Finite Group Databasebeta

← #13105108

Subgroups — #13105108

Order 1600.

590 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C5×#4022C211yesmaximal, characteristic, complemented
3C5×#4018C211yesmaximal, characteristic, complemented
4#11717041C211yesmaximal, characteristic
5#11717039C211yesmaximal, characteristic, complemented
6D5×#685C211yesmaximal, characteristic, complemented
7#11717016C211yesmaximal, characteristic, complemented
8#11717007C211yesmaximal, characteristic, complemented
9#11716999C211yesmaximal, characteristic, complemented
10#11716984C211yesmaximal, characteristic, complemented
11#11716977C211yesmaximal, characteristic, complemented
12#11716971C211yesmaximal, characteristic, complemented
13#11716970C211yesmaximal, characteristic, complemented
14#11716968C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C4⋊(C5⋊D5)C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C4×C5⋊D5C2211yescharacteristic, complemented
20C2×C4×C5×D5C2211yescharacteristic
21C2×#6039C2211yescharacteristic, complemented
22C2×#6038C2211yescharacteristic
23C2×#6038C2211yescharacteristic
24C2×#6036C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C42×C52C2211yescharacteristic, complemented
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic, complemented
29C4⋊#1039C2211yescharacteristic, complemented
30C5×#685C2211yescharacteristic
31C5×#685C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#683C2211yescharacteristic, complemented
36C5×#683C2211yescharacteristic, complemented
37C5×#683C2211yescharacteristic, complemented
38C5×#683C2211yescharacteristic, complemented
39C5×#683C2211yescharacteristic, complemented
40C4×C5×Dic5C2211yescharacteristic, complemented
41#87781C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87779C2211yescharacteristic, complemented
46#87779C2211yescharacteristic, complemented
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50Dic52C2211yescharacteristic, complemented
51#66039—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
65C4⋊(C5⋊D5)—22no—
66C4×C5⋊D5—41no—
67C4×C5⋊D5—41no—
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C4×C5×D5—22no—
75#6039—41no—
76#6038—22no—
77#6038—41no—
78#6036—22no—
79D5×Dic5—41no—
80C5×#126—51no—
81C5×#117—51no—
82#4075—51no—
83C2×#685—51no—
84#4044—51no—
85#4043—51no—
86#4042—51no—
87#4042—51no—
88#4040—51no—
89#4040—51no—
90#4040—51no—
91#4040—51no—
92#4039—51no—
93#4038—51no—
94#4031—51no—
95#4029—51no—
96#4028—51no—
97#4028—51no—
98#4027—51no—
99D5×C22⋊C4—51no—
100D5×C22⋊C4—51no—
101D5×C22⋊C4—51no—
102#4025—51no—
103#4024—51no—
104#4022D511yescharacteristic, complemented
105#4021—51no—
106C4×D20—51no—
107C4×D20—51no—
108C4×D20—102no—
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#4112yes—
118C2×C5×D5—21no—
119C2×C5×D5—21no—
120C2×C5×D5—21no—
121C2×C5×D5—21no—
122C2×C5×D5—21no—
123C2×C5×D5#4112yes—
124D52—44no—
125C4×C52—21no—
126C4×C52—21no—
127C4×C52—21no—
128#1039—21no—
129#1039—21no—
130C5×Dic5—21no—
131C5×Dic5—21no—
132C5×Dic5—21no—
133C5×Dic5—21no—
134C5×Dic5—21no—
135C5×Dic5—21no—
136C23×D5—51no—
137C23×D5—51no—
138C22×C4×C5—51no—
139C2×C22⋊D5—51no—
140C2×C22⋊D5—51no—
141C22×Dic5—51no—
142C2×D20—51no—
143C2×D20—51no—
144C2×D20—51no—
145C2×D20—51no—
146C2×D20—102no—
147C2×C4×D5—51no—
148C2×C4×D5—51no—
149C2×C4×D5—51no—
150C2×C4×D5—51no—
151C2×C4×D5—51no—
152C2×C4×D5—51no—
153C2×C4×D5—51no—
154C2×C4×D5—102no—
155C2×C4×D5—102no—
156C2×C4×D5C2×D511yescharacteristic
157C2×Dic10—51no—
158C2×Dic10—51no—
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C4⋊C4—51no—
162C5×C4⋊C4—51no—
163C5×C4⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C5×C22⋊C4—51no—
167C5×C22⋊C4—51no—
168C5×C22⋊C4—51no—
169C42×C5—22no—
170C42×C5—51no—
171C42×C5C2×D511yescharacteristic, complemented
172C42×C5C2×D511yescharacteristic, complemented
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5—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—51no—
188#685—51no—
189#685—51no—
190#685—51no—
191#685—102no—
192#685—102no—
193#685C2×D511yescharacteristic
194#685C2×D511yescharacteristic
195#685C2×D511yescharacteristic
196#685C2×D511yescharacteristic
197#685C2×D511yescharacteristic
198C4⋊Dic5—51no—
199C4⋊Dic5—51no—
200C4⋊Dic5—51no—
201C4⋊Dic5—102no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683C2×D511yescharacteristic, complemented
206#683C2×D511yescharacteristic, complemented
207#683C2×D511yescharacteristic, complemented
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—22no—
218C5⋊D5—41no—
219C5×D5—22no—
220C5×D5—41no—
221C23×C5—51no—
222C23×C5—51no—
223C22×D5—51no—
224C22×D5—51no—
225C22×D5—51no—
226C22×D5—51no—
227C22×D5—102no—
228C22×D5—102no—
229C22×D5—102no—
230C22×D5—102no—
231C22×D5—102no—
232C22×D5—102no—
233C22×D5—201no—
234C22×D5—201no—
235C22×D5C22×D511yescharacteristic
236C22×D5C22×D511yescharacteristic
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—51no—
246C2×C4×C5—102no—
247C2×C4×C5C22×D511yescharacteristic
248C2×C4×C5C22×D511yescharacteristic
249C2×C4×C5C22×D511yescharacteristic
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C22⋊D5—102no—
254C22⋊D5—201no—
255C2×Dic5—51no—
256C2×Dic5—51no—
257C2×Dic5—51no—
258C2×Dic5—51no—
259C2×Dic5—102no—
260C2×Dic5—102no—
261C2×Dic5—201no—
262C2×Dic5C22×D511yescharacteristic
263C2×Dic5C22×D511yescharacteristic
264C2×Dic5C22×D511yescharacteristic
265C2×Dic5C22×D511yescharacteristic
266C2×Dic5C22×D511yescharacteristic
267C2×Dic5C22×D511yescharacteristic
268D20—102no—
269D20—102no—
270D20—108no—
271D20—102no—
272D20—201no—
273C4×D5—22no—
274C4×D5—102no—
275C4×D5—102no—
276C4×D5—201no—
277C4×D5—201no—
278C4×D5—201no—
279C4×D5—201no—
280C4×D5—201no—
281C4×D5—204no—
282C4×D5—204no—
283Dic10—201no—
284Dic10—201no—
285#126—51no—
286#126—251no—
287#124—251no—
288#123—251no—
289#123—251no—
290#123—251no—
291C22⋊Q8—251no—
292C22⋊Q8—251no—
293#120—251no—
294C4×D4—251no—
295C4×D4—251no—
296#117—51no—
297#117—251no—
298C2×C22⋊C4—251no—
299C2×C22⋊C4—251no—
300C52#55111yescharacteristic, complemented, Sylow
301C22×C5—22no—
302C22×C5—52no—
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—52no—
316C2×D5—52no—
317C2×D5—104no—
318C2×D5—101no—
319C2×D5—101no—
320C2×D5—101no—
321C2×D5—101no—
322C2×D5—104no—
323C2×D5—101no—
324C2×D5—101no—
325C2×D5—101no—
326C2×D5—101no—
327C2×D5—101no—
328C2×D5—101no—
329C2×D5—202no—
330C2×D5—202no—
331C2×D5—202no—
332C2×D5—202no—
333C2×D5—204no—
334C2×D5—204no—
335C2×D5#70912yes—
336C4×C5—24no—
337C4×C5—24no—
338C4×C5—21no—
339C4×C5—21no—
340C4×C5—21no—
341C4×C5—21no—
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—42no—
345C4×C5—101no—
346C4×C5—101no—
347C4×C5—101no—
348C4×C5—101no—
349C4×C5—101no—
350C4×C5—101no—
351Dic5—21no—
352Dic5—21no—
353Dic5—21no—
354Dic5—21no—
355Dic5—21no—
356Dic5—21no—
357Dic5—101no—
358Dic5—101no—
359Dic5—101no—
360Dic5—101no—
361Dic5—202no—
362Dic5—202no—
363C24—251no—
364C2×Q8—251no—
365C2×D4—251no—
366C2×D4—251no—
367C2×D4—251no—
368C22×C4—51no—
369C22×C4—251no—
370C22×C4—251no—
371C22×C4—251no—
372C22×C4—251no—
373C4⋊C4—51no—
374C4⋊C4—51no—
375C4⋊C4—51no—
376C4⋊C4—51no—
377C4⋊C4—51no—
378C4⋊C4—251no—
379C4⋊C4—251no—
380C4⋊C4—251no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—51no—
384C22⋊C4—51no—
385C22⋊C4—51no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C22⋊C4—251no—
391C22⋊C4—251no—
392C22⋊C4—251no—
393C22⋊C4—251no—
394C22⋊C4—251no—
395C42—51no—
396C42—251no—
397C42D5211yescharacteristic, complemented
398C2×C5—22no—
399C2×C5—22no—
400C2×C5—22no—
401C2×C5—102no—
402C2×C5—201no—
403C2×C5C2×#70911yescharacteristic
404C2×C5D5×#4111yescharacteristic
405C2×C5D5×#4111yescharacteristic
406C2×C5D5×#4111yescharacteristic
407C2×C5#414911yescharacteristic
408C2×C5#414911yescharacteristic
409D5—22no—
410D5—41no—
411D5—102no—
412D5—102no—
413D5—204no—
414D5—204no—
415D5—201no—
416D5—201no—
417C23—51no—
418C23—51no—
419C23—251no—
420C23—251no—
421C23—502no—
422C23—502no—
423C23—1001no—
424Q8—1001no—
425D4—502no—
426D4—502no—
427D4—1001no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—102no—
435C2×C4—251no—
436C2×C4—251no—
437C2×C4—502no—
438C2×C4—1001no—
439C2×C4—1001no—
440C2×C4—1001no—
441C2×C4C2×D5211yescharacteristic
442C2×C4C2×D5211yescharacteristic
443C2×C4C2×D5211yescharacteristic
444C5—22no—
445C5#6586111yescharacteristic, complemented
446C5#6603911yescharacteristic, complemented
447C22—52no—
448C22—101no—
449C22—101no—
450C22—101no—
451C22—101no—
452C22—101no—
453C22—252no—
454C22—501no—
455C22—501no—
456C22—501no—
457C22—501no—
458C22—501no—
459C22—1004no—
460C22C22×D5211yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—101no—
465C4—101no—
466C4—101no—
467C4—101no—
468C4—101no—
469C4—101no—
470C4—501no—
471C4—501no—
472C2—102no—
473C2—201no—
474C2—502no—
475C2—1001no—
476C2D5×#70911yescharacteristic
477C2#1171747111yescharacteristic
478C2#1171747311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radical#11717041C2
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. C4×C4⋊(C5⋊D5)C2semidirect
2. C5×#4022C2semidirect
3. C5×#4018C2semidirect
5. #11717039C2semidirect
6. D5×#685C2semidirect
7. #11717016C2semidirect
8. #11717007C2semidirect
9. #11716999C2semidirect
10. #11716984C2semidirect
11. #11716977C2semidirect
12. #11716971C2semidirect
13. #11716970C2semidirect
14. #11716968C2semidirect
15. #11716954C2semidirect
19. C2×C4×C5⋊D5C22semidirect
21. C2×#6039C22semidirect
26. C42×C52C22semidirect
28. #87808C22semidirect
29. C4⋊#1039C22semidirect
35. C5×#683C22semidirect
36. C5×#683C22semidirect
37. C5×#683C22semidirect
38. C5×#683C22semidirect
39. C5×#683C22semidirect
40. C4×C5×Dic5C22semidirect
41. #87781C22semidirect
42. #87780C22semidirect
45. #87779C22semidirect
46. #87779C22semidirect
50. Dic52C22semidirect
104. #4022D5semidirect
110. #4018D5semidirect
171. C42×C5C2×D5semidirect
172. C42×C5C2×D5semidirect
205. #683C2×D5semidirect
206. #683C2×D5semidirect
207. #683C2×D5semidirect
208. #683C2×D5semidirect
209. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
300. C52#551semidirect
397. C42D52semidirect
445. C5#65861semidirect
446. C5#66039semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C5⋊D5)2
C5×#40222
C5×#40182
#117170412
#117170392
D5×#6852
#117170162
#117170072
#117169992
#117169842
#117169772
#117169712
#117169702
#117169682
#117169542
#660395
#658615