Finite Group Databasebeta

← #13106113

Subgroups — #13106113

Order 1600.

530 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C4×C5×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×C5×#683C211yesmaximal, characteristic, complemented
4#11717096C211yesmaximal, characteristic
5#11717084C211yesmaximal, characteristic, complemented
6#11717082C211yesmaximal, characteristic, complemented
7#11717079C211yesmaximal, characteristic, complemented
8#11716989C211yesmaximal, characteristic, complemented
9#11716987C211yesmaximal, characteristic, complemented
10#11716986C211yesmaximal, characteristic, complemented
11#11716984C211yesmaximal, characteristic, complemented
12#11716980C211yesmaximal, characteristic, complemented
13#11716971C211yesmaximal, characteristic, complemented
14#11716969C211yesmaximal, characteristic, complemented
15#11716968C211yesmaximal, characteristic, complemented
16C22×C4×C52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic
22C2×C4×C5×D5C2211yescharacteristic
23C2×#6039C2211yescharacteristic
24C2×#6038C2211yescharacteristic
25C22⋊#1039C2211yescharacteristic
26#87808C2211yescharacteristic
27#87806C2211yescharacteristic
28C4×#1039C2211yescharacteristic
29C5×C22⋊Dic5C2211yescharacteristic
30C5×#685C2211yescharacteristic
31C5×#683C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37#87781C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87780C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87779C2211yescharacteristic
46#87779C2211yescharacteristic
47#87779C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50#87778C2211yescharacteristic
51#65898—51nomaximal
52#65849—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C2×C4×C52—41no—
57C2×C4×C52C2311yescharacteristic
58C2×C4×C52C2311yescharacteristic
59C22⋊(C5⋊D5)—22no—
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C2×#1039C2311yescharacteristic
63C4×C5⋊D5—41no—
64C5×C22⋊D5—22no—
65C2×C5×Dic5—22no—
66C2×C5×Dic5—41no—
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C4×C5×D5—41no—
75#6039—41no—
76#6038—41no—
77C4×C5×D4—51no—
78C2×C5×C4⋊C4—51no—
79#4085—51no—
80#4081—51no—
81#4076—51no—
82#4075—51no—
83#4075—51no—
84#4075—51no—
85C4×C22⋊D5—51no—
86C4×C22⋊D5—51no—
87C4×C22⋊D5—102no—
88C4×C22⋊D5D511yescharacteristic, complemented
89#4070—51no—
90#4070—51no—
91C2×#683D511yescharacteristic, complemented
92#4044—51no—
93#4044—51no—
94#4042—51no—
95#4042—51no—
96#4042—51no—
97#4040—51no—
98#4040—51no—
99#4035—51no—
100#4035—51no—
101#4035—51no—
102#4032—51no—
103#4029—51no—
104#4025—51no—
105#4024—51no—
106#4022—51no—
107#4016—51no—
108C22×C52—21no—
109C22×C52—21no—
110C22×C52#4112yes—
111C22×C52C2411yescharacteristic
112C2×C5⋊D5—21no—
113C2×C5⋊D5—21no—
114C2×C5⋊D5—21no—
115C2×C5×D5—21no—
116C2×C5×D5—21no—
117C2×C5×D5—21no—
118C4×C52—21no—
119C4×C52—21no—
120#1039—21no—
121#1039—21no—
122#1039—21no—
123C5×Dic5—21no—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5—21no—
128C5×Dic5—21no—
129C5×Dic5—21no—
130C2×C5×D4—51no—
131C22×C4×C5—22no—
132C22×C4×C5—51no—
133C22×C4×C5—51no—
134C22×C4×C5—51no—
135C22×C4×C5C2×D511yescharacteristic
136C22×C4×C5C2×D511yescharacteristic
137C2×C22⋊D5—51no—
138C2×C22⋊D5—51no—
139C2×C22⋊D5—51no—
140C2×C22⋊D5—102no—
141C2×C22⋊D5C2×D511yescharacteristic
142C22×Dic5C2×D511yescharacteristic
143C22×Dic5C2×D511yescharacteristic
144C2×D20—51no—
145C2×C4×D5—51no—
146C2×C4×D5—51no—
147C2×C4×D5—102no—
148C2×C4×D5C2×D511yescharacteristic
149C2×Dic10—51no—
150C2×Dic10—51no—
151C5×C4⋊C4—51no—
152C5×C4⋊C4—51no—
153C5×C4⋊C4—51no—
154C5×C4⋊C4—51no—
155C5×C4⋊C4—51no—
156C5×C22⋊C4—51no—
157C5×C22⋊C4—51no—
158C42×C5—51no—
159C22⋊Dic5—51no—
160C22⋊Dic5—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163C22⋊Dic5—51no—
164C22⋊Dic5—102no—
165C22⋊Dic5C2×D511yescharacteristic
166#685—51no—
167#685—51no—
168#685—51no—
169#685—51no—
170#685—51no—
171#685—51no—
172#685—51no—
173#685—51no—
174#685—51no—
175#685—51no—
176#685—51no—
177#685—102no—
178#685C2×D511yescharacteristic
179C4⋊Dic5—51no—
180C4⋊Dic5—51no—
181C4⋊Dic5—51no—
182C4⋊Dic5—51no—
183C4⋊Dic5—51no—
184#683—51no—
185#683—51no—
186#683—51no—
187#683—51no—
188#683—51no—
189#683—51no—
190#683—51no—
191#683—51no—
192#683—51no—
193#683—51no—
194#683—51no—
195#683—51no—
196#683—51no—
197#683—102no—
198#683C2×D511yescharacteristic
199#683C2×D511yescharacteristic
200#683C2×D511yescharacteristic
201#683C2×D511yescharacteristic
202#683C2×D511yescharacteristic
203C4×Dic5—51no—
204C4×Dic5—51no—
205C4×Dic5—102no—
206C4×Dic5C2×D511yescharacteristic
207#539—251noSylow
208C2×C52—22no—
209C2×C52C2×#4111yescharacteristic
210C2×C52H(5,𝔽2)11yescharacteristic
211C2×C522−1+411yescharacteristic
212C5⋊D5—41no—
213C5×D5—41no—
214C23×C5—22no—
215C23×C5—51no—
216C23×C5C22×D511yescharacteristic
217C23×C5C22×D511yescharacteristic
218C22×D5—51no—
219C22×D5—51no—
220C22×D5—102no—
221C22×D5C22×D511yescharacteristic
222C5×D4—102no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—41no—
226C2×C4×C5—41no—
227C2×C4×C5—44no—
228C2×C4×C5—51no—
229C2×C4×C5—51no—
230C2×C4×C5—51no—
231C2×C4×C5—51no—
232C2×C4×C5—51no—
233C2×C4×C5—51no—
234C2×C4×C5—51no—
235C2×C4×C5—102no—
236C2×C4×C5—201no—
237C2×C4×C5—201no—
238C2×C4×C5C22×D511yescharacteristic
239C2×C4×C5C22×D511yescharacteristic
240C2×C4×C5C22×D511yescharacteristic
241C2×C4×C5C22×D511yescharacteristic
242C22⋊D5—22no—
243C22⋊D5—108no—
244C22⋊D5—102no—
245C22⋊D5—102no—
246C22⋊D5—201no—
247C2×Dic5—22no—
248C2×Dic5—41no—
249C2×Dic5—51no—
250C2×Dic5—51no—
251C2×Dic5—51no—
252C2×Dic5—51no—
253C2×Dic5—51no—
254C2×Dic5—51no—
255C2×Dic5—102no—
256C2×Dic5—102no—
257C2×Dic5—102no—
258C2×Dic5C22×D511yescharacteristic
259C2×Dic5C22×D511yescharacteristic
260C2×Dic5C22×D511yescharacteristic
261C2×Dic5C22×D511yescharacteristic
262C2×Dic5C22×D511yescharacteristic
263C2×Dic5C22×D511yescharacteristic
264C2×Dic5C22×D511yescharacteristic
265D20—201no—
266C4×D5—41no—
267C4×D5—201no—
268C4×D5—201no—
269C4×D5—204no—
270Dic10—201no—
271Dic10—201no—
272#126—251no—
273#126—251no—
274#125—251no—
275#125—251no—
276#123—251no—
277#123—251no—
278#123—251no—
279#123—251no—
280C22⋊Q8—251no—
281C22⋊Q8—251no—
282C22⋊Q8—251no—
283#121—251no—
284C4×D4—51no—
285C4×D4—251no—
286C2×C4⋊C4—51no—
287C52#53911yescharacteristic, complemented, Sylow
288C22×C5—21no—
289C22×C5—21no—
290C22×C5—21no—
291C22×C5—21no—
292C22×C5—24no—
293C22×C5—22no—
294C22×C5—42no—
295C22×C5—42no—
296C22×C5—101no—
297C22×C5—101no—
298C22×C5—101no—
299C22×C5#70912yes—
300C22×C5#70912yes—
301C22×C5C23×D511yescharacteristic
302C22×C5C23×D511yescharacteristic
303C2×D5—21no—
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—104no—
307C2×D5—101no—
308C2×D5—101no—
309C2×D5—101no—
310C2×D5—101no—
311C2×D5—101no—
312C2×D5—101no—
313C2×D5—202no—
314C2×D5—202no—
315C4×C5—24no—
316C4×C5—21no—
317C4×C5—21no—
318C4×C5—21no—
319C4×C5—21no—
320C4×C5—42no—
321C4×C5—101no—
322C4×C5—101no—
323C4×C5—101no—
324C4×C5—101no—
325C4×C5—101no—
326C4×C5—101no—
327C4×C5—101no—
328Dic5—21no—
329Dic5—21no—
330Dic5—21no—
331Dic5—21no—
332Dic5—21no—
333Dic5—21no—
334Dic5—21no—
335Dic5—104no—
336Dic5—101no—
337Dic5—101no—
338Dic5—101no—
339Dic5—101no—
340Dic5—101no—
341Dic5—101no—
342Dic5—202no—
343Dic5—202no—
344C2×Q8—251no—
345C2×D4—51no—
346C2×D4—251no—
347C2×D4—251no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—251no—
352C22×C4D5211yescharacteristic
353C4⋊C4—51no—
354C4⋊C4—51no—
355C4⋊C4—51no—
356C4⋊C4—51no—
357C4⋊C4—51no—
358C4⋊C4—251no—
359C4⋊C4—251no—
360C4⋊C4—251no—
361C4⋊C4—251no—
362C4⋊C4—251no—
363C4⋊C4—251no—
364C4⋊C4—251no—
365C4⋊C4—251no—
366C4⋊C4—251no—
367C22⋊C4—51no—
368C22⋊C4—51no—
369C22⋊C4—251no—
370C22⋊C4—251no—
371C22⋊C4—251no—
372C22⋊C4—251no—
373C22⋊C4—251no—
374C22⋊C4—251no—
375C22⋊C4—251no—
376C22⋊C4—251no—
377C42—51no—
378C42—251no—
379C2×C5—22no—
380C2×C5—22no—
381C2×C5—22no—
382C2×C5—22no—
383C2×C5—22no—
384C2×C5—44no—
385C2×C5—201no—
386C2×C5C2×#70911yescharacteristic
387C2×C5C2×#70911yescharacteristic
388C2×C5#414411yescharacteristic
389C2×C5#414411yescharacteristic
390C2×C5#414711yescharacteristic
391C2×C5#415011yescharacteristic
392D5—41no—
393D5—204no—
394D5—201no—
395D5—201no—
396C23—51no—
397C23—251no—
398C23C2×D5211yescharacteristic
399Q8—1001no—
400D4—102no—
401D4—502no—
402D4—1001no—
403C2×C4—41no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—51no—
407C2×C4—51no—
408C2×C4—51no—
409C2×C4—51no—
410C2×C4—51no—
411C2×C4—102no—
412C2×C4—201no—
413C2×C4—201no—
414C2×C4—251no—
415C2×C4—251no—
416C2×C4—251no—
417C2×C4—1001no—
418C2×C4C2×D5211yescharacteristic
419C2×C4C2×D5211yescharacteristic
420C5—22no—
421C5#6584911yescharacteristic, complemented
422C5#6589811yescharacteristic, complemented
423C22—21no—
424C22—21no—
425C22—101no—
426C22—101no—
427C22—101no—
428C22—501no—
429C22—501no—
430C22—501no—
431C22#8787312yes—
432C22C22×D5211yescharacteristic
433C4—21no—
434C4—21no—
435C4—101no—
436C4—101no—
437C4—101no—
438C4—101no—
439C4—101no—
440C4—101no—
441C4—101no—
442C4—501no—
443C4—501no—
444C4—501no—
445C2—22no—
446C2—201no—
447C2—1001no—
448C2#1171746211yescharacteristic
449C2C2×#8787311yescharacteristic
450C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radical#11717096C2
hypercenterC22×C4D52
hypocenterC52#539
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×C22⋊(C5⋊D5)C2semidirect
2. C4×C5×C22⋊D5C2semidirect
3. C2×C5×#683C2semidirect
5. #11717084C2semidirect
6. #11717082C2semidirect
7. #11717079C2semidirect
8. #11716989C2semidirect
9. #11716987C2semidirect
10. #11716986C2semidirect
11. #11716984C2semidirect
12. #11716980C2semidirect
13. #11716971C2semidirect
14. #11716969C2semidirect
15. #11716968C2semidirect
88. C4×C22⋊D5D5semidirect
91. C2×#683D5semidirect
287. C52#539semidirect
421. C5#65849semidirect
422. C5#65898semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊(C5⋊D5)2
C4×C5×C22⋊D52
C2×C5×#6832
#117170962
#117170842
#117170822
#117170792
#117169892
#117169872
#117169862
#117169842
#117169802
#117169712
#117169692
#117169682
#658985
#658495