Finite Group Databasebeta

← #12065210

Subgroups — #12065210

Order 1280.

1210 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2#10611066C212yesmaximal, complemented
3C5×#1464C211yesmaximal, characteristic, complemented
4#10602365C211yesmaximal, characteristic, complemented
5#10601623C212yesmaximal, complemented
6#10598937C212yesmaximal, complemented
7D5×#324C212yesmaximal, complemented
8#10598914C212yesmaximal, complemented
9#10598912C212yesmaximal, complemented
10C22×#709C411yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C22×C4⋊Dic5C411yescharacteristic
13#65857—24no—
14#65857—24no—
15C2×#4038C411yescharacteristic
16C2×#4038C411yescharacteristic
17C2×D5×C4⋊C4C2211yescharacteristic
18C2×D5×C4⋊C4C2211yescharacteristic
19C2×#4027C411yescharacteristic
20C2×#4027C411yescharacteristic
21C2×#4027C2211yescharacteristic
22C2×#4027C2211yescharacteristic
23C2×C4×D20C411yescharacteristic
24C2×C4×D20C2211yescharacteristic
25C2×C4×Dic10C411yescharacteristic
26C2×C4×Dic10C2211yescharacteristic
27C5×#432C2212yes—
28#65215C2212yes—
29#65125—24no—
30#65015C2214yes—
31D5×C22⋊C8C2214yes—
32#64995—24no—
33C5×#324C2212yes—
34C5×#323C2212yes—
35#64748C2212yes—
36#64745C2212yes—
37#64689C2214yes—
38#64689C2214yes—
39#17368—51nomaximal, Sylow
40C23×C4×C5C2×C411yescharacteristic
41C22×C22⋊D5C2×C411yescharacteristic
42C22×C22⋊D5C2×C411yescharacteristic
43C23×Dic5C2×C411yescharacteristic
44C23×Dic5C2×C411yescharacteristic
45C2×#709—24no—
46C2×#709—22no—
47C2×#709—22no—
48C2×#709—24no—
49C22×D20C2×C411yescharacteristic
50C22×C4×D5C2×C411yescharacteristic
51C22×C4×D5C2×C411yescharacteristic
52C22×C4×D5C2×C411yescharacteristic
53C22×C4×D5C2×C411yescharacteristic
54C22×C4×D5C2311yescharacteristic
55C22×C4×D5C2311yescharacteristic
56C22×Dic10C2×C411yescharacteristic
57C5×#117—22no—
58C5×#117—22no—
59C2×C5×C4⋊C4C2×C411yescharacteristic
60C2×C5×C4⋊C4C2311yescharacteristic
61C2×C5×C22⋊C4C2×C411yescharacteristic
62C2×C5×C22⋊C4C2311yescharacteristic
63C2×C42×C5C2×C411yescharacteristic
64C2×C42×C5C2311yescharacteristic
65C2×#685C2×C411yescharacteristic
66C2×#685C2×C411yescharacteristic
67C2×#685C2×C411yescharacteristic
68C2×#685C2×C411yescharacteristic
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5C2×C411yescharacteristic
72C2×C4⋊Dic5C2×C411yescharacteristic
73C2×C4⋊Dic5C2×C411yescharacteristic
74C2×C4⋊Dic5C2×C411yescharacteristic
75C2×#683C2×C411yescharacteristic
76C2×#683C2×C411yescharacteristic
77C2×#683C2311yescharacteristic
78C2×#683C2311yescharacteristic
79C2×C4×Dic5C2×C411yescharacteristic
80C2×C4×Dic5C2×C411yescharacteristic
81C2×C4×Dic5C2×C411yescharacteristic
82C2×C4×Dic5C2×C411yescharacteristic
83C2×#676—24no—
84C2×C8×D5—24no—
85#4038—24no—
86#4038—24no—
87D5×C4⋊C4—24no—
88D5×C4⋊C4—24no—
89#4027—24no—
90#4027—24no—
91#4027—24no—
92#4027—24no—
93C4×D20—22no—
94C4×D20—22no—
95C4×D20D414yes—
96C4×D20D414yes—
97C4×Dic10—22no—
98C4×Dic10—22no—
99C4×Dic10—22no—
100C4×Dic10—22no—
101C5×C4⋊C8—22no—
102C5×C22⋊C8C2314yes—
103C5×#97—22no—
104C22⋊#168C2314yes—
105#3952—24no—
106#3952—24no—
107C4⋊#168—22no—
108#3935—22no—
109C2×#520—51no—
110#2973—52no—
111#1467—52no—
112#1465—52no—
113#1464—51no—
114#1464D511yescharacteristic, complemented
115#1463—52no—
116#1459—52no—
117C2×#324—52no—
118C24×C5C22×C411yescharacteristic
119C23×D5C22×C411yescharacteristic
120C23×D5C22×C411yescharacteristic
121C22×C4×C5—22no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5C22×C411yescharacteristic
126C22×C4×C5C22×C411yescharacteristic
127C22×C4×C5C22×C411yescharacteristic
128C22×C4×C5C22×C411yescharacteristic
129C22×C4×C5C22×C411yescharacteristic
130C22×C4×C5C2411yescharacteristic
131C2×C22⋊D5—22no—
132C2×C22⋊D5—22no—
133C2×C22⋊D5—22no—
134C2×C22⋊D5—22no—
135C2×C22⋊D5—22no—
136C2×C22⋊D5M1614yescomplemented
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—81no—
140C22×Dic5—81no—
141C22×Dic5C22×C411yescharacteristic
142C22×Dic5C22×C411yescharacteristic
143C22×Dic5C22×C411yescharacteristic
144C22×Dic5C22×C411yescharacteristic
145C22×Dic5C22×C411yescharacteristic
146C22×Dic5C22×C411yescharacteristic
147#709—216no—
148#709—216no—
149C2×D20—22no—
150C2×D20—22no—
151C2×D20C22⋊C412yes—
152C2×D20C22⋊C412yes—
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5—22no—
157C2×C4×D5—22no—
158C2×C4×D5—22no—
159C2×C4×D5—22no—
160C2×C4×D5—81no—
161C2×C4×D5—81no—
162C2×C4×D5—81no—
163C2×C4×D5—81no—
164C2×C4×D5M1614yescomplemented
165C2×C4×D5C2×D414yes—
166C2×C4×D5C2×D414yes—
167C2×Dic10—21no—
168C2×Dic10—21no—
169C2×Dic10—22no—
170C2×Dic10—22no—
171C2×C5×C8—24no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—22no—
174C5×C22⋊C4—22no—
175C5×C22⋊C4—22no—
176C42×C5—21no—
177C42×C5—21no—
178C42×C5C2×D412yes—
179C42×C5C2×D412yes—
180#685—22no—
181#685—22no—
182#685C22⋊C414yes—
183#685C22⋊C414yes—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—22no—
187C4⋊Dic5—22no—
188C4⋊Dic5C22⋊C412yes—
189C4⋊Dic5C22⋊C412yes—
190#683—22no—
191#683—22no—
192#683—22no—
193#683—22no—
194C4×Dic5—22no—
195C4×Dic5—22no—
196C4×Dic5—22no—
197C4×Dic5—22no—
198C2×#168—24no—
199#676—84no—
200C8×D5—84no—
201C22×#41—51no—
202#520—104no—
203#520—104no—
204C2×C4×Q8—51no—
205C2×C4×Q8—51no—
206C2×C4×D4—51no—
207C2×C4×D4—51no—
208C2×C4×D4—51no—
209C2×C4×D4—51no—
210C2×C4×D4—51no—
211C2×C4×D4—51no—
212C2×#117—51no—
213C2×#117—51no—
214C2×#117C2×D511yescharacteristic
215C22×C4⋊C4—51no—
216C22×C4⋊C4—51no—
217C22×C4⋊C4—51no—
218#435—104no—
219#435—104no—
220#432—52no—
221#432C2×D512yes—
222#407—54no—
223C2×C22⋊C8—54no—
224#324—52no—
225#324—54no—
226#324—54no—
227#324C2×D512yes—
228#323—52no—
229#323C2×D512yes—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—41no—
233C23×C5—41no—
234C23×C5C2×M1612yes—
235C23×C5C23×C411yescharacteristic
236C22×D5—21no—
237C22×D5—21no—
238C22×D5—41no—
239C22×D5—41no—
240C22×D5—41no—
241C22×D5—41no—
242C22×D5C2×C22⋊C412yes—
243C22×D5C2×C22⋊C412yes—
244C22×D5C2×M1612yes—
245C22×D5C2×M1612yes—
246C2×C4×C5—24no—
247C2×C4×C5—24no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5—41no—
258C2×C4×C5C2×C22⋊C412yes—
259C2×C4×C5C2×C22⋊C411yescharacteristic
260C2×C4×C5C2×C22⋊C411yescharacteristic
261C2×C4×C5C2×M1612yes—
262C2×C4×C5C22×D412yes—
263C22⋊D5—28no—
264C22⋊D5—28no—
265C2×Dic5—21no—
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—21no—
271C2×Dic5—41no—
272C2×Dic5—41no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275C2×Dic5—82no—
276C2×Dic5—82no—
277C2×Dic5C2×C22⋊C412yes—
278C2×Dic5C2×C22⋊C412yes—
279C2×Dic5C2×M1612yes—
280C2×Dic5C2×M1612yes—
281D20—24no—
282D20—24no—
283C4×D5—28no—
284C4×D5—28no—
285C4×D5—44no—
286C4×D5—44no—
287C4×D5—82no—
288C4×D5—82no—
289Dic10—24no—
290Dic10—24no—
291C5×C8—44no—
292#168—44no—
293C2×#41—104no—
294C2×#41—102no—
295C2×#41—102no—
296C2×#41—104no—
297C22×Q8—51no—
298C22×D4—51no—
299C22×D4—51no—
300C22×D4—51no—
301C23×C4—51no—
302C23×C4—51no—
303C23×C4—51no—
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4—51no—
308C23×C4—51no—
309C23×C4C4×D511yescharacteristic
310C2×M16—104no—
311C22×C8—104no—
312C4×Q8—102no—
313C4×Q8—102no—
314C4×Q8—102no—
315C4×Q8—102no—
316C4×D4—54no—
317C4×D4—54no—
318C4×D4—102no—
319C4×D4—102no—
320C4×D4—104no—
321C4×D4—104no—
322C4×D4—104no—
323C4×D4—104no—
324#117—22no—
325#117—22no—
326#117—104no—
327#117—104no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—104no—
337C2×C4⋊C4—104no—
338C2×C4⋊C4—102no—
339C2×C4⋊C4—102no—
340C2×C4⋊C4C4×D511yescharacteristic
341C2×C4⋊C4C22×D511yescharacteristic
342C2×C22⋊C4—51no—
343C2×C22⋊C4—51no—
344C2×C22⋊C4—51no—
345C2×C22⋊C4—51no—
346C2×C22⋊C4C4×D511yescharacteristic
347C2×C22⋊C4C22×D511yescharacteristic
348C2×C42—51no—
349C2×C42—51no—
350C2×C42—51no—
351C2×C42—51no—
352C2×C42C4×D511yescharacteristic
353C2×C42C22×D511yescharacteristic
354C4⋊C8—22no—
355C4⋊C8—102no—
356C22⋊C8—54no—
357C22⋊C8—104no—
358C22⋊C8—104no—
359C22⋊C8C22×D514yes—
360#97—22no—
361#97—102no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—22no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—41no—
368C22×C5—41no—
369C22×C5C22×C22⋊C411yescharacteristic
370C22×C5C22×M1611yescharacteristic
371C22×C5C22×M1611yescharacteristic
372C2×D5—22no—
373C2×D5—22no—
374C2×D5—22no—
375C2×D5—22no—
376C2×D5—22no—
377C2×D5—42no—
378C2×D5—42no—
379C2×D5—42no—
380C2×D5—42no—
381C2×D5#40714yes—
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—22no—
385C4×C5—22no—
386C4×C5—22no—
387C4×C5—41no—
388C4×C5—41no—
389Dic5—22no—
390Dic5—22no—
391Dic5—22no—
392Dic5—22no—
393Dic5—41no—
394Dic5—41no—
395C24—51no—
396C24—51no—
397C24C2×C4×D511yescharacteristic
398#41—1016no—
399#41—1016no—
400C2×Q8—102no—
401C2×Q8—102no—
402C2×Q8—101no—
403C2×Q8—101no—
404C2×D4—54no—
405C2×D4—52no—
406C2×D4—52no—
407C2×D4—102no—
408C2×D4—102no—
409C2×D4—102no—
410C2×D4—102no—
411C2×D4—102no—
412C2×D4—102no—
413C2×D4—102no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—54no—
423C22×C4—54no—
424C22×C4—54no—
425C22×C4—51no—
426C22×C4—51no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—102no—
430C22×C4—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—401no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4—401no—
440C22×C4—401no—
441C22×C4—401no—
442C22×C4C2×C4×D511yescharacteristic
443C22×C4C2×C4×D511yescharacteristic
444C22×C4C2×C4×D511yescharacteristic
445C22×C4C2×C4×D511yescharacteristic
446C22×C4C2×C4×D511yescharacteristic
447C22×C4C23×D511yescharacteristic
448M16—404no—
449C2×C8—24no—
450C2×C8—104no—
451C2×C8—404no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—52no—
455C4⋊C4—52no—
456C4⋊C4—102no—
457C4⋊C4—102no—
458C4⋊C4—102no—
459C4⋊C4—102no—
460C4⋊C4—102no—
461C4⋊C4—102no—
462C4⋊C4—101no—
463C4⋊C4—101no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—54no—
467C22⋊C4—54no—
468C22⋊C4—102no—
469C22⋊C4—102no—
470C42—21no—
471C42—21no—
472C42—102no—
473C42—102no—
474C42—102no—
475C42—102no—
476C42D4×D512yes—
477C42D4×D512yes—
478C2×C5—21no—
479C2×C5—21no—
480C2×C5—42no—
481C2×C5C2×#40711yescharacteristic
482C2×C5#288311yescharacteristic
483C2×C5#288911yescharacteristic
484D5—24no—
485D5—42no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23—41no—
490C23—52no—
491C23—52no—
492C23—52no—
493C23—52no—
494C23—101no—
495C23—101no—
496C23—201no—
497C23—201no—
498C23—201no—
499C23—201no—
500C23D5×M1612yes—
501C23C22×C4×D511yescharacteristic
502Q8—104no—
503Q8—104no—
504D4—104no—
505D4—104no—
506D4—108no—
507D4—108no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
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—108no—
525C2×C4—108no—
526C2×C4—101no—
527C2×C4—101no—
528C2×C4—101no—
529C2×C4—101no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—204no—
533C2×C4—204no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4—201no—
537C2×C4—201no—
538C2×C4—402no—
539C2×C4—402no—
540C2×C4—402no—
541C2×C4—402no—
542C2×C4D5×C22⋊C411yescharacteristic
543C2×C4D5×C22⋊C411yescharacteristic
544C2×C4D5×C22⋊C412yes—
545C2×C4D5×M1612yes—
546C2×C4C2×D4×D512yes—
547C8—44no—
548C8—204no—
549C5#1736811yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—54no—
558C22—102no—
559C22—102no—
560C22—102no—
561C22—102no—
562C22—102no—
563C22—202no—
564C22—202no—
565C22—202no—
566C22—202no—
567C22C2×D5×C22⋊C411yescharacteristic
568C22C2×D5×M1611yescharacteristic
569C22C2×D5×M1611yescharacteristic
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—102no—
578C4—102no—
579C4—102no—
580C4—102no—
581C4—201no—
582C4—201no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—104no—
587C2—202no—
588C2D5×#40711yescharacteristic
589C2#1061042511yescharacteristic
590C2#1061044611yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1464C2
socleC22×C5C22×C22⋊C4
prime radicalC22×#709C4
cosocleC22×C4×C5C24
hypercenter#1464D5
hypocenterC5#17368
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #10611066C2semidirect
3. C5×#1464C2semidirect
4. #10602365C2semidirect
5. #10601623C2semidirect
6. #10598937C2semidirect
7. D5×#324C2semidirect
8. #10598914C2semidirect
9. #10598912C2semidirect
114. #1464D5semidirect
136. C2×C22⋊D5M16semidirect
164. C2×C4×D5M16semidirect
549. C5#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#658572
#106110662
C5×#14642
#106023652
#106016232
#105989372
D5×#3242
#105989142
#105989122
#173685