Finite Group Databasebeta

← #13105106

Subgroups — #13105106

Order 1600.

556 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717196C211yesmaximal, characteristic, complemented
2C4×C5×D20C211yesmaximal, characteristic, complemented
3C5×#4018C211yesmaximal, characteristic, complemented
4#11717040C211yesmaximal, characteristic
5D5×#685C211yesmaximal, characteristic, complemented
6D5×#685C211yesmaximal, characteristic, complemented
7#11717023C211yesmaximal, characteristic, complemented
8#11717014C211yesmaximal, characteristic, complemented
9#11717002C211yesmaximal, characteristic, complemented
10#11716997C211yesmaximal, characteristic, complemented
11#11716995C211yesmaximal, characteristic, complemented
12#11716970C211yesmaximal, characteristic, complemented
13#11716969C211yesmaximal, characteristic, complemented
14#11716968C211yesmaximal, characteristic, complemented
15#11716966C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C5×D20C2211yescharacteristic
18C2×C4×C5×D5C2211yescharacteristic
19C2×C4×C5×D5C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic, complemented
21C2×#6039C2211yescharacteristic, complemented
22C2×#6038C2211yescharacteristic
23C2×#6037C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C42×C52C2211yescharacteristic, complemented
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic
29#87808C2211yescharacteristic
30#87806C2211yescharacteristic, complemented
31#87806C2211yescharacteristic, complemented
32#87806C2211yescharacteristic, complemented
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#685C2211yescharacteristic
36C5×#685C2211yescharacteristic, complemented
37C5×C4⋊Dic5C2211yescharacteristic, complemented
38C5×#683C2211yescharacteristic, complemented
39C5×#683C2211yescharacteristic, complemented
40C4×C5×Dic5C2211yescharacteristic, complemented
41#87781C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65890—51nomaximal
52#65861—51nomaximal
53C22×C5⋊D5C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C2×D52—22no—
58C2×D52—22no—
59C2×D52—41no—
60C2×C4×C52C2311yescharacteristic
61C2×C4×C52C2311yescharacteristic
62C2×C4×C52C2311yescharacteristic
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C5×D20—22no—
72C4×C5×D5—22no—
73C4×C5×D5—41no—
74C4×C5×D5—41no—
75#6039—41no—
76#6038—41no—
77#6037—22no—
78D5×Dic5—22no—
79D5×Dic5—41no—
80C4×C5×D4—51no—
81C5×#117—51no—
82#4083—51no—
83#4081—51no—
84#4075—51no—
85#4075—51no—
86C4×C22⋊D5—51no—
87C2×#685—51no—
88C2×#685—51no—
89#4070—51no—
90#4044—51no—
91#4044—51no—
92#4042—51no—
93#4042—51no—
94#4040—51no—
95#4040—51no—
96#4031—51no—
97#4029—51no—
98#4028—51no—
99#4027—51no—
100D5×C22⋊C4—51no—
101D5×C22⋊C4—51no—
102#4024—51no—
103#4023—51no—
104#4022—51no—
105#4022—51no—
106#4022—102no—
107#4021—51no—
108C4×D20D511yescharacteristic, complemented
109#4018—51no—
110#4018D511yescharacteristic, complemented
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—
118C2×C5×D5—21no—
119C2×C5×D5—21no—
120C2×C5×D5—21no—
121C2×C5×D5—21no—
122C2×C5×D5#4112yes—
123C2×C5×D5#4112yes—
124D52—44no—
125C4×C52—21no—
126C4×C52—21no—
127C4×C52—21no—
128#1039—21no—
129#1039—21no—
130#1039—21no—
131C5×Dic5—21no—
132C5×Dic5—21no—
133C5×Dic5—21no—
134C5×Dic5—21no—
135C5×Dic5—21no—
136C23×D5—51no—
137C23×D5—51no—
138C2×C5×D4—51no—
139C22×C4×C5—51no—
140C22×C4×C5—51no—
141C22×C4×C5—51no—
142C2×C22⋊D5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—51no—
145C22×Dic5—51no—
146C22×Dic5—51no—
147C2×D20—51no—
148C2×D20C2×D511yescharacteristic
149C2×C4×D5—51no—
150C2×C4×D5—51no—
151C2×C4×D5C2×D511yescharacteristic
152C2×C4×D5C2×D511yescharacteristic
153C2×C4×D5C2×D511yescharacteristic, complemented
154C2×Dic10—51no—
155C2×Dic10—51no—
156C5×C4⋊C4—51no—
157C5×C4⋊C4—51no—
158C5×C4⋊C4—51no—
159C5×C22⋊C4—51no—
160C5×C22⋊C4—51no—
161C5×C22⋊C4—51no—
162C5×C22⋊C4—51no—
163C42×C5—22no—
164C42×C5—51no—
165C42×C5C2×D511yescharacteristic, complemented
166C42×C5C2×D511yescharacteristic, complemented
167C22⋊Dic5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172#685—51no—
173#685—51no—
174#685—51no—
175#685—51no—
176#685—51no—
177#685—51no—
178#685—51no—
179#685—51no—
180#685—51no—
181#685—51no—
182#685—51no—
183#685—51no—
184#685—51no—
185#685—51no—
186#685—51no—
187#685—102no—
188#685—102no—
189#685—102no—
190#685C2×D511yescharacteristic
191#685C2×D511yescharacteristic
192#685C2×D511yescharacteristic
193#685C2×D511yescharacteristic, complemented
194C4⋊Dic5—51no—
195C4⋊Dic5C2×D511yescharacteristic, complemented
196#683—51no—
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683—102no—
206#683—102no—
207#683—102no—
208#683C2×D511yescharacteristic, complemented
209#683C2×D511yescharacteristic, complemented
210C4×Dic5—51no—
211C4×Dic5—51no—
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52C2×#4111yescharacteristic
215C2×C52C2×#4111yescharacteristic
216C2×C52H(5,𝔽2)11yescharacteristic
217C5⋊D5—41no—
218C5×D5—22no—
219C5×D5—22no—
220C5×D5—41no—
221C23×C5—51no—
222C23×C5—51no—
223C23×C5—51no—
224C22×D5—51no—
225C22×D5—51no—
226C22×D5—102no—
227C22×D5—102no—
228C22×D5—102no—
229C22×D5—102no—
230C22×D5—102no—
231C22×D5—201no—
232C22×D5—201no—
233C22×D5C22×D511yescharacteristic
234C22×D5C22×D511yescharacteristic
235C22×D5C22×D511yescharacteristic
236C5×D4—102no—
237C2×C4×C5—22no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—51no—
241C2×C4×C5—51no—
242C2×C4×C5—51no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—102no—
246C2×C4×C5—201no—
247C2×C4×C5—201no—
248C2×C4×C5C22×D511yescharacteristic
249C2×C4×C5C22×D511yescharacteristic
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C2×C4×C5C22×D511yescharacteristic
254C22⋊D5—102no—
255C22⋊D5—102no—
256C22⋊D5—201no—
257C2×Dic5—51no—
258C2×Dic5—51no—
259C2×Dic5—51no—
260C2×Dic5—51no—
261C2×Dic5—51no—
262C2×Dic5—51no—
263C2×Dic5—102no—
264C2×Dic5—102no—
265C2×Dic5—102no—
266C2×Dic5—102no—
267C2×Dic5—201no—
268C2×Dic5C22×D511yescharacteristic
269C2×Dic5C22×D511yescharacteristic
270C2×Dic5C22×D511yescharacteristic
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273D20—22no—
274D20—201no—
275C4×D5—22no—
276C4×D5—41no—
277C4×D5—41no—
278C4×D5—102no—
279C4×D5—201no—
280Dic10—201no—
281Dic10—201no—
282#126—251no—
283#126—251no—
284#124—251no—
285#123—251no—
286#123—251no—
287#123—251no—
288C22⋊Q8—251no—
289C22⋊Q8—251no—
290#120—251no—
291C4×D4—51no—
292C4×D4—251no—
293#117—51no—
294#117—251no—
295C2×C22⋊C4—251no—
296C2×C22⋊C4—251no—
297C52#55111yescharacteristic, complemented, Sylow
298C22×C5—22no—
299C22×C5—52no—
300C22×C5—52no—
301C22×C5—101no—
302C22×C5—101no—
303C22×C5—101no—
304C22×C5—101no—
305C22×C5—101no—
306C22×C5—101no—
307C22×C5—101no—
308C22×C5C23×D511yescharacteristic
309C22×C5C23×D511yescharacteristic
310C2×D5—21no—
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—21no—
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—21no—
317C2×D5—101no—
318C2×D5—101no—
319C2×D5—101no—
320C2×D5—101no—
321C2×D5—101no—
322C2×D5—101no—
323C2×D5—202no—
324C2×D5—202no—
325C2×D5—202no—
326C2×D5—204no—
327C2×D5—204no—
328C2×D5#70912yes—
329C2×D5#70912yes—
330C4×C5—21no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—21no—
336C4×C5—42no—
337C4×C5—42no—
338C4×C5—42no—
339C4×C5—101no—
340C4×C5—101no—
341C4×C5—101no—
342C4×C5—101no—
343C4×C5—101no—
344Dic5—21no—
345Dic5—21no—
346Dic5—21no—
347Dic5—21no—
348Dic5—21no—
349Dic5—101no—
350Dic5—101no—
351Dic5—101no—
352Dic5—101no—
353Dic5—101no—
354Dic5—101no—
355Dic5—202no—
356Dic5—202no—
357Dic5—202no—
358C24—251no—
359C2×Q8—251no—
360C2×D4—51no—
361C2×D4—251no—
362C2×D4—251no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—251no—
367C22×C4—251no—
368C4⋊C4—51no—
369C4⋊C4—51no—
370C4⋊C4—51no—
371C4⋊C4—251no—
372C4⋊C4—251no—
373C4⋊C4—251no—
374C4⋊C4—251no—
375C4⋊C4—251no—
376C22⋊C4—51no—
377C22⋊C4—51no—
378C22⋊C4—51no—
379C22⋊C4—51no—
380C22⋊C4—251no—
381C22⋊C4—251no—
382C22⋊C4—251no—
383C22⋊C4—251no—
384C22⋊C4—251no—
385C22⋊C4—251no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C42—51no—
391C42—251no—
392C42D5211yescharacteristic, complemented
393C2×C5—22no—
394C2×C5—22no—
395C2×C5—22no—
396C2×C5—102no—
397C2×C5—102no—
398C2×C5—201no—
399C2×C5C2×#70911yescharacteristic
400C2×C5C2×#70911yescharacteristic
401C2×C5#414411yescharacteristic
402C2×C5D5×#4111yescharacteristic
403C2×C5D5×#4111yescharacteristic
404C2×C5#414911yescharacteristic
405D5—22no—
406D5—22no—
407D5—41no—
408D5—201no—
409D5—201no—
410D5—402no—
411C23—51no—
412C23—51no—
413C23—51no—
414C23—251no—
415C23—502no—
416C23—502no—
417C23—1001no—
418Q8—1001no—
419D4—102no—
420D4—502no—
421D4—1001no—
422C2×C4—51no—
423C2×C4—51no—
424C2×C4—51no—
425C2×C4—51no—
426C2×C4—51no—
427C2×C4—102no—
428C2×C4—201no—
429C2×C4—201no—
430C2×C4—251no—
431C2×C4—251no—
432C2×C4—251no—
433C2×C4—502no—
434C2×C4—1001no—
435C2×C4C2×D5211yescharacteristic
436C2×C4C2×D5211yescharacteristic
437C2×C4C2×D5211yescharacteristic
438C5—22no—
439C5#6586111yescharacteristic, complemented
440C5#6589011yescharacteristic, complemented
441C22—52no—
442C22—52no—
443C22—101no—
444C22—101no—
445C22—101no—
446C22—101no—
447C22—101no—
448C22—101no—
449C22—101no—
450C22—501no—
451C22—501no—
452C22—501no—
453C22—1004no—
454C22C22×D5211yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—101no—
459C4—101no—
460C4—101no—
461C4—101no—
462C4—101no—
463C4—501no—
464C4—501no—
465C4—501no—
466C2—102no—
467C2—102no—
468C2—201no—
469C2—1001no—
470C2D5×#70911yescharacteristic
471C2D5×#70911yescharacteristic
472C2#1171747211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radical#11717040C2
hypercenterC42D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11717196C2semidirect
2. C4×C5×D20C2semidirect
3. C5×#4018C2semidirect
5. D5×#685C2semidirect
6. D5×#685C2semidirect
7. #11717023C2semidirect
8. #11717014C2semidirect
9. #11717002C2semidirect
10. #11716997C2semidirect
11. #11716995C2semidirect
12. #11716970C2semidirect
13. #11716969C2semidirect
14. #11716968C2semidirect
15. #11716966C2semidirect
20. C2×C4×C5×D5C22semidirect
21. C2×#6039C22semidirect
26. C42×C52C22semidirect
30. #87806C22semidirect
31. #87806C22semidirect
32. #87806C22semidirect
36. C5×#685C22semidirect
37. C5×C4⋊Dic5C22semidirect
38. C5×#683C22semidirect
39. C5×#683C22semidirect
40. C4×C5×Dic5C22semidirect
41. #87781C22semidirect
42. #87780C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. Dic52C22semidirect
108. C4×D20D5semidirect
110. #4018D5semidirect
153. C2×C4×D5C2×D5semidirect
165. C42×C5C2×D5semidirect
166. C42×C5C2×D5semidirect
193. #685C2×D5semidirect
195. C4⋊Dic5C2×D5semidirect
208. #683C2×D5semidirect
209. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
297. C52#551semidirect
392. C42D52semidirect
439. C5#65861semidirect
440. C5#65890semidirect

Maximal subgroups

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

subgroupindex
#117171962
C4×C5×D202
C5×#40182
#117170402
D5×#6852
#117170232
#117170142
#117170022
#117169972
#117169952
#117169702
#117169692
#117169682
#117169662
#658905
#658615