Finite Group Databasebeta

← #12069421

Subgroups — #12069421

Order 1280.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3100C211yesmaximal, characteristic
2#10615011C211yesmaximal, characteristic, complemented
3#10614032C211yesmaximal, characteristic, complemented
4#10602594C211yesmaximal, characteristic, complemented
5#10602392C211yesmaximal, characteristic, complemented
6#10601673C212yesmaximal, complemented
7#10601672C212yesmaximal, complemented
8#10601671C212yesmaximal, complemented
9D8×#168C212yesmaximal, complemented
10#10601544C211yesmaximal, characteristic, complemented
11#10601536C211yesmaximal, characteristic, complemented
12C5×#532C2211yescharacteristic
13C2×C4×C5×D4C2211yescharacteristic
14C5×#480C412yes—
15C5×#474C412yes—
16C5×#459C412yes—
17C5×#447C412yes—
18C4×C5×D8C2212yes—
19C4×C5×D8C2212yes—
20C5×#431C2211yescharacteristic
21#65523—22no—
22C2×C22⋊#168—22no—
23#65301C2212yescomplemented
24#65300—22no—
25D4×#168—22no—
26D4×#168C2212yes—
27D4×#168C2212yes—
28D4×#168C2212yescomplemented
29#65215C2211yescharacteristic
30#65214C2211yescharacteristic
31#65214C2211yescharacteristic
32C4×#681C2211yescharacteristic
33D4⋊#168C2212yes—
34D4⋊#168C2212yes—
35D4⋊#168C2212yescomplemented
36D4⋊#168C2212yescomplemented
37#64678C2212yescomplemented
38#64677C2212yescomplemented
39#64675C2211yescharacteristic, complemented
40#64674C2211yescharacteristic, complemented
41#64674C2211yescharacteristic, complemented
42C8×#168C2211yescharacteristic, complemented
43#17922—51nomaximal, Sylow
44C2×C5×#41—21no—
45C22×C5×D4C2×C411yescharacteristic
46C23×C4×C5—21no—
47C2×C5×D8C2×C412yes—
48C2×C5×D8C2×C412yes—
49C5×C4⋊Q8C2×C411yescharacteristic
50C5×#127C2×C411yescharacteristic
51C5×#124—21no—
52C5×#121—21no—
53C5×#121C2×C412yes—
54C4×C5×D4—21no—
55C4×C5×D4—21no—
56C4×C5×D4—21no—
57C4×C5×D4C2×C412yes—
58C4×C5×D4C2312yes—
59C4×C5×D4C2312yes—
60C2×C5×C4⋊C4C2×C411yescharacteristic
61C2×C5×C22⋊C4—21no—
62C2×C42×C5C2311yescharacteristic
63C2×#681—21no—
64C2×#681—22no—
65C22×#168—22no—
66C22×#168—22no—
67C22×#168—22no—
68C5×#107C2×C412yes—
69C5×#107C2×C412yes—
70C5×C4⋊C8C2×C412yes—
71C5×#102C2×C412yes—
72C5×#102C2×C412yes—
73C5×#102C2×C412yes—
74C5×#102C2×C412yes—
75C5×C22⋊C8C2×C412yes—
76C4×C5×C8C2311yescharacteristic
77C4×C5×C8C2311yescharacteristic
78C22⋊#168—21no—
79C22⋊#168—21no—
80C22⋊#168—22no—
81C22⋊#168—22no—
82C22⋊#168—22no—
83C22⋊#168—22no—
84C22⋊#168D412yes—
85C4⋊#168D412yes—
86C4⋊#168C2312yes—
87C4⋊#168C2312yes—
88#3935—21no—
89#3935—21no—
90C4×#168—21no—
91C4×#168C2311yescharacteristic
92C4×#168C2311yescharacteristic
93C4×#168C2311yescharacteristic
94C4×#168C2311yescharacteristic
95#3100D511yescharacteristic
96#2981—51no—
97#2972—51no—
98#1586—52no—
99#1581—52no—
100#1580—52no—
101C8×D8—52no—
102#1572—51no—
103#1490—51no—
104#1484—51no—
105#1453—51no—
106C24×C5—21no—
107C5×#41—82no—
108C2×C5×Q8—21no—
109C2×C5×D4—24no—
110C2×C5×D4—22no—
111C2×C5×D4—21no—
112C2×C5×D4—21no—
113C2×C5×D4—21no—
114C2×C5×D4C22×C412yes—
115C2×C5×D4C22×C412yes—
116C22×C4×C5—21no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—82no—
121C22×C4×C5C22×C411yescharacteristic
122C22×C4×C5C2×D411yescharacteristic
123C22×C4×C5#4111yescharacteristic
124C5×D8—24no—
125C5×D8—24no—
126C2×C5×C8C22×C412yes—
127C2×C5×C8C22×C412yes—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C22×C412yes—
131C5×C4⋊C4C22×C412yes—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—22no—
134C5×C22⋊C4—22no—
135C5×C22⋊C4—22no—
136C42×C5C22×C411yescharacteristic
137C42×C5C2×D411yescharacteristic
138C42×C5#4111yescharacteristic
139C42×C5C2411yescharacteristic
140#681—42no—
141#681—82no—
142C2×#168—21no—
143C2×#168—22no—
144C2×#168—22no—
145C2×#168—21no—
146C2×#168—82no—
147C2×#168—82no—
148C2×#168—82no—
149C2×#168C2×D412yes—
150C2×#168C2×D412yes—
151#532C2×D511yescharacteristic
152C2×C4×D4C2×D511yescharacteristic
153#480Dic512yes—
154#474Dic512yes—
155#459Dic512yes—
156#447Dic512yes—
157C4×D8C2×D512yes—
158C4×D8C2×D512yes—
159#436—52no—
160#435—102no—
161C8×D4—52no—
162C8×D4—52no—
163C8×D4—52no—
164C8×D4—102no—
165#432—51no—
166#431—51no—
167#431—51no—
168#431C2×D511yescharacteristic
169#408—102no—
170C2×C22⋊C8—102no—
171C4×M16—51no—
172#335—52no—
173#334—52no—
174D4⋊C8—52no—
175D4⋊C8—52no—
176D4⋊C8—52no—
177D4⋊C8—52no—
178#322—51no—
179#322—51no—
180#322—51no—
181C82—51no—
182C23×C5—24no—
183C23×C5—22no—
184C23×C5—22no—
185C23×C5—81no—
186C23×C5—81no—
187C23×C5C4×D411yescharacteristic
188C5×Q8—81no—
189C5×D4—24no—
190C5×D4—22no—
191C5×D4—22no—
192C5×D4—22no—
193C5×D4—81no—
194C5×D4—81no—
195C5×D4—82no—
196C5×D4#13114yes—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—22no—
200C2×C4×C5—22no—
201C2×C4×C5—41no—
202C2×C4×C5—42no—
203C2×C4×C5—42no—
204C2×C4×C5—42no—
205C2×C4×C5—42no—
206C2×C4×C5—82no—
207C2×C4×C5—81no—
208C2×C4×C5C4×D411yescharacteristic
209C2×C4×C5C4×D411yescharacteristic
210C2×C4×C5C4×D411yescharacteristic
211C2×C4×C5C23×C411yescharacteristic
212C2×C4×C5C22×D411yescharacteristic
213C2×C4×C5C2×#4111yescharacteristic
214C5×C8—22no—
215C5×C8—22no—
216#168—22no—
217#168—22no—
218#168—22no—
219#168—41no—
220#168—42no—
221#168—42no—
222C2×#41—21no—
223C22×D4C2×Dic511yescharacteristic
224C23×C4—21no—
225C2×D8C2×Dic512yes—
226C2×D8C2×Dic512yes—
227C2×M16—102no—
228C2×M16—101no—
229C22×C8—102no—
230C22×C8—102no—
231C22×C8—102no—
232C4⋊Q8C2×Dic511yescharacteristic
233#127C2×Dic511yescharacteristic
234#124—21no—
235#121—21no—
236#121C2×Dic512yes—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4C2×Dic512yes—
241C4×D4C22×D512yes—
242C4×D4C22×D512yes—
243C2×C4⋊C4C2×Dic511yescharacteristic
244C2×C22⋊C4—21no—
245C2×C42C22×D511yescharacteristic
246#107C2×Dic512yes—
247#107C2×Dic512yes—
248C4⋊C8—52no—
249C4⋊C8—52no—
250C4⋊C8—52no—
251C4⋊C8C2×Dic512yes—
252#102C2×Dic512yes—
253#102C2×Dic512yes—
254#102C2×Dic512yes—
255#102C2×Dic512yes—
256C22⋊C8—52no—
257C22⋊C8—102no—
258C22⋊C8—102no—
259C22⋊C8—102no—
260C22⋊C8—102no—
261C22⋊C8—101no—
262C22⋊C8—101no—
263C22⋊C8C2×Dic512yes—
264#97—101no—
265#97—101no—
266C4×C8—51no—
267C4×C8—51no—
268C4×C8—51no—
269C4×C8—51no—
270C4×C8—101no—
271C4×C8C22×D511yescharacteristic
272C4×C8C22×D511yescharacteristic
273C22×C5—21no—
274C22×C5—24no—
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—42no—
278C22×C5—42no—
279C22×C5—42no—
280C22×C5—42no—
281C22×C5—42no—
282C22×C5—84no—
283C22×C5C2×C4×D411yescharacteristic
284C4×C5—21no—
285C4×C5—21no—
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—42no—
289C4×C5—42no—
290C4×C5C2×#13112yes—
291C4×C5#57612yes—
292C24—21no—
293#41—82no—
294C2×Q8—21no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C22×Dic512yes—
301C2×D4C22×Dic512yes—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—82no—
307C22×C4D4×D511yescharacteristic
308C22×C4#71111yescharacteristic
309C22×C4C22×Dic511yescharacteristic
310D8—24no—
311D8—24no—
312M16—202no—
313M16—402no—
314C2×C8—52no—
315C2×C8—52no—
316C2×C8—101no—
317C2×C8—102no—
318C2×C8—102no—
319C2×C8—101no—
320C2×C8—402no—
321C2×C8—402no—
322C2×C8—402no—
323C2×C8C22×Dic512yes—
324C2×C8C22×Dic512yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4C22×Dic512yes—
328C4⋊C4C22×Dic512yes—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C42D4×D511yescharacteristic
334C42#71111yescharacteristic
335C42C22×Dic511yescharacteristic
336C42C23×D511yescharacteristic
337C2×C5—41no—
338C2×C5—44no—
339C2×C5—82no—
340C2×C5#292911yescharacteristic
341C2×C5#294811yescharacteristic
342C2×C5#295611yescharacteristic
343C23—24no—
344C23—22no—
345C23—22no—
346C23—81no—
347C23—81no—
348C23D4×Dic511yescharacteristic
349Q8—81no—
350D4—24no—
351D4—22no—
352D4—22no—
353D4—22no—
354D4—81no—
355D4—81no—
356D4—82no—
357D4#409414yes—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4D4×Dic511yescharacteristic
370C2×C4D4×Dic511yescharacteristic
371C2×C4D4×Dic511yescharacteristic
372C2×C4C2×D4×D511yescharacteristic
373C2×C4C2×#71111yescharacteristic
374C2×C4C23×Dic511yescharacteristic
375C8—22no—
376C8—22no—
377C8—102no—
378C8—102no—
379C8—102no—
380C8—201no—
381C8—202no—
382C8—202no—
383C5#1792211yescharacteristic, complemented, Sylow
384C22—21no—
385C22—21no—
386C22—24no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—84no—
394C22C2×D4×Dic511yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—42no—
401C4#6609112yes—
402C4C2×#409412yes—
403C2—41no—
404C2—44no—
405C2—82no—
406C2#1061464711yescharacteristic
407C2#1061555411yescharacteristic
408C2#1061573111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
Fitting, prime radicalC5×#3100C2
socleC22×C5C2×C4×D4
cosocleC42×C5C24
hypercenter#3100D5
hypocenterC5#17922
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #10615011C2semidirect
3. #10614032C2semidirect
4. #10602594C2semidirect
5. #10602392C2semidirect
6. #10601673C2semidirect
7. #10601672C2semidirect
8. #10601671C2semidirect
9. D8×#168C2semidirect
10. #10601544C2semidirect
11. #10601536C2semidirect
23. #65301C22semidirect
28. D4×#168C22semidirect
35. D4⋊#168C22semidirect
36. D4⋊#168C22semidirect
37. #64678C22semidirect
38. #64677C22semidirect
39. #64675C22semidirect
40. #64674C22semidirect
41. #64674C22semidirect
42. C8×#168C22semidirect
383. C5#17922semidirect

Maximal subgroups

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

subgroupindex
C5×#31002
#106150112
#106140322
#106025942
#106023922
#106016732
#106016722
#106016712
D8×#1682
#106015442
#106015362
#179225