Finite Group Databasebeta

← #13105474

Subgroups — #13105474

Order 1600.

444 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717218C211yesmaximal, characteristic
2C5×#4035C211yesmaximal, characteristic, complemented
3C5×#4033C211yesmaximal, characteristic, complemented
4C4×#6039C211yesmaximal, characteristic, complemented
5#11717009C211yesmaximal, characteristic
6#11717008C211yesmaximal, characteristic
7#11716989C211yesmaximal, characteristic, complemented
8#11716988C211yesmaximal, characteristic, complemented
9#11716987C211yesmaximal, characteristic
10#11716987C211yesmaximal, characteristic
11#11716987C211yesmaximal, characteristic
12#11716986C211yesmaximal, characteristic, complemented
13#11716963C211yesmaximal, characteristic, complemented
14#11716957C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic
16C2×C5×Dic10C2211yescharacteristic
17C2×#6039C2211yescharacteristic
18C2×#6036C2211yescharacteristic
19C52×C4⋊C4C2211yescharacteristic
20#87808C2211yescharacteristic
21#87808C2211yescharacteristic
22#87808C2211yescharacteristic
23C4⋊#1039C2211yescharacteristic
24#87806C2211yescharacteristic
25C4×#1039C2211yescharacteristic
26C5×C4⋊Dic5C2211yescharacteristic
27C5×#683C2211yescharacteristic
28C5×#683C2211yescharacteristic
29C5×#683C2211yescharacteristic
30C5×#683C2211yescharacteristic
31C5×#683C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C4×C5×Dic5C2211yescharacteristic
34C4×C5×Dic5C2211yescharacteristic
35C4×C5×Dic5C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37#87781C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87779C2211yescharacteristic
48#87779C2211yescharacteristic
49#87779C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66030—51nomaximal
52#65917—51nomaximal
53C22×C5⋊D5C2311yescharacteristic
54C2×C4×C52C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×C5×Dic5C2311yescharacteristic
61C2×C5×Dic5C2311yescharacteristic
62C2×C5×Dic5C2311yescharacteristic
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C5×Dic10—22no—
69#6039—22no—
70#6036—22no—
71C5×#125—51no—
72C4×C5×Q8—51no—
73#4092—51no—
74#4090—51no—
75#4044—51no—
76#4044—51no—
77#4044—51no—
78#4044—51no—
79#4044—51no—
80#4044—102no—
81#4043—51no—
82#4042—51no—
83#4042—51no—
84#4038—51no—
85#4036—51no—
86#4036—51no—
87#4035—51no—
88#4035—51no—
89#4035—51no—
90#4035—51no—
91#4035D511yescharacteristic, complemented
92#4034—51no—
93#4033D511yescharacteristic, complemented
94#4022—51no—
95#4022—51no—
96#4022—51no—
97#4018—51no—
98#4016—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—
112C5×Dic5—21no—
113C5×Dic5—21no—
114C5×Dic5—21no—
115C5×Dic5—21no—
116C5×Dic5—21no—
117C5×Dic5—21no—
118C5×Dic5—21no—
119C5×Dic5#4112yes—
120C2×C5×Q8—51no—
121C2×C4×D5—51no—
122C2×C4×D5—51no—
123C2×Dic10—51no—
124C2×Dic10—51no—
125C2×Dic10C2×D511yescharacteristic
126C5×C4⋊C4—22no—
127C5×C4⋊C4—51no—
128C5×C4⋊C4—51no—
129C5×C4⋊C4—51no—
130C5×C4⋊C4—51no—
131C5×C4⋊C4—51no—
132C5×C4⋊C4—51no—
133C5×C4⋊C4—51no—
134C5×C4⋊C4C2×D511yescharacteristic
135C5×C4⋊C4C2×D511yescharacteristic
136C42×C5—51no—
137C42×C5—51no—
138C42×C5—51no—
139C42×C5—51no—
140#685—51no—
141#685—51no—
142#685—51no—
143#685—51no—
144#685—51no—
145#685—51no—
146#685—51no—
147#685—51no—
148#685—51no—
149#685—51no—
150#685—51no—
151#685—51no—
152#685—102no—
153#685—102no—
154#685—102no—
155C4⋊Dic5—51no—
156C4⋊Dic5—51no—
157C4⋊Dic5—51no—
158C4⋊Dic5—51no—
159C4⋊Dic5—51no—
160C4⋊Dic5—51no—
161C4⋊Dic5—51no—
162C4⋊Dic5—51no—
163C4⋊Dic5—102no—
164C4⋊Dic5C2×D511yescharacteristic
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—51no—
180#683—51no—
181#683—102no—
182#683C2×D511yescharacteristic
183#683C2×D511yescharacteristic
184#683C2×D511yescharacteristic
185#683C2×D511yescharacteristic
186#683C2×D511yescharacteristic
187#683C2×D511yescharacteristic
188C4×Dic5—51no—
189C4×Dic5—51no—
190C4×Dic5—51no—
191C4×Dic5—51no—
192C4×Dic5—102no—
193C4×Dic5C2×D511yescharacteristic
194C4×Dic5C2×D511yescharacteristic
195C4×Dic5C2×D511yescharacteristic
196C4×Dic5C2×D511yescharacteristic
197#541—251noSylow
198C2×C52C2×#4111yescharacteristic
199C2×C522−1+411yescharacteristic
200C2×C522−1+411yescharacteristic
201C5⋊D5—41no—
202C22×D5—51no—
203C22×D5—51no—
204C22×D5—102no—
205C5×Q8—102no—
206C2×C4×C5—22no—
207C2×C4×C5—22no—
208C2×C4×C5—22no—
209C2×C4×C5—51no—
210C2×C4×C5—51no—
211C2×C4×C5—51no—
212C2×C4×C5—51no—
213C2×C4×C5—51no—
214C2×C4×C5—51no—
215C2×C4×C5—51no—
216C2×C4×C5—51no—
217C2×C4×C5C22×D511yescharacteristic
218C2×C4×C5C22×D511yescharacteristic
219C2×C4×C5C22×D511yescharacteristic
220C2×C4×C5C22×D511yescharacteristic
221C2×C4×C5C22×D511yescharacteristic
222C2×C4×C5C22×D511yescharacteristic
223C2×Dic5—51no—
224C2×Dic5—51no—
225C2×Dic5—51no—
226C2×Dic5—51no—
227C2×Dic5—51no—
228C2×Dic5—51no—
229C2×Dic5—102no—
230C2×Dic5—102no—
231C2×Dic5—102no—
232C2×Dic5C22×D511yescharacteristic
233C2×Dic5C22×D511yescharacteristic
234C2×Dic5C22×D511yescharacteristic
235C2×Dic5C22×D511yescharacteristic
236C2×Dic5C22×D511yescharacteristic
237C2×Dic5C22×D511yescharacteristic
238C2×Dic5C22×D511yescharacteristic
239C2×Dic5C22×D511yescharacteristic
240C4×D5—102no—
241C4×D5—102no—
242Dic10—22no—
243Dic10—102no—
244Dic10—102no—
245C4⋊Q8—251no—
246#126—251no—
247#126—251no—
248#126—251no—
249#126—251no—
250#125—51no—
251#125—251no—
252#125—251no—
253#125—251no—
254#125—251no—
255C22⋊Q8—251no—
256C22⋊Q8—251no—
257C4×Q8—51no—
258C4×Q8—251no—
259#117—251no—
260C52#54111yescharacteristic, complemented, Sylow
261C22×C5—22no—
262C22×C5C23×D511yescharacteristic
263C22×C5C23×D511yescharacteristic
264C2×D5—101no—
265C2×D5—101no—
266C2×D5—101no—
267C2×D5—101no—
268C2×D5—101no—
269C2×D5—101no—
270C2×D5—202no—
271C2×D5—202no—
272C2×D5—202no—
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—42no—
282C4×C5—52no—
283C4×C5—101no—
284C4×C5—101no—
285C4×C5—101no—
286C4×C5—101no—
287C4×C5—101no—
288C4×C5—101no—
289C4×C5—101no—
290Dic5—21no—
291Dic5—21no—
292Dic5—21no—
293Dic5—21no—
294Dic5—21no—
295Dic5—21no—
296Dic5—21no—
297Dic5—101no—
298Dic5—101no—
299Dic5—101no—
300Dic5—101no—
301Dic5—101no—
302Dic5—101no—
303Dic5—202no—
304Dic5—202no—
305Dic5—202no—
306Dic5#70912yes—
307C2×Q8—51no—
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—251no—
331C22⋊C4—251no—
332C22⋊C4—251no—
333C22⋊C4—251no—
334C22⋊C4—251no—
335C22⋊C4—251no—
336C42—51no—
337C42—51no—
338C42—51no—
339C42—51no—
340C42—251no—
341C42—251no—
342C2×C5—22no—
343C2×C5—22no—
344C2×C5—22no—
345C2×C5C2×#70911yescharacteristic
346C2×C5#414711yescharacteristic
347C2×C5#414711yescharacteristic
348C2×C5D5×#4111yescharacteristic
349C2×C5#415011yescharacteristic
350C2×C5#415011yescharacteristic
351D5—201no—
352D5—201no—
353D5—402no—
354C23—251no—
355Q8—102no—
356Q8—502no—
357C2×C4—51no—
358C2×C4—51no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—51no—
362C2×C4—51no—
363C2×C4—51no—
364C2×C4—51no—
365C2×C4—251no—
366C2×C4—251no—
367C2×C4—251no—
368C2×C4—502no—
369C2×C4C2×D5211yescharacteristic
370C2×C4C2×D5211yescharacteristic
371C2×C4C2×D5211yescharacteristic
372C5—22no—
373C5#6591711yescharacteristic, complemented
374C5#6603011yescharacteristic, complemented
375C22—501no—
376C22—501no—
377C22—501no—
378C22C22×D5211yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—52no—
383C4—101no—
384C4—101no—
385C4—101no—
386C4—101no—
387C4—101no—
388C4—101no—
389C4—101no—
390C4—501no—
391C4—501no—
392C4—501no—
393C2—1001no—
394C2D5×#70911yescharacteristic
395C2#1171747411yescharacteristic
396C2#1171748111yescharacteristic
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
2. C5×#4035C2semidirect
3. C5×#4033C2semidirect
4. C4×#6039C2semidirect
7. #11716989C2semidirect
8. #11716988C2semidirect
12. #11716986C2semidirect
13. #11716963C2semidirect
14. #11716957C2semidirect
91. #4035D5semidirect
93. #4033D5semidirect
260. C52#541semidirect
373. C5#65917semidirect
374. C5#66030semidirect

Maximal subgroups

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

subgroupindex
#117172182
C5×#40352
C5×#40332
C4×#60392
#117170092
#117170082
#117169892
#117169882
#117169872
#117169862
#117169632
#117169572
#117169542
#660305
#659175