Finite Group Databasebeta

← #12121313

Subgroups — #12121313

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#478C211yesmaximal, characteristic
2#10614756C211yesmaximal, characteristic, complemented
3D4×C4⋊Dic5C211yesmaximal, characteristic
4#10602534C211yesmaximal, characteristic, complemented
5#10602402C211yesmaximal, characteristic
6#10601406C211yesmaximal, characteristic, complemented
7#10601402C211yesmaximal, characteristic, complemented
8#10601353C211yesmaximal, characteristic, complemented
9#10601351C211yesmaximal, characteristic, complemented
10#10601260C211yesmaximal, characteristic, complemented
11#10601247C211yesmaximal, characteristic, complemented
12#10600963C211yesmaximal, characteristic
13#10600963C211yesmaximal, characteristic
14#10600945C211yesmaximal, characteristic
15Dic5×#102C211yesmaximal, characteristic
16C2×C5×#125C2211yescharacteristic
17C2×C4×C5×D4C2211yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C2×D4×Dic5C2211yescharacteristic
20C22×C4⋊Dic5—21no—
21C5×#478C418yescomplemented
22C2×C5×#107C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×#3964C2211yescharacteristic
28C2×#3964C2211yescharacteristic
29#65501—21no—
30C2×#3949C2211yescharacteristic
31C2×#3949C2211yescharacteristic
32C2×#3947C2211yescharacteristic
33C2×C8×Dic5C2211yescharacteristic
34#65274—21no—
35#65274C2211yescharacteristic
36#65274C2211yescharacteristic
37#65274C2211yescharacteristic
38#65274C2211yescharacteristic
39#65270—21no—
40#65270—21no—
41C4⋊C4×Dic5—21no—
42C4⋊C4×Dic5C2211yescharacteristic
43C4⋊C4×Dic5C2211yescharacteristic
44#65239—21no—
45#65239—21no—
46#65224C2211yescharacteristic
47C4×C4⋊Dic5C2211yescharacteristic
48C4×C4⋊Dic5C2211yescharacteristic
49C2×#3935C2211yescharacteristic
50#64771C2211yescharacteristic
51#64771C2211yescharacteristic
52#64770C2211yescharacteristic
53#64770C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#22978—51nomaximal, Sylow
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×Dic5—21no—
64C23×Dic5—21no—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C5×#125—22no—
68C5×#125C2×C414yescomplemented
69C4×C5×D4—22no—
70C4×C5×D4C2×C414yes—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C42×C5C2311yescharacteristic
78C2×C22⋊Dic5—21no—
79C2×C22⋊Dic5—21no—
80D4×Dic5—22no—
81D4×Dic5—22no—
82D4×Dic5—41no—
83D4×Dic5—41no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—21no—
90C2×C4⋊Dic5—44no—
91C2×C4⋊Dic5D411yescharacteristic
92C2×C4⋊Dic5D411yescharacteristic
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4⋊Dic5C2311yescharacteristic
95C2×C4⋊Dic5C2311yescharacteristic
96C2×C4⋊Dic5C2311yescharacteristic
97C2×C4×Dic5—21no—
98C2×C4×Dic5—21no—
99C2×C4×Dic5—21no—
100C2×C4×Dic5C2311yescharacteristic
101C2×C4×Dic5C2311yescharacteristic
102C22×#168C2311yescharacteristic
103C22×#168C2311yescharacteristic
104C5×#107C2×C414yescomplemented
105C5×#106C2×C414yescomplemented
106C5×C4⋊C8C2×C414yescomplemented
107C5×#102C2×C414yes—
108C5×#102C2×C414yes—
109#3964—22no—
110#3964Q814yes—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963—21no—
116#3963—21no—
117#3963—21no—
118#3963D411yescharacteristic
119#3963D411yescharacteristic
120#3949—41no—
121#3949—41no—
122#3947—41no—
123C8×Dic5—41no—
124#3935—22no—
125C2×#478D511yescharacteristic
126#2370—51no—
127D4×C4⋊C4—51no—
128#1992—51no—
129#1948—51no—
130#1943—51no—
131#1941—51no—
132#1920—51no—
133#1918—51no—
134#1864—51no—
135#1864—51no—
136#1771—51no—
137#1762—51no—
138#1762—51no—
139C4×#102—51no—
140C24×C5—21no—
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4C22×C412yes—
145C2×C5×D4C2×Q812yes—
146C2×C5×D4#4112yes—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—24no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2411yescharacteristic
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—42no—
161C22×Dic5—42no—
162C22×Dic5—42no—
163C22×Dic5—42no—
164C22×Dic5—81no—
165C22×Dic5—81no—
166C22×Dic5C2×D411yescharacteristic
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8C22×C412yes—
175C2×C5×C8C22×C412yes—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—24no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4C22×C412yes—
182C5×C4⋊C4C22×C412yes—
183C5×C4⋊C4C22×C412yes—
184C5×C22⋊C4—24no—
185C42×C5C22×C412yes—
186C42×C5#4112yes—
187C22⋊Dic5—42no—
188C22⋊Dic5—42no—
189C4⋊Dic5—21no—
190C4⋊Dic5—22no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—41no—
198C4⋊Dic5—42no—
199C4⋊Dic5—42no—
200C4⋊Dic5—42no—
201C4⋊Dic5—42no—
202C4⋊Dic5—81no—
203C4⋊Dic5—81no—
204C4⋊Dic5C2×Q812yes—
205C4×Dic5—21no—
206C4×Dic5—21no—
207C4×Dic5—21no—
208C4×Dic5—21no—
209C4×Dic5—42no—
210C4×Dic5—81no—
211C4×Dic5—81no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168C2×Q812yes—
218C2×#125C2×D511yescharacteristic
219C2×C4×D4—51no—
220C2×C4×D4—51no—
221C2×C4×D4C2×D511yescharacteristic
222C22×C4⋊C4—101no—
223#478Dic518yescomplemented
224C2×#107C2×D511yescharacteristic
225C2×#106—51no—
226C2×#106—51no—
227C2×#106C2×D511yescharacteristic
228C2×C4⋊C8C2×D511yescharacteristic
229C2×#102—51no—
230C2×#102—51no—
231C2×#102C2×D511yescharacteristic
232C2×#102C2×D511yescharacteristic
233C2×#97—51no—
234C2×#97—51no—
235C2×C4×C8—51no—
236#389—51no—
237#389—51no—
238#389—51no—
239#389—51no—
240#389—101no—
241#387—101no—
242#387—101no—
243#385—101no—
244#385—101no—
245#384—51no—
246#380—101no—
247C4×C4⋊C4—51no—
248C4×C4⋊C4—51no—
249C4×C4⋊C4—51no—
250C4×C4⋊C4—51no—
251C4×C4⋊C4—101no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340—51no—
260#336—51no—
261#336—51no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—22no—
265C23×C5—41no—
266C23×C5—41no—
267C23×C5—41no—
268C23×C5—41no—
269C23×C5C22×D411yescharacteristic
270C5×D4—22no—
271C5×D4—22no—
272C5×D4—22no—
273C5×D4C4×Q814yes—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—48no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5C4×D412yes—
289C2×C4×C5C4×D412yes—
290C2×C4×C5C23×C411yescharacteristic
291C2×C4×C5C22×Q811yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5H(5,𝔽2)11yescharacteristic
295C2×C4×C52−1+411yescharacteristic
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—22no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—84no—
320C2×Dic5—84no—
321C2×Dic5AGL(1,ℤ/8)12yes—
322C5×C8—22no—
323C5×C8—22no—
324#168—22no—
325#168—41no—
326C22×D4C22×D511yescharacteristic
327C23×C4—21no—
328C23×C4—101no—
329C23×C4—101no—
330C22×C8—51no—
331C22×C8—51no—
332C22×C8C22×D511yescharacteristic
333C22×C8C22×D511yescharacteristic
334#125—22no—
335#125C2×Dic514yescomplemented
336C4×D4—22no—
337C4×D4—102no—
338C4×D4—102no—
339C4×D4—201no—
340C4×D4—201no—
341C4×D4C2×Dic514yes—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4—101no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4—101no—
356C2×C4⋊C4—204no—
357C2×C4⋊C4C22×D511yescharacteristic
358C2×C4⋊C4C22×D511yescharacteristic
359C2×C4⋊C4C22×D511yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—101no—
363C2×C42—51no—
364C2×C42—51no—
365C2×C42—101no—
366C2×C42—101no—
367C2×C42—101no—
368C2×C42C22×D511yescharacteristic
369#107C2×Dic514yescomplemented
370#106—201no—
371#106—201no—
372#106C2×Dic514yescomplemented
373C4⋊C8C2×Dic514yescomplemented
374#102—54no—
375#102—102no—
376#102C2×Dic514yes—
377#102C2×Dic514yes—
378#97—102no—
379#97—201no—
380C4×C8—201no—
381#95—51no—
382#95—51no—
383#95—101no—
384#95—101no—
385#95—101no—
386#95—101no—
387#95—101no—
388#95—101no—
389#95—101no—
390C22×C5—24no—
391C22×C5—42no—
392C22×C5—42no—
393C22×C5—42no—
394C22×C5—42no—
395C22×C5—42no—
396C22×C5—42no—
397C22×C5C2×C4×D411yescharacteristic
398C22×C5#54611yescharacteristic
399C22×C5D4×Q811yescharacteristic
400C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C5#57511yescharacteristic
402C22×C5#57711yescharacteristic
403C22×C5#57711yescharacteristic
404C4×C5—22no—
405C4×C5—22no—
406C4×C5—42no—
407C4×C5—42no—
408C4×C5—42no—
409C4×C5C2×C4×Q811yescharacteristic
410C4×C5#52011yescharacteristic
411C4×C5#55211yescharacteristic
412C4×C5#55411yescharacteristic
413Dic5—22no—
414Dic5—42no—
415Dic5—41no—
416Dic5—41no—
417Dic5—41no—
418Dic5—81no—
419Dic5—81no—
420Dic5—81no—
421C24—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4Q8×D512yes—
426C2×D4#71312yes—
427C2×D4C22×Dic512yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4—202no—
443C22×C4—202no—
444C22×C4—202no—
445C22×C4—202no—
446C22×C4—401no—
447C22×C4—401no—
448C22×C4D4×D511yescharacteristic
449C22×C4D4×D511yescharacteristic
450C22×C4C23×D511yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—52no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8C22×Dic512yes—
462C2×C8C22×Dic512yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—52no—
469C4⋊C4—102no—
470C4⋊C4—101no—
471C4⋊C4—101no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—201no—
478C4⋊C4—202no—
479C4⋊C4—202no—
480C4⋊C4—202no—
481C4⋊C4—202no—
482C4⋊C4—401no—
483C4⋊C4—401no—
484C4⋊C4C22×Dic512yes—
485C4⋊C4C22×Dic512yes—
486C4⋊C4C22×Dic512yes—
487C22⋊C4—24no—
488C22⋊C4—202no—
489C22⋊C4—202no—
490C42—101no—
491C42—101no—
492C42—101no—
493C42—101no—
494C42—202no—
495C42—401no—
496C42—401no—
497C42#71112yes—
498C42C22×Dic512yes—
499C2×C5—44no—
500C2×C5#236011yescharacteristic
501C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
502C2×C5#295211yescharacteristic
503C2×C5#330711yescharacteristic
504C2×C5#331011yescharacteristic
505C2×C5#338411yescharacteristic
506C2×C5#338811yescharacteristic
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D511yescharacteristic
515D4—22no—
516D4—22no—
517D4—22no—
518D4Q8×Dic514yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—52no—
534C2×C4—101no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—101no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—102no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—404no—
558C2×C4—404no—
559C2×C4D4×Dic512yes—
560C2×C4D4×Dic512yes—
561C2×C4C2×#71111yescharacteristic
562C2×C4C2×Q8×D511yescharacteristic
563C2×C4C2×#71311yescharacteristic
564C2×C4#414911yescharacteristic
565C2×C4#415011yescharacteristic
566C2×C4C23×Dic511yescharacteristic
567C8—22no—
568C8—22no—
569C8—102no—
570C8—201no—
571C5#2297811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22#6596411yescharacteristic
580C22#6596611yescharacteristic
581C22#6610411yescharacteristic
582C22#6610411yescharacteristic
583C22D5×AGL(1,ℤ/8)11yescharacteristic
584C22#6610711yescharacteristic
585C22C2×D4×Dic511yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—102no—
592C4—202no—
593C4—201no—
594C4—201no—
595C4—201no—
596C4—401no—
597C4—401no—
598C4—401no—
599C4#6596911yescharacteristic
600C4#6597611yescharacteristic
601C4C2×Q8×Dic511yescharacteristic
602C4#6616111yescharacteristic
603C2—44no—
604C2#1061288911yescharacteristic
605C2#1061289211yescharacteristic
606C2#1061291911yescharacteristic
607C2#1061292011yescharacteristic
608C2#1061447811yescharacteristic
609C2#1061563011yescharacteristic
610C2Dic5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#478C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenterC2×#478D5
hypocenterC5#22978
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #10614756C2semidirect
4. #10602534C2semidirect
6. #10601406C2semidirect
7. #10601402C2semidirect
8. #10601353C2semidirect
9. #10601351C2semidirect
10. #10601260C2semidirect
11. #10601247C2semidirect
21. C5×#478C4semidirect
68. C5×#125C2×C4semidirect
104. C5×#107C2×C4semidirect
105. C5×#106C2×C4semidirect
106. C5×C4⋊C8C2×C4semidirect
223. #478Dic5semidirect
335. #125C2×Dic5semidirect
369. #107C2×Dic5semidirect
372. #106C2×Dic5semidirect
373. C4⋊C8C2×Dic5semidirect
571. C5#22978semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4782
#106147562
D4×C4⋊Dic52
#106025342
#106024022
#106014062
#106014022
#106013532
#106013512
#106012602
#106012472
#106009632
#106009452
Dic5×#1022
#229785