Finite Group Databasebeta

← #12109860

Subgroups — #12109860

Order 1280.

902 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64994C211yesmaximal, characteristic
2D5×#384C211yesmaximal, characteristic
3#10605878C211yesmaximal, characteristic
4C5×#1847C211yesmaximal, characteristic, complemented
5#10601425C212yesmaximal
6#10601408C212yesmaximal, complemented
7#10601372C212yesmaximal
8#10601356C212yesmaximal, complemented
9#10600608C211yesmaximal, characteristic, complemented
10#10600368C211yesmaximal, characteristic, complemented
11#10600343C211yesmaximal, characteristic, complemented
12C2×D5×C4⋊C4—21no—
13C2×D5×C4⋊C4—21no—
14C2×D5×C4⋊C4C2212yes—
15C2×C42×D5C2211yescharacteristic
16C2×C5×#107C2212yes—
17C2×C5×#106C2212yes—
18C2×C5×#97C2211yescharacteristic
19C5×#384C2211yescharacteristic
20C5×#382C2211yescharacteristic
21C2×#3952C2212yes—
22C2×#3946C2212yes—
23#65276C2211yescharacteristic
24#65276C2211yescharacteristic
25#65265—21no—
26#65265—21no—
27#65265C2212yes—
28C4⋊C4×Dic5C2211yescharacteristic
29C4⋊C4×Dic5C2211yescharacteristic
30C2×#3940C2212yes—
31C2×#3939C2212yes—
32#65225C2211yescharacteristic
33#65224C2211yescharacteristic
34C2×#3935C2211yescharacteristic
35#64994C418yescomplemented
36#64955—21no—
37#64953—21no—
38#64940—21no—
39#64937—21no—
40#64771C2212yes—
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×C5C2311yescharacteristic
56C2×#685—21no—
57C2×#685—21no—
58C2×C4⋊Dic5—21no—
59C2×C4⋊Dic5—21no—
60C2×C4⋊Dic5C2312yes—
61C2×C4⋊Dic5C2311yescharacteristic
62C2×C4⋊Dic5C2311yescharacteristic
63C2×#683—22no—
64C2×#683—22no—
65C2×#683—22no—
66C2×C4×Dic5—21no—
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×#107—42no—
79C5×#106—42no—
80C5×#97C2×C414yescomplemented
81C5×#95—21no—
82C5×#95—21no—
83#3963—21no—
84#3963—21no—
85#3963—21no—
86#3963—21no—
87#3963—21no—
88#3963—21no—
89#3952C2×C418yes—
90#3946C2×C418yescomplemented
91#3940—42no—
92#3939—42no—
93#3935C2×C414yescomplemented
94C2×#432—51no—
95C2×#384—51no—
96#2283—51no—
97#1948—52no—
98#1947—52no—
99#1853—52no—
100#1852—52no—
101#1847—51no—
102#1847D511yescharacteristic, complemented
103#1836—51no—
104#1836—51no—
105C23×D5C2×D411yescharacteristic
106C22×C4×C5—22no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
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—22no—
123C2×C4×D5—22no—
124C2×C4×D5—44no—
125C2×C4×D5—44no—
126C2×C4×D5—44no—
127C2×C4×D5C4⋊C414yes—
128C2×C4×D5C4⋊C414yes—
129C2×C4×D5C22×C412yes—
130C2×C4×D5C2×Q812yes—
131C2×C5×C8—22no—
132C2×C5×C8—22no—
133C2×C5×C8C22×C414yes—
134C5×C4⋊C4—22no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—42no—
139C5×C4⋊C4—42no—
140C42×C5C22×C412yes—
141C42×C5C2×Q812yes—
142#685—42no—
143#685—42no—
144C4⋊Dic5—22no—
145C4⋊Dic5—22no—
146C4⋊Dic5—22no—
147C4⋊Dic5—22no—
148C4⋊Dic5—42no—
149C4⋊Dic5—42no—
150#683—44no—
151#683—44no—
152#683—44no—
153C4×Dic5—42no—
154C4×Dic5—42no—
155C4×Dic5C4⋊C414yes—
156C4×Dic5C4⋊C414yes—
157C4×Dic5C22×C412yes—
158C4×Dic5C2×Q812yes—
159C2×#168—22no—
160C2×#168—22no—
161C2×#168C22×C414yes—
162C22×C4⋊C4—52no—
163C22×C4⋊C4—101no—
164C22×C4⋊C4—101no—
165C22×C42—51no—
166#432—58no—
167C2×#107—52no—
168C2×#107C2×D512yes—
169C2×#106—52no—
170C2×#106C2×D512yes—
171C2×C4⋊C8—52no—
172C2×C22⋊C8—52no—
173C2×#97—51no—
174C2×#97C2×D511yescharacteristic
175#384—52no—
176#384—51no—
177#384—101no—
178#384—101no—
179#384C2×D511yescharacteristic
180#383—101no—
181#382—51no—
182#382C2×D511yescharacteristic
183#381—101no—
184#380—51no—
185#380—51no—
186C4×C4⋊C4—51no—
187C4×C4⋊C4—51no—
188C4×C22⋊C4—101no—
189#376—101no—
190#340—52no—
191#340—52no—
192#340—52no—
193#340—52no—
194C23×C5C22×D411yescharacteristic
195C22×D5—21no—
196C22×D5—21no—
197C22×D5—21no—
198C22×D5—21no—
199C22×D5C2×C4⋊C412yes—
200C22×D5AGL(1,ℤ/8)12yes—
201C22×D5#13712yes—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5—21no—
205C2×C4×C5—21no—
206C2×C4×C5—42no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—42no—
211C2×C4×C5—42no—
212C2×C4×C5—41no—
213C2×C4×C5—41no—
214C2×C4×C5—41no—
215C2×C4×C5C2×C4⋊C412yes—
216C2×C4×C5C2×C4⋊C412yes—
217C2×C4×C5C23×C411yescharacteristic
218C2×C4×C5C22×Q811yescharacteristic
219C2×C4×C5H(5,𝔽2)12yes—
220C2×C4×C52−1+412yes—
221C2×Dic5—21no—
222C2×Dic5—21no—
223C2×Dic5—21no—
224C2×Dic5—21no—
225C2×Dic5—42no—
226C2×Dic5—41no—
227C2×Dic5—41no—
228C2×Dic5—41no—
229C2×Dic5—42no—
230C2×Dic5—42no—
231C2×Dic5—41no—
232C2×Dic5—41no—
233C2×Dic5—41no—
234C2×Dic5C2×C4⋊C412yes—
235C2×Dic5C2×C4⋊C412yes—
236C2×Dic5C2×C4⋊C412yes—
237C2×Dic5AGL(1,ℤ/8)12yes—
238C2×Dic5#13712yes—
239C4×D5—28no—
240C4×D5—24no—
241C4×D5—24no—
242C4×D5—28no—
243C4×D5—88no—
244C5×C8—24no—
245#168—24no—
246C23×C4—51no—
247C23×C4—51no—
248C23×C4—51no—
249C23×C4—102no—
250C22×C8—52no—
251C22×C8C22×D512yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—52no—
255C2×C4⋊C4—51no—
256C2×C4⋊C4—51no—
257C2×C4⋊C4—102no—
258C2×C4⋊C4—102no—
259C2×C4⋊C4—102no—
260C2×C4⋊C4—101no—
261C2×C4⋊C4—101no—
262C2×C4⋊C4—202no—
263C2×C4⋊C4—202no—
264C2×C4⋊C4—204no—
265C2×C4⋊C4—204no—
266C2×C4⋊C4C22×D512yes—
267C2×C4⋊C4C22×D511yescharacteristic
268C2×C4⋊C4C22×D511yescharacteristic
269C2×C22⋊C4—101no—
270C2×C22⋊C4—101no—
271C2×C42—51no—
272C2×C42—51no—
273C2×C42—51no—
274C2×C42—54no—
275C2×C42—54no—
276C2×C42—101no—
277C2×C42—101no—
278C2×C42C22×D511yescharacteristic
279#107—42no—
280#107—202no—
281#106—42no—
282#106—202no—
283C4⋊C8—58no—
284C22⋊C8—58no—
285#97—54no—
286#97C4×D514yescomplemented
287#95—21no—
288#95—21no—
289#95—101no—
290#95—101no—
291#95—101no—
292#95—101no—
293#95—101no—
294#95—101no—
295C22×C5C22×C4⋊C411yescharacteristic
296C22×C5#53511yescharacteristic
297C22×C5#53611yescharacteristic
298C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C5C2×#13711yescharacteristic
301C22×C5C2×#13711yescharacteristic
302C2×D5—22no—
303C2×D5—22no—
304C2×D5—22no—
305C2×D5—22no—
306C2×D5—22no—
307C2×D5—22no—
308C2×D5#42814yes—
309C4×C5—22no—
310C4×C5—22no—
311C4×C5—82no—
312C4×C5—81no—
313C4×C5—81no—
314C4×C5#52012yes—
315C4×C5#54412yes—
316Dic5—22no—
317Dic5—22no—
318Dic5—82no—
319Dic5—81no—
320Dic5—81no—
321Dic5#42814yes—
322C24—51no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—52no—
330C22×C4—52no—
331C22×C4—54no—
332C22×C4—54no—
333C22×C4—102no—
334C22×C4—102no—
335C22×C4—102no—
336C22×C4—102no—
337C22×C4—102no—
338C22×C4—102no—
339C22×C4—101no—
340C22×C4—101no—
341C22×C4—102no—
342C22×C4—204no—
343C22×C4—204no—
344C22×C4—204no—
345C22×C4D4×D511yescharacteristic
346C22×C4D4×D511yescharacteristic
347C22×C4C23×D511yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—54no—
351C2×C8—102no—
352C2×C8—102no—
353C2×C8C2×C4×D514yes—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—102no—
361C4⋊C4—102no—
362C4⋊C4—102no—
363C4⋊C4—102no—
364C4⋊C4—202no—
365C4⋊C4—202no—
366C4⋊C4—204no—
367C4⋊C4—204no—
368C4⋊C4—204no—
369C22⋊C4—202no—
370C22⋊C4—202no—
371C42—52no—
372C42—52no—
373C42—54no—
374C42—54no—
375C42—202no—
376C42—202no—
377C42C2×C4×D512yes—
378C42Q8×D512yes—
379C2×C5#231711yescharacteristic
380C2×C5C2×#42811yescharacteristic
381C2×C5C2×#42811yescharacteristic
382C2×C5#321711yescharacteristic
383C2×C5#321811yescharacteristic
384C2×C5#323811yescharacteristic
385C2×C5#323911yescharacteristic
386D5—24no—
387C23—52no—
388C23—52no—
389C23—52no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23C2×D4×D511yescharacteristic
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—52no—
409C2×C4—52no—
410C2×C4—52no—
411C2×C4—52no—
412C2×C4—52no—
413C2×C4—108no—
414C2×C4—108no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—101no—
419C2×C4—104no—
420C2×C4—104no—
421C2×C4—202no—
422C2×C4—201no—
423C2×C4—201no—
424C2×C4—201no—
425C2×C4—201no—
426C2×C4—202no—
427C2×C4—202no—
428C2×C4—201no—
429C2×C4—201no—
430C2×C4—408no—
431C2×C4D5×C4⋊C412yes—
432C2×C4D5×C4⋊C412yes—
433C2×C4C22×C4×D511yescharacteristic
434C2×C4#414412yes—
435C2×C4C2×Q8×D511yescharacteristic
436C2×C4#414712yes—
437C8—24no—
438C8—104no—
439C5#2198711yescharacteristic, complemented, Sylow
440C22—54no—
441C22—102no—
442C22—102no—
443C22—102no—
444C22—102no—
445C22—102no—
446C22—102no—
447C22C2×D5×C4⋊C411yescharacteristic
448C22#6604911yescharacteristic
449C22#6605811yescharacteristic
450C22D5×AGL(1,ℤ/8)11yescharacteristic
451C22D5×AGL(1,ℤ/8)11yescharacteristic
452C22D5×#13711yescharacteristic
453C22D5×#13711yescharacteristic
454C4—22no—
455C4—22no—
456C4—54no—
457C4—82no—
458C4—81no—
459C4—81no—
460C4—102no—
461C4—102no—
462C4—402no—
463C4—401no—
464C4—401no—
465C4#6583812yes—
466C4#6606312yes—
467C2—104no—
468C2#1060609011yescharacteristic
469C2D5×#42811yescharacteristic
470C2D5×#42811yescharacteristic
471C2#1061358211yescharacteristic
472C2#1061360711yescharacteristic
473C2#1061371311yescharacteristic
474C2#1061371411yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1847C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1847D5
hypocenterC5#21987
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. C5×#1847C2semidirect
6. #10601408C2semidirect
8. #10601356C2semidirect
9. #10600608C2semidirect
10. #10600368C2semidirect
11. #10600343C2semidirect
35. #64994C4semidirect
80. C5×#97C2×C4semidirect
90. #3946C2×C4semidirect
93. #3935C2×C4semidirect
102. #1847D5semidirect
286. #97C4×D5semidirect
439. C5#21987semidirect

Maximal subgroups

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

subgroupindex
C2×#649942
D5×#3842
#106058782
C5×#18472
#106014252
#106014082
#106013722
#106013562
#106006082
#106003682
#106003432
#219875