Finite Group Databasebeta

← #12108311

Subgroups — #12108311

Order 1280.

1570 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4043C211yesmaximal, characteristic
2C2×D5×C22⋊C8C211yesmaximal, characteristic
3D5×#380C211yesmaximal, characteristic
4C5×#1824C211yesmaximal, characteristic, complemented
5#10602347C211yesmaximal, characteristic, complemented
6#10600756C211yesmaximal, characteristic, complemented
7#10600318C211yesmaximal, characteristic, complemented
8#10599675C214yesmaximal, complemented
9#10599671C214yesmaximal, complemented
10C23×C4×D5C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×#685—21no—
14C22×C4⋊Dic5—21no—
15C22×C8×D5C2212yes—
16C2×#4043—21no—
17C2×#4043—21no—
18C2×#4043—24no—
19C2×#4043C418yescomplemented
20C2×#4043C2212yes—
21C2×#4043C2212yes—
22C2×D5×C4⋊C4C2212yes—
23C2×D5×C22⋊C4—21no—
24C2×C5×C22⋊C8C2211yescharacteristic
25C5×#380C2211yescharacteristic
26#65501C2211yescharacteristic
27C2×C22⋊#168C2211yescharacteristic
28C2×#3952C2212yes—
29C2×#3948C2214yescomplemented
30#65276C2212yes—
31C2×#3942C2214yescomplemented
32#65231—21no—
33D5×C22⋊C8—24no—
34#64956—21no—
35D5×#95—21no—
36C5×#340C2214yescomplemented
37#64749C2214yescomplemented
38#21759—51nomaximal, Sylow
39C24×D5D411yescharacteristic
40C23×C4×C5—21no—
41C23×C4×C5C2311yescharacteristic
42C23×Dic5—21no—
43C23×Dic5D411yescharacteristic
44C22×C4×D5—22no—
45C22×C4×D5—44no—
46C22×C4×D5C2×C414yes—
47C22×C4×D5D411yescharacteristic
48C22×C4×D5D411yescharacteristic
49C22×C4×D5D414yes—
50C22×C4×D5C2311yescharacteristic
51C22×C4×D5C2311yescharacteristic
52C22×Dic10—21no—
53C22×Dic10—22no—
54C22×Dic10—44no—
55C22×Dic10C2×C414yescomplemented
56C22×Dic10C2312yes—
57C22×C5×C8C2312yes—
58C2×C5×C4⋊C4—21no—
59C2×C5×C4⋊C4—22no—
60C2×C5×C4⋊C4C2×C414yescomplemented
61C2×C5×C4⋊C4C2312yes—
62C2×C5×C4⋊C4C2312yes—
63C2×C5×C22⋊C4—21no—
64C2×C22⋊Dic5—21no—
65C2×#685—28no—
66C2×#685—22no—
67C2×#685—22no—
68C2×#685—22no—
69C2×C4⋊Dic5—28no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—22no—
72C2×C4⋊Dic5C2312yes—
73C2×#683—22no—
74C22×#168C2312yes—
75C2×C8×D5—24no—
76C2×C8×D5—44no—
77#4043—28no—
78#4043—28no—
79#4043—24no—
80#4043—24no—
81#4043C2×C418yes—
82#4043C2×C418yescomplemented
83D5×C4⋊C4—44no—
84D5×C22⋊C4—44no—
85C5×C22⋊C8—22no—
86C5×#95—21no—
87#3963—21no—
88#3963—22no—
89C22⋊#168—22no—
90#3952—24no—
91#3948—44no—
92#3942—44no—
93C22×C22⋊Q8—51no—
94C22×C22⋊C8—51no—
95C2×#380—51no—
96#1892—54no—
97#1872—54no—
98#1824—51no—
99#1824D511yescharacteristic, complemented
100#1787—51no—
101#1787—51no—
102C24×C5C2×D411yescharacteristic
103C23×D5—42no—
104C23×D5—42no—
105C23×D5C22⋊C414yes—
106C23×D5SD1614yes—
107C23×D5Q1614yes—
108C23×D5C2×D411yescharacteristic
109C23×D5C2×D411yescharacteristic
110C22×C4×C5—24no—
111C22×C4×C5—24no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—44no—
115C22×C4×C5C22×C412yes—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D412yes—
118C22×C4×C5C2411yescharacteristic
119C22×Dic5—24no—
120C22×Dic5—22no—
121C22×Dic5—22no—
122C22×Dic5—42no—
123C22×Dic5—44no—
124C22×Dic5C22⋊C414yes—
125C22×Dic5C2×D411yescharacteristic
126C22×Dic5C2×D411yescharacteristic
127C2×C4×D5—24no—
128C2×C4×D5—24no—
129C2×C4×D5—48no—
130C2×C4×D5—48no—
131C2×C4×D5—42no—
132C2×C4×D5—42no—
133C2×C4×D5—44no—
134C2×C4×D5—44no—
135C2×C4×D5—44no—
136C2×C4×D5C22⋊C414yes—
137C2×C4×D5C22⋊C414yes—
138C2×C4×D5C22×C412yes—
139C2×C4×D5C22×C412yes—
140C2×C4×D5C2×D412yes—
141C2×C4×D5C2×D412yes—
142C2×Dic10—24no—
143C2×Dic10—24no—
144C2×Dic10—24no—
145C2×Dic10—22no—
146C2×Dic10—48no—
147C2×Dic10—44no—
148C2×Dic10—48no—
149C2×Dic10C22×C414yes—
150C2×C5×C8—22no—
151C2×C5×C8—24no—
152C5×C4⋊C4—24no—
153C5×C4⋊C4—22no—
154C5×C4⋊C4—24no—
155C5×C4⋊C4C22×C414yes—
156C5×C22⋊C4—42no—
157C22⋊Dic5—42no—
158#685—28no—
159#685—28no—
160#685—44no—
161C4⋊Dic5—28no—
162C4⋊Dic5—28no—
163C4⋊Dic5—24no—
164#683—44no—
165C2×#168—22no—
166C2×#168—24no—
167C8×D5—48no—
168C23×Q8—51no—
169C24×C4—51no—
170C23×C8—52no—
171C2×C22⋊Q8—58no—
172C2×C22⋊Q8—52no—
173C2×C22⋊Q8—52no—
174C2×C22⋊Q8—101no—
175C2×C22⋊Q8—101no—
176C2×C22⋊Q8—104no—
177C22×C4⋊C4—52no—
178C22×C4⋊C4—101no—
179C22×C4⋊C4C2×D511yescharacteristic
180C22×C22⋊C4—101no—
181C22×C22⋊C4—101no—
182C2×#103—54no—
183C2×#103—54no—
184C2×C22⋊C8—51no—
185C2×C22⋊C8—52no—
186C2×C22⋊C8—104no—
187C2×C22⋊C8C2×D511yescharacteristic
188#380—51no—
189#380—52no—
190#380—101no—
191#380—101no—
192#380C2×D511yescharacteristic
193C2×#95—101no—
194#340—54no—
195#340C2×D514yescomplemented
196C23×C5—22no—
197C23×C5—22no—
198C23×C5C2×C22⋊C412yes—
199C23×C5C2×SD1612yes—
200C23×C5C2×Q1612yes—
201C23×C5C22×D411yescharacteristic
202C22×D5—22no—
203C22×D5—22no—
204C22×D5—22no—
205C22×D5—22no—
206C22×D5—44no—
207C22×D5—44no—
208C22×D5—48no—
209C22×D5—48no—
210C22×D5#103116yescomplemented
211C22×D5C2×C22⋊C412yes—
212C22×D5C2×C22⋊C412yes—
213C22×D5C2×SD1612yes—
214C22×D5C2×SD1612yes—
215C22×D5C2×Q1612yes—
216C22×D5C2×Q1612yes—
217C2×C4×C5—22no—
218C2×C4×C5—24no—
219C2×C4×C5—24no—
220C2×C4×C5—24no—
221C2×C4×C5—48no—
222C2×C4×C5—44no—
223C2×C4×C5—42no—
224C2×C4×C5—42no—
225C2×C4×C5—42no—
226C2×C4×C5C2×C22⋊C412yes—
227C2×C4×C5C23×C411yescharacteristic
228C2×C4×C5C22×D411yescharacteristic
229C2×C4×C5H(5,𝔽2)14yes—
230C2×Dic5—22no—
231C2×Dic5—22no—
232C2×Dic5—24no—
233C2×Dic5—44no—
234C2×Dic5—48no—
235C2×Dic5—44no—
236C2×Dic5—42no—
237C2×Dic5—42no—
238C2×Dic5—42no—
239C2×Dic5C2×C22⋊C412yes—
240C2×Dic5C2×C22⋊C412yes—
241C4×D5—24no—
242C4×D5—24no—
243C4×D5—24no—
244C4×D5—24no—
245C4×D5—44no—
246C4×D5—44no—
247C4×D5—88no—
248Dic10—24no—
249Dic10—24no—
250Dic10—48no—
251Dic10—44no—
252C5×C8—42no—
253#168—42no—
254C25—51no—
255C22×Q8—54no—
256C22×Q8—52no—
257C22×Q8—101no—
258C22×Q8—102no—
259C22×Q8—204no—
260C23×C4—21no—
261C23×C4—51no—
262C23×C4—51no—
263C23×C4—51no—
264C23×C4—54no—
265C23×C4—51no—
266C23×C4—51no—
267C23×C4—54no—
268C23×C4—102no—
269C23×C4—101no—
270C23×C4—204no—
271C23×C4C22×D511yescharacteristic
272C22×C8—52no—
273C22×C8—104no—
274C22×C8—204no—
275C22×C8C22×D512yes—
276C22⋊Q8—58no—
277C22⋊Q8—58no—
278C22⋊Q8—104no—
279C22⋊Q8—104no—
280C22⋊Q8—108no—
281C22⋊Q8—108no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—52no—
285C2×C4⋊C4—102no—
286C2×C4⋊C4—102no—
287C2×C4⋊C4—108no—
288C2×C4⋊C4—102no—
289C2×C4⋊C4—204no—
290C2×C4⋊C4C4×D514yescomplemented
291C2×C4⋊C4C22×D512yes—
292C2×C4⋊C4C22×D512yes—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—102no—
295C2×C22⋊C4—102no—
296C2×C22⋊C4—108no—
297C2×C22⋊C4—102no—
298C2×C22⋊C4—101no—
299C2×C22⋊C4—204no—
300#103—204no—
301#103—204no—
302C22⋊C8—22no—
303C22⋊C8—104no—
304C22⋊C8—102no—
305#95—21no—
306#95—102no—
307#95—101no—
308C22×C5—22no—
309C22×C5—22no—
310C22×C5—24no—
311C22×C5—24no—
312C22×C5C2×#10314yes—
313C22×C5C22×C22⋊C411yescharacteristic
314C22×C5#53412yes—
315C22×C5C22×SD1611yescharacteristic
316C22×C5C22×Q1611yescharacteristic
317C22×C5#57512yes—
318C2×D5—22no—
319C2×D5—22no—
320C2×D5—22no—
321C2×D5—22no—
322C2×D5—24no—
323C2×D5—24no—
324C2×D5—24no—
325C2×D5—24no—
326C2×D5—416no—
327C2×D5C2×#10314yes—
328C2×D5C2×#10314yes—
329C4×C5—22no—
330C4×C5—44no—
331C4×C5—82no—
332C4×C5#51812yes—
333C4×C5#53412yes—
334Dic5—22no—
335Dic5—22no—
336Dic5—44no—
337Dic5—82no—
338C24—51no—
339C24—51no—
340C24—54no—
341C24—54no—
342C24—54no—
343C24—202no—
344C24—202no—
345C24D4×D511yescharacteristic
346C2×Q8—54no—
347C2×Q8—102no—
348C2×Q8—104no—
349C2×Q8—104no—
350C2×Q8—104no—
351C2×Q8—208no—
352C2×Q8—204no—
353C2×Q8—208no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—51no—
360C22×C4—51no—
361C22×C4—54no—
362C22×C4—52no—
363C22×C4—52no—
364C22×C4—52no—
365C22×C4—52no—
366C22×C4—54no—
367C22×C4—54no—
368C22×C4—104no—
369C22×C4—104no—
370C22×C4—102no—
371C22×C4—104no—
372C22×C4—102no—
373C22×C4—208no—
374C22×C4—208no—
375C22×C4—202no—
376C22×C4—202no—
377C22×C4—204no—
378C22×C4—204no—
379C22×C4—204no—
380C22×C4—202no—
381C22×C4—204no—
382C22×C4C2×C4×D512yes—
383C22×C4D4×D512yes—
384C22×C4D4×D511yescharacteristic
385C22×C4C23×D511yescharacteristic
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—102no—
389C2×C8—104no—
390C2×C8—208no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—108no—
395C4⋊C4—108no—
396C4⋊C4—104no—
397C4⋊C4—204no—
398C4⋊C4C2×C4×D514yes—
399C22⋊C4—42no—
400C22⋊C4—108no—
401C22⋊C4—108no—
402C22⋊C4—204no—
403C22⋊C4—202no—
404C2×C5—24no—
405C2×C5#230811yescharacteristic
406C2×C5C22×#10311yescharacteristic
407C2×C5#289611yescharacteristic
408C2×C5#318712yes—
409C2×C5#318912yes—
410D5—24no—
411D5—24no—
412C23—22no—
413C23—22no—
414C23—52no—
415C23—52no—
416C23—52no—
417C23—52no—
418C23—52no—
419C23—52no—
420C23—516no—
421C23—102no—
422C23—102no—
423C23—102no—
424C23—102no—
425C23—204no—
426C23—208no—
427C23—204no—
428C23—208no—
429C23D5×C22⋊C412yes—
430C23D5×SD1612yes—
431C23D5×Q1612yes—
432C23C2×D4×D511yescharacteristic
433Q8—104no—
434Q8—104no—
435Q8—208no—
436Q8—204no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—52no—
447C2×C4—52no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—104no—
451C2×C4—104no—
452C2×C4—104no—
453C2×C4—104no—
454C2×C4—104no—
455C2×C4—204no—
456C2×C4—204no—
457C2×C4—204no—
458C2×C4—208no—
459C2×C4—204no—
460C2×C4—202no—
461C2×C4—202no—
462C2×C4—202no—
463C2×C4—408no—
464C2×C4D5×C22⋊C412yes—
465C2×C4C22×C4×D511yescharacteristic
466C2×C4C2×D4×D511yescharacteristic
467C2×C4#414914yes—
468C8—42no—
469C8—202no—
470C5#2175911yescharacteristic, complemented, Sylow
471C22—22no—
472C22—24no—
473C22—22no—
474C22—24no—
475C22—54no—
476C22—54no—
477C22—102no—
478C22—102no—
479C22—104no—
480C22—104no—
481C22—102no—
482C22—102no—
483C22—104no—
484C22—104no—
485C22—2016no—
486C22D5×#10314yes—
487C22C2×D5×C22⋊C411yescharacteristic
488C22#6598712yes—
489C22C2×D5×SD1611yescharacteristic
490C22C2×D5×Q1611yescharacteristic
491C22#6610712yes—
492C4—22no—
493C4—44no—
494C4—82no—
495C4—102no—
496C4—102no—
497C4—204no—
498C4—402no—
499C4#6585512yes—
500C4#6598712yes—
501C2—24no—
502C2—104no—
503C2—104no—
504C2#1060585011yescharacteristic
505C2C2×D5×#10311yescharacteristic
506C2#1061065511yescharacteristic
507C2#1061306012yes—
508C2#1061321912yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1824C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenter#1824D5
hypocenterC5#21759
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. C5×#1824C2semidirect
5. #10602347C2semidirect
6. #10600756C2semidirect
7. #10600318C2semidirect
8. #10599675C2semidirect
9. #10599671C2semidirect
19. C2×#4043C4semidirect
29. C2×#3948C22semidirect
31. C2×#3942C22semidirect
36. C5×#340C22semidirect
37. #64749C22semidirect
55. C22×Dic10C2×C4semidirect
60. C2×C5×C4⋊C4C2×C4semidirect
82. #4043C2×C4semidirect
99. #1824D5semidirect
195. #340C2×D5semidirect
210. C22×D5#103semidirect
290. C2×C4⋊C4C4×D5semidirect
470. C5#21759semidirect

Maximal subgroups

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

subgroupindex
C22×#40432
C2×D5×C22⋊C82
D5×#3802
C5×#18242
#106023472
#106007562
#106003182
#105996752
#105996712
#217595