Finite Group Databasebeta

← #13105047

Subgroups — #13105047

Order 1600.

528 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717196C211yesmaximal, characteristic, complemented
2C5×#4022C211yesmaximal, characteristic, complemented
3C42×C5×D5C211yesmaximal, characteristic, complemented
4#11717024C211yesmaximal, characteristic
5#11717023C211yesmaximal, characteristic, complemented
6#11717022C211yesmaximal, characteristic, complemented
7#11716998C211yesmaximal, characteristic, complemented
8#11716997C211yesmaximal, characteristic, complemented
9#11716995C211yesmaximal, characteristic, complemented
10#11716986C211yesmaximal, characteristic, complemented
11#11716970C211yesmaximal, characteristic, complemented
12#11716969C211yesmaximal, characteristic, complemented
13#11716968C211yesmaximal, characteristic, complemented
14#11716955C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C2×C4×C5×D5C2211yescharacteristic
17C2×C4×C5×D5C2211yescharacteristic
18C2×C4×C5×D5C2211yescharacteristic
19C2×#6039C2211yescharacteristic
20C2×#6038C2211yescharacteristic
21C2×#6038C2211yescharacteristic
22C2×#6037C2211yescharacteristic
23C2×#6036C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C42×C52C2211yescharacteristic
26#87808C2211yescharacteristic
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic
29#87806C2211yescharacteristic
30#87806C2211yescharacteristic
31#87806C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C5×#683C2211yescharacteristic
38C4×C5×Dic5C2211yescharacteristic
39C4×C5×Dic5C2211yescharacteristic
40C4×C5×Dic5C2211yescharacteristic
41#87781C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87779C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66040—51nomaximal
52#65813—51nomaximal
53C22×C5⋊D5C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×C4×C52C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C2×C5×Dic5C2311yescharacteristic
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C4×C5×D5—22no—
69C4×C5×D5—22no—
70C4×C5×D5—22no—
71#6039—22no—
72#6038—22no—
73#6038—22no—
74#6037—22no—
75#6036—22no—
76D5×Dic5—22no—
77C5×#126—51no—
78C2×C42×C5—51no—
79#4076—51no—
80#4075—51no—
81#4075—51no—
82C4×C22⋊D5—51no—
83C4×C22⋊D5—51no—
84#4070—51no—
85#4042—51no—
86#4040—51no—
87#4039—51no—
88#4038—51no—
89#4035—51no—
90#4033—51no—
91#4031—51no—
92#4030—51no—
93#4029—51no—
94#4027—51no—
95#4027—51no—
96#4023—51no—
97#4022—51no—
98#4022—51no—
99#4022—102no—
100#4022D511yescharacteristic, complemented
101#4021—51no—
102C4×D20—51no—
103#4018—51no—
104#4018—51no—
105C42×D5D511yescharacteristic, complemented
106#4016—51no—
107C4×Dic10—51no—
108C22×C52C2411yescharacteristic
109C2×C5⋊D5—21no—
110C2×C5⋊D5—21no—
111C2×C5⋊D5—21no—
112C2×C5×D5—21no—
113C2×C5×D5—21no—
114C2×C5×D5—21no—
115C2×C5×D5—21no—
116C2×C5×D5—21no—
117C2×C5×D5#4112yes—
118C4×C52—21no—
119C4×C52—21no—
120C4×C52—21no—
121#1039—21no—
122#1039—21no—
123#1039—21no—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5—21no—
128C5×Dic5#4112yes—
129C5×Dic5#4112yes—
130C22×C4×C5—51no—
131C22×C4×C5—51no—
132C22×C4×C5—51no—
133C2×C22⋊D5—51no—
134C2×C22⋊D5—51no—
135C2×C22⋊D5—51no—
136C2×C22⋊D5—51no—
137C22×Dic5—51no—
138C2×D20—51no—
139C2×D20—51no—
140C2×C4×D5—51no—
141C2×C4×D5—51no—
142C2×C4×D5—51no—
143C2×C4×D5C2×D511yescharacteristic
144C2×C4×D5C2×D511yescharacteristic
145C2×C4×D5C2×D511yescharacteristic
146C2×Dic10—51no—
147C2×Dic10—51no—
148C5×C4⋊C4—51no—
149C5×C4⋊C4—51no—
150C5×C4⋊C4—51no—
151C5×C22⋊C4—51no—
152C5×C22⋊C4—51no—
153C5×C22⋊C4—51no—
154C42×C5—22no—
155C42×C5—51no—
156C42×C5—51no—
157C42×C5—51no—
158C42×C5C2×D511yescharacteristic
159C42×C5C2×D511yescharacteristic
160C22⋊Dic5—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163#685—51no—
164#685—51no—
165#685—51no—
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—102no—
175#685—102no—
176#685—102no—
177#685C2×D511yescharacteristic
178#685C2×D511yescharacteristic
179#685C2×D511yescharacteristic
180C4⋊Dic5—51no—
181C4⋊Dic5—51no—
182C4⋊Dic5—51no—
183#683—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—102no—
195#683—102no—
196#683—102no—
197#683C2×D511yescharacteristic
198#683C2×D511yescharacteristic
199#683C2×D511yescharacteristic
200C4×Dic5—51no—
201C4×Dic5—51no—
202C4×Dic5—51no—
203C4×Dic5—51no—
204C4×Dic5C2×D511yescharacteristic
205C4×Dic5C2×D511yescharacteristic
206C4×Dic5C2×D511yescharacteristic
207#529—251noSylow
208C2×C52C2×#4111yescharacteristic
209C2×C52C2×#4111yescharacteristic
210C2×C52C2×#4111yescharacteristic
211C5⋊D5—41no—
212C5×D5—22no—
213C5×D5—41no—
214C23×C5—51no—
215C23×C5—51no—
216C22×D5—51no—
217C22×D5—51no—
218C22×D5—102no—
219C22×D5C22×D511yescharacteristic
220C22×D5C22×D511yescharacteristic
221C2×C4×C5—22no—
222C2×C4×C5—22no—
223C2×C4×C5—22no—
224C2×C4×C5—51no—
225C2×C4×C5—51no—
226C2×C4×C5—51no—
227C2×C4×C5—51no—
228C2×C4×C5—51no—
229C2×C4×C5—51no—
230C2×C4×C5—102no—
231C2×C4×C5—102no—
232C2×C4×C5—102no—
233C2×C4×C5C22×D511yescharacteristic
234C2×C4×C5C22×D511yescharacteristic
235C2×C4×C5C22×D511yescharacteristic
236C2×C4×C5C22×D511yescharacteristic
237C2×C4×C5C22×D511yescharacteristic
238C2×C4×C5C22×D511yescharacteristic
239C22⋊D5—102no—
240C22⋊D5—102no—
241C22⋊D5—102no—
242C22⋊D5—102no—
243C2×Dic5—51no—
244C2×Dic5—51no—
245C2×Dic5—51no—
246C2×Dic5—51no—
247C2×Dic5—51no—
248C2×Dic5—51no—
249C2×Dic5—102no—
250C2×Dic5—102no—
251C2×Dic5—102no—
252C2×Dic5—102no—
253C2×Dic5C22×D511yescharacteristic
254C2×Dic5C22×D511yescharacteristic
255C2×Dic5C22×D511yescharacteristic
256C2×Dic5C22×D511yescharacteristic
257C2×Dic5C22×D511yescharacteristic
258C2×Dic5C22×D511yescharacteristic
259D20—102no—
260D20—102no—
261C4×D5—22no—
262C4×D5—22no—
263C4×D5—22no—
264C4×D5—102no—
265C4×D5—102no—
266C4×D5—102no—
267Dic10—102no—
268Dic10—102no—
269#126—51no—
270#126—251no—
271#125—251no—
272#124—251no—
273#123—251no—
274#123—251no—
275C22⋊Q8—251no—
276#121—251no—
277C4×Q8—251no—
278C4×D4—251no—
279C4×D4—251no—
280C4×D4—251no—
281#117—251no—
282#117—251no—
283C2×C42—51no—
284C52#52911yescharacteristic, complemented, Sylow
285C22×C5—22no—
286C22×C5—52no—
287C22×C5—101no—
288C22×C5—101no—
289C22×C5—101no—
290C22×C5—101no—
291C22×C5—101no—
292C22×C5C23×D511yescharacteristic
293C22×C5C23×D511yescharacteristic
294C2×D5—21no—
295C2×D5—21no—
296C2×D5—21no—
297C2×D5—21no—
298C2×D5—21no—
299C2×D5—101no—
300C2×D5—101no—
301C2×D5—101no—
302C2×D5—101no—
303C2×D5—101no—
304C2×D5—101no—
305C2×D5—202no—
306C2×D5—202no—
307C2×D5—202no—
308C2×D5#70912yes—
309C4×C5—21no—
310C4×C5—21no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—21no—
314C4×C5—21no—
315C4×C5—42no—
316C4×C5—42no—
317C4×C5—42no—
318C4×C5—52no—
319C4×C5—52no—
320C4×C5—101no—
321C4×C5—101no—
322C4×C5—101no—
323C4×C5—101no—
324Dic5—21no—
325Dic5—21no—
326Dic5—21no—
327Dic5—21no—
328Dic5—101no—
329Dic5—101no—
330Dic5—101no—
331Dic5—101no—
332Dic5—101no—
333Dic5—101no—
334Dic5—202no—
335Dic5—202no—
336Dic5—202no—
337Dic5#70912yes—
338Dic5#70912yes—
339C2×Q8—251no—
340C2×D4—251no—
341C2×D4—251no—
342C2×D4—251no—
343C22×C4—51no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—251no—
347C22×C4—251no—
348C4⋊C4—51no—
349C4⋊C4—51no—
350C4⋊C4—51no—
351C4⋊C4—251no—
352C4⋊C4—251no—
353C4⋊C4—251no—
354C4⋊C4—251no—
355C4⋊C4—251no—
356C4⋊C4—251no—
357C4⋊C4—251no—
358C22⋊C4—51no—
359C22⋊C4—51no—
360C22⋊C4—51no—
361C22⋊C4—251no—
362C22⋊C4—251no—
363C22⋊C4—251no—
364C22⋊C4—251no—
365C22⋊C4—251no—
366C22⋊C4—251no—
367C22⋊C4—251no—
368C42—51no—
369C42—51no—
370C42—51no—
371C42—251no—
372C42—251no—
373C42D5211yescharacteristic
374C2×C5—22no—
375C2×C5—22no—
376C2×C5—22no—
377C2×C5—102no—
378C2×C5—201no—
379C2×C5C2×#70911yescharacteristic
380C2×C5C2×#70911yescharacteristic
381C2×C5C2×#70911yescharacteristic
382C2×C5D5×#4111yescharacteristic
383C2×C5D5×#4111yescharacteristic
384C2×C5D5×#4111yescharacteristic
385D5—22no—
386D5—41no—
387D5—201no—
388D5—201no—
389D5—402no—
390C23—51no—
391C23—51no—
392C23—251no—
393Q8—502no—
394D4—502no—
395D4—502no—
396D4—502no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—51no—
403C2×C4—102no—
404C2×C4—102no—
405C2×C4—102no—
406C2×C4—251no—
407C2×C4—251no—
408C2×C4—251no—
409C2×C4—502no—
410C2×C4—502no—
411C2×C4C2×D5211yescharacteristic
412C2×C4C2×D5211yescharacteristic
413C2×C4C2×D5211yescharacteristic
414C5—22no—
415C5#6581311yescharacteristic, complemented
416C5#6604011yescharacteristic, complemented
417C22—52no—
418C22—101no—
419C22—101no—
420C22—101no—
421C22—101no—
422C22—101no—
423C22—501no—
424C22—501no—
425C22—501no—
426C22C22×D5211yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—52no—
431C4—52no—
432C4—101no—
433C4—101no—
434C4—101no—
435C4—101no—
436C4—501no—
437C4—501no—
438C4—501no—
439C2—102no—
440C2—201no—
441C2—1001no—
442C2D5×#70911yescharacteristic
443C2D5×#70911yescharacteristic
444C2D5×#70911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radical#11717024C2
hypercenterC42D52
hypocenterC52#529
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11717196C2semidirect
2. C5×#4022C2semidirect
3. C42×C5×D5C2semidirect
5. #11717023C2semidirect
6. #11717022C2semidirect
7. #11716998C2semidirect
8. #11716997C2semidirect
9. #11716995C2semidirect
10. #11716986C2semidirect
11. #11716970C2semidirect
12. #11716969C2semidirect
13. #11716968C2semidirect
14. #11716955C2semidirect
15. #11716954C2semidirect
100. #4022D5semidirect
105. C42×D5D5semidirect
284. C52#529semidirect
415. C5#65813semidirect
416. C5#66040semidirect

Maximal subgroups

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

subgroupindex
#117171962
C5×#40222
C42×C5×D52
#117170242
#117170232
#117170222
#117169982
#117169972
#117169952
#117169862
#117169702
#117169692
#117169682
#117169552
#117169542
#660405
#658135