Finite Group Databasebeta

← #12030656

Subgroups — #12030656

Order 1280.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4044C211yesmaximal, characteristic
2C2×C42×Dic5C211yesmaximal, characteristic, complemented
3C2×#64940C211yesmaximal, characteristic, complemented
4C5×#1436C211yesmaximal, characteristic, complemented
5#10602357C211yesmaximal, characteristic, complemented
6#10602330C211yesmaximal, characteristic
7#10600201C212yesmaximal
8#10600201C212yesmaximal
9#10600201C211yesmaximal, characteristic
10#10600200C212yesmaximal, complemented
11#10600199C212yesmaximal, complemented
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C42×C5C2211yescharacteristic
14C22×#685C2212yes—
15C22×#685C2212yes—
16C22×#685C2211yescharacteristic
17C22×#685C2211yescharacteristic
18C22×#685C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×#683C2211yescharacteristic
21C22×C4×Dic5C2212yes—
22C22×C4×Dic5C2212yes—
23C22×C4×Dic5C2211yescharacteristic
24C22×C4×Dic5C2211yescharacteristic
25C2×#4044—24no—
26C2×#4044—24no—
27C2×#4044C418yescomplemented
28C2×C5×#95C2212yes—
29C2×C5×#95C2212yes—
30C2×C5×#95C2211yescharacteristic
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2211yescharacteristic
38C42×Dic5—24no—
39#64940—24no—
40#14711—51nomaximal, Sylow
41C24×D5C2311yescharacteristic
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C23×C4×C5C2312yes—
47C23×Dic5C2312yes—
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2311yescharacteristic
51C23×Dic5C2312yes—
52C2×C5×C4⋊C4—22no—
53C2×C5×C4⋊C4—22no—
54C2×C5×C4⋊C4C2×C414yes—
55C2×C42×C5—22no—
56C2×C42×C5—24no—
57C2×#685—22no—
58C2×#685—22no—
59C2×#685—22no—
60C2×#685—22no—
61C2×#685—22no—
62C2×#685—22no—
63C2×#685—24no—
64C2×#685—24no—
65C2×#685—24no—
66C2×#685—24no—
67C2×#685—24no—
68C2×#685—24no—
69C2×#685C2×C414yes—
70C2×#685C2×C414yes—
71C2×#685C2×C414yes—
72C2×C4⋊Dic5—22no—
73C2×C4⋊Dic5—22no—
74C2×C4⋊Dic5C2×C414yes—
75C2×#683—22no—
76C2×#683—22no—
77C2×#683C2×C414yes—
78C2×C4×Dic5—22no—
79C2×C4×Dic5—24no—
80C2×C4×Dic5—24no—
81C2×C4×Dic5—24no—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—24no—
84C2×C4×Dic5—24no—
85C2×C4×Dic5—24no—
86C2×C4×Dic5C2×C414yes—
87C2×C4×Dic5D414yes—
88C2×C4×Dic5D414yes—
89#4044—232no—
90C5×#95—22no—
91C5×#95—24no—
92C5×#95—24no—
93#3963—22no—
94#3963—24no—
95#3963—24no—
96#3963—24no—
97#3963—24no—
98#3963—24no—
99#3963—24no—
100C22×#126—51no—
101C2×#383—51no—
102C2×C43—51no—
103#1437—51no—
104#1437—51no—
105#1437—52no—
106#1437—52no—
107#1436—52no—
108#1436—52no—
109#1436—51no—
110#1436D511yescharacteristic, complemented
111C24×C5C2411yescharacteristic
112C23×D5—21no—
113C23×D5—21no—
114C23×D5—22no—
115C23×D5—22no—
116C23×D5—22no—
117C23×D5—22no—
118C23×D5—22no—
119C23×D5—22no—
120C23×D5C22×C412yes—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
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×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5C22×C412yes—
143C22×C4×C5C22×C412yes—
144C22×C4×C5C22×C412yes—
145C22×C4×C5C2×D412yes—
146C22×C4×C5C2×D412yes—
147C22×C4×C5#4114yes—
148C22×C4×C5#4114yes—
149C22×C4×C5#4114yes—
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—21no—
159C22×Dic5—21no—
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×Dic5C22×C412yes—
170C22×Dic5C22×C412yes—
171C22×Dic5C22×C412yes—
172C22×Dic5C2×D412yes—
173C22×Dic5C2×D412yes—
174C22×Dic5C2×D412yes—
175C22×Dic5C2×D412yes—
176C22×Dic5#4114yes—
177C22×Dic5#4114yes—
178C22×Dic5#4114yes—
179C5×C4⋊C4—28no—
180C42×C5—28no—
181#685—28no—
182#685—28no—
183#685—28no—
184#685—48no—
185#685—48no—
186C4⋊Dic5—28no—
187#683—28no—
188C4×Dic5—216no—
189C4×Dic5—216no—
190C4×Dic5—28no—
191C4×Dic5C22⋊C4116yescomplemented
192C2×#126—58no—
193C2×#126—104no—
194C2×#126—104no—
195C22×C4⋊C4—51no—
196C22×C4⋊C4—51no—
197C22×C4⋊C4C2×D511yescharacteristic
198C22×C22⋊C4—51no—
199C22×C22⋊C4—51no—
200C22×C22⋊C4—51no—
201C22×C22⋊C4—52no—
202C22×C22⋊C4—52no—
203C22×C42—51no—
204C22×C42—52no—
205C22×C42—52no—
206C22×C42—51no—
207C22×C42C2×D511yescharacteristic
208#383—104no—
209C2×#95—52no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95—52no—
213C2×#95—52no—
214C2×#95—52no—
215C2×#95—51no—
216C2×#95C2×D512yes—
217C2×#95C2×D512yes—
218C2×#95C2×D511yescharacteristic
219C43—104no—
220C23×C5C23×C411yescharacteristic
221C23×C5C22×D411yescharacteristic
222C23×C5C22×D411yescharacteristic
223C23×C5C2×#4112yes—
224C23×C5C2×#4112yes—
225C23×C5C2×#4112yes—
226C23×C5C2×#4112yes—
227C23×C5C2×#4112yes—
228C23×C5C2×#4112yes—
229C22×D5—22no—
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—24no—
233C22×D5—24no—
234C22×D5—42no—
235C22×D5—41no—
236C22×D5—42no—
237C22×D5—41no—
238C22×D5—42no—
239C22×D5—42no—
240C22×D5—42no—
241C22×D5—42no—
242C22×D5—42no—
243C22×D5—42no—
244C22×D5—42no—
245C22×D5—42no—
246C22×D5—41no—
247C22×D5—42no—
248C22×D5—42no—
249C22×D5—41no—
250C2×C4×C5—24no—
251C2×C4×C5—22no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—22no—
255C2×C4×C5—24no—
256C2×C4×C5—24no—
257C2×C4×C5—24no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—42no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
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×C5—42no—
277C2×C4×C5C2×C22⋊C414yes—
278C2×C4×C5#12618yescomplemented
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—24no—
282C2×Dic5—24no—
283C2×Dic5—24no—
284C2×Dic5—24no—
285C2×Dic5—24no—
286C2×Dic5—24no—
287C2×Dic5—24no—
288C2×Dic5—24no—
289C2×Dic5—24no—
290C2×Dic5—22no—
291C2×Dic5—24no—
292C2×Dic5—22no—
293C2×Dic5—22no—
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×Dic5C2×C22⋊C414yes—
306C2×Dic5C2×C22⋊C414yes—
307C2×Dic5#12618yescomplemented
308C25—51no—
309C23×C4—51no—
310C23×C4—51no—
311C23×C4—52no—
312C23×C4—51no—
313C23×C4—52no—
314C23×C4C22×D512yes—
315C23×C4C22×D511yescharacteristic
316C23×C4C22×D511yescharacteristic
317C23×C4C22×D511yescharacteristic
318C23×C4C22×D512yes—
319#126—1032no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—54no—
323C2×C4⋊C4—54no—
324C2×C4⋊C4—102no—
325C2×C4⋊C4—102no—
326C2×C4⋊C4—102no—
327C2×C4⋊C4—102no—
328C2×C4⋊C4C4×D514yes—
329C2×C22⋊C4—54no—
330C2×C22⋊C4—54no—
331C2×C22⋊C4—54no—
332C2×C22⋊C4—102no—
333C2×C22⋊C4—102no—
334C2×C22⋊C4—102no—
335C2×C22⋊C4—102no—
336C2×C22⋊C4—102no—
337C2×C22⋊C4—102no—
338C2×C22⋊C4—104no—
339C2×C22⋊C4—104no—
340C2×C22⋊C4—104no—
341C2×C22⋊C4—104no—
342C2×C22⋊C4—104no—
343C2×C22⋊C4—104no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—54no—
347C2×C42—54no—
348C2×C42—54no—
349C2×C42—104no—
350C2×C42—104no—
351C2×C42—104no—
352C2×C42—104no—
353C2×C42—104no—
354C2×C42—104no—
355C2×C42—102no—
356C2×C42—104no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—104no—
361#95—104no—
362#95—104no—
363#95—104no—
364#95—104no—
365#95—102no—
366#95—104no—
367C22×C5C22×C22⋊C411yescharacteristic
368C22×C5C4×#4111yescharacteristic
369C22×C5C4×#4111yescharacteristic
370C22×C5C4×#4112yes—
371C22×C5C4×#4112yes—
372C22×C5#52512yes—
373C22×C5#52511yescharacteristic
374C22×C5#52512yes—
375C22×C5#52511yescharacteristic
376C22×C5C2×#12612yes—
377C22×C5C2×#12612yes—
378C22×C5#52912yes—
379C22×C5#52912yes—
380C22×C5#52912yes—
381C22×C5#52912yes—
382C22×C5#52912yes—
383C22×C5#52912yes—
384C22×C5#53211yescharacteristic
385C22×C5#53212yes—
386C22×C5#53212yes—
387C22×C5#53211yescharacteristic
388C2×D5—44no—
389C2×D5—44no—
390C2×D5—42no—
391C2×D5—42no—
392C2×D5—42no—
393C2×D5—82no—
394C2×D5—82no—
395C2×D5—82no—
396C2×D5—82no—
397C4×C5—28no—
398C4×C5—24no—
399C4×C5—42no—
400C4×C5—42no—
401C4×C5—82no—
402Dic5—28no—
403Dic5—24no—
404Dic5—24no—
405Dic5—42no—
406Dic5—82no—
407C24—52no—
408C24—102no—
409C24—101no—
410C24—101no—
411C24—102no—
412C24—102no—
413C24—102no—
414C24—102no—
415C24—102no—
416C24C23×D511yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
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—52no—
439C22×C4—52no—
440C22×C4—52no—
441C22×C4—52no—
442C22×C4—52no—
443C22×C4—52no—
444C22×C4—54no—
445C22×C4—54no—
446C22×C4—54no—
447C22×C4—52no—
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—101no—
457C22×C4—101no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4C2×C4×D512yes—
468C22×C4C2×C4×D512yes—
469C22×C4C2×C4×D512yes—
470C22×C4#70914yes—
471C22×C4D4×D512yes—
472C22×C4D4×D512yes—
473C22×C4#71114yes—
474C22×C4#71314yes—
475C4⋊C4—28no—
476C4⋊C4—108no—
477C4⋊C4—108no—
478C22⋊C4—108no—
479C22⋊C4—108no—
480C22⋊C4—108no—
481C22⋊C4—208no—
482C22⋊C4—208no—
483C42—28no—
484C42—516no—
485C42—1016no—
486C42—1016no—
487C42—108no—
488C2×C5#229711yescharacteristic
489C2×C5#229711yescharacteristic
490C2×C5#229711yescharacteristic
491C2×C5C4×#12612yes—
492C2×C5C4×#12612yes—
493C2×C5#285212yes—
494C2×C5#285212yes—
495C2×C5#286012yes—
496C2×C5#286012yes—
497D5—82no—
498C23—102no—
499C23—102no—
500C23—102no—
501C23—104no—
502C23—104no—
503C23—201no—
504C23—202no—
505C23—202no—
506C23—202no—
507C23—202no—
508C23—202no—
509C23—202no—
510C23—201no—
511C23—202no—
512C23—202no—
513C23—202no—
514C23—202no—
515C23—201no—
516C23—201no—
517C23—202no—
518C23—202no—
519C23C22×C4×D511yescharacteristic
520C23C2×#70912yes—
521C23C2×D4×D511yescharacteristic
522C23C2×D4×D511yescharacteristic
523C23C2×#71112yes—
524C23C2×#71312yes—
525C23D5×#4112yes—
526C23D5×#4112yes—
527C23D5×#4112yes—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—54no—
556C2×C4—54no—
557C2×C4—58no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—102no—
561C2×C4—104no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—104no—
565C2×C4—104no—
566C2×C4—104no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—104no—
570C2×C4—102no—
571C2×C4—104no—
572C2×C4—102no—
573C2×C4—104no—
574C2×C4—201no—
575C2×C4—202no—
576C2×C4—201no—
577C2×C4—202no—
578C2×C4—202no—
579C2×C4—202no—
580C2×C4—202no—
581C2×C4—202no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4D5×C22⋊C414yes—
585C2×C4#404418yescomplemented
586C5#1471111yescharacteristic, complemented, Sylow
587C22—204no—
588C22—202no—
589C22—204no—
590C22—202no—
591C22—202no—
592C22—402no—
593C22—402no—
594C22—402no—
595C22—402no—
596C22C4×#70911yescharacteristic
597C22C2×D5×C22⋊C411yescharacteristic
598C22C2×#404412yes—
599C22#6585611yescharacteristic
600C22#6586212yes—
601C22C4×#71112yes—
602C22#6587212yes—
603C22#6588112yes—
604C22#6588212yes—
605C22C4×#71312yes—
606C22#6591412yes—
607C22#6592011yescharacteristic
608C22#6593012yes—
609C22#6595812yes—
610C22#6596112yes—
611C22#6598312yes—
612C22#6599011yescharacteristic
613C22#6600211yescharacteristic
614C22#6602512yes—
615C22D5×#12612yes—
616C22#6605911yescharacteristic
617C4—24no—
618C4—28no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—108no—
623C4—104no—
624C4—104no—
625C4—202no—
626C4—402no—
627C2—402no—
628C2#1060529411yescharacteristic
629C2C4×#404412yes—
630C2#1060631812yes—
631C2#1060638911yescharacteristic
632C2#1060667312yes—
633C2#1060717712yes—
634C2#1060755112yes—
635C2#1060792212yes—
636C2#1060820911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1436C2
socle, cosocleC24×C5C24
prime radicalC24×D5C23
hypercenter#1436D5
hypocenterC5#14711
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
2. C2×C42×Dic5C2semidirect
3. C2×#64940C2semidirect
4. C5×#1436C2semidirect
5. #10602357C2semidirect
10. #10600200C2semidirect
11. #10600199C2semidirect
27. C2×#4044C4semidirect
110. #1436D5semidirect
191. C4×Dic5C22⋊C4semidirect
278. C2×C4×C5#126semidirect
307. C2×Dic5#126semidirect
585. C2×C4#4044semidirect
586. C5#14711semidirect

Maximal subgroups

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

subgroupindex
C22×#40442
C2×C42×Dic52
C2×#649402
C5×#14362
#106023572
#106023302
#106002012
#106002002
#106001992
#147115