Finite Group Databasebeta

← #12110516

Subgroups — #12110516

Order 1280.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#391C211yesmaximal, characteristic, complemented
2C2×#65516C211yesmaximal, characteristic
3C4×#4076C211yesmaximal, characteristic
4#10602925C211yesmaximal, characteristic, complemented
5#10602353C211yesmaximal, characteristic
6#10602348C211yesmaximal, characteristic, complemented
7#10602322C211yesmaximal, characteristic, complemented
8#10600480C212yesmaximal, complemented
9#10600472C212yesmaximal, complemented
10#10600468C212yesmaximal, complemented
11#10600460C212yesmaximal, complemented
12C23×C5×Q8C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×#688C2212yes—
16C2×#4076C2211yescharacteristic
17C2×C4×C22⋊D5—21no—
18C2×C4×D20C2211yescharacteristic
19C5×#391—21no—
20C5×#391—21no—
21C5×#391C2212yes—
22C5×#391C2212yes—
23C2×C5×#95—21no—
24C2×C5×#95—21no—
25#65516C418yescomplemented
26C2×#3967C2212yes—
27C2×#3967C2212yes—
28C2×#3967C2212yes—
29#65502—21no—
30#65501C2211yescharacteristic
31C2×C22⋊#168C2211yescharacteristic
32C2×C22⋊#168C2211yescharacteristic
33C2×#3941C2212yes—
34C2×#3939C2212yes—
35#65229—21no—
36#65228C2211yescharacteristic
37C4×#685—21no—
38C4×C4⋊Dic5C2211yescharacteristic
39#65217C2211yescharacteristic
40C2×#3935C2212yes—
41#64749C2212yes—
42#64749C2212yes—
43#64744C2212yes—
44#22047—51nomaximal, Sylow
45C22×C5×Q8—21no—
46C22×C5×Q8—22no—
47C22×C5×Q8—42no—
48C22×C5×Q8—42no—
49C22×C5×Q8C2×C414yescomplemented
50C22×C5×Q8C2312yes—
51C23×C4×C5—21no—
52C23×C4×C5—21no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C22×C22⋊D5—21no—
57C22×D20C2311yescharacteristic
58C22×C4×D5—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4—42no—
61C2×C5×C4⋊C4C2312yes—
62C2×C42×C5—42no—
63C2×C42×C5D411yescharacteristic
64C2×C42×C5D411yescharacteristic
65C2×C42×C5C2311yescharacteristic
66C2×C42×C5C2311yescharacteristic
67C2×C22⋊Dic5—21no—
68C2×#688—42no—
69C2×#688—42no—
70C2×#688C2×C418yescomplemented
71#4076—22no—
72#4076C2×C414yes—
73C4×C22⋊D5—44no—
74C2×#685—21no—
75C2×#685—21no—
76C2×#685—21no—
77C2×#685—21no—
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×#683—21no—
82C2×C4×Dic5—21no—
83C22×#168C2312yes—
84C22×#168C2312yes—
85C4×D20—42no—
86C5×#95—22no—
87C5×#95—22no—
88C5×#95—22no—
89C5×#95—22no—
90#3967—42no—
91#3967—42no—
92#3967C2×C418yescomplemented
93#3963—21no—
94#3963—21no—
95#3963—21no—
96C22⋊#168—22no—
97C22⋊#168C2×C414yescomplemented
98#3941—42no—
99#3939—42no—
100#3935—42no—
101C2×#448—51no—
102C4×#121—51no—
103C2×#391D511yescharacteristic, complemented
104#1973—52no—
105#1935—52no—
106#1928—52no—
107#1877—52no—
108#1874—51no—
109#1852—51no—
110#1787—51no—
111#1749—51no—
112C24×C5C2×D411yescharacteristic
113C23×D5—21no—
114C2×C5×Q8—24no—
115C2×C5×Q8—22no—
116C2×C5×Q8—22no—
117C2×C5×Q8—22no—
118C2×C5×Q8—22no—
119C2×C5×Q8—22no—
120C2×C5×Q8—48no—
121C2×C5×Q8—42no—
122C2×C5×Q8—42no—
123C2×C5×Q8—48no—
124C2×C5×Q8C22×C414yes—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—24no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—41no—
134C22×C4×C5—41no—
135C22×C4×C5—41no—
136C22×C4×C5—42no—
137C22×C4×C5—42no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—42no—
141C22×C4×C5C22×C412yes—
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5#4112yes—
148C22×C4×C5C2411yescharacteristic
149C2×C22⋊D5—24no—
150C2×C22⋊D5—42no—
151C2×C22⋊D5—42no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C2×D20—22no—
156C2×D20—22no—
157C2×D20—21no—
158C2×D20C22×C412yes—
159C2×C4×D5—42no—
160C2×C4×D5—42no—
161C2×C4×D5—42no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—42no—
165C42×C5—22no—
166C42×C5—22no—
167C42×C5—41no—
168C42×C5—41no—
169C22⋊Dic5—42no—
170#688—48no—
171#685—24no—
172#685—42no—
173#685—42no—
174#685—42no—
175C4⋊Dic5—21no—
176C4⋊Dic5—22no—
177C4⋊Dic5—22no—
178C4⋊Dic5C22×C412yes—
179#683—42no—
180C4×Dic5—42no—
181C2×#168—22no—
182C2×#168—22no—
183C2×#168—22no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168C22×C414yes—
187C23×Q8C2×D511yescharacteristic
188C22×SD16—52no—
189C2×#121—51no—
190C2×C4×D4—51no—
191C2×C4×D4—101no—
192C22×C4⋊C4C2×D511yescharacteristic
193C22×C42C2×D511yescharacteristic
194#448—58no—
195C2×#107—52no—
196C2×#103—52no—
197C2×#103—52no—
198C2×#103—52no—
199C2×#102—52no—
200C2×C22⋊C8—51no—
201C2×C22⋊C8—51no—
202C2×#97—52no—
203#391—21no—
204#391—21no—
205#391C2×D512yes—
206#391C2×D512yes—
207#390—51no—
208#389—51no—
209#388—101no—
210#386—101no—
211#380—51no—
212C4×C4⋊C4—51no—
213C4×C22⋊C4—101no—
214C2×#95—21no—
215C2×#95—21no—
216#340—52no—
217#340—52no—
218#336—52no—
219C23×C5—21no—
220C23×C5—21no—
221C23×C5—21no—
222C23×C5—21no—
223C23×C5C4×D412yes—
224C23×C5AGL(1,ℤ/8)12yes—
225C23×C5#13712yes—
226C23×C5C22×D411yescharacteristic
227C22×D5—22no—
228C22×D5—41no—
229C22×D5—41no—
230C22×D5—41no—
231C22×D5—41no—
232C22×D5—41no—
233C22×D5—41no—
234C5×Q8—24no—
235C5×Q8—24no—
236C5×Q8—48no—
237C5×Q8—42no—
238C5×Q8—42no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—24no—
244C2×C4×C5—24no—
245C2×C4×C5—24no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—21no—
255C2×C4×C5—44no—
256C2×C4×C5—44no—
257C2×C4×C5—48no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—84no—
264C2×C4×C5C4×D412yes—
265C2×C4×C5#13712yes—
266C2×C4×C5C23×C411yescharacteristic
267C2×C4×C5C2×#4111yescharacteristic
268C2×C4×C5H(5,𝔽2)12yes—
269C2×C4×C5H(5,𝔽2)12yes—
270C22⋊D5—48no—
271C2×Dic5—22no—
272C2×Dic5—41no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275C2×Dic5—41no—
276C2×Dic5—41no—
277C2×Dic5—41no—
278C2×Dic5—41no—
279C2×Dic5—41no—
280D20—24no—
281D20—42no—
282C4×D5—84no—
283#168—24no—
284#168—42no—
285C22×Q8—22no—
286C22×Q8—21no—
287C22×Q8—42no—
288C22×Q8—42no—
289C22×Q8C4×D514yescomplemented
290C22×Q8C22×D512yes—
291C22×D4—51no—
292C22×D4—101no—
293C23×C4—21no—
294C23×C4—21no—
295C23×C4—101no—
296C23×C4C22⋊D511yescharacteristic
297C23×C4C22⋊D511yescharacteristic
298C23×C4C22×D511yescharacteristic
299C2×SD16—58no—
300C2×SD16—202no—
301C2×SD16—202no—
302C22×C8—52no—
303C22×C8—52no—
304#121—54no—
305#121—102no—
306C4×D4—202no—
307C4×D4—204no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—51no—
311C2×C4⋊C4—51no—
312C2×C4⋊C4—51no—
313C2×C4⋊C4—101no—
314C2×C4⋊C4C22×D512yes—
315C2×C22⋊C4—101no—
316C2×C22⋊C4—101no—
317C2×C22⋊C4—101no—
318C2×C22⋊C4—101no—
319C2×C22⋊C4—101no—
320C2×C42—42no—
321C2×C42—101no—
322C2×C42C22⋊D511yescharacteristic
323C2×C42C22⋊D511yescharacteristic
324C2×C42C22×D511yescharacteristic
325C2×C42C22×D511yescharacteristic
326#107—202no—
327#103—58no—
328#103—202no—
329#103—202no—
330#102—202no—
331C22⋊C8—54no—
332C22⋊C8—102no—
333#97—202no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—101no—
339#95—101no—
340#95—101no—
341C22×C5—22no—
342C22×C5—22no—
343C22×C5—22no—
344C22×C5—22no—
345C22×C5—22no—
346C22×C5—22no—
347C22×C5#44014yes—
348C22×C5C2×C4×D411yescharacteristic
349C22×C5#53411yescharacteristic
350C22×C5#53511yescharacteristic
351C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C5C2×#13711yescharacteristic
353C22×C5C2×#13711yescharacteristic
354C22×C5#57511yescharacteristic
355C2×D5—42no—
356C2×D5—42no—
357C2×D5—42no—
358C2×D5—81no—
359C2×D5—81no—
360C2×D5—81no—
361C2×D5—81no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—44no—
365C4×C5—41no—
366C4×C5—41no—
367C4×C5—41no—
368C4×C5—82no—
369C4×C5#51812yes—
370C4×C5#53812yes—
371Dic5—42no—
372Dic5—81no—
373Dic5—81no—
374C24—101no—
375C24C2×C22⋊D511yescharacteristic
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8C2×C4×D514yes—
387C2×D4—52no—
388C2×D4—104no—
389C2×D4—101no—
390C2×D4—102no—
391C2×D4—102no—
392C2×D4—202no—
393C2×D4—202no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—101no—
411C22×C4—101no—
412C22×C4—101no—
413C22×C4—202no—
414C22×C4—202no—
415C22×C4—202no—
416C22×C4C2×C4×D512yes—
417C22×C4#70912yes—
418C22×C4C2×C22⋊D511yescharacteristic
419C22×C4C2×C22⋊D511yescharacteristic
420C22×C4C2×C22⋊D511yescharacteristic
421C22×C4C2×C22⋊D511yescharacteristic
422C22×C4C2×C22⋊D511yescharacteristic
423C22×C4C23×D511yescharacteristic
424SD16—208no—
425C2×C8—54no—
426C2×C8—102no—
427C2×C8—102no—
428C2×C8—102no—
429C2×C8—102no—
430C2×C8—102no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—52no—
435C4⋊C4—101no—
436C4⋊C4—102no—
437C4⋊C4—102no—
438C4⋊C4—202no—
439C22⋊C4—104no—
440C22⋊C4—202no—
441C22⋊C4—202no—
442C22⋊C4—202no—
443C22⋊C4—202no—
444C42—22no—
445C42—22no—
446C42—41no—
447C42—41no—
448C42—202no—
449C2×C5—24no—
450C2×C5#232111yescharacteristic
451C2×C5C2×#44011yescharacteristic
452C2×C5C4×#13711yescharacteristic
453C2×C5#319311yescharacteristic
454C2×C5#319611yescharacteristic
455C2×C5#321311yescharacteristic
456C2×C5#321811yescharacteristic
457D5—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—102no—
463C23—201no—
464C23—201no—
465C23—201no—
466C23—201no—
467C23—201no—
468C23—201no—
469C23C4×C22⋊D512yes—
470C23#408812yes—
471C23#409512yes—
472C23C22×C22⋊D511yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—104no—
479D4—208no—
480D4—202no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—84no—
506C2×C4—102no—
507C2×C4—201no—
508C2×C4—201no—
509C2×C4—201no—
510C2×C4—201no—
511C2×C4—201no—
512C2×C4—201no—
513C2×C4—201no—
514C2×C4—201no—
515C2×C4—404no—
516C2×C4C4×C22⋊D512yes—
517C2×C4#408812yes—
518C2×C4C22×C4×D511yescharacteristic
519C2×C4C2×#70911yescharacteristic
520C2×C4#414412yes—
521C2×C4#414412yes—
522C8—104no—
523C8—202no—
524C5#2204711yescharacteristic, complemented, Sylow
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—202no—
532C22—202no—
533C22—202no—
534C22—401no—
535C22—401no—
536C22—401no—
537C22—401no—
538C22#6531514yes—
539C22C2×C4×C22⋊D511yescharacteristic
540C22#6613811yescharacteristic
541C22#6613911yescharacteristic
542C22C2×#408811yescharacteristic
543C22C2×#408811yescharacteristic
544C22C2×#409511yescharacteristic
545C22#6615611yescharacteristic
546C4—22no—
547C4—22no—
548C4—44no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—82no—
553C4—202no—
554C4—401no—
555C4—401no—
556C4#6582012yes—
557C4#6582812yes—
558C2—24no—
559C2—402no—
560C2#1061382711yescharacteristic
561C2C4×#408811yescharacteristic
562C2C2×#6531511yescharacteristic
563C2#1061446911yescharacteristic
564C2#1061447011yescharacteristic
565C2#1061467811yescharacteristic
566C2#1061494511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#391C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
cosocleC22×C4×C5C24
hypercenterC2×#391D5
hypocenterC5#22047
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C2×C5×#391C2semidirect
4. #10602925C2semidirect
6. #10602348C2semidirect
7. #10602322C2semidirect
8. #10600480C2semidirect
9. #10600472C2semidirect
10. #10600468C2semidirect
11. #10600460C2semidirect
25. #65516C4semidirect
49. C22×C5×Q8C2×C4semidirect
70. C2×#688C2×C4semidirect
92. #3967C2×C4semidirect
97. C22⋊#168C2×C4semidirect
103. C2×#391D5semidirect
289. C22×Q8C4×D5semidirect
524. C5#22047semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3912
C2×#655162
C4×#40762
#106029252
#106023532
#106023482
#106023222
#106004802
#106004722
#106004682
#106004602
#220475