Finite Group Databasebeta

← #10613193

Subgroups — #10613193

Order 640.

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

normal subgroups only

This group has 334 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
1#65958C211yesmaximal, characteristic
2C2×#4036C211yesmaximal, characteristic, complemented
3C5×#484C211yesmaximal, characteristic, complemented
4#65332C211yesmaximal, characteristic, complemented
5#65257C211yesmaximal, characteristic
6#65180C212yesmaximal, complemented
7#65167C212yesmaximal, complemented
8#65089C212yesmaximal, complemented
9#65085C212yesmaximal, complemented
10#65025C211yesmaximal, characteristic
11#65007C211yesmaximal, characteristic, complemented
12C5×C22⋊Q8C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14#4093C2211yescharacteristic
15Q8×Dic5C2211yescharacteristic
16#4076C2211yescharacteristic
17C2×C4⋊Dic5—21no—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×#683—21no—
20C2×C4×Dic5C2211yescharacteristic
21#4044—21no—
22#4039C2211yescharacteristic
23#4038—21no—
24#4036—21no—
25#4036—21no—
26#4036C2212yes—
27#4036C2212yes—
28#4036C2211yescharacteristic
29#4032—21no—
30#4027—21no—
31C5×#107C2212yes—
32C5×#103C2212yes—
33C5×C22⋊C8C2211yescharacteristic
34#3967C2212yes—
35C22⋊#168C2211yescharacteristic
36#3953C2212yes—
37#3950C2212yes—
38#3946C2212yes—
39C8×Dic5C2212yes—
40#3941C2212yes—
41#3940C2212yes—
42#3940C2212yes—
43#3232—51nomaximal, Sylow
44C2×C5×Q8C2311yescharacteristic
45C22×C4×C5—21no—
46C22×C4×C5C2311yescharacteristic
47C2×C22⋊D5—21no—
48C22×Dic5—21no—
49C22×Dic5D411yescharacteristic
50C22×Dic5D411yescharacteristic
51C2×D20C2311yescharacteristic
52C2×C4×D5—21no—
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4—21no—
56C5×C4⋊C4C2312yes—
57C5×C4⋊C4C2311yescharacteristic
58C5×C22⋊C4—21no—
59#685—21no—
60#685—21no—
61#685—21no—
62#685—21no—
63C4⋊Dic5—21no—
64C4⋊Dic5—21no—
65C4⋊Dic5—21no—
66C4⋊Dic5—22no—
67C4⋊Dic5—22no—
68C4⋊Dic5C2311yescharacteristic
69C4⋊Dic5C2312yes—
70#683—21no—
71#683—22no—
72#683—22no—
73C4×Dic5—21no—
74C4×Dic5D411yescharacteristic
75C4×Dic5D411yescharacteristic
76C4×Dic5C2311yescharacteristic
77C4×Dic5C2311yescharacteristic
78C2×#168C2312yes—
79#529—51no—
80C2×#125—51no—
81#499—52no—
82#488—52no—
83#484D511yescharacteristic, complemented
84#483—51no—
85#481—51no—
86#480—51no—
87#479—52no—
88#478—52no—
89#431—51no—
90C23×C5C2×D411yescharacteristic
91C22×D5—21no—
92C5×Q8—22no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—42no—
98C2×C4×C5C2×D411yescharacteristic
99C2×C4×C5C2411yescharacteristic
100C22⋊D5—42no—
101C2×Dic5—22no—
102C2×Dic5—21no—
103C2×Dic5—22no—
104C2×Dic5—41no—
105C2×Dic5—42no—
106C2×Dic5C2×D411yescharacteristic
107C2×Dic5C2×D411yescharacteristic
108C2×Dic5C2×D411yescharacteristic
109C2×Dic5C2×D411yescharacteristic
110D20—22no—
111C4×D5—42no—
112C5×C8—22no—
113#168—22no—
114#126—101no—
115#125—52no—
116#125—52no—
117#125—51no—
118#125—101no—
119#125—101no—
120#124—51no—
121#123—101no—
122C22⋊Q8C2×D511yescharacteristic
123#121—51no—
124C4×Q8—51no—
125C4×D4—51no—
126C4×D4—101no—
127#117—101no—
128C2×C4⋊C4—51no—
129C2×C4⋊C4—101no—
130C2×C4⋊C4—101no—
131C2×C4⋊C4C2×D511yescharacteristic
132C2×C42—51no—
133#107—52no—
134#107C2×D512yes—
135#106—52no—
136#106—52no—
137C4⋊C8—52no—
138#103—52no—
139#103C2×D512yes—
140#102—52no—
141#102—52no—
142C22⋊C8—51no—
143C22⋊C8C2×D511yescharacteristic
144C4×C8—52no—
145C22×C5—21no—
146C22×C5—21no—
147C22×C5#13512yes—
148C22×C5C22×D411yescharacteristic
149C2×D5—41no—
150C2×D5—41no—
151C2×D5—41no—
152C4×C5—21no—
153C4×C5—41no—
154C4×C5—41no—
155C4×C5—42no—
156C4×C52−1+412yes—
157Dic5—21no—
158Dic5—21no—
159Dic5—21no—
160Dic5—21no—
161Dic5—42no—
162Dic5—41no—
163C2×Q8C22×D511yescharacteristic
164C2×D4—51no—
165C2×D4—101no—
166C22×C4—21no—
167C22×C4—51no—
168C22×C4—51no—
169C22×C4—101no—
170C22×C4—101no—
171C22×C4C22×D511yescharacteristic
172C2×C8—52no—
173C2×C8C22×D512yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—52no—
177C4⋊C4—51no—
178C4⋊C4—102no—
179C4⋊C4—102no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—101no—
183C4⋊C4—101no—
184C4⋊C4—101no—
185C4⋊C4—101no—
186C4⋊C4C22×D512yes—
187C4⋊C4C22×D511yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—101no—
190C22⋊C4—101no—
191C22⋊C4—101no—
192C22⋊C4—101no—
193C42—51no—
194C42—51no—
195C42—51no—
196C42—51no—
197C42—101no—
198C2×C5—22no—
199C2×C5#53611yescharacteristic
200C2×C5C2×#13511yescharacteristic
201C2×C5#57511yescharacteristic
202D5—81no—
203C23—101no—
204C23D4×D511yescharacteristic
205Q8—22no—
206D4—102no—
207D4—202no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—51no—
214C2×C4—51no—
215C2×C4—51no—
216C2×C4—51no—
217C2×C4—102no—
218C2×C4—101no—
219C2×C4—102no—
220C2×C4—202no—
221C2×C4—201no—
222C2×C4—202no—
223C2×C4D4×D511yescharacteristic
224C2×C4C23×D511yescharacteristic
225C8—22no—
226C8—102no—
227C5#323211yescharacteristic, complemented, Sylow
228C22—21no—
229C22—21no—
230C22—201no—
231C22—201no—
232C22—201no—
233C22#406512yes—
234C22C2×D4×D511yescharacteristic
235C4—21no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—101no—
240C4—101no—
241C4—101no—
242C4—101no—
243C4—202no—
244C4—201no—
245C4#415012yes—
246C2—22no—
247C2—401no—
248C2#6598211yescharacteristic
249C2C2×#406511yescharacteristic
250C2#6610711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#484C2
socleC22×C5C22×D4
prime radical#4076C22
hypercenter#484D5
hypocenterC5#3232
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#4036C2semidirect
3. C5×#484C2semidirect
4. #65332C2semidirect
6. #65180C2semidirect
7. #65167C2semidirect
8. #65089C2semidirect
9. #65085C2semidirect
11. #65007C2semidirect
83. #484D5semidirect
227. C5#3232semidirect

Maximal subgroups

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

subgroupindex
#659582
C2×#40362
C5×#4842
#653322
#652572
#651802
#651672
#650892
#650852
#650252
#650072
#32325