Finite Group Databasebeta

← #10613716

Subgroups — #10613716

Order 640.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 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#66062C211yesmaximal, characteristic, complemented
2#66025C211yesmaximal, characteristic
3C5×#501C211yesmaximal, characteristic, complemented
4#65378C211yesmaximal, characteristic, complemented
5#65354C211yesmaximal, characteristic, complemented
6#65174C212yesmaximal
7#65166C212yesmaximal, complemented
8#65160C212yesmaximal, complemented
9#65156C212yesmaximal, complemented
10#65003C211yesmaximal, characteristic, complemented
11#64994C211yesmaximal, characteristic, complemented
12C5×C4⋊Q8C2211yescharacteristic
13C5×#125C2211yescharacteristic
14#4092—21no—
15Q8×Dic5—21no—
16#4044—21no—
17#4043—21no—
18#4043C2211yescharacteristic
19#4041C2211yescharacteristic
20#4040—21no—
21#4039—21no—
22#4039C2211yescharacteristic
23#4038—21no—
24#4038C2212yes—
25#4036C2212yes—
26#4033—21no—
27#4033C2211yescharacteristic
28#4021C2211yescharacteristic
29C42×D5C2211yescharacteristic
30#4015C2211yescharacteristic
31C5×#107C2212yes—
32C5×#106C2212yes—
33C5×#97C2211yescharacteristic
34#3953C2212yes—
35#3952C2212yes—
36#3948C2212yes—
37#3946C2212yes—
38#3942C2212yes—
39#3941C2212yes—
40#3940C2212yes—
41#3939C2212yes—
42#3935C2211yescharacteristic
43#3240—51nomaximal, Sylow
44C2×C5×Q8—21no—
45C2×D20—21no—
46C2×D20C2311yescharacteristic
47C2×C4×D5—21no—
48C2×C4×D5D411yescharacteristic
49C2×C4×D5D411yescharacteristic
50C2×C4×D5C2311yescharacteristic
51C2×Dic10—21no—
52C2×Dic10C2311yescharacteristic
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4—21no—
56C5×C4⋊C4—21no—
57C5×C4⋊C4C2311yescharacteristic
58C5×C4⋊C4C2311yescharacteristic
59C5×C4⋊C4C2312yes—
60C42×C5C2311yescharacteristic
61#685—22no—
62#685—21no—
63#685—21no—
64#685—21no—
65#685—21no—
66#685—21no—
67C4⋊Dic5—21no—
68C4⋊Dic5—22no—
69C4⋊Dic5—21no—
70C4⋊Dic5C2312yes—
71#683—22no—
72#683—21no—
73C4×Dic5—22no—
74C4×Dic5—21no—
75C4×Dic5D411yescharacteristic
76C4×Dic5D411yescharacteristic
77C4×Dic5C2311yescharacteristic
78C2×#168C2312yes—
79#533—51no—
80#529—51no—
81#501—51no—
82#501D511yescharacteristic, complemented
83#489—51no—
84#489—51no—
85#485—52no—
86#482—52no—
87#479—52no—
88#478—52no—
89#432—51no—
90C22×D5—21no—
91C22×D5C2×D411yescharacteristic
92C5×Q8—42no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5C2×D411yescharacteristic
97C2×C4×C5C2×D411yescharacteristic
98C2×C4×C5C2411yescharacteristic
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5C2×D411yescharacteristic
102C2×Dic5C2×D411yescharacteristic
103C2×Dic5C2×D411yescharacteristic
104D20—22no—
105D20—42no—
106C4×D5—22no—
107C4×D5—41no—
108C4×D5—41no—
109C4×D5—42no—
110Dic10—22no—
111Dic10—42no—
112C5×C8—22no—
113#168—22no—
114C4⋊Q8—51no—
115C4⋊Q8C2×D511yescharacteristic
116#126—101no—
117#125—52no—
118#125C2×D511yescharacteristic
119#124—51no—
120#123—101no—
121C22⋊Q8—51no—
122C22⋊Q8—101no—
123C22⋊Q8—101no—
124#121—51no—
125C4×Q8—51no—
126C4×Q8—101no—
127C4×Q8—101no—
128C4×D4—51no—
129C4×D4—101no—
130#117—52no—
131#117—101no—
132C2×C42—51no—
133#107—52no—
134#107C2×D512yes—
135#106—52no—
136#106C2×D512yes—
137C4⋊C8—52no—
138#103—52no—
139#103—52no—
140#102—52no—
141#102—52no—
142C22⋊C8—52no—
143#97—51no—
144#97C2×D511yescharacteristic
145C22×C5C22×D411yescharacteristic
146C2×D5—21no—
147C2×D5—21no—
148C2×D5—21no—
149C2×D5—41no—
150C2×D5—41no—
151C2×D5—41no—
152C4×C5—21no—
153C4×C5—21no—
154C4×C5—41no—
155C4×C5—41no—
156C4×C5—42no—
157C4×C52−1+412yes—
158Dic5—21no—
159Dic5—21no—
160Dic5—21no—
161Dic5—41no—
162Dic5—42no—
163C2×Q8—21no—
164C2×Q8—51no—
165C2×Q8—101no—
166C2×D4—51no—
167C2×D4—101no—
168C22×C4—51no—
169C22×C4—51no—
170C22×C4—51no—
171C22×C4—101no—
172C2×C8—52no—
173C2×C8C22×D512yes—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—52no—
178C4⋊C4—101no—
179C4⋊C4—101no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—101no—
183C4⋊C4C22×D512yes—
184C4⋊C4C22×D511yescharacteristic
185C4⋊C4C22×D511yescharacteristic
186C22⋊C4—101no—
187C22⋊C4—101no—
188C22⋊C4—102no—
189C22⋊C4—101no—
190C22⋊C4—101no—
191C22⋊C4—101no—
192C42—51no—
193C42—51no—
194C42—51no—
195C42—102no—
196C42—101no—
197C42C22×D511yescharacteristic
198C2×C5#53611yescharacteristic
199C2×C5#57511yescharacteristic
200C2×C5#57511yescharacteristic
201D5—41no—
202D5—81no—
203C23—51no—
204C23—101no—
205Q8—42no—
206Q8—102no—
207Q8—202no—
208D4—102no—
209D4—202no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—51no—
214C2×C4—51no—
215C2×C4—51no—
216C2×C4—102no—
217C2×C4—101no—
218C2×C4—102no—
219C2×C4—201no—
220C2×C4—201no—
221C2×C4—202no—
222C2×C4D4×D511yescharacteristic
223C2×C4D4×D511yescharacteristic
224C2×C4C23×D511yescharacteristic
225C8—22no—
226C8—102no—
227C5#324011yescharacteristic, complemented, Sylow
228C22—101no—
229C22—101no—
230C22—101no—
231C22—201no—
232C22—201no—
233C22—201no—
234C22C2×D4×D511yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—101no—
241C4—101no—
242C4—101no—
243C4—201no—
244C4—202no—
245C4#414712yes—
246C2—201no—
247C2—401no—
248C2#6605811yescharacteristic
249C2#6610711yescharacteristic
250C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#501C2
socleC22×C5C22×D4
prime radical#4041C22
hypercenter#501D5
hypocenterC5#3240
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#501C2semidirect
4. #65378C2semidirect
5. #65354C2semidirect
7. #65166C2semidirect
8. #65160C2semidirect
9. #65156C2semidirect
10. #65003C2semidirect
11. #64994C2semidirect
82. #501D5semidirect
227. C5#3240semidirect

Maximal subgroups

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

subgroupindex
#660622
#660252
C5×#5012
#653782
#653542
#651742
#651662
#651602
#651562
#650032
#649942
#32405