Finite Group Databasebeta

← #13105156

Subgroups — #13105156

Order 1600.

496 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#11717199C211yesmaximal, characteristic, complemented
2C5×#4029C211yesmaximal, characteristic, complemented
3C5×#4027C211yesmaximal, characteristic, complemented
4#11717088C211yesmaximal, characteristic
5#11717087C211yesmaximal, characteristic, complemented
6#11717082C211yesmaximal, characteristic, complemented
7#11717079C211yesmaximal, characteristic, complemented
8C4×#6037C211yesmaximal, characteristic, complemented
9#11717014C211yesmaximal, characteristic, complemented
10#11717003C211yesmaximal, characteristic, complemented
11D5×#683C211yesmaximal, characteristic, complemented
12#11716989C211yesmaximal, characteristic, complemented
13#11716988C211yesmaximal, characteristic, complemented
14#11716978C211yesmaximal, characteristic, complemented
15#11716969C211yesmaximal, characteristic, complemented
16C2×C5×C22⋊D5C2211yescharacteristic
17C2×C5×C22⋊D5C2211yescharacteristic
18C22×C5×Dic5C2211yescharacteristic
19C2×C4×C5×D5C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic
21C2×#6039C2211yescharacteristic
22C2×#6037C2211yescharacteristic
23C2×D5×Dic5C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C52×C22⋊C4C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27C22⋊#1039C2211yescharacteristic
28C4⋊#1039C2211yescharacteristic
29#87806C2211yescharacteristic
30#87806C2211yescharacteristic
31C4×#1039C2211yescharacteristic
32C5×C22⋊Dic5C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×C4⋊Dic5C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C5×#683C2211yescharacteristic
38C4×C5×Dic5C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87781C2211yescharacteristic
42#87781C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50#87778C2211yescharacteristic
51#65994—51nomaximal
52#65898—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C5×C22⋊D5—22no—
63C5×C22⋊D5—41no—
64C2×C5×Dic5—41no—
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C4×C5×D5—41no—
72C4×C5×D5—41no—
73#6039—41no—
74#6037—22no—
75D5×Dic5—22no—
76D5×Dic5—41no—
77C5×#123—51no—
78C4×C5×D4—51no—
79#4085—51no—
80#4084—51no—
81#4081—51no—
82#4081—51no—
83#4075—51no—
84C4×C22⋊D5—51no—
85C4×C22⋊D5—51no—
86#4070—51no—
87#4070—51no—
88C2×#683—51no—
89#4042—51no—
90#4042—51no—
91D5×C4⋊C4—51no—
92#4036—51no—
93#4035—51no—
94#4035—51no—
95#4032—51no—
96#4029—51no—
97#4029—51no—
98#4029D511yescharacteristic, complemented
99#4027D511yescharacteristic, complemented
100#4025—51no—
101#4025—51no—
102#4025—51no—
103#4025—102no—
104#4024—51no—
105#4024—51no—
106#4022—51no—
107#4016—51no—
108C22×C52—21no—
109C22×C52—21no—
110C22×C52—21no—
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#4112yes—
118C4×C52—21no—
119C4×C52—21no—
120#1039—21no—
121#1039—21no—
122#1039—21no—
123#1039—21no—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5—21no—
128C5×Dic5—21no—
129C5×Dic5—21no—
130C2×C5×D4—51no—
131C2×C5×D4—51no—
132C22×C4×C5—51no—
133C22×C4×C5—51no—
134C22×C4×C5—51no—
135C2×C22⋊D5—51no—
136C2×C22⋊D5—51no—
137C2×C22⋊D5C2×D511yescharacteristic
138C2×C22⋊D5C2×D511yescharacteristic
139C22×Dic5—51no—
140C22×Dic5—51no—
141C22×Dic5C2×D511yescharacteristic
142C2×C4×D5—51no—
143C2×C4×D5—51no—
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—22no—
152C5×C22⋊C4—51no—
153C5×C22⋊C4—51no—
154C5×C22⋊C4—51no—
155C5×C22⋊C4C2×D511yescharacteristic
156C5×C22⋊C4C2×D511yescharacteristic
157C42×C5—51no—
158C22⋊Dic5—51no—
159C22⋊Dic5—51no—
160C22⋊Dic5—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163C22⋊Dic5—51no—
164C22⋊Dic5—51no—
165C22⋊Dic5—51no—
166C22⋊Dic5—102no—
167C22⋊Dic5—102no—
168C22⋊Dic5C2×D511yescharacteristic
169#685—51no—
170#685—51no—
171#685—51no—
172#685—51no—
173#685C2×D511yescharacteristic
174#685C2×D511yescharacteristic
175C4⋊Dic5—51no—
176C4⋊Dic5—51no—
177C4⋊Dic5—51no—
178C4⋊Dic5—51no—
179C4⋊Dic5—51no—
180C4⋊Dic5—51no—
181C4⋊Dic5—102no—
182C4⋊Dic5C2×D511yescharacteristic
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—51no—
195#683—51no—
196#683—51no—
197#683—51no—
198#683—51no—
199#683—102no—
200#683—102no—
201#683C2×D511yescharacteristic
202#683C2×D511yescharacteristic
203C4×Dic5—51no—
204C4×Dic5—51no—
205C4×Dic5—102no—
206C4×Dic5C2×D511yescharacteristic
207#539—251noSylow
208C2×C52—41no—
209C2×C52C2×#4111yescharacteristic
210C2×C52H(5,𝔽2)11yescharacteristic
211C2×C522−1+411yescharacteristic
212C5×D5—22no—
213C5×D5—41no—
214C23×C5—22no—
215C23×C5—51no—
216C23×C5—51no—
217C23×C5C22×D511yescharacteristic
218C23×C5C22×D511yescharacteristic
219C22×D5C22×D511yescharacteristic
220C22×D5C22×D511yescharacteristic
221C5×D4—102no—
222C5×D4—201no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—51no—
226C2×C4×C5—51no—
227C2×C4×C5—51no—
228C2×C4×C5—51no—
229C2×C4×C5—51no—
230C2×C4×C5—51no—
231C2×C4×C5—201no—
232C2×C4×C5—201no—
233C2×C4×C5—201no—
234C2×C4×C5C22×D511yescharacteristic
235C2×C4×C5C22×D511yescharacteristic
236C2×C4×C5C22×D511yescharacteristic
237C2×C4×C5C22×D511yescharacteristic
238C22⋊D5—22no—
239C22⋊D5—41no—
240C22⋊D5—102no—
241C22⋊D5—102no—
242C2×Dic5—41no—
243C2×Dic5—51no—
244C2×Dic5—51no—
245C2×Dic5—51no—
246C2×Dic5—51no—
247C2×Dic5—51no—
248C2×Dic5—51no—
249C2×Dic5—51no—
250C2×Dic5—51no—
251C2×Dic5—102no—
252C2×Dic5—102no—
253C2×Dic5—102no—
254C2×Dic5—102no—
255C2×Dic5—102no—
256C2×Dic5—201no—
257C2×Dic5C22×D511yescharacteristic
258C2×Dic5C22×D511yescharacteristic
259C2×Dic5C22×D511yescharacteristic
260C2×Dic5C22×D511yescharacteristic
261C2×Dic5C22×D511yescharacteristic
262C2×Dic5C22×D511yescharacteristic
263C4×D5—41no—
264C4×D5—41no—
265C4×D5—102no—
266C4×D5—201no—
267Dic10—201no—
268Dic10—201no—
269#126—251no—
270#126—251no—
271#125—251no—
272#125—251no—
273#123—51no—
274#123—251no—
275#123—251no—
276#123—251no—
277C22⋊Q8—251no—
278C22⋊Q8—251no—
279C22⋊Q8—251no—
280#121—251no—
281C4×D4—51no—
282C4×D4—251no—
283C2×C4⋊C4—251no—
284C52#53911yescharacteristic, complemented, Sylow
285C22×C5—21no—
286C22×C5—21no—
287C22×C5—21no—
288C22×C5—21no—
289C22×C5—21no—
290C22×C5—21no—
291C22×C5—22no—
292C22×C5—42no—
293C22×C5—42no—
294C22×C5—42no—
295C22×C5—52no—
296C22×C5—101no—
297C22×C5—101no—
298C22×C5—101no—
299C22×C5—101no—
300C22×C5—101no—
301C22×C5C23×D511yescharacteristic
302C22×C5C23×D511yescharacteristic
303C2×D5—21no—
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5#70912yes—
309C4×C5—21no—
310C4×C5—21no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—42no—
314C4×C5—42no—
315C4×C5—101no—
316C4×C5—101no—
317C4×C5—101no—
318C4×C5—101no—
319C4×C5—101no—
320C4×C5—101no—
321Dic5—21no—
322Dic5—21no—
323Dic5—21no—
324Dic5—21no—
325Dic5—21no—
326Dic5—21no—
327Dic5—101no—
328Dic5—101no—
329Dic5—101no—
330Dic5—101no—
331Dic5—101no—
332Dic5—101no—
333Dic5—101no—
334Dic5—101no—
335Dic5—202no—
336Dic5—202no—
337Dic5—202no—
338Dic5—202no—
339C2×Q8—251no—
340C2×D4—51no—
341C2×D4—51no—
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—
358C4⋊C4—251no—
359C4⋊C4—251no—
360C4⋊C4—251no—
361C4⋊C4—251no—
362C22⋊C4—51no—
363C22⋊C4—51no—
364C22⋊C4—51no—
365C22⋊C4—251no—
366C22⋊C4—251no—
367C22⋊C4—251no—
368C22⋊C4—251no—
369C22⋊C4—251no—
370C22⋊C4—251no—
371C22⋊C4D5211yescharacteristic
372C42—51no—
373C42—251no—
374C2×C5—22no—
375C2×C5—22no—
376C2×C5—22no—
377C2×C5—41no—
378C2×C5—41no—
379C2×C5—82no—
380C2×C5—102no—
381C2×C5—201no—
382C2×C5C2×#70911yescharacteristic
383C2×C5#414411yescharacteristic
384C2×C5#414411yescharacteristic
385C2×C5D5×#4111yescharacteristic
386C2×C5#415011yescharacteristic
387C2×C5#415011yescharacteristic
388D5—22no—
389D5—41no—
390C23—51no—
391C23—51no—
392C23C2×D5211yescharacteristic
393Q8—1001no—
394D4—102no—
395D4—201no—
396D4—502no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—51no—
403C2×C4—201no—
404C2×C4—201no—
405C2×C4—201no—
406C2×C4—251no—
407C2×C4—251no—
408C2×C4—251no—
409C2×C4—251no—
410C2×C4—502no—
411C2×C4—1001no—
412C2×C4C2×D5211yescharacteristic
413C2×C4C2×D5211yescharacteristic
414C5—22no—
415C5#6589811yescharacteristic, complemented
416C5#6599411yescharacteristic, complemented
417C22—21no—
418C22—21no—
419C22—21no—
420C22—52no—
421C22—101no—
422C22—101no—
423C22—101no—
424C22—101no—
425C22—101no—
426C22C22×D5211yescharacteristic
427C4—21no—
428C4—21no—
429C4—101no—
430C4—101no—
431C4—101no—
432C4—101no—
433C4—101no—
434C4—101no—
435C4—501no—
436C4—501no—
437C4—501no—
438C4—501no—
439C2—41no—
440C2—102no—
441C2—201no—
442C2D5×#70911yescharacteristic
443C2#1171747411yescharacteristic
444C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717088C2
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. #11717199C2semidirect
2. C5×#4029C2semidirect
3. C5×#4027C2semidirect
5. #11717087C2semidirect
6. #11717082C2semidirect
7. #11717079C2semidirect
8. C4×#6037C2semidirect
9. #11717014C2semidirect
10. #11717003C2semidirect
11. D5×#683C2semidirect
12. #11716989C2semidirect
13. #11716988C2semidirect
14. #11716978C2semidirect
15. #11716969C2semidirect
98. #4029D5semidirect
99. #4027D5semidirect
284. C52#539semidirect
415. C5#65898semidirect
416. C5#65994semidirect

Maximal subgroups

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

subgroupindex
#117171992
C5×#40292
C5×#40272
#117170882
#117170872
#117170822
#117170792
C4×#60372
#117170142
#117170032
D5×#6832
#117169892
#117169882
#117169782
#117169692
#659945
#658985