Finite Group Databasebeta

← #13106105

Subgroups — #13106105

Order 1600.

628 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#87808C211yesmaximal, characteristic, complemented
2C4×C5×C22⋊D5C211yesmaximal, characteristic, complemented
3C5×#4070C211yesmaximal, characteristic, complemented
4#11717099C211yesmaximal, characteristic
5#11717092C211yesmaximal, characteristic, complemented
6#11717080C211yesmaximal, characteristic, complemented
7#11717075C211yesmaximal, characteristic, complemented
8#11717025C211yesmaximal, characteristic, complemented
9#11717022C211yesmaximal, characteristic, complemented
10#11717008C211yesmaximal, characteristic, complemented
11#11716987C211yesmaximal, characteristic, complemented
12#11716972C211yesmaximal, characteristic, complemented
13#11716968C211yesmaximal, characteristic, complemented
14#11716966C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C23×C5⋊D5C2211yescharacteristic
17C22×C4×C52C2211yescharacteristic
18C22×#1039C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic, complemented
21C2×C5×Dic10C2211yescharacteristic, complemented
22C2×#6038C2211yescharacteristic
23C2×#6038C2211yescharacteristic
24C2×#6036C2211yescharacteristic
25C2×#6036C2211yescharacteristic
26#87808C2211yescharacteristic
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic
29#87808C2211yescharacteristic
30C5×C22⋊Dic5C2211yescharacteristic
31C5×C22⋊Dic5C2211yescharacteristic
32C5×C22⋊Dic5C2211yescharacteristic
33C5×#685C2211yescharacteristic, complemented
34C5×C4⋊Dic5C2211yescharacteristic, complemented
35C5×#683C2211yescharacteristic, complemented
36C5×#683C2211yescharacteristic, complemented
37C5×#683C2211yescharacteristic, complemented
38C4×C5×Dic5C2211yescharacteristic, complemented
39#87781C2211yescharacteristic, complemented
40#87781C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87779C2211yescharacteristic
46#87779C2211yescharacteristic
47#87778C2211yescharacteristic, complemented
48#87778C2211yescharacteristic, complemented
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65971—51nomaximal
52#65890—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5—22no—
55C22×C5⋊D5—22no—
56C22×C5⋊D5—41no—
57C22×C5⋊D5C2311yescharacteristic
58C22×C5⋊D5C2311yescharacteristic
59C22×C5×D5C2311yescharacteristic
60C2×C4×C52—41no—
61C2×C4×C52C2311yescharacteristic
62C2×C4×C52C2311yescharacteristic
63C2×#1039—22no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C5×C22⋊D5—22no—
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C4×C5×D5—41no—
75C5×Dic10—41no—
76#6038—22no—
77#6038—41no—
78#6036—22no—
79#6036—41no—
80C5×C22⋊Q8—51no—
81C4×C5×D4—51no—
82#4093—51no—
83#4092—51no—
84#4083—51no—
85#4081—51no—
86#4075—51no—
87C4×C22⋊D5—51no—
88C4×C22⋊D5D511yescharacteristic, complemented
89C2×#685—51no—
90C2×#685—51no—
91C2×#685—102no—
92#4070D511yescharacteristic, complemented
93#4044—51no—
94#4044—51no—
95#4044—51no—
96#4042—51no—
97#4040—51no—
98#4040—51no—
99#4039—51no—
100#4038—51no—
101#4032—51no—
102#4031—51no—
103#4029—51no—
104#4028—51no—
105D5×C22⋊C4—51no—
106D5×C22⋊C4—51no—
107#4023—51no—
108#4023—51no—
109#4022—51no—
110#4018—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—44no—
121C2×C5⋊D5#4112yes—
122C2×C5×D5—21no—
123C2×C5×D5—21no—
124C2×C5×D5—21no—
125C4×C52—21no—
126C4×C52—21no—
127#1039—21no—
128#1039—21no—
129C5×Dic5—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—
138C23×D5—102no—
139C2×C5×Q8—51no—
140C2×C5×D4—51no—
141C22×C4×C5—22no—
142C22×C4×C5—51no—
143C22×C4×C5C2×D511yescharacteristic
144C22×C4×C5C2×D511yescharacteristic
145C2×C22⋊D5—51no—
146C2×C22⋊D5—51no—
147C2×C22⋊D5C2×D511yescharacteristic
148C22×Dic5—51no—
149C22×Dic5—51no—
150C22×Dic5—102no—
151C2×D20—51no—
152C2×D20—51no—
153C2×C4×D5—51no—
154C2×C4×D5—51no—
155C2×C4×D5—51no—
156C2×C4×D5—51no—
157C2×C4×D5C2×D511yescharacteristic, complemented
158C2×Dic10C2×D511yescharacteristic, complemented
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C4⋊C4—51no—
162C5×C4⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C5×C22⋊C4—51no—
167C42×C5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5C2×D511yescharacteristic
171C22⋊Dic5C2×D511yescharacteristic
172C22⋊Dic5C2×D511yescharacteristic
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—51no—
188#685—51no—
189#685—51no—
190#685—51no—
191#685—102no—
192#685—102no—
193#685—102no—
194#685—102no—
195#685C2×D511yescharacteristic, complemented
196C4⋊Dic5—51no—
197C4⋊Dic5—51no—
198C4⋊Dic5C2×D511yescharacteristic, complemented
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683C2×D511yescharacteristic, complemented
206#683C2×D511yescharacteristic, complemented
207#683C2×D511yescharacteristic, complemented
208C4×Dic5—51no—
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5—51no—
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52—22no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—22no—
219C5⋊D5—41no—
220C5×D5—41no—
221C23×C5—22no—
222C23×C5—51no—
223C23×C5C22×D511yescharacteristic
224C23×C5C22×D511yescharacteristic
225C22×D5—51no—
226C22×D5—51no—
227C22×D5—51no—
228C22×D5—51no—
229C22×D5—102no—
230C22×D5—102no—
231C22×D5—102no—
232C22×D5—102no—
233C22×D5—108no—
234C22×D5—102no—
235C22×D5—102no—
236C22×D5—204no—
237C22×D5—201no—
238C22×D5—201no—
239C22×D5—204no—
240C22×D5C22×D511yescharacteristic
241C5×Q8—201no—
242C5×D4—102no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—44no—
248C2×C4×C5—51no—
249C2×C4×C5—51no—
250C2×C4×C5—51no—
251C2×C4×C5—51no—
252C2×C4×C5—51no—
253C2×C4×C5—51no—
254C2×C4×C5—51no—
255C2×C4×C5—201no—
256C2×C4×C5C22×D511yescharacteristic
257C2×C4×C5C22×D511yescharacteristic
258C2×C4×C5C22×D511yescharacteristic
259C2×C4×C5C22×D511yescharacteristic
260C22⋊D5—22no—
261C22⋊D5—102no—
262C22⋊D5—201no—
263C2×Dic5—51no—
264C2×Dic5—51no—
265C2×Dic5—51no—
266C2×Dic5—51no—
267C2×Dic5—102no—
268C2×Dic5—102no—
269C2×Dic5—102no—
270C2×Dic5—102no—
271C2×Dic5—204no—
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C2×Dic5C22×D511yescharacteristic
277C2×Dic5C22×D511yescharacteristic
278C2×Dic5C22×D511yescharacteristic
279D20—102no—
280D20—201no—
281C4×D5—41no—
282C4×D5—102no—
283C4×D5—102no—
284C4×D5—201no—
285C4×D5—201no—
286Dic10—41no—
287#126—251no—
288#126—251no—
289#124—251no—
290#123—251no—
291#123—251no—
292#123—251no—
293C22⋊Q8—51no—
294C22⋊Q8—251no—
295#120—251no—
296C4×D4—51no—
297C4×D4—251no—
298#117—251no—
299#117—251no—
300C2×C22⋊C4—251no—
301C2×C22⋊C4—251no—
302C52#55111yescharacteristic, complemented, Sylow
303C22×C5—24no—
304C22×C5—24no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—21no—
309C22×C5—24no—
310C22×C5—22no—
311C22×C5—101no—
312C22×C5—101no—
313C22×C5—101no—
314C22×C5#70912yes—
315C22×C5#71312yes—
316C22×C5C23×D511yescharacteristic
317C22×C5C23×D511yescharacteristic
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—52no—
322C2×D5—52no—
323C2×D5—104no—
324C2×D5—101no—
325C2×D5—101no—
326C2×D5—101no—
327C2×D5—101no—
328C2×D5—104no—
329C2×D5—101no—
330C2×D5—101no—
331C2×D5—101no—
332C2×D5—101no—
333C2×D5—101no—
334C2×D5—101no—
335C2×D5—202no—
336C2×D5—202no—
337C2×D5—202no—
338C2×D5—202no—
339C2×D5—204no—
340C2×D5—204no—
341C2×D5—2016no—
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—21no—
345C4×C5—21no—
346C4×C5—42no—
347C4×C5—42no—
348C4×C5—101no—
349C4×C5—101no—
350C4×C5—101no—
351C4×C5—101no—
352C4×C5—101no—
353C4×C5—101no—
354C4×C5—101no—
355Dic5—21no—
356Dic5—21no—
357Dic5—21no—
358Dic5—21no—
359Dic5—21no—
360Dic5—21no—
361Dic5—21no—
362Dic5—101no—
363Dic5—101no—
364Dic5—101no—
365Dic5—101no—
366Dic5—202no—
367Dic5—202no—
368C24—251no—
369C2×Q8—51no—
370C2×D4—51no—
371C2×D4—251no—
372C2×D4—251no—
373C22×C4—51no—
374C22×C4—251no—
375C22×C4—251no—
376C22×C4—251no—
377C22×C4D5211yescharacteristic
378C4⋊C4—51no—
379C4⋊C4—51no—
380C4⋊C4—51no—
381C4⋊C4—51no—
382C4⋊C4—251no—
383C4⋊C4—251no—
384C4⋊C4—251no—
385C4⋊C4—251no—
386C22⋊C4—51no—
387C22⋊C4—51no—
388C22⋊C4—51no—
389C22⋊C4—51no—
390C22⋊C4—251no—
391C22⋊C4—251no—
392C22⋊C4—251no—
393C22⋊C4—251no—
394C22⋊C4—251no—
395C22⋊C4—251no—
396C22⋊C4—251no—
397C22⋊C4—251no—
398C22⋊C4—251no—
399C22⋊C4—251no—
400C42—51no—
401C42—251no—
402C42—251no—
403C2×C5—28no—
404C2×C5—22no—
405C2×C5—22no—
406C2×C5—22no—
407C2×C5—22no—
408C2×C5—22no—
409C2×C5—201no—
410C2×C5C2×#70911yescharacteristic
411C2×C5#414411yescharacteristic
412C2×C5#414411yescharacteristic
413C2×C5C2×#71311yescharacteristic
414C2×C5D5×#4111yescharacteristic
415C2×C5D5×#4111yescharacteristic
416D5—41no—
417D5—102no—
418D5—102no—
419D5—204no—
420D5—204no—
421D5—201no—
422D5—201no—
423C23—51no—
424C23—251no—
425C23—251no—
426C23—502no—
427C23—502no—
428C23—1001no—
429C23C2×D5211yescharacteristic
430Q8—201no—
431D4—102no—
432D4—502no—
433D4—1001no—
434C2×C4—41no—
435C2×C4—51no—
436C2×C4—51no—
437C2×C4—51no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—201no—
443C2×C4—251no—
444C2×C4—251no—
445C2×C4—502no—
446C2×C4—502no—
447C2×C4—1001no—
448C2×C4C2×D5211yescharacteristic
449C2×C4C2×D5211yescharacteristic
450C5—22no—
451C5#6589011yescharacteristic, complemented
452C5#6597111yescharacteristic, complemented
453C22—21no—
454C22—21no—
455C22—101no—
456C22—101no—
457C22—101no—
458C22—252no—
459C22—501no—
460C22—501no—
461C22—501no—
462C22—501no—
463C22—501no—
464C22—1004no—
465C22#8787412yes—
466C22C22×D5211yescharacteristic
467C4—21no—
468C4—21no—
469C4—101no—
470C4—101no—
471C4—101no—
472C4—101no—
473C4—101no—
474C4—101no—
475C4—101no—
476C4—501no—
477C4—501no—
478C2—22no—
479C2—201no—
480C2—502no—
481C2—1001no—
482C2C2×#8787411yescharacteristic
483C2#1171747111yescharacteristic
484C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radical#11717099C2
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. C2×#87808C2semidirect
2. C4×C5×C22⋊D5C2semidirect
3. C5×#4070C2semidirect
5. #11717092C2semidirect
6. #11717080C2semidirect
7. #11717075C2semidirect
8. #11717025C2semidirect
9. #11717022C2semidirect
10. #11717008C2semidirect
11. #11716987C2semidirect
12. #11716972C2semidirect
13. #11716968C2semidirect
14. #11716966C2semidirect
15. #11716954C2semidirect
20. C2×C4×C5×D5C22semidirect
21. C2×C5×Dic10C22semidirect
33. C5×#685C22semidirect
34. C5×C4⋊Dic5C22semidirect
35. C5×#683C22semidirect
36. C5×#683C22semidirect
37. C5×#683C22semidirect
38. C4×C5×Dic5C22semidirect
39. #87781C22semidirect
40. #87781C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
47. #87778C22semidirect
48. #87778C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
88. C4×C22⋊D5D5semidirect
92. #4070D5semidirect
157. C2×C4×D5C2×D5semidirect
158. C2×Dic10C2×D5semidirect
195. #685C2×D5semidirect
198. C4⋊Dic5C2×D5semidirect
205. #683C2×D5semidirect
206. #683C2×D5semidirect
207. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
302. C52#551semidirect
451. C5#65890semidirect
452. C5#65971semidirect

Maximal subgroups

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

subgroupindex
C2×#878082
C4×C5×C22⋊D52
C5×#40702
#117170992
#117170922
#117170802
#117170752
#117170252
#117170222
#117170082
#117169872
#117169722
#117169682
#117169662
#117169542
#659715
#658905