Finite Group Databasebeta

← #13106528

Subgroups — #13106528

Order 1600.

570 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717266C211yesmaximal, characteristic, complemented
2C5×#4093C211yesmaximal, characteristic, complemented
3C5×#4092C211yesmaximal, characteristic, complemented
4#11717040C211yesmaximal, characteristic
5#11717039C211yesmaximal, characteristic, complemented
6D5×#685C211yesmaximal, characteristic, complemented
7#11717003C211yesmaximal, characteristic, complemented
8#11717002C211yesmaximal, characteristic, complemented
9#11716999C211yesmaximal, characteristic, complemented
10#11716997C211yesmaximal, characteristic, complemented
11#11716995C211yesmaximal, characteristic, complemented
12#11716977C211yesmaximal, characteristic, complemented
13#11716976C211yesmaximal, characteristic, complemented
14#11716972C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C52×Q8C2211yescharacteristic, complemented
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C5×D20C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic
21C2×#6038C2211yescharacteristic
22C2×#6037C2211yescharacteristic
23C2×#6036C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic, complemented
26#87808C2211yescharacteristic
27#87808C2211yescharacteristic
28C4⋊#1039C2211yescharacteristic, complemented
29#87806C2211yescharacteristic, complemented
30#87806C2211yescharacteristic, complemented
31C5×#685C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#685C2211yescharacteristic, complemented
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#87778C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic, complemented
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#66013—51nomaximal
52#65971—51nomaximal
53C22×C5⋊D5C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C2×D52—22no—
58C2×D52—22no—
59C2×D52—41no—
60C52×Q8—41no—
61C2×C4×C52C2311yescharacteristic
62C2×C4×C52C2311yescharacteristic
63C2×C4×C52C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C4×C5⋊D5—41no—
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C5×D20—41no—
74C4×C5×D5—41no—
75#6038—22no—
76#6037—22no—
77#6036—22no—
78D5×Dic5—22no—
79D5×Dic5—41no—
80C5×#124—51no—
81C5×C22⋊Q8—51no—
82#4093D511yescharacteristic, complemented
83#4092—51no—
84#4092—51no—
85#4092—102no—
86#4092D511yescharacteristic, complemented
87#4083—51no—
88#4081—51no—
89C4×C22⋊D5—51no—
90C2×#685—51no—
91#4044—51no—
92#4044—51no—
93#4040—51no—
94#4040—51no—
95#4039—51no—
96#4038—51no—
97#4032—51no—
98#4029—51no—
99#4029—51no—
100#4028—51no—
101#4027—51no—
102#4027—51no—
103D5×C22⋊C4—51no—
104D5×C22⋊C4—51no—
105D5×C22⋊C4—51no—
106#4025—51no—
107#4025—51no—
108#4023—51no—
109#4018—51no—
110#4018—51no—
111C22×C52C2411yescharacteristic
112C2×C5⋊D5—21no—
113C2×C5⋊D5—21no—
114C2×C5⋊D5#4112yes—
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#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×Q8—22no—
139C2×C5×Q8C2×D511yescharacteristic, complemented
140C2×C5×Q8C2×D511yescharacteristic, complemented
141C2×C5×D4—51no—
142C22×C4×C5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—51no—
145C2×C22⋊D5—51no—
146C22×Dic5—51no—
147C22×Dic5—51no—
148C2×D20—51no—
149C2×D20C2×D511yescharacteristic
150C2×C4×D5—51no—
151C2×C4×D5—51no—
152C2×C4×D5—51no—
153C2×C4×D5—51no—
154C2×C4×D5—51no—
155C2×C4×D5—51no—
156C2×C4×D5—102no—
157C2×C4×D5C2×D511yescharacteristic
158C5×C4⋊C4—51no—
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C22⋊C4—51no—
162C5×C22⋊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⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—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—102no—
185#685—102no—
186#685C2×D511yescharacteristic
187#685C2×D511yescharacteristic
188#685C2×D511yescharacteristic
189#685C2×D511yescharacteristic
190#685C2×D511yescharacteristic, complemented
191#685C2×D511yescharacteristic, complemented
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—51no—
195C4⋊Dic5—102no—
196C4⋊Dic5C2×D511yescharacteristic, complemented
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—102no—
205#683—102no—
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×C52C2×#4111yescharacteristic
215C2×C52C2×#4111yescharacteristic
216C2×C52H(5,𝔽2)11yescharacteristic
217C5⋊D5—22no—
218C5×D5—22no—
219C5×D5—41no—
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×Q8—41no—
237C5×Q8—41no—
238C5×Q8—44no—
239C5×D4—201no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
248C2×C4×C5—201no—
249C2×C4×C5C22×D511yescharacteristic
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C2×C4×C5C22×D511yescharacteristic
254C2×C4×C5C22×D511yescharacteristic
255C22⋊D5—102no—
256C22⋊D5—102no—
257C22⋊D5—102no—
258C2×Dic5—51no—
259C2×Dic5—51no—
260C2×Dic5—51no—
261C2×Dic5—51no—
262C2×Dic5—51no—
263C2×Dic5—51no—
264C2×Dic5—102no—
265C2×Dic5—102no—
266C2×Dic5—102no—
267C2×Dic5—102no—
268C2×Dic5—201no—
269C2×Dic5C22×D511yescharacteristic
270C2×Dic5C22×D511yescharacteristic
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274D20—41no—
275D20—102no—
276C4×D5—41no—
277C4×D5—102no—
278C4×D5—102no—
279C4×D5—102no—
280C4×D5—201no—
281C4×D5—201no—
282C4×D5—201no—
283C4×D5—204no—
284#126—251no—
285#126—251no—
286#124—51no—
287#123—251no—
288#123—251no—
289#123—251no—
290C22⋊Q8—51no—
291C22⋊Q8—251no—
292#120—251no—
293C4×D4—251no—
294C4×D4—251no—
295#117—251no—
296#117—251no—
297C2×C22⋊C4—251no—
298C2×C22⋊C4—251no—
299C52#55111yescharacteristic, complemented, Sylow
300C22×C5—22no—
301C22×C5—52no—
302C22×C5—101no—
303C22×C5—101no—
304C22×C5—101no—
305C22×C5—101no—
306C22×C5—101no—
307C22×C5—101no—
308C22×C5—101no—
309C22×C5—101no—
310C22×C5C23×D511yescharacteristic
311C22×C5C23×D511yescharacteristic
312C2×D5—21no—
313C2×D5—21no—
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—21no—
317C2×D5—21no—
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—52no—
321C2×D5—52no—
322C2×D5—104no—
323C2×D5—101no—
324C2×D5—101no—
325C2×D5—101no—
326C2×D5—101no—
327C2×D5—202no—
328C2×D5—202no—
329C2×D5—204no—
330C2×D5—204no—
331C2×D5#71312yes—
332C4×C5—24no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—21no—
338C4×C5—21no—
339C4×C5—42no—
340C4×C5—42no—
341C4×C5—101no—
342C4×C5—101no—
343C4×C5—101no—
344C4×C5—101no—
345C4×C5—101no—
346Dic5—21no—
347Dic5—21no—
348Dic5—21no—
349Dic5—21no—
350Dic5—21no—
351Dic5—101no—
352Dic5—101no—
353Dic5—101no—
354Dic5—101no—
355Dic5—101no—
356Dic5—101no—
357Dic5—202no—
358Dic5—202no—
359Dic5—202no—
360C24—251no—
361C2×Q8D5211yescharacteristic, complemented
362C2×D4—51no—
363C2×D4—251no—
364C2×D4—251no—
365C22×C4—51no—
366C22×C4—251no—
367C22×C4—251no—
368C22×C4—251no—
369C22×C4—251no—
370C4⋊C4—51no—
371C4⋊C4—51no—
372C4⋊C4—51no—
373C4⋊C4—251no—
374C4⋊C4—251no—
375C4⋊C4—251no—
376C4⋊C4—251no—
377C4⋊C4—251no—
378C22⋊C4—51no—
379C22⋊C4—51no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—51no—
384C22⋊C4—251no—
385C22⋊C4—251no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C22⋊C4—251no—
391C22⋊C4—251no—
392C42—51no—
393C42—251no—
394C42—251no—
395C2×C5—22no—
396C2×C5—22no—
397C2×C5—22no—
398C2×C5—102no—
399C2×C5—201no—
400C2×C5—201no—
401C2×C5#414411yescharacteristic
402C2×C5#414411yescharacteristic
403C2×C5C2×#71311yescharacteristic
404C2×C5D5×#4111yescharacteristic
405C2×C5D5×#4111yescharacteristic
406C2×C5D5×#4111yescharacteristic
407D5—22no—
408D5—41no—
409D5—41no—
410D5—102no—
411D5—102no—
412D5—204no—
413C23—51no—
414C23—51no—
415C23—51no—
416C23—251no—
417C23—502no—
418C23—502no—
419C23—1001no—
420Q8—41no—
421D4—201no—
422D4—502no—
423D4—502no—
424C2×C4—51no—
425C2×C4—51no—
426C2×C4—51no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—201no—
430C2×C4—251no—
431C2×C4—251no—
432C2×C4—251no—
433C2×C4—502no—
434C2×C4—502no—
435C2×C4—1001no—
436C2×C4—1001no—
437C2×C4C2×D5211yescharacteristic
438C2×C4C2×D5211yescharacteristic
439C2×C4C2×D5211yescharacteristic
440C5—22no—
441C5#6597111yescharacteristic, complemented
442C5#6601311yescharacteristic, complemented
443C22—52no—
444C22—101no—
445C22—101no—
446C22—101no—
447C22—101no—
448C22—101no—
449C22—101no—
450C22—101no—
451C22—101no—
452C22—252no—
453C22—501no—
454C22—501no—
455C22—1004no—
456C22C22×D5211yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—101no—
461C4—101no—
462C4—101no—
463C4—101no—
464C4—101no—
465C4—501no—
466C4—501no—
467C4—501no—
468C2—102no—
469C2—201no—
470C2—201no—
471C2—502no—
472C2D5×#71311yescharacteristic
473C2#1171748411yescharacteristic
474C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radical#11717040C2
hypercenterC2×Q8D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11717266C2semidirect
2. C5×#4093C2semidirect
3. C5×#4092C2semidirect
5. #11717039C2semidirect
6. D5×#685C2semidirect
7. #11717003C2semidirect
8. #11717002C2semidirect
9. #11716999C2semidirect
10. #11716997C2semidirect
11. #11716995C2semidirect
12. #11716977C2semidirect
13. #11716976C2semidirect
14. #11716972C2semidirect
15. #11716954C2semidirect
17. C2×C52×Q8C22semidirect
25. C2×D5×Dic5C22semidirect
28. C4⋊#1039C22semidirect
29. #87806C22semidirect
30. #87806C22semidirect
35. C5×#685C22semidirect
36. C5×#685C22semidirect
37. C5×C4⋊Dic5C22semidirect
38. C5×#683C22semidirect
39. C5×#683C22semidirect
40. C4×C5×Dic5C22semidirect
41. #87781C22semidirect
42. #87780C22semidirect
48. #87778C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
82. #4093D5semidirect
86. #4092D5semidirect
139. C2×C5×Q8C2×D5semidirect
140. C2×C5×Q8C2×D5semidirect
190. #685C2×D5semidirect
191. #685C2×D5semidirect
196. C4⋊Dic5C2×D5semidirect
206. #683C2×D5semidirect
207. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
299. C52#551semidirect
361. C2×Q8D52semidirect
441. C5#65971semidirect
442. C5#66013semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×#40932
C5×#40922
#117170402
#117170392
D5×#6852
#117170032
#117170022
#117169992
#117169972
#117169952
#117169772
#117169762
#117169722
#117169542
#660135
#659715