Finite Group Databasebeta

← #29505608

Subgroups — #29505608

Order 15296.

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#21440008C212yesmaximal, complemented
2#21440007C214yesmaximal, complemented
3#21440006C214yesmaximal, complemented
4#21440005C216yesmaximal, complemented
5C2×#16838548C216yesmaximal, complemented
6#41×D239C218yesmaximal, complemented
7C2×#41×C239C211yesmaximal, characteristic, complemented
8#16838549C2218yescomplemented
9#16838548C22112yescomplemented
10#16838548C2214yescomplemented
11#16838548C22112yescomplemented
12#16838548C22112yescomplemented
13#16838547C22124yescomplemented
14C2×Dic478C2216yescomplemented
15Q8×D239C2218yescomplemented
16C2×D956C2216yescomplemented
17C2×C22⋊D239C22112yescomplemented
18D4×D239C22124yescomplemented
19C2×C4×D239C22112yescomplemented
20#41×C239C2214yescomplemented
21#41×C239C2214yescomplemented
22C2×Q8×C239C2211yescharacteristic, complemented
23C2×D4×C239C2213yescomplemented
24C22×C4×C239C2213yescomplemented
25C22×D239C23112yes—
26Dic478C23112yes—
27Dic478C2314yes—
28D956C23112yes—
29D956C2314yes—
30C22⋊D239C23124yes—
31C22⋊D239C2318yes—
32C2×Dic239C23112yes—
33C4×D239C2318yes—
34C4×D239C23124yes—
35C23×C239C2313yes—
36Q8×C239C2314yes—
37D4×C239C23112yes—
38C2×C4×C239C2311yescharacteristic
39C2×C4×C239C2313yes—
40C2×C4×C239C2316yes—
41C2×C4×C239C2316yes—
42C22×C239—43no—
43C22×C239C2411yescharacteristic
44C22×C239C2416yes—
45C2×D239—412no—
46C2×D239C2418yes—
47C4×C239C2411yescharacteristic
48C4×C239C2416yes—
49C4×C239C2411yescharacteristic
50Dic239C2418yes—
51C2×C239—26no—
52C2×C239—21no—
53C2×C239C2511yescharacteristic
54D239—28no—
55C239#58511yescharacteristic, complemented, Sylow
56#585—2391nomaximal, Sylow
572−1+4—2394no—
582−1+4—2392no—
59H(5,𝔽2)—2394no—
60H(5,𝔽2)—2396no—
61C2×#41—2398no—
62C2×#41—2396no—
63C2×#41D23911yescharacteristic, complemented
64#41—2398no—
65#41—23924no—
66#41—23912no—
67#41—2394no—
68#41—23912no—
69#41—23912no—
70#41C2×D23914yescomplemented
71#41C2×D23914yescomplemented
72C2×Q8—2398no—
73C2×Q8—2396no—
74C2×Q8C2×D23911yescharacteristic, complemented
75C2×D4—23912no—
76C2×D4—23924no—
77C2×D4—2396no—
78C2×D4C2×D23913yescomplemented
79C22×C4—23912no—
80C22×C4C2×D23913yescomplemented
81C23—23912no—
82C23C22×D23913yes—
83Q8—23912no—
84Q8—2394no—
85Q8C22×D23914yes—
86D4—23924no—
87D4—2398no—
88D4—23912no—
89D4—2394no—
90D4C22×D239112yes—
91C2×C4—23912no—
92C2×C4—23924no—
93C2×C4—2398no—
94C2×C4C22×D23916yes—
95C2×C4C22×D23913yes—
96C2×C4C22×D23916yes—
97C2×C4C22×D23911yescharacteristic
98C22—43no—
99C22—95612no—
100C22—2398no—
101C22C23×D23916yes—
102C22C23×D23911yescharacteristic
103C4—2398no—
104C4C23×D23916yes—
105C4C23×D23911yescharacteristic
106C4C23×D23911yescharacteristic
107C2—21no—
108C2—26no—
109C2—4788no—
110C2C24×D23911yescharacteristic
subgroupG/H
centerC4C23×D239
commutator, socle, cosocleC2×C239C25
FrattiniC2C24×D239
FittingC2×#41×C239C2
hypercenterC2×#41D239
hypocenterC239#585
LCSG ▷ C2×C239 ▷ C239
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #21440008C2semidirect
2. #21440007C2semidirect
3. #21440006C2semidirect
4. #21440005C2semidirect
5. C2×#16838548C2semidirect
6. #41×D239C2semidirect
7. C2×#41×C239C2semidirect
8. #16838549C22semidirect
9. #16838548C22semidirect
10. #16838548C22semidirect
11. #16838548C22semidirect
12. #16838548C22semidirect
13. #16838547C22semidirect
14. C2×Dic478C22semidirect
15. Q8×D239C22semidirect
16. C2×D956C22semidirect
17. C2×C22⋊D239C22semidirect
18. D4×D239C22semidirect
19. C2×C4×D239C22semidirect
20. #41×C239C22semidirect
21. #41×C239C22semidirect
22. C2×Q8×C239C22semidirect
23. C2×D4×C239C22semidirect
24. C22×C4×C239C22semidirect
55. C239#585semidirect
63. C2×#41D239semidirect
70. #41C2×D239semidirect
71. #41C2×D239semidirect
74. C2×Q8C2×D239semidirect
78. C2×D4C2×D239semidirect
80. C22×C4C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214400082
#214400072
#214400062
#214400052
C2×#168385482
#41×D2392
C2×#41×C2392
#585239