Finite Group Databasebeta

← #13099697

Subgroups — #13099697

Order 1600.

1215 conjugacy classes of subgroups; only the 511 normal subgroups are shown.

normal subgroups only

This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 511 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
1C25×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11716569C216yesmaximal, complemented
3#41×D25C216yesmaximal, complemented
4#11716563C219yesmaximal, complemented
5C2×#87746C219yesmaximal, complemented
6#41×C25C2216yescomplemented
7C2×D4×C25C2219yescomplemented
8C2×C22⋊D25C22118yescomplemented
9C22×Dic25C2216yescomplemented
10#87748C2212yescomplemented
11Q8×D25C2216yescomplemented
12#87746C22118yescomplemented
13#87746C22136yescomplemented
14D4×D25C22118yescomplemented
15#87744C22118yescomplemented
16C2×C4×D25C2219yescomplemented
17C2×Dic50C2219yescomplemented
18#66285—51nomaximal
19C23×C25C2316yes—
20C22×D25C2319yes—
21Q8×C25C2312yes—
22D4×C25C23118yes—
23C2×C4×C25C2319yes—
24C22⋊D25C23136yes—
25C2×Dic25C23118yes—
26C2×Dic25C2319yes—
27D100C2316yes—
28C4×D25C2316yes—
29C4×D25C23118yes—
30Dic50C23118yes—
31C5×H(5,𝔽2)D511yescharacteristic
32#4150—56no—
33D5×#41—56no—
34#4144—59no—
35C2×#711—59no—
36C22×C25—46no—
37C22×C25C2419yes—
38C2×D25—49no—
39C2×D25C2416yes—
40C4×C25C2416yes—
41Dic25C2419yes—
42Dic25C2411yescharacteristic
43C5×#41C2×D516yes—
44C2×C5×D4C2×D519yes—
45C2×C22⋊D5—518no—
46C22×Dic5—56no—
47#713—52no—
48Q8×D5—56no—
49#711—518no—
50#711—536no—
51D4×D5—518no—
52#709—518no—
53C2×C4×D5—59no—
54C2×Dic10—59no—
55#585—251noSylow
56C2×C25—29no—
57C2×C25C2511yescharacteristic
58D25—26no—
59C23×C5C22×D516yes—
60C22×D5—59no—
61C5×Q8C22×D512yes—
62C5×D4C22×D5118yes—
63C2×C4×C5C22×D519yes—
64C22⋊D5—536no—
65C2×Dic5—518no—
66C2×Dic5—59no—
67D20—56no—
68C4×D5—56no—
69C4×D5—518no—
70Dic10—518no—
712−1+4—256no—
72H(5,𝔽2)—259no—
73H(5,𝔽2)D2511yescharacteristic, complemented
74C2×#41—256no—
75C2×#41—259no—
76C25#58511yescharacteristic, complemented, Sylow
77C22×C5—46no—
78C22×C5C23×D519yes—
79C2×D5—56no—
80C2×D5—209no—
81C4×C5C23×D516yes—
82Dic5—59no—
83Dic5—51no—
84#41—252no—
85#41—2518no—
86#41—2536no—
87#41—2518no—
88#41C2×D2516yescomplemented
89C2×Q8—256no—
90C2×Q8—259no—
91C2×D4—2518no—
92C2×D4—2518no—
93C2×D4C2×D2519yescomplemented
94C22×C4—256no—
95C22×C4—259no—
96C2×C5—29no—
97C2×C5C24×D511yescharacteristic
98D5—106no—
99C23—259no—
100C23C22×D2516yes—
101Q8—2518no—
102Q8C22×D2512yes—
103D4—2536no—
104D4—256no—
105D4C22×D25118yes—
106C2×C4—259no—
107C2×C4—2518no—
108C2×C4—256no—
109C2×C4—2518no—
110C2×C4C22×D2519yes—
111C5#6628511yescharacteristic
112C22—46no—
113C22—256no—
114C22—1009no—
115C22C23×D2519yes—
116C4—251no—
117C4—259no—
118C4C23×D2516yes—
119C2—29no—
120C2—506no—
121C2C24×D2511yescharacteristic
subgroupG/H
centerC2C24×D25
commutator, cosocleC2×C25C25
Frattini, socleC2×C5C24×D5
FittingC25×H(5,𝔽2)C2
hypercenterH(5,𝔽2)D25
hypocenterC25#585
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C25×H(5,𝔽2)C2semidirect
2. #11716569C2semidirect
3. #41×D25C2semidirect
4. #11716563C2semidirect
5. C2×#87746C2semidirect
6. #41×C25C22semidirect
7. C2×D4×C25C22semidirect
8. C2×C22⋊D25C22semidirect
9. C22×Dic25C22semidirect
10. #87748C22semidirect
11. Q8×D25C22semidirect
12. #87746C22semidirect
13. #87746C22semidirect
14. D4×D25C22semidirect
15. #87744C22semidirect
16. C2×C4×D25C22semidirect
17. C2×Dic50C22semidirect
73. H(5,𝔽2)D25semidirect
76. C25#585semidirect
88. #41C2×D25semidirect
93. C2×D4C2×D25semidirect

Maximal subgroups

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

subgroupindex
C25×H(5,𝔽2)2
#117165692
#41×D252
#117165632
C2×#877462
#662855