Finite Group Databasebeta

← #12115543

Subgroups — #12115543

Order 1280.

1258 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66064C211yesmaximal, characteristic
2C2×#65123C211yesmaximal, characteristic
3C4⋊C4×D20C211yesmaximal, characteristic
4C5×#1987C211yesmaximal, characteristic, complemented
5#10602580C211yesmaximal, characteristic, complemented
6#10601278C211yesmaximal, characteristic, complemented
7C4×#3942C211yesmaximal, characteristic, complemented
8#10599675C212yesmaximal, complemented
9#10599671C212yesmaximal, complemented
10#10599665C212yesmaximal, complemented
11D5×#340C212yesmaximal, complemented
12C2×C5×C4⋊Q8C2211yescharacteristic
13C2×#4092—22no—
14C22×C8×D5C2212yes—
15#66064—24no—
16#66064C418yescomplemented
17C2×#4043C2212yes—
18C2×#4039—21no—
19C2×D5×C4⋊C4C2212yes—
20C2×D5×C4⋊C4C2212yes—
21C2×#4036—21no—
22C2×C4×D20—21no—
23C2×C4×D20C2211yescharacteristic
24C2×C4×Dic10C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C4×C5×C4⋊C4C2211yescharacteristic
27C2×#3952C2212yes—
28C2×#3948C2212yes—
29#65276C2212yes—
30C2×#3942C2211yescharacteristic
31C2×#3942C2211yescharacteristic
32#65224C2211yescharacteristic
33#65217—21no—
34C2×C4×#168C2211yescharacteristic
35#65123—24no—
36#64957—22no—
37C5×#340C2212yes—
38C5×#340C2212yes—
39#64771C2212yes—
40#64749C2211yescharacteristic
41#64749C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#22487—51nomaximal, Sylow
45C22×C5×Q8—21no—
46C22×D20D411yescharacteristic
47C22×C4×D5—22no—
48C22×C4×D5—22no—
49C22×C4×D5C2312yes—
50C22×Dic10C2311yescharacteristic
51C22×C5×C8C2312yes—
52C5×C4⋊Q8—22no—
53C5×C4⋊Q8C2×C414yescomplemented
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4C2311yescharacteristic
56C2×C5×C4⋊C4C2312yes—
57C2×C5×C4⋊C4C2311yescharacteristic
58C2×C42×C5—21no—
59C2×C42×C5C2311yescharacteristic
60#4092—216no—
61C2×#685—22no—
62C2×#685—22no—
63C2×#685D412yes—
64C2×C4⋊Dic5—22no—
65C2×C4⋊Dic5—21no—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5D411yescharacteristic
68C2×C4⋊Dic5C2312yes—
69C2×C4⋊Dic5C2311yescharacteristic
70C2×#683—21no—
71C2×#683—22no—
72C2×#683—22no—
73C2×C4×Dic5—21no—
74C22×#168C2311yescharacteristic
75C22×#168C2311yescharacteristic
76C2×C8×D5—24no—
77C2×C8×D5—24no—
78C2×C8×D5—42no—
79#4043—24no—
80#4043C2×C418yescomplemented
81#4039—28no—
82D5×C4⋊C4—24no—
83D5×C4⋊C4—24no—
84D5×C4⋊C4—42no—
85D5×C4⋊C4C2×C418yescomplemented
86#4036—28no—
87C4×D20—44no—
88C4×D20C2×C414yes—
89C4×D20D414yes—
90C4×Dic10—22no—
91C4×Dic10C2×C414yescomplemented
92C5×C4⋊C8—22no—
93C5×#95—21no—
94#3963—22no—
95#3952—24no—
96#3948—42no—
97#3942—22no—
98#3942—22no—
99C4×#168—22no—
100C2×#554—51no—
101C2×C8×D4—51no—
102D4×C4⋊C4—51no—
103#1990—51no—
104#1987D511yescharacteristic, complemented
105#1892—52no—
106#1872—52no—
107#1824—52no—
108#1803—51no—
109C4×#103—51no—
110C2×#340—52no—
111C23×D5C2×D412yes—
112C2×C5×Q8—24no—
113C2×C5×Q8—81no—
114C2×C5×Q8—81no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5C2×D411yescharacteristic
119C22×C4×C5C2×D411yescharacteristic
120C22×C4×C5C2411yescharacteristic
121C22×Dic5—21no—
122C22×Dic5—21no—
123C22×Dic5—22no—
124C22×Dic5C2×D412yes—
125C2×D20C22⋊C414yes—
126C2×D20SD1614yes—
127C2×D20Q1614yes—
128C2×C4×D5—22no—
129C2×C4×D5—22no—
130C2×C4×D5—22no—
131C2×C4×D5—22no—
132C2×C4×D5—28no—
133C2×C4×D5—44no—
134C2×C4×D5—44no—
135C2×C4×D5—82no—
136C2×C4×D5—82no—
137C2×C4×D5—82no—
138C2×C4×D5C22×C414yes—
139C2×C4×D5C2×D414yes—
140C2×Dic10—21no—
141C2×Dic10—21no—
142C2×Dic10—21no—
143C2×Dic10—21no—
144C2×Dic10—21no—
145C2×Dic10C22×C412yes—
146C2×C5×C8—22no—
147C2×C5×C8—22no—
148C2×C5×C8—22no—
149C5×C4⋊C4—24no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4C22×C414yes—
155C5×C4⋊C4#4112yes—
156C42×C5—42no—
157C42×C5C22×C412yes—
158C42×C5C2×D412yes—
159#685—28no—
160#685—44no—
161#685C22⋊C418yes—
162C4⋊Dic5—21no—
163C4⋊Dic5—28no—
164C4⋊Dic5—22no—
165C4⋊Dic5—22no—
166C4⋊Dic5—42no—
167C4⋊Dic5—81no—
168C4⋊Dic5C22⋊C414yes—
169C4⋊Dic5C22×C412yes—
170#683—24no—
171#683—28no—
172#683—44no—
173C4×Dic5—24no—
174C2×#168—21no—
175C2×#168—21no—
176C2×#168—21no—
177C2×#168—21no—
178C2×#168—21no—
179C2×#168#4112yes—
180C8×D5—48no—
181C23×C8—52no—
182#554—58no—
183#554—104no—
184C2×C4⋊Q8C2×D511yescharacteristic
185C2×#125—101no—
186C2×C22⋊Q8—52no—
187C2×C22⋊Q8—102no—
188C2×C4×Q8—51no—
189C2×C4×D4—51no—
190C2×C4×D4—101no—
191C2×C4×D4—101no—
192C22×C4⋊C4—52no—
193C22×C4⋊C4—52no—
194C8×D4—104no—
195C2×C4⋊C8C2×D511yescharacteristic
196C2×#103—51no—
197C2×#103—51no—
198C2×#103—52no—
199C2×C22⋊C8—52no—
200C2×C4×C8—51no—
201#389—101no—
202#385—102no—
203#384—51no—
204#380—52no—
205C4×C4⋊C4C2×D511yescharacteristic
206#340—52no—
207#340—51no—
208#340—52no—
209#340—52no—
210#340—51no—
211#340C2×D512yes—
212#340C2×D512yes—
213C23×C5C22×D411yescharacteristic
214C22×D5—22no—
215C22×D5—22no—
216C22×D5—22no—
217C22×D5—22no—
218C22×D5C2×C22⋊C414yes—
219C22×D5C2×SD1614yes—
220C22×D5C2×Q1614yes—
221C5×Q8—84no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—24no—
225C2×C4×C5—22no—
226C2×C4×C5—41no—
227C2×C4×C5—41no—
228C2×C4×C5—42no—
229C2×C4×C5—42no—
230C2×C4×C5—41no—
231C2×C4×C5—41no—
232C2×C4×C5—41no—
233C2×C4×C5C2×C22⋊C412yes—
234C2×C4×C5C2×C22⋊C412yes—
235C2×C4×C5C2×SD1612yes—
236C2×C4×C5C2×Q1612yes—
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—22no—
246C2×Dic5—22no—
247C2×Dic5—41no—
248C2×Dic5—41no—
249C2×Dic5—41no—
250C2×Dic5—41no—
251C2×Dic5—42no—
252C2×Dic5—42no—
253C2×Dic5—42no—
254C2×Dic5C2×C22⋊C414yes—
255D20#103116yescomplemented
256C4×D5—24no—
257C4×D5—24no—
258C4×D5—24no—
259C4×D5—24no—
260C4×D5—88no—
261C4×D5—88no—
262Dic10—22no—
263Dic10—22no—
264Dic10—41no—
265Dic10—41no—
266C5×C8—42no—
267#168—22no—
268#168—41no—
269C22×Q8—21no—
270C22×Q8—51no—
271C22×D4—51no—
272C23×C4—52no—
273C23×C4—102no—
274C23×C4—102no—
275C22×C8—51no—
276C22×C8—51no—
277C22×C8—104no—
278C22×C8—104no—
279C22×C8—202no—
280C22×C8C22×D512yes—
281C4⋊Q8—22no—
282C4⋊Q8C4×D514yescomplemented
283#125—108no—
284C22⋊Q8—58no—
285C22⋊Q8—1016no—
286C22⋊Q8—104no—
287C4×Q8—54no—
288C4×Q8—102no—
289C4×D4—54no—
290C4×D4—54no—
291C4×D4—108no—
292C4×D4—204no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—51no—
295C2×C4⋊C4—51no—
296C2×C4⋊C4—58no—
297C2×C4⋊C4—52no—
298C2×C4⋊C4—104no—
299C2×C4⋊C4—102no—
300C2×C4⋊C4—102no—
301C2×C4⋊C4—101no—
302C2×C4⋊C4—104no—
303C2×C4⋊C4—102no—
304C2×C4⋊C4—101no—
305C2×C4⋊C4—101no—
306C2×C4⋊C4—202no—
307C2×C4⋊C4C22×D511yescharacteristic
308C2×C4⋊C4C22×D512yes—
309C2×C4⋊C4C22×D511yescharacteristic
310C2×C22⋊C4—52no—
311C2×C22⋊C4—102no—
312C2×C22⋊C4—102no—
313C2×C42—21no—
314C2×C42—101no—
315C2×C42C22×D511yescharacteristic
316C4⋊C8—22no—
317#103—102no—
318#103—102no—
319#103—202no—
320C22⋊C8—104no—
321C4×C8—102no—
322#95—21no—
323#95—102no—
324C22×C5C22×C22⋊C411yescharacteristic
325C22×C5#54611yescharacteristic
326C22×C5#54711yescharacteristic
327C22×C5C22×SD1611yescharacteristic
328C22×C5C22×Q1611yescharacteristic
329C22×C5#57611yescharacteristic
330C22×C5#57711yescharacteristic
331C2×D5—24no—
332C2×D5—24no—
333C2×D5—24no—
334C2×D5—24no—
335C2×D5—24no—
336C2×D5—24no—
337C2×D5C2×#10318yes—
338C4×C5—22no—
339C4×C5—44no—
340C4×C5—42no—
341C4×C5—81no—
342C4×C5C2×#10314yes—
343C4×C5C4×#4111yescharacteristic
344C4×C5#52011yescharacteristic
345C4×C5#54611yescharacteristic
346C4×C5#54711yescharacteristic
347Dic5—24no—
348Dic5—42no—
349Dic5—42no—
350Dic5—82no—
351C24—52no—
352C2×Q8—24no—
353C2×Q8—52no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—101no—
357C2×Q8—101no—
358C2×Q8—101no—
359C2×Q8—101no—
360C2×Q8—101no—
361C2×D4—54no—
362C2×D4—54no—
363C2×D4—54no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—52no—
368C22×C4—54no—
369C22×C4—54no—
370C22×C4—108no—
371C22×C4—102no—
372C22×C4—102no—
373C22×C4—102no—
374C22×C4—102no—
375C22×C4—101no—
376C22×C4—101no—
377C22×C4—102no—
378C22×C4—204no—
379C22×C4—204no—
380C22×C4—402no—
381C22×C4—402no—
382C22×C4—402no—
383C22×C4D4×D511yescharacteristic
384C22×C4D4×D511yescharacteristic
385C22×C4C23×D511yescharacteristic
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—52no—
390C2×C8—101no—
391C2×C8—101no—
392C2×C8—101no—
393C2×C8—101no—
394C2×C8—101no—
395C2×C8—208no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—54no—
402C4⋊C4—52no—
403C4⋊C4—108no—
404C4⋊C4—108no—
405C4⋊C4—101no—
406C4⋊C4—104no—
407C4⋊C4—102no—
408C4⋊C4—102no—
409C4⋊C4—204no—
410C4⋊C4—202no—
411C4⋊C4—401no—
412C4⋊C4C2×C4×D514yes—
413C4⋊C4#71312yes—
414C22⋊C4—58no—
415C22⋊C4—108no—
416C22⋊C4—204no—
417C42—42no—
418C42—104no—
419C42C2×C4×D512yes—
420C42D4×D512yes—
421C2×C5#233911yescharacteristic
422C2×C5C22×#10311yescharacteristic
423C2×C5#289711yescharacteristic
424C2×C5#329511yescharacteristic
425C2×C5#329911yescharacteristic
426C2×C5#333511yescharacteristic
427C2×C5#333811yescharacteristic
428D5—28no—
429C23—54no—
430C23—54no—
431C23—54no—
432C23—102no—
433C23—102no—
434C23—102no—
435C23—102no—
436C23C2×D4×D511yescharacteristic
437Q8—84no—
438Q8—102no—
439Q8—102no—
440Q8—201no—
441Q8—201no—
442D4—516no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—54no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—104no—
458C2×C4—104no—
459C2×C4—104no—
460C2×C4—104no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—201no—
464C2×C4—201no—
465C2×C4—201no—
466C2×C4—201no—
467C2×C4—202no—
468C2×C4—202no—
469C2×C4—202no—
470C2×C4—408no—
471C2×C4—408no—
472C2×C4D5×C22⋊C412yes—
473C2×C4D5×C22⋊C412yes—
474C2×C4D5×SD1612yes—
475C2×C4D5×Q1612yes—
476C2×C4C22×C4×D511yescharacteristic
477C2×C4C2×D4×D511yescharacteristic
478C2×C4C2×#71311yescharacteristic
479C2×C4#414711yescharacteristic
480C2×C4D5×#4111yescharacteristic
481C2×C4#414911yescharacteristic
482C8—42no—
483C8—102no—
484C8—201no—
485C5#2248711yescharacteristic, complemented, Sylow
486C22—58no—
487C22—104no—
488C22—104no—
489C22—104no—
490C22—104no—
491C22—104no—
492C22—104no—
493C22C2×D5×C22⋊C411yescharacteristic
494C22#6596411yescharacteristic
495C22#6596511yescharacteristic
496C22C2×D5×SD1611yescharacteristic
497C22C2×D5×Q1611yescharacteristic
498C22#6610811yescharacteristic
499C22#6610911yescharacteristic
500C4—22no—
501C4—42no—
502C4—44no—
503C4—81no—
504C4—104no—
505C4—202no—
506C4—202no—
507C4—402no—
508C4D5×#10314yes—
509C4C4×#71311yescharacteristic
510C4#6590811yescharacteristic
511C4#6601011yescharacteristic
512C4#6606011yescharacteristic
513C2—108no—
514C2#1060834411yescharacteristic
515C2C2×D5×#10311yescharacteristic
516C2#1061065611yescharacteristic
517C2#1061280611yescharacteristic
518C2#1061281211yescharacteristic
519C2#1061285011yescharacteristic
520C2#1061285611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1987C2
socleC23×C5C22×D4
prime radicalC22×D20D4
cosocleC22×C4×C5C24
hypercenter#1987D5
hypocenterC5#22487
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
4. C5×#1987C2semidirect
5. #10602580C2semidirect
6. #10601278C2semidirect
7. C4×#3942C2semidirect
8. #10599675C2semidirect
9. #10599671C2semidirect
10. #10599665C2semidirect
11. D5×#340C2semidirect
16. #66064C4semidirect
53. C5×C4⋊Q8C2×C4semidirect
80. #4043C2×C4semidirect
85. D5×C4⋊C4C2×C4semidirect
91. C4×Dic10C2×C4semidirect
104. #1987D5semidirect
255. D20#103semidirect
282. C4⋊Q8C4×D5semidirect
485. C5#22487semidirect

Maximal subgroups

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

subgroupindex
C2×#660642
C2×#651232
C4⋊C4×D202
C5×#19872
#106025802
#106012782
C4×#39422
#105996752
#105996712
#105996652
D5×#3402
#224875