Finite Group Databasebeta

← #12106911

Subgroups — #12106911

Order 1280.

948 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
1C4×D4×Dic5C211yesmaximal, characteristic
2C2×#65055C211yesmaximal, characteristic
3#10607697C211yesmaximal, characteristic, complemented
4C5×#1862C211yesmaximal, characteristic
5#10602400C211yesmaximal, characteristic
6#10601409C211yesmaximal, characteristic, complemented
7Dic5×#107C211yesmaximal, characteristic, complemented
8#10600947C211yesmaximal, characteristic
9Dic5×#102C211yesmaximal, characteristic
10Dic5×#102C211yesmaximal, characteristic
11#10600394C211yesmaximal, characteristic, complemented
12#10599595C211yesmaximal, characteristic, complemented
13#10599577C211yesmaximal, characteristic, complemented
14#10599417C211yesmaximal, characteristic, complemented
15#10599395C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C2×D4×Dic5C2211yescharacteristic
18C2×D4×Dic5C2211yescharacteristic
19C22×C4×Dic5—21no—
20C2×#4036C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#102C2211yescharacteristic
23C2×C5×#102C2211yescharacteristic
24C5×#389C2211yescharacteristic
25C2×#3964C2211yescharacteristic
26C2×#3964C2211yescharacteristic
27C4×C22⋊Dic5—21no—
28C2×#3950C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×C8×Dic5C2211yescharacteristic
32C2×C8×Dic5C2211yescharacteristic
33#65267—21no—
34#65266C2211yescharacteristic
35C4⋊C4×Dic5—21no—
36C4⋊C4×Dic5C2211yescharacteristic
37C4⋊C4×Dic5C2211yescharacteristic
38C4⋊C4×Dic5C2211yescharacteristic
39C4⋊C4×Dic5C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3940C2211yescharacteristic
42C22⋊C4×Dic5—21no—
43C22⋊C4×Dic5—21no—
44C4×C4⋊Dic5C2211yescharacteristic
45C42×Dic5C2211yescharacteristic
46#65217C2211yescharacteristic
47#65055C418yescomplemented
48#64942—21no—
49#64938—21no—
50C5×#340C2211yescharacteristic
51C5×#336C2211yescharacteristic
52#64771C2211yescharacteristic
53#64771C2211yescharacteristic
54#64771C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60C4×#478—51nomaximal, Sylow
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×Dic5—21no—
64C23×Dic5—21no—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C4×C5×D4—22no—
68C4×C5×D4—22no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C22⋊C4—21no—
74C2×C42×C5C2311yescharacteristic
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—21no—
77D4×Dic5—22no—
78D4×Dic5—22no—
79D4×Dic5C2×C414yes—
80D4×Dic5Q814yes—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×#683—21no—
87C2×#683—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5—44no—
94C2×C4×Dic5D411yescharacteristic
95C2×C4×Dic5D411yescharacteristic
96C2×C4×Dic5D411yescharacteristic
97C2×C4×Dic5D411yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C2×C4×Dic5C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C22×#168C2311yescharacteristic
102#4036—22no—
103#4036C2×C414yescomplemented
104C5×#107—22no—
105C5×#102—22no—
106C5×#102C2×C414yes—
107C5×#95—21no—
108#3964—22no—
109#3964C2×C414yes—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3963—21no—
116#3963—21no—
117#3963—21no—
118#3950C2×C414yescomplemented
119#3946—22no—
120#3946C2×C414yescomplemented
121C8×Dic5—22no—
122C8×Dic5—22no—
123#3940—22no—
124#3940C2×C414yescomplemented
125C2×#478—51no—
126C4×#125—51no—
127C42×D4—51no—
128#1989—51no—
129#1949—51no—
130#1921—51no—
131#1919—51no—
132#1864—51no—
133#1862D511yescharacteristic
134#1804—51no—
135C4×#107—51no—
136C4×#106—51no—
137C4×C4⋊C8—51no—
138C4×#102—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—24no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
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—24no—
160C22×Dic5—24no—
161C22×Dic5—21no—
162C22×Dic5—42no—
163C22×Dic5—42no—
164C22×Dic5—42no—
165C22×Dic5—42no—
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×C8—21no—
175C2×C5×C8C22×C412yes—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—42no—
181C5×C4⋊C4C22×C412yes—
182C5×C4⋊C4#4112yes—
183C5×C22⋊C4—24no—
184C42×C5—21no—
185C42×C5—21no—
186C22⋊Dic5—24no—
187C22⋊Dic5—24no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5—24no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5C22×C412yes—
195C4⋊Dic5C22×C412yes—
196C4⋊Dic5C2×Q812yes—
197#683—24no—
198#683—42no—
199C4×Dic5—22no—
200C4×Dic5—22no—
201C4×Dic5—22no—
202C4×Dic5—41no—
203C4×Dic5—42no—
204C4×Dic5—42no—
205C4×Dic5—42no—
206C4×Dic5—42no—
207C4×Dic5—42no—
208C4×Dic5C22×C412yes—
209C4×Dic5C2×Q812yes—
210C4×Dic5#4112yes—
211C4×Dic5#4112yes—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168C22×C412yes—
218C2×#125—51no—
219C2×C4×D4—51no—
220C2×C4×D4—51no—
221C2×C4×D4C2×D511yescharacteristic
222C22×C42—101no—
223#478—58no—
224C2×#107—51no—
225C2×#107C2×D511yescharacteristic
226C2×#106—51no—
227C2×#106—51no—
228C2×C4⋊C8—51no—
229C2×C4⋊C8—51no—
230C2×#102—51no—
231C2×#102—51no—
232C2×#102C2×D511yescharacteristic
233C2×#102C2×D511yescharacteristic
234C2×C4×C8—51no—
235C2×C4×C8—51no—
236#389—51no—
237#389C2×D511yescharacteristic
238#387—101no—
239#387—101no—
240#382—51no—
241C4×C4⋊C4—51no—
242C4×C4⋊C4—51no—
243C4×C4⋊C4—51no—
244C4×C4⋊C4—51no—
245C4×C4⋊C4—51no—
246C4×C4⋊C4—101no—
247C4×C4⋊C4—101no—
248C4×C22⋊C4—101no—
249C4×C22⋊C4—101no—
250C4×C22⋊C4—101no—
251C43—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340C2×D511yescharacteristic
260#336—51no—
261#336C2×D511yescharacteristic
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—22no—
277C2×C4×C5—21no—
278C2×C4×C5—22no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
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×C5—41no—
289C2×C4×C5#13512yes—
290C2×C4×C5C23×C411yescharacteristic
291C2×C4×C5C22×Q811yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5C2×#4111yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—48no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—48no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5C4×D412yes—
317C2×Dic5C4×D412yes—
318C2×Dic5C22⋊Q812yes—
319C2×Dic5C22⋊Q812yes—
320C2×Dic5#13512yes—
321C2×Dic5AGL(1,ℤ/8)12yes—
322C5×C8—22no—
323C5×C8—41no—
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—54no—
335#125—102no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—54no—
339C4×D4—54no—
340C4×D4—102no—
341C4×D4—102no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—51no—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4—101no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D511yescharacteristic
352C2×C4⋊C4C22×D511yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—101no—
355C2×C22⋊C4—101no—
356C2×C42—51no—
357C2×C42—51no—
358C2×C42—51no—
359C2×C42—51no—
360C2×C42—51no—
361C2×C42—51no—
362C2×C42—101no—
363C2×C42—101no—
364C2×C42—101no—
365C2×C42—101no—
366C2×C42—101no—
367C2×C42—204no—
368C2×C42C22×D511yescharacteristic
369#107—22no—
370#107—54no—
371#106—54no—
372#106—102no—
373C4⋊C8—54no—
374C4⋊C8—102no—
375#102—22no—
376#102—54no—
377#102—102no—
378#102C4×D514yes—
379C4×C8—102no—
380C4×C8—102no—
381#95—21no—
382#95—101no—
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×C5C2×C22⋊Q811yescharacteristic
399C22×C5#52511yescharacteristic
400C22×C5C2×#13511yescharacteristic
401C22×C5C2×#13511yescharacteristic
402C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
403C22×C5#57511yescharacteristic
404C4×C5—22no—
405C4×C5—41no—
406C4×C5—42no—
407C4×C5—42no—
408C4×C5—81no—
409C4×C5C2×C4×Q811yescharacteristic
410C4×C5C4×#4111yescharacteristic
411C4×C5#52911yescharacteristic
412C4×C5#53311yescharacteristic
413Dic5—22no—
414Dic5—22no—
415Dic5—22no—
416Dic5—42no—
417Dic5—42no—
418Dic5—42no—
419Dic5—81no—
420Dic5#47814yescomplemented
421C24—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4C2×C4×D512yes—
426C2×D4#71112yes—
427C2×D4Q8×D512yes—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—104no—
439C22×C4—104no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4—101no—
444C22×C4—202no—
445C22×C4—202no—
446C22×C4—202no—
447C22×C4—202no—
448C22×C4D4×D511yescharacteristic
449C22×C4D4×D511yescharacteristic
450C22×C4C23×D511yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—52no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8C2×C4×D512yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—52no—
469C4⋊C4—52no—
470C4⋊C4—52no—
471C4⋊C4—104no—
472C4⋊C4—104no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—101no—
478C4⋊C4—202no—
479C4⋊C4C2×C4×D512yes—
480C4⋊C4#71112yes—
481C22⋊C4—24no—
482C22⋊C4—104no—
483C22⋊C4—104no—
484C42—21no—
485C42—21no—
486C42—52no—
487C42—52no—
488C42—52no—
489C42—52no—
490C42—102no—
491C42—102no—
492C42—102no—
493C42—201no—
494C42—202no—
495C42—202no—
496C42—202no—
497C42—202no—
498C42—202no—
499C2×C5—44no—
500C2×C5C4×C22⋊Q811yescharacteristic
501C2×C5C4×#13511yescharacteristic
502C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
503C2×C5C2×#47811yescharacteristic
504C2×C5#307611yescharacteristic
505C2×C5#310211yescharacteristic
506C2×C5#310511yescharacteristic
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D511yescharacteristic
515D4—22no—
516D4—22no—
517D4—22no—
518D4#403314yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
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—41no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—52no—
537C2×C4—52no—
538C2×C4—52no—
539C2×C4—52no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—101no—
546C2×C4—102no—
547C2×C4—102no—
548C2×C4—102no—
549C2×C4—208no—
550C2×C4—208no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4#405812yes—
561C2×C4C22×C4×D511yescharacteristic
562C2×C4C2×#71111yescharacteristic
563C2×C4C2×#71111yescharacteristic
564C2×C4C2×Q8×D511yescharacteristic
565C2×C4D5×#4111yescharacteristic
566C2×C4D5×#4111yescharacteristic
567C8—22no—
568C8—41no—
569C8—102no—
570C8—201no—
571C5C4×#47811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C4×D4×D511yescharacteristic
580C22#6592011yescharacteristic
581C22D5×C22⋊Q811yescharacteristic
582C22C2×#405811yescharacteristic
583C22D5×#13511yescharacteristic
584C22D5×AGL(1,ℤ/8)11yescharacteristic
585C22#6611111yescharacteristic
586C4—22no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—54no—
591C4—81no—
592C4—102no—
593C4—102no—
594C4—102no—
595C4—202no—
596C4—202no—
597C4—202no—
598C4—401no—
599C4C2×#403311yescharacteristic
600C4C4×#71111yescharacteristic
601C4#6595611yescharacteristic
602C4#6598311yescharacteristic
603C2—44no—
604C2#1060611211yescharacteristic
605C2C4×#405811yescharacteristic
606C2#1061150411yescharacteristic
607C2#1061172011yescharacteristic
608C2#1061194311yescharacteristic
609C2#1061269511yescharacteristic
610C2D5×#47811yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1862C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
cosocleC22×C4×C5C24
hypercenter#1862D5
hypocenterC5C4×#478
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
3. #10607697C2semidirect
6. #10601409C2semidirect
7. Dic5×#107C2semidirect
11. #10600394C2semidirect
12. #10599595C2semidirect
13. #10599577C2semidirect
14. #10599417C2semidirect
15. #10599395C2semidirect
47. #65055C4semidirect
103. #4036C2×C4semidirect
118. #3950C2×C4semidirect
120. #3946C2×C4semidirect
124. #3940C2×C4semidirect
420. Dic5#478semidirect
571. C5C4×#478semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Dic52
C2×#650552
#106076972
C5×#18622
#106024002
#106014092
Dic5×#1072
#106009472
Dic5×#1022
#106003942
#105995952
#105995772
#105994172
#105993952
C4×#4785