Finite Group Databasebeta

← #10614359

Subgroups — #10614359

Order 640.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#121C211yesmaximal, characteristic, complemented
2#66125C211yesmaximal, characteristic, complemented
3#65507C211yesmaximal, characteristic, complemented
4#65506C211yesmaximal, characteristic, complemented
5#65496C211yesmaximal, characteristic, complemented
6#65331C212yesmaximal, complemented
7#65328C212yesmaximal, complemented
8#65326C212yesmaximal, complemented
9#65324C212yesmaximal, complemented
10#65260C211yesmaximal, characteristic, complemented
11#65256C211yesmaximal, characteristic, complemented
12C22×C5×D4—21no—
13C22×C5×D4C2211yescharacteristic
14C23×C4×C5C2211yescharacteristic
15C2×#709C2211yescharacteristic, complemented
16C5×#121—21no—
17C5×#121—21no—
18C5×#121C2212yes—
19C5×#121C2212yes—
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4—21no—
22#4099—21no—
23C2×#687C2212yescomplemented
24C2×#686C2212yescomplemented
25#4076C2211yescharacteristic
26#4076C2211yescharacteristic, complemented
27#4075—21no—
28C4×C22⋊D5—21no—
29C4×C22⋊D5—21no—
30#4072C2211yescharacteristic, complemented
31#4070C2211yescharacteristic
32#4070C2211yescharacteristic, complemented
33C2×#681C2212yescomplemented
34#3964C2212yescomplemented
35#3964C2212yescomplemented
36C22⋊#168C2211yescharacteristic
37C22⋊#168C2211yescharacteristic
38C22⋊#168C2212yescomplemented
39#3942C2212yescomplemented
40#3941C2212yescomplemented
41#3940C2212yescomplemented
42#3939C2212yescomplemented
43#3194—51nomaximal, Sylow
44C24×C5—21no—
45C24×C5D411yescharacteristic
46C2×C5×D4—22no—
47C2×C5×D4—22no—
48C2×C5×D4—21no—
49C2×C5×D4—42no—
50C2×C5×D4—44no—
51C2×C5×D4C2312yes—
52C22×C4×C5—21no—
53C22×C4×C5—42no—
54C22×C4×C5D411yescharacteristic
55C22×C4×C5D411yescharacteristic
56C22×C4×C5D411yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C2×C22⋊D5—21no—
61C2×C22⋊D5—21no—
62C2×C22⋊D5—21no—
63#709—42no—
64C2×D20C2311yescharacteristic
65C2×C4×D5—21no—
66C2×Dic10C2311yescharacteristic
67C5×C4⋊C4—21no—
68C5×C4⋊C4C2312yes—
69C5×C22⋊C4—22no—
70C5×C22⋊C4—22no—
71C22⋊Dic5—21no—
72C22⋊Dic5—21no—
73C22⋊Dic5—21no—
74#687—42no—
75#686—42no—
76#685—21no—
77#685—21no—
78C4⋊Dic5C2311yescharacteristic
79C4⋊Dic5C2311yescharacteristic
80#683—21no—
81#683—21no—
82C4×Dic5—21no—
83#681—42no—
84C2×#168C2312yes—
85C2×#168C2312yes—
86#525—51no—
87C2×#121D511yescharacteristic, complemented
88#483—51no—
89#482—52no—
90#480—51no—
91#469—52no—
92#468—52no—
93#450—51no—
94#449—52no—
95#447—51no—
96#407—51no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—41no—
100C23×C5—41no—
101C23×C5—42no—
102C23×C5—42no—
103C23×C5—42no—
104C23×C5C2×D411yescharacteristic
105C23×C5C2×D411yescharacteristic
106C23×C5C2×D411yescharacteristic
107C22×D5—21no—
108C5×D4—22no—
109C5×D4—22no—
110C5×D4—42no—
111C5×D4—44no—
112C5×D4—44no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—41no—
118C2×C4×C5—41no—
119C2×C4×C5—41no—
120C2×C4×C5—42no—
121C2×C4×C5C2×D411yescharacteristic
122C2×C4×C5C2×D411yescharacteristic
123C2×C4×C5C2×D411yescharacteristic
124C2×C4×C5C2411yescharacteristic
125C22⋊D5—42no—
126C22⋊D5—42no—
127C22⋊D5—42no—
128C2×Dic5—21no—
129C2×Dic5—21no—
130C2×Dic5—21no—
131D20—22no—
132C4×D5—42no—
133Dic10—22no—
134#168—22no—
135#168—22no—
136C2×#41—51no—
137C22×D4—21no—
138C22×D4C2×D511yescharacteristic
139C23×C4C2×D511yescharacteristic
140C2×SD16—52no—
141C2×D8—52no—
142C2×M16—52no—
143#123—101no—
144C22⋊Q8—51no—
145C22⋊Q8—51no—
146#121—21no—
147#121—21no—
148#121—51no—
149#121—51no—
150#121C2×D512yes—
151#121C2×D512yes—
152#120—101no—
153C4×D4—101no—
154C4×D4—101no—
155#117—51no—
156C2×C4⋊C4C2×D511yescharacteristic
157C2×C22⋊C4—21no—
158#107—52no—
159#106—52no—
160#103—52no—
161#102—52no—
162#102—52no—
163#102—52no—
164C22⋊C8—51no—
165C22⋊C8—51no—
166C22⋊C8—52no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—44no—
176C22×C5—42no—
177C22×C5—42no—
178C22×C5—42no—
179C22×C5—84no—
180C22×C5AGL(1,ℤ/8)12yes—
181C22×C5C22×D411yescharacteristic
182C2×D5—41no—
183C2×D5—41no—
184C2×D5—41no—
185C4×C5—21no—
186C4×C5—21no—
187C4×C5—21no—
188C4×C5—42no—
189C4×C5H(5,𝔽2)12yes—
190Dic5—41no—
191Dic5—41no—
192Dic5—41no—
193C24—21no—
194C24C22⋊D511yescharacteristic
195#41—202no—
196C2×Q8—51no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—51no—
203C2×D4—101no—
204C2×D4—101no—
205C2×D4—101no—
206C2×D4C22×D512yes—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—101no—
210C22×C4C22⋊D511yescharacteristic
211C22×C4C22⋊D511yescharacteristic
212C22×C4C22⋊D511yescharacteristic
213C22×C4C22×D511yescharacteristic
214C22×C4C22×D511yescharacteristic
215C22×C4C22×D511yescharacteristic
216SD16—202no—
217D8—202no—
218M16—202no—
219C2×C8—52no—
220C2×C8—52no—
221C4⋊C4—21no—
222C4⋊C4—51no—
223C4⋊C4—51no—
224C4⋊C4—101no—
225C4⋊C4—101no—
226C4⋊C4C22×D512yes—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—101no—
230C22⋊C4—101no—
231C22⋊C4—101no—
232C22⋊C4—101no—
233C22⋊C4—101no—
234C42—101no—
235C2×C5—22no—
236C2×C5—41no—
237C2×C5—41no—
238C2×C5—82no—
239C2×C5#53411yescharacteristic
240C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C5#57511yescharacteristic
242D5—81no—
243C23—22no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—101no—
251C23C2×C22⋊D511yescharacteristic
252C23C2×C22⋊D511yescharacteristic
253C23C2×C22⋊D511yescharacteristic
254Q8—102no—
255D4—22no—
256D4—22no—
257D4—42no—
258D4—44no—
259D4—44no—
260D4—102no—
261D4—202no—
262D4—202no—
263D4—202no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—101no—
273C2×C4—101no—
274C2×C4—101no—
275C2×C4—202no—
276C2×C4C2×C22⋊D511yescharacteristic
277C2×C4C2×C22⋊D511yescharacteristic
278C2×C4C2×C22⋊D511yescharacteristic
279C2×C4C23×D511yescharacteristic
280C8—102no—
281C8—102no—
282C5#319411yescharacteristic, complemented, Sylow
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—84no—
296C22—201no—
297C22—201no—
298C22—201no—
299C22#407812yes—
300C22C22×C22⋊D511yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—201no—
306C4—201no—
307C4—201no—
308C4#414412yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2—401no—
314C2C2×#407811yescharacteristic
315C2#6613811yescharacteristic
316C2#6615611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC2×C5×#121C2
socleC22×C5C22×D4
hypercenterC2×#121D5
hypocenterC5#3194
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C5×#121C2semidirect
2. #66125C2semidirect
3. #65507C2semidirect
4. #65506C2semidirect
5. #65496C2semidirect
6. #65331C2semidirect
7. #65328C2semidirect
8. #65326C2semidirect
9. #65324C2semidirect
10. #65260C2semidirect
11. #65256C2semidirect
15. C2×#709C22semidirect
23. C2×#687C22semidirect
24. C2×#686C22semidirect
26. #4076C22semidirect
30. #4072C22semidirect
32. #4070C22semidirect
33. C2×#681C22semidirect
34. #3964C22semidirect
35. #3964C22semidirect
38. C22⋊#168C22semidirect
39. #3942C22semidirect
40. #3941C22semidirect
41. #3940C22semidirect
42. #3939C22semidirect
87. C2×#121D5semidirect
282. C5#3194semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
#661252
#655072
#655062
#654962
#653312
#653282
#653262
#653242
#652602
#652562
#31945