Finite Group Databasebeta

← #41003676

Subgroups — #41003676

Order 26944.

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#27212414C212yesmaximal, complemented
2#27212413C214yesmaximal, complemented
3#27212412C214yesmaximal, complemented
4#27212411C216yesmaximal, complemented
5C2×#20269771C216yesmaximal, complemented
6#41×D421C218yesmaximal, complemented
7C2×#41×C421C211yesmaximal, characteristic, complemented
8#20269772C2218yescomplemented
9#20269771C22112yescomplemented
10#20269771C2214yescomplemented
11#20269771C22112yescomplemented
12#20269771C22112yescomplemented
13#20269770C22124yescomplemented
14C2×Dic842C2216yescomplemented
15Q8×D421C2218yescomplemented
16C2×D1684C2216yescomplemented
17C2×C22⋊D421C22112yescomplemented
18D4×D421C22124yescomplemented
19C2×C4×D421C22112yescomplemented
20#41×C421C2214yescomplemented
21#41×C421C2214yescomplemented
22C2×Q8×C421C2211yescharacteristic, complemented
23C2×D4×C421C2213yescomplemented
24C22×C4×C421C2213yescomplemented
25C22×D421C23112yes—
26Dic842C23112yes—
27Dic842C2314yes—
28D1684C23112yes—
29D1684C2314yes—
30C22⋊D421C23124yes—
31C22⋊D421C2318yes—
32C2×Dic421C23112yes—
33C4×D421C2318yes—
34C4×D421C23124yes—
35C23×C421C2313yes—
36Q8×C421C2314yes—
37D4×C421C23112yes—
38C2×C4×C421C2311yescharacteristic
39C2×C4×C421C2313yes—
40C2×C4×C421C2316yes—
41C2×C4×C421C2316yes—
42C22×C421—43no—
43C22×C421C2411yescharacteristic
44C22×C421C2416yes—
45C2×D421—412no—
46C2×D421C2418yes—
47C4×C421C2411yescharacteristic
48C4×C421C2416yes—
49C4×C421C2411yescharacteristic
50Dic421C2418yes—
51C2×C421—26no—
52C2×C421—21no—
53C2×C421C2511yescharacteristic
54D421—28no—
55C421#58511yescharacteristic, complemented, Sylow
56#585—4211nomaximal, Sylow
572−1+4—4214no—
582−1+4—4212no—
59H(5,𝔽2)—4214no—
60H(5,𝔽2)—4216no—
61C2×#41—4218no—
62C2×#41—4216no—
63C2×#41D42111yescharacteristic, complemented
64#41—4218no—
65#41—42124no—
66#41—42112no—
67#41—4214no—
68#41—42112no—
69#41—42112no—
70#41C2×D42114yescomplemented
71#41C2×D42114yescomplemented
72C2×Q8—4218no—
73C2×Q8—4216no—
74C2×Q8C2×D42111yescharacteristic, complemented
75C2×D4—42112no—
76C2×D4—42124no—
77C2×D4—4216no—
78C2×D4C2×D42113yescomplemented
79C22×C4—42112no—
80C22×C4C2×D42113yescomplemented
81C23—42112no—
82C23C22×D42113yes—
83Q8—42112no—
84Q8—4214no—
85Q8C22×D42114yes—
86D4—42124no—
87D4—4218no—
88D4—42112no—
89D4—4214no—
90D4C22×D421112yes—
91C2×C4—42112no—
92C2×C4—42124no—
93C2×C4—4218no—
94C2×C4C22×D42116yes—
95C2×C4C22×D42113yes—
96C2×C4C22×D42116yes—
97C2×C4C22×D42111yescharacteristic
98C22—43no—
99C22—168412no—
100C22—4218no—
101C22C23×D42116yes—
102C22C23×D42111yescharacteristic
103C4—4218no—
104C4C23×D42116yes—
105C4C23×D42111yescharacteristic
106C4C23×D42111yescharacteristic
107C2—21no—
108C2—26no—
109C2—8428no—
110C2C24×D42111yescharacteristic
subgroupG/H
centerC4C23×D421
commutator, socle, cosocleC2×C421C25
FrattiniC2C24×D421
FittingC2×#41×C421C2
hypercenterC2×#41D421
hypocenterC421#585
LCSG ▷ C2×C421 ▷ C421
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #27212414C2semidirect
2. #27212413C2semidirect
3. #27212412C2semidirect
4. #27212411C2semidirect
5. C2×#20269771C2semidirect
6. #41×D421C2semidirect
7. C2×#41×C421C2semidirect
8. #20269772C22semidirect
9. #20269771C22semidirect
10. #20269771C22semidirect
11. #20269771C22semidirect
12. #20269771C22semidirect
13. #20269770C22semidirect
14. C2×Dic842C22semidirect
15. Q8×D421C22semidirect
16. C2×D1684C22semidirect
17. C2×C22⋊D421C22semidirect
18. D4×D421C22semidirect
19. C2×C4×D421C22semidirect
20. #41×C421C22semidirect
21. #41×C421C22semidirect
22. C2×Q8×C421C22semidirect
23. C2×D4×C421C22semidirect
24. C22×C4×C421C22semidirect
55. C421#585semidirect
63. C2×#41D421semidirect
70. #41C2×D421semidirect
71. #41C2×D421semidirect
74. C2×Q8C2×D421semidirect
78. C2×D4C2×D421semidirect
80. C22×C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
#272124142
#272124132
#272124122
#272124112
C2×#202697712
#41×D4212
C2×#41×C4212
#585421