Finite Group Databasebeta

← #12155827

Subgroups — #12155827

Order 1280.

1260 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×Q8C211yesmaximal, characteristic
2C2×#65371C211yesmaximal, characteristic, complemented
3C2×#65311C211yesmaximal, characteristic, complemented
4#10614114C211yesmaximal, characteristic
5#10602925C212yesmaximal, complemented
6#10602925C212yesmaximal, complemented
7#10602571C211yesmaximal, characteristic, complemented
8#10602488C212yesmaximal, complemented
9#10602487C212yesmaximal, complemented
10#10602424C211yesmaximal, characteristic, complemented
11#10602408C211yesmaximal, characteristic
12C23×C5×Q8C2212yes—
13C5×D4×Q8—24no—
14C5×D4×Q8—24no—
15C2×C5×C4⋊Q8—21no—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C22⋊Q8—22no—
18C2×C5×C22⋊Q8C2212yes—
19C2×C4×C5×Q8C2211yescharacteristic
20C2×C4×C5×D4—21no—
21C2×C4×C5×D4C2211yescharacteristic
22C2×#4016C2211yescharacteristic
23C2×#3967C2212yes—
24C2×#3967C2212yes—
25C2×#3967C2211yescharacteristic
26C2×#3967C2211yescharacteristic
27C2×#3967C2212yes—
28C2×#3967C2212yes—
29C2×#3967C2212yes—
30#65511—21no—
31#65501C2212yes—
32#65501C2212yes—
33C2×C22⋊#168C2212yes—
34C2×C22⋊#168C2212yes—
35#65371—24no—
36#65311—24no—
37#65271—21no—
38C2×#3940C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40#65234—22no—
41C4×C4⋊Dic5C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43C2×#3935C2211yescharacteristic
44#64749C2212yes—
45#25894—51nomaximal, Sylow
46C22×C5×Q8—24no—
47C22×C5×Q8—24no—
48C22×C5×Q8—22no—
49C22×C5×Q8—24no—
50C22×C5×Q8—24no—
51C22×C5×Q8—42no—
52C22×C5×Q8—42no—
53C22×C5×Q8C2312yes—
54C22×C5×Q8C2311yescharacteristic
55C22×C5×D4D411yescharacteristic
56C23×C4×C5—22no—
57C23×C4×C5C2312yes—
58C5×C4⋊Q8—28no—
59C5×C4⋊Q8—22no—
60C5×C4⋊Q8D414yes—
61C5×C22⋊Q8—216no—
62C5×C22⋊Q8—24no—
63C5×C22⋊Q8—24no—
64C4×C5×Q8—22no—
65C4×C5×Q8D414yes—
66C4×C5×D4—28no—
67C4×C5×D4—22no—
68C4×C5×D4—22no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—22no—
73C2×C5×C4⋊C4D411yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4D412yes—
77C2×C42×C5—21no—
78C2×C42×C5C2311yescharacteristic
79C2×C22⋊Dic5—22no—
80C2×C22⋊Dic5—22no—
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2312yes—
84C2×#683—21no—
85C2×#683—21no—
86C2×C4×Dic5—21no—
87C22×#168C2312yes—
88C22×#168C2311yescharacteristic
89C22×#168C2311yescharacteristic
90#4016—22no—
91#4016—22no—
92#3967—22no—
93#3967—22no—
94#3967—24no—
95#3967—24no—
96#3967—24no—
97#3967—42no—
98#3967—42no—
99#3963—21no—
100#3963—22no—
101#3963—22no—
102C22⋊#168—42no—
103C22⋊#168—42no—
104#3940—22no—
105#3939—22no—
106C4⋊#168—22no—
107#3935—22no—
108C2×D4×Q8D511yescharacteristic
109C2×#491—51no—
110C2×#479—51no—
111#2562—51no—
112#2036—52no—
113#2035—52no—
114#1981—51no—
115#1955—51no—
116#1787—52no—
117#1787—52no—
118#1763—51no—
119C24×C5C2×D412yes—
120C2×C5×Q8—28no—
121C2×C5×Q8—21no—
122C2×C5×Q8—22no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8—22no—
126C2×C5×Q8—22no—
127C2×C5×Q8—22no—
128C2×C5×Q8—44no—
129C2×C5×Q8—44no—
130C2×C5×Q8—48no—
131C2×C5×Q8—48no—
132C2×C5×Q8—48no—
133C2×C5×Q8—48no—
134C2×C5×Q8C2×D414yes—
135C2×C5×Q8C2×D412yes—
136C2×C5×Q8#4114yes—
137C2×C5×D4—24no—
138C2×C5×D4—22no—
139C22×C4×C5—21no—
140C22×C4×C5—28no—
141C22×C4×C5—21no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2411yescharacteristic
155C22×Dic5—22no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C5×C4⋊C4—28no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4C2×D412yes—
165C5×C22⋊C4—28no—
166C5×C22⋊C4—24no—
167C42×C5—24no—
168C42×C5C2×D412yes—
169C42×C5C2×D412yes—
170C22⋊Dic5—44no—
171C22⋊Dic5—82no—
172C4⋊Dic5—21no—
173C4⋊Dic5—21no—
174C4⋊Dic5—21no—
175C4⋊Dic5—21no—
176C4⋊Dic5—22no—
177C4⋊Dic5—22no—
178#683—42no—
179#683—42no—
180C4×Dic5—81no—
181C2×#168—22no—
182C2×#168—21no—
183C2×#168—21no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—22no—
187C2×#168—22no—
188C23×Q8C2×D512yes—
189D4×Q8—24no—
190D4×Q8—24no—
191C2×C4⋊Q8—21no—
192C2×C4⋊Q8C2×D511yescharacteristic
193C2×#125—51no—
194C2×C22⋊Q8—22no—
195C2×C22⋊Q8C2×D512yes—
196C2×C4×Q8C2×D511yescharacteristic
197C2×C4×D4—21no—
198C2×C4×D4C2×D511yescharacteristic
199#491—104no—
200#479—104no—
201C2×#107—51no—
202C2×#106—51no—
203C2×C4⋊C8—51no—
204C2×#103—51no—
205C2×#103—51no—
206C2×#103—52no—
207C2×#103—52no—
208C2×#103—52no—
209C2×#103—52no—
210C2×#103—52no—
211C2×C22⋊C8—52no—
212C2×C22⋊C8—52no—
213C2×#97—51no—
214#400—101no—
215#394—101no—
216#388—102no—
217#380—52no—
218#380—52no—
219C4×C4⋊C4—51no—
220#340—52no—
221C23×C5—22no—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5#13714yes—
228C23×C5C22×D411yescharacteristic
229C5×Q8—22no—
230C5×Q8—24no—
231C5×Q8—24no—
232C5×Q8—24no—
233C5×Q8—416no—
234C5×Q8—42no—
235C5×Q8—42no—
236C5×Q8#12414yes—
237C5×D4—28no—
238C2×C4×C5—22no—
239C2×C4×C5—24no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—22no—
244C2×C4×C5—24no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—416no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5#12012yes—
258C2×C4×C5#12012yes—
259C2×C4×C5C22×D411yescharacteristic
260C2×C4×C5C22×D411yescharacteristic
261C2×C4×C5C2×#4112yes—
262C2×C4×C5H(5,𝔽2)12yes—
263C2×Dic5—41no—
264C2×Dic5—41no—
265C2×Dic5—42no—
266C2×Dic5—42no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271#168—41no—
272#168—41no—
273#168—42no—
274C22×Q8—24no—
275C22×Q8—24no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—24no—
279C22×Q8—42no—
280C22×Q8—42no—
281C22×Q8C22×D512yes—
282C22×Q8C22×D511yescharacteristic
283C22×D4C22⋊D511yescharacteristic
284C23×C4—22no—
285C23×C4C22×D512yes—
286C22×C8—52no—
287C22×C8—51no—
288C22×C8—51no—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8C22⋊D514yes—
292#125—102no—
293#125—102no—
294C22⋊Q8—216no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C4×Q8—22no—
298C4×Q8C22⋊D514yes—
299C4×D4—28no—
300C4×D4—22no—
301C4×D4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—51no—
307C2×C4⋊C4—51no—
308C2×C4⋊C4—52no—
309C2×C4⋊C4—101no—
310C2×C4⋊C4—101no—
311C2×C4⋊C4C22⋊D511yescharacteristic
312C2×C4⋊C4C22×D511yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—102no—
315C2×C22⋊C4—102no—
316C2×C22⋊C4C22⋊D512yes—
317C2×C42—21no—
318C2×C42—101no—
319C2×C42C22×D511yescharacteristic
320#107—102no—
321#106—102no—
322C4⋊C8—102no—
323#103—104no—
324#103—102no—
325#103—102no—
326#103—104no—
327#103—104no—
328#103—202no—
329#103—202no—
330C22⋊C8—202no—
331C22⋊C8—202no—
332#97—102no—
333#95—101no—
334#95—102no—
335#95—102no—
336C22×C5—24no—
337C22×C5—24no—
338C22×C5—24no—
339C22×C5—42no—
340C22×C5—42no—
341C22×C5—42no—
342C22×C5—42no—
343C22×C5C2×#12011yescharacteristic
344C22×C5#54612yes—
345C22×C5C2×#13712yes—
346C22×C5#57711yescharacteristic
347C22×C5#57811yescharacteristic
348C4×C5—24no—
349C4×C5—22no—
350C4×C5—22no—
351C4×C5—42no—
352C4×C5—44no—
353C4×C5—42no—
354C4×C5C2×#12411yescharacteristic
355C4×C5#53511yescharacteristic
356C4×C5#54612yes—
357Dic5—81no—
358Dic5—81no—
359Dic5—82no—
360C24C2×C22⋊D512yes—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8#71114yes—
376C2×Q8C2×C22⋊D514yes—
377C2×Q8C2×C22⋊D512yes—
378C2×D4—24no—
379C2×D4—22no—
380C22×C4—21no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—102no—
393C22×C4—101no—
394C22×C4—101no—
395C22×C4C2×C22⋊D512yes—
396C22×C4C2×C22⋊D511yescharacteristic
397C22×C4C2×C22⋊D511yescharacteristic
398C22×C4C23×D511yescharacteristic
399C2×C8—102no—
400C2×C8—102no—
401C2×C8—102no—
402C2×C8—102no—
403C2×C8—102no—
404C2×C8—101no—
405C2×C8—101no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—102no—
413C4⋊C4—102no—
414C4⋊C4—101no—
415C4⋊C4—101no—
416C4⋊C4—101no—
417C4⋊C4—101no—
418C4⋊C4—202no—
419C4⋊C4—202no—
420C4⋊C4C2×C22⋊D512yes—
421C22⋊C4—28no—
422C22⋊C4—24no—
423C22⋊C4—204no—
424C22⋊C4—402no—
425C42—24no—
426C42—401no—
427C42C2×C22⋊D512yes—
428C42C2×C22⋊D512yes—
429C2×C5—44no—
430C2×C5#243511yescharacteristic
431C2×C5#300411yescharacteristic
432C2×C5#301311yescharacteristic
433C2×C5#331212yes—
434C2×C5#332012yes—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23#408814yes—
442C23C22×C22⋊D511yescharacteristic
443Q8—22no—
444Q8—24no—
445Q8—24no—
446Q8—24no—
447Q8—416no—
448Q8—42no—
449Q8—42no—
450Q8#408214yes—
451D4—28no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—416no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—202no—
472C2×C4—202no—
473C2×C4—202no—
474C2×C4—202no—
475C2×C4—201no—
476C2×C4—201no—
477C2×C4—201no—
478C2×C4—201no—
479C2×C4#409912yes—
480C2×C4#409912yes—
481C2×C4C2×#71112yes—
482C2×C4#414412yes—
483C2×C4C22×C22⋊D511yescharacteristic
484C2×C4C22×C22⋊D511yescharacteristic
485C8—201no—
486C8—201no—
487C8—202no—
488C5#2589411yescharacteristic, complemented, Sylow
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22#6614012yes—
497C22C2×#408812yes—
498C22#6617111yescharacteristic
499C22#6617211yescharacteristic
500C22C2×#409911yescharacteristic
501C4—22no—
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—401no—
508C4—401no—
509C4—402no—
510C4C2×#408211yescharacteristic
511C4#6613911yescharacteristic
512C4#6614012yes—
513C2—44no—
514C2#1061528312yes—
515C2#1061528412yes—
516C2#1061578111yescharacteristic
517C2#1061579011yescharacteristic
518C2#1061584511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×D4×Q8C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×D4×Q8D5
hypocenterC5#25894
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#65371C2semidirect
3. C2×#65311C2semidirect
5. #10602925C2semidirect
6. #10602925C2semidirect
7. #10602571C2semidirect
8. #10602488C2semidirect
9. #10602487C2semidirect
10. #10602424C2semidirect
488. C5#25894semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
C2×#653712
C2×#653112
#106141142
#106029252
#106025712
#106024882
#106024872
#106024242
#106024082
#258945