Finite Group Databasebeta

← #12115746

Subgroups — #12115746

Order 1280.

1154 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65913C211yesmaximal, characteristic
2C2×#65127C211yesmaximal, characteristic
3#10606055C211yesmaximal, characteristic
4C5×#1803C211yesmaximal, characteristic, complemented
5C4×#3967C211yesmaximal, characteristic, complemented
6#10601252C211yesmaximal, characteristic, complemented
7#10601064C212yesmaximal, complemented
8#10601062C212yesmaximal, complemented
9#10600375C211yesmaximal, characteristic, complemented
10#10599664C212yesmaximal, complemented
11#10599660C212yesmaximal, complemented
12C2×C4×C5×Q8C2211yescharacteristic
13C2×#4092C2212yes—
14C22×#676C2212yes—
15#65913—24no—
16#65913C418yescomplemented
17C2×#4042—22no—
18C2×#4038C2212yes—
19C2×D5×C4⋊C4C2212yes—
20C2×C4×D20—21no—
21C2×C4×D20C2211yescharacteristic
22C2×#4016—21no—
23C2×#4015C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×#103C2212yes—
26C5×#384C2211yescharacteristic
27C2×#3967C2211yescharacteristic
28C2×#3967C2211yescharacteristic
29C2×#3952C2212yes—
30#65279—22no—
31#65277—21no—
32#65276C2212yes—
33C4⋊C4×Dic5—21no—
34C4×C4⋊Dic5C2211yescharacteristic
35C2×C4×#168C2211yescharacteristic
36#65127—24no—
37C5×#340C2212yes—
38#64771C2212yes—
39#64771C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#64749C2211yescharacteristic
43#64749C2211yescharacteristic
44#22502—51nomaximal, Sylow
45C22×C5×Q8C2311yescharacteristic
46C22×D20D411yescharacteristic
47C22×C4×D5—22no—
48C22×C4×D5C2312yes—
49C22×Dic10—21no—
50C22×C5×C8C2312yes—
51C4×C5×Q8—22no—
52C4×C5×Q8C2×C414yes—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4—21no—
56C2×C5×C4⋊C4C2312yes—
57C2×C5×C4⋊C4C2311yescharacteristic
58C2×C42×C5—21no—
59C2×C42×C5C2311yescharacteristic
60#4092—24no—
61#4092C2×C418yes—
62C2×#685—22no—
63C2×#685—22no—
64C2×#685—22no—
65C2×#685D412yes—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5D411yescharacteristic
68C2×C4⋊Dic5C2311yescharacteristic
69C2×C4⋊Dic5C2312yes—
70C2×C4⋊Dic5C2311yescharacteristic
71C2×#683—22no—
72C2×#683—22no—
73C2×C4×Dic5—21no—
74C2×C4×Dic5—22no—
75C22×#168C2311yescharacteristic
76C22×#168C2311yescharacteristic
77C2×#676—24no—
78C2×#676—24no—
79C2×#676—42no—
80#4042—216no—
81#4038—24no—
82#4038—42no—
83D5×C4⋊C4—24no—
84D5×C4⋊C4C2×C418yes—
85C4×D20—28no—
86C4×D20C2×C414yes—
87C4×D20D414yes—
88#4016—28no—
89#4015—22no—
90#4015C2×C414yes—
91C5×C4⋊C8—22no—
92C5×#103—42no—
93#3967—22no—
94#3967—22no—
95#3963—21no—
96#3963—22no—
97#3952—24no—
98C4×#168—22no—
99C2×#554—51no—
100C2×#436—51no—
101#2354—51no—
102#1990—51no—
103#1987—51no—
104#1894—52no—
105#1873—52no—
106#1823—52no—
107#1803D511yescharacteristic, complemented
108C4×#103—51no—
109#1736—52no—
110C23×D5C2×D412yes—
111C2×C5×Q8—21no—
112C2×C5×Q8—21no—
113C2×C5×Q8—21no—
114C2×C5×Q8—21no—
115C2×C5×Q8—21no—
116C2×C5×Q8C22×C412yes—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5—22no—
120C22×C4×C5C2×D411yescharacteristic
121C22×C4×C5C2×D411yescharacteristic
122C22×C4×C5C2411yescharacteristic
123C22×Dic5—22no—
124C22×Dic5—21no—
125C22×Dic5—21no—
126C22×Dic5C2×D412yes—
127C2×D20—22no—
128C2×D20—22no—
129C2×D20C22⋊C414yes—
130C2×C4×D5—22no—
131C2×C4×D5—22no—
132C2×C4×D5—22no—
133C2×C4×D5—22no—
134C2×C4×D5—28no—
135C2×C4×D5—82no—
136C2×C4×D5—82no—
137C2×C4×D5C22×C414yes—
138C2×C4×D5C2×D414yes—
139C2×Dic10—24no—
140C2×Dic10—81no—
141C2×Dic10—81no—
142C2×C5×C8—22no—
143C2×C5×C8—22no—
144C2×C5×C8—22no—
145C5×C4⋊C4—24no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—22no—
148C5×C4⋊C4—22no—
149C5×C4⋊C4—42no—
150C5×C4⋊C4—81no—
151C5×C4⋊C4C22×C412yes—
152C42×C5—24no—
153C42×C5C22×C412yes—
154C42×C5C2×D412yes—
155#685—28no—
156#685—44no—
157#685—82no—
158#685C22⋊C418yes—
159C4⋊Dic5—24no—
160C4⋊Dic5—22no—
161C4⋊Dic5—21no—
162C4⋊Dic5—21no—
163C4⋊Dic5—21no—
164C4⋊Dic5C22⋊C414yes—
165C4⋊Dic5C22×C414yes—
166C4⋊Dic5#4112yes—
167#683—28no—
168#683—28no—
169C4×Dic5—44no—
170C4×Dic5—42no—
171C2×#168—21no—
172C2×#168—21no—
173C2×#168—21no—
174C2×#168—21no—
175C2×#168—21no—
176C2×#168#4112yes—
177#676—48no—
178C22×M16—52no—
179#554—58no—
180#554—104no—
181C2×C4⋊Q8—51no—
182C2×#125—101no—
183C2×C22⋊Q8—52no—
184C2×C22⋊Q8—102no—
185C2×C4×Q8C2×D511yescharacteristic
186C2×C4×D4—51no—
187C2×C4×D4—101no—
188C2×#117—52no—
189C22×C4⋊C4—52no—
190#436—104no—
191C2×C4⋊C8C2×D511yescharacteristic
192C2×#103—51no—
193C2×#103—51no—
194C2×#103C2×D512yes—
195C2×C22⋊C8—52no—
196C2×C4×C8—51no—
197#390—101no—
198#388—102no—
199#384C2×D511yescharacteristic
200#380—52no—
201C4×C4⋊C4—51no—
202C4×C4⋊C4—101no—
203#340—52no—
204#340—52no—
205#340—51no—
206#340—52no—
207#340—52no—
208#340—51no—
209#340C2×D512yes—
210C23×C5C22×D411yescharacteristic
211C22×D5—22no—
212C22×D5—22no—
213C22×D5—22no—
214C22×D5—22no—
215C22×D5—22no—
216C22×D5—22no—
217C22×D5C2×C22⋊C414yes—
218C5×Q8—22no—
219C5×Q8—22no—
220C5×Q8—41no—
221C5×Q8—41no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—22no—
225C2×C4×C5—22no—
226C2×C4×C5—41no—
227C2×C4×C5—41no—
228C2×C4×C5—41no—
229C2×C4×C5—41no—
230C2×C4×C5—42no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5C2×C22⋊C412yes—
234C2×C4×C5C2×C22⋊C412yes—
235C2×C4×C5#13712yes—
236C2×C4×C5#13712yes—
237C2×C4×C5C23×C411yescharacteristic
238C2×C4×C5C22×D411yescharacteristic
239C2×C4×C5C2×#4111yescharacteristic
240C2×C4×C5C2×#4111yescharacteristic
241C2×C4×C5H(5,𝔽2)11yescharacteristic
242C2×C4×C52−1+411yescharacteristic
243C2×Dic5—22no—
244C2×Dic5—22no—
245C2×Dic5—24no—
246C2×Dic5—22no—
247C2×Dic5—41no—
248C2×Dic5—42no—
249C2×Dic5—42no—
250C2×Dic5—41no—
251C2×Dic5—41no—
252C2×Dic5—41no—
253C2×Dic5—41no—
254C2×Dic5C2×C22⋊C414yes—
255D20—28no—
256C4×D5—24no—
257C4×D5—24no—
258C4×D5—24no—
259C4×D5—24no—
260C4×D5—88no—
261Dic10—84no—
262C5×C8—42no—
263#168—22no—
264#168—41no—
265C22×Q8—101no—
266C22×Q8C22×D511yescharacteristic
267C22×D4—51no—
268C23×C4—52no—
269C23×C4—102no—
270C2×M16—104no—
271C2×M16—104no—
272C2×M16—202no—
273C22×C8—51no—
274C22×C8—51no—
275C22×C8C22×D512yes—
276C4⋊Q8—54no—
277C4⋊Q8—102no—
278#125—108no—
279C22⋊Q8—58no—
280C22⋊Q8—1016no—
281C22⋊Q8—104no—
282C4×Q8—22no—
283C4×Q8C4×D514yes—
284C4×D4—54no—
285C4×D4—54no—
286C4×D4—108no—
287#117—104no—
288#117—202no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—51no—
293C2×C4⋊C4—52no—
294C2×C4⋊C4—51no—
295C2×C4⋊C4—58no—
296C2×C4⋊C4—51no—
297C2×C4⋊C4—104no—
298C2×C4⋊C4—102no—
299C2×C4⋊C4—102no—
300C2×C4⋊C4—101no—
301C2×C4⋊C4C22×D512yes—
302C2×C4⋊C4C22×D511yescharacteristic
303C2×C22⋊C4—52no—
304C2×C22⋊C4—102no—
305C2×C22⋊C4—102no—
306C2×C22⋊C4—102no—
307C2×C42—21no—
308C2×C42—102no—
309C2×C42—101no—
310C2×C42C22×D511yescharacteristic
311C4⋊C8—22no—
312#103—42no—
313#103—102no—
314#103—102no—
315C22⋊C8—104no—
316C4×C8—102no—
317#95—102no—
318#95—101no—
319C22×C5C22×C22⋊C411yescharacteristic
320C22×C5#54611yescharacteristic
321C22×C5#54711yescharacteristic
322C22×C5C2×#13711yescharacteristic
323C22×C5C2×#13711yescharacteristic
324C22×C5#57611yescharacteristic
325C22×C5#57711yescharacteristic
326C2×D5—24no—
327C2×D5—24no—
328C2×D5—24no—
329C2×D5—42no—
330C2×D5—42no—
331C2×D5—42no—
332C2×D5—42no—
333C4×C5—22no—
334C4×C5—42no—
335C4×C5—42no—
336C4×C5—82no—
337C4×C5#41914yes—
338C4×C5C4×#4111yescharacteristic
339C4×C5#52011yescharacteristic
340C4×C5#54611yescharacteristic
341C4×C5#54711yescharacteristic
342Dic5—24no—
343Dic5—42no—
344Dic5—44no—
345Dic5—81no—
346C24—52no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—104no—
353C2×Q8—401no—
354C2×Q8—401no—
355C2×Q8C2×C4×D512yes—
356C2×D4—54no—
357C2×D4—102no—
358C2×D4—102no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—52no—
363C22×C4—54no—
364C22×C4—54no—
365C22×C4—108no—
366C22×C4—102no—
367C22×C4—102no—
368C22×C4—102no—
369C22×C4—102no—
370C22×C4—101no—
371C22×C4—102no—
372C22×C4—101no—
373C22×C4—402no—
374C22×C4—402no—
375C22×C4D4×D511yescharacteristic
376C22×C4D4×D511yescharacteristic
377C22×C4C23×D511yescharacteristic
378M16—208no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—52no—
383C2×C8—101no—
384C2×C8—101no—
385C2×C8—101no—
386C2×C8—101no—
387C2×C8—101no—
388C4⋊C4—21no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—54no—
394C4⋊C4—54no—
395C4⋊C4—52no—
396C4⋊C4—81no—
397C4⋊C4—108no—
398C4⋊C4—108no—
399C4⋊C4—102no—
400C4⋊C4—101no—
401C4⋊C4—104no—
402C4⋊C4—101no—
403C4⋊C4—101no—
404C4⋊C4C2×C4×D512yes—
405C22⋊C4—58no—
406C22⋊C4—108no—
407C22⋊C4—204no—
408C22⋊C4—402no—
409C42—24no—
410C42—204no—
411C42—202no—
412C42C2×C4×D512yes—
413C42D4×D512yes—
414C2×C5#233911yescharacteristic
415C2×C5C2×#41911yescharacteristic
416C2×C5#289711yescharacteristic
417C2×C5#333111yescharacteristic
418C2×C5#333311yescharacteristic
419C2×C5#335611yescharacteristic
420C2×C5#335711yescharacteristic
421D5—44no—
422C23—54no—
423C23—102no—
424C23—102no—
425C23—102no—
426C23—102no—
427C23—102no—
428C23—102no—
429C23C2×D4×D511yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—404no—
435D4—108no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—54no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—104no—
451C2×C4—104no—
452C2×C4—104no—
453C2×C4—104no—
454C2×C4—104no—
455C2×C4—102no—
456C2×C4—201no—
457C2×C4—202no—
458C2×C4—202no—
459C2×C4—201no—
460C2×C4—201no—
461C2×C4—201no—
462C2×C4—201no—
463C2×C4—408no—
464C2×C4D5×C22⋊C412yes—
465C2×C4D5×C22⋊C412yes—
466C2×C4#406112yes—
467C2×C4#406412yes—
468C2×C4C22×C4×D511yescharacteristic
469C2×C4C2×D4×D511yescharacteristic
470C2×C4#414411yescharacteristic
471C2×C4#414711yescharacteristic
472C2×C4D5×#4111yescharacteristic
473C2×C4D5×#4111yescharacteristic
474C8—42no—
475C8—102no—
476C8—201no—
477C5#2250211yescharacteristic, complemented, Sylow
478C22—104no—
479C22—104no—
480C22—104no—
481C22—202no—
482C22—202no—
483C22—202no—
484C22—202no—
485C22C2×D5×C22⋊C411yescharacteristic
486C22#6594611yescharacteristic
487C22#6596511yescharacteristic
488C22C2×#406111yescharacteristic
489C22C2×#406411yescharacteristic
490C22#6610811yescharacteristic
491C22#6611211yescharacteristic
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—104no—
497C4—202no—
498C4—204no—
499C4—401no—
500C4#6509114yes—
501C4#6583811yescharacteristic
502C4#6585611yescharacteristic
503C4#6594611yescharacteristic
504C4#6596511yescharacteristic
505C2—204no—
506C2#1060573911yescharacteristic
507C2C2×#6509111yescharacteristic
508C2#1061061311yescharacteristic
509C2#1061213111yescharacteristic
510C2#1061222711yescharacteristic
511C2#1061276411yescharacteristic
512C2#1061286111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1803C2
socleC23×C5C22×D4
prime radicalC22×D20D4
cosocleC22×C4×C5C24
hypercenter#1803D5
hypocenterC5#22502
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. C5×#1803C2semidirect
5. C4×#3967C2semidirect
6. #10601252C2semidirect
7. #10601064C2semidirect
8. #10601062C2semidirect
9. #10600375C2semidirect
10. #10599664C2semidirect
11. #10599660C2semidirect
16. #65913C4semidirect
107. #1803D5semidirect
477. C5#22502semidirect

Maximal subgroups

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

subgroupindex
C2×#659132
C2×#651272
#106060552
C5×#18032
C4×#39672
#106012522
#106010642
#106010622
#106003752
#105996642
#105996602
#225025