Finite Group Databasebeta

← #13105395

Subgroups — #13105395

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
1#11717201C211yesmaximal, characteristic, complemented
2C5×#4031C211yesmaximal, characteristic, complemented
3C5×#4029C211yesmaximal, characteristic, complemented
4#11717101C211yesmaximal, characteristic
5#11717093C211yesmaximal, characteristic, complemented
6#11717091C211yesmaximal, characteristic, complemented
7#11717080C211yesmaximal, characteristic, complemented
8#11717039C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717006C211yesmaximal, characteristic, complemented
11#11716990C211yesmaximal, characteristic, complemented
12#11716980C211yesmaximal, characteristic, complemented
13#11716976C211yesmaximal, characteristic, complemented
14#11716973C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C22×#1039C2211yescharacteristic, complemented
19C2×C4×C5⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C4×C5×D5C2211yescharacteristic
23C2×C5×Dic10C2211yescharacteristic, complemented
24C2×#6036C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C52×C22⋊C4C2211yescharacteristic, complemented
27#87808C2211yescharacteristic
28#87806C2211yescharacteristic, complemented
29C4×#1039C2211yescharacteristic, complemented
30C5×C22⋊Dic5C2211yescharacteristic, complemented
31C5×C22⋊Dic5C2211yescharacteristic, complemented
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×C4⋊Dic5C2211yescharacteristic, complemented
36C5×#683C2211yescharacteristic, complemented
37C4×C5×Dic5C2211yescharacteristic, complemented
38#87780C2211yescharacteristic, complemented
39#87780C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50Dic52C2211yescharacteristic, complemented
51#66013—51nomaximal
52#65992—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C2×D52—22no—
58C2×D52—22no—
59C2×D52—41no—
60C2×C4×C52C2311yescharacteristic
61C2×C4×C52C2311yescharacteristic
62C22⋊(C5⋊D5)—22no—
63C2×#1039—41no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C4×C5⋊D5—41no—
68C5×C22⋊D5—22no—
69C5×C22⋊D5—41no—
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C4×C5×D5—22no—
77C5×Dic10—41no—
78#6036—22no—
79D5×Dic5—41no—
80C5×#124—51no—
81C5×#123—51no—
82#4092—51no—
83#4092—51no—
84#4083—51no—
85#4083—51no—
86#4081—51no—
87D4×Dic5—51no—
88C2×#685—51no—
89#4072—51no—
90#4044—51no—
91#4044—51no—
92#4043—51no—
93#4038—51no—
94#4032—51no—
95#4031D511yescharacteristic, complemented
96#4029—51no—
97#4029—51no—
98#4029D511yescharacteristic, complemented
99#4027—51no—
100#4027—51no—
101#4027—51no—
102#4027—102no—
103D5×C22⋊C4—51no—
104D5×C22⋊C4—51no—
105D5×C22⋊C4—51no—
106#4025—51no—
107#4025—51no—
108#4024—51no—
109#4018—51no—
110#4018—51no—
111C22×C52—21no—
112C22×C52—21no—
113C22×C52—21no—
114C22×C52C2411yescharacteristic
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—
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—
136C23×D5—51no—
137C23×D5—51no—
138C2×C5×Q8—51no—
139C2×C5×D4—51no—
140C2×C5×D4—51no—
141C22×C4×C5—51no—
142C2×C22⋊D5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—102no—
145C2×C22⋊D5C2×D511yescharacteristic
146C2×C22⋊D5C2×D511yescharacteristic
147C22×Dic5—51no—
148C22×Dic5—51no—
149C22×Dic5—51no—
150C22×Dic5—102no—
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
158C2×Dic10C2×D511yescharacteristic, complemented
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C22⋊C4—22no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C5×C22⋊C4—51no—
167C5×C22⋊C4C2×D511yescharacteristic, complemented
168C5×C22⋊C4C2×D511yescharacteristic, complemented
169C42×C5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5C2×D511yescharacteristic, complemented
176C22⋊Dic5C2×D511yescharacteristic, complemented
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—102no—
189#685C2×D511yescharacteristic
190#685C2×D511yescharacteristic
191#685C2×D511yescharacteristic
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—51no—
195C4⋊Dic5—51no—
196C4⋊Dic5—51no—
197C4⋊Dic5C2×D511yescharacteristic, complemented
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#683C2×D511yescharacteristic, complemented
207C4×Dic5—51no—
208C4×Dic5—51no—
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5—102no—
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52—41no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—22no—
219C5×D5—22no—
220C5×D5—41no—
221C23×C5—22no—
222C23×C5—51no—
223C23×C5—51no—
224C23×C5C22×D511yescharacteristic
225C23×C5C22×D511yescharacteristic
226C22×D5—51no—
227C22×D5—51no—
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
237C5×Q8—201no—
238C5×D4—102no—
239C5×D4—201no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—51no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
248C2×C4×C5—102no—
249C2×C4×C5C22×D511yescharacteristic
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C22⋊D5—22no—
254C22⋊D5—41no—
255C22⋊D5—102no—
256C22⋊D5—102no—
257C22⋊D5—108no—
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—201no—
268C2×Dic5—201no—
269C2×Dic5—201no—
270C2×Dic5—204no—
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C2×Dic5C22×D511yescharacteristic
277C4×D5—22no—
278C4×D5—102no—
279C4×D5—102no—
280C4×D5—201no—
281C4×D5—201no—
282C4×D5—204no—
283C4×D5—201no—
284Dic10—41no—
285#126—251no—
286#126—251no—
287#124—51no—
288#123—51no—
289#123—251no—
290#123—251no—
291C22⋊Q8—251no—
292C22⋊Q8—251no—
293#120—251no—
294C4×D4—251no—
295C4×D4—251no—
296#117—251no—
297#117—251no—
298C2×C22⋊C4—251no—
299C2×C22⋊C4—251no—
300C52#55111yescharacteristic, complemented, Sylow
301C22×C5—24no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—22no—
309C22×C5—42no—
310C22×C5—42no—
311C22×C5—52no—
312C22×C5—101no—
313C22×C5—101no—
314C22×C5—101no—
315C22×C5—101no—
316C22×C5—101no—
317C22×C5C23×D511yescharacteristic
318C22×C5C23×D511yescharacteristic
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—52no—
325C2×D5—52no—
326C2×D5—104no—
327C2×D5—101no—
328C2×D5—101no—
329C2×D5—101no—
330C2×D5—101no—
331C2×D5—202no—
332C2×D5—202no—
333C2×D5—204no—
334C2×D5—204no—
335C2×D5#71112yes—
336C4×C5—21no—
337C4×C5—21no—
338C4×C5—21no—
339C4×C5—21no—
340C4×C5—42no—
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—104no—
356Dic5—101no—
357Dic5—101no—
358Dic5—101no—
359Dic5—101no—
360Dic5—101no—
361Dic5—101no—
362Dic5—202no—
363C24—251no—
364C2×Q8—51no—
365C2×D4—51no—
366C2×D4—51no—
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—251no—
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—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C22⋊C4—251no—
391C22⋊C4—251no—
392C22⋊C4—251no—
393C22⋊C4—251no—
394C22⋊C4D5211yescharacteristic, complemented
395C42—51no—
396C42—251no—
397C42—251no—
398C2×C5—22no—
399C2×C5—22no—
400C2×C5—22no—
401C2×C5—44no—
402C2×C5—41no—
403C2×C5—41no—
404C2×C5—102no—
405C2×C5—201no—
406C2×C5C2×#71111yescharacteristic
407C2×C5#414411yescharacteristic
408C2×C5D5×#4111yescharacteristic
409C2×C5D5×#4111yescharacteristic
410C2×C5D5×#4111yescharacteristic
411C2×C5#414911yescharacteristic
412D5—22no—
413D5—41no—
414D5—102no—
415D5—102no—
416D5—204no—
417C23—51no—
418C23—51no—
419C23—251no—
420C23—502no—
421C23—502no—
422C23—1001no—
423C23C2×D5211yescharacteristic
424Q8—201no—
425D4—102no—
426D4—201no—
427D4—502no—
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—251no—
438C2×C4—502no—
439C2×C4—1001no—
440C2×C4—1001no—
441C2×C4—1001no—
442C2×C4C2×D5211yescharacteristic
443C2×C4C2×D5211yescharacteristic
444C5—22no—
445C5#6599211yescharacteristic, complemented
446C5#6601311yescharacteristic, complemented
447C22—21no—
448C22—21no—
449C22—21no—
450C22—52no—
451C22—101no—
452C22—101no—
453C22—101no—
454C22—101no—
455C22—101no—
456C22—252no—
457C22—501no—
458C22—501no—
459C22—1004no—
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—41no—
473C2—102no—
474C2—201no—
475C2—502no—
476C2#1171747111yescharacteristic
477C2D5×#71111yescharacteristic
478C2#1171747911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717101C2
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. #11717201C2semidirect
2. C5×#4031C2semidirect
3. C5×#4029C2semidirect
5. #11717093C2semidirect
6. #11717091C2semidirect
7. #11717080C2semidirect
8. #11717039C2semidirect
9. D5×#685C2semidirect
10. #11717006C2semidirect
11. #11716990C2semidirect
12. #11716980C2semidirect
13. #11716976C2semidirect
14. #11716973C2semidirect
15. #11716954C2semidirect
18. C22×#1039C22semidirect
23. C2×C5×Dic10C22semidirect
26. C52×C22⋊C4C22semidirect
28. #87806C22semidirect
29. C4×#1039C22semidirect
30. C5×C22⋊Dic5C22semidirect
31. C5×C22⋊Dic5C22semidirect
35. C5×C4⋊Dic5C22semidirect
36. C5×#683C22semidirect
37. C4×C5×Dic5C22semidirect
38. #87780C22semidirect
39. #87780C22semidirect
40. #87780C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
50. Dic52C22semidirect
95. #4031D5semidirect
98. #4029D5semidirect
158. C2×Dic10C2×D5semidirect
167. C5×C22⋊C4C2×D5semidirect
168. C5×C22⋊C4C2×D5semidirect
175. C22⋊Dic5C2×D5semidirect
176. C22⋊Dic5C2×D5semidirect
197. C4⋊Dic5C2×D5semidirect
206. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
300. C52#551semidirect
394. C22⋊C4D52semidirect
445. C5#65992semidirect
446. C5#66013semidirect

Maximal subgroups

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

subgroupindex
#117172012
C5×#40312
C5×#40292
#117171012
#117170932
#117170912
#117170802
#117170392
D5×#6852
#117170062
#117169902
#117169802
#117169762
#117169732
#117169542
#660135
#659925