Finite Group Databasebeta

← #13105157

Subgroups — #13105157

Order 1600.

556 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717199C211yesmaximal, characteristic, complemented
2C5×#4031C211yesmaximal, characteristic, complemented
3C5×D5×C22⋊C4C211yesmaximal, characteristic, complemented
4#11717094C211yesmaximal, characteristic
5#11717087C211yesmaximal, characteristic, complemented
6D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
7#11717074C211yesmaximal, characteristic, complemented
8#11717023C211yesmaximal, characteristic, complemented
9#11717020C211yesmaximal, characteristic, complemented
10#11717006C211yesmaximal, characteristic, complemented
11#11717003C211yesmaximal, characteristic, complemented
12#11716997C211yesmaximal, characteristic, complemented
13#11716995C211yesmaximal, characteristic, complemented
14#11716974C211yesmaximal, characteristic, complemented
15#11716973C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C2×C5×C22⋊D5C2211yescharacteristic
18C2×C4×C5×D5C2211yescharacteristic
19C2×C4×C5×D5C2211yescharacteristic
20C2×C5×Dic10C2211yescharacteristic, complemented
21C2×#6037C2211yescharacteristic
22C2×#6037C2211yescharacteristic
23C2×D5×Dic5C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic, complemented
26C52×C22⋊C4C2211yescharacteristic
27C22⋊#1039C2211yescharacteristic
28C22⋊#1039C2211yescharacteristic
29C4⋊#1039C2211yescharacteristic, complemented
30#87806C2211yescharacteristic, complemented
31#87806C2211yescharacteristic, complemented
32C4×#1039C2211yescharacteristic, complemented
33C5×C22⋊Dic5C2211yescharacteristic
34C5×C22⋊Dic5C2211yescharacteristic
35C5×#685C2211yescharacteristic
36C5×#685C2211yescharacteristic
37C5×#685C2211yescharacteristic, complemented
38C5×#685C2211yescharacteristic, complemented
39C4×C5×Dic5C2211yescharacteristic, complemented
40#87781C2211yescharacteristic, complemented
41#87781C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87780C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#66013—51nomaximal
52#65829—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×C52C2311yescharacteristic
61C2×C4×C52C2311yescharacteristic
62C2×#1039C2311yescharacteristic
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C5×C22⋊D5—41no—
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C4×C5×D5—22no—
73C4×C5×D5—41no—
74C5×Dic10—41no—
75#6037—22no—
76#6037—22no—
77D5×Dic5—22no—
78D5×Dic5—41no—
79D5×Dic5—41no—
80C5×#124—51no—
81C2×C5×C22⋊C4—51no—
82#4099—51no—
83C2×C22⋊Dic5—51no—
84#4092—51no—
85#4092—51no—
86#4083—51no—
87#4081—51no—
88#4075—51no—
89C4×C22⋊D5—51no—
90#4072—51no—
91#4070—51no—
92#4032—51no—
93#4031D511yescharacteristic, complemented
94#4029—51no—
95#4029—51no—
96#4029—51no—
97#4027—51no—
98#4027—51no—
99#4027—51no—
100D5×C22⋊C4—51no—
101D5×C22⋊C4D511yescharacteristic, complemented
102#4025—51no—
103#4025—51no—
104#4025—51no—
105#4025—51no—
106#4025—102no—
107#4024—51no—
108#4023—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—21no—
118C2×C5×D5—21no—
119C2×C5×D5—21no—
120C2×C5×D5—21no—
121C2×C5×D5—21no—
122C2×C5×D5—44no—
123C2×C5×D5#4112yes—
124C2×C5×D5#4112yes—
125C4×C52—21no—
126C4×C52—21no—
127#1039—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—
136C24×C5—51no—
137C23×D5C2×D511yescharacteristic
138C2×C5×Q8—51no—
139C2×C5×D4—51no—
140C22×C4×C5—51no—
141C22×C4×C5—51no—
142C2×C22⋊D5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—51no—
145C2×C22⋊D5—51no—
146C2×C22⋊D5C2×D511yescharacteristic
147C22×Dic5—51no—
148C22×Dic5—51no—
149C22×Dic5—51no—
150C2×C4×D5—51no—
151C2×C4×D5—51no—
152C2×C4×D5—51no—
153C2×C4×D5C2×D511yescharacteristic
154C2×C4×D5C2×D511yescharacteristic
155C2×Dic10C2×D511yescharacteristic, complemented
156C5×C22⋊C4—22no—
157C5×C22⋊C4—51no—
158C5×C22⋊C4—51no—
159C5×C22⋊C4—51no—
160C5×C22⋊C4—51no—
161C5×C22⋊C4—51no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4C2×D511yescharacteristic
164C5×C22⋊C4C2×D511yescharacteristic
165C42×C5—51no—
166C22⋊Dic5—51no—
167C22⋊Dic5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5—102no—
176C22⋊Dic5—102no—
177C22⋊Dic5C2×D511yescharacteristic
178C22⋊Dic5C2×D511yescharacteristic
179#685—51no—
180#685—51no—
181#685—51no—
182#685—51no—
183#685—51no—
184#685C2×D511yescharacteristic
185#685C2×D511yescharacteristic
186#685C2×D511yescharacteristic, complemented
187#685C2×D511yescharacteristic, complemented
188C4⋊Dic5—51no—
189C4⋊Dic5—51no—
190C4⋊Dic5—51no—
191C4⋊Dic5—51no—
192C4⋊Dic5—51no—
193C4⋊Dic5—102no—
194#683—51no—
195#683—51no—
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—102no—
206#683—102no—
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×C5—51no—
225C23×C5—102no—
226C23×C5—102no—
227C23×C5—201no—
228C23×C5C22×D511yescharacteristic
229C23×C5C22×D511yescharacteristic
230C22×D5—22no—
231C22×D5—22no—
232C22×D5—41no—
233C22×D5C22×D511yescharacteristic
234C22×D5C22×D511yescharacteristic
235C22×D5C22×D511yescharacteristic
236C5×Q8—201no—
237C5×D4—201no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—51no—
241C2×C4×C5—51no—
242C2×C4×C5—51no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—102no—
246C2×C4×C5—201no—
247C2×C4×C5C22×D511yescharacteristic
248C2×C4×C5C22×D511yescharacteristic
249C2×C4×C5C22×D511yescharacteristic
250C2×C4×C5C22×D511yescharacteristic
251C22⋊D5—41no—
252C22⋊D5—102no—
253C22⋊D5—102no—
254C22⋊D5—102no—
255C22⋊D5—102no—
256C2×Dic5—51no—
257C2×Dic5—51no—
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—102no—
269C2×Dic5—201no—
270C2×Dic5—201no—
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C4×D5—22no—
277C4×D5—41no—
278C4×D5—102no—
279C4×D5—201no—
280C4×D5—201no—
281Dic10—41no—
282#126—251no—
283#126—251no—
284#124—51no—
285#123—251no—
286#123—251no—
287#123—251no—
288C22⋊Q8—251no—
289C22⋊Q8—251no—
290#120—251no—
291C4×D4—251no—
292C4×D4—251no—
293#117—251no—
294#117—251no—
295C2×C22⋊C4—51no—
296C2×C22⋊C4—251no—
297C52#55111yescharacteristic, complemented, Sylow
298C22×C5—21no—
299C22×C5—21no—
300C22×C5—21no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—22no—
305C22×C5—42no—
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—101no—
315C22×C5—101no—
316C22×C5—101no—
317C22×C5—204no—
318C22×C5C23×D511yescharacteristic
319C22×C5C23×D511yescharacteristic
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—21no—
326C2×D5—21no—
327C2×D5—44no—
328C2×D5#70912yes—
329C2×D5#71112yes—
330C4×C5—21no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—42no—
335C4×C5—42no—
336C4×C5—101no—
337C4×C5—101no—
338C4×C5—101no—
339C4×C5—101no—
340C4×C5—101no—
341Dic5—21no—
342Dic5—21no—
343Dic5—21no—
344Dic5—21no—
345Dic5—21no—
346Dic5—101no—
347Dic5—101no—
348Dic5—101no—
349Dic5—101no—
350Dic5—101no—
351Dic5—101no—
352Dic5—101no—
353Dic5—101no—
354Dic5—202no—
355Dic5—202no—
356Dic5—202no—
357Dic5—202no—
358C24—51no—
359C2×Q8—51no—
360C2×D4—51no—
361C2×D4—251no—
362C2×D4—251no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—251no—
366C22×C4—251no—
367C22×C4—251no—
368C4⋊C4—251no—
369C4⋊C4—251no—
370C4⋊C4—251no—
371C4⋊C4—251no—
372C4⋊C4—251no—
373C4⋊C4—251no—
374C4⋊C4—251no—
375C4⋊C4—251no—
376C22⋊C4—51no—
377C22⋊C4—51no—
378C22⋊C4—51no—
379C22⋊C4—51no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—251no—
383C22⋊C4—251no—
384C22⋊C4—251no—
385C22⋊C4—251no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4D5211yescharacteristic
390C42—51no—
391C42—251no—
392C42—251no—
393C2×C5—22no—
394C2×C5—22no—
395C2×C5—22no—
396C2×C5—41no—
397C2×C5—41no—
398C2×C5—82no—
399C2×C5—102no—
400C2×C5—102no—
401C2×C5—201no—
402C2×C5C2×#70911yescharacteristic
403C2×C5C2×#71111yescharacteristic
404C2×C5#414411yescharacteristic
405C2×C5#414411yescharacteristic
406C2×C5D5×#4111yescharacteristic
407C2×C5D5×#4111yescharacteristic
408D5—22no—
409D5—22no—
410D5—41no—
411C23—51no—
412C23—51no—
413C23—51no—
414C23—102no—
415C23—102no—
416C23—201no—
417C23C2×D5211yescharacteristic
418Q8—201no—
419D4—201no—
420D4—502no—
421D4—502no—
422C2×C4—51no—
423C2×C4—51no—
424C2×C4—51no—
425C2×C4—51no—
426C2×C4—51no—
427C2×C4—102no—
428C2×C4—201no—
429C2×C4—251no—
430C2×C4—251no—
431C2×C4—251no—
432C2×C4—251no—
433C2×C4—502no—
434C2×C4—1001no—
435C2×C4—1001no—
436C2×C4C2×D5211yescharacteristic
437C2×C4C2×D5211yescharacteristic
438C5—22no—
439C5#6582911yescharacteristic, complemented
440C5#6601311yescharacteristic, complemented
441C22—21no—
442C22—21no—
443C22—21no—
444C22—52no—
445C22—52no—
446C22—101no—
447C22—101no—
448C22—101no—
449C22—101no—
450C22—101no—
451C22—101no—
452C22—101no—
453C22—204no—
454C22C22×D5211yescharacteristic
455C4—21no—
456C4—21no—
457C4—101no—
458C4—101no—
459C4—101no—
460C4—101no—
461C4—101no—
462C4—501no—
463C4—501no—
464C4—501no—
465C4—501no—
466C2—41no—
467C2—102no—
468C2—102no—
469C2—201no—
470C2D5×#70911yescharacteristic
471C2D5×#71111yescharacteristic
472C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717094C2
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. #11717199C2semidirect
2. C5×#4031C2semidirect
3. C5×D5×C22⋊C4C2semidirect
5. #11717087C2semidirect
6. D5×C22⋊Dic5C2semidirect
7. #11717074C2semidirect
8. #11717023C2semidirect
9. #11717020C2semidirect
10. #11717006C2semidirect
11. #11717003C2semidirect
12. #11716997C2semidirect
13. #11716995C2semidirect
14. #11716974C2semidirect
15. #11716973C2semidirect
20. C2×C5×Dic10C22semidirect
25. C2×D5×Dic5C22semidirect
29. C4⋊#1039C22semidirect
30. #87806C22semidirect
31. #87806C22semidirect
32. C4×#1039C22semidirect
37. C5×#685C22semidirect
38. C5×#685C22semidirect
39. C4×C5×Dic5C22semidirect
40. #87781C22semidirect
41. #87781C22semidirect
42. #87780C22semidirect
43. #87780C22semidirect
44. #87780C22semidirect
49. #87778C22semidirect
50. Dic52C22semidirect
93. #4031D5semidirect
101. D5×C22⋊C4D5semidirect
155. C2×Dic10C2×D5semidirect
186. #685C2×D5semidirect
187. #685C2×D5semidirect
212. C4×Dic5C2×D5semidirect
297. C52#551semidirect
439. C5#65829semidirect
440. C5#66013semidirect

Maximal subgroups

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

subgroupindex
#117171992
C5×#40312
C5×D5×C22⋊C42
#117170942
#117170872
D5×C22⋊Dic52
#117170742
#117170232
#117170202
#117170062
#117170032
#117169972
#117169952
#117169742
#117169732
#660135
#658295