Finite Group Databasebeta

← #10611920

Subgroups — #10611920

Order 640.

410 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 115 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×#4092C211yesmaximal, characteristic
2D5×C22⋊Q8C211yesmaximal, characteristic
3C5×#451C211yesmaximal, characteristic, complemented
4#65517C211yesmaximal, characteristic, complemented
5#65478C212yesmaximal, complemented
6#65475C212yesmaximal, complemented
7#65332C211yesmaximal, characteristic, complemented
8#65105C212yesmaximal, complemented
9#65094C212yesmaximal, complemented
10#65015C211yesmaximal, characteristic
11#65010C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C2×Q8×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C2×C5×Q16C2212yescomplemented
16C5×C22⋊Q8C2211yescharacteristic
17#4092—21no—
18#4092—21no—
19#4092C2212yes—
20#4092C2212yes—
21#4092C2211yescharacteristic
22C2×#689C2212yescomplemented
23C2×#685—21no—
24C2×C4⋊Dic5C2211yescharacteristic
25#4070C2211yescharacteristic
26C2×#683—21no—
27C2×#676C2212yes—
28#4043C2211yescharacteristic
29#4042—21no—
30D5×C4⋊C4—21no—
31D5×C4⋊C4C2211yescharacteristic
32D5×C22⋊C4—21no—
33#4024—21no—
34C5×#103C2212yescomplemented
35C5×C22⋊C8C2211yescharacteristic
36#3967C2212yescomplemented
37#3967C2212yescomplemented
38C22⋊#168C2211yescharacteristic
39#3952C2212yes—
40#3949C2212yescomplemented
41#3948C2212yescomplemented
42#3940C2212yescomplemented
43#3195—51nomaximal, Sylow
44C23×D5D411yescharacteristic
45C2×C5×Q8—21no—
46C2×C5×Q8—42no—
47C2×C5×Q8C2311yescharacteristic
48C2×C5×Q8C2312yes—
49C22×C4×C5—21no—
50C22×C4×C5C2311yescharacteristic
51C22×Dic5—21no—
52C22×Dic5D411yescharacteristic
53Q8×D5—42no—
54C2×C4×D5—21no—
55C2×C4×D5—21no—
56C2×C4×D5—42no—
57C2×C4×D5D411yescharacteristic
58C2×C4×D5D411yescharacteristic
59C2×C4×D5C2311yescharacteristic
60C2×C4×D5C2311yescharacteristic
61C2×Dic10—21no—
62C2×Dic10C2311yescharacteristic
63C5×Q16—42no—
64C2×C5×C8C2312yes—
65C5×C4⋊C4—21no—
66C5×C4⋊C4C2311yescharacteristic
67C5×C22⋊C4—21no—
68C22⋊Dic5—21no—
69#689—42no—
70#685—21no—
71#685—21no—
72#685—22no—
73#685—22no—
74C4⋊Dic5—21no—
75C4⋊Dic5—21no—
76C4⋊Dic5C2312yes—
77C4⋊Dic5C2311yescharacteristic
78#683—21no—
79#683—21no—
80#683—21no—
81#683—22no—
82#683—22no—
83C2×#168C2312yes—
84#676—42no—
85C2×C22⋊Q8—51no—
86C2×C22⋊Q8—51no—
87#483—51no—
88#483—52no—
89#483—51no—
90#470—52no—
91#470—52no—
92#451—51no—
93#451—52no—
94#451D511yescharacteristic, complemented
95#407—51no—
96C23×C5C2×D411yescharacteristic
97C22×D5—22no—
98C22×D5—22no—
99C22×D5—41no—
100C22×D5C2×D411yescharacteristic
101C22×D5C2×D411yescharacteristic
102C5×Q8—22no—
103C5×Q8—22no—
104C5×Q8—22no—
105C5×Q8—42no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—42no—
111C2×C4×C5C2×D411yescharacteristic
112C2×C4×C5C2411yescharacteristic
113C2×Dic5—21no—
114C2×Dic5—22no—
115C2×Dic5—21no—
116C2×Dic5—41no—
117C2×Dic5—42no—
118C2×Dic5C2×D411yescharacteristic
119C2×Dic5C2×D411yescharacteristic
120C4×D5—22no—
121C4×D5—22no—
122C4×D5—41no—
123C4×D5—41no—
124C4×D5—42no—
125C4×D5—42no—
126Dic10—22no—
127Dic10—42no—
128C5×C8—22no—
129#168—22no—
130C22×Q8—51no—
131C22×Q8C2×D511yescharacteristic
132C23×C4—51no—
133C2×Q16—52no—
134C2×Q16C2×D512yescomplemented
135C2×M16—52no—
136C22⋊Q8—52no—
137C22⋊Q8—51no—
138C22⋊Q8—52no—
139C22⋊Q8—51no—
140C22⋊Q8—51no—
141C22⋊Q8—101no—
142C22⋊Q8—101no—
143C22⋊Q8—101no—
144C22⋊Q8—101no—
145C22⋊Q8C2×D511yescharacteristic
146C2×C4⋊C4—51no—
147C2×C4⋊C4—51no—
148C2×C4⋊C4—101no—
149C2×C4⋊C4—101no—
150C2×C22⋊C4—101no—
151C2×C22⋊C4—101no—
152#106—52no—
153#106—52no—
154#103—52no—
155#103—52no—
156#103—52no—
157#103C2×D512yescomplemented
158C22⋊C8—51no—
159C22⋊C8—52no—
160C22⋊C8C2×D511yescharacteristic
161C22×C5—21no—
162C22×C5—21no—
163C22×C5#13712yes—
164C22×C5C22×D411yescharacteristic
165C2×D5—21no—
166C2×D5—21no—
167C2×D5—21no—
168C2×D5—21no—
169C2×D5—21no—
170C2×D5—44no—
171C2×D5#13712yes—
172C4×C5—21no—
173C4×C5—41no—
174C4×C5—42no—
175C4×C5—41no—
176C4×C5H(5,𝔽2)12yes—
177Dic5—21no—
178Dic5—21no—
179Dic5—41no—
180Dic5—41no—
181Dic5—42no—
182C24—51no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—51no—
186C2×Q8—101no—
187C2×Q8—202no—
188C2×Q8C22×D511yescharacteristic
189C2×Q8C22×D512yes—
190C22×C4—21no—
191C22×C4—51no—
192C22×C4—51no—
193C22×C4—51no—
194C22×C4—51no—
195C22×C4—51no—
196C22×C4—101no—
197C22×C4—101no—
198C22×C4—101no—
199C22×C4—202no—
200C22×C4C22×D511yescharacteristic
201Q16—42no—
202Q16—202no—
203M16—202no—
204C2×C8—52no—
205C2×C8C22×D512yes—
206C4⋊C4—21no—
207C4⋊C4—51no—
208C4⋊C4—52no—
209C4⋊C4—101no—
210C4⋊C4—101no—
211C4⋊C4—102no—
212C4⋊C4—102no—
213C4⋊C4—101no—
214C4⋊C4—101no—
215C4⋊C4—101no—
216C4⋊C4C22×D511yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—101no—
219C22⋊C4—101no—
220C22⋊C4—102no—
221C22⋊C4—102no—
222C22⋊C4—101no—
223C2×C5—22no—
224C2×C5#53411yescharacteristic
225C2×C5C2×#13711yescharacteristic
226C2×C5C2×#13711yescharacteristic
227D5—22no—
228D5—41no—
229C23—51no—
230C23—51no—
231C23—102no—
232C23—102no—
233C23—201no—
234C23D4×D511yescharacteristic
235Q8—22no—
236Q8—22no—
237Q8—22no—
238Q8—42no—
239Q8—102no—
240Q8—202no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—51no—
247C2×C4—51no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—101no—
252C2×C4—101no—
253C2×C4—201no—
254C2×C4—201no—
255C2×C4—202no—
256C2×C4—202no—
257C2×C4—201no—
258C2×C4—202no—
259C2×C4D4×D511yescharacteristic
260C2×C4C23×D511yescharacteristic
261C8—22no—
262C8—102no—
263C5#319511yescharacteristic, complemented, Sylow
264C22—21no—
265C22—21no—
266C22—52no—
267C22—101no—
268C22—101no—
269C22—101no—
270C22—101no—
271C22—101no—
272C22—204no—
273C22#406412yes—
274C22C2×D4×D511yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—101no—
280C4—101no—
281C4—202no—
282C4—201no—
283C4—201no—
284C4#414412yes—
285C2—22no—
286C2—102no—
287C2—201no—
288C2#6592311yescharacteristic
289C2C2×#406411yescharacteristic
290C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#451C2
socleC22×C5C22×D4
prime radicalC23×D5D4
hypercenter#451D5
hypocenterC5#3195
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. C5×#451C2semidirect
4. #65517C2semidirect
5. #65478C2semidirect
6. #65475C2semidirect
7. #65332C2semidirect
8. #65105C2semidirect
9. #65094C2semidirect
11. #65010C2semidirect
15. C2×C5×Q16C22semidirect
22. C2×#689C22semidirect
34. C5×#103C22semidirect
36. #3967C22semidirect
37. #3967C22semidirect
40. #3949C22semidirect
41. #3948C22semidirect
42. #3940C22semidirect
94. #451D5semidirect
134. C2×Q16C2×D5semidirect
157. #103C2×D5semidirect
263. C5#3195semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
D5×C22⋊Q82
C5×#4512
#655172
#654782
#654752
#653322
#651052
#650942
#650152
#650102
#31955