Finite Group Databasebeta

← #13106008

Subgroups — #13106008

Order 1600.

530 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717244C211yesmaximal, characteristic, complemented
2C4×C5×C22⋊D5C211yesmaximal, characteristic
3C2×C5×#683C211yesmaximal, characteristic, complemented
4Dic5×C22⋊D5C211yesmaximal, characteristic
5Dic5×C22⋊D5C211yesmaximal, characteristic
6#11717082C211yesmaximal, characteristic, complemented
7#11717082C211yesmaximal, characteristic, complemented
8#11717015C211yesmaximal, characteristic, complemented
9#11717007C211yesmaximal, characteristic, complemented
10D5×#683C211yesmaximal, characteristic, complemented
11#11716986C211yesmaximal, characteristic, complemented
12#11716969C211yesmaximal, characteristic, complemented
13#11716963C211yesmaximal, characteristic, complemented
14#11716958C211yesmaximal, characteristic, complemented
15#11716955C211yesmaximal, characteristic, complemented
16C22×C4×C52C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×C5×C22⋊D5C2211yescharacteristic
19C22×C5×Dic5C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C2×C4×C5×D5C2211yescharacteristic
22C2×#6039C2211yescharacteristic
23C2×#6039C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27C22⋊#1039C2211yescharacteristic
28C4⋊#1039C2211yescharacteristic
29#87806C2211yescharacteristic
30#87806C2211yescharacteristic
31C5×C22⋊Dic5C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C5×#683C2211yescharacteristic
38C4×C5×Dic5C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65873—51nomaximal
52#65845—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C2×C4×C52—41no—
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62#6045—41no—
63C5×C22⋊D5Q814yescomplemented
64C2×C5×Dic5—22no—
65C2×C5×Dic5—41no—
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C4×C5×D5—41no—
74#6039—22no—
75#6039—41no—
76D5×Dic5—22no—
77D5×Dic5—41no—
78C4×C5×D4—51no—
79C2×C5×C4⋊C4—51no—
80D4×Dic5—51no—
81D4×Dic5—51no—
82C4×C22⋊D5—51no—
83C4×C22⋊D5—51no—
84C4×C22⋊D5D511yescharacteristic
85#4070—51no—
86#4070—51no—
87#4070—51no—
88#4070—51no—
89#4070—51no—
90#4070—102no—
91C2×#683—51no—
92C2×#683D511yescharacteristic, complemented
93#4043—51no—
94#4042—51no—
95#4042—51no—
96D5×C4⋊C4—51no—
97#4036—51no—
98#4035—51no—
99#4035—51no—
100#4034—51no—
101#4034—51no—
102#4033—51no—
103#4024—51no—
104#4024—51no—
105#4024—51no—
106#4024—51no—
107#4016—51no—
108C4×Dic10—51no—
109C22×C52—21no—
110C22×C52—21no—
111C22×C52C2×Q812yes—
112C22×C52C2411yescharacteristic
113C2×C5×D5—21no—
114C2×C5×D5—21no—
115C2×C5×D5C2×Q812yes—
116C4×C52—21no—
117C4×C52—21no—
118#1039—21no—
119#1039—21no—
120#1039—21no—
121#1039—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5C2×Q812yes—
128C5×Dic5#4112yes—
129C2×C5×D4—51no—
130C22×C4×C5—22no—
131C22×C4×C5—51no—
132C22×C4×C5—51no—
133C22×C4×C5—51no—
134C22×C4×C5C2×D511yescharacteristic
135C22×C4×C5C2×D511yescharacteristic
136C2×C22⋊D5C2×D511yescharacteristic
137C22×Dic5—51no—
138C22×Dic5—51no—
139C22×Dic5C2×D511yescharacteristic
140C22×Dic5C2×D511yescharacteristic
141C2×C4×D5—51no—
142C2×C4×D5—51no—
143C2×C4×D5C2×D511yescharacteristic
144C2×Dic10—51no—
145C2×Dic10—51no—
146C2×Dic10—51no—
147C2×Dic10—51no—
148C2×Dic10—51no—
149C2×Dic10—51no—
150C2×Dic10—102no—
151C5×C4⋊C4—51no—
152C5×C4⋊C4—51no—
153C5×C4⋊C4—51no—
154C5×C4⋊C4—51no—
155C5×C4⋊C4—51no—
156C5×C22⋊C4—51no—
157C5×C22⋊C4—51no—
158C42×C5—51no—
159C22⋊Dic5—51no—
160C22⋊Dic5—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163C22⋊Dic5—51no—
164C22⋊Dic5—51no—
165C22⋊Dic5—102no—
166C22⋊Dic5—102no—
167C22⋊Dic5C2×D511yescharacteristic
168#685—51no—
169#685—51no—
170#685C2×D511yescharacteristic
171C4⋊Dic5—51no—
172C4⋊Dic5—51no—
173C4⋊Dic5—51no—
174C4⋊Dic5—51no—
175C4⋊Dic5—51no—
176C4⋊Dic5—51no—
177C4⋊Dic5—51no—
178C4⋊Dic5—51no—
179C4⋊Dic5—102no—
180#683—51no—
181#683—51no—
182#683—51no—
183#683—51no—
184#683—51no—
185#683—51no—
186#683—51no—
187#683—51no—
188#683—51no—
189#683—51no—
190#683—51no—
191#683—51no—
192#683—51no—
193#683—51no—
194#683—102no—
195#683—102no—
196#683C2×D511yescharacteristic
197#683C2×D511yescharacteristic
198#683C2×D511yescharacteristic
199#683C2×D511yescharacteristic
200#683C2×D511yescharacteristic
201C4×Dic5—51no—
202C4×Dic5—51no—
203C4×Dic5—51no—
204C4×Dic5—51no—
205C4×Dic5C2×D511yescharacteristic
206#554—251noSylow
207C2×C52—22no—
208C2×C52C22×Q811yescharacteristic
209C2×C52C2×#4111yescharacteristic
210C2×C52H(5,𝔽2)11yescharacteristic
211C5×D5—22no—
212C23×C5—22no—
213C23×C5—51no—
214C23×C5C22×D511yescharacteristic
215C23×C5C22×D511yescharacteristic
216C22×D5C22×D511yescharacteristic
217C5×D4—54no—
218C2×C4×C5—22no—
219C2×C4×C5—22no—
220C2×C4×C5—41no—
221C2×C4×C5—41no—
222C2×C4×C5—44no—
223C2×C4×C5—51no—
224C2×C4×C5—51no—
225C2×C4×C5—51no—
226C2×C4×C5—51no—
227C2×C4×C5—51no—
228C2×C4×C5—51no—
229C2×C4×C5—51no—
230C2×C4×C5—102no—
231C2×C4×C5—201no—
232C2×C4×C5—201no—
233C2×C4×C5C22×D511yescharacteristic
234C2×C4×C5C22×D511yescharacteristic
235C2×C4×C5C22×D511yescharacteristic
236C2×C4×C5C22×D511yescharacteristic
237C22⋊D5Dic1014yescomplemented
238C2×Dic5—22no—
239C2×Dic5—41no—
240C2×Dic5—51no—
241C2×Dic5—51no—
242C2×Dic5—51no—
243C2×Dic5—51no—
244C2×Dic5—51no—
245C2×Dic5—51no—
246C2×Dic5—51no—
247C2×Dic5—51no—
248C2×Dic5—102no—
249C2×Dic5—102no—
250C2×Dic5—102no—
251C2×Dic5—102no—
252C2×Dic5—102no—
253C2×Dic5—201no—
254C2×Dic5C22×D511yescharacteristic
255C2×Dic5C22×D511yescharacteristic
256C2×Dic5C22×D511yescharacteristic
257C2×Dic5C22×D511yescharacteristic
258C2×Dic5C22×D511yescharacteristic
259C2×Dic5C22×D511yescharacteristic
260C2×Dic5C22×D511yescharacteristic
261C4×D5—41no—
262C4×D5—102no—
263C4×D5—201no—
264Dic10—102no—
265Dic10—102no—
266Dic10—201no—
267Dic10—201no—
268Dic10—201no—
269Dic10—201no—
270Dic10—204no—
271C4⋊Q8—251no—
272#125—251no—
273#125—251no—
274C22⋊Q8—251no—
275C22⋊Q8—251no—
276C22⋊Q8—251no—
277C22⋊Q8—251no—
278C22⋊Q8—251no—
279C22⋊Q8—251no—
280C4×Q8—251no—
281C4×D4—51no—
282C4×D4—251no—
283C4×D4—251no—
284C2×C4⋊C4—51no—
285C2×C4⋊C4—251no—
286C52#55411yescharacteristic, complemented, Sylow
287C22×C5—21no—
288C22×C5—21no—
289C22×C5—21no—
290C22×C5—21no—
291C22×C5—24no—
292C22×C5—22no—
293C22×C5—42no—
294C22×C5—42no—
295C22×C5—52no—
296C22×C5—101no—
297C22×C5—101no—
298C22×C5C2×Dic1012yes—
299C22×C5Q8×D512yes—
300C22×C5C23×D511yescharacteristic
301C22×C5C23×D511yescharacteristic
302C2×D5—21no—
303C2×D5—21no—
304C2×D5C2×Dic1012yes—
305C4×C5—24no—
306C4×C5—21no—
307C4×C5—21no—
308C4×C5—21no—
309C4×C5—21no—
310C4×C5—42no—
311C4×C5—52no—
312C4×C5—52no—
313C4×C5—101no—
314C4×C5—101no—
315C4×C5—101no—
316C4×C5—101no—
317C4×C5—101no—
318Dic5—21no—
319Dic5—21no—
320Dic5—21no—
321Dic5—21no—
322Dic5—21no—
323Dic5—101no—
324Dic5—101no—
325Dic5—101no—
326Dic5—101no—
327Dic5—101no—
328Dic5—101no—
329Dic5—101no—
330Dic5—101no—
331Dic5—202no—
332Dic5—202no—
333Dic5—202no—
334Dic5—202no—
335Dic5C2×Dic1012yes—
336Dic5#70912yes—
337C2×Q8—251no—
338C2×Q8—251no—
339C2×Q8—251no—
340C2×D4—51no—
341C22×C4—51no—
342C22×C4—51no—
343C22×C4—51no—
344C22×C4—251no—
345C22×C4—251no—
346C22×C4D5211yescharacteristic
347C4⋊C4—51no—
348C4⋊C4—51no—
349C4⋊C4—51no—
350C4⋊C4—51no—
351C4⋊C4—51no—
352C4⋊C4—251no—
353C4⋊C4—251no—
354C4⋊C4—251no—
355C4⋊C4—251no—
356C4⋊C4—251no—
357C4⋊C4—251no—
358C4⋊C4—251no—
359C4⋊C4—251no—
360C4⋊C4—251no—
361C4⋊C4—251no—
362C4⋊C4—251no—
363C22⋊C4—51no—
364C22⋊C4—51no—
365C22⋊C4—251no—
366C22⋊C4—251no—
367C22⋊C4—251no—
368C22⋊C4—251no—
369C42—51no—
370C42—251no—
371C42—251no—
372C2×C5—22no—
373C2×C5—22no—
374C2×C5—22no—
375C2×C5—22no—
376C2×C5—22no—
377C2×C5—44no—
378C2×C5—102no—
379C2×C5C22×Dic1011yescharacteristic
380C2×C5C2×#70911yescharacteristic
381C2×C5#414411yescharacteristic
382C2×C5#414411yescharacteristic
383C2×C5C2×Q8×D511yescharacteristic
384C2×C5D5×#4111yescharacteristic
385D5—22no—
386C23—51no—
387C23C2×D5211yescharacteristic
388Q8—502no—
389Q8—1001no—
390Q8—1001no—
391D4—54no—
392C2×C4—41no—
393C2×C4—51no—
394C2×C4—51no—
395C2×C4—51no—
396C2×C4—51no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—102no—
401C2×C4—201no—
402C2×C4—201no—
403C2×C4—251no—
404C2×C4—251no—
405C2×C4—251no—
406C2×C4—251no—
407C2×C4—502no—
408C2×C4—1001no—
409C2×C4C2×D5211yescharacteristic
410C2×C4C2×D5211yescharacteristic
411C5—22no—
412C5#6584511yescharacteristic, complemented
413C5#6587311yescharacteristic, complemented
414C22—21no—
415C22—21no—
416C22—52no—
417C22—101no—
418C22—101no—
419C22D5×Dic1012yes—
420C22C22×D5211yescharacteristic
421C4—21no—
422C4—21no—
423C4—52no—
424C4—52no—
425C4—101no—
426C4—101no—
427C4—101no—
428C4—101no—
429C4—101no—
430C4—501no—
431C4—501no—
432C4—501no—
433C4—501no—
434C2—22no—
435C2—102no—
436C2C2×D5×Dic1011yescharacteristic
437C2D5×#70911yescharacteristic
438C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC2×C5×C22⋊D5C22
hypercenterC22×C4D52
hypocenterC52#554
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11717244C2semidirect
3. C2×C5×#683C2semidirect
6. #11717082C2semidirect
7. #11717082C2semidirect
8. #11717015C2semidirect
9. #11717007C2semidirect
10. D5×#683C2semidirect
11. #11716986C2semidirect
12. #11716969C2semidirect
13. #11716963C2semidirect
14. #11716958C2semidirect
15. #11716955C2semidirect
63. C5×C22⋊D5Q8semidirect
92. C2×#683D5semidirect
237. C22⋊D5Dic10semidirect
286. C52#554semidirect
412. C5#65845semidirect
413. C5#65873semidirect

Maximal subgroups

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

subgroupindex
#117172442
C4×C5×C22⋊D52
C2×C5×#6832
Dic5×C22⋊D52
#117170822
#117170152
#117170072
D5×#6832
#117169862
#117169692
#117169632
#117169582
#117169552
#658735
#658455