Finite Group Databasebeta

← #13105187

Subgroups — #13105187

Order 1600.

542 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717198C211yesmaximal, characteristic, complemented
2C5×#4032C211yesmaximal, characteristic, complemented
3C5×#4027C211yesmaximal, characteristic, complemented
4#11717098C211yesmaximal, characteristic
5Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
6#11717079C211yesmaximal, characteristic, complemented
7C2×Dic52C211yesmaximal, characteristic, complemented
8#11717031C211yesmaximal, characteristic, complemented
9#11716997C211yesmaximal, characteristic, complemented
10#11716995C211yesmaximal, characteristic, complemented
11#11716977C211yesmaximal, characteristic, complemented
12#11716976C211yesmaximal, characteristic, complemented
13#11716965C211yesmaximal, characteristic, complemented
14#11716962C211yesmaximal, characteristic, complemented
15#11716959C211yesmaximal, characteristic, complemented
16C22×#1039C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×C5×C22⋊D5C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic
22C2×C4×C5×D5C2211yescharacteristic
23C2×#6037C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C52×C22⋊C4C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic
27C4⋊#1039C2211yescharacteristic
28#87806C2211yescharacteristic
29#87806C2211yescharacteristic
30C5×#685C2211yescharacteristic
31C5×#685C2211yescharacteristic
32C5×#685C2211yescharacteristic
33C5×C4⋊Dic5C2211yescharacteristic
34C5×C4⋊Dic5C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37#87781C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87780C2211yescharacteristic
41#87778C2211yescharacteristic
42#87778C2211yescharacteristic
43#87778C2211yescharacteristic
44#87778C2211yescharacteristic
45#87778C2211yescharacteristic
46Dic52C2211yescharacteristic
47Dic52C2211yescharacteristic
48Dic52C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65983—51nomaximal
52#65872—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×#1039—22no—
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C2×#1039C2311yescharacteristic
63#6045—22no—
64C5×C22⋊D5—22no—
65C5×C22⋊D5—22no—
66C2×C5×Dic5—22no—
67C2×C5×Dic5—22no—
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C4×C5×D5—22no—
75#6037—22no—
76D5×Dic5—22no—
77C5×#123—51no—
78C4×C5×D4—51no—
79#4085—51no—
80#4085—51no—
81#4081—51no—
82D4×Dic5—51no—
83#4075—51no—
84C4×C22⋊D5—51no—
85C4×C22⋊D5—51no—
86#4072—51no—
87C2×C4×Dic5—51no—
88C2×C4×Dic5—51no—
89#4044—51no—
90#4044—51no—
91#4038—51no—
92#4035—51no—
93#4035—51no—
94#4033—51no—
95#4032D511yescharacteristic, complemented
96#4031—51no—
97#4031—51no—
98#4027—51no—
99#4027D511yescharacteristic, complemented
100#4025—51no—
101#4025—51no—
102#4024—51no—
103#4024—51no—
104#4024—102no—
105#4023—51no—
106#4018—51no—
107C4×Dic10—51no—
108C22×C52—21no—
109C22×C52—21no—
110C22×C52#4112yes—
111C22×C52C2411yescharacteristic
112C2×C5×D5—21no—
113C2×C5×D5—21no—
114C2×C5×D5—21no—
115C2×C5×D5—21no—
116C2×C5×D5—21no—
117C2×C5×D5—21no—
118C4×C52—21no—
119C4×C52—21no—
120#1039—21no—
121#1039—21no—
122#1039—21no—
123#1039#4112yes—
124C5×Dic5—21no—
125C5×Dic5—21no—
126C5×Dic5—21no—
127C5×Dic5—21no—
128C5×Dic5—21no—
129C5×Dic5#4112yes—
130C2×C5×D4—51no—
131C2×C5×D4—51no—
132C22×C4×C5—51no—
133C22×C4×C5—51no—
134C22×C4×C5—51no—
135C2×C22⋊D5—51no—
136C2×C22⋊D5—51no—
137C2×C22⋊D5C2×D511yescharacteristic
138C2×C22⋊D5C2×D511yescharacteristic
139C22×Dic5—51no—
140C22×Dic5—51no—
141C22×Dic5—51no—
142C22×Dic5—102no—
143C22×Dic5C2×D511yescharacteristic
144C22×Dic5C2×D511yescharacteristic
145C2×C4×D5—51no—
146C2×C4×D5C2×D511yescharacteristic
147C2×Dic10—51no—
148C2×Dic10—51no—
149C2×Dic10—102no—
150C5×C4⋊C4—51no—
151C5×C4⋊C4—51no—
152C5×C4⋊C4—51no—
153C5×C22⋊C4—22no—
154C5×C22⋊C4—51no—
155C5×C22⋊C4—51no—
156C5×C22⋊C4—51no—
157C5×C22⋊C4C2×D511yescharacteristic
158C5×C22⋊C4C2×D511yescharacteristic
159C42×C5—51no—
160C22⋊Dic5—51no—
161C22⋊Dic5—51no—
162C22⋊Dic5—51no—
163C22⋊Dic5—51no—
164C22⋊Dic5—51no—
165C22⋊Dic5—51no—
166C22⋊Dic5—51no—
167C22⋊Dic5—102no—
168#685—51no—
169#685—51no—
170#685—51no—
171#685—51no—
172#685—51no—
173#685C2×D511yescharacteristic
174#685C2×D511yescharacteristic
175#685C2×D511yescharacteristic
176C4⋊Dic5—51no—
177C4⋊Dic5—51no—
178C4⋊Dic5—51no—
179C4⋊Dic5—102no—
180C4⋊Dic5C2×D511yescharacteristic
181C4⋊Dic5C2×D511yescharacteristic
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—102no—
194#683—102no—
195#683C2×D511yescharacteristic
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—51no—
199C4×Dic5—51no—
200C4×Dic5—51no—
201C4×Dic5—51no—
202C4×Dic5—51no—
203C4×Dic5—51no—
204C4×Dic5—51no—
205C4×Dic5—51no—
206C4×Dic5C2×D511yescharacteristic
207#529—251noSylow
208C2×C52—22no—
209C2×C52C2×#4111yescharacteristic
210C2×C52C2×#4111yescharacteristic
211C2×C52C2×#4111yescharacteristic
212C5×D5—41no—
213C5×D5—41no—
214C23×C5—22no—
215C23×C5—51no—
216C23×C5—51no—
217C23×C5C22×D511yescharacteristic
218C23×C5C22×D511yescharacteristic
219C22×D5C22×D511yescharacteristic
220C22×D5C22×D511yescharacteristic
221C5×D4—102no—
222C5×D4—102no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
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—102no—
232C2×C4×C5—102no—
233C2×C4×C5—102no—
234C2×C4×C5C22×D511yescharacteristic
235C2×C4×C5C22×D511yescharacteristic
236C2×C4×C5C22×D511yescharacteristic
237C2×C4×C5C22×D511yescharacteristic
238C22⋊D5—22no—
239C22⋊D5—22no—
240C22⋊D5—102no—
241C22⋊D5—102no—
242C2×Dic5—22no—
243C2×Dic5—22no—
244C2×Dic5—51no—
245C2×Dic5—51no—
246C2×Dic5—51no—
247C2×Dic5—51no—
248C2×Dic5—51no—
249C2×Dic5—51no—
250C2×Dic5—51no—
251C2×Dic5—51no—
252C2×Dic5—102no—
253C2×Dic5—102no—
254C2×Dic5—102no—
255C2×Dic5—102no—
256C2×Dic5—102no—
257C2×Dic5—102no—
258C2×Dic5—102no—
259C2×Dic5—204no—
260C2×Dic5C22×D511yescharacteristic
261C2×Dic5C22×D511yescharacteristic
262C2×Dic5C22×D511yescharacteristic
263C2×Dic5C22×D511yescharacteristic
264C2×Dic5C22×D511yescharacteristic
265C2×Dic5C22×D511yescharacteristic
266C4×D5—22no—
267C4×D5—102no—
268Dic10—102no—
269Dic10—102no—
270Dic10—204no—
271#126—251no—
272#126—251no—
273#125—251no—
274#124—251no—
275#123—51no—
276#123—251no—
277C22⋊Q8—251no—
278#121—251no—
279C4×Q8—251no—
280C4×D4—51no—
281C4×D4—251no—
282C4×D4—251no—
283#117—251no—
284#117—251no—
285C2×C42—251no—
286C52#52911yescharacteristic, 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—101no—
296C22×C5—101no—
297C22×C5—101no—
298C22×C5—101no—
299C22×C5—101no—
300C22×C5—101no—
301C22×C5#70912yes—
302C22×C5#71112yes—
303C22×C5C23×D511yescharacteristic
304C22×C5C23×D511yescharacteristic
305C2×D5—21no—
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—21no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—21no—
314C4×C5—21no—
315C4×C5—42no—
316C4×C5—42no—
317C4×C5—52no—
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—52no—
329Dic5—52no—
330Dic5—104no—
331Dic5—101no—
332Dic5—101no—
333Dic5—101no—
334Dic5—101no—
335Dic5—101no—
336Dic5—101no—
337Dic5—202no—
338Dic5—202no—
339Dic5—202no—
340Dic5#71112yes—
341C2×Q8—251no—
342C2×D4—51no—
343C2×D4—51no—
344C2×D4—251no—
345C22×C4—51no—
346C22×C4—51no—
347C22×C4—51no—
348C22×C4—251no—
349C22×C4—251no—
350C4⋊C4—51no—
351C4⋊C4—51no—
352C4⋊C4—51no—
353C4⋊C4—251no—
354C4⋊C4—251no—
355C4⋊C4—251no—
356C4⋊C4—251no—
357C4⋊C4—251no—
358C4⋊C4—251no—
359C4⋊C4—251no—
360C22⋊C4—51no—
361C22⋊C4—51no—
362C22⋊C4—51no—
363C22⋊C4—251no—
364C22⋊C4—251no—
365C22⋊C4—251no—
366C22⋊C4—251no—
367C22⋊C4—251no—
368C22⋊C4—251no—
369C22⋊C4D5211yescharacteristic
370C42—51no—
371C42—251no—
372C42—251no—
373C42—251no—
374C42—251no—
375C42—251no—
376C2×C5—22no—
377C2×C5—22no—
378C2×C5—22no—
379C2×C5—22no—
380C2×C5—22no—
381C2×C5—44no—
382C2×C5—201no—
383C2×C5—201no—
384C2×C5C2×#70911yescharacteristic
385C2×C5C2×#71111yescharacteristic
386C2×C5C2×#71111yescharacteristic
387C2×C5D5×#4111yescharacteristic
388C2×C5D5×#4111yescharacteristic
389C2×C5D5×#4111yescharacteristic
390D5—41no—
391D5—41no—
392C23—51no—
393C23—51no—
394C23C2×D5211yescharacteristic
395Q8—502no—
396D4—102no—
397D4—102no—
398D4—502no—
399C2×C4—51no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—51no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—102no—
406C2×C4—102no—
407C2×C4—102no—
408C2×C4—251no—
409C2×C4—251no—
410C2×C4—251no—
411C2×C4—251no—
412C2×C4—502no—
413C2×C4—502no—
414C2×C4C2×D5211yescharacteristic
415C2×C4C2×D5211yescharacteristic
416C5—22no—
417C5#6587211yescharacteristic, complemented
418C5#6598311yescharacteristic, complemented
419C22—21no—
420C22—21no—
421C22—101no—
422C22—101no—
423C22—101no—
424C22—101no—
425C22—101no—
426C22—101no—
427C22#8787012yes—
428C22C22×D5211yescharacteristic
429C4—21no—
430C4—21no—
431C4—52no—
432C4—101no—
433C4—101no—
434C4—101no—
435C4—101no—
436C4—101no—
437C4—252no—
438C4—501no—
439C4—501no—
440C4—501no—
441C2—22no—
442C2—201no—
443C2—201no—
444C2C2×#8787011yescharacteristic
445C2D5×#71111yescharacteristic
446C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717098C2
hypercenterC22⋊C4D52
hypocenterC52#529
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717198C2semidirect
2. C5×#4032C2semidirect
3. C5×#4027C2semidirect
5. Dic5×C22⋊D5C2semidirect
6. #11717079C2semidirect
7. C2×Dic52C2semidirect
8. #11717031C2semidirect
9. #11716997C2semidirect
10. #11716995C2semidirect
11. #11716977C2semidirect
12. #11716976C2semidirect
13. #11716965C2semidirect
14. #11716962C2semidirect
15. #11716959C2semidirect
95. #4032D5semidirect
99. #4027D5semidirect
286. C52#529semidirect
417. C5#65872semidirect
418. C5#65983semidirect

Maximal subgroups

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

subgroupindex
#117171982
C5×#40322
C5×#40272
#117170982
Dic5×C22⋊D52
#117170792
C2×Dic522
#117170312
#117169972
#117169952
#117169772
#117169762
#117169652
#117169622
#117169592
#659835
#658725