Finite Group Databasebeta

← #564×D97 #27155878

Subgroups — #564×D97 #27155878

Order 12416.

896 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C97⋊#564C2115yesmaximal, complemented
2#125×D97C2115yesmaximal
3#564×C97C211yesmaximal, characteristic, complemented
4#15690802C22115yes—
5#15690801C22130yes—
6#15690800C22160yes—
7C4⋊C4×D97C22130yes—
8C42×D97C2215yes—
9#125×C97C22115yes—
10C2×C4×D97C23115yes—
11C4⋊Dic97C23130yes—
12#13095490C23160yes—
13C4×Dic97C23115yes—
14C4⋊C4×C97C23130yes—
15C42×C97C2315yes—
16C22×D97C2411yescharacteristic
17C2×C4×C97C24115yes—
18C2×Dic97C24115yes—
19C4×D97—230no—
20C22×C97C2511yescharacteristic
21C2×D972−1+416yes—
22C4×C97—215no—
23Dic97—215no—
24C2×C97C2×2−1+413yes—
25D97#56414yesdirect product factor
26C2×#564—971nomaximal, Sylow
27C97C2×#56411yescharacteristic, complemented, Sylow
28#564—9715no—
29#564D9711yescharacteristic, direct product factor
30C2×#125—9715no—
31#125—9760no—
32#125—9730no—
33#125—9715no—
34#125C2×D97115yes—
35C2×C4⋊C4—9730no—
36C2×C42—975no—
37C22×C4—9715no—
38C4⋊C4—9760no—
39C4⋊C4—9730no—
40C4⋊C4C22×D97130yes—
41C42—9715no—
42C42C22×D9715yes—
43C23—971no—
44C2×C4—9715no—
45C2×C4—19430no—
46C2×C4C23×D97115yes—
47C22—976no—
48C22C24×D9711yescharacteristic
49C4—215no—
50C4—19415no—
51C2—974no—
52C22−1+4×D9713yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#564×C97C2
prime radicalC22×D97C24
hypercenter#564D97
hypocenterC97C2×#564
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
1. C97⋊#564C2semidirect
3. #564×C97C2semidirect
25. D97#564direct (N × H)
27. C97C2×#564semidirect
29. #564D97direct (N × H)

Maximal subgroups

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

subgroupindex
C97⋊#5642
#125×D972
#564×C972
C2×#56497