Finite Group Databasebeta

← #12099116

Subgroups — #12099116

Order 1280.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65838C211yesmaximal, characteristic
2C2×#65399C211yesmaximal, characteristic
3#10613917C211yesmaximal, characteristic
4C2×C5×#340C211yesmaximal, characteristic, complemented
5#10602762C211yesmaximal, characteristic, complemented
6#10602351C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10599666C212yesmaximal, complemented
9#10599665C212yesmaximal, complemented
10#10599664C212yesmaximal, complemented
11#10599606C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×C4×C22⋊D5—22no—
17C2×#4072C2211yescharacteristic
18C2×#4072C2211yescharacteristic
19C22×#681C2211yescharacteristic
20#65838—24no—
21#65838C418yescomplemented
22C2×#4039—21no—
23C2×#4038C2212yes—
24C2×D5×C4⋊C4C2212yes—
25C2×#4033—21no—
26#65502—22no—
27#65399—24no—
28C2×#3952C2212yes—
29C2×#3952C2212yes—
30#65277—21no—
31#65276C2212yes—
32#65276C2212yes—
33#65263—21no—
34C5×#340C2212yes—
35C5×#340C2212yes—
36#64771C2211yescharacteristic
37#64771C2211yescharacteristic
38#64771C2211yescharacteristic
39#64771C2211yescharacteristic
40#64749C2212yes—
41#64749C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#20703—51nomaximal, Sylow
45C23×C4×C5—21no—
46C23×C4×C5—21no—
47C23×C4×C5C2311yescharacteristic
48C22×C22⋊D5D412yes—
49C2×#709—22no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709C2×C414yes—
54C2×#709D414yes—
55C22×D20D411yescharacteristic
56C22×C4×D5—22no—
57C22×C4×D5C2312yes—
58C22×Dic10D411yescharacteristic
59C22×C5×C8—22no—
60C22×C5×C8—22no—
61C22×C5×C8—22no—
62C22×C5×C8C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2×C414yes—
70C2×C5×C4⋊C4C2312yes—
71C2×C5×C4⋊C4C2312yes—
72C2×C22⋊Dic5—21no—
73C2×C22⋊Dic5—21no—
74C4×C22⋊D5—216no—
75C2×#685—22no—
76C2×#685—22no—
77C2×#685—22no—
78#4072—22no—
79#4072—22no—
80#4072—22no—
81#4072C2×C414yes—
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×#683—22no—
87C2×C4×Dic5—21no—
88C2×C4×Dic5—22no—
89C2×C4×Dic5—21no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C22×#168C2312yes—
94#4039—28no—
95#4038—24no—
96#4038C2×C418yescomplemented
97D5×C4⋊C4—24no—
98D5×C4⋊C4C2×C418yescomplemented
99#4033—28no—
100#3963—22no—
101#3963—22no—
102#3952—24no—
103#3952—24no—
104C2×#520—51no—
105C2×#408—51no—
106#2311—51no—
107#1825—52no—
108#1825—52no—
109#1824—52no—
110#1823—52no—
111#1736—51no—
112#1736—51no—
113#1735—51no—
114C2×#340D511yescharacteristic, complemented
115C24×C5C2×D411yescharacteristic
116C23×D5C2×D412yes—
117C22×C4×C5—22no—
118C22×C4×C5—24no—
119C22×C4×C5—22no—
120C22×C4×C5—42no—
121C22×C4×C5—81no—
122C22×C4×C5—81no—
123C22×C4×C5—81no—
124C22×C4×C5—81no—
125C22×C4×C5C22×C412yes—
126C22×C4×C5C2×D411yescharacteristic
127C22×C4×C5C2×D412yes—
128C22×C4×C5#4112yes—
129C22×C4×C5#4112yes—
130C22×C4×C5#4112yes—
131C22×C4×C5#4112yes—
132C22×C4×C5C2411yescharacteristic
133C2×C22⋊D5—24no—
134C2×C22⋊D5—24no—
135C2×C22⋊D5C22⋊C418yes—
136C22×Dic5—21no—
137C22×Dic5—21no—
138C22×Dic5—21no—
139C22×Dic5—21no—
140C22×Dic5C2×D412yes—
141#709—28no—
142#709—28no—
143#709—28no—
144#709—28no—
145C2×D20—22no—
146C2×D20—22no—
147C2×D20C22⋊C414yes—
148C2×C4×D5—22no—
149C2×C4×D5—22no—
150C2×C4×D5—22no—
151C2×C4×D5—22no—
152C2×C4×D5—28no—
153C2×C4×D5—82no—
154C2×C4×D5—82no—
155C2×C4×D5C22×C414yes—
156C2×C4×D5C2×D414yes—
157C2×Dic10—22no—
158C2×Dic10—22no—
159C2×Dic10C22⋊C414yes—
160C2×C5×C8—21no—
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8—44no—
167C5×C4⋊C4—24no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—42no—
172C5×C4⋊C4C22×C414yes—
173C22⋊Dic5—24no—
174C22⋊Dic5—42no—
175#685—28no—
176#685—82no—
177#685—82no—
178C4⋊Dic5—21no—
179C4⋊Dic5—21no—
180C4⋊Dic5—21no—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5C22×C412yes—
185C4⋊Dic5C22×C412yes—
186#683—28no—
187C4×Dic5—24no—
188C4×Dic5—28no—
189C4×Dic5—42no—
190#681—44no—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168—22no—
194C22×#41—51no—
195C22×M16—51no—
196C23×C8C2×D511yescharacteristic
197#520—58no—
198#520—104no—
199C2×C4×Q8—101no—
200C2×C4×D4—101no—
201C2×C4×D4—102no—
202C2×#117—51no—
203C2×#117—51no—
204C2×#117—52no—
205C22×C4⋊C4—52no—
206C22×C4⋊C4C2×D511yescharacteristic
207C22×C4⋊C4C2×D511yescharacteristic
208#408—104no—
209C2×C22⋊C8—52no—
210C2×C22⋊C8—52no—
211#391—101no—
212#390—101no—
213#386—102no—
214#380—52no—
215#380—52no—
216#340—51no—
217#340—51no—
218#340—51no—
219#340—51no—
220#340—52no—
221#340—52no—
222#340—52no—
223#340—52no—
224#340C2×D512yes—
225#340C2×D512yes—
226C23×C5—21no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5C2×C22⋊C412yes—
231C23×C5#12312yes—
232C23×C5#12312yes—
233C23×C5C22×D411yescharacteristic
234C22×D5—22no—
235C22×D5—22no—
236C22×D5—22no—
237C22×D5—22no—
238C22×D5—22no—
239C22×D5—22no—
240C22×D5C2×C22⋊C414yes—
241C2×C4×C5—24no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—84no—
248C2×C4×C5—84no—
249C2×C4×C5C2×C22⋊C412yes—
250C2×C4×C5#11714yes—
251C2×C4×C5#11714yes—
252C2×C4×C5#12314yes—
253C2×C4×C5#12312yes—
254C2×C4×C5#12314yes—
255C2×C4×C5#12312yes—
256C2×C4×C5C23×C411yescharacteristic
257C2×C4×C5C22×D411yescharacteristic
258C2×C4×C5C2×#4111yescharacteristic
259C2×C4×C5C2×#4111yescharacteristic
260C2×C4×C5C2×#4111yescharacteristic
261C2×C4×C5C2×#4111yescharacteristic
262C22⋊D5—216no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—22no—
267C2×Dic5—41no—
268C2×Dic5—41no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—41no—
272C2×Dic5—41no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275C2×Dic5—41no—
276C2×Dic5—41no—
277C2×Dic5C2×C22⋊C414yes—
278D20—28no—
279C4×D5—24no—
280C4×D5—24no—
281C4×D5—24no—
282C4×D5—24no—
283C4×D5—88no—
284Dic10—28no—
285C5×C8—41no—
286C5×C8—41no—
287#168—42no—
288C2×#41—54no—
289C2×#41—54no—
290C2×#41—102no—
291C2×#41—102no—
292C2×#41—102no—
293C2×#41—102no—
294C22×Q8—51no—
295C22×D4—52no—
296C22×D4—51no—
297C23×C4—21no—
298C23×C4—21no—
299C23×C4—52no—
300C23×C4—102no—
301C23×C4C22×D511yescharacteristic
302C2×M16—102no—
303C2×M16—102no—
304C2×M16—102no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8—52no—
309C22×C8C22×D511yescharacteristic
310C22×C8C22×D511yescharacteristic
311C4×Q8—108no—
312C4×D4—108no—
313C4×D4—1016no—
314#117—54no—
315#117—58no—
316#117—104no—
317#117—102no—
318#117—102no—
319#117—102no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4—58no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—104no—
331C2×C4⋊C4—102no—
332C2×C4⋊C4C4×D514yes—
333C2×C4⋊C4C22×D512yes—
334C2×C4⋊C4C22×D512yes—
335C2×C22⋊C4—102no—
336C2×C22⋊C4—101no—
337C2×C22⋊C4—101no—
338C2×C22⋊C4—102no—
339C2×C22⋊C4—102no—
340C2×C42—102no—
341C2×C42—101no—
342C2×C42—101no—
343C22⋊C8—104no—
344C22⋊C8—104no—
345#95—102no—
346#95—102no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5#38114yes—
354C22×C5C22×C22⋊C411yescharacteristic
355C22×C5C2×#12311yescharacteristic
356C22×C5C2×#12311yescharacteristic
357C22×C5#57611yescharacteristic
358C22×C5#57711yescharacteristic
359C22×C5#57711yescharacteristic
360C22×C5#57811yescharacteristic
361C2×D5—24no—
362C2×D5—24no—
363C2×D5—24no—
364C2×D5—42no—
365C2×D5—42no—
366C2×D5—42no—
367C2×D5—42no—
368C4×C5—44no—
369C4×C5—82no—
370C4×C5#38114yes—
371C4×C5C2×#11711yescharacteristic
372C4×C5C2×#11711yescharacteristic
373C4×C5C2×#12311yescharacteristic
374C4×C5C2×#12311yescharacteristic
375Dic5—24no—
376Dic5—42no—
377Dic5—42no—
378Dic5—81no—
379Dic5—81no—
380C24—52no—
381C24D4×D511yescharacteristic
382#41—108no—
383#41—108no—
384#41—108no—
385#41—108no—
386C2×Q8—54no—
387C2×Q8—102no—
388C2×Q8—102no—
389C2×D4—58no—
390C2×D4—54no—
391C2×D4—104no—
392C2×D4—104no—
393C2×D4—102no—
394C2×D4—102no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—52no—
400C22×C4—54no—
401C22×C4—54no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—108no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—101no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4—101no—
415C22×C4—402no—
416C22×C4—402no—
417C22×C4C2×C4×D512yes—
418C22×C4#70912yes—
419C22×C4#70912yes—
420C22×C4D4×D512yes—
421C22×C4D4×D511yescharacteristic
422C22×C4#71112yes—
423C22×C4#71312yes—
424C22×C4C23×D511yescharacteristic
425M16—204no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—44no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—52no—
442C4⋊C4—52no—
443C4⋊C4—108no—
444C4⋊C4—101no—
445C4⋊C4—101no—
446C4⋊C4—101no—
447C4⋊C4—101no—
448C4⋊C4—101no—
449C4⋊C4—101no—
450C4⋊C4C2×C4×D514yes—
451C22⋊C4—108no—
452C22⋊C4—104no—
453C22⋊C4—202no—
454C22⋊C4—402no—
455C22⋊C4—402no—
456C42—108no—
457C42—104no—
458C42—202no—
459C2×C5—24no—
460C2×C5C2×#38111yescharacteristic
461C2×C5#289711yescharacteristic
462C2×C5#289811yescharacteristic
463C2×C5#309611yescharacteristic
464C2×C5#309711yescharacteristic
465C2×C5#309811yescharacteristic
466C2×C5#309911yescharacteristic
467D5—44no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—54no—
473C23—102no—
474C23—102no—
475C23—102no—
476C23—102no—
477C23—102no—
478C23—102no—
479C23D5×C22⋊C412yes—
480C23#402912yes—
481C23#404012yes—
482C23C2×D4×D511yescharacteristic
483Q8—108no—
484D4—1016no—
485D4—108no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—54no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—102no—
496C2×C4—102no—
497C2×C4—104no—
498C2×C4—104no—
499C2×C4—104no—
500C2×C4—104no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—201no—
504C2×C4—201no—
505C2×C4—201no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—201no—
509C2×C4—201no—
510C2×C4—201no—
511C2×C4—201no—
512C2×C4—201no—
513C2×C4—408no—
514C2×C4#401814yes—
515C2×C4D5×C22⋊C412yes—
516C2×C4#402914yes—
517C2×C4#402912yes—
518C2×C4#403814yes—
519C2×C4#404014yes—
520C2×C4#404012yes—
521C2×C4C22×C4×D511yescharacteristic
522C2×C4C2×#70911yescharacteristic
523C2×C4C2×#70911yescharacteristic
524C2×C4C2×D4×D511yescharacteristic
525C2×C4C2×#71111yescharacteristic
526C2×C4C2×#71311yescharacteristic
527C8—41no—
528C8—41no—
529C8—202no—
530C5#2070311yescharacteristic, complemented, Sylow
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—104no—
538C22—104no—
539C22—104no—
540C22—202no—
541C22—202no—
542C22—202no—
543C22—202no—
544C22#6495314yes—
545C22C2×D5×C22⋊C411yescharacteristic
546C22C2×#402911yescharacteristic
547C22C2×#404011yescharacteristic
548C22#6609111yescharacteristic
549C22#6609611yescharacteristic
550C22#6609611yescharacteristic
551C22#6610011yescharacteristic
552C4—44no—
553C4—82no—
554C4—104no—
555C4—202no—
556C4—202no—
557C4—401no—
558C4—401no—
559C4#6495314yes—
560C4C2×#401811yescharacteristic
561C4C2×#402911yescharacteristic
562C4C2×#403811yescharacteristic
563C4C2×#404011yescharacteristic
564C2—24no—
565C2—204no—
566C2C2×#6495311yescharacteristic
567C2#1061047311yescharacteristic
568C2#1061048711yescharacteristic
569C2#1061052111yescharacteristic
570C2#1061053511yescharacteristic
571C2#1061084311yescharacteristic
572C2#1061087211yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#340C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#340D5
hypocenterC5#20703
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. C2×C5×#340C2semidirect
5. #10602762C2semidirect
6. #10602351C2semidirect
7. #10602351C2semidirect
8. #10599666C2semidirect
9. #10599665C2semidirect
10. #10599664C2semidirect
11. #10599606C2semidirect
21. #65838C4semidirect
96. #4038C2×C4semidirect
98. D5×C4⋊C4C2×C4semidirect
114. C2×#340D5semidirect
530. C5#20703semidirect

Maximal subgroups

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

subgroupindex
C2×#658382
C2×#653992
#106139172
C2×C5×#3402
#106027622
#106023512
#105996662
#105996652
#105996642
#105996062
#207035