Finite Group Databasebeta

← #13106021

Subgroups — #13106021

Order 1600.

550 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic
2C5×#4070C211yesmaximal, characteristic, complemented
3C2×C5×#683C211yesmaximal, characteristic, complemented
4#11717082C211yesmaximal, characteristic, complemented
5#11717082C211yesmaximal, characteristic, complemented
6#11717070C211yesmaximal, characteristic
7#11717070C211yesmaximal, characteristic
8#11717009C211yesmaximal, characteristic, complemented
9#11717008C211yesmaximal, characteristic, complemented
10#11717001C211yesmaximal, characteristic, complemented
11#11716989C211yesmaximal, characteristic, complemented
12#11716984C211yesmaximal, characteristic, complemented
13#11716965C211yesmaximal, characteristic, complemented
14#11716957C211yesmaximal, characteristic, complemented
15#11716956C211yesmaximal, characteristic, complemented
16C22×C4×C52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C22×C5×Dic5C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C2×C5×Dic10C2211yescharacteristic
22C2×#6039C2211yescharacteristic
23C2×#6039C2211yescharacteristic
24C2×#6036C2211yescharacteristic
25C2×#6036C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27#87808C2211yescharacteristic
28#87806C2211yescharacteristic
29C4×#1039C2211yescharacteristic
30C5×C22⋊Dic5C2211yescharacteristic
31C5×C22⋊Dic5C2211yescharacteristic
32C5×C4⋊Dic5C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C5×#683C2211yescharacteristic
38C5×#683C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87781C2211yescharacteristic
42#87781C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87779C2211yescharacteristic
48#87779C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65970—51nomaximal
52#65845—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C2×C4×C52—41no—
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C22⋊(C5⋊D5)Q814yescomplemented
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C4×C5⋊D5—41no—
63C2×C5×Dic5—22no—
64C2×C5×Dic5—41no—
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C5×Dic10—41no—
74#6039—22no—
75#6039—41no—
76#6036—22no—
77#6036—41no—
78C5×C22⋊Q8—51no—
79C2×C5×C4⋊C4—51no—
80#4092—51no—
81#4090—51no—
82C4×C22⋊D5—51no—
83C4×C22⋊D5—51no—
84C4×C22⋊D5—51no—
85C4×C22⋊D5—51no—
86C4×C22⋊D5—102no—
87#4070—51no—
88#4070—51no—
89#4070D511yescharacteristic, complemented
90C2×#683D511yescharacteristic, complemented
91#4043—51no—
92#4042—51no—
93#4042—51no—
94#4042—51no—
95#4042—51no—
96D5×C4⋊C4—51no—
97D5×C4⋊C4—51no—
98#4035—51no—
99#4035—51no—
100#4035—51no—
101#4035—51no—
102#4034—51no—
103#4033—51no—
104#4033—51no—
105#4027—51no—
106#4027—51no—
107#4024—51no—
108#4024—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#1039C2×Q812yes—
121C5×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5—21no—
128C5×Dic5#4112yes—
129C2×C5×Q8—51no—
130C22×C4×C5—22no—
131C22×C4×C5—51no—
132C22×C4×C5—51no—
133C22×C4×C5C2×D511yescharacteristic
134C22×C4×C5C2×D511yescharacteristic
135C2×C22⋊D5—51no—
136C2×C22⋊D5—51no—
137C2×C22⋊D5—102no—
138C22×Dic5C2×D511yescharacteristic
139C22×Dic5C2×D511yescharacteristic
140C2×C4×D5—51no—
141C2×C4×D5—51no—
142C2×C4×D5—51no—
143C2×C4×D5—51no—
144C2×C4×D5—51no—
145C2×C4×D5—51no—
146C2×C4×D5—102no—
147C2×Dic10—51no—
148C2×Dic10—51no—
149C2×Dic10—51no—
150C2×Dic10—51no—
151C2×Dic10C2×D511yescharacteristic
152C5×C4⋊C4—51no—
153C5×C4⋊C4—51no—
154C5×C4⋊C4—51no—
155C5×C4⋊C4—51no—
156C5×C4⋊C4—51no—
157C5×C4⋊C4—51no—
158C5×C4⋊C4—51no—
159C5×C22⋊C4—51no—
160C5×C22⋊C4—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163C22⋊Dic5—102no—
164C22⋊Dic5C2×D511yescharacteristic
165C22⋊Dic5C2×D511yescharacteristic
166#685—51no—
167#685—51no—
168#685—51no—
169#685—51no—
170#685—51no—
171#685—51no—
172#685—102no—
173C4⋊Dic5—51no—
174C4⋊Dic5—51no—
175C4⋊Dic5—51no—
176C4⋊Dic5—51no—
177C4⋊Dic5C2×D511yescharacteristic
178#683—51no—
179#683—51no—
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—102no—
193#683C2×D511yescharacteristic
194#683C2×D511yescharacteristic
195#683C2×D511yescharacteristic
196#683C2×D511yescharacteristic
197#683C2×D511yescharacteristic
198#683C2×D511yescharacteristic
199C4×Dic5—51no—
200C4×Dic5—51no—
201C4×Dic5—51no—
202C4×Dic5—51no—
203C4×Dic5—51no—
204C4×Dic5—51no—
205C4×Dic5—102no—
206#554—251noSylow
207C2×C52—22no—
208C2×C52C22×Q811yescharacteristic
209C2×C52C2×#4111yescharacteristic
210C2×C52H(5,𝔽2)11yescharacteristic
211C5⋊D5—22no—
212C23×C5—22no—
213C23×C5C22×D511yescharacteristic
214C23×C5C22×D511yescharacteristic
215C22×D5—51no—
216C22×D5—51no—
217C22×D5—102no—
218C5×Q8—201no—
219C2×C4×C5—22no—
220C2×C4×C5—22no—
221C2×C4×C5—41no—
222C2×C4×C5—41no—
223C2×C4×C5—44no—
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—51no—
231C2×C4×C5—51no—
232C2×C4×C5—102no—
233C2×C4×C5—201no—
234C2×C4×C5C22×D511yescharacteristic
235C2×C4×C5C22×D511yescharacteristic
236C2×C4×C5C22×D511yescharacteristic
237C2×C4×C5C22×D511yescharacteristic
238C22⋊D5—54no—
239C22⋊D5—54no—
240C22⋊D5—108no—
241C2×Dic5—22no—
242C2×Dic5—41no—
243C2×Dic5—51no—
244C2×Dic5—51no—
245C2×Dic5—51no—
246C2×Dic5—51no—
247C2×Dic5—51no—
248C2×Dic5—51no—
249C2×Dic5—102no—
250C2×Dic5—102no—
251C2×Dic5—102no—
252C2×Dic5C22×D511yescharacteristic
253C2×Dic5C22×D511yescharacteristic
254C2×Dic5C22×D511yescharacteristic
255C2×Dic5C22×D511yescharacteristic
256C2×Dic5C22×D511yescharacteristic
257C2×Dic5C22×D511yescharacteristic
258C2×Dic5C22×D511yescharacteristic
259C2×Dic5C22×D511yescharacteristic
260C4×D5—102no—
261C4×D5—102no—
262C4×D5—201no—
263C4×D5—201no—
264C4×D5—201no—
265C4×D5—201no—
266C4×D5—204no—
267Dic10—41no—
268Dic10—102no—
269Dic10—102no—
270Dic10—201no—
271Dic10—201no—
272C4⋊Q8—251no—
273#125—251no—
274#125—251no—
275C22⋊Q8—51no—
276C22⋊Q8—251no—
277C22⋊Q8—251no—
278C22⋊Q8—251no—
279C22⋊Q8—251no—
280C22⋊Q8—251no—
281C4×Q8—251no—
282C4×D4—251no—
283C4×D4—251no—
284C4×D4—251no—
285C2×C4⋊C4—51no—
286C2×C4⋊C4—251no—
287C52#55411yescharacteristic, complemented, Sylow
288C22×C5—21no—
289C22×C5—21no—
290C22×C5—21no—
291C22×C5—21no—
292C22×C5—24no—
293C22×C5—22no—
294C22×C5—42no—
295C22×C5—42no—
296C22×C5Q8×D512yes—
297C22×C5Q8×D512yes—
298C22×C5C23×D511yescharacteristic
299C22×C5C23×D511yescharacteristic
300C2×D5—52no—
301C2×D5—52no—
302C2×D5—104no—
303C2×D5—101no—
304C2×D5—101no—
305C2×D5—101no—
306C2×D5—101no—
307C2×D5—202no—
308C2×D5—202no—
309C4×C5—24no—
310C4×C5—21no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—21no—
314C4×C5—42no—
315C4×C5—52no—
316C4×C5—101no—
317C4×C5—101no—
318C4×C5—101no—
319C4×C5—101no—
320C4×C5—101no—
321C4×C5—101no—
322C4×C5—101no—
323Dic5—21no—
324Dic5—21no—
325Dic5—21no—
326Dic5—21no—
327Dic5—21no—
328Dic5—21no—
329Dic5—21no—
330Dic5—52no—
331Dic5—52no—
332Dic5—104no—
333Dic5—101no—
334Dic5—101no—
335Dic5—101no—
336Dic5—101no—
337Dic5—202no—
338Dic5—202no—
339Dic5#70912yes—
340C2×Q8—51no—
341C2×Q8—251no—
342C2×Q8—251no—
343C2×D4—251no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—251no—
347C22×C4—251no—
348C22×C4—251no—
349C22×C4D5211yescharacteristic
350C4⋊C4—51no—
351C4⋊C4—51no—
352C4⋊C4—51no—
353C4⋊C4—51no—
354C4⋊C4—51no—
355C4⋊C4—51no—
356C4⋊C4—51no—
357C4⋊C4—251no—
358C4⋊C4—251no—
359C4⋊C4—251no—
360C4⋊C4—251no—
361C4⋊C4—251no—
362C4⋊C4—251no—
363C4⋊C4—251no—
364C4⋊C4—251no—
365C4⋊C4—251no—
366C22⋊C4—51no—
367C22⋊C4—51no—
368C22⋊C4—251no—
369C22⋊C4—251no—
370C22⋊C4—251no—
371C22⋊C4—251no—
372C42—251no—
373C42—251no—
374C42—251no—
375C2×C5—22no—
376C2×C5—22no—
377C2×C5—22no—
378C2×C5—22no—
379C2×C5—22no—
380C2×C5—44no—
381C2×C5C2×#70911yescharacteristic
382C2×C5#414411yescharacteristic
383C2×C5#414411yescharacteristic
384C2×C5C2×Q8×D511yescharacteristic
385C2×C5C2×Q8×D511yescharacteristic
386C2×C5D5×#4111yescharacteristic
387D5—102no—
388D5—102no—
389D5—204no—
390C23—251no—
391C23C2×D5211yescharacteristic
392Q8—201no—
393Q8—502no—
394Q8—1001no—
395D4—254no—
396C2×C4—41no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—51no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—102no—
406C2×C4—201no—
407C2×C4—251no—
408C2×C4—251no—
409C2×C4—251no—
410C2×C4—502no—
411C2×C4—1001no—
412C2×C4—1001no—
413C2×C4C2×D5211yescharacteristic
414C2×C4C2×D5211yescharacteristic
415C5—22no—
416C5#6584511yescharacteristic, complemented
417C5#6597011yescharacteristic, complemented
418C22—21no—
419C22—21no—
420C22—252no—
421C22—501no—
422C22—501no—
423C22#8787212yes—
424C22C22×D5211yescharacteristic
425C4—21no—
426C4—21no—
427C4—52no—
428C4—101no—
429C4—101no—
430C4—101no—
431C4—101no—
432C4—101no—
433C4—101no—
434C4—101no—
435C4—252no—
436C4—501no—
437C4—501no—
438C2—22no—
439C2—502no—
440C2C2×#8787211yescharacteristic
441C2D5×#70911yescharacteristic
442C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC2×C22⋊(C5⋊D5)C22
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
2. C5×#4070C2semidirect
3. C2×C5×#683C2semidirect
4. #11717082C2semidirect
5. #11717082C2semidirect
8. #11717009C2semidirect
9. #11717008C2semidirect
10. #11717001C2semidirect
11. #11716989C2semidirect
12. #11716984C2semidirect
13. #11716965C2semidirect
14. #11716957C2semidirect
15. #11716956C2semidirect
58. C22⋊(C5⋊D5)Q8semidirect
89. #4070D5semidirect
90. C2×#683D5semidirect
287. C52#554semidirect
416. C5#65845semidirect
417. C5#65970semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊(C5⋊D5)2
C5×#40702
C2×C5×#6832
#117170822
#117170702
#117170092
#117170082
#117170012
#117169892
#117169842
#117169652
#117169572
#117169562
#659705
#658455