Finite Group Databasebeta

← #10611789

Subgroups — #10611789

Order 640.

444 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 113 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×#4093C211yesmaximal, characteristic, complemented
2#65930C211yesmaximal, characteristic, complemented
3C5×#448C211yesmaximal, characteristic, complemented
4#65516C211yesmaximal, characteristic, complemented
5#65465C212yesmaximal, complemented
6#65454C212yesmaximal, complemented
7#65325C211yesmaximal, characteristic, complemented
8#65108C212yesmaximal, complemented
9#65084C212yesmaximal, complemented
10#65021C211yesmaximal, characteristic, complemented
11#65008C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C2×#711—21no—
14C22×D20C2211yescharacteristic
15C2×C5×SD16C2212yescomplemented
16C5×#121C2211yescharacteristic
17#4093—21no—
18#4093—21no—
19#4093C2212yes—
20#4093C2212yes—
21C2×#688C2212yescomplemented
22#4086C2211yescharacteristic, complemented
23#4085—21no—
24D4×Dic5C2211yescharacteristic, complemented
25C2×#686C2212yes—
26#4076C2211yescharacteristic
27C2×#685—21no—
28C2×#685—21no—
29C2×C4×Dic5C2211yescharacteristic
30C2×D40C2212yes—
31#4039C2211yescharacteristic, complemented
32#4034C2211yescharacteristic, complemented
33#4031—21no—
34#4027—21no—
35C5×#103C2212yescomplemented
36C5×C22⋊C8C2211yescharacteristic
37#3967C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3953C2212yes—
40#3947C2212yescomplemented
41#3946C2212yescomplemented
42#3941C2212yes—
43#3213—51nomaximal, Sylow
44C23×D5—21no—
45C2×C5×Q8—21no—
46C2×C5×Q8—42no—
47C2×C5×Q8C2312yes—
48C2×C5×D4—21no—
49C2×C5×D4C2311yescharacteristic
50C22×C4×C5—21no—
51C22×C4×C5C2311yescharacteristic
52C2×C22⋊D5—21no—
53C2×C22⋊D5—21no—
54C2×C22⋊D5—21no—
55C22×Dic5—21no—
56C22×Dic5D411yescharacteristic
57C22×Dic5D411yescharacteristic
58#711—44no—
59C2×D20—21no—
60C2×D20—42no—
61C2×D20C2312yes—
62C2×D20C2311yescharacteristic
63C2×C4×D5—21no—
64C2×Dic10—21no—
65C5×SD16—42no—
66C2×C5×C8C2312yes—
67C5×C4⋊C4C2311yescharacteristic
68C5×C22⋊C4—21no—
69C22⋊Dic5—21no—
70#688—42no—
71#686—42no—
72#685—21no—
73#685—21no—
74#685—22no—
75#685—22no—
76#685—22no—
77#685—22no—
78C4⋊Dic5C2311yescharacteristic
79#683—21no—
80C4×Dic5D411yescharacteristic
81C4×Dic5D411yescharacteristic
82C4×Dic5C2311yescharacteristic
83C4×Dic5C2311yescharacteristic
84C2×#168C2312yes—
85D40—42no—
86#532—51no—
87C2×#124—51no—
88#496—52no—
89#489—52no—
90#463—52no—
91#463—52no—
92#448—51no—
93#448D511yescharacteristic, complemented
94#447—51no—
95#447—51no—
96#432—51no—
97C23×C5—21no—
98C23×C5C2×D411yescharacteristic
99C22×D5—22no—
100C22×D5—21no—
101C22×D5—42no—
102C22×D5—42no—
103C22×D5—42no—
104C5×Q8—22no—
105C5×Q8—22no—
106C5×Q8—42no—
107C5×D4—22no—
108C5×D4—42no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—42no—
113C2×C4×C5C2×D411yescharacteristic
114C2×C4×C5C2411yescharacteristic
115C22⋊D5—42no—
116C22⋊D5—42no—
117C22⋊D5—42no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—41no—
121C2×Dic5—42no—
122C2×Dic5C2×D411yescharacteristic
123C2×Dic5C2×D411yescharacteristic
124C2×Dic5C2×D411yescharacteristic
125C2×Dic5C2×D411yescharacteristic
126D20—22no—
127D20—22no—
128D20—22no—
129D20—42no—
130C4×D5—42no—
131Dic10—42no—
132C5×C8—22no—
133#168—22no—
134C2×#41—101no—
135C22×Q8C2×D511yescharacteristic
136C22×D4—51no—
137C2×SD16—52no—
138C2×SD16C2×D512yescomplemented
139C2×D8—52no—
140C2×D8—52no—
141C4⋊Q8—51no—
142#127—51no—
143#124—52no—
144#124—52no—
145#124—101no—
146#124—101no—
147#124—101no—
148#121—51no—
149#121—101no—
150#121C2×D511yescharacteristic
151C4×D4—51no—
152C4×D4—51no—
153C4×D4—101no—
154C2×C22⋊C4—101no—
155C2×C22⋊C4—101no—
156C2×C42—51no—
157C4⋊C8—52no—
158#103—52no—
159#103C2×D512yescomplemented
160#102—52no—
161#102—52no—
162C22⋊C8—51no—
163C22⋊C8C2×D511yescharacteristic
164#97—52no—
165C22×C5—21no—
166C22×C5—21no—
167C22×C5—41no—
168C22×C5—41no—
169C22×C5—41no—
170C22×C5AGL(1,ℤ/8)12yes—
171C22×C5C22×D411yescharacteristic
172C2×D5—42no—
173C2×D5—41no—
174C2×D5—41no—
175C2×D5—42no—
176C2×D5—41no—
177C2×D5—42no—
178C2×D5—84no—
179C4×C5—21no—
180C4×C5—42no—
181C4×C5—41no—
182C4×C5H(5,𝔽2)12yes—
183Dic5—21no—
184Dic5—21no—
185Dic5—21no—
186Dic5—21no—
187Dic5—41no—
188C24—101no—
189#41—204no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—101no—
193C2×Q8C22×D512yes—
194C2×D4—21no—
195C2×D4—52no—
196C2×D4—51no—
197C2×D4—101no—
198C2×D4—101no—
199C2×D4—101no—
200C2×D4—101no—
201C2×D4—202no—
202C2×D4C22×D511yescharacteristic
203C22×C4—21no—
204C22×C4—51no—
205C22×C4—51no—
206C22×C4—101no—
207C22×C4—101no—
208C22×C4C22×D511yescharacteristic
209SD16—42no—
210SD16—202no—
211D8—202no—
212D8—202no—
213C2×C8—52no—
214C2×C8C22×D512yes—
215C4⋊C4—51no—
216C4⋊C4—101no—
217C4⋊C4C22×D511yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—102no—
220C22⋊C4—102no—
221C22⋊C4—102no—
222C22⋊C4—102no—
223C22⋊C4—101no—
224C22⋊C4—101no—
225C22⋊C4—101no—
226C42—51no—
227C42—51no—
228C42—51no—
229C42—51no—
230C2×C5—22no—
231C2×C5—81no—
232C2×C5#53511yescharacteristic
233C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
234C2×C5#57511yescharacteristic
235D5—81no—
236D5—82no—
237C23—21no—
238C23—102no—
239C23—101no—
240C23—202no—
241C23—202no—
242C23—202no—
243C23D4×D511yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—42no—
247Q8—202no—
248D4—22no—
249D4—42no—
250D4—102no—
251D4—102no—
252D4—102no—
253D4—202no—
254D4—202no—
255D4—202no—
256D4—202no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—51no—
262C2×C4—51no—
263C2×C4—51no—
264C2×C4—51no—
265C2×C4—102no—
266C2×C4—101no—
267C2×C4—202no—
268C2×C4—201no—
269C2×C4—202no—
270C2×C4D4×D511yescharacteristic
271C2×C4C23×D511yescharacteristic
272C8—22no—
273C8—102no—
274C5#321311yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—201no—
281C22—202no—
282C22—201no—
283C22—202no—
284C22—202no—
285C22—201no—
286C22—404no—
287C22#406012yes—
288C22C2×D4×D511yescharacteristic
289C4—21no—
290C4—41no—
291C4—42no—
292C4—101no—
293C4—101no—
294C4—101no—
295C4—101no—
296C4—201no—
297C4#414412yes—
298C2—22no—
299C2—81no—
300C2—402no—
301C2—401no—
302C2#6592611yescharacteristic
303C2C2×#406011yescharacteristic
304C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#448C2
socleC22×C5C22×D4
hypercenter#448D5
hypocenterC5#3213
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#4093C2semidirect
2. #65930C2semidirect
3. C5×#448C2semidirect
4. #65516C2semidirect
5. #65465C2semidirect
6. #65454C2semidirect
7. #65325C2semidirect
8. #65108C2semidirect
9. #65084C2semidirect
10. #65021C2semidirect
11. #65008C2semidirect
15. C2×C5×SD16C22semidirect
21. C2×#688C22semidirect
22. #4086C22semidirect
24. D4×Dic5C22semidirect
31. #4039C22semidirect
32. #4034C22semidirect
35. C5×#103C22semidirect
37. #3967C22semidirect
40. #3947C22semidirect
41. #3946C22semidirect
93. #448D5semidirect
138. C2×SD16C2×D5semidirect
159. #103C2×D5semidirect
274. C5#3213semidirect

Maximal subgroups

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

subgroupindex
C2×#40932
#659302
C5×#4482
#655162
#654652
#654542
#653252
#651082
#650842
#650212
#650082
#32135