Finite Group Databasebeta

← #12068589

Subgroups — #12068589

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2990C211yesmaximal, characteristic
2#10614253C211yesmaximal, characteristic
3#41×#168C211yesmaximal, characteristic
4#10601590C212yesmaximal, complemented
5#10601587C212yesmaximal, complemented
6#10601541C211yesmaximal, characteristic, complemented
7#10601538C211yesmaximal, characteristic, complemented
8#10601288C211yesmaximal, characteristic, complemented
9#10601283C211yesmaximal, characteristic, complemented
10#10601216C212yesmaximal, complemented
11C2×#64700C212yesmaximal, complemented
12C4×C5×#41C2211yescharacteristic
13C5×#446—21no—
14C5×#435—22no—
15C5×C8×D4—21no—
16C5×#432C2212yes—
17C5×#431C2211yescharacteristic
18C5×#423C2211yescharacteristic
19C2×C5×C4⋊C8C2212yes—
20#65313—21no—
21Q8×#168C411yescharacteristic
22Q8×#168C411yescharacteristic
23#65300—22no—
24D4×#168—21no—
25D4×#168C412yes—
26D4×#168C411yescharacteristic
27D4×#168C412yes—
28D4×#168C411yescharacteristic
29#65215C2212yes—
30#65214C2211yescharacteristic
31#65214C2212yes—
32#65214C2211yescharacteristic
33#65213C2211yescharacteristic
34C2×C4⋊#168C2212yes—
35C2×C4×#168C2211yescharacteristic
36C2×C4×#168C2212yes—
37#64700C2212yes—
38#64700C2212yes—
39#64700C2211yescharacteristic
40#64700C2211yescharacteristic
41#64700C2212yes—
42#64700C2212yes—
43#64699C2211yescharacteristic
44#64699C2212yes—
45#64699C2211yescharacteristic
46#64699C2212yes—
47#17834—51nomaximal, Sylow
48C2×C5×#41Q811yescharacteristic
49C2×C5×M16—21no—
50C22×C5×C8—22no—
51C4×C5×Q8C2×C411yescharacteristic
52C4×C5×Q8D411yescharacteristic
53C4×C5×D4C2×C412yes—
54C4×C5×D4C2×C411yescharacteristic
55C4×C5×D4D411yescharacteristic
56C4×C5×D4D412yes—
57C5×#117Q811yescharacteristic
58C5×#117Q812yes—
59C2×C42×C5C2312yes—
60C2×C42×C5C2311yescharacteristic
61C2×#681—21no—
62C22×#168—22no—
63C22×#168C2×C411yescharacteristic
64C22×#168C2×C412yes—
65C22×#168C2×C411yescharacteristic
66C22×#168C2×C412yes—
67C5×C4⋊C8—22no—
68C5×C4⋊C8—21no—
69C5×C4⋊C8C2311yescharacteristic
70C5×C4⋊C8C2312yes—
71C5×C4⋊C8C2311yescharacteristic
72C5×C22⋊C8—21no—
73C5×C22⋊C8—22no—
74C5×#97—22no—
75C4×C5×C8—21no—
76C22⋊#168—22no—
77C22⋊#168—21no—
78C22⋊#168C2×C412yes—
79C22⋊#168C2×C412yes—
80C22⋊#168C2×C412yes—
81C4⋊#168—21no—
82C4⋊#168—22no—
83C4⋊#168C2×C412yes—
84C4⋊#168C2×C412yes—
85C4⋊#168C2×C411yescharacteristic
86C4⋊#168C2×C411yescharacteristic
87C4⋊#168C2311yescharacteristic
88C4⋊#168C2311yescharacteristic
89C4⋊#168C2312yes—
90#3935—22no—
91C4×#168—21no—
92C4×#168C2×C411yescharacteristic
93C4×#168C2×C412yes—
94C4×#168C2×C411yescharacteristic
95C4×#168C2×C412yes—
96C4×#168C2311yescharacteristic
97C4×#168C2311yescharacteristic
98C4×#168C2312yes—
99#2990—51no—
100#2990D511yescharacteristic
101C8×#41—51no—
102#1574—51no—
103#1574—51no—
104#1573—52no—
105#1573—52no—
106#1568—52no—
107#1566—51no—
108#1564—51no—
109C2×#334—52no—
110C5×#41—24no—
111C2×C5×Q8C4⋊C411yescharacteristic
112C2×C5×D4C4⋊C412yes—
113C2×C5×D4C4⋊C411yescharacteristic
114C22×C4×C5C22×C411yescharacteristic
115C22×C4×C5C22×C412yes—
116C22×C4×C5C2×D412yes—
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2×Q812yes—
119C22×C4×C5C2×Q811yescharacteristic
120C5×M16—81no—
121C2×C5×C8—21no—
122C2×C5×C8—22no—
123C2×C5×C8—21no—
124C2×C5×C8—82no—
125C5×C4⋊C4C4⋊C412yes—
126C5×C4⋊C4C4⋊C411yescharacteristic
127C5×C4⋊C4C4⋊C412yes—
128C5×C4⋊C4C4⋊C411yescharacteristic
129C5×C22⋊C4C4⋊C412yes—
130C5×C22⋊C4C4⋊C412yes—
131C5×C22⋊C4C4⋊C412yes—
132C42×C5C22×C412yes—
133C42×C5C22×C411yescharacteristic
134C42×C5C2×D411yescharacteristic
135C42×C5C2×D412yes—
136C42×C5C2×Q812yes—
137C42×C5C2×Q811yescharacteristic
138C42×C5C2411yescharacteristic
139#681—81no—
140C2×#168—24no—
141C2×#168—22no—
142C2×#168—21no—
143C2×#168—21no—
144C2×#168—22no—
145C2×#168—42no—
146C2×#168—41no—
147C2×#168—82no—
148C2×#168C22×C411yescharacteristic
149C2×#168C22×C411yescharacteristic
150C2×#168C22×C412yes—
151C2×#168C22×C411yescharacteristic
152C2×#168C22×C412yes—
153C2×#168C22×C411yescharacteristic
154C4×#41C2×D511yescharacteristic
155#446—21no—
156#446—101no—
157C8×Q8—51no—
158C8×Q8—51no—
159#435—22no—
160#435—102no—
161C8×D4—21no—
162C8×D4—52no—
163C8×D4—51no—
164C8×D4—52no—
165C8×D4—51no—
166C8×D4—101no—
167#432—52no—
168#432C2×D512yes—
169#431—52no—
170#431—51no—
171#431—51no—
172#431C2×D511yescharacteristic
173#423—51no—
174#423C2×D511yescharacteristic
175C2×C4⋊C8—52no—
176C2×C4⋊C8C2×D512yes—
177C2×C4×C8—52no—
178C2×C4×C8—51no—
179#335—51no—
180#335—51no—
181#335—52no—
182#335—52no—
183#334—51no—
184#334—52no—
185#334—52no—
186#334—52no—
187#334—51no—
188#334—52no—
189C23×C5C2×C4⋊C411yescharacteristic
190C23×C5C2×C4⋊C412yes—
191C5×Q8—22no—
192C5×D4—24no—
193C5×D4#10814yes—
194C2×C4×C5—24no—
195C2×C4×C5—22no—
196C2×C4×C5—21no—
197C2×C4×C5—22no—
198C2×C4×C5—21no—
199C2×C4×C5—22no—
200C2×C4×C5—21no—
201C2×C4×C5—22no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5C2×C4⋊C412yes—
205C2×C4×C5C2×C4⋊C412yes—
206C2×C4×C5C2×C4⋊C411yescharacteristic
207C2×C4×C5C2×C4⋊C411yescharacteristic
208C2×C4×C5C2×C4⋊C412yes—
209C2×C4×C5C2×C4⋊C411yescharacteristic
210C2×C4×C5C23×C411yescharacteristic
211C2×C4×C5C22×D411yescharacteristic
212C2×C4×C5C22×Q811yescharacteristic
213C5×C8—41no—
214C5×C8—41no—
215C5×C8—42no—
216#168—22no—
217#168—21no—
218#168—21no—
219#168—22no—
220#168—21no—
221#168—41no—
222#168—41no—
223#168—42no—
224#168#13112yes—
225C2×#41Dic1011yescharacteristic
226C2×M16—21no—
227C2×M16—101no—
228C22×C8—22no—
229C22×C8—52no—
230C22×C8—51no—
231C22×C8—52no—
232C22×C8—51no—
233C22×C8—102no—
234C4×Q8C4×D511yescharacteristic
235C4×Q8C22⋊D511yescharacteristic
236C4×D4C4×D512yes—
237C4×D4C4×D511yescharacteristic
238C4×D4C22⋊D511yescharacteristic
239C4×D4C22⋊D512yes—
240#117Dic1011yescharacteristic
241#117Dic1012yes—
242C2×C42C22×D512yes—
243C2×C42C22×D511yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—52no—
247C4⋊C8—52no—
248C4⋊C8—51no—
249C4⋊C8—51no—
250C4⋊C8—51no—
251C4⋊C8—52no—
252C4⋊C8—51no—
253C4⋊C8—102no—
254C4⋊C8—101no—
255C4⋊C8C22×D511yescharacteristic
256C4⋊C8C22×D512yes—
257C4⋊C8C22×D511yescharacteristic
258C22⋊C8—21no—
259C22⋊C8—22no—
260C22⋊C8—52no—
261C22⋊C8—52no—
262C22⋊C8—52no—
263C22⋊C8—101no—
264C22⋊C8—102no—
265#97—22no—
266#97—102no—
267C4×C8—21no—
268C4×C8—52no—
269C4×C8—51no—
270C4×C8—52no—
271C4×C8—51no—
272C4×C8—51no—
273C4×C8—52no—
274C4×C8—51no—
275C4×C8—101no—
276C22×C5—22no—
277C22×C5—22no—
278C22×C5—21no—
279C22×C5—21no—
280C22×C5—21no—
281C22×C5C2×#10814yes—
282C22×C5C22×C4⋊C411yescharacteristic
283C4×C5—22no—
284C4×C5—21no—
285C4×C5—22no—
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—22no—
289C4×C5C2×#10812yes—
290C4×C5C2×#13111yescharacteristic
291C4×C5#56811yescharacteristic
292C4×C5#57711yescharacteristic
293C4×C5#57711yescharacteristic
294#41—24no—
295C2×Q8#68311yescharacteristic
296C2×D4#68312yes—
297C2×D4#68311yescharacteristic
298C22×C4C2×Dic1012yes—
299C22×C4C2×Dic1011yescharacteristic
300C22×C4C2×C4×D511yescharacteristic
301C22×C4C2×C4×D512yes—
302C22×C4C2×C22⋊D512yes—
303C22×C4C2×C22⋊D511yescharacteristic
304M16—81no—
305M16—401no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—51no—
310C2×C8—51no—
311C2×C8—52no—
312C2×C8—51no—
313C2×C8—52no—
314C2×C8—51no—
315C2×C8—82no—
316C2×C8—104no—
317C2×C8—102no—
318C2×C8—101no—
319C2×C8—101no—
320C2×C8—102no—
321C2×C8—202no—
322C2×C8—201no—
323C2×C8—402no—
324C4⋊C4#68312yes—
325C4⋊C4#68311yescharacteristic
326C4⋊C4#68312yes—
327C4⋊C4#68311yescharacteristic
328C22⋊C4#68312yes—
329C22⋊C4#68312yes—
330C22⋊C4#68312yes—
331C42C2×Dic1012yes—
332C42C2×Dic1011yescharacteristic
333C42C2×C4×D512yes—
334C42C2×C4×D511yescharacteristic
335C42C2×C22⋊D511yescharacteristic
336C42C2×C22⋊D512yes—
337C42C23×D511yescharacteristic
338C2×C5—24no—
339C2×C5—41no—
340C2×C5#290611yescharacteristic
341C2×C5#291211yescharacteristic
342C2×C5C22×#10811yescharacteristic
343C23C2×#68311yescharacteristic
344C23C2×#68312yes—
345Q8—22no—
346D4—24no—
347D4#395414yes—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4C2×#68312yes—
359C2×C4C2×#68311yescharacteristic
360C2×C4C2×#68311yescharacteristic
361C2×C4C2×#68312yes—
362C2×C4C2×#68312yes—
363C2×C4C2×#68311yescharacteristic
364C2×C4C22×Dic1011yescharacteristic
365C2×C4C22×C4×D511yescharacteristic
366C2×C4C22×C22⋊D511yescharacteristic
367C8—41no—
368C8—42no—
369C8—41no—
370C8—52no—
371C8—102no—
372C8—101no—
373C8—101no—
374C8—102no—
375C8—101no—
376C8—201no—
377C8—201no—
378C8—202no—
379C5#1783411yescharacteristic, complemented, Sylow
380C22—22no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22C2×#395414yes—
386C22C22×#68311yescharacteristic
387C4—22no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—22no—
392C4—21no—
393C4C2×#395412yes—
394C4D5×#13111yescharacteristic
395C4#6608411yescharacteristic
396C4#6617011yescharacteristic
397C4#6617111yescharacteristic
398C2—24no—
399C2—41no—
400C2#1061429211yescharacteristic
401C2C22×#395411yescharacteristic
402C2#1061551411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2990C2
socleC22×C5C22×C4⋊C4
prime radicalC2×C5×#41Q8
cosocleC42×C5C24
hypercenter#2990D5
hypocenterC5#17834
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2990

Complements

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

NH ≅ G/Ntype
4. #10601590C2semidirect
5. #10601587C2semidirect
6. #10601541C2semidirect
7. #10601538C2semidirect
8. #10601288C2semidirect
9. #10601283C2semidirect
10. #10601216C2semidirect
11. C2×#64700C2semidirect
379. C5#17834semidirect

Maximal subgroups

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

subgroupindex
C5×#29902
#106142532
#41×#1682
#106015902
#106015872
#106015412
#106015382
#106012882
#106012832
#106012162
C2×#647002
#178345