Finite Group Databasebeta

← #12030015

Subgroups — #12030015

Order 1280.

1326 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
1C2×C5×#399C211yesmaximal, characteristic
2C2×#65274C211yesmaximal, characteristic, complemented
3C2×#65240C211yesmaximal, characteristic
4#10602362C211yesmaximal, characteristic, complemented
5#10602358C212yesmaximal, complemented
6#10602358C212yesmaximal, complemented
7#10602341C212yesmaximal
8#10602339C211yesmaximal, characteristic
9#10602339C212yesmaximal
10#10600205C211yesmaximal, characteristic, complemented
11#10600194C211yesmaximal, characteristic, complemented
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2212yes—
14C22×C5×C22⋊C4C2211yescharacteristic
15C22×C5×C22⋊C4C2212yes—
16C22×C22⋊Dic5C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2212yes—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4⋊Dic5C2211yescharacteristic
21C22×C4⋊Dic5C2212yes—
22C22×C4×Dic5C2211yescharacteristic
23C5×#399C418yescomplemented
24C2×C5×#95C2211yescharacteristic
25C2×#3963C2211yescharacteristic
26C2×#3963C2211yescharacteristic
27C2×#3963C2211yescharacteristic
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36#65274—24no—
37#65240—24no—
38#14628—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C23×C4×C5C2312yes—
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×Dic5C2312yes—
45C23×Dic5C2311yescharacteristic
46C23×Dic5C2311yescharacteristic
47C23×Dic5C2312yes—
48C23×Dic5C2311yescharacteristic
49C23×Dic5C2311yescharacteristic
50C2×C5×C4⋊C4—22no—
51C2×C5×C4⋊C4—22no—
52C2×C5×C4⋊C4—42no—
53C2×C5×C4⋊C4—42no—
54C2×C5×C4⋊C4C2×C414yes—
55C2×C5×C4⋊C4C2×C418yes—
56C2×C5×C22⋊C4—22no—
57C2×C5×C22⋊C4—22no—
58C2×C5×C22⋊C4—42no—
59C2×C5×C22⋊C4—42no—
60C2×C5×C22⋊C4C2×C414yes—
61C2×C5×C22⋊C4C2×C418yes—
62C2×C22⋊Dic5—22no—
63C2×C22⋊Dic5—22no—
64C2×C22⋊Dic5—42no—
65C2×C22⋊Dic5—42no—
66C2×C22⋊Dic5—42no—
67C2×C22⋊Dic5—42no—
68C2×C22⋊Dic5—42no—
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—24no—
72C2×C4⋊Dic5—42no—
73C2×C4⋊Dic5—42no—
74C2×C4⋊Dic5—42no—
75C2×C4⋊Dic5—42no—
76C2×C4×Dic5—42no—
77C2×C4×Dic5D414yes—
78C5×#95C2×C414yes—
79#3963—22no—
80#3963—22no—
81#3963—22no—
82#3963—24no—
83#3963—42no—
84#3963—42no—
85#3963—42no—
86#3963—42no—
87#3963—42no—
88#3963—42no—
89#3963—42no—
90C2×#399D511yescharacteristic
91C2×#389—51no—
92C2×#388—51no—
93#1445—51no—
94#1442—51no—
95#1441—52no—
96#1441—52no—
97#1440—52no—
98#1439—52no—
99#1439—51no—
100#1434—51no—
101C24×C5—21no—
102C24×C5—21no—
103C24×C5—22no—
104C24×C5—21no—
105C24×C5—22no—
106C24×C5—22no—
107C24×C5—21no—
108C24×C5—21no—
109C24×C5—22no—
110C24×C5—21no—
111C24×C5C22×C412yes—
112C24×C5C2411yescharacteristic
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
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—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5C22×C412yes—
132C22×C4×C5C22×C414yes—
133C22×C4×C5C22×C414yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5#4112yes—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—21no—
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—21no—
143C22×Dic5—21no—
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—21no—
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—21no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—21no—
163C22×Dic5—22no—
164C22×Dic5—21no—
165C22×Dic5—22no—
166C22×Dic5—21no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5C2×D412yes—
170C22×Dic5C2×D412yes—
171C5×C4⋊C4—28no—
172C5×C4⋊C4—48no—
173C5×C22⋊C4—28no—
174C5×C22⋊C4—48no—
175C22⋊Dic5—44no—
176C22⋊Dic5—44no—
177C22⋊Dic5—48no—
178C4⋊Dic5—48no—
179C4⋊Dic5—44no—
180C4⋊Dic5—44no—
181C4×Dic5—44no—
182C22×C4⋊C4—52no—
183C22×C4⋊C4—51no—
184C22×C4⋊C4—51no—
185C22×C4⋊C4C2×D511yescharacteristic
186C22×C4⋊C4C2×D512yes—
187C22×C22⋊C4—51no—
188C22×C22⋊C4—51no—
189C22×C22⋊C4—52no—
190C22×C22⋊C4C2×D511yescharacteristic
191C22×C22⋊C4C2×D512yes—
192C22×C42—51no—
193#399Dic518yescomplemented
194#389—104no—
195#388—104no—
196C2×#95—51no—
197C2×#95—52no—
198C2×#95—51no—
199C2×#95—51no—
200C2×#95—52no—
201C2×#95—52no—
202C2×#95—52no—
203C2×#95—52no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—52no—
207C2×#95C2×D511yescharacteristic
208C23×C5—22no—
209C23×C5—24no—
210C23×C5—22no—
211C23×C5—24no—
212C23×C5—22no—
213C23×C5—41no—
214C23×C5—41no—
215C23×C5—41no—
216C23×C5—41no—
217C23×C5—41no—
218C23×C5—41no—
219C23×C5—42no—
220C23×C5—42no—
221C23×C5—41no—
222C23×C5—41no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—42no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5C23×C411yescharacteristic
232C23×C5C22×D411yescharacteristic
233C23×C5C2×#4111yescharacteristic
234C23×C5H(5,𝔽2)12yes—
235C23×C5H(5,𝔽2)12yes—
236C23×C5H(5,𝔽2)12yes—
237C23×C5H(5,𝔽2)11yescharacteristic
238C23×C52−1+411yescharacteristic
239C23×C52−1+411yescharacteristic
240C23×C52−1+412yes—
241C23×C52−1+411yescharacteristic
242C2×C4×C5—24no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—24no—
246C2×C4×C5—22no—
247C2×C4×C5—24no—
248C2×C4×C5—24no—
249C2×C4×C5—24no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—41no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5C4×D414yes—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—22no—
270C2×Dic5—22no—
271C2×Dic5—22no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—41no—
278C2×Dic5—41no—
279C2×Dic5—42no—
280C2×Dic5—41no—
281C2×Dic5—41no—
282C2×Dic5—41no—
283C2×Dic5—41no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—42no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—42no—
290C2×Dic5—42no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—42no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C25C22×D511yescharacteristic
301C23×C4—51no—
302C23×C4—51no—
303C23×C4—52no—
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—52no—
307C23×C4C22×D511yescharacteristic
308C23×C4C22×D512yes—
309C23×C4C22×D512yes—
310C23×C4C22×D511yescharacteristic
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—102no—
316C2×C4⋊C4—104no—
317C2×C4⋊C4—102no—
318C2×C4⋊C4—202no—
319C2×C4⋊C4—202no—
320C2×C4⋊C4—202no—
321C2×C4⋊C4—202no—
322C2×C4⋊C4C2×Dic514yes—
323C2×C4⋊C4C2×Dic518yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—102no—
329C2×C22⋊C4—102no—
330C2×C22⋊C4—202no—
331C2×C22⋊C4—202no—
332C2×C22⋊C4—202no—
333C2×C22⋊C4—202no—
334C2×C22⋊C4—202no—
335C2×C22⋊C4C2×Dic514yes—
336C2×C22⋊C4C2×Dic518yes—
337C2×C42—54no—
338C2×C42—202no—
339#95—102no—
340#95—104no—
341#95—102no—
342#95—102no—
343#95—202no—
344#95—202no—
345#95—202no—
346#95—202no—
347#95—202no—
348#95—202no—
349#95—202no—
350#95C2×Dic514yes—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—42no—
354C22×C5—44no—
355C22×C5—44no—
356C22×C5—81no—
357C22×C5—81no—
358C22×C5—81no—
359C22×C5—82no—
360C22×C5—82no—
361C22×C5—81no—
362C22×C5C2×C4×D411yescharacteristic
363C22×C5#51812yes—
364C22×C5#51911yescharacteristic
365C22×C5#52012yes—
366C22×C5#52011yescharacteristic
367C22×C5#53411yescharacteristic
368C22×C5#53511yescharacteristic
369C22×C5#53511yescharacteristic
370C22×C5#53611yescharacteristic
371C22×C5#53611yescharacteristic
372C22×C5#53711yescharacteristic
373C22×C5#53912yes—
374C22×C5#54012yes—
375C22×C5#54111yescharacteristic
376C22×C5#54211yescharacteristic
377C22×C5#56012yes—
378C22×C5#56212yes—
379C22×C5#56212yes—
380C22×C5#56212yes—
381C22×C5#56212yes—
382C22×C5#56312yes—
383C22×C5#56312yes—
384C22×C5#56312yes—
385C4×C5—24no—
386C4×C5—42no—
387C4×C5—44no—
388C4×C5—44no—
389Dic5—42no—
390Dic5—42no—
391Dic5—81no—
392Dic5—81no—
393Dic5—82no—
394Dic5—82no—
395C24—21no—
396C24—21no—
397C24—22no—
398C24—22no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—22no—
403C24—21no—
404C24—22no—
405C24C22×Dic512yes—
406C24C23×D511yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—52no—
426C22×C4—52no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—101no—
435C22×C4—101no—
436C22×C4—101no—
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—101no—
446C22×C4—101no—
447C22×C4—102no—
448C22×C4—102no—
449C22×C4—101no—
450C22×C4—102no—
451C22×C4—102no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—101no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4D4×D512yes—
460C22×C4#71112yes—
461C22×C4C22×Dic514yes—
462C22×C4C22×Dic512yes—
463C22×C4C22×Dic514yes—
464C22×C4C22×Dic512yes—
465C4⋊C4—28no—
466C4⋊C4—48no—
467C4⋊C4—204no—
468C4⋊C4—204no—
469C4⋊C4—208no—
470C22⋊C4—28no—
471C22⋊C4—48no—
472C22⋊C4—204no—
473C22⋊C4—204no—
474C22⋊C4—208no—
475C42—204no—
476C2×C5—82no—
477C2×C5#232211yescharacteristic
478C2×C5#232311yescharacteristic
479C2×C5#232811yescharacteristic
480C2×C5#237612yes—
481C2×C5#237812yes—
482C2×C5#281311yescharacteristic
483C2×C5#281611yescharacteristic
484C2×C5#282011yescharacteristic
485C2×C5#282111yescharacteristic
486C2×C5#283612yes—
487C2×C5#283712yes—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23C2×D4×D511yescharacteristic
512C23C2×#71111yescharacteristic
513C23#414412yes—
514C23#414412yes—
515C23#414711yescharacteristic
516C23#414711yescharacteristic
517C23#414911yescharacteristic
518C23#414912yes—
519C23#415011yescharacteristic
520C23#415012yes—
521C23C23×Dic511yescharacteristic
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—102no—
548C2×C4—102no—
549C2×C4—102no—
550C2×C4—102no—
551C2×C4—202no—
552C2×C4—202no—
553C2×C4—202no—
554C2×C4—202no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—202no—
558C2×C4—202no—
559C2×C4—201no—
560C2×C4—201no—
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×C4—201no—
570C2×C4—201no—
571C2×C4—202no—
572C2×C4—202no—
573C2×C4—202no—
574C2×C4—202no—
575C2×C4—202no—
576C2×C4—202no—
577C2×C4—202no—
578C2×C4—202no—
579C2×C4D4×Dic514yes—
580C5#1462811yescharacteristic, complemented, Sylow
581C22—42no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22#6595911yescharacteristic
593C22#6596211yescharacteristic
594C22#6596311yescharacteristic
595C22#6597711yescharacteristic
596C22#6598012yes—
597C22#6598012yes—
598C22#6598112yes—
599C22#6598112yes—
600C22#6598211yescharacteristic
601C22#6598512yes—
602C22#6598711yescharacteristic
603C22#6599912yes—
604C22#6600012yes—
605C22#6600112yes—
606C22#6600112yes—
607C22#6604911yescharacteristic
608C22#6604911yescharacteristic
609C22#6605312yes—
610C22C2×D4×Dic511yescharacteristic
611C22#6613212yes—
612C22#6614611yescharacteristic
613C22#6616112yes—
614C22#6616111yescharacteristic
615C4—24no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—202no—
620C4—202no—
621C4—401no—
622C4—401no—
623C4—402no—
624C4—402no—
625C2—82no—
626C2#1060982111yescharacteristic
627C2#1060982211yescharacteristic
628C2#1060982311yescharacteristic
629C2#1060982612yes—
630C2#1060982712yes—
631C2#1060985611yescharacteristic
632C2#1061447511yescharacteristic
633C2#1061448112yes—
634C2#1061456711yescharacteristic
635C2#1061457112yes—
636C2#1061490311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#399C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#399D5
hypocenterC5#14628
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×#65274C2semidirect
4. #10602362C2semidirect
5. #10602358C2semidirect
6. #10602358C2semidirect
10. #10600205C2semidirect
11. #10600194C2semidirect
23. C5×#399C4semidirect
193. #399Dic5semidirect
580. C5#14628semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3992
C2×#652742
C2×#652402
#106023622
#106023582
#106023412
#106023392
#106002052
#106001942
#146285