Finite Group Databasebeta

← #12967833

Subgroups — #12967833

Order 1280.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#541C211yesmaximal, characteristic, complemented
2#10617632C211yesmaximal, characteristic
3#10615182C214yesmaximal, complemented
4#10615178C214yesmaximal, complemented
5#10615177C218yesmaximal, complemented
6#10614940C212yesmaximal, complemented
7#10614748C212yesmaximal, complemented
8#10614466C212yesmaximal
9#10614458C212yesmaximal, complemented
10C4×#4088C214yesmaximal, complemented
11#10613940C211yesmaximal, characteristic, complemented
12C22×#709C2211yescharacteristic
13C5×#541—22no—
14C5×#541—22no—
15C5×#541C2214yes—
16C5×#541C2214yescomplemented
17C2×C5×C4⋊Q8C2211yescharacteristic, complemented
18C2×C5×#126—22no—
19C2×C5×#125—22no—
20C2×C5×#125C2211yescharacteristic, complemented
21C2×C5×C22⋊Q8C2212yes—
22C2×C4×C5×Q8C2212yescomplemented
23C2×C5×#117C2211yescharacteristic
24C2×#4088C2214yes—
25C2×#4072C2211yescharacteristic
26#65857C2214yes—
27#65856—22no—
28#65855C2212yes—
29D5×#117—22no—
30#65850C2212yes—
31#65838—22no—
32#65820—22no—
33C4×#709C2212yes—
34C4×#709C2212yes—
35#65514C2214yescomplemented
36#65496C2212yes—
37#65379C2218yescomplemented
38#65372C2218yescomplemented
39#65355C2218yescomplemented
40#65352C2218yescomplemented
41#65339C2214yes—
42#65336C2218yes—
43#65335C2214yes—
44#65334C2218yes—
45#65333C2214yescomplemented
46#65332C2214yescomplemented
47#65319C2218yescomplemented
48C4×#689C2218yescomplemented
49#65315C2218yescomplemented
50C4×#688C2218yescomplemented
51#65289C2214yes—
52#65289C2214yescomplemented
53#65255C2214yes—
54#65253C2214yescomplemented
55#65213C2212yescomplemented
56C4×#681C2212yescomplemented
57#60569—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C23×C4×C5C2311yescharacteristic
60C22×C22⋊D5—21no—
61C2×#709—21no—
62C2×#709—21no—
63C2×#709—28no—
64C2×#709C2312yes—
65C2×#709C2312yes—
66C22×D20C2311yescharacteristic
67C22×C4×D5—21no—
68C22×Dic10C2311yescharacteristic
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8C2314yes—
71C5×#126—28no—
72C5×#126—28no—
73C5×#125—28no—
74C5×#125—28no—
75C5×#125—22no—
76C5×#125C2314yes—
77C5×C22⋊Q8—22no—
78C5×C22⋊Q8—22no—
79C5×C22⋊Q8C2314yes—
80C5×C22⋊Q8C2314yes—
81C4×C5×Q8—24no—
82C4×C5×Q8C2318yes—
83C5×#117D412yes—
84C5×#117D412yes—
85C5×#117C2312yes—
86C5×#117C2312yes—
87C2×C5×C4⋊C4—22no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4—22no—
90C2×C5×C4⋊C4—22no—
91C2×C5×C4⋊C4D412yes—
92C2×C5×C4⋊C4C2312yes—
93C2×C5×C22⋊C4—22no—
94C2×C5×C22⋊C4D412yes—
95C2×C42×C5—22no—
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C2×C22⋊Dic5—21no—
99C2×#689C2318yes—
100#4088—216no—
101C2×#688C2318yes—
102C4×C22⋊D5—24no—
103C4×C22⋊D5—24no—
104#4072—21no—
105#4072—21no—
106#4072C2312yes—
107#4072C2312yes—
108C2×C4⋊Dic5C2312yes—
109C2×C4×Dic5—21no—
110C2×#681C2314yes—
111#4039—24no—
112#4038—24no—
113D5×C4⋊C4—24no—
114#4033—24no—
115#4027—24no—
116#4027—24no—
117D5×C22⋊C4—24no—
118#4023—24no—
119C4×D20C2314yes—
120C4×D20C2314yes—
121#4018—22no—
122#4018—22no—
123C42×D5—22no—
124C42×D5—22no—
125C4×Dic10C2314yes—
126C4×Dic10C2314yes—
127#3967C2318yes—
128#3967C2318yes—
129C22⋊#168C2314yes—
130C22⋊#168C2314yes—
131#3942C2318yes—
132#3941C2318yes—
133#3940C2318yes—
134#3939C2318yes—
135C4⋊#168C2318yes—
136#3935C2314yes—
137C4×#168C2314yes—
138C2×#541D511yescharacteristic, complemented
139#3428—51no—
140#3127—54no—
141#3123—54no—
142#3120—58no—
143#3095—52no—
144#3080—52no—
145#3039—52no—
146#3007—52no—
147C4×#137—54no—
148#2921—51no—
149C24×C5C2×D411yescharacteristic
150C23×D5—21no—
151C2×C5×Q8—22no—
152C2×C5×Q8—28no—
153C2×C5×Q8C2414yes—
154C22×C4×C5—24no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D412yes—
161C22×C4×C5C2×D412yes—
162C22×C4×C5C2411yescharacteristic
163C22×C4×C5C2411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C2×C22⋊D5—28no—
166C2×C22⋊D5—22no—
167C2×C22⋊D5—22no—
168C22×Dic5—21no—
169C22×Dic5—22no—
170#709—28no—
171#709—28no—
172#709—24no—
173#709—24no—
174#709—24no—
175#709#4118yes—
176C2×D20—21no—
177C2×D20—24no—
178C2×D20C2412yes—
179C2×C4×D5—28no—
180C2×C4×D5—22no—
181C2×C4×D5—22no—
182C2×C4×D5—24no—
183C2×C4×D5—24no—
184C2×Dic10—21no—
185C2×Dic10—24no—
186C2×Dic10C2412yes—
187C5×C4⋊C4—28no—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—28no—
191C5×C4⋊C4—28no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4C2×D414yes—
194C5×C4⋊C4C2×D414yes—
195C5×C4⋊C4C2414yes—
196C5×C22⋊C4—24no—
197C5×C22⋊C4—24no—
198C5×C22⋊C4C2×D414yes—
199C5×C22⋊C4C2×D414yes—
200C42×C5—28no—
201C42×C5C2×D412yes—
202C42×C5C2×D412yes—
203C42×C5C2412yes—
204C42×C5C2412yes—
205C22⋊Dic5—22no—
206C22⋊Dic5—22no—
207#689—216no—
208#688—216no—
209#685—24no—
210#685—24no—
211C4⋊Dic5—22no—
212C4⋊Dic5C2414yes—
213#683—24no—
214#683—24no—
215C4×Dic5—24no—
216C4×Dic5—24no—
217C4×Dic5—22no—
218C4×Dic5—22no—
219#681—28no—
220C2×#168C2418yes—
221C22×#41—51no—
222C2×#137—54no—
223#541—22no—
224#541—22no—
225#541C2×D514yes—
226#541C2×D514yescomplemented
227C2×C4⋊Q8C2×D511yescharacteristic, complemented
228C2×#126—22no—
229C2×#125—22no—
230C2×#125C2×D511yescharacteristic, complemented
231C2×C22⋊Q8C2×D512yes—
232#520—54no—
233#520—102no—
234#519—52no—
235#518—52no—
236#518—102no—
237C4×#41—52no—
238C4×#41—52no—
239C4×#41—102no—
240C2×C4×Q8C2×D512yescomplemented
241C2×#117—51no—
242C2×#117—102no—
243C2×#117C2×D511yescharacteristic
244#485—58no—
245#484—54no—
246#483—54no—
247#479—58no—
248#478—58no—
249#462—58no—
250#461—58no—
251#452—58no—
252#451—54no—
253#450—54no—
254#441—58no—
255#440—58no—
256C4×Q16—58no—
257C4×SD16—58no—
258#433—54no—
259#433—54no—
260#428—54no—
261#423—52no—
262#419—54no—
263#417—54no—
264#407—52no—
265C4×M16—52no—
266C23×C5—22no—
267C23×C5—42no—
268C23×C5C22×D411yescharacteristic
269C23×C5C22×D411yescharacteristic
270C23×C5C22×D411yescharacteristic
271C22×D5—22no—
272C22×D5—24no—
273C22×D5—82no—
274C5×Q8—28no—
275C5×Q8—44no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—24no—
282C2×C4×C5—24no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×C4×C5—44no—
286C2×C4×C5—44no—
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×D414yes—
289C2×C4×C5C22×D414yes—
290C2×C4×C5C22×D411yescharacteristic
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5C2×#4114yes—
293C2×C4×C5C2511yescharacteristic
294C22⋊D5—28no—
295C22⋊D5—28no—
296C2×Dic5—24no—
297C2×Dic5—22no—
298C2×Dic5—24no—
299C2×Dic5—82no—
300D20—24no—
301D20—22no—
302D20C2×#4114yes—
303C4×D5—28no—
304C4×D5—28no—
305C4×D5—84no—
306C4×D5—84no—
307Dic10—24no—
308Dic10—22no—
309Dic10C2×#4114yes—
310#168—28no—
311C2×#41—52no—
312C2×#41—52no—
313C2×#41—108no—
314C2×#41—101no—
315C2×#41—101no—
316C22×Q8—51no—
317C22×Q8C22×D512yes—
318C22×D4—51no—
319C22×D4—101no—
320C23×C4—101no—
321C23×C4C22×D511yescharacteristic
322#137—1016no—
323C2×Q16—58no—
324C2×SD16—58no—
325C2×M16—54no—
326C4⋊Q8—22no—
327C4⋊Q8C22×D514yes—
328#126—28no—
329#126—28no—
330#125—28no—
331#125—28no—
332#125—22no—
333#125C22×D514yes—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8C22×D514yes—
337C22⋊Q8C22×D514yes—
338C4×Q8—24no—
339C4×Q8—54no—
340C4×Q8—54no—
341C4×Q8—104no—
342C4×Q8C22×D518yes—
343C4×D4—54no—
344C4×D4—54no—
345C4×D4—104no—
346C4×D4—104no—
347C4×D4—104no—
348C4×D4—104no—
349C4×D4—104no—
350#117—52no—
351#117—52no—
352#117—101no—
353#117—101no—
354#117—104no—
355#117—104no—
356#117—102no—
357#117—102no—
358#117C22⋊D512yes—
359#117C22⋊D512yes—
360#117C22×D512yes—
361#117C22×D512yes—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—52no—
367C2×C4⋊C4—104no—
368C2×C4⋊C4C22⋊D512yes—
369C2×C4⋊C4C22×D512yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—101no—
372C2×C22⋊C4—104no—
373C2×C22⋊C4C22⋊D512yes—
374C2×C42—22no—
375C2×C42—101no—
376C2×C42—102no—
377C2×C42—102no—
378C2×C42C22×D511yescharacteristic
379C2×C42C22×D511yescharacteristic
380#107—58no—
381#106—58no—
382C4⋊C8—58no—
383#103—58no—
384#103—58no—
385#103—58no—
386#102—58no—
387C22⋊C8—54no—
388C22⋊C8—54no—
389#97—54no—
390C4×C8—54no—
391C22×C5—22no—
392C22×C5—22no—
393C22×C5—22no—
394C22×C5—21no—
395C22×C5—21no—
396C22×C5—82no—
397C22×C5#52512yes—
398C22×C5C23×D411yescharacteristic
399C2×D5—24no—
400C2×D5—44no—
401C2×D5—84no—
402C4×C5—24no—
403C4×C5—24no—
404C4×C5—21no—
405C4×C5—21no—
406C4×C5—44no—
407C4×C5—44no—
408C4×C5#52512yes—
409C4×C5C22×#4112yes—
410Dic5—24no—
411Dic5—44no—
412C24—101no—
413C24C2×C22⋊D511yescharacteristic
414#41—58no—
415#41—104no—
416#41—104no—
417#41—108no—
418#41—108no—
419#41—104no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—52no—
423C2×Q8—104no—
424C2×Q8—101no—
425C2×Q8C23×D514yes—
426C2×D4—52no—
427C2×D4—104no—
428C2×D4—108no—
429C2×D4—102no—
430C2×D4—102no—
431C2×D4—101no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—108no—
436C22×C4—101no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4—104no—
441C22×C4—104no—
442C22×C4C2×C22⋊D511yescharacteristic
443C22×C4C2×C22⋊D511yescharacteristic
444C22×C4C2×C22⋊D511yescharacteristic
445C22×C4C2×C22⋊D512yes—
446C22×C4C2×C22⋊D512yes—
447C22×C4C23×D511yescharacteristic
448C22×C4C23×D511yescharacteristic
449C22×C4C23×D511yescharacteristic
450Q16—1016no—
451SD16—1016no—
452M16—108no—
453C2×C8—58no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—22no—
460C4⋊C4—54no—
461C4⋊C4—102no—
462C4⋊C4—104no—
463C4⋊C4—104no—
464C4⋊C4C2×C22⋊D514yes—
465C4⋊C4C2×C22⋊D514yes—
466C4⋊C4C23×D514yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—102no—
470C22⋊C4—102no—
471C22⋊C4—104no—
472C22⋊C4—104no—
473C22⋊C4C2×C22⋊D514yes—
474C22⋊C4C2×C22⋊D514yes—
475C42—28no—
476C42—102no—
477C42—102no—
478C42—104no—
479C42—104no—
480C42C2×C22⋊D512yes—
481C42C2×C22⋊D512yes—
482C42C23×D512yes—
483C42C23×D512yes—
484C2×C5—22no—
485C2×C5—41no—
486C2×C5—41no—
487C2×C5C2×#52511yescharacteristic
488C2×C5#358612yes—
489D5—44no—
490C23—22no—
491C23—42no—
492C23—104no—
493C23—102no—
494C23—402no—
495C23C22×C22⋊D511yescharacteristic
496C23C22×C22⋊D511yescharacteristic
497C23C22×C22⋊D511yescharacteristic
498Q8—28no—
499Q8—44no—
500Q8—54no—
501Q8—104no—
502Q8—102no—
503D4—54no—
504D4—104no—
505D4—108no—
506D4—108no—
507D4—102no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—108no—
520C2×C4—104no—
521C2×C4—102no—
522C2×C4—108no—
523C2×C4—104no—
524C2×C4—404no—
525C2×C4—404no—
526C2×C4—402no—
527C2×C4D5×#4114yes—
528C2×C4C22×C22⋊D514yes—
529C2×C4C22×C22⋊D511yescharacteristic
530C2×C4C22×C22⋊D514yes—
531C2×C4C22×C22⋊D511yescharacteristic
532C2×C4C22×C22⋊D511yescharacteristic
533C2×C4C24×D511yescharacteristic
534C8—108no—
535C5#6056911yescharacteristic, complemented, Sylow
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—82no—
542C22—104no—
543C22—204no—
544C22—404no—
545C22#6616212yes—
546C22C23×C22⋊D511yescharacteristic
547C4—21no—
548C4—21no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—104no—
554C4—204no—
555C4#6616212yes—
556C4C2×D5×#4112yes—
557C2—22no—
558C2—41no—
559C2—41no—
560C2—204no—
561C2C2×#6616211yescharacteristic
562C2#1061918812yes—
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#541C2
socleC22×C5C23×D4
prime radicalC22×#709C22
hypercenterC2×#541D5
hypocenterC5#60569
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C5×#541C2semidirect
3. #10615182C2semidirect
4. #10615178C2semidirect
5. #10615177C2semidirect
6. #10614940C2semidirect
7. #10614748C2semidirect
9. #10614458C2semidirect
10. C4×#4088C2semidirect
11. #10613940C2semidirect
16. C5×#541C22semidirect
17. C2×C5×C4⋊Q8C22semidirect
20. C2×C5×#125C22semidirect
22. C2×C4×C5×Q8C22semidirect
35. #65514C22semidirect
37. #65379C22semidirect
38. #65372C22semidirect
39. #65355C22semidirect
40. #65352C22semidirect
45. #65333C22semidirect
46. #65332C22semidirect
47. #65319C22semidirect
48. C4×#689C22semidirect
49. #65315C22semidirect
50. C4×#688C22semidirect
52. #65289C22semidirect
54. #65253C22semidirect
55. #65213C22semidirect
56. C4×#681C22semidirect
138. C2×#541D5semidirect
226. #541C2×D5semidirect
227. C2×C4⋊Q8C2×D5semidirect
230. C2×#125C2×D5semidirect
240. C2×C4×Q8C2×D5semidirect
535. C5#60569semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5412
#106176322
#106151822
#106151782
#106151772
#106149402
#106147482
#106144662
#106144582
C4×#40882
#106139402
#605695