Finite Group Databasebeta

← #10612759

Subgroups — #10612759

Order 640.

388 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 123 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#66060C211yesmaximal, characteristic
2#65913C211yesmaximal, characteristic, complemented
3C5×#475C211yesmaximal, characteristic, complemented
4#65368C211yesmaximal, characteristic, complemented
5C4×#688C211yesmaximal, characteristic, complemented
6#65152C212yesmaximal, complemented
7D5×#106C212yesmaximal, complemented
8#65131C211yesmaximal, characteristic, complemented
9#65123C211yesmaximal, characteristic, complemented
10#65096C212yesmaximal, complemented
11#65094C212yesmaximal, complemented
12C2×#713—21no—
13C5×C4⋊Q8C2211yescharacteristic
14C4×C5×Q8C2211yescharacteristic
15#4092—21no—
16#4092C2212yes—
17C2×#688C2211yescharacteristic
18C2×C8×D5C2212yes—
19#4042—22no—
20#4041C2212yes—
21#4040—22no—
22D5×C4⋊C4C2212yes—
23D5×C4⋊C4C2212yes—
24C4×D20—21no—
25C4×D20C2211yescharacteristic
26C4×D20C2211yescharacteristic
27#4016—21no—
28#4015C2211yescharacteristic
29C5×#106C2212yes—
30C5×C4⋊C8C2211yescharacteristic
31C5×#103C2212yes—
32#3967C2211yescharacteristic
33#3953C2212yes—
34#3952C2212yes—
35#3949C2212yes—
36#3941C2211yescharacteristic
37#3940C2211yescharacteristic
38#3940C2212yes—
39#3940C2212yes—
40C4×#168C2211yescharacteristic
41#3335—51nomaximal, Sylow
42C2×C5×Q8—21no—
43C2×C5×Q8C2311yescharacteristic
44#713—44no—
45C2×D20—22no—
46C2×D20D411yescharacteristic
47C2×D20C2311yescharacteristic
48C2×C4×D5—22no—
49C2×C4×D5—21no—
50C2×C4×D5—22no—
51C2×C4×D5C2312yes—
52C2×Dic10—21no—
53C2×C5×C8C2312yes—
54C5×C4⋊C4—21no—
55C5×C4⋊C4—21no—
56C5×C4⋊C4C2311yescharacteristic
57C5×C4⋊C4C2312yes—
58C42×C5—21no—
59C42×C5C2311yescharacteristic
60#688—22no—
61#685—22no—
62#685—21no—
63#685—22no—
64#685D412yes—
65C4⋊Dic5—21no—
66C4⋊Dic5D411yescharacteristic
67C4⋊Dic5C2312yes—
68C4⋊Dic5C2311yescharacteristic
69#683—22no—
70#683—22no—
71#683—22no—
72C2×#168C2311yescharacteristic
73C2×#168C2311yescharacteristic
74C8×D5—24no—
75#554—51no—
76#547—51no—
77#498—51no—
78#483—52no—
79#481—52no—
80#476—51no—
81#475D511yescharacteristic, complemented
82#465—52no—
83C4×SD16—51no—
84C8×D4—51no—
85C2×#106—52no—
86C22×D5—21no—
87C22×D5C2×D412yes—
88C5×Q8—21no—
89C5×Q8—21no—
90C5×Q8—42no—
91C2×C4×C5—21no—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5C2×D411yescharacteristic
95C2×C4×C5C2×D411yescharacteristic
96C2×C4×C5C2411yescharacteristic
97C2×Dic5—22no—
98C2×Dic5—21no—
99C2×Dic5C2×D412yes—
100D20—21no—
101D20—21no—
102D20—44no—
103D20SD1614yescomplemented
104C4×D5—22no—
105C4×D5—22no—
106C4×D5—44no—
107C4×D5—42no—
108C4×D5—82no—
109Dic10—81no—
110C5×C8—22no—
111#168—21no—
112#168#4112yescomplemented
113C2×#41—101no—
114C2×SD16—51no—
115C22×C8—52no—
116C4⋊Q8—51no—
117C4⋊Q8C2×D511yescharacteristic
118#125—101no—
119#123—102no—
120C22⋊Q8—52no—
121C22⋊Q8—102no—
122C22⋊Q8—101no—
123#121—52no—
124C4×Q8C2×D511yescharacteristic
125C4×D4—51no—
126C4×D4—51no—
127C4×D4—101no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—52no—
130#106—52no—
131#106—51no—
132#106—52no—
133#106—52no—
134#106C2×D512yes—
135C4⋊C8C2×D511yescharacteristic
136#103—51no—
137#103C2×D512yes—
138#102—52no—
139#102—51no—
140C22⋊C8—52no—
141C4×C8—51no—
142C22×C5C22×D411yescharacteristic
143C2×D5—22no—
144C2×D5—22no—
145C2×D5—41no—
146C2×D5—41no—
147C2×D5—41no—
148C2×D5C2×SD1614yes—
149C4×C5—21no—
150C4×C5—41no—
151C4×C5—42no—
152C4×C5—41no—
153C4×C5C2×SD1612yes—
154C4×C5C2×#4111yescharacteristic
155C4×C52−1+411yescharacteristic
156Dic5—22no—
157Dic5—41no—
158Dic5—42no—
159#41—204no—
160C2×Q8—21no—
161C2×Q8—101no—
162C2×Q8C22×D511yescharacteristic
163C2×D4—51no—
164C2×D4—51no—
165C2×D4—102no—
166C22×C4—52no—
167C22×C4—102no—
168C22×C4—102no—
169C22×C4—101no—
170SD16—102no—
171C2×C8—51no—
172C2×C8—51no—
173C2×C8—104no—
174C2×C8C22×D512yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—51no—
178C4⋊C4—52no—
179C4⋊C4—51no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4—101no—
184C4⋊C4C22×D511yescharacteristic
185C4⋊C4C22×D512yes—
186C22⋊C4—52no—
187C22⋊C4—102no—
188C22⋊C4—102no—
189C22⋊C4—101no—
190C42—21no—
191C42C22×D511yescharacteristic
192C2×C5#54711yescharacteristic
193C2×C5C22×SD1611yescharacteristic
194C2×C5#57711yescharacteristic
195D5—24no—
196D5—81no—
197C23—52no—
198C23—101no—
199Q8—21no—
200Q8—21no—
201Q8—42no—
202Q8—401no—
203D4—54no—
204D4—101no—
205D4—101no—
206D4—204no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—52no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—101no—
215C2×C4—204no—
216C2×C4—202no—
217C2×C4—402no—
218C2×C4D4×D511yescharacteristic
219C2×C4D4×D511yescharacteristic
220C2×C4C23×D511yescharacteristic
221C8—22no—
222C8—52no—
223C8—101no—
224C5#333511yescharacteristic, complemented, Sylow
225C22—54no—
226C22—102no—
227C22—102no—
228C22—201no—
229C22—201no—
230C22—201no—
231C22C2×D4×D511yescharacteristic
232C4—21no—
233C4—42no—
234C4—41no—
235C4—41no—
236C4—102no—
237C4—201no—
238C4—202no—
239C4D5×SD1612yes—
240C4#414711yescharacteristic
241C4D5×#4111yescharacteristic
242C2—104no—
243C2—401no—
244C2#6596511yescharacteristic
245C2C2×D5×SD1611yescharacteristic
246C2#6611211yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#475C2
socleC22×C5C22×D4
prime radical#66060C2
hypercenter#475D5
hypocenterC5#3335
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #65913C2semidirect
3. C5×#475C2semidirect
4. #65368C2semidirect
5. C4×#688C2semidirect
6. #65152C2semidirect
7. D5×#106C2semidirect
8. #65131C2semidirect
9. #65123C2semidirect
10. #65096C2semidirect
11. #65094C2semidirect
81. #475D5semidirect
103. D20SD16semidirect
112. #168#41semidirect
224. C5#3335semidirect

Maximal subgroups

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

subgroupindex
#660602
#659132
C5×#4752
#653682
C4×#6882
#651522
D5×#1062
#651312
#651232
#650962
#650942
#33355