Finite Group Databasebeta

← #13105217

Subgroups — #13105217

Order 1600.

558 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717198C211yesmaximal, characteristic, complemented
2C5×D5×C22⋊C4C211yesmaximal, characteristic
3C5×#4024C211yesmaximal, characteristic, complemented
4#11717100C211yesmaximal, characteristic, complemented
5D5×C22⋊Dic5C211yesmaximal, characteristic
6#11717082C211yesmaximal, characteristic, complemented
7C2×#87778C211yesmaximal, characteristic
8#11717015C211yesmaximal, characteristic, complemented
9#11717014C211yesmaximal, characteristic, complemented
10#11717007C211yesmaximal, characteristic, complemented
11#11717006C211yesmaximal, characteristic, complemented
12#11716975C211yesmaximal, characteristic, complemented
13#11716974C211yesmaximal, characteristic, complemented
14#11716969C211yesmaximal, characteristic, complemented
15#11716969C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×#1039C2211yescharacteristic
18C2×#6045C2211yescharacteristic, complemented
19C22×C5×Dic5C2211yescharacteristic
20C2×C4×C5×D5C2211yescharacteristic
21C2×C4×C5×D5C2211yescharacteristic
22C2×C5×Dic10C2211yescharacteristic, complemented
23C2×#6039C2211yescharacteristic, complemented
24C2×#6039C2211yescharacteristic, complemented
25C2×D5×Dic5C2211yescharacteristic
26C2×D5×Dic5C2211yescharacteristic
27C52×C22⋊C4C2211yescharacteristic
28C22⋊#1039C2211yescharacteristic
29C4⋊#1039C2211yescharacteristic, complemented
30#87806C2211yescharacteristic, complemented
31#87806C2211yescharacteristic, complemented
32C5×C22⋊Dic5C2211yescharacteristic
33C5×C22⋊Dic5C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#685C2211yescharacteristic
36C5×C4⋊Dic5C2211yescharacteristic, complemented
37C5×#683C2211yescharacteristic, complemented
38C5×#683C2211yescharacteristic, complemented
39#87781C2211yescharacteristic, complemented
40#87781C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87780C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50#87778C2211yescharacteristic
51#65968—51nomaximal
52#65817—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5×D5—41no—
55C22×C5×D5—41no—
56C22×C5×D5Q814yescomplemented
57C22×C5×D5C2311yescharacteristic
58C22×C5×D5C2311yescharacteristic
59C2×C4×C52C2311yescharacteristic
60C2×C4×C52C2311yescharacteristic
61C2×#1039—41no—
62C2×#1039C2311yescharacteristic
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66#6045—41no—
67C2×C5×Dic5—41no—
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C4×C5×D5—41no—
75C4×C5×D5—41no—
76C5×Dic10—41no—
77#6039—41no—
78#6039—41no—
79D5×Dic5—41no—
80D5×Dic5—41no—
81C5×C22⋊Q8—51no—
82C2×C5×C22⋊C4—51no—
83C2×C22⋊Dic5—51no—
84C2×C22⋊Dic5—51no—
85#4092—51no—
86#4092—51no—
87C2×#685—51no—
88#4070—51no—
89#4070—51no—
90#4070—51no—
91#4070—51no—
92#4070—51no—
93#4043—51no—
94#4043—51no—
95#4042—51no—
96#4042—51no—
97#4042—51no—
98#4042—51no—
99D5×C22⋊C4—51no—
100D5×C22⋊C4D511yescharacteristic
101#4024—51no—
102#4024—51no—
103#4024—51no—
104#4024—51no—
105#4024—51no—
106#4024—51no—
107#4024—51no—
108#4024—51no—
109#4024—51no—
110#4024—102no—
111#4024D511yescharacteristic, complemented
112C22×C52—21no—
113C22×C52—21no—
114C22×C52C2×Q812yes—
115C22×C52C2411yescharacteristic
116C2×C5×D5—21no—
117C2×C5×D5—21no—
118C2×C5×D5—21no—
119C2×C5×D5—21no—
120C2×C5×D5—44no—
121C2×C5×D5C2×Q812yes—
122C2×C5×D5C2×Q812yes—
123C4×C52—21no—
124C4×C52—21no—
125#1039—21no—
126#1039—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—
135C24×C5—51no—
136C23×D5C2×D511yescharacteristic
137C2×C5×Q8—51no—
138C22×C4×C5—51no—
139C22×C4×C5—51no—
140C22×C4×C5—51no—
141C22×Dic5—51no—
142C22×Dic5—51no—
143C22×Dic5—51no—
144C22×Dic5—51no—
145C22×Dic5—102no—
146C22×Dic5C2×D511yescharacteristic
147C2×C4×D5—51no—
148C2×C4×D5—51no—
149C2×C4×D5C2×D511yescharacteristic
150C2×C4×D5C2×D511yescharacteristic
151C2×Dic10—51no—
152C2×Dic10—51no—
153C2×Dic10—51no—
154C2×Dic10—51no—
155C2×Dic10—51no—
156C2×Dic10—51no—
157C2×Dic10—102no—
158C2×Dic10C2×D511yescharacteristic, complemented
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C4⋊C4—51no—
162C5×C22⋊C4—22no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4—51no—
166C5×C22⋊C4—51no—
167C5×C22⋊C4C2×D511yescharacteristic
168C5×C22⋊C4C2×D511yescharacteristic
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5—51no—
176C22⋊Dic5—51no—
177C22⋊Dic5—102no—
178C22⋊Dic5C2×D511yescharacteristic
179C22⋊Dic5C2×D511yescharacteristic
180#685—51no—
181#685—51no—
182#685—51no—
183#685—51no—
184#685—51no—
185#685—51no—
186#685C2×D511yescharacteristic
187#685C2×D511yescharacteristic
188C4⋊Dic5—51no—
189C4⋊Dic5—51no—
190C4⋊Dic5—51no—
191C4⋊Dic5—51no—
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—102no—
195C4⋊Dic5C2×D511yescharacteristic, complemented
196#683—51no—
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683—51no—
206#683—51no—
207#683—51no—
208#683—102no—
209#683—102no—
210#683C2×D511yescharacteristic, complemented
211#683C2×D511yescharacteristic, complemented
212#543—251noSylow
213C2×C52—22no—
214C2×C52C22×Q811yescharacteristic
215C2×C52H(5,𝔽2)11yescharacteristic
216C2×C52H(5,𝔽2)11yescharacteristic
217C5×D5—22no—
218C5×D5—22no—
219C23×C5—22no—
220C23×C5—51no—
221C23×C5—51no—
222C23×C5—54no—
223C23×C5—201no—
224C23×C5—201no—
225C23×C5C22×D511yescharacteristic
226C23×C5C22×D511yescharacteristic
227C22×D5—41no—
228C22×D5—41no—
229C22×D5Dic1014yescomplemented
230C22×D5C22×D511yescharacteristic
231C22×D5C22×D511yescharacteristic
232C5×Q8—201no—
233C2×C4×C5—22no—
234C2×C4×C5—22no—
235C2×C4×C5—51no—
236C2×C4×C5—51no—
237C2×C4×C5—51no—
238C2×C4×C5—51no—
239C2×C4×C5—51no—
240C2×C4×C5—51no—
241C2×C4×C5—201no—
242C2×C4×C5—201no—
243C2×C4×C5—201no—
244C2×C4×C5C22×D511yescharacteristic
245C2×C4×C5C22×D511yescharacteristic
246C2×C4×C5C22×D511yescharacteristic
247C2×C4×C5C22×D511yescharacteristic
248C2×Dic5—41no—
249C2×Dic5—51no—
250C2×Dic5—51no—
251C2×Dic5—51no—
252C2×Dic5—51no—
253C2×Dic5—51no—
254C2×Dic5—51no—
255C2×Dic5—51no—
256C2×Dic5—51no—
257C2×Dic5—102no—
258C2×Dic5—102no—
259C2×Dic5—102no—
260C2×Dic5—102no—
261C2×Dic5—201no—
262C2×Dic5—201no—
263C2×Dic5—201no—
264C2×Dic5—201no—
265C2×Dic5—204no—
266C2×Dic5C22×D511yescharacteristic
267C2×Dic5C22×D511yescharacteristic
268C2×Dic5C22×D511yescharacteristic
269C2×Dic5C22×D511yescharacteristic
270C2×Dic5C22×D511yescharacteristic
271C2×Dic5C22×D511yescharacteristic
272C4×D5—41no—
273C4×D5—41no—
274C4×D5—201no—
275C4×D5—201no—
276Dic10—41no—
277Dic10—201no—
278Dic10—201no—
279Dic10—201no—
280Dic10—201no—
281Dic10—201no—
282Dic10—201no—
283Dic10—204no—
284C22⋊Q8—51no—
285C22⋊Q8—251no—
286C22⋊Q8—251no—
287C22⋊Q8—251no—
288C22⋊Q8—251no—
289C22⋊Q8—251no—
290C22⋊Q8—251no—
291C22⋊Q8—251no—
292C22⋊Q8—251no—
293C22⋊Q8—251no—
294C22⋊Q8—251no—
295C22⋊Q8—251no—
296C2×C22⋊C4—51no—
297C2×C22⋊C4—251no—
298C2×C22⋊C4—251no—
299C52#54311yescharacteristic, complemented, Sylow
300C22×C5—21no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—24no—
305C22×C5—22no—
306C22×C5—42no—
307C22×C5—42no—
308C22×C5—52no—
309C22×C5—52no—
310C22×C5—101no—
311C22×C5—101no—
312C22×C5—101no—
313C22×C5—101no—
314C22×C5—204no—
315C22×C5C2×Dic1012yes—
316C22×C5Q8×D512yes—
317C22×C5C23×D511yescharacteristic
318C22×C5C23×D511yescharacteristic
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—44no—
324C2×D5C2×Dic1012yes—
325C2×D5C2×Dic1012yes—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—21no—
330C4×C5—42no—
331C4×C5—42no—
332C4×C5—101no—
333C4×C5—101no—
334C4×C5—101no—
335C4×C5—101no—
336C4×C5—101no—
337C4×C5—101no—
338Dic5—21no—
339Dic5—21no—
340Dic5—21no—
341Dic5—21no—
342Dic5—21no—
343Dic5—21no—
344Dic5—104no—
345Dic5—101no—
346Dic5—101no—
347Dic5—101no—
348Dic5—101no—
349Dic5—101no—
350Dic5—101no—
351Dic5—101no—
352Dic5—101no—
353Dic5—202no—
354Dic5—202no—
355Dic5—202no—
356C24—51no—
357C2×Q8—51no—
358C2×Q8—251no—
359C2×Q8—251no—
360C2×Q8—251no—
361C22×C4—51no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—251no—
365C22×C4—251no—
366C22×C4—251no—
367C4⋊C4—51no—
368C4⋊C4—51no—
369C4⋊C4—51no—
370C4⋊C4—251no—
371C4⋊C4—251no—
372C4⋊C4—251no—
373C4⋊C4—251no—
374C4⋊C4—251no—
375C4⋊C4—251no—
376C4⋊C4—251no—
377C4⋊C4—251no—
378C4⋊C4—251no—
379C22⋊C4—51no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—251no—
384C22⋊C4—251no—
385C22⋊C4—251no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C22⋊C4D5211yescharacteristic
391C2×C5—22no—
392C2×C5—22no—
393C2×C5—22no—
394C2×C5—22no—
395C2×C5—22no—
396C2×C5—44no—
397C2×C5—102no—
398C2×C5—102no—
399C2×C5C22×Dic1011yescharacteristic
400C2×C5#414411yescharacteristic
401C2×C5#414411yescharacteristic
402C2×C5#414411yescharacteristic
403C2×C5C2×Q8×D511yescharacteristic
404C2×C5#414911yescharacteristic
405D5—22no—
406D5—22no—
407C23—51no—
408C23—51no—
409C23—54no—
410C23—201no—
411C23—201no—
412C23C2×D5211yescharacteristic
413Q8—201no—
414Q8—1001no—
415Q8—1001no—
416Q8—1001no—
417C2×C4—51no—
418C2×C4—51no—
419C2×C4—51no—
420C2×C4—51no—
421C2×C4—51no—
422C2×C4—51no—
423C2×C4—201no—
424C2×C4—201no—
425C2×C4—201no—
426C2×C4—251no—
427C2×C4—251no—
428C2×C4—251no—
429C2×C4—251no—
430C2×C4—1001no—
431C2×C4—1001no—
432C2×C4—1001no—
433C2×C4C2×D5211yescharacteristic
434C2×C4C2×D5211yescharacteristic
435C5—22no—
436C5#6581711yescharacteristic, complemented
437C5#6596811yescharacteristic, complemented
438C22—21no—
439C22—21no—
440C22—52no—
441C22—52no—
442C22—101no—
443C22—101no—
444C22—101no—
445C22—101no—
446C22—204no—
447C22D5×Dic1012yes—
448C22C22×D5211yescharacteristic
449C4—21no—
450C4—21no—
451C4—101no—
452C4—101no—
453C4—101no—
454C4—101no—
455C4—101no—
456C4—101no—
457C4—501no—
458C4—501no—
459C4—501no—
460C4—501no—
461C2—22no—
462C2—102no—
463C2—102no—
464C2C2×D5×Dic1011yescharacteristic
465C2#1171747811yescharacteristic
466C2#1171747911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC23×C5×D5C22
hypercenterC22⋊C4D52
hypocenterC52#543
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717198C2semidirect
3. C5×#4024C2semidirect
4. #11717100C2semidirect
6. #11717082C2semidirect
8. #11717015C2semidirect
9. #11717014C2semidirect
10. #11717007C2semidirect
11. #11717006C2semidirect
12. #11716975C2semidirect
13. #11716974C2semidirect
14. #11716969C2semidirect
15. #11716969C2semidirect
18. C2×#6045C22semidirect
22. C2×C5×Dic10C22semidirect
23. C2×#6039C22semidirect
24. C2×#6039C22semidirect
29. C4⋊#1039C22semidirect
30. #87806C22semidirect
31. #87806C22semidirect
36. C5×C4⋊Dic5C22semidirect
37. C5×#683C22semidirect
38. C5×#683C22semidirect
39. #87781C22semidirect
40. #87781C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
43. #87780C22semidirect
44. #87780C22semidirect
56. C22×C5×D5Q8semidirect
111. #4024D5semidirect
158. C2×Dic10C2×D5semidirect
195. C4⋊Dic5C2×D5semidirect
210. #683C2×D5semidirect
211. #683C2×D5semidirect
229. C22×D5Dic10semidirect
299. C52#543semidirect
436. C5#65817semidirect
437. C5#65968semidirect

Maximal subgroups

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

subgroupindex
#117171982
C5×D5×C22⋊C42
C5×#40242
#117171002
D5×C22⋊Dic52
#117170822
C2×#877782
#117170152
#117170142
#117170072
#117170062
#117169752
#117169742
#117169692
#659685
#658175