Finite Group Databasebeta

← #10599772

Subgroups — #10599772

Order 640.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#65834C211yesmaximal, characteristic
2#65410C212yesmaximal
3#65266C211yesmaximal, characteristic
4C5×#343C211yesmaximal, characteristic
5#64775C212yesmaximal
6C2×C5×M16C2212yes—
7C2×C5×C4⋊C4C2211yescharacteristic
8#4072C2212yes—
9C2×C4⋊Dic5C2211yescharacteristic
10#4070—22no—
11C2×C4×Dic5C2211yescharacteristic
12#4036—22no—
13#4034—22no—
14#3963—22no—
15#3950—22no—
16#3949—22no—
17#2061—51nomaximal, Sylow
18C22×C4×C5D412yes—
19C22×C4×C5C2311yescharacteristic
20C22×Dic5—21no—
21C22×Dic5—21no—
22C2×Dic10—22no—
23C5×M16—22no—
24C5×M16Q814yescomplemented
25C2×C5×C8—22no—
26C5×C4⋊C4—22no—
27C22⋊Dic5—22no—
28C4⋊Dic5—22no—
29C4⋊Dic5—22no—
30C4⋊Dic5D412yes—
31C4⋊Dic5D412yes—
32#683—24no—
33C4×Dic5—22no—
34C4×Dic5—21no—
35C4×Dic5Q812yescomplemented
36#544—51no—
37#428—52no—
38#382—51no—
39#343D511yescharacteristic
40#339—52no—
41C23×C5C2×D411yescharacteristic
42C2×C4×C5—22no—
43C2×C4×C5—42no—
44C2×C4×C5C2×D412yes—
45C2×C4×C5C2×Q812yes—
46C2×C4×C5C2×Q811yescharacteristic
47C2×C4×C5#4111yescharacteristic
48C2×Dic5—22no—
49C2×Dic5—22no—
50C2×Dic5—22no—
51C2×Dic5—41no—
52C2×Dic5—41no—
53C2×Dic5—42no—
54C2×Dic5—41no—
55Dic10—82no—
56C5×C8—24no—
57C2×M16C2×D512yes—
58C4⋊Q8—102no—
59#125—102no—
60C22⋊Q8—102no—
61#117—52no—
62C2×C4⋊C4—51no—
63C2×C4⋊C4C2×D511yescharacteristic
64C2×C42—51no—
65#107—102no—
66#106—102no—
67#95—102no—
68C22×C5—21no—
69C22×C5—21no—
70C22×C5#12011yescharacteristic
71C22×C5C22⋊Q811yescharacteristic
72C22×C5C4⋊Q811yescharacteristic
73C4×C5—42no—
74C4×C5C22⋊Q812yes—
75C4×C5C4⋊Q812yes—
76Dic5—42no—
77Dic5—42no—
78Dic5—42no—
79Dic5—81no—
80C2×Q8—102no—
81C22×C4—101no—
82C22×C4—101no—
83C22×C4D2012yes—
84C22×C4C22×D511yescharacteristic
85M16—22no—
86M16Dic1014yescomplemented
87C2×C8—22no—
88C4⋊C4—22no—
89C4⋊C4—52no—
90C4⋊C4—52no—
91C4⋊C4—102no—
92C4⋊C4—104no—
93C4⋊C4—102no—
94C22⋊C4—102no—
95C42—52no—
96C42—102no—
97C42—101no—
98C2×C5—21no—
99C2×C5—21no—
100C2×C5#39511yescharacteristic
101C2×C5#45411yescharacteristic
102C2×C5#45511yescharacteristic
103C23C2×D2011yescharacteristic
104Q8—402no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—102no—
108C2×C4—102no—
109C2×C4—102no—
110C2×C4—201no—
111C2×C4—201no—
112C2×C4—202no—
113C2×C4—201no—
114C2×C4C2×Dic1012yes—
115C2×C4D4×D512yes—
116C2×C4#71111yescharacteristic
117C2×C4Q8×D511yescharacteristic
118C8—24no—
119C5#206111yescharacteristic, complemented, Sylow
120C22—21no—
121C22—21no—
122C22#401511yescharacteristic
123C22#402811yescharacteristic
124C22#404311yescharacteristic
125C4—42no—
126C4—202no—
127C4—202no—
128C4—202no—
129C4—401no—
130C4#402412yes—
131C4#403412yes—
132C2—21no—
133C2—21no—
134C2#6494511yescharacteristic
135C2#6503511yescharacteristic
136C2#6504211yescharacteristic
subgroupG/H
centerC22#4028
commutator, cosocleC22×C4×C5C23
FrattiniC22×C4C22×D5
FittingC5×#343C2
socleC22×C5#120
prime radicalC23×C5C2×D4
hypercenter#343D5
hypocenterC5#2061
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
24. C5×M16Q8semidirect
35. C4×Dic5Q8semidirect
86. M16Dic10semidirect
119. C5#2061semidirect

Maximal subgroups

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

subgroupindex
#658342
#654102
#652662
C5×#3432
#647752
#20615