Finite Group Databasebeta

← #12109871

Subgroups — #12109871

Order 1280.

830 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65124C211yesmaximal, characteristic
2#10606049C211yesmaximal, characteristic
3#10605484C211yesmaximal, characteristic
4C5×#1947C211yesmaximal, characteristic, complemented
5#10601426C212yesmaximal
6#10601401C212yesmaximal, complemented
7#10601247C211yesmaximal, characteristic, complemented
8#10600360C211yesmaximal, characteristic, complemented
9#10600260C211yesmaximal, characteristic, complemented
10#10599669C212yesmaximal
11#10599642C212yesmaximal, complemented
12C2×#4038C2212yes—
13C2×#4018—21no—
14C2×C42×D5C2211yescharacteristic
15C2×C5×#107C2212yes—
16C2×C5×C4⋊C8C2211yescharacteristic
17C5×#384C2211yescharacteristic
18C4×C5×C4⋊C4C2211yescharacteristic
19C2×#3952C2212yes—
20C2×#3949C2212yes—
21C2×#3947C2212yes—
22#65276—21no—
23#65276—21no—
24#65276C2211yescharacteristic
25#65276C2211yescharacteristic
26#65266C2212yes—
27#65265C2211yescharacteristic
28#65265C2211yescharacteristic
29C4⋊C4×Dic5—21no—
30C4⋊C4×Dic5—21no—
31C2×#3940C2212yes—
32C2×#3939C2212yes—
33#65224C2211yescharacteristic
34C4×C4⋊Dic5C2211yescharacteristic
35C4×#683—21no—
36C2×C4⋊#168C2211yescharacteristic
37#65124C418yescomplemented
38#64956—21no—
39#64939—21no—
40C5×#340C2212yes—
41#64771C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#21988—51nomaximal, Sylow
45C22×C4×D5D411yescharacteristic
46C22×C4×D5D411yescharacteristic
47C22×C4×D5C2311yescharacteristic
48C22×C5×C8C2312yes—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4C2311yescharacteristic
52C2×C5×C4⋊C4C2311yescharacteristic
53C2×C5×C4⋊C4C2312yes—
54C2×C42×C5—21no—
55C2×C42×C5C2311yescharacteristic
56C2×#685—21no—
57C2×#685—21no—
58C2×#685—22no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5C2311yescharacteristic
62C2×C4⋊Dic5C2312yes—
63C2×C4⋊Dic5C2311yescharacteristic
64C2×#683—21no—
65C2×#683—21no—
66C2×#683—21no—
67C2×#683—21no—
68C2×C4×Dic5—21no—
69C2×C4×Dic5—22no—
70C2×C4×Dic5D411yescharacteristic
71C2×C4×Dic5D411yescharacteristic
72C2×C4×Dic5C2311yescharacteristic
73C22×#168C2312yes—
74#4038—42no—
75#4038—42no—
76#4018—44no—
77C42×D5C2×C414yes—
78C42×D5Q814yes—
79C5×#107—42no—
80C5×C4⋊C8C2×C414yes—
81C5×#95—21no—
82#3963—21no—
83#3963—21no—
84#3963—21no—
85#3963—22no—
86#3963—22no—
87#3952C2×C418yes—
88#3949—42no—
89#3947C2×C418yes—
90#3940—42no—
91#3939—42no—
92C4⋊#168C2×C414yes—
93C2×#432—51no—
94#2285—51no—
95#2285—51no—
96#1948—51no—
97#1948—51no—
98#1947—51no—
99#1947D511yescharacteristic, complemented
100#1854—52no—
101#1854—52no—
102#1847—52no—
103#1836—52no—
104C23×D5C2×D411yescharacteristic
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—22no—
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5C2×D411yescharacteristic
110C22×C4×C5C2411yescharacteristic
111C22×Dic5—21no—
112C22×Dic5—21no—
113C22×Dic5—22no—
114C22×Dic5C2×D411yescharacteristic
115C22×Dic5C2×D411yescharacteristic
116C22×Dic5C2×D411yescharacteristic
117C2×C4×D5—22no—
118C2×C4×D5—22no—
119C2×C4×D5—21no—
120C2×C4×D5—21no—
121C2×C4×D5—21no—
122C2×C4×D5—21no—
123C2×C4×D5—41no—
124C2×C4×D5—41no—
125C2×C4×D5C4⋊C414yes—
126C2×C4×D5C4⋊C414yes—
127C2×C4×D5C22×C412yes—
128C2×C4×D5C2×Q812yes—
129C2×C5×C8—22no—
130C2×C5×C8—22no—
131C2×C5×C8C22×C414yes—
132C5×C4⋊C4—22no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4—42no—
140C42×C5—42no—
141C42×C5C22×C412yes—
142C42×C5C2×Q812yes—
143#685—42no—
144#685—42no—
145#685—44no—
146C4⋊Dic5—21no—
147C4⋊Dic5—22no—
148C4⋊Dic5—21no—
149C4⋊Dic5—22no—
150C4⋊Dic5—21no—
151C4⋊Dic5—21no—
152C4⋊Dic5—42no—
153C4⋊Dic5—42no—
154#683—42no—
155#683—42no—
156#683—42no—
157#683—42no—
158C4×Dic5—44no—
159C4×Dic5—42no—
160C4×Dic5C4⋊C414yes—
161C4×Dic5C4⋊C414yes—
162C4×Dic5C22×C412yes—
163C4×Dic5C2×Q812yes—
164C2×#168—22no—
165C2×#168—22no—
166C2×#168C22×C414yes—
167C2×#117—52no—
168C2×#117—101no—
169C22×C42—51no—
170#432—58no—
171C2×#107—52no—
172C2×#107C2×D512yes—
173C2×#106—52no—
174C2×#106—52no—
175C2×C4⋊C8—51no—
176C2×C4⋊C8C2×D511yescharacteristic
177C2×C22⋊C8—52no—
178C2×#97—52no—
179#384—51no—
180#384—51no—
181#384—51no—
182#384C2×D511yescharacteristic
183#382—52no—
184#382—101no—
185#380—51no—
186#380—51no—
187#380—101no—
188#380—101no—
189#380—101no—
190C4×C4⋊C4—51no—
191C4×C4⋊C4—101no—
192C4×C4⋊C4—101no—
193C4×C4⋊C4—101no—
194C4×C4⋊C4C2×D511yescharacteristic
195#340—52no—
196#340—52no—
197#340—52no—
198#340C2×D512yes—
199C23×C5C22×D411yescharacteristic
200C22×D5—21no—
201C22×D5—21no—
202C22×D5—21no—
203C22×D5—21no—
204C22×D5—21no—
205C22×D5—21no—
206C22×D5C2×C4⋊C412yes—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—42no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5—41no—
213C2×C4×C5—41no—
214C2×C4×C5—42no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—42no—
218C2×C4×C5C2×C4⋊C412yes—
219C2×C4×C5C2×C4⋊C412yes—
220C2×C4×C5AGL(1,ℤ/8)12yes—
221C2×C4×C5#13712yes—
222C2×C4×C5C23×C411yescharacteristic
223C2×C4×C5C22×Q811yescharacteristic
224C2×C4×C5H(5,𝔽2)11yescharacteristic
225C2×C4×C5H(5,𝔽2)11yescharacteristic
226C2×C4×C52−1+411yescharacteristic
227C2×C4×C52−1+411yescharacteristic
228C2×Dic5—21no—
229C2×Dic5—21no—
230C2×Dic5—21no—
231C2×Dic5—21no—
232C2×Dic5—21no—
233C2×Dic5—21no—
234C2×Dic5—42no—
235C2×Dic5—41no—
236C2×Dic5—41no—
237C2×Dic5—41no—
238C2×Dic5—41no—
239C2×Dic5—42no—
240C2×Dic5—41no—
241C2×Dic5—41no—
242C2×Dic5—42no—
243C2×Dic5C2×C4⋊C412yes—
244C2×Dic5C2×C4⋊C412yes—
245C2×Dic5C2×C4⋊C412yes—
246C4×D5—22no—
247C4×D5—22no—
248C4×D5—22no—
249C4×D5—22no—
250C4×D5—28no—
251C4×D5—44no—
252C5×C8—24no—
253#168—24no—
254C23×C4—51no—
255C23×C4—51no—
256C23×C4—51no—
257C22×C8—52no—
258C22×C8C22×D512yes—
259#117—204no—
260#117—202no—
261#117—202no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—52no—
265C2×C4⋊C4—51no—
266C2×C4⋊C4—51no—
267C2×C4⋊C4—101no—
268C2×C4⋊C4—101no—
269C2×C4⋊C4—101no—
270C2×C4⋊C4—101no—
271C2×C4⋊C4—101no—
272C2×C4⋊C4—101no—
273C2×C4⋊C4C22×D512yes—
274C2×C4⋊C4C22×D511yescharacteristic
275C2×C4⋊C4C22×D511yescharacteristic
276C2×C22⋊C4—102no—
277C2×C22⋊C4—101no—
278C2×C22⋊C4—101no—
279C2×C42—21no—
280C2×C42—51no—
281C2×C42—51no—
282C2×C42—51no—
283C2×C42—54no—
284C2×C42—54no—
285C2×C42—102no—
286C2×C42—101no—
287C2×C42C22×D511yescharacteristic
288#107—42no—
289#107—202no—
290#106—202no—
291#106—202no—
292C4⋊C8—54no—
293C4⋊C8C4×D514yes—
294C22⋊C8—58no—
295#97—58no—
296#95—21no—
297#95—101no—
298#95—101no—
299#95—102no—
300#95—102no—
301#95—101no—
302C22×C5C22×C4⋊C411yescharacteristic
303C22×C5#53511yescharacteristic
304C22×C5#53611yescharacteristic
305C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C5C2×#13711yescharacteristic
307C22×C5#57511yescharacteristic
308C22×C5#57511yescharacteristic
309C2×D5—22no—
310C2×D5—22no—
311C2×D5—22no—
312C2×D5—41no—
313C2×D5—41no—
314C2×D5—41no—
315C2×D5—41no—
316C4×C5—22no—
317C4×C5—82no—
318C4×C5—81no—
319C4×C5—81no—
320C4×C5#42814yes—
321C4×C5#52011yescharacteristic
322C4×C5#52011yescharacteristic
323C4×C5#54411yescharacteristic
324C4×C5#54411yescharacteristic
325Dic5—22no—
326Dic5—22no—
327Dic5—22no—
328Dic5—81no—
329Dic5—82no—
330Dic5—81no—
331C24—51no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—51no—
336C22×C4—51no—
337C22×C4—51no—
338C22×C4—52no—
339C22×C4—52no—
340C22×C4—54no—
341C22×C4—54no—
342C22×C4—101no—
343C22×C4—101no—
344C22×C4—101no—
345C22×C4—101no—
346C22×C4—102no—
347C22×C4—102no—
348C22×C4—101no—
349C22×C4—101no—
350C22×C4—102no—
351C22×C4—201no—
352C22×C4—201no—
353C22×C4D4×D511yescharacteristic
354C22×C4D4×D511yescharacteristic
355C22×C4C23×D511yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—54no—
359C2×C8—102no—
360C2×C8—102no—
361C2×C8C2×C4×D514yes—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—101no—
371C4⋊C4—102no—
372C4⋊C4—101no—
373C4⋊C4—102no—
374C4⋊C4—101no—
375C4⋊C4—101no—
376C4⋊C4—202no—
377C4⋊C4—202no—
378C4⋊C4—202no—
379C4⋊C4—202no—
380C4⋊C4—202no—
381C4⋊C4—202no—
382C22⋊C4—202no—
383C22⋊C4—202no—
384C22⋊C4—204no—
385C42—42no—
386C42—52no—
387C42—52no—
388C42—54no—
389C42—54no—
390C42—204no—
391C42—202no—
392C42C2×C4×D512yes—
393C42Q8×D512yes—
394C2×C5#231711yescharacteristic
395C2×C5C2×#42811yescharacteristic
396C2×C5#291111yescharacteristic
397C2×C5#321411yescharacteristic
398C2×C5#321611yescharacteristic
399C2×C5#323511yescharacteristic
400C2×C5#323711yescharacteristic
401D5—42no—
402C23—52no—
403C23—101no—
404C23—101no—
405C23—101no—
406C23—101no—
407C23—101no—
408C23—101no—
409C23C2×D4×D511yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—52no—
422C2×C4—52no—
423C2×C4—52no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—102no—
431C2×C4—102no—
432C2×C4—102no—
433C2×C4—108no—
434C2×C4—102no—
435C2×C4—204no—
436C2×C4—201no—
437C2×C4—201no—
438C2×C4—202no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—202no—
442C2×C4—202no—
443C2×C4—201no—
444C2×C4—201no—
445C2×C4D5×C4⋊C412yes—
446C2×C4D5×C4⋊C412yes—
447C2×C4#405712yes—
448C2×C4#406412yes—
449C2×C4C22×C4×D511yescharacteristic
450C2×C4#414411yescharacteristic
451C2×C4C2×Q8×D511yescharacteristic
452C2×C4#414711yescharacteristic
453C2×C4#414911yescharacteristic
454C2×C4#415011yescharacteristic
455C8—24no—
456C8—104no—
457C5#2198811yescharacteristic, complemented, Sylow
458C22—102no—
459C22—102no—
460C22—102no—
461C22—201no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22C2×D5×C4⋊C411yescharacteristic
466C22#6594111yescharacteristic
467C22#6596211yescharacteristic
468C22C2×#405711yescharacteristic
469C22C2×#406411yescharacteristic
470C22#6610711yescharacteristic
471C22#6611111yescharacteristic
472C4—22no—
473C4—81no—
474C4—82no—
475C4—81no—
476C4—102no—
477C4—102no—
478C4—102no—
479C4—401no—
480C4—401no—
481C4—402no—
482C4#6517014yes—
483C4#6588011yescharacteristic
484C4#6590811yescharacteristic
485C4#6602311yescharacteristic
486C4#6606311yescharacteristic
487C2—202no—
488C2#1060769911yescharacteristic
489C2C2×#6517011yescharacteristic
490C2#1061093911yescharacteristic
491C2#1061206911yescharacteristic
492C2#1061221311yescharacteristic
493C2#1061270211yescharacteristic
494C2#1061284511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1947C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1947D5
hypocenterC5#21988
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. C5×#1947C2semidirect
6. #10601401C2semidirect
7. #10601247C2semidirect
8. #10600360C2semidirect
9. #10600260C2semidirect
11. #10599642C2semidirect
37. #65124C4semidirect
99. #1947D5semidirect
457. C5#21988semidirect

Maximal subgroups

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

subgroupindex
C2×#651242
#106060492
#106054842
C5×#19472
#106014262
#106014012
#106012472
#106003602
#106002602
#105996692
#105996422
#219885