Finite Group Databasebeta

← #12109861

Subgroups — #12109861

Order 1280.

902 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65124C211yesmaximal, characteristic
2D5×#384C211yesmaximal, characteristic
3#10605483C211yesmaximal, characteristic
4C5×#1948C211yesmaximal, characteristic, complemented
5#10601372C212yesmaximal
6#10601349C212yesmaximal, complemented
7#10601246C211yesmaximal, characteristic, complemented
8#10600360C211yesmaximal, characteristic, complemented
9#10600260C211yesmaximal, characteristic, complemented
10#10599667C212yesmaximal
11#10599642C212yesmaximal, complemented
12C2×D5×C4⋊C4—21no—
13C2×D5×C4⋊C4—21no—
14C2×D5×C4⋊C4C2212yes—
15C2×C42×D5C2211yescharacteristic
16C2×C5×#106C2212yes—
17C2×C5×C4⋊C8C2211yescharacteristic
18C5×#384C2211yescharacteristic
19C4×C5×C4⋊C4C2211yescharacteristic
20C2×#3952C2212yes—
21C2×#3950C2212yes—
22C2×#3947C2212yes—
23#65276C2211yescharacteristic
24#65276C2211yescharacteristic
25#65266C2211yescharacteristic
26#65266C2211yescharacteristic
27#65265—21no—
28#65265—21no—
29#65265C2212yes—
30C2×#3940C2212yes—
31C2×#3939C2212yes—
32C4×#685—21no—
33#65224C2211yescharacteristic
34C4×C4⋊Dic5C2211yescharacteristic
35#65221—21no—
36C2×C4⋊#168C2211yescharacteristic
37#65124C418yescomplemented
38#64953—21no—
39#64940—21no—
40C5×#340C2212yes—
41#64771C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#21987—51nomaximal, Sylow
45C22×C4×D5—22no—
46C22×C4×D5D411yescharacteristic
47C22×C4×D5D411yescharacteristic
48C22×C4×D5C2311yescharacteristic
49C22×C5×C8C2312yes—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4C2311yescharacteristic
53C2×C5×C4⋊C4C2311yescharacteristic
54C2×C5×C4⋊C4C2312yes—
55C2×C42×C5—21no—
56C2×C42×C5C2311yescharacteristic
57C2×#685—21no—
58C2×#685—21no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5C2311yescharacteristic
62C2×C4⋊Dic5C2312yes—
63C2×C4⋊Dic5C2311yescharacteristic
64C2×#683—22no—
65C2×#683—22no—
66C2×#683—22no—
67C2×C4×Dic5—21no—
68C2×C4×Dic5D411yescharacteristic
69C2×C4×Dic5D411yescharacteristic
70C2×C4×Dic5C2311yescharacteristic
71C22×#168C2312yes—
72D5×C4⋊C4—42no—
73D5×C4⋊C4—42no—
74D5×C4⋊C4—44no—
75D5×C4⋊C4—44no—
76C42×D5C2×C414yes—
77C42×D5Q814yes—
78C5×#106—42no—
79C5×C4⋊C8C2×C414yescomplemented
80C5×#95—21no—
81#3963—21no—
82#3963—21no—
83#3963—21no—
84#3963—21no—
85#3963—21no—
86#3963—21no—
87#3963—21no—
88#3952C2×C418yes—
89#3950—42no—
90#3947C2×C418yescomplemented
91#3940—42no—
92#3939—42no—
93C4⋊#168C2×C414yescomplemented
94C2×#432—51no—
95C2×#384—51no—
96#2283—51no—
97#1948—51no—
98#1948D511yescharacteristic, complemented
99#1947—51no—
100#1947—51no—
101#1853—52no—
102#1852—52no—
103#1847—52no—
104#1836—52no—
105C23×D5C2×D411yescharacteristic
106C22×C4×C5—21no—
107C22×C4×C5—21no—
108C22×C4×C5—22no—
109C22×C4×C5C2×D411yescharacteristic
110C22×C4×C5C2×D411yescharacteristic
111C22×C4×C5C2411yescharacteristic
112C22×Dic5—22no—
113C22×Dic5—21no—
114C22×Dic5—21no—
115C22×Dic5C2×D411yescharacteristic
116C22×Dic5C2×D411yescharacteristic
117C22×Dic5C2×D411yescharacteristic
118C2×C4×D5—22no—
119C2×C4×D5—22no—
120C2×C4×D5—22no—
121C2×C4×D5—22no—
122C2×C4×D5—21no—
123C2×C4×D5—21no—
124C2×C4×D5—21no—
125C2×C4×D5—21no—
126C2×C4×D5—44no—
127C2×C4×D5—44no—
128C2×C4×D5—44no—
129C2×C4×D5C4⋊C414yes—
130C2×C4×D5C4⋊C414yes—
131C2×C4×D5C22×C412yes—
132C2×C4×D5C2×Q812yes—
133C2×C5×C8—22no—
134C2×C5×C8—22no—
135C2×C5×C8C22×C414yes—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—22no—
140C5×C4⋊C4—21no—
141C5×C4⋊C4—21no—
142C5×C4⋊C4—42no—
143C5×C4⋊C4—42no—
144C42×C5—42no—
145C42×C5C22×C412yes—
146C42×C5C2×Q812yes—
147#685—42no—
148#685—42no—
149C4⋊Dic5—21no—
150C4⋊Dic5—22no—
151C4⋊Dic5—21no—
152C4⋊Dic5—22no—
153C4⋊Dic5—21no—
154C4⋊Dic5—21no—
155C4⋊Dic5—42no—
156C4⋊Dic5—42no—
157#683—44no—
158#683—44no—
159#683—44no—
160C4×Dic5—42no—
161C4×Dic5C4⋊C414yes—
162C4×Dic5C4⋊C414yes—
163C4×Dic5C22×C412yes—
164C4×Dic5C2×Q812yes—
165C2×#168—22no—
166C2×#168—22no—
167C2×#168C22×C414yes—
168C22×C4⋊C4—52no—
169C22×C4⋊C4—101no—
170C22×C4⋊C4—101no—
171C22×C42—51no—
172#432—58no—
173C2×#107—52no—
174C2×#107—52no—
175C2×#106—52no—
176C2×#106C2×D512yes—
177C2×C4⋊C8—51no—
178C2×C4⋊C8C2×D511yescharacteristic
179C2×C22⋊C8—52no—
180C2×#97—52no—
181#384—52no—
182#384—51no—
183#384—101no—
184#384—101no—
185#384C2×D511yescharacteristic
186#383—101no—
187#382—51no—
188#382—51no—
189#381—101no—
190#380—51no—
191#380—51no—
192C4×C4⋊C4—51no—
193C4×C4⋊C4C2×D511yescharacteristic
194C4×C22⋊C4—101no—
195#376—101no—
196#340—52no—
197#340—52no—
198#340—52no—
199#340C2×D512yes—
200C23×C5C22×D411yescharacteristic
201C22×D5—21no—
202C22×D5—21no—
203C22×D5—21no—
204C22×D5—21no—
205C22×D5C2×C4⋊C412yes—
206C22×D5AGL(1,ℤ/8)12yes—
207C22×D5#13712yes—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—42no—
211C2×C4×C5—41no—
212C2×C4×C5—41no—
213C2×C4×C5—42no—
214C2×C4×C5—42no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—41no—
218C2×C4×C5—41no—
219C2×C4×C5C2×C4⋊C412yes—
220C2×C4×C5C2×C4⋊C412yes—
221C2×C4×C5AGL(1,ℤ/8)12yes—
222C2×C4×C5#13712yes—
223C2×C4×C5C23×C411yescharacteristic
224C2×C4×C5C22×Q811yescharacteristic
225C2×C4×C5H(5,𝔽2)11yescharacteristic
226C2×C4×C5H(5,𝔽2)11yescharacteristic
227C2×C4×C52−1+411yescharacteristic
228C2×C4×C52−1+411yescharacteristic
229C2×Dic5—21no—
230C2×Dic5—21no—
231C2×Dic5—21no—
232C2×Dic5—21no—
233C2×Dic5—21no—
234C2×Dic5—21no—
235C2×Dic5—42no—
236C2×Dic5—41no—
237C2×Dic5—41no—
238C2×Dic5—41no—
239C2×Dic5—41no—
240C2×Dic5—42no—
241C2×Dic5—42no—
242C2×Dic5—41no—
243C2×Dic5—41no—
244C2×Dic5C2×C4⋊C412yes—
245C2×Dic5C2×C4⋊C412yes—
246C2×Dic5C2×C4⋊C412yes—
247C4×D5—28no—
248C4×D5—22no—
249C4×D5—22no—
250C4×D5—22no—
251C4×D5—22no—
252C4×D5—28no—
253C4×D5—88no—
254C5×C8—24no—
255#168—24no—
256C23×C4—51no—
257C23×C4—51no—
258C23×C4—51no—
259C23×C4—102no—
260C22×C8—52no—
261C22×C8C22×D512yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—52no—
265C2×C4⋊C4—51no—
266C2×C4⋊C4—51no—
267C2×C4⋊C4—102no—
268C2×C4⋊C4—102no—
269C2×C4⋊C4—102no—
270C2×C4⋊C4—101no—
271C2×C4⋊C4—101no—
272C2×C4⋊C4—202no—
273C2×C4⋊C4—202no—
274C2×C4⋊C4—204no—
275C2×C4⋊C4—204no—
276C2×C4⋊C4C22×D512yes—
277C2×C4⋊C4C22×D511yescharacteristic
278C2×C4⋊C4C22×D511yescharacteristic
279C2×C22⋊C4—101no—
280C2×C22⋊C4—101no—
281C2×C42—21no—
282C2×C42—51no—
283C2×C42—51no—
284C2×C42—51no—
285C2×C42—54no—
286C2×C42—54no—
287C2×C42—101no—
288C2×C42C22×D511yescharacteristic
289#107—202no—
290#107—202no—
291#106—42no—
292#106—202no—
293C4⋊C8—54no—
294C4⋊C8C4×D514yescomplemented
295C22⋊C8—58no—
296#97—58no—
297#95—21no—
298#95—101no—
299#95—101no—
300#95—101no—
301#95—101no—
302#95—101no—
303#95—101no—
304#95—101no—
305C22×C5C22×C4⋊C411yescharacteristic
306C22×C5#53511yescharacteristic
307C22×C5#53611yescharacteristic
308C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C5C2×#13711yescharacteristic
311C22×C5C2×#13711yescharacteristic
312C2×D5—22no—
313C2×D5—22no—
314C2×D5—22no—
315C2×D5—22no—
316C2×D5—22no—
317C2×D5—22no—
318C2×D5#42814yes—
319C4×C5—22no—
320C4×C5—82no—
321C4×C5—81no—
322C4×C5—81no—
323C4×C5#42814yes—
324C4×C5#52011yescharacteristic
325C4×C5#52011yescharacteristic
326C4×C5#54411yescharacteristic
327C4×C5#54411yescharacteristic
328Dic5—22no—
329Dic5—22no—
330Dic5—22no—
331Dic5—82no—
332Dic5—81no—
333Dic5—81no—
334C24—51no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—51no—
339C22×C4—51no—
340C22×C4—51no—
341C22×C4—52no—
342C22×C4—52no—
343C22×C4—54no—
344C22×C4—54no—
345C22×C4—102no—
346C22×C4—102no—
347C22×C4—101no—
348C22×C4—101no—
349C22×C4—101no—
350C22×C4—101no—
351C22×C4—102no—
352C22×C4—102no—
353C22×C4—101no—
354C22×C4—101no—
355C22×C4—102no—
356C22×C4—204no—
357C22×C4—204no—
358C22×C4—204no—
359C22×C4D4×D511yescharacteristic
360C22×C4D4×D511yescharacteristic
361C22×C4C23×D511yescharacteristic
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—54no—
365C2×C8—102no—
366C2×C8—102no—
367C2×C8C2×C4×D514yes—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—101no—
377C4⋊C4—102no—
378C4⋊C4—101no—
379C4⋊C4—102no—
380C4⋊C4—101no—
381C4⋊C4—101no—
382C4⋊C4—202no—
383C4⋊C4—204no—
384C4⋊C4—204no—
385C4⋊C4—204no—
386C4⋊C4—202no—
387C22⋊C4—202no—
388C22⋊C4—202no—
389C42—42no—
390C42—52no—
391C42—52no—
392C42—54no—
393C42—54no—
394C42—202no—
395C42C2×C4×D512yes—
396C42Q8×D512yes—
397C2×C5#231711yescharacteristic
398C2×C5C2×#42811yescharacteristic
399C2×C5C2×#42811yescharacteristic
400C2×C5#321711yescharacteristic
401C2×C5#321811yescharacteristic
402C2×C5#323811yescharacteristic
403C2×C5#323911yescharacteristic
404D5—24no—
405C23—52no—
406C23—52no—
407C23—52no—
408C23—101no—
409C23—101no—
410C23—101no—
411C23—101no—
412C23C2×D4×D511yescharacteristic
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—52no—
425C2×C4—52no—
426C2×C4—52no—
427C2×C4—108no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—102no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—108no—
438C2×C4—102no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—202no—
442C2×C4—201no—
443C2×C4—202no—
444C2×C4—201no—
445C2×C4—202no—
446C2×C4—201no—
447C2×C4—201no—
448C2×C4—408no—
449C2×C4D5×C4⋊C412yes—
450C2×C4D5×C4⋊C412yes—
451C2×C4#406012yes—
452C2×C4#406112yes—
453C2×C4C22×C4×D511yescharacteristic
454C2×C4#414411yescharacteristic
455C2×C4C2×Q8×D511yescharacteristic
456C2×C4#414711yescharacteristic
457C2×C4#414911yescharacteristic
458C2×C4#415011yescharacteristic
459C8—24no—
460C8—104no—
461C5#2198711yescharacteristic, complemented, Sylow
462C22—54no—
463C22—102no—
464C22—102no—
465C22—102no—
466C22—102no—
467C22—102no—
468C22—102no—
469C22C2×D5×C4⋊C411yescharacteristic
470C22#6594111yescharacteristic
471C22#6596211yescharacteristic
472C22C2×#406011yescharacteristic
473C22C2×#406111yescharacteristic
474C22D5×AGL(1,ℤ/8)11yescharacteristic
475C22D5×#13711yescharacteristic
476C4—22no—
477C4—81no—
478C4—82no—
479C4—81no—
480C4—102no—
481C4—102no—
482C4—102no—
483C4—401no—
484C4—402no—
485C4—401no—
486C4#6515014yes—
487C4#6588011yescharacteristic
488C4#6590811yescharacteristic
489C4#6602311yescharacteristic
490C4#6606311yescharacteristic
491C2—104no—
492C2#1060769911yescharacteristic
493C2C2×#6515011yescharacteristic
494C2D5×#42811yescharacteristic
495C2#1061212111yescharacteristic
496C2#1061216611yescharacteristic
497C2#1061274911yescharacteristic
498C2#1061279811yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1948C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1948D5
hypocenterC5#21987
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. C5×#1948C2semidirect
6. #10601349C2semidirect
7. #10601246C2semidirect
8. #10600360C2semidirect
9. #10600260C2semidirect
11. #10599642C2semidirect
37. #65124C4semidirect
79. C5×C4⋊C8C2×C4semidirect
90. #3947C2×C4semidirect
93. C4⋊#168C2×C4semidirect
98. #1948D5semidirect
294. C4⋊C8C4×D5semidirect
461. C5#21987semidirect

Maximal subgroups

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

subgroupindex
C2×#651242
D5×#3842
#106054832
C5×#19482
#106013722
#106013492
#106012462
#106003602
#106002602
#105996672
#105996422
#219875