Finite Group Databasebeta

← #13105524

Subgroups — #13105524

Order 1600.

452 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717208C211yesmaximal, characteristic, complemented
2C5×#4044C211yesmaximal, characteristic
3C5×#4035C211yesmaximal, characteristic, complemented
4#11717015C211yesmaximal, characteristic
5#11717015C211yesmaximal, characteristic
6#11716996C211yesmaximal, characteristic
7#11716989C211yesmaximal, characteristic, complemented
8#11716986C211yesmaximal, characteristic, complemented
9#11716978C211yesmaximal, characteristic
10#11716977C211yesmaximal, characteristic
11#11716976C211yesmaximal, characteristic
12#11716964C211yesmaximal, characteristic, complemented
13#11716963C211yesmaximal, characteristic, complemented
14#11716962C211yesmaximal, characteristic, complemented
15#11716962C211yesmaximal, characteristic, complemented
16C2×#6045C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×D5×Dic5C2211yescharacteristic
19C52×C4⋊C4C2211yescharacteristic
20C4⋊#1039C2211yescharacteristic
21#87806C2211yescharacteristic
22#87806C2211yescharacteristic
23C4×#1039C2211yescharacteristic
24C5×#685C2211yescharacteristic
25C5×#685C2211yescharacteristic
26C5×#685C2211yescharacteristic
27C5×C4⋊Dic5C2211yescharacteristic
28C5×C4⋊Dic5C2211yescharacteristic
29C5×#683C2211yescharacteristic
30C5×#683C2211yescharacteristic
31C5×#683C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C4×C5×Dic5C2211yescharacteristic
35C4×C5×Dic5C2211yescharacteristic
36#87781C2211yescharacteristic
37#87781C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87780C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48Dic52C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66035—51nomaximal
52#66030—51nomaximal
53C22×C5×D5C2311yescharacteristic
54C2×C4×C52C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61#6045—22no—
62#6045—22no—
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70D5×Dic5—22no—
71C5×#126—51no—
72C5×#125—51no—
73#4044—51no—
74#4044—51no—
75#4044D511yescharacteristic
76#4042—51no—
77#4042—51no—
78#4038—51no—
79#4036—51no—
80#4036—51no—
81#4035—51no—
82#4035—51no—
83#4035—51no—
84#4035—51no—
85#4035—51no—
86#4035D511yescharacteristic, complemented
87#4034—51no—
88#4034—51no—
89#4034—102no—
90#4033—51no—
91#4033—51no—
92#4025—51no—
93#4025—51no—
94#4025—51no—
95#4024—51no—
96#4024—51no—
97#4023—51no—
98#4022—51no—
99#4016—51no—
100C4×Dic10—51no—
101C4×Dic10—51no—
102C22×C52C2411yescharacteristic
103C2×C5×D5—21no—
104C2×C5×D5—21no—
105C2×C5×D5—21no—
106C4×C52—21no—
107C4×C52—21no—
108C4×C52—21no—
109#1039—21no—
110#1039—21no—
111#1039—21no—
112#1039#4112yes—
113C5×Dic5—21no—
114C5×Dic5—21no—
115C5×Dic5—21no—
116C5×Dic5—21no—
117C5×Dic5—21no—
118C5×Dic5—21no—
119C5×Dic5—21no—
120C22×Dic5—51no—
121C2×C4×D5—51no—
122C2×Dic10—51no—
123C2×Dic10—51no—
124C2×Dic10—51no—
125C2×Dic10—51no—
126C2×Dic10—102no—
127C2×Dic10—102no—
128C5×C4⋊C4—22no—
129C5×C4⋊C4—51no—
130C5×C4⋊C4—51no—
131C5×C4⋊C4—51no—
132C5×C4⋊C4—51no—
133C5×C4⋊C4—51no—
134C5×C4⋊C4—51no—
135C5×C4⋊C4—51no—
136C5×C4⋊C4C2×D511yescharacteristic
137C5×C4⋊C4C2×D511yescharacteristic
138C5×C22⋊C4—51no—
139C5×C22⋊C4—51no—
140C5×C22⋊C4—51no—
141C42×C5—51no—
142C42×C5—51no—
143C22⋊Dic5—51no—
144C22⋊Dic5—51no—
145C22⋊Dic5—51no—
146#685—51no—
147#685—51no—
148#685—51no—
149#685C2×D511yescharacteristic
150#685C2×D511yescharacteristic
151#685C2×D511yescharacteristic
152C4⋊Dic5—51no—
153C4⋊Dic5—51no—
154C4⋊Dic5—51no—
155C4⋊Dic5—51no—
156C4⋊Dic5—51no—
157C4⋊Dic5—51no—
158C4⋊Dic5—51no—
159C4⋊Dic5—102no—
160C4⋊Dic5C2×D511yescharacteristic
161C4⋊Dic5C2×D511yescharacteristic
162#683—51no—
163#683—51no—
164#683—51no—
165#683—51no—
166#683—51no—
167#683—51no—
168#683—51no—
169#683—51no—
170#683—51no—
171#683—51no—
172#683—51no—
173#683—51no—
174#683—51no—
175#683—51no—
176#683—51no—
177#683—51no—
178#683—51no—
179#683—102no—
180#683—102no—
181#683C2×D511yescharacteristic
182#683C2×D511yescharacteristic
183#683C2×D511yescharacteristic
184#683C2×D511yescharacteristic
185#683C2×D511yescharacteristic
186C4×Dic5—51no—
187C4×Dic5—51no—
188C4×Dic5—51no—
189C4×Dic5—51no—
190C4×Dic5—51no—
191C4×Dic5—51no—
192C4×Dic5—51no—
193C4×Dic5—51no—
194C4×Dic5—102no—
195C4×Dic5C2×D511yescharacteristic
196C4×Dic5C2×D511yescharacteristic
197#541—251noSylow
198C2×C52C2×#4111yescharacteristic
199C2×C522−1+411yescharacteristic
200C2×C522−1+411yescharacteristic
201C5×D5—41no—
202C23×C5—51no—
203C22×D5C22×D511yescharacteristic
204C2×C4×C5—22no—
205C2×C4×C5—22no—
206C2×C4×C5—22no—
207C2×C4×C5—51no—
208C2×C4×C5—51no—
209C2×C4×C5—51no—
210C2×C4×C5—51no—
211C2×C4×C5—51no—
212C2×C4×C5—51no—
213C2×C4×C5—51no—
214C2×C4×C5C22×D511yescharacteristic
215C2×C4×C5C22×D511yescharacteristic
216C2×C4×C5C22×D511yescharacteristic
217C2×C4×C5C22×D511yescharacteristic
218C2×C4×C5C22×D511yescharacteristic
219C2×C4×C5C22×D511yescharacteristic
220C2×Dic5—51no—
221C2×Dic5—51no—
222C2×Dic5—51no—
223C2×Dic5—51no—
224C2×Dic5—51no—
225C2×Dic5—51no—
226C2×Dic5—51no—
227C2×Dic5—51no—
228C2×Dic5—102no—
229C2×Dic5—102no—
230C2×Dic5—102no—
231C2×Dic5—102no—
232C2×Dic5—102no—
233C2×Dic5C22×D511yescharacteristic
234C2×Dic5C22×D511yescharacteristic
235C2×Dic5C22×D511yescharacteristic
236C2×Dic5C22×D511yescharacteristic
237C2×Dic5C22×D511yescharacteristic
238C2×Dic5C22×D511yescharacteristic
239C2×Dic5C22×D511yescharacteristic
240C4×D5—102no—
241Dic10—102no—
242Dic10—102no—
243Dic10—102no—
244Dic10—102no—
245Dic10—204no—
246Dic10—204no—
247C4⋊Q8—251no—
248#126—51no—
249#126—251no—
250#126—251no—
251#126—251no—
252#125—51no—
253#125—251no—
254#125—251no—
255#125—251no—
256#125—251no—
257C22⋊Q8—251no—
258C22⋊Q8—251no—
259C4×Q8—251no—
260C4×Q8—251no—
261#117—251no—
262C52#54111yescharacteristic, complemented, Sylow
263C22×C5—22no—
264C22×C5—101no—
265C22×C5—101no—
266C22×C5—101no—
267C22×C5C23×D511yescharacteristic
268C22×C5C23×D511yescharacteristic
269C2×D5—21no—
270C2×D5—21no—
271C2×D5—21no—
272C4×C5—24no—
273C4×C5—21no—
274C4×C5—21no—
275C4×C5—21no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—21no—
279C4×C5—42no—
280C4×C5—42no—
281C4×C5—101no—
282C4×C5—101no—
283C4×C5—101no—
284C4×C5—101no—
285C4×C5—101no—
286C4×C5—101no—
287C4×C5—101no—
288Dic5—21no—
289Dic5—21no—
290Dic5—21no—
291Dic5—21no—
292Dic5—21no—
293Dic5—21no—
294Dic5—21no—
295Dic5—52no—
296Dic5—52no—
297Dic5—104no—
298Dic5—104no—
299Dic5—101no—
300Dic5—101no—
301Dic5—101no—
302Dic5—101no—
303Dic5—101no—
304Dic5—101no—
305Dic5—202no—
306Dic5—202no—
307C2×Q8—251no—
308C2×Q8—251no—
309C22×C4—251no—
310C4⋊C4—51no—
311C4⋊C4—51no—
312C4⋊C4—51no—
313C4⋊C4—51no—
314C4⋊C4—51no—
315C4⋊C4—51no—
316C4⋊C4—51no—
317C4⋊C4—251no—
318C4⋊C4—251no—
319C4⋊C4—251no—
320C4⋊C4—251no—
321C4⋊C4—251no—
322C4⋊C4—251no—
323C4⋊C4—251no—
324C4⋊C4—251no—
325C4⋊C4—251no—
326C4⋊C4—251no—
327C4⋊C4—251no—
328C4⋊C4—251no—
329C4⋊C4D5211yescharacteristic
330C22⋊C4—51no—
331C22⋊C4—51no—
332C22⋊C4—51no—
333C22⋊C4—251no—
334C22⋊C4—251no—
335C22⋊C4—251no—
336C42—51no—
337C42—51no—
338C42—251no—
339C42—251no—
340C42—251no—
341C42—251no—
342C2×C5—22no—
343C2×C5—22no—
344C2×C5—22no—
345C2×C5—201no—
346C2×C5#414711yescharacteristic
347C2×C5D5×#4111yescharacteristic
348C2×C5D5×#4111yescharacteristic
349C2×C5#415011yescharacteristic
350C2×C5#415011yescharacteristic
351C2×C5#415011yescharacteristic
352D5—41no—
353C23—51no—
354Q8—502no—
355Q8—502no—
356C2×C4—51no—
357C2×C4—51no—
358C2×C4—51no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—251no—
364C2×C4—251no—
365C2×C4—251no—
366C2×C4—251no—
367C2×C4—502no—
368C2×C4C2×D5211yescharacteristic
369C2×C4C2×D5211yescharacteristic
370C2×C4C2×D5211yescharacteristic
371C5—22no—
372C5#6603011yescharacteristic, complemented
373C5#6603511yescharacteristic, complemented
374C22—101no—
375C22—101no—
376C22—101no—
377C22C22×D5211yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—101no—
382C4—101no—
383C4—101no—
384C4—101no—
385C4—101no—
386C4—101no—
387C4—101no—
388C4—252no—
389C4—501no—
390C4—501no—
391C4—501no—
392C2—201no—
393C2#1171746311yescharacteristic
394C2#1171747711yescharacteristic
395C2#1171748011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC22×C5×D5C23
hypercenterC4⋊C4D52
hypocenterC52#541
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717208C2semidirect
3. C5×#4035C2semidirect
7. #11716989C2semidirect
8. #11716986C2semidirect
12. #11716964C2semidirect
13. #11716963C2semidirect
14. #11716962C2semidirect
15. #11716962C2semidirect
86. #4035D5semidirect
262. C52#541semidirect
372. C5#66030semidirect
373. C5#66035semidirect

Maximal subgroups

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

subgroupindex
#117172082
C5×#40442
C5×#40352
#117170152
#117169962
#117169892
#117169862
#117169782
#117169772
#117169762
#117169642
#117169632
#117169622
#660355
#660305