Finite Group Databasebeta

← #13105235

Subgroups — #13105235

Order 1600.

808 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6038C211yesmaximal, characteristic
2C22⋊C4×C5⋊D5C211yesmaximal, characteristic, complemented
3C5×D5×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×#4023C211yesmaximal, characteristic, complemented
5#11717092C211yesmaximal, characteristic, complemented
6D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
7#11717073C211yesmaximal, characteristic, complemented
8#11717022C212yesmaximal, complemented
9C4×#6038C212yesmaximal, complemented
10#11716999C212yesmaximal, complemented
11#11716998C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C23×C5×D5C2211yescharacteristic
14C2×C4×C5⋊D5C2212yescomplemented
15C22×C5×Dic5C2211yescharacteristic
16C2×C4×C5×D5C2212yescomplemented
17C2×#6038C418yescomplemented
18C2×#6038C2211yescharacteristic
19C2×#6038C2211yescharacteristic
20C2×#6038C2211yescharacteristic
21C2×#6038C2211yescharacteristic
22C2×#6036C2212yescomplemented
23C2×D5×Dic5C2212yescomplemented
24C52×C22⋊C4C2211yescharacteristic
25C22⋊#1039C2211yescharacteristic
26#87808C2212yescomplemented
27C5×C22⋊Dic5C2211yescharacteristic
28C5×C22⋊Dic5C2211yescharacteristic
29C5×#685C2212yescomplemented
30C5×#683C2212yescomplemented
31C4×C5×Dic5C2212yescomplemented
32#87780C2212yescomplemented
33#87779C2212yescomplemented
34#87778C2212yescomplemented
35Dic52C2212yescomplemented
36#65855—51nomaximal
37#65820—51nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5—42no—
40C22×C5⋊D5C2×C414yescomplemented
41C22×C5⋊D5C2311yescharacteristic
42C22×C5⋊D5C2311yescharacteristic
43C22×C5×D5—42no—
44C22×C5×D5C2×C414yescomplemented
45C22×C5×D5C2311yescharacteristic
46C22×C5×D5C2311yescharacteristic
47C2×C4×C52C2312yes—
48C2×#1039C2312yes—
49C4×C5⋊D5—42no—
50C2×C5×Dic5C2×C414yescomplemented
51C2×C5×Dic5C2312yes—
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2311yescharacteristic
54C2×C5×Dic5C2311yescharacteristic
55C4×C5×D5—42no—
56#6038C2×C414yescomplemented
57#6038C2×C414yescomplemented
58#6038C2×C414yescomplemented
59#6038C2×C414yescomplemented
60#6036—42no—
61D5×Dic5—42no—
62C22×C22⋊D5—51no—
63C22×D20—51no—
64C5×#117—51no—
65C2×C5×C22⋊C4—51no—
66C2×C22⋊Dic5—51no—
67C4×C22⋊D5—52no—
68C4×C22⋊D5—52no—
69#4072—51no—
70#4039—52no—
71#4039—52no—
72#4027—52no—
73#4027—52no—
74D5×C22⋊C4—51no—
75D5×C22⋊C4—51no—
76D5×C22⋊C4—51no—
77D5×C22⋊C4—51no—
78D5×C22⋊C4—51no—
79D5×C22⋊C4—102no—
80D5×C22⋊C4D511yescharacteristic, complemented
81#4023—51no—
82#4023D511yescharacteristic, complemented
83C4×D20—52no—
84C4×D20—52no—
85C22×C52—22no—
86C22×C52C22×C412yes—
87C22×C52C2411yescharacteristic
88C2×C5⋊D5—22no—
89C2×C5⋊D5—22no—
90C2×C5⋊D5—44no—
91C2×C5⋊D5C22×C412yes—
92C2×C5⋊D5C22×C412yes—
93C2×C5×D5—22no—
94C2×C5×D5—22no—
95C2×C5×D5—44no—
96C2×C5×D5C22×C412yes—
97C2×C5×D5C22×C412yes—
98C4×C52—22no—
99#1039—22no—
100C5×Dic5—22no—
101C5×Dic5—22no—
102C5×Dic5C22×C412yes—
103C5×Dic5C22×C412yes—
104C24×C5—51no—
105C23×D5—51no—
106C23×D5—51no—
107C23×D5—102no—
108C23×D5C2×D511yescharacteristic
109C22×C4×C5—51no—
110C22×C4×C5—52no—
111C2×C22⋊D5—58no—
112C2×C22⋊D5—51no—
113C2×C22⋊D5—51no—
114C2×C22⋊D5—51no—
115C2×C22⋊D5—51no—
116C22×Dic5—52no—
117C22×Dic5C2×D511yescharacteristic
118C2×D20—58no—
119C2×D20—51no—
120C2×D20—51no—
121C2×D20—51no—
122C2×D20—51no—
123C2×C4×D5—52no—
124C2×C4×D5—52no—
125C2×C4×D5—52no—
126C2×C4×D5—52no—
127C2×C4×D5—52no—
128C2×C4×D5—104no—
129C2×C4×D5C2×D512yescomplemented
130C5×C4⋊C4—52no—
131C5×C22⋊C4—22no—
132C5×C22⋊C4—51no—
133C5×C22⋊C4—51no—
134C5×C22⋊C4—52no—
135C5×C22⋊C4C2×D511yescharacteristic
136C5×C22⋊C4C2×D511yescharacteristic
137C42×C5—52no—
138C22⋊Dic5—51no—
139C22⋊Dic5—51no—
140C22⋊Dic5—52no—
141C22⋊Dic5—102no—
142C22⋊Dic5C2×D511yescharacteristic
143C22⋊Dic5C2×D511yescharacteristic
144#685—52no—
145#685—52no—
146#685—52no—
147#685—52no—
148#685—52no—
149#685—104no—
150#685C2×D512yescomplemented
151C4⋊Dic5—52no—
152#683—52no—
153#683C2×D512yescomplemented
154C4×Dic5—52no—
155C4×Dic5—52no—
156C4×Dic5C2×D512yescomplemented
157#518—251noSylow
158C2×C52—22no—
159C2×C52C23×C411yescharacteristic
160C2×C52H(5,𝔽2)12yes—
161C5⋊D5—22no—
162C5⋊D5—22no—
163C5×D5—22no—
164C5×D5—22no—
165C23×C5—22no—
166C23×C5—51no—
167C23×C5—51no—
168C23×C5—54no—
169C23×C5—202no—
170C23×C5C22×D511yescharacteristic
171C23×C5C22×D511yescharacteristic
172C22×D5—42no—
173C22×D5—51no—
174C22×D5—51no—
175C22×D5—51no—
176C22×D5—51no—
177C22×D5—54no—
178C22×D5—54no—
179C22×D5—102no—
180C22×D5—102no—
181C22×D5—108no—
182C22×D5—202no—
183C22×D5—202no—
184C22×D5—208no—
185C22×D5C4×D514yescomplemented
186C22×D5C22×D511yescharacteristic
187C22×D5C22×D511yescharacteristic
188C2×C4×C5—24no—
189C2×C4×C5—54no—
190C2×C4×C5—51no—
191C2×C4×C5—51no—
192C2×C4×C5—52no—
193C2×C4×C5—52no—
194C2×C4×C5—202no—
195C2×C4×C5C22×D512yes—
196C2×C4×C5C22×D512yes—
197C22⋊D5—54no—
198C22⋊D5—54no—
199C22⋊D5—54no—
200C22⋊D5—54no—
201C2×Dic5—52no—
202C2×Dic5—52no—
203C2×Dic5—104no—
204C2×Dic5—202no—
205C2×Dic5C4×D514yescomplemented
206C2×Dic5C22×D512yes—
207C2×Dic5C22×D512yes—
208C2×Dic5C22×D511yescharacteristic
209C2×Dic5C22×D511yescharacteristic
210D20—54no—
211D20—54no—
212D20—54no—
213D20—54no—
214C4×D5—42no—
215C4×D5—202no—
216C4×D5—202no—
217C4×D5—202no—
218C4×D5—202no—
219C4×D5—202no—
220C4×D5—208no—
221C22×D4—251no—
222C4×D4—252no—
223C4×D4—252no—
224C4×D4—252no—
225C4×D4—252no—
226#117—51no—
227#117—251no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—251no—
230C2×C22⋊C4—251no—
231C2×C22⋊C4—251no—
232C52#51811yescharacteristic, complemented, Sylow
233C22×C5—22no—
234C22×C5—22no—
235C22×C5—24no—
236C22×C5—22no—
237C22×C5—44no—
238C22×C5—52no—
239C22×C5—52no—
240C22×C5—102no—
241C22×C5—102no—
242C22×C5—204no—
243C22×C5C2×C4×D512yes—
244C22×C5C2×C4×D512yes—
245C22×C5C23×D511yescharacteristic
246C22×C5C23×D511yescharacteristic
247C2×D5—22no—
248C2×D5—22no—
249C2×D5—44no—
250C2×D5—52no—
251C2×D5—52no—
252C2×D5—52no—
253C2×D5—52no—
254C2×D5—104no—
255C2×D5—104no—
256C2×D5—108no—
257C2×D5—102no—
258C2×D5—102no—
259C2×D5—102no—
260C2×D5—102no—
261C2×D5—204no—
262C2×D5—204no—
263C2×D5—204no—
264C2×D5—2016no—
265C2×D5C2×C4×D512yes—
266C2×D5C2×C4×D512yes—
267C4×C5—22no—
268C4×C5—22no—
269C4×C5—44no—
270C4×C5—52no—
271C4×C5—52no—
272C4×C5—102no—
273C4×C5—102no—
274Dic5—22no—
275Dic5—22no—
276Dic5—102no—
277Dic5—102no—
278Dic5—204no—
279Dic5C2×C4×D512yes—
280Dic5C2×C4×D512yes—
281C24—51no—
282C24—251no—
283C2×D4—251no—
284C2×D4—251no—
285C2×D4—251no—
286C2×D4—251no—
287C2×D4—258no—
288C22×C4—51no—
289C22×C4—52no—
290C22×C4—252no—
291C22×C4—252no—
292C22×C4—252no—
293C4⋊C4—52no—
294C4⋊C4—252no—
295C22⋊C4—51no—
296C22⋊C4—51no—
297C22⋊C4—52no—
298C22⋊C4—252no—
299C22⋊C4—252no—
300C22⋊C4—251no—
301C22⋊C4—252no—
302C22⋊C4D5211yescharacteristic
303C42—52no—
304C42—252no—
305C2×C5—22no—
306C2×C5—22no—
307C2×C5—22no—
308C2×C5—24no—
309C2×C5—44no—
310C2×C5—102no—
311C2×C5—102no—
312C2×C5C22×C4×D511yescharacteristic
313C2×C5C22×C4×D511yescharacteristic
314C2×C5#414412yes—
315C2×C5#414912yes—
316D5—22no—
317D5—22no—
318D5—108no—
319D5—102no—
320D5—102no—
321D5—102no—
322D5—102no—
323D5—204no—
324C23—51no—
325C23—51no—
326C23—54no—
327C23—202no—
328C23—251no—
329C23—251no—
330C23—254no—
331C23—1002no—
332C23C2×D5211yescharacteristic
333D4—254no—
334D4—254no—
335D4—254no—
336D4—254no—
337C2×C4—54no—
338C2×C4—51no—
339C2×C4—51no—
340C2×C4—52no—
341C2×C4—52no—
342C2×C4—202no—
343C2×C4—252no—
344C2×C4—1002no—
345C2×C4—1002no—
346C2×C4—1002no—
347C2×C4C2×D5212yes—
348C5—22no—
349C5#6582011yescharacteristic, complemented
350C5#6585511yescharacteristic, complemented
351C22—22no—
352C22—52no—
353C22—52no—
354C22—102no—
355C22—102no—
356C22—204no—
357C22—252no—
358C22—252no—
359C22—502no—
360C22—502no—
361C22—1004no—
362C22C4×D5212yes—
363C22C22×D5211yescharacteristic
364C4—22no—
365C4—52no—
366C4—52no—
367C4—102no—
368C4—102no—
369C4—502no—
370C2—22no—
371C2—102no—
372C2—102no—
373C2—502no—
374C2—502no—
375C2C2×C4×D5211yescharacteristic
376C2#1171747912yes—
subgroupG/H
center, FrattiniC22C22×D52
commutatorC2×C52C23×C4
FittingC52×C22⋊C4C22
socle, cosocleC22×C52C24
prime radicalC22×#6038C2
hypercenterC22⋊C4D52
hypocenterC52#518
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×C5⋊D5C2semidirect
3. C5×D5×C22⋊C4C2semidirect
4. C5×#4023C2semidirect
5. #11717092C2semidirect
6. D5×C22⋊Dic5C2semidirect
7. #11717073C2semidirect
8. #11717022C2semidirect
9. C4×#6038C2semidirect
10. #11716999C2semidirect
11. #11716998C2semidirect
14. C2×C4×C5⋊D5C22semidirect
16. C2×C4×C5×D5C22semidirect
17. C2×#6038C4semidirect
22. C2×#6036C22semidirect
23. C2×D5×Dic5C22semidirect
26. #87808C22semidirect
29. C5×#685C22semidirect
30. C5×#683C22semidirect
31. C4×C5×Dic5C22semidirect
32. #87780C22semidirect
33. #87779C22semidirect
34. #87778C22semidirect
35. Dic52C22semidirect
40. C22×C5⋊D5C2×C4semidirect
44. C22×C5×D5C2×C4semidirect
50. C2×C5×Dic5C2×C4semidirect
56. #6038C2×C4semidirect
57. #6038C2×C4semidirect
58. #6038C2×C4semidirect
59. #6038C2×C4semidirect
80. D5×C22⋊C4D5semidirect
82. #4023D5semidirect
129. C2×C4×D5C2×D5semidirect
150. #685C2×D5semidirect
153. #683C2×D5semidirect
156. C4×Dic5C2×D5semidirect
185. C22×D5C4×D5semidirect
205. C2×Dic5C4×D5semidirect
232. C52#518semidirect
349. C5#65820semidirect
350. C5#65855semidirect

Maximal subgroups

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

subgroupindex
C22×#60382
C22⋊C4×C5⋊D52
C5×D5×C22⋊C42
C5×#40232
#117170922
D5×C22⋊Dic52
#117170732
#117170222
C4×#60382
#117169992
#117169982
#658555
#658205