Finite Group Databasebeta

← #27136095

Subgroups — #27136095

Order 12352.

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#20259947C212yesmaximal, complemented
2#20259946C214yesmaximal, complemented
3#20259945C214yesmaximal, complemented
4#20259944C216yesmaximal, complemented
5C2×#15690555C216yesmaximal, complemented
6#41×D193C218yesmaximal, complemented
7C2×#41×C193C211yesmaximal, characteristic, complemented
8#15690556C2218yescomplemented
9#15690555C22112yescomplemented
10#15690555C2214yescomplemented
11#15690555C22112yescomplemented
12#15690555C22112yescomplemented
13#15690554C22124yescomplemented
14C2×Dic386C2216yescomplemented
15Q8×D193C2218yescomplemented
16C2×D772C2216yescomplemented
17C2×C22⋊D193C22112yescomplemented
18D4×D193C22124yescomplemented
19C2×C4×D193C22112yescomplemented
20#41×C193C2214yescomplemented
21#41×C193C2214yescomplemented
22C2×Q8×C193C2211yescharacteristic, complemented
23C2×D4×C193C2213yescomplemented
24C22×C4×C193C2213yescomplemented
25C22×D193C23112yes—
26Dic386C23112yes—
27Dic386C2314yes—
28D772C23112yes—
29D772C2314yes—
30C22⋊D193C23124yes—
31C22⋊D193C2318yes—
32C2×Dic193C23112yes—
33C4×D193C2318yes—
34C4×D193C23124yes—
35C23×C193C2313yes—
36Q8×C193C2314yes—
37D4×C193C23112yes—
38C2×C4×C193C2311yescharacteristic
39C2×C4×C193C2313yes—
40C2×C4×C193C2316yes—
41C2×C4×C193C2316yes—
42C22×C193—43no—
43C22×C193C2411yescharacteristic
44C22×C193C2416yes—
45C2×D193—412no—
46C2×D193C2418yes—
47C4×C193C2411yescharacteristic
48C4×C193C2416yes—
49C4×C193C2411yescharacteristic
50Dic193C2418yes—
51C2×C193—26no—
52C2×C193—21no—
53C2×C193C2511yescharacteristic
54D193—28no—
55C193#58511yescharacteristic, complemented, Sylow
56#585—1931nomaximal, Sylow
572−1+4—1934no—
582−1+4—1932no—
59H(5,𝔽2)—1934no—
60H(5,𝔽2)—1936no—
61C2×#41—1938no—
62C2×#41—1936no—
63C2×#41D19311yescharacteristic, complemented
64#41—1938no—
65#41—19324no—
66#41—19312no—
67#41—1934no—
68#41—19312no—
69#41—19312no—
70#41C2×D19314yescomplemented
71#41C2×D19314yescomplemented
72C2×Q8—1938no—
73C2×Q8—1936no—
74C2×Q8C2×D19311yescharacteristic, complemented
75C2×D4—19312no—
76C2×D4—19324no—
77C2×D4—1936no—
78C2×D4C2×D19313yescomplemented
79C22×C4—19312no—
80C22×C4C2×D19313yescomplemented
81C23—19312no—
82C23C22×D19313yes—
83Q8—19312no—
84Q8—1934no—
85Q8C22×D19314yes—
86D4—19324no—
87D4—1938no—
88D4—19312no—
89D4—1934no—
90D4C22×D193112yes—
91C2×C4—19312no—
92C2×C4—19324no—
93C2×C4—1938no—
94C2×C4C22×D19316yes—
95C2×C4C22×D19313yes—
96C2×C4C22×D19316yes—
97C2×C4C22×D19311yescharacteristic
98C22—43no—
99C22—77212no—
100C22—1938no—
101C22C23×D19316yes—
102C22C23×D19311yescharacteristic
103C4—1938no—
104C4C23×D19316yes—
105C4C23×D19311yescharacteristic
106C4C23×D19311yescharacteristic
107C2—21no—
108C2—26no—
109C2—3868no—
110C2C24×D19311yescharacteristic
subgroupG/H
centerC4C23×D193
commutator, socle, cosocleC2×C193C25
FrattiniC2C24×D193
FittingC2×#41×C193C2
hypercenterC2×#41D193
hypocenterC193#585
LCSG ▷ C2×C193 ▷ C193
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20259947C2semidirect
2. #20259946C2semidirect
3. #20259945C2semidirect
4. #20259944C2semidirect
5. C2×#15690555C2semidirect
6. #41×D193C2semidirect
7. C2×#41×C193C2semidirect
8. #15690556C22semidirect
9. #15690555C22semidirect
10. #15690555C22semidirect
11. #15690555C22semidirect
12. #15690555C22semidirect
13. #15690554C22semidirect
14. C2×Dic386C22semidirect
15. Q8×D193C22semidirect
16. C2×D772C22semidirect
17. C2×C22⋊D193C22semidirect
18. D4×D193C22semidirect
19. C2×C4×D193C22semidirect
20. #41×C193C22semidirect
21. #41×C193C22semidirect
22. C2×Q8×C193C22semidirect
23. C2×D4×C193C22semidirect
24. C22×C4×C193C22semidirect
55. C193#585semidirect
63. C2×#41D193semidirect
70. #41C2×D193semidirect
71. #41C2×D193semidirect
74. C2×Q8C2×D193semidirect
78. C2×D4C2×D193semidirect
80. C22×C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599472
#202599462
#202599452
#202599442
C2×#156905552
#41×D1932
C2×#41×C1932
#585193