#106029202
#106023402
#106023382
#106002152
#106002022
#106001992
#144495