Finite Group Databasebeta

← #27161915

Subgroups — #27161915

Order 12608.

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#20263374C212yesmaximal, complemented
2#20263373C214yesmaximal, complemented
3#20263372C214yesmaximal, complemented
4#20263371C216yesmaximal, complemented
5C2×#15691358C216yesmaximal, complemented
6#41×D197C218yesmaximal, complemented
7C2×#41×C197C211yesmaximal, characteristic, complemented
8#15691359C2218yescomplemented
9#15691358C22112yescomplemented
10#15691358C2214yescomplemented
11#15691358C22112yescomplemented
12#15691358C22112yescomplemented
13#15691357C22124yescomplemented
14C2×Dic394C2216yescomplemented
15Q8×D197C2218yescomplemented
16C2×D788C2216yescomplemented
17C2×C22⋊D197C22112yescomplemented
18D4×D197C22124yescomplemented
19C2×C4×D197C22112yescomplemented
20#41×C197C2214yescomplemented
21#41×C197C2214yescomplemented
22C2×Q8×C197C2211yescharacteristic, complemented
23C2×D4×C197C2213yescomplemented
24C22×C4×C197C2213yescomplemented
25C22×D197C23112yes—
26Dic394C23112yes—
27Dic394C2314yes—
28D788C23112yes—
29D788C2314yes—
30C22⋊D197C23124yes—
31C22⋊D197C2318yes—
32C2×Dic197C23112yes—
33C4×D197C2318yes—
34C4×D197C23124yes—
35C23×C197C2313yes—
36Q8×C197C2314yes—
37D4×C197C23112yes—
38C2×C4×C197C2311yescharacteristic
39C2×C4×C197C2313yes—
40C2×C4×C197C2316yes—
41C2×C4×C197C2316yes—
42C22×C197—43no—
43C22×C197C2411yescharacteristic
44C22×C197C2416yes—
45C2×D197—412no—
46C2×D197C2418yes—
47C4×C197C2411yescharacteristic
48C4×C197C2416yes—
49C4×C197C2411yescharacteristic
50Dic197C2418yes—
51C2×C197—26no—
52C2×C197—21no—
53C2×C197C2511yescharacteristic
54D197—28no—
55C197#58511yescharacteristic, complemented, Sylow
56#585—1971nomaximal, Sylow
572−1+4—1974no—
582−1+4—1972no—
59H(5,𝔽2)—1974no—
60H(5,𝔽2)—1976no—
61C2×#41—1978no—
62C2×#41—1976no—
63C2×#41D19711yescharacteristic, complemented
64#41—1978no—
65#41—19724no—
66#41—19712no—
67#41—1974no—
68#41—19712no—
69#41—19712no—
70#41C2×D19714yescomplemented
71#41C2×D19714yescomplemented
72C2×Q8—1978no—
73C2×Q8—1976no—
74C2×Q8C2×D19711yescharacteristic, complemented
75C2×D4—19712no—
76C2×D4—19724no—
77C2×D4—1976no—
78C2×D4C2×D19713yescomplemented
79C22×C4—19712no—
80C22×C4C2×D19713yescomplemented
81C23—19712no—
82C23C22×D19713yes—
83Q8—19712no—
84Q8—1974no—
85Q8C22×D19714yes—
86D4—19724no—
87D4—1978no—
88D4—19712no—
89D4—1974no—
90D4C22×D197112yes—
91C2×C4—19712no—
92C2×C4—19724no—
93C2×C4—1978no—
94C2×C4C22×D19716yes—
95C2×C4C22×D19713yes—
96C2×C4C22×D19716yes—
97C2×C4C22×D19711yescharacteristic
98C22—43no—
99C22—78812no—
100C22—1978no—
101C22C23×D19716yes—
102C22C23×D19711yescharacteristic
103C4—1978no—
104C4C23×D19716yes—
105C4C23×D19711yescharacteristic
106C4C23×D19711yescharacteristic
107C2—21no—
108C2—26no—
109C2—3948no—
110C2C24×D19711yescharacteristic
subgroupG/H
centerC4C23×D197
commutator, socle, cosocleC2×C197C25
FrattiniC2C24×D197
FittingC2×#41×C197C2
hypercenterC2×#41D197
hypocenterC197#585
LCSG ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20263374C2semidirect
2. #20263373C2semidirect
3. #20263372C2semidirect
4. #20263371C2semidirect
5. C2×#15691358C2semidirect
6. #41×D197C2semidirect
7. C2×#41×C197C2semidirect
8. #15691359C22semidirect
9. #15691358C22semidirect
10. #15691358C22semidirect
11. #15691358C22semidirect
12. #15691358C22semidirect
13. #15691357C22semidirect
14. C2×Dic394C22semidirect
15. Q8×D197C22semidirect
16. C2×D788C22semidirect
17. C2×C22⋊D197C22semidirect
18. D4×D197C22semidirect
19. C2×C4×D197C22semidirect
20. #41×C197C22semidirect
21. #41×C197C22semidirect
22. C2×Q8×C197C22semidirect
23. C2×D4×C197C22semidirect
24. C22×C4×C197C22semidirect
55. C197#585semidirect
63. C2×#41D197semidirect
70. #41C2×D197semidirect
71. #41C2×D197semidirect
74. C2×Q8C2×D197semidirect
78. C2×D4C2×D197semidirect
80. C22×C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633742
#202633732
#202633722
#202633712
C2×#156913582
#41×D1972
C2×#41×C1972
#585197