Finite Group Databasebeta

← #12068541

Subgroups — #12068541

Order 1280.

610 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2925C211yesmaximal, characteristic
2#41×#168C211yesmaximal, characteristic
3#10614246C211yesmaximal, characteristic
4#10601589C212yesmaximal, complemented
5#10601587C212yesmaximal, complemented
6#10601540C212yesmaximal, complemented
7#10601538C212yesmaximal, complemented
8#10601286C211yesmaximal, characteristic, complemented
9#10601284C211yesmaximal, characteristic, complemented
10#10601214C211yesmaximal, characteristic, complemented
11C2×#64699C211yesmaximal, characteristic, complemented
12C4×C5×#41C2211yescharacteristic
13C5×#433—22no—
14C5×#432C2212yes—
15C5×#431C2212yes—
16C5×#423C2211yescharacteristic
17C2×C5×C4⋊C8C2211yescharacteristic
18C5×#408—21no—
19C5×#405—21no—
20#65523—21no—
21Q8×#168C411yescharacteristic
22Q8×#168C411yescharacteristic
23D4×#168C412yes—
24D4×#168C411yescharacteristic
25D4×#168C412yes—
26D4×#168C411yescharacteristic
27#65289—22no—
28#65286—21no—
29#65215C2212yes—
30#65214C2212yes—
31#65214C2212yes—
32#65214C2211yescharacteristic
33#65213C2211yescharacteristic
34C2×C4⋊#168C2211yescharacteristic
35C2×C4×#168C2212yes—
36C2×C4×#168C2211yescharacteristic
37#64700C2212yes—
38#64700C2212yes—
39#64700C2212yes—
40#64700C2212yes—
41#64699C2211yescharacteristic
42#64699C2212yes—
43#64699C2212yes—
44#64699C2211yescharacteristic
45#64699C2211yescharacteristic
46#64699C2211yescharacteristic
47#17829—51nomaximal, Sylow
48C2×C5×#41D411yescharacteristic
49C2×C5×M16—21no—
50C22×C5×C8—21no—
51C4×C5×Q8C2×C411yescharacteristic
52C4×C5×Q8Q811yescharacteristic
53C4×C5×D4C2×C412yes—
54C4×C5×D4C2×C411yescharacteristic
55C4×C5×D4Q812yes—
56C4×C5×D4Q811yescharacteristic
57C5×#117D411yescharacteristic
58C5×#117D412yes—
59C2×C42×C5C2311yescharacteristic
60C2×C42×C5C2312yes—
61C2×#681—21no—
62C22×#168—21no—
63C22×#168C2×C412yes—
64C22×#168C2×C411yescharacteristic
65C22×#168C2×C412yes—
66C22×#168C2×C411yescharacteristic
67C5×C4⋊C8—21no—
68C5×C4⋊C8—21no—
69C5×C4⋊C8C2311yescharacteristic
70C5×C4⋊C8C2311yescharacteristic
71C5×C4⋊C8C2312yes—
72C5×C22⋊C8—22no—
73C5×C22⋊C8—22no—
74C5×#97—22no—
75C4×C5×C8—22no—
76C22⋊#168—22no—
77C22⋊#168—22no—
78C22⋊#168C2×C412yes—
79C22⋊#168C2×C412yes—
80C22⋊#168C2×C412yes—
81C4⋊#168—21no—
82C4⋊#168—21no—
83C4⋊#168C2×C412yes—
84C4⋊#168C2×C412yes—
85C4⋊#168C2×C411yescharacteristic
86C4⋊#168C2×C411yescharacteristic
87C4⋊#168C2311yescharacteristic
88C4⋊#168C2311yescharacteristic
89C4⋊#168C2312yes—
90#3935—22no—
91C4×#168—22no—
92C4×#168C2×C412yes—
93C4×#168C2×C411yescharacteristic
94C4×#168C2×C412yes—
95C4×#168C2×C411yescharacteristic
96C4×#168C2311yescharacteristic
97C4×#168C2311yescharacteristic
98C4×#168C2312yes—
99C8×#41—51no—
100#2925—51no—
101#2925D511yescharacteristic
102#1572—52no—
103#1572—52no—
104#1570—51no—
105#1570—51no—
106#1568—52no—
107#1567—51no—
108#1564—52no—
109C2×#335—51no—
110C5×#41—22no—
111C5×#41—22no—
112C2×C5×Q8C4⋊C411yescharacteristic
113C2×C5×D4C4⋊C412yes—
114C2×C5×D4C4⋊C411yescharacteristic
115C22×C4×C5C22×C412yes—
116C22×C4×C5C22×C411yescharacteristic
117C22×C4×C5C2×D412yes—
118C22×C4×C5C2×D411yescharacteristic
119C22×C4×C5C2×Q812yes—
120C22×C4×C5C2×Q811yescharacteristic
121C5×M16—42no—
122C2×C5×C8—21no—
123C2×C5×C8—21no—
124C2×C5×C8—22no—
125C2×C5×C8—42no—
126C5×C4⋊C4C4⋊C412yes—
127C5×C4⋊C4C4⋊C411yescharacteristic
128C5×C4⋊C4C4⋊C412yes—
129C5×C4⋊C4C4⋊C411yescharacteristic
130C5×C22⋊C4C4⋊C412yes—
131C5×C22⋊C4C4⋊C412yes—
132C5×C22⋊C4C4⋊C412yes—
133C42×C5C22×C412yes—
134C42×C5C22×C411yescharacteristic
135C42×C5C2×D412yes—
136C42×C5C2×D411yescharacteristic
137C42×C5C2×Q812yes—
138C42×C5C2×Q811yescharacteristic
139C42×C5C2411yescharacteristic
140#681—42no—
141C2×#168—22no—
142C2×#168—24no—
143C2×#168—21no—
144C2×#168—22no—
145C2×#168—21no—
146C2×#168—41no—
147C2×#168—42no—
148C2×#168—42no—
149C2×#168C22×C411yescharacteristic
150C2×#168C22×C411yescharacteristic
151C2×#168C22×C412yes—
152C2×#168C22×C411yescharacteristic
153C2×#168C22×C412yes—
154C2×#168C22×C411yescharacteristic
155C4×#41C2×D511yescharacteristic
156C8×Q8—51no—
157C8×Q8—51no—
158C8×D4—51no—
159C8×D4—52no—
160C8×D4—51no—
161C8×D4—52no—
162#433—22no—
163#433—102no—
164#432—52no—
165#432C2×D512yes—
166#431—51no—
167#431—52no—
168#431—52no—
169#431C2×D512yes—
170#423—51no—
171#423C2×D511yescharacteristic
172C2×C4⋊C8—51no—
173C2×C4⋊C8C2×D511yescharacteristic
174#408—21no—
175#408—101no—
176#405—21no—
177#405—101no—
178C2×C4×C8—51no—
179C2×C4×C8—52no—
180#335—51no—
181#335—51no—
182#335—51no—
183#335—51no—
184#335—52no—
185#335—52no—
186#334—52no—
187#334—52no—
188#334—52no—
189#334—52no—
190C23×C5C2×C4⋊C412yes—
191C23×C5C2×C4⋊C411yescharacteristic
192C5×Q8—22no—
193C5×D4—24no—
194C5×D4—22no—
195C2×C4×C5—21no—
196C2×C4×C5—22no—
197C2×C4×C5—21no—
198C2×C4×C5—22no—
199C2×C4×C5—22no—
200C2×C4×C5—22no—
201C2×C4×C5—41no—
202C2×C4×C5—42no—
203C2×C4×C5C2×C4⋊C412yes—
204C2×C4×C5C2×C4⋊C411yescharacteristic
205C2×C4×C5C2×C4⋊C412yes—
206C2×C4×C5C2×C4⋊C412yes—
207C2×C4×C5C2×C4⋊C411yescharacteristic
208C2×C4×C5C2×C4⋊C411yescharacteristic
209C2×C4×C5#13512yes—
210C2×C4×C5#13512yes—
211C2×C4×C5C23×C411yescharacteristic
212C2×C4×C5C22×D411yescharacteristic
213C2×C4×C5C22×Q811yescharacteristic
214C5×C8—41no—
215C5×C8—41no—
216C5×C8—42no—
217#168—21no—
218#168—22no—
219#168—21no—
220#168—21no—
221#168—22no—
222#168—41no—
223#168—41no—
224#168—42no—
225#168#13112yes—
226C2×#41C22⋊D511yescharacteristic
227C2×M16—21no—
228C2×M16—101no—
229C22×C8—21no—
230C22×C8—51no—
231C22×C8—52no—
232C22×C8—51no—
233C22×C8—52no—
234C22×C8—101no—
235C4×Q8Dic1011yescharacteristic
236C4×Q8C4×D511yescharacteristic
237C4×D4Dic1012yes—
238C4×D4Dic1011yescharacteristic
239C4×D4C4×D512yes—
240C4×D4C4×D511yescharacteristic
241#117C22⋊D511yescharacteristic
242#117C22⋊D512yes—
243C2×C42C22×D511yescharacteristic
244C2×C42C22×D512yes—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—51no—
248C4⋊C8—51no—
249C4⋊C8—52no—
250C4⋊C8—52no—
251C4⋊C8—51no—
252C4⋊C8—51no—
253C4⋊C8—52no—
254C4⋊C8—101no—
255C4⋊C8—101no—
256C4⋊C8C22×D511yescharacteristic
257C4⋊C8C22×D511yescharacteristic
258C4⋊C8C22×D512yes—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—52no—
262C22⋊C8—52no—
263C22⋊C8—52no—
264C22⋊C8—102no—
265C22⋊C8—102no—
266#97—22no—
267#97—102no—
268C4×C8—22no—
269C4×C8—51no—
270C4×C8—52no—
271C4×C8—51no—
272C4×C8—52no—
273C4×C8—52no—
274C4×C8—51no—
275C4×C8—51no—
276C4×C8—102no—
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—22no—
280C22×C5—22no—
281C22×C5—22no—
282C22×C5#42712yes—
283C22×C5C22×C4⋊C411yescharacteristic
284C4×C5—21no—
285C4×C5—22no—
286C4×C5—22no—
287C4×C5—21no—
288C4×C5—21no—
289C4×C5—22no—
290C4×C5#42712yes—
291C4×C5C2×#13111yescharacteristic
292C4×C5#56811yescharacteristic
293C4×C5C2×#13511yescharacteristic
294C4×C5C2×#13511yescharacteristic
295#41—22no—
296#41—22no—
297C2×Q8#68311yescharacteristic
298C2×D4#68312yes—
299C2×D4#68311yescharacteristic
300C22×C4C2×Dic1012yes—
301C22×C4C2×Dic1011yescharacteristic
302C22×C4C2×C4×D512yes—
303C22×C4C2×C4×D511yescharacteristic
304C22×C4C2×C22⋊D512yes—
305C22×C4C2×C22⋊D511yescharacteristic
306M16—42no—
307M16—202no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—42no—
312C2×C8—51no—
313C2×C8—51no—
314C2×C8—52no—
315C2×C8—51no—
316C2×C8—52no—
317C2×C8—51no—
318C2×C8—102no—
319C2×C8—104no—
320C2×C8—101no—
321C2×C8—102no—
322C2×C8—101no—
323C2×C8—201no—
324C2×C8—202no—
325C2×C8—202no—
326C4⋊C4#68312yes—
327C4⋊C4#68311yescharacteristic
328C4⋊C4#68312yes—
329C4⋊C4#68311yescharacteristic
330C22⋊C4#68312yes—
331C22⋊C4#68312yes—
332C22⋊C4#68312yes—
333C42C2×Dic1012yes—
334C42C2×Dic1011yescharacteristic
335C42C2×C4×D512yes—
336C42C2×C4×D511yescharacteristic
337C42C2×C22⋊D512yes—
338C42C2×C22⋊D511yescharacteristic
339C42C23×D511yescharacteristic
340C2×C5—22no—
341C2×C5—42no—
342C2×C5#290611yescharacteristic
343C2×C5C2×#42711yescharacteristic
344C2×C5#291611yescharacteristic
345C23C2×#68312yes—
346C23C2×#68311yescharacteristic
347Q8—22no—
348D4—24no—
349D4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4C2×#68312yes—
359C2×C4C2×#68312yes—
360C2×C4C2×#68311yescharacteristic
361C2×C4C2×#68312yes—
362C2×C4C2×#68311yescharacteristic
363C2×C4C2×#68311yescharacteristic
364C2×C4#409612yes—
365C2×C4#409612yes—
366C2×C4C22×Dic1011yescharacteristic
367C2×C4C22×C4×D511yescharacteristic
368C2×C4C22×C22⋊D511yescharacteristic
369C8—41no—
370C8—41no—
371C8—42no—
372C8—52no—
373C8—102no—
374C8—101no—
375C8—101no—
376C8—102no—
377C8—101no—
378C8—201no—
379C8—201no—
380C8—202no—
381C5#1782911yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22#6528712yes—
388C22C22×#68311yescharacteristic
389C4—22no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4#6528712yes—
396C4D5×#13111yescharacteristic
397C4#6608411yescharacteristic
398C4C2×#409611yescharacteristic
399C4C2×#409611yescharacteristic
400C2—22no—
401C2—42no—
402C2C2×#6528711yescharacteristic
403C2#1061551411yescharacteristic
404C2#1061554111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2925C2
socleC22×C5C22×C4⋊C4
prime radicalC2×C5×#41D4
cosocleC42×C5C24
hypercenter#2925D5
hypocenterC5#17829
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2925

Complements

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

NH ≅ G/Ntype
4. #10601589C2semidirect
5. #10601587C2semidirect
6. #10601540C2semidirect
7. #10601538C2semidirect
8. #10601286C2semidirect
9. #10601284C2semidirect
10. #10601214C2semidirect
11. C2×#64699C2semidirect
381. C5#17829semidirect

Maximal subgroups

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

subgroupindex
C5×#29252
#41×#1682
#106142462
#106015892
#106015872
#106015402
#106015382
#106012862
#106012842
#106012142
C2×#646992
#178295