Finite Group Databasebeta

← #13098916

Subgroups — #13098916

Order 1600.

375 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C25×#120C211yesmaximal, characteristic, complemented
2#11716515C211yesmaximal, characteristic, complemented
3C2×C22⋊Dic25C211yesmaximal, characteristic, complemented
4#11716501C211yesmaximal, characteristic, complemented
5#11716501C211yesmaximal, characteristic, complemented
6#11716500C211yesmaximal, characteristic, complemented
7#11716498C211yesmaximal, characteristic, complemented
8#11716497C211yesmaximal, characteristic, complemented
9D4×Dic25C211yesmaximal, characteristic, complemented
10#11716447C211yesmaximal, characteristic, complemented
11#11716445C211yesmaximal, characteristic, complemented
12#11716443C211yesmaximal, characteristic, complemented
13#11716441C211yesmaximal, characteristic, complemented
14#11716441C211yesmaximal, characteristic, complemented
15#11716440C211yesmaximal, characteristic, complemented
16C24×C25C2211yescharacteristic
17C2×D4×C25C2211yescharacteristic
18C2×D4×C25C2211yescharacteristic
19C2×D4×C25C2211yescharacteristic
20C2×C22⋊D25C2211yescharacteristic
21C2×C22⋊D25C2211yescharacteristic
22C2×C22⋊D25C2211yescharacteristic
23C2×C22⋊D25C2211yescharacteristic, complemented
24C22×Dic25C2211yescharacteristic
25C22×Dic25C2211yescharacteristic
26C22×Dic25C2211yescharacteristic, complemented
27C2×C4×D25C2211yescharacteristic, complemented
28C2×Dic50C2211yescharacteristic, complemented
29C22⋊C4×C25C2211yescharacteristic
30C22⋊C4×C25C2211yescharacteristic
31C22⋊C4×C25C2211yescharacteristic
32C22⋊Dic25C2211yescharacteristic
33C22⋊Dic25C2211yescharacteristic
34C22⋊Dic25C2211yescharacteristic
35C22⋊Dic25C2211yescharacteristic
36C22⋊Dic25C2211yescharacteristic
37C22⋊Dic25C2211yescharacteristic
38C22⋊Dic25C2211yescharacteristic
39C22⋊Dic25C2211yescharacteristic, complemented
40C22⋊Dic25C2211yescharacteristic, complemented
41#87720C2211yescharacteristic, complemented
42#87720C2211yescharacteristic, complemented
43C4⋊Dic25C2211yescharacteristic, complemented
44C4⋊Dic25C2211yescharacteristic, complemented
45#87718C2211yescharacteristic, complemented
46#87718C2211yescharacteristic, complemented
47#87718C2211yescharacteristic, complemented
48#87718C2211yescharacteristic, complemented
49C4×Dic25C2211yescharacteristic, complemented
50C4×Dic25C2211yescharacteristic, complemented
51#65927—51nomaximal
52C23×C25—22no—
53C23×C25—41no—
54C23×C25—41no—
55C23×C25C2311yescharacteristic
56C23×C25C2311yescharacteristic
57C23×C25C2311yescharacteristic
58C23×C25C2311yescharacteristic
59C22×D25C2311yescharacteristic
60D4×C25—22no—
61D4×C25—41no—
62D4×C25—41no—
63C2×C4×C25C2311yescharacteristic
64C2×C4×C25C2311yescharacteristic
65C2×C4×C25C2311yescharacteristic
66C22⋊D25—22no—
67C22⋊D25—41no—
68C22⋊D25—41no—
69C22⋊D25—41no—
70C2×Dic25—41no—
71C2×Dic25—41no—
72C2×Dic25—41no—
73C2×Dic25C2311yescharacteristic
74C2×Dic25C2311yescharacteristic
75C2×Dic25C2311yescharacteristic
76C2×Dic25C2311yescharacteristic
77C2×Dic25C2311yescharacteristic
78C2×Dic25C2311yescharacteristic
79C2×Dic25C2311yescharacteristic
80C4×D25—41no—
81Dic50—41no—
82C5×#120D511yescharacteristic
83#4099—51no—
84C2×C22⋊Dic5—51no—
85#4085—51no—
86#4085—51no—
87#4084—51no—
88#4082—51no—
89#4081—51no—
90D4×Dic5—51no—
91#4031—51no—
92#4029—51no—
93#4027—51no—
94#4025—51no—
95#4025—51no—
96#4024—51no—
97C22×C25—21no—
98C22×C25—21no—
99C22×C25—21no—
100C22×C25—21no—
101C22×C25—21no—
102C22×C25—21no—
103C22×C25—21no—
104C22×C25—21no—
105C22×C25—21no—
106C22×C25—21no—
107C22×C25—21no—
108C22×C25—44no—
109C22×C25#4112yes—
110C22×C25C2411yescharacteristic
111C2×D25—21no—
112C2×D25—21no—
113C2×D25—21no—
114C4×C25—21no—
115C4×C25—21no—
116C4×C25—21no—
117Dic25—21no—
118Dic25—21no—
119Dic25—21no—
120Dic25—21no—
121Dic25—21no—
122Dic25—21no—
123Dic25—21no—
124C24×C5C2×D511yescharacteristic
125C2×C5×D4C2×D511yescharacteristic
126C2×C5×D4C2×D511yescharacteristic
127C2×C5×D4C2×D511yescharacteristic
128C2×C22⋊D5—51no—
129C2×C22⋊D5—51no—
130C2×C22⋊D5—51no—
131C2×C22⋊D5—51no—
132C22×Dic5—51no—
133C22×Dic5—51no—
134C22×Dic5—51no—
135C2×C4×D5—51no—
136C2×Dic10—51no—
137C5×C22⋊C4C2×D511yescharacteristic
138C5×C22⋊C4C2×D511yescharacteristic
139C5×C22⋊C4C2×D511yescharacteristic
140C22⋊Dic5—51no—
141C22⋊Dic5—51no—
142C22⋊Dic5—51no—
143C22⋊Dic5—51no—
144C22⋊Dic5—51no—
145C22⋊Dic5—51no—
146C22⋊Dic5—51no—
147C22⋊Dic5—51no—
148C22⋊Dic5—51no—
149#685—51no—
150#685—51no—
151C4⋊Dic5—51no—
152C4⋊Dic5—51no—
153#683—51no—
154#683—51no—
155#683—51no—
156#683—51no—
157C4×Dic5—51no—
158C4×Dic5—51no—
159#538—251noSylow
160C2×C25—22no—
161C2×C25—41no—
162C2×C25—41no—
163C2×C25—41no—
164C2×C25C2×#4111yescharacteristic
165C2×C25H(5,𝔽2)11yescharacteristic
166C2×C25H(5,𝔽2)11yescharacteristic
167D25—41no—
168C23×C5—22no—
169C23×C5—41no—
170C23×C5—41no—
171C23×C5C22×D511yescharacteristic
172C23×C5C22×D511yescharacteristic
173C23×C5C22×D511yescharacteristic
174C23×C5C22×D511yescharacteristic
175C22×D5—51no—
176C5×D4—22no—
177C5×D4—41no—
178C5×D4—41no—
179C2×C4×C5C22×D511yescharacteristic
180C2×C4×C5C22×D511yescharacteristic
181C2×C4×C5C22×D511yescharacteristic
182C22⋊D5—102no—
183C22⋊D5—201no—
184C22⋊D5—201no—
185C22⋊D5—201no—
186C2×Dic5—51no—
187C2×Dic5—51no—
188C2×Dic5—51no—
189C2×Dic5—51no—
190C2×Dic5—51no—
191C2×Dic5—51no—
192C2×Dic5—51no—
193C2×Dic5—201no—
194C2×Dic5—201no—
195C2×Dic5—201no—
196C4×D5—201no—
197Dic10—201no—
198#126—251no—
199#126—251no—
200#124—251no—
201#124—251no—
202#123—251no—
203#123—251no—
204C22⋊Q8—251no—
205#121—251no—
206#121—251no—
207#121—251no—
208#120—251no—
209#120D2511yescharacteristic, complemented
210C4×D4—251no—
211C4×D4—251no—
212C2×C22⋊C4—251no—
213C25#53811yescharacteristic, complemented, Sylow
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—44no—
226C22×C5#71112yes—
227C22×C5C23×D511yescharacteristic
228C2×D5—101no—
229C2×D5—101no—
230C2×D5—101no—
231C4×C5—21no—
232C4×C5—21no—
233C4×C5—21no—
234Dic5—101no—
235Dic5—101no—
236Dic5—101no—
237Dic5—101no—
238Dic5—101no—
239Dic5—101no—
240Dic5—101no—
241C24C2×D2511yescharacteristic
242C2×Q8—251no—
243C2×D4—251no—
244C2×D4—251no—
245C2×D4—251no—
246C2×D4—251no—
247C2×D4C2×D2511yescharacteristic
248C2×D4C2×D2511yescharacteristic
249C2×D4C2×D2511yescharacteristic
250C22×C4—251no—
251C22×C4—251no—
252C22×C4—251no—
253C22×C4—251no—
254C4⋊C4—251no—
255C4⋊C4—251no—
256C4⋊C4—251no—
257C4⋊C4—251no—
258C4⋊C4—251no—
259C4⋊C4—251no—
260C22⋊C4—251no—
261C22⋊C4—251no—
262C22⋊C4—251no—
263C22⋊C4—251no—
264C22⋊C4—251no—
265C22⋊C4—251no—
266C22⋊C4—251no—
267C22⋊C4—251no—
268C22⋊C4—251no—
269C22⋊C4—251no—
270C22⋊C4—251no—
271C22⋊C4C2×D2511yescharacteristic
272C22⋊C4C2×D2511yescharacteristic
273C22⋊C4C2×D2511yescharacteristic
274C42—251no—
275C42—251no—
276C2×C5—22no—
277C2×C5—41no—
278C2×C5—41no—
279C2×C5—41no—
280C2×C5C2×#71111yescharacteristic
281C2×C5#414411yescharacteristic
282C2×C5#414411yescharacteristic
283D5—201no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—251no—
288C23C22×D2511yescharacteristic
289C23C22×D2511yescharacteristic
290C23C22×D2511yescharacteristic
291C23C22×D2511yescharacteristic
292Q8—1001no—
293D4—22no—
294D4—41no—
295D4—41no—
296D4—502no—
297D4—1001no—
298D4—1001no—
299D4—1001no—
300C2×C4—251no—
301C2×C4—251no—
302C2×C4—251no—
303C2×C4—251no—
304C2×C4—251no—
305C2×C4—251no—
306C2×C4—251no—
307C2×C4—1001no—
308C2×C4—1001no—
309C2×C4—1001no—
310C2×C4—1001no—
311C2×C4C22×D2511yescharacteristic
312C2×C4C22×D2511yescharacteristic
313C2×C4C22×D2511yescharacteristic
314C5#6592711yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—44no—
327C22—501no—
328C22—501no—
329C22—501no—
330C22#8774612yes—
331C22C23×D2511yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—501no—
336C4—501no—
337C4—501no—
338C4—501no—
339C4—501no—
340C4—501no—
341C4—501no—
342C2—22no—
343C2—41no—
344C2—41no—
345C2—41no—
346C2—1001no—
347C2C2×#8774611yescharacteristic
348C2#1171656311yescharacteristic
349C2#1171656311yescharacteristic
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×#120C2
hypercenter#120D25
hypocenterC25#538
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C25×#120C2semidirect
2. #11716515C2semidirect
3. C2×C22⋊Dic25C2semidirect
4. #11716501C2semidirect
5. #11716501C2semidirect
6. #11716500C2semidirect
7. #11716498C2semidirect
8. #11716497C2semidirect
9. D4×Dic25C2semidirect
10. #11716447C2semidirect
11. #11716445C2semidirect
12. #11716443C2semidirect
13. #11716441C2semidirect
14. #11716441C2semidirect
15. #11716440C2semidirect
23. C2×C22⋊D25C22semidirect
26. C22×Dic25C22semidirect
27. C2×C4×D25C22semidirect
28. C2×Dic50C22semidirect
39. C22⋊Dic25C22semidirect
40. C22⋊Dic25C22semidirect
41. #87720C22semidirect
42. #87720C22semidirect
43. C4⋊Dic25C22semidirect
44. C4⋊Dic25C22semidirect
45. #87718C22semidirect
46. #87718C22semidirect
47. #87718C22semidirect
48. #87718C22semidirect
49. C4×Dic25C22semidirect
50. C4×Dic25C22semidirect
209. #120D25semidirect
213. C25#538semidirect

Maximal subgroups

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

subgroupindex
C25×#1202
#117165152
C2×C22⋊Dic252
#117165012
#117165002
#117164982
#117164972
D4×Dic252
#117164472
#117164452
#117164432
#117164412
#117164402
#659275