Finite Group Databasebeta

← #12104194

Subgroups — #12104194

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic5×#117C211yesmaximal, characteristic
2C2×#65008C211yesmaximal, characteristic
3#10605594C211yesmaximal, characteristic
4C5×#1854C211yesmaximal, characteristic
5#10602369C211yesmaximal, characteristic
6#10601409C211yesmaximal, characteristic, complemented
7#10601398C211yesmaximal, characteristic, complemented
8#10601357C211yesmaximal, characteristic, complemented
9#10601346C211yesmaximal, characteristic, complemented
10#10600722C211yesmaximal, characteristic
11#10600299C211yesmaximal, characteristic
12#10599591C211yesmaximal, characteristic, complemented
13#10599590C211yesmaximal, characteristic, complemented
14#10599579C211yesmaximal, characteristic, complemented
15#10599579C211yesmaximal, characteristic, complemented
16C2×C5×#117C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#106C2211yescharacteristic
21C2×C5×C22⋊C8C2211yescharacteristic
22C5×#380C2211yescharacteristic
23#65501C2211yescharacteristic
24C2×C22⋊#168C2211yescharacteristic
25C2×#3950C2211yescharacteristic
26C2×#3949C2211yescharacteristic
27C2×#3947C2211yescharacteristic
28C2×#3947C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31#65266C2211yescharacteristic
32#65266C2211yescharacteristic
33C4⋊C4×Dic5C2211yescharacteristic
34C4⋊C4×Dic5C2211yescharacteristic
35C4⋊C4×Dic5C2211yescharacteristic
36C4⋊C4×Dic5C2211yescharacteristic
37C4⋊C4×Dic5C2211yescharacteristic
38C4⋊C4×Dic5C2211yescharacteristic
39C2×#3940C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3939C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43#65232—21no—
44C22⋊C4×Dic5—21no—
45C22⋊C4×Dic5—21no—
46C22⋊C4×Dic5—21no—
47#65221—21no—
48C42×Dic5—21no—
49#65008C418yescomplemented
50#64940—21no—
51#64937—21no—
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54#64771C2211yescharacteristic
55#64771C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#21366—51nomaximal, Sylow
61C23×C4×C5C2311yescharacteristic
62C23×Dic5D411yescharacteristic
63C23×Dic5D411yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×#117—22no—
67C5×#117—22no—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—21no—
77#4072—22no—
78#4072—22no—
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4×Dic5—21no—
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×Dic5C2×C414yes—
91C2×C4×Dic5D411yescharacteristic
92C2×C4×Dic5D411yescharacteristic
93C2×C4×Dic5Q814yes—
94C2×C4×Dic5C2311yescharacteristic
95C2×C4×Dic5C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C22×#168C2311yescharacteristic
98C5×#107—22no—
99C5×#106—22no—
100C5×C22⋊C8C2×C414yes—
101C5×#95—21no—
102#3963—21no—
103#3963—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113C22⋊#168C2×C414yes—
114#3950—22no—
115#3949—22no—
116#3947C2×C414yes—
117#3947C2×C414yes—
118#3946C2×C414yes—
119#3946C2×C414yes—
120#3940—22no—
121#3940—22no—
122#3939—22no—
123#3939—22no—
124C2×#432—51no—
125#2283—51no—
126C4×#117—51no—
127#1949—51no—
128#1949—51no—
129#1949—51no—
130#1949—51no—
131#1854—51no—
132#1854—51no—
133#1854—51no—
134#1854D511yescharacteristic
135#1776—51no—
136#1776—51no—
137#1776—51no—
138#1776—51no—
139C24×C5C2×D411yescharacteristic
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×Q812yes—
151C22×C4×C5C2411yescharacteristic
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—41no—
157C22×Dic5—41no—
158C22×Dic5—41no—
159C22×Dic5—41no—
160C22×Dic5C4⋊C414yes—
161C22×Dic5C4⋊C414yes—
162C22×Dic5C2×D411yescharacteristic
163C22×Dic5C2×D411yescharacteristic
164C22×Dic5C2×D411yescharacteristic
165C22×Dic5C2×D411yescharacteristic
166C2×C5×C8—21no—
167C2×C5×C8—21no—
168C2×C5×C8—21no—
169C2×C5×C8—21no—
170C2×C5×C8C22×C412yes—
171C2×C5×C8C22×C412yes—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4#4112yes—
179C5×C4⋊C4#4112yes—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—42no—
182C42×C5—24no—
183C22⋊Dic5—24no—
184C22⋊Dic5—42no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5#4112yes—
192C4⋊Dic5#4112yes—
193C4×Dic5—24no—
194C4×Dic5—42no—
195C4×Dic5—42no—
196C4×Dic5—42no—
197C4×Dic5—42no—
198C4×Dic5—42no—
199C4×Dic5—42no—
200C4×Dic5C4⋊C414yes—
201C4×Dic5C4⋊C414yes—
202C4×Dic5C22×C412yes—
203C4×Dic5C22×C412yes—
204C4×Dic5C2×Q812yes—
205C4×Dic5C2×Q812yes—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168C22×C412yes—
211C2×#168C22×C412yes—
212C2×#117—51no—
213C2×#117C2×D511yescharacteristic
214C22×C42—51no—
215#432—58no—
216C2×#107—51no—
217C2×#107—51no—
218C2×#107—51no—
219C2×#107C2×D511yescharacteristic
220C2×#106—51no—
221C2×#106—51no—
222C2×#106—51no—
223C2×#106C2×D511yescharacteristic
224C2×C4⋊C8—51no—
225C2×C4⋊C8—51no—
226C2×C22⋊C8—51no—
227C2×C22⋊C8C2×D511yescharacteristic
228C2×#97—51no—
229C2×#97—51no—
230#383—101no—
231#382—51no—
232#382—51no—
233#381—101no—
234#380—51no—
235#380C2×D511yescharacteristic
236C4×C4⋊C4—51no—
237C4×C4⋊C4—51no—
238C4×C4⋊C4—51no—
239C4×C4⋊C4—51no—
240C4×C4⋊C4—51no—
241C4×C4⋊C4—51no—
242C4×C22⋊C4—101no—
243C4×C22⋊C4—101no—
244C4×C22⋊C4—101no—
245#376—101no—
246#376—101no—
247C43—101no—
248#340—51no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340C2×D511yescharacteristic
255#340C2×D511yescharacteristic
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—21no—
262C23×C5C2×C4⋊C412yes—
263C23×C5C22×D411yescharacteristic
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5C2×C4⋊C412yes—
283C2×C4×C5C23×C411yescharacteristic
284C2×C4×C5C22×Q811yescharacteristic
285C2×C4×C5C2×#4111yescharacteristic
286C2×C4×C5C2×#4111yescharacteristic
287C2×C4×C5C2×#4111yescharacteristic
288C2×C4×C5C2×#4111yescharacteristic
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—44no—
300C2×Dic5—44no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5C2×C4⋊C412yes—
312C2×Dic5C2×C4⋊C412yes—
313C2×Dic5C2×C4⋊C412yes—
314C2×Dic5C2×C4⋊C412yes—
315C5×C8—22no—
316C5×C8—22no—
317#168—22no—
318#168—22no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4C22×D511yescharacteristic
322C22×C8—51no—
323C22×C8—51no—
324C22×C8C22×D511yescharacteristic
325C22×C8C22×D511yescharacteristic
326#117—22no—
327#117—22no—
328#117—102no—
329#117—102no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4C22×D511yescharacteristic
335C2×C4⋊C4C22×D511yescharacteristic
336C2×C4⋊C4C22×D511yescharacteristic
337C2×C4⋊C4C22×D511yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—101no—
341C2×C22⋊C4—101no—
342C2×C42—21no—
343C2×C42—54no—
344C2×C42—51no—
345C2×C42—51no—
346C2×C42—54no—
347C2×C42—51no—
348C2×C42—51no—
349C2×C42—101no—
350C2×C42—101no—
351C2×C42—101no—
352C2×C42—101no—
353C2×C42—101no—
354C2×C42—101no—
355C2×C42—101no—
356#107—22no—
357#107—102no—
358#107—102no—
359#107—102no—
360#106—22no—
361#106—102no—
362#106—102no—
363#106—102no—
364C4⋊C8—54no—
365C4⋊C8—54no—
366C22⋊C8—54no—
367C22⋊C8C4×D514yes—
368#97—54no—
369#97—54no—
370#95—21no—
371#95—101no—
372#95—101no—
373#95—101no—
374#95—101no—
375#95—101no—
376#95—101no—
377#95—101no—
378#95—101no—
379#95—101no—
380#95—101no—
381#95—101no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5C22×C4⋊C411yescharacteristic
390C22×C5#52511yescharacteristic
391C22×C5#52511yescharacteristic
392C22×C5#57511yescharacteristic
393C22×C5#57511yescharacteristic
394C22×C5#57511yescharacteristic
395C22×C5#57511yescharacteristic
396C4×C5—22no—
397C4×C5—42no—
398C4×C5—42no—
399C4×C5—81no—
400C4×C5—81no—
401C4×C5C4×#4111yescharacteristic
402C4×C5C4×#4111yescharacteristic
403C4×C5#53311yescharacteristic
404C4×C5#53311yescharacteristic
405Dic5—22no—
406Dic5—22no—
407Dic5—22no—
408Dic5—22no—
409Dic5—42no—
410Dic5—42no—
411Dic5—81no—
412Dic5—81no—
413C24D4×D511yescharacteristic
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—51no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—51no—
426C22×C4—54no—
427C22×C4—54no—
428C22×C4—101no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—201no—
433C22×C4—201no—
434C22×C4—201no—
435C22×C4—201no—
436C22×C4C2×C4×D512yes—
437C22×C4D4×D511yescharacteristic
438C22×C4Q8×D512yes—
439C22×C4C23×D511yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—52no—
445C2×C8—52no—
446C2×C8—101no—
447C2×C8—101no—
448C2×C8—101no—
449C2×C8—101no—
450C2×C8C2×C4×D512yes—
451C2×C8C2×C4×D512yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—52no—
459C4⋊C4—52no—
460C4⋊C4—101no—
461C4⋊C4—101no—
462C4⋊C4—101no—
463C4⋊C4—101no—
464C4⋊C4—101no—
465C4⋊C4—101no—
466C4⋊C4#71112yes—
467C4⋊C4#71112yes—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—104no—
471C22⋊C4—202no—
472C42—24no—
473C42—52no—
474C42—52no—
475C42—52no—
476C42—52no—
477C42—54no—
478C42—54no—
479C42—104no—
480C42—202no—
481C42—202no—
482C42—202no—
483C42—202no—
484C42—202no—
485C42—202no—
486C2×C5—42no—
487C2×C5#229611yescharacteristic
488C2×C5#291111yescharacteristic
489C2×C5#291111yescharacteristic
490C2×C5#311111yescharacteristic
491C2×C5#311111yescharacteristic
492C2×C5#311111yescharacteristic
493C2×C5#311111yescharacteristic
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23D5×C4⋊C412yes—
501C23C2×D4×D511yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—52no—
521C2×C4—52no—
522C2×C4—52no—
523C2×C4—52no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—101no—
527C2×C4—101no—
528C2×C4—101no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—102no—
533C2×C4—102no—
534C2×C4—204no—
535C2×C4—204no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—201no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4D5×C4⋊C412yes—
547C2×C4C22×C4×D511yescharacteristic
548C2×C4C2×#71111yescharacteristic
549C2×C4C2×#71111yescharacteristic
550C2×C4C2×Q8×D511yescharacteristic
551C2×C4D5×#4111yescharacteristic
552C2×C4D5×#4111yescharacteristic
553C8—22no—
554C8—22no—
555C8—102no—
556C8—102no—
557C5#2136611yescharacteristic, complemented, Sylow
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×D5×C4⋊C411yescharacteristic
566C22#6592011yescharacteristic
567C22#6599011yescharacteristic
568C22#6610211yescharacteristic
569C22#6610211yescharacteristic
570C22#6610711yescharacteristic
571C22#6611111yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—102no—
578C4—102no—
579C4—102no—
580C4—102no—
581C4—202no—
582C4—202no—
583C4—401no—
584C4—401no—
585C4C4×#71111yescharacteristic
586C4C4×#71111yescharacteristic
587C4#6595611yescharacteristic
588C4#6595611yescharacteristic
589C2—42no—
590C2#1060610911yescharacteristic
591C2#1061090811yescharacteristic
592C2#1061093911yescharacteristic
593C2#1061180111yescharacteristic
594C2#1061194511yescharacteristic
595C2#1061308111yescharacteristic
596C2#1061322511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1854C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenter#1854D5
hypocenterC5#21366
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
6. #10601409C2semidirect
7. #10601398C2semidirect
8. #10601357C2semidirect
9. #10601346C2semidirect
12. #10599591C2semidirect
13. #10599590C2semidirect
14. #10599579C2semidirect
15. #10599579C2semidirect
49. #65008C4semidirect
557. C5#21366semidirect

Maximal subgroups

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

subgroupindex
Dic5×#1172
C2×#650082
#106055942
C5×#18542
#106023692
#106014092
#106013982
#106013572
#106013462
#106007222
#106002992
#105995912
#105995902
#105995792
#213665