Finite Group Databasebeta

← #13106001

Subgroups — #13106001

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×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×C5×#685C211yesmaximal, characteristic, complemented
3C5×#4070C211yesmaximal, characteristic, complemented
4#11717095C211yesmaximal, characteristic
5D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
6#11717084C211yesmaximal, characteristic, complemented
7#11717073C211yesmaximal, characteristic, complemented
8#11717025C211yesmaximal, characteristic, complemented
9#11717011C211yesmaximal, characteristic, complemented
10#11717006C211yesmaximal, characteristic, complemented
11#11716999C211yesmaximal, characteristic, complemented
12#11716996C211yesmaximal, characteristic, complemented
13#11716980C211yesmaximal, characteristic, complemented
14#11716976C211yesmaximal, characteristic, complemented
15#11716971C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×C4×C52C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C4×C5⋊D5C2211yescharacteristic, complemented
20C22×C5×Dic5C2211yescharacteristic
21C2×C5×Dic10C2211yescharacteristic, complemented
22C2×#6038C2211yescharacteristic
23C2×#6038C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27#87808C2211yescharacteristic, complemented
28#87806C2211yescharacteristic, complemented
29C4×#1039C2211yescharacteristic, complemented
30C5×C22⋊Dic5C2211yescharacteristic
31C5×C22⋊Dic5C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#685C2211yescharacteristic
36C5×C4⋊Dic5C2211yescharacteristic, complemented
37C5×#683C2211yescharacteristic, complemented
38C5×#683C2211yescharacteristic, complemented
39#87780C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic, complemented
44#87779C2211yescharacteristic, complemented
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65971—51nomaximal
52#65829—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C22×C5×D5—22no—
56C22×C5×D5—22no—
57C22×C5×D5—41no—
58C22×C5×D5C2311yescharacteristic
59C22×C5×D5C2311yescharacteristic
60C2×C4×C52—41no—
61C2×C4×C52C2311yescharacteristic
62C2×C4×C52C2311yescharacteristic
63C22⋊(C5⋊D5)—22no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C4×C5⋊D5—41no—
68C2×C5×Dic5—22no—
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C5×Dic10—41no—
76#6038—22no—
77#6038—41no—
78D5×Dic5—22no—
79D5×Dic5—41no—
80C5×C22⋊Q8—51no—
81C2×C5×C22⋊C4—51no—
82#4099—51no—
83C2×C22⋊Dic5—51no—
84#4093—51no—
85#4092—51no—
86#4075—51no—
87C4×C22⋊D5—51no—
88C4×C22⋊D5—51no—
89C4×C22⋊D5—102no—
90C2×#685D511yescharacteristic, complemented
91#4072—51no—
92#4070D511yescharacteristic, complemented
93#4044—51no—
94#4044—51no—
95#4040—51no—
96#4040—51no—
97#4039—51no—
98#4038—51no—
99#4032—51no—
100#4032—51no—
101#4031—51no—
102#4029—51no—
103#4028—51no—
104#4027—51no—
105D5×C22⋊C4—51no—
106#4025—51no—
107#4025—51no—
108#4024—51no—
109#4023—51no—
110#4023—51no—
111C22×C52—21no—
112C22×C52—21no—
113C22×C52#4112yes—
114C22×C52C2411yescharacteristic
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—21no—
123C2×C5×D5—44no—
124C2×C5×D5#4112yes—
125C4×C52—21no—
126C4×C52—21no—
127#1039—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—
136C24×C5—51no—
137C23×D5C2×D511yescharacteristic
138C2×C5×Q8—51no—
139C22×C4×C5—22no—
140C22×C4×C5—51no—
141C22×C4×C5C2×D511yescharacteristic
142C22×C4×C5C2×D511yescharacteristic
143C2×C22⋊D5—51no—
144C2×C22⋊D5—51no—
145C2×C22⋊D5—51no—
146C2×C22⋊D5—51no—
147C2×C22⋊D5—102no—
148C22×Dic5—51no—
149C22×Dic5—51no—
150C22×Dic5C2×D511yescharacteristic
151C2×D20—51no—
152C2×D20—51no—
153C2×C4×D5—51no—
154C2×C4×D5—51no—
155C2×C4×D5—51no—
156C2×C4×D5—51no—
157C2×C4×D5—102no—
158C2×Dic10C2×D511yescharacteristic, complemented
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C4⋊C4—51no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C5×C22⋊C4—51no—
167C5×C22⋊C4—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—102no—
175C22⋊Dic5C2×D511yescharacteristic
176C22⋊Dic5C2×D511yescharacteristic
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#685C2×D511yescharacteristic
189#685C2×D511yescharacteristic
190#685C2×D511yescharacteristic
191#685C2×D511yescharacteristic
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—51no—
195C4⋊Dic5—51no—
196C4⋊Dic5C2×D511yescharacteristic, complemented
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—102no—
204#683C2×D511yescharacteristic, complemented
205#683C2×D511yescharacteristic, complemented
206C4×Dic5—51no—
207C4×Dic5—51no—
208C4×Dic5—51no—
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5—51no—
212C4×Dic5—102no—
213#551—251noSylow
214C2×C52—22no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—41no—
219C5×D5—22no—
220C5×D5—41no—
221C23×C5—22no—
222C23×C5—51no—
223C23×C5—51no—
224C23×C5—102no—
225C23×C5—102no—
226C23×C5—201no—
227C23×C5C22×D511yescharacteristic
228C23×C5C22×D511yescharacteristic
229C22×D5—22no—
230C22×D5—22no—
231C22×D5—41no—
232C22×D5—51no—
233C22×D5—51no—
234C22×D5—102no—
235C22×D5C22×D511yescharacteristic
236C22×D5C22×D511yescharacteristic
237C5×Q8—201no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—44no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
248C2×C4×C5—51no—
249C2×C4×C5—102no—
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C2×C4×C5C22×D511yescharacteristic
254C22⋊D5—102no—
255C22⋊D5—108no—
256C22⋊D5—102no—
257C22⋊D5—102no—
258C22⋊D5—201no—
259C2×Dic5—22no—
260C2×Dic5—51no—
261C2×Dic5—51no—
262C2×Dic5—51no—
263C2×Dic5—51no—
264C2×Dic5—51no—
265C2×Dic5—51no—
266C2×Dic5—102no—
267C2×Dic5—102no—
268C2×Dic5—102no—
269C2×Dic5—102no—
270C2×Dic5—201no—
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C2×Dic5C22×D511yescharacteristic
277D20—102no—
278D20—201no—
279C4×D5—102no—
280C4×D5—201no—
281C4×D5—201no—
282C4×D5—201no—
283C4×D5—204no—
284Dic10—41no—
285#126—251no—
286#126—251no—
287#124—251no—
288#123—251no—
289#123—251no—
290#123—251no—
291C22⋊Q8—51no—
292C22⋊Q8—251no—
293#120—251no—
294C4×D4—251no—
295C4×D4—251no—
296#117—251no—
297#117—251no—
298C2×C22⋊C4—51no—
299C2×C22⋊C4—251no—
300C52#55111yescharacteristic, complemented, Sylow
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—21no—
305C22×C5—24no—
306C22×C5—22no—
307C22×C5—42no—
308C22×C5—42no—
309C22×C5—52no—
310C22×C5—101no—
311C22×C5—101no—
312C22×C5—101no—
313C22×C5—101no—
314C22×C5—101no—
315C22×C5—204no—
316C22×C5#71112yes—
317C22×C5#71312yes—
318C22×C5C23×D511yescharacteristic
319C22×C5C23×D511yescharacteristic
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—44no—
326C2×D5—104no—
327C2×D5—101no—
328C2×D5—101no—
329C2×D5—101no—
330C2×D5—101no—
331C2×D5—101no—
332C2×D5—101no—
333C2×D5—202no—
334C2×D5—202no—
335C2×D5#70912yes—
336C4×C5—24no—
337C4×C5—21no—
338C4×C5—21no—
339C4×C5—21no—
340C4×C5—21no—
341C4×C5—42no—
342C4×C5—101no—
343C4×C5—101no—
344C4×C5—101no—
345C4×C5—101no—
346C4×C5—101no—
347C4×C5—101no—
348Dic5—21no—
349Dic5—21no—
350Dic5—21no—
351Dic5—21no—
352Dic5—21no—
353Dic5—21no—
354Dic5—104no—
355Dic5—101no—
356Dic5—101no—
357Dic5—101no—
358Dic5—101no—
359Dic5—101no—
360Dic5—101no—
361Dic5—202no—
362Dic5—202no—
363C24—51no—
364C2×Q8—51no—
365C2×D4—251no—
366C2×D4—251no—
367C2×D4—251no—
368C22×C4—51no—
369C22×C4—251no—
370C22×C4—251no—
371C22×C4—251no—
372C22×C4D5211yescharacteristic
373C4⋊C4—51no—
374C4⋊C4—51no—
375C4⋊C4—51no—
376C4⋊C4—251no—
377C4⋊C4—251no—
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—51no—
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—251no—
396C42—251no—
397C42—251no—
398C2×C5—22no—
399C2×C5—22no—
400C2×C5—22no—
401C2×C5—22no—
402C2×C5—22no—
403C2×C5—44no—
404C2×C5—102no—
405C2×C5—201no—
406C2×C5C2×#70911yescharacteristic
407C2×C5C2×#71111yescharacteristic
408C2×C5#414411yescharacteristic
409C2×C5#414411yescharacteristic
410C2×C5C2×#71311yescharacteristic
411C2×C5D5×#4111yescharacteristic
412D5—22no—
413D5—41no—
414D5—204no—
415D5—201no—
416D5—201no—
417C23—51no—
418C23—51no—
419C23—102no—
420C23—102no—
421C23—201no—
422C23—251no—
423C23C2×D5211yescharacteristic
424Q8—201no—
425D4—502no—
426D4—502no—
427D4—1001no—
428C2×C4—41no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—51no—
435C2×C4—102no—
436C2×C4—251no—
437C2×C4—251no—
438C2×C4—251no—
439C2×C4—502no—
440C2×C4—1001no—
441C2×C4—1001no—
442C2×C4C2×D5211yescharacteristic
443C2×C4C2×D5211yescharacteristic
444C5—22no—
445C5#6582911yescharacteristic, complemented
446C5#6597111yescharacteristic, complemented
447C22—21no—
448C22—21no—
449C22—52no—
450C22—101no—
451C22—101no—
452C22—101no—
453C22—101no—
454C22—101no—
455C22—204no—
456C22—501no—
457C22—501no—
458C22—501no—
459C22#8787112yes—
460C22C22×D5211yescharacteristic
461C4—21no—
462C4—21no—
463C4—101no—
464C4—101no—
465C4—101no—
466C4—101no—
467C4—101no—
468C4—101no—
469C4—501no—
470C4—501no—
471C4—501no—
472C2—22no—
473C2—102no—
474C2—201no—
475C2—1001no—
476C2C2×#8787111yescharacteristic
477C2D5×#70911yescharacteristic
478C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radical#11717095C2
hypercenterC22×C4D52
hypocenterC52#551
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. C2×C5×#685C2semidirect
3. C5×#4070C2semidirect
5. D5×C22⋊Dic5C2semidirect
6. #11717084C2semidirect
7. #11717073C2semidirect
8. #11717025C2semidirect
9. #11717011C2semidirect
10. #11717006C2semidirect
11. #11716999C2semidirect
12. #11716996C2semidirect
13. #11716980C2semidirect
14. #11716976C2semidirect
15. #11716971C2semidirect
19. C2×C4×C5⋊D5C22semidirect
21. C2×C5×Dic10C22semidirect
27. #87808C22semidirect
28. #87806C22semidirect
29. C4×#1039C22semidirect
36. C5×C4⋊Dic5C22semidirect
37. C5×#683C22semidirect
38. C5×#683C22semidirect
39. #87780C22semidirect
40. #87780C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
43. #87779C22semidirect
44. #87779C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
90. C2×#685D5semidirect
92. #4070D5semidirect
158. C2×Dic10C2×D5semidirect
196. C4⋊Dic5C2×D5semidirect
204. #683C2×D5semidirect
205. #683C2×D5semidirect
300. C52#551semidirect
445. C5#65829semidirect
446. C5#65971semidirect

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
C2×C5×#6852
C5×#40702
#117170952
D5×C22⋊Dic52
#117170842
#117170732
#117170252
#117170112
#117170062
#117169992
#117169962
#117169802
#117169762
#117169712
#659715
#658295