Finite Group Databasebeta

← #12109567

Subgroups — #12109567

Order 1280.

1266 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65276C211yesmaximal, characteristic
2C2×D5×C22⋊C8C211yesmaximal, characteristic
3D5×#380C211yesmaximal, characteristic
4C5×#1852C211yesmaximal, characteristic, complemented
5#10602348C211yesmaximal, characteristic, complemented
6#10601425C212yesmaximal, complemented
7#10601423C212yesmaximal, complemented
8#10600720C211yesmaximal, characteristic, complemented
9#10600297C211yesmaximal, characteristic, complemented
10#10599667C212yesmaximal, complemented
11#10599662C212yesmaximal, complemented
12C23×C4×D5C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×#685—21no—
15C22×C4⋊Dic5C2211yescharacteristic
16C22×C8×D5C2212yes—
17C2×D5×C4⋊C4C2211yescharacteristic
18C2×D5×C4⋊C4C2211yescharacteristic
19C2×D5×C22⋊C4—21no—
20C2×C5×#107C2212yescomplemented
21C2×C5×C22⋊C8C2211yescharacteristic
22C5×#380C2211yescharacteristic
23#65501C2211yescharacteristic
24C2×#3963—21no—
25C2×C22⋊#168C2211yescharacteristic
26C2×#3952C2212yes—
27C2×#3950C2212yescomplemented
28#65276—21no—
29#65276—21no—
30#65276C2212yes—
31#65276C2212yes—
32#65276C2211yescharacteristic
33#65276C2211yescharacteristic
34C2×#3939C2212yescomplemented
35C2×#3939C2212yescomplemented
36#65231—21no—
37D5×C22⋊C8C418yescomplemented
38#64956—21no—
39D5×#95—21no—
40C5×#340C2212yescomplemented
41#64771C2212yescomplemented
42#64749C2212yescomplemented
43#64749C2212yescomplemented
44#21940—51nomaximal, Sylow
45C24×D5D411yescharacteristic
46C23×C4×C5—21no—
47C23×C4×C5C2311yescharacteristic
48C23×Dic5—21no—
49C23×Dic5D411yescharacteristic
50C22×C4×D5—21no—
51C22×C4×D5—21no—
52C22×C4×D5—42no—
53C22×C4×D5—42no—
54C22×C4×D5C2×C414yes—
55C22×C4×D5D411yescharacteristic
56C22×C4×D5D411yescharacteristic
57C22×C4×D5Q814yes—
58C22×C4×D5C2311yescharacteristic
59C22×C4×D5C2311yescharacteristic
60C22×C5×C8C2312yes—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—42no—
63C2×C5×C4⋊C4C2311yescharacteristic
64C2×C5×C4⋊C4C2311yescharacteristic
65C2×C5×C4⋊C4C2312yes—
66C2×C5×C22⋊C4—21no—
67C2×C22⋊Dic5—21no—
68C2×#685—21no—
69C2×#685—21no—
70C2×#685—22no—
71C2×#685—22no—
72C2×#685—44no—
73C2×C4⋊Dic5—21no—
74C2×C4⋊Dic5—42no—
75C2×C4⋊Dic5C2311yescharacteristic
76C2×C4⋊Dic5C2312yes—
77C2×C4⋊Dic5C2311yescharacteristic
78C2×#683—21no—
79C2×#683—21no—
80C22×#168C2312yes—
81C2×C8×D5—42no—
82C2×C8×D5—42no—
83C2×C8×D5C2×C418yescomplemented
84D5×C4⋊C4—42no—
85D5×C4⋊C4—42no—
86D5×C22⋊C4—44no—
87C5×#107—42no—
88C5×C22⋊C8C2×C414yescomplemented
89C5×#95—21no—
90#3963—21no—
91#3963—21no—
92#3963—21no—
93#3963—22no—
94#3963—22no—
95C22⋊#168C2×C414yescomplemented
96#3952C2×C418yescomplemented
97#3950—42no—
98#3939—42no—
99#3939—42no—
100C22×C22⋊C8—51no—
101C2×#380—51no—
102C2×#380—51no—
103#1852—52no—
104#1852—51no—
105#1852—51no—
106#1852—52no—
107#1852—51no—
108#1852D511yescharacteristic, complemented
109#1808—52no—
110#1808—52no—
111C24×C5C2×D411yescharacteristic
112C23×D5—41no—
113C23×D5—41no—
114C23×D5—41no—
115C23×D5—41no—
116C23×D5C4⋊C414yes—
117C23×D5D814yes—
118C23×D5Q1614yes—
119C23×D5C2×D411yescharacteristic
120C23×D5C2×D411yescharacteristic
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—42no—
127C22×C4×C5—42no—
128C22×C4×C5—42no—
129C22×C4×C5C22×C412yes—
130C22×C4×C5C2×D411yescharacteristic
131C22×C4×C5C2×Q812yes—
132C22×C4×C5C2411yescharacteristic
133C22×Dic5—21no—
134C22×Dic5—21no—
135C22×Dic5—22no—
136C22×Dic5—41no—
137C22×Dic5—41no—
138C22×Dic5—42no—
139C22×Dic5—42no—
140C22×Dic5—42no—
141C22×Dic5C4⋊C414yes—
142C22×Dic5C2×D411yescharacteristic
143C22×Dic5C2×D411yescharacteristic
144C2×C4×D5—22no—
145C2×C4×D5—22no—
146C2×C4×D5—22no—
147C2×C4×D5—22no—
148C2×C4×D5—48no—
149C2×C4×D5—48no—
150C2×C4×D5—41no—
151C2×C4×D5—41no—
152C2×C4×D5—41no—
153C2×C4×D5—41no—
154C2×C4×D5—42no—
155C2×C4×D5—42no—
156C2×C4×D5—42no—
157C2×C4×D5—42no—
158C2×C4×D5—42no—
159C2×C4×D5—42no—
160C2×C4×D5C4⋊C414yes—
161C2×C4×D5C4⋊C414yes—
162C2×C4×D5C22×C412yes—
163C2×C4×D5C22×C412yes—
164C2×C4×D5C2×Q812yes—
165C2×C4×D5C2×Q812yes—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8C22×C414yes—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—42no—
174C5×C22⋊C4—42no—
175C22⋊Dic5—42no—
176#685—42no—
177#685—42no—
178#685—44no—
179#685—44no—
180C4⋊Dic5—22no—
181C4⋊Dic5—22no—
182C4⋊Dic5—22no—
183C4⋊Dic5—22no—
184C4⋊Dic5—42no—
185#683—42no—
186#683—42no—
187C2×#168—22no—
188C2×#168—22no—
189C2×#168C22×C414yes—
190C8×D5—48no—
191C24×C4—51no—
192C23×C8—52no—
193C22×C4⋊C4—51no—
194C22×C4⋊C4—51no—
195C22×C4⋊C4—51no—
196C22×C4⋊C4C2×D511yescharacteristic
197C22×C22⋊C4—101no—
198C22×C22⋊C4—101no—
199C2×#107—52no—
200C2×#107—52no—
201C2×#107—52no—
202C2×#107C2×D512yescomplemented
203C2×C22⋊C8—51no—
204C2×C22⋊C8—52no—
205C2×C22⋊C8—58no—
206C2×C22⋊C8C2×D511yescharacteristic
207#380—51no—
208#380—51no—
209#380—51no—
210#380—52no—
211#380—52no—
212#380—101no—
213#380—101no—
214#380—101no—
215#380—101no—
216#380C2×D511yescharacteristic
217C2×#95—101no—
218C2×#95—101no—
219#340—52no—
220#340—52no—
221#340—52no—
222#340C2×D512yescomplemented
223C23×C5—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—21no—
227C23×C5C2×C4⋊C412yes—
228C23×C5C2×D812yes—
229C23×C5C2×Q1612yes—
230C23×C5C22×D411yescharacteristic
231C22×D5—21no—
232C22×D5—21no—
233C22×D5—21no—
234C22×D5—21no—
235C22×D5—21no—
236C22×D5—21no—
237C22×D5—21no—
238C22×D5—21no—
239C22×D5—44no—
240C22×D5—44no—
241C22×D5—44no—
242C22×D5—44no—
243C22×D5—44no—
244C22×D5—44no—
245C22×D5#107116yescomplemented
246C22×D5C2×C4⋊C412yes—
247C22×D5C2×C4⋊C412yes—
248C22×D5C2×D812yes—
249C22×D5C2×D812yes—
250C22×D5C2×Q1612yes—
251C22×D5C2×Q1612yes—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—41no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—84no—
266C2×C4×C5C2×C4⋊C412yes—
267C2×C4×C5C23×C411yescharacteristic
268C2×C4×C5C22×Q811yescharacteristic
269C2×C4×C5H(5,𝔽2)12yes—
270C2×C4×C5H(5,𝔽2)12yes—
271C2×Dic5—21no—
272C2×Dic5—21no—
273C2×Dic5—21no—
274C2×Dic5—21no—
275C2×Dic5—44no—
276C2×Dic5—41no—
277C2×Dic5—41no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—41no—
281C2×Dic5—41no—
282C2×Dic5—41no—
283C2×Dic5—41no—
284C2×Dic5—42no—
285C2×Dic5—84no—
286C2×Dic5C2×C4⋊C412yes—
287C2×Dic5C2×C4⋊C412yes—
288C4×D5—24no—
289C4×D5—24no—
290C4×D5—24no—
291C4×D5—24no—
292C4×D5—44no—
293C4×D5—44no—
294C4×D5—84no—
295C4×D5—84no—
296C5×C8—24no—
297#168—24no—
298C25—51no—
299C23×C4—21no—
300C23×C4—51no—
301C23×C4—51no—
302C23×C4—51no—
303C23×C4—54no—
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—54no—
307C23×C4—101no—
308C23×C4—101no—
309C23×C4—101no—
310C23×C4—202no—
311C23×C4—202no—
312C23×C4C22×D511yescharacteristic
313C22×C8—52no—
314C22×C8—58no—
315C22×C8—202no—
316C22×C8—202no—
317C22×C8C22×D512yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—51no—
321C2×C4⋊C4—52no—
322C2×C4⋊C4—51no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—101no—
325C2×C4⋊C4—101no—
326C2×C4⋊C4—202no—
327C2×C4⋊C4—202no—
328C2×C4⋊C4—202no—
329C2×C4⋊C4C22×D511yescharacteristic
330C2×C4⋊C4C22×D511yescharacteristic
331C2×C4⋊C4C22×D512yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—101no—
334C2×C22⋊C4—101no—
335C2×C22⋊C4—102no—
336C2×C22⋊C4—102no—
337C2×C22⋊C4—101no—
338C2×C22⋊C4—204no—
339C2×C22⋊C4—204no—
340#107—42no—
341#107—202no—
342#107—202no—
343#107—202no—
344C22⋊C8—54no—
345C22⋊C8—58no—
346C22⋊C8C4×D514yescomplemented
347#95—21no—
348#95—101no—
349#95—101no—
350#95—102no—
351#95—102no—
352#95—101no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5—22no—
358C22×C5—22no—
359C22×C5C2×#10714yes—
360C22×C5C22×C4⋊C411yescharacteristic
361C22×C5#53411yescharacteristic
362C22×C5#53411yescharacteristic
363C22×C5C22×D811yescharacteristic
364C22×C5C22×Q1611yescharacteristic
365C22×C5#57511yescharacteristic
366C22×C5#57511yescharacteristic
367C2×D5—22no—
368C2×D5—22no—
369C2×D5—22no—
370C2×D5—22no—
371C2×D5—22no—
372C2×D5—22no—
373C2×D5—22no—
374C2×D5—22no—
375C2×D5—22no—
376C2×D5—22no—
377C2×D5—22no—
378C2×D5—22no—
379C2×D5—416no—
380C2×D5C2×#10714yes—
381C2×D5C2×#10714yes—
382C4×C5—22no—
383C4×C5—81no—
384C4×C5—81no—
385C4×C5—82no—
386C4×C5#51812yes—
387C4×C5#54312yes—
388Dic5—22no—
389Dic5—22no—
390Dic5—81no—
391Dic5—81no—
392Dic5—82no—
393C24—51no—
394C24—51no—
395C24—54no—
396C24—54no—
397C24—54no—
398C24—201no—
399C24—201no—
400C24—201no—
401C24—201no—
402C24D4×D511yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—54no—
414C22×C4—52no—
415C22×C4—52no—
416C22×C4—52no—
417C22×C4—52no—
418C22×C4—54no—
419C22×C4—54no—
420C22×C4—102no—
421C22×C4—102no—
422C22×C4—102no—
423C22×C4—102no—
424C22×C4—101no—
425C22×C4—102no—
426C22×C4—101no—
427C22×C4—208no—
428C22×C4—208no—
429C22×C4—201no—
430C22×C4—201no—
431C22×C4—201no—
432C22×C4—201no—
433C22×C4—202no—
434C22×C4—202no—
435C22×C4—202no—
436C22×C4—202no—
437C22×C4—202no—
438C22×C4—202no—
439C22×C4—201no—
440C22×C4—201no—
441C22×C4—202no—
442C22×C4—202no—
443C22×C4—202no—
444C22×C4C2×C4×D512yes—
445C22×C4D4×D511yescharacteristic
446C22×C4Q8×D512yes—
447C22×C4C23×D511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—54no—
451C2×C8—102no—
452C2×C8—102no—
453C2×C8—208no—
454C2×C8C2×C4×D514yes—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—102no—
461C4⋊C4—102no—
462C4⋊C4—102no—
463C4⋊C4—102no—
464C4⋊C4—202no—
465C4⋊C4—202no—
466C4⋊C4—202no—
467C22⋊C4—42no—
468C22⋊C4—202no—
469C22⋊C4—202no—
470C22⋊C4—204no—
471C22⋊C4—204no—
472C22⋊C4—202no—
473C2×C5—24no—
474C2×C5#231511yescharacteristic
475C2×C5C22×#10711yescharacteristic
476C2×C5#291111yescharacteristic
477C2×C5#318611yescharacteristic
478C2×C5#318611yescharacteristic
479C2×C5#318911yescharacteristic
480C2×C5#318911yescharacteristic
481D5—24no—
482D5—24no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—52no—
488C23—52no—
489C23—52no—
490C23—52no—
491C23—52no—
492C23—52no—
493C23—516no—
494C23—101no—
495C23—101no—
496C23—101no—
497C23—101no—
498C23—101no—
499C23—101no—
500C23—101no—
501C23—101no—
502C23—204no—
503C23—204no—
504C23—204no—
505C23—204no—
506C23—204no—
507C23—204no—
508C23D5×C4⋊C412yes—
509C23D5×D812yes—
510C23D5×Q1612yes—
511C23C2×D4×D511yescharacteristic
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—52no—
526C2×C4—52no—
527C2×C4—84no—
528C2×C4—101no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—104no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—104no—
536C2×C4—204no—
537C2×C4—204no—
538C2×C4—204no—
539C2×C4—201no—
540C2×C4—202no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—202no—
545C2×C4—201no—
546C2×C4—202no—
547C2×C4—201no—
548C2×C4—404no—
549C2×C4—404no—
550C2×C4—404no—
551C2×C4D5×C4⋊C412yes—
552C2×C4C22×C4×D511yescharacteristic
553C2×C4#414412yes—
554C2×C4C2×Q8×D511yescharacteristic
555C2×C4#414912yes—
556C8—24no—
557C8—104no—
558C5#2194011yescharacteristic, complemented, Sylow
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—54no—
566C22—54no—
567C22—102no—
568C22—102no—
569C22—102no—
570C22—102no—
571C22—102no—
572C22—102no—
573C22—102no—
574C22—102no—
575C22—102no—
576C22—102no—
577C22—102no—
578C22—102no—
579C22—2016no—
580C22D5×#10714yes—
581C22C2×D5×C4⋊C411yescharacteristic
582C22#6592311yescharacteristic
583C22#6598711yescharacteristic
584C22C2×D5×D811yescharacteristic
585C22C2×D5×Q1611yescharacteristic
586C22#6610711yescharacteristic
587C22#6611111yescharacteristic
588C4—22no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—102no—
593C4—102no—
594C4—401no—
595C4—401no—
596C4—402no—
597C4#6587912yes—
598C4#6596812yes—
599C2—24no—
600C2—104no—
601C2—104no—
602C2#1060632211yescharacteristic
603C2C2×D5×#10711yescharacteristic
604C2#1061093911yescharacteristic
605C2#1061173711yescharacteristic
606C2#1061191811yescharacteristic
607C2#1061303011yescharacteristic
608C2#1061321911yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1852C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenter#1852D5
hypocenterC5#21940
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. C5×#1852C2semidirect
5. #10602348C2semidirect
6. #10601425C2semidirect
7. #10601423C2semidirect
8. #10600720C2semidirect
9. #10600297C2semidirect
10. #10599667C2semidirect
11. #10599662C2semidirect
20. C2×C5×#107C22semidirect
27. C2×#3950C22semidirect
34. C2×#3939C22semidirect
35. C2×#3939C22semidirect
37. D5×C22⋊C8C4semidirect
40. C5×#340C22semidirect
41. #64771C22semidirect
42. #64749C22semidirect
43. #64749C22semidirect
83. C2×C8×D5C2×C4semidirect
88. C5×C22⋊C8C2×C4semidirect
95. C22⋊#168C2×C4semidirect
96. #3952C2×C4semidirect
108. #1852D5semidirect
202. C2×#107C2×D5semidirect
222. #340C2×D5semidirect
245. C22×D5#107semidirect
346. C22⋊C8C4×D5semidirect
558. C5#21940semidirect

Maximal subgroups

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

subgroupindex
C2×#652762
C2×D5×C22⋊C82
D5×#3802
C5×#18522
#106023482
#106014252
#106014232
#106007202
#106002972
#105996672
#105996622
#219405