Finite Group Databasebeta

← #27164854

Subgroups — #27164854

Order 12736.

810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 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#20264003C212yesmaximal, complemented
2#20264002C214yesmaximal, complemented
3#20264001C214yesmaximal, complemented
4#20264000C216yesmaximal, complemented
5C2×#15691886C216yesmaximal, complemented
6#41×D199C218yesmaximal, complemented
7C2×#41×C199C211yesmaximal, characteristic, complemented
8#15691887C2218yescomplemented
9#15691886C22112yescomplemented
10#15691886C2214yescomplemented
11#15691886C22112yescomplemented
12#15691886C22112yescomplemented
13#15691885C22124yescomplemented
14C2×Dic398C2216yescomplemented
15Q8×D199C2218yescomplemented
16C2×D796C2216yescomplemented
17C2×C22⋊D199C22112yescomplemented
18D4×D199C22124yescomplemented
19C2×C4×D199C22112yescomplemented
20#41×C199C2214yescomplemented
21#41×C199C2214yescomplemented
22C2×Q8×C199C2211yescharacteristic, complemented
23C2×D4×C199C2213yescomplemented
24C22×C4×C199C2213yescomplemented
25C22×D199C23112yes—
26Dic398C23112yes—
27Dic398C2314yes—
28D796C23112yes—
29D796C2314yes—
30C22⋊D199C23124yes—
31C22⋊D199C2318yes—
32C2×Dic199C23112yes—
33C4×D199C2318yes—
34C4×D199C23124yes—
35C23×C199C2313yes—
36Q8×C199C2314yes—
37D4×C199C23112yes—
38C2×C4×C199C2311yescharacteristic
39C2×C4×C199C2313yes—
40C2×C4×C199C2316yes—
41C2×C4×C199C2316yes—
42C22×C199—43no—
43C22×C199C2411yescharacteristic
44C22×C199C2416yes—
45C2×D199—412no—
46C2×D199C2418yes—
47C4×C199C2411yescharacteristic
48C4×C199C2416yes—
49C4×C199C2411yescharacteristic
50Dic199C2418yes—
51C2×C199—26no—
52C2×C199—21no—
53C2×C199C2511yescharacteristic
54D199—28no—
55C199#58511yescharacteristic, complemented, Sylow
56#585—1991nomaximal, Sylow
572−1+4—1994no—
582−1+4—1992no—
59H(5,𝔽2)—1994no—
60H(5,𝔽2)—1996no—
61C2×#41—1998no—
62C2×#41—1996no—
63C2×#41D19911yescharacteristic, complemented
64#41—1998no—
65#41—19924no—
66#41—19912no—
67#41—1994no—
68#41—19912no—
69#41—19912no—
70#41C2×D19914yescomplemented
71#41C2×D19914yescomplemented
72C2×Q8—1998no—
73C2×Q8—1996no—
74C2×Q8C2×D19911yescharacteristic, complemented
75C2×D4—19912no—
76C2×D4—19924no—
77C2×D4—1996no—
78C2×D4C2×D19913yescomplemented
79C22×C4—19912no—
80C22×C4C2×D19913yescomplemented
81C23—19912no—
82C23C22×D19913yes—
83Q8—19912no—
84Q8—1994no—
85Q8C22×D19914yes—
86D4—19924no—
87D4—1998no—
88D4—19912no—
89D4—1994no—
90D4C22×D199112yes—
91C2×C4—19912no—
92C2×C4—19924no—
93C2×C4—1998no—
94C2×C4C22×D19916yes—
95C2×C4C22×D19913yes—
96C2×C4C22×D19916yes—
97C2×C4C22×D19911yescharacteristic
98C22—43no—
99C22—79612no—
100C22—1998no—
101C22C23×D19916yes—
102C22C23×D19911yescharacteristic
103C4—1998no—
104C4C23×D19916yes—
105C4C23×D19911yescharacteristic
106C4C23×D19911yescharacteristic
107C2—21no—
108C2—26no—
109C2—3988no—
110C2C24×D19911yescharacteristic
subgroupG/H
centerC4C23×D199
commutator, socle, cosocleC2×C199C25
FrattiniC2C24×D199
FittingC2×#41×C199C2
hypercenterC2×#41D199
hypocenterC199#585
LCSG ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20264003C2semidirect
2. #20264002C2semidirect
3. #20264001C2semidirect
4. #20264000C2semidirect
5. C2×#15691886C2semidirect
6. #41×D199C2semidirect
7. C2×#41×C199C2semidirect
8. #15691887C22semidirect
9. #15691886C22semidirect
10. #15691886C22semidirect
11. #15691886C22semidirect
12. #15691886C22semidirect
13. #15691885C22semidirect
14. C2×Dic398C22semidirect
15. Q8×D199C22semidirect
16. C2×D796C22semidirect
17. C2×C22⋊D199C22semidirect
18. D4×D199C22semidirect
19. C2×C4×D199C22semidirect
20. #41×C199C22semidirect
21. #41×C199C22semidirect
22. C2×Q8×C199C22semidirect
23. C2×D4×C199C22semidirect
24. C22×C4×C199C22semidirect
55. C199#585semidirect
63. C2×#41D199semidirect
70. #41C2×D199semidirect
71. #41C2×D199semidirect
74. C2×Q8C2×D199semidirect
78. C2×D4C2×D199semidirect
80. C22×C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640032
#202640022
#202640012
#202640002
C2×#156918862
#41×D1992
C2×#41×C1992
#585199