Finite Group Databasebeta

← #12065211

Subgroups — #12065211

Order 1280.

1210 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65880C211yesmaximal, characteristic
2C5×#2973C211yesmaximal, characteristic, complemented
3#10614033C211yesmaximal, characteristic, complemented
4#10601623C212yesmaximal, complemented
5#10601619C212yesmaximal, complemented
6#10600764C212yesmaximal, complemented
7#10600761C212yesmaximal, complemented
8C2×#64689C212yesmaximal, complemented
9#10600712C212yesmaximal, complemented
10C22×#711C411yescharacteristic
11C2×C4×C5×D4C2211yescharacteristic
12C2×D4×Dic5C411yescharacteristic
13C2×C4×C22⋊D5C2211yescharacteristic
14C2×C4×C22⋊D5C2211yescharacteristic
15C22×#683C2211yescharacteristic
16C22×#683C2211yescharacteristic
17#65880—24no—
18#65880—24no—
19C2×D5×C4⋊C4C411yescharacteristic
20C2×#4033C411yescharacteristic
21C2×#4027C411yescharacteristic
22C2×#4027C411yescharacteristic
23C2×#4023C411yescharacteristic
24C2×#4023C411yescharacteristic
25C2×#4018C2211yescharacteristic
26C2×C4×Dic10C2211yescharacteristic
27C5×#435—24no—
28C5×#432C2212yes—
29C5×#407C2212yes—
30C2×C5×C22⋊C8C2212yes—
31#65496C2212yes—
32C2×C22⋊#168C2212yes—
33#65300—24no—
34#65215C2212yes—
35#64689C2214yes—
36#64689C2214yes—
37#64689C2214yes—
38#64688C2214yes—
39#17368—51nomaximal, Sylow
40C22×C5×D4C2×C411yescharacteristic
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2311yescharacteristic
43C22×C22⋊D5C2×C411yescharacteristic
44C22×C22⋊D5C2×C411yescharacteristic
45C23×Dic5C2×C411yescharacteristic
46C23×Dic5C2×C411yescharacteristic
47C23×Dic5C2×C411yescharacteristic
48C23×Dic5C2×C411yescharacteristic
49C2×#711—24no—
50C2×#711—24no—
51C2×#711—24no—
52C22×C4×D5C2×C411yescharacteristic
53C22×C4×D5C2×C411yescharacteristic
54C22×C4×D5C2×C411yescharacteristic
55C22×Dic10C2×C411yescharacteristic
56C2×C5×M16—22no—
57C22×C5×C8—22no—
58C4×C5×D4D414yes—
59C4×C5×D4D414yes—
60C2×C5×C4⋊C4C2×C411yescharacteristic
61C2×C5×C22⋊C4C2×C411yescharacteristic
62C2×C5×C22⋊C4C2×C411yescharacteristic
63C2×C42×C5C2311yescharacteristic
64C2×C22⋊Dic5C2×C411yescharacteristic
65C2×C22⋊Dic5C2×C411yescharacteristic
66D4×Dic5—24no—
67C4×C22⋊D5—22no—
68C4×C22⋊D5—22no—
69C4×C22⋊D5—22no—
70C4×C22⋊D5—22no—
71C2×#685C2×C411yescharacteristic
72C2×#685C2311yescharacteristic
73C2×C4⋊Dic5C2×C411yescharacteristic
74C2×#683—22no—
75C2×#683—22no—
76C2×#683—22no—
77C2×#683—22no—
78C2×#683C2×C411yescharacteristic
79C2×#683C2×C411yescharacteristic
80C2×#683C2×C411yescharacteristic
81C2×#683C2×C411yescharacteristic
82C2×#683C2×C411yescharacteristic
83C2×#683C2311yescharacteristic
84C2×#683C2311yescharacteristic
85C2×#683C2311yescharacteristic
86C2×C4×Dic5C2×C411yescharacteristic
87C2×C4×Dic5C2×C411yescharacteristic
88C2×C4×Dic5C2×C411yescharacteristic
89C2×C4×Dic5C2×C411yescharacteristic
90C2×C4×Dic5C2×C411yescharacteristic
91C2×#681—22no—
92C22×#168—22no—
93D5×C4⋊C4—24no—
94#4033—24no—
95#4027—24no—
96#4027—24no—
97#4023—24no—
98#4023—24no—
99#4018—22no—
100#4018—22no—
101C4×Dic10—22no—
102C4×Dic10—22no—
103C5×C4⋊C8—22no—
104C5×C22⋊C8—22no—
105C5×C22⋊C8—22no—
106C5×C22⋊C8C2314yes—
107C5×#97—22no—
108C22⋊#168—22no—
109C22⋊#168—22no—
110C22⋊#168C2314yes—
111C4⋊#168—22no—
112#3935—22no—
113C2×#520—51no—
114#2973—51no—
115#2973D511yescharacteristic, complemented
116#1467—52no—
117#1465—52no—
118#1464—52no—
119#1463—52no—
120#1459—52no—
121C2×#324—52no—
122C24×C5C22×C411yescharacteristic
123C24×C5C22×C411yescharacteristic
124C23×D5C22×C411yescharacteristic
125C2×C5×D4—22no—
126C2×C5×D4—22no—
127C2×C5×D4C22⋊C414yes—
128C22×C4×C5—81no—
129C22×C4×C5—81no—
130C22×C4×C5C22×C411yescharacteristic
131C22×C4×C5C22×C411yescharacteristic
132C22×C4×C5C22×C411yescharacteristic
133C22×C4×C5C22×C411yescharacteristic
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2411yescharacteristic
139C2×C22⋊D5—22no—
140C2×C22⋊D5—22no—
141C2×C22⋊D5—22no—
142C2×C22⋊D5—22no—
143C2×C22⋊D5—22no—
144C2×C22⋊D5M1614yescomplemented
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—81no—
153C22×Dic5—81no—
154C22×Dic5M1614yescomplemented
155C22×Dic5C22×C411yescharacteristic
156C22×Dic5C22×C411yescharacteristic
157C22×Dic5C22×C411yescharacteristic
158C22×Dic5C22×C411yescharacteristic
159C22×Dic5C22×C411yescharacteristic
160C22×Dic5C22×C411yescharacteristic
161C22×Dic5C22×C411yescharacteristic
162#711—232no—
163C2×C4×D5—22no—
164C2×C4×D5—22no—
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—81no—
169C2×C4×D5—81no—
170C2×Dic10—22no—
171C2×Dic10—22no—
172C2×Dic10—22no—
173C5×M16—82no—
174C2×C5×C8—24no—
175C2×C5×C8—82no—
176C5×C4⋊C4C22⋊C414yes—
177C5×C22⋊C4C22⋊C414yes—
178C5×C22⋊C4C22⋊C414yes—
179C42×C5C2×D412yes—
180C42×C5C2×D412yes—
181C22⋊Dic5—22no—
182C22⋊Dic5—22no—
183#685—22no—
184#685—21no—
185#685—21no—
186C4⋊Dic5—22no—
187#683—22no—
188#683—22no—
189#683—22no—
190#683—22no—
191#683—22no—
192#683—21no—
193#683—21no—
194#683—21no—
195#683—21no—
196#683—21no—
197#683—21no—
198C4×Dic5—22no—
199C4×Dic5—22no—
200C4×Dic5—22no—
201C4×Dic5—22no—
202C4×Dic5—22no—
203#681—82no—
204C2×#168—24no—
205C2×#168—82no—
206C22×#41—51no—
207#520—104no—
208#520—104no—
209C2×C4×Q8—51no—
210C2×C4×Q8—51no—
211C2×C4×D4—51no—
212C2×C4×D4—51no—
213C2×C4×D4—51no—
214C2×C4×D4—51no—
215C2×C4×D4—51no—
216C2×C4×D4C2×D511yescharacteristic
217C2×#117—51no—
218C2×#117—51no—
219C2×#117—51no—
220C22×C4⋊C4—51no—
221C22×C4⋊C4—51no—
222C22×C4⋊C4—51no—
223#435—24no—
224#435—104no—
225#432—52no—
226#432C2×D512yes—
227#407—52no—
228#407C2×D512yes—
229C2×C22⋊C8—52no—
230C2×C22⋊C8C2×D512yes—
231#324—54no—
232#324—54no—
233#324—54no—
234#323—54no—
235C23×C5—21no—
236C23×C5—21no—
237C23×C5—41no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—41no—
241C23×C5C2×C22⋊C412yes—
242C23×C5C2×C22⋊C412yes—
243C23×C5C2×M1612yes—
244C23×C5C2×M1612yes—
245C23×C5C23×C411yescharacteristic
246C22×D5—21no—
247C22×D5—21no—
248C22×D5—41no—
249C22×D5—41no—
250C22×D5C2×M1612yes—
251C5×D4—28no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5C2×C22⋊C412yes—
263C2×C4×C5C2×C22⋊C412yes—
264C2×C4×C5C2×C22⋊C412yes—
265C2×C4×C5C2×C22⋊C412yes—
266C2×C4×C5C22×D411yescharacteristic
267C2×C4×C5C22×D411yescharacteristic
268C22⋊D5—28no—
269C22⋊D5—28no—
270C2×Dic5—28no—
271C2×Dic5—21no—
272C2×Dic5—21no—
273C2×Dic5—21no—
274C2×Dic5—21no—
275C2×Dic5—21no—
276C2×Dic5—21no—
277C2×Dic5—28no—
278C2×Dic5—21no—
279C2×Dic5—21no—
280C2×Dic5—21no—
281C2×Dic5—21no—
282C2×Dic5—41no—
283C2×Dic5—41no—
284C2×Dic5—41no—
285C2×Dic5—41no—
286C2×Dic5—82no—
287C2×Dic5—82no—
288C2×Dic5C2×M1612yes—
289C2×Dic5C2×M1612yes—
290C2×Dic5C2×M1612yes—
291C4×D5—28no—
292C4×D5—82no—
293C4×D5—82no—
294Dic10—28no—
295C5×C8—44no—
296#168—44no—
297C2×#41—104no—
298C2×#41—104no—
299C2×#41—104no—
300C22×Q8—51no—
301C22×D4—51no—
302C22×D4—51no—
303C22×D4C4×D511yescharacteristic
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4—51no—
308C23×C4—51no—
309C23×C4—51no—
310C23×C4—51no—
311C23×C4C22×D511yescharacteristic
312C23×C4C22×D511yescharacteristic
313C2×M16—22no—
314C2×M16—102no—
315C22×C8—22no—
316C22×C8—102no—
317C4×Q8—104no—
318C4×Q8—102no—
319C4×Q8—102no—
320C4×D4—104no—
321C4×D4—102no—
322C4×D4—102no—
323C4×D4—102no—
324C4×D4—102no—
325C4×D4—104no—
326C4×D4—104no—
327C4×D4D2014yes—
328C4×D4C22⋊D514yes—
329#117—104no—
330#117—104no—
331#117—102no—
332#117—102no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—51no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—51no—
342C2×C4⋊C4—102no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—104no—
347C2×C4⋊C4C4×D511yescharacteristic
348C2×C22⋊C4—51no—
349C2×C22⋊C4—51no—
350C2×C22⋊C4—51no—
351C2×C22⋊C4—51no—
352C2×C22⋊C4C4×D511yescharacteristic
353C2×C22⋊C4C4×D511yescharacteristic
354C2×C42—51no—
355C2×C42—51no—
356C2×C42—51no—
357C2×C42—51no—
358C2×C42—51no—
359C2×C42C22×D511yescharacteristic
360C4⋊C8—22no—
361C4⋊C8—102no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—54no—
365C22⋊C8—102no—
366C22⋊C8—102no—
367C22⋊C8C22×D514yes—
368#97—22no—
369#97—102no—
370C22×C5—21no—
371C22×C5—22no—
372C22×C5—21no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5#40714yes—
382C22×C5C22×C22⋊C411yescharacteristic
383C22×C5C22×M1611yescharacteristic
384C22×C5C22×M1611yescharacteristic
385C2×D5—22no—
386C2×D5—42no—
387C2×D5—41no—
388C2×D5—41no—
389C2×D5—42no—
390C4×C5—22no—
391C4×C5—21no—
392C4×C5—21no—
393C4×C5—22no—
394C4×C5—22no—
395C4×C5—22no—
396Dic5—22no—
397Dic5—22no—
398Dic5—22no—
399Dic5—41no—
400Dic5—41no—
401Dic5—41no—
402Dic5—41no—
403C24—51no—
404C24C2×C4×D511yescharacteristic
405C24C2×C4×D511yescharacteristic
406#41—1032no—
407C2×Q8—102no—
408C2×Q8—102no—
409C2×Q8—102no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—54no—
413C2×D4—102no—
414C2×D4—102no—
415C2×D4—102no—
416C2×D4—102no—
417C2×D4—102no—
418C2×D4#68514yes—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—54no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—51no—
426C22×C4—51no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—102no—
430C22×C4—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4—102no—
441C22×C4—401no—
442C22×C4—401no—
443C22×C4—401no—
444C22×C4—401no—
445C22×C4C2×C4×D511yescharacteristic
446C22×C4C2×C4×D511yescharacteristic
447C22×C4C2×C4×D511yescharacteristic
448C22×C4C2×C4×D511yescharacteristic
449C22×C4C2×D2012yes—
450C22×C4C2×D2012yes—
451C22×C4C2×C22⋊D512yes—
452C22×C4C2×C22⋊D512yes—
453C22×C4C23×D511yescharacteristic
454M16—82no—
455M16—402no—
456C2×C8—24no—
457C2×C8—82no—
458C2×C8—104no—
459C2×C8—402no—
460C4⋊C4—101no—
461C4⋊C4—101no—
462C4⋊C4—101no—
463C4⋊C4—101no—
464C4⋊C4—101no—
465C4⋊C4—101no—
466C4⋊C4—102no—
467C4⋊C4—102no—
468C4⋊C4—102no—
469C4⋊C4—102no—
470C4⋊C4—102no—
471C4⋊C4—102no—
472C4⋊C4#68514yes—
473C22⋊C4—102no—
474C22⋊C4—102no—
475C22⋊C4—101no—
476C22⋊C4—101no—
477C22⋊C4—102no—
478C22⋊C4#68514yes—
479C22⋊C4#68514yes—
480C42—102no—
481C42—102no—
482C42—102no—
483C42—102no—
484C42—102no—
485C42C2×D2012yes—
486C42C2×C22⋊D512yes—
487C2×C5—21no—
488C2×C5—21no—
489C2×C5—24no—
490C2×C5—42no—
491C2×C5C2×#40711yescharacteristic
492C2×C5#288311yescharacteristic
493C2×C5#288911yescharacteristic
494D5—42no—
495C23—21no—
496C23—21no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—52no—
502C23—101no—
503C23—101no—
504C23—201no—
505C23—201no—
506C23D5×M1612yes—
507C23D5×M1612yes—
508C23C2×#68512yes—
509C23C2×#68512yes—
510C23C22×C4×D511yescharacteristic
511Q8—108no—
512D4—28no—
513D4—108no—
514D4—108no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—52no—
526C2×C4—52no—
527C2×C4—52no—
528C2×C4—108no—
529C2×C4—108no—
530C2×C4—101no—
531C2×C4—101no—
532C2×C4—101no—
533C2×C4—101no—
534C2×C4—101no—
535C2×C4—101no—
536C2×C4—108no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—101no—
540C2×C4—101no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—402no—
546C2×C4—402no—
547C2×C4—402no—
548C2×C4—402no—
549C2×C4C2×#68512yes—
550C2×C4C2×#68512yes—
551C2×C4C2×#68512yes—
552C2×C4C2×#68512yes—
553C2×C4C22×D2011yescharacteristic
554C2×C4C22×C22⋊D511yescharacteristic
555C8—44no—
556C8—204no—
557C5#1736811yescharacteristic, complemented, Sylow
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—102no—
570C22—202no—
571C22—202no—
572C22—201no—
573C22—201no—
574C22#6541514yes—
575C22C2×D5×M1611yescharacteristic
576C22C2×D5×M1611yescharacteristic
577C22C22×#68511yescharacteristic
578C4—21no—
579C4—21no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—102no—
585C4—102no—
586C4—102no—
587C4—201no—
588C4—201no—
589C4—201no—
590C4—201no—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—42no—
595C2—202no—
596C2#1061427711yescharacteristic
597C2C2×#6541511yescharacteristic
598C2#1061554611yescharacteristic
subgroupG/H
centerC22C22×#685
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2973C2
socleC22×C5C22×C22⋊C4
prime radicalC22×#711C4
cosocleC22×C4×C5C24
hypercenter#2973D5
hypocenterC5#17368
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. C5×#2973C2semidirect
3. #10614033C2semidirect
4. #10601623C2semidirect
5. #10601619C2semidirect
6. #10600764C2semidirect
7. #10600761C2semidirect
8. C2×#64689C2semidirect
9. #10600712C2semidirect
115. #2973D5semidirect
144. C2×C22⋊D5M16semidirect
154. C22×Dic5M16semidirect
557. C5#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#658802
C5×#29732
#106140332
#106016232
#106016192
#106007642
#106007612
C2×#646892
#106007122
#173685