Finite Group Databasebeta

← #41001197

Subgroups — #41001197

Order 26816.

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#27211766C212yesmaximal, complemented
2#27211765C214yesmaximal, complemented
3#27211764C214yesmaximal, complemented
4#27211763C216yesmaximal, complemented
5C2×#20269411C216yesmaximal, complemented
6#41×D419C218yesmaximal, complemented
7C2×#41×C419C211yesmaximal, characteristic, complemented
8#20269412C2218yescomplemented
9#20269411C22112yescomplemented
10#20269411C2214yescomplemented
11#20269411C22112yescomplemented
12#20269411C22112yescomplemented
13#20269410C22124yescomplemented
14C2×Dic838C2216yescomplemented
15Q8×D419C2218yescomplemented
16C2×D1676C2216yescomplemented
17C2×C22⋊D419C22112yescomplemented
18D4×D419C22124yescomplemented
19C2×C4×D419C22112yescomplemented
20#41×C419C2214yescomplemented
21#41×C419C2214yescomplemented
22C2×Q8×C419C2211yescharacteristic, complemented
23C2×D4×C419C2213yescomplemented
24C22×C4×C419C2213yescomplemented
25C22×D419C23112yes—
26Dic838C23112yes—
27Dic838C2314yes—
28D1676C23112yes—
29D1676C2314yes—
30C22⋊D419C23124yes—
31C22⋊D419C2318yes—
32C2×Dic419C23112yes—
33C4×D419C2318yes—
34C4×D419C23124yes—
35C23×C419C2313yes—
36Q8×C419C2314yes—
37D4×C419C23112yes—
38C2×C4×C419C2311yescharacteristic
39C2×C4×C419C2313yes—
40C2×C4×C419C2316yes—
41C2×C4×C419C2316yes—
42C22×C419—43no—
43C22×C419C2411yescharacteristic
44C22×C419C2416yes—
45C2×D419—412no—
46C2×D419C2418yes—
47C4×C419C2411yescharacteristic
48C4×C419C2416yes—
49C4×C419C2411yescharacteristic
50Dic419C2418yes—
51C2×C419—26no—
52C2×C419—21no—
53C2×C419C2511yescharacteristic
54D419—28no—
55C419#58511yescharacteristic, complemented, Sylow
56#585—4191nomaximal, Sylow
572−1+4—4194no—
582−1+4—4192no—
59H(5,𝔽2)—4194no—
60H(5,𝔽2)—4196no—
61C2×#41—4198no—
62C2×#41—4196no—
63C2×#41D41911yescharacteristic, complemented
64#41—4198no—
65#41—41924no—
66#41—41912no—
67#41—4194no—
68#41—41912no—
69#41—41912no—
70#41C2×D41914yescomplemented
71#41C2×D41914yescomplemented
72C2×Q8—4198no—
73C2×Q8—4196no—
74C2×Q8C2×D41911yescharacteristic, complemented
75C2×D4—41912no—
76C2×D4—41924no—
77C2×D4—4196no—
78C2×D4C2×D41913yescomplemented
79C22×C4—41912no—
80C22×C4C2×D41913yescomplemented
81C23—41912no—
82C23C22×D41913yes—
83Q8—41912no—
84Q8—4194no—
85Q8C22×D41914yes—
86D4—41924no—
87D4—4198no—
88D4—41912no—
89D4—4194no—
90D4C22×D419112yes—
91C2×C4—41912no—
92C2×C4—41924no—
93C2×C4—4198no—
94C2×C4C22×D41916yes—
95C2×C4C22×D41913yes—
96C2×C4C22×D41916yes—
97C2×C4C22×D41911yescharacteristic
98C22—43no—
99C22—167612no—
100C22—4198no—
101C22C23×D41916yes—
102C22C23×D41911yescharacteristic
103C4—4198no—
104C4C23×D41916yes—
105C4C23×D41911yescharacteristic
106C4C23×D41911yescharacteristic
107C2—21no—
108C2—26no—
109C2—8388no—
110C2C24×D41911yescharacteristic
subgroupG/H
centerC4C23×D419
commutator, socle, cosocleC2×C419C25
FrattiniC2C24×D419
FittingC2×#41×C419C2
hypercenterC2×#41D419
hypocenterC419#585
LCSG ▷ C2×C419 ▷ C419
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27211766C2semidirect
2. #27211765C2semidirect
3. #27211764C2semidirect
4. #27211763C2semidirect
5. C2×#20269411C2semidirect
6. #41×D419C2semidirect
7. C2×#41×C419C2semidirect
8. #20269412C22semidirect
9. #20269411C22semidirect
10. #20269411C22semidirect
11. #20269411C22semidirect
12. #20269411C22semidirect
13. #20269410C22semidirect
14. C2×Dic838C22semidirect
15. Q8×D419C22semidirect
16. C2×D1676C22semidirect
17. C2×C22⋊D419C22semidirect
18. D4×D419C22semidirect
19. C2×C4×D419C22semidirect
20. #41×C419C22semidirect
21. #41×C419C22semidirect
22. C2×Q8×C419C22semidirect
23. C2×D4×C419C22semidirect
24. C22×C4×C419C22semidirect
55. C419#585semidirect
63. C2×#41D419semidirect
70. #41C2×D419semidirect
71. #41C2×D419semidirect
74. C2×Q8C2×D419semidirect
78. C2×D4C2×D419semidirect
80. C22×C4C2×D419semidirect

Maximal subgroups

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

subgroupindex
#272117662
#272117652
#272117642
#272117632
C2×#202694112
#41×D4192
C2×#41×C4192
#585419