Finite Group Databasebeta

← #37555022

Subgroups — #37555022

Order 23488.

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#26044309C212yesmaximal, complemented
2#26044308C214yesmaximal, complemented
3#26044307C214yesmaximal, complemented
4#26044306C216yesmaximal, complemented
5C2×#19154577C216yesmaximal, complemented
6#41×D367C218yesmaximal, complemented
7C2×#41×C367C211yesmaximal, characteristic, complemented
8#19154578C2218yescomplemented
9#19154577C22112yescomplemented
10#19154577C2214yescomplemented
11#19154577C22112yescomplemented
12#19154577C22112yescomplemented
13#19154576C22124yescomplemented
14C2×Dic734C2216yescomplemented
15Q8×D367C2218yescomplemented
16C2×D1468C2216yescomplemented
17C2×C22⋊D367C22112yescomplemented
18D4×D367C22124yescomplemented
19C2×C4×D367C22112yescomplemented
20#41×C367C2214yescomplemented
21#41×C367C2214yescomplemented
22C2×Q8×C367C2211yescharacteristic, complemented
23C2×D4×C367C2213yescomplemented
24C22×C4×C367C2213yescomplemented
25C22×D367C23112yes—
26Dic734C23112yes—
27Dic734C2314yes—
28D1468C23112yes—
29D1468C2314yes—
30C22⋊D367C23124yes—
31C22⋊D367C2318yes—
32C2×Dic367C23112yes—
33C4×D367C2318yes—
34C4×D367C23124yes—
35C23×C367C2313yes—
36Q8×C367C2314yes—
37D4×C367C23112yes—
38C2×C4×C367C2311yescharacteristic
39C2×C4×C367C2313yes—
40C2×C4×C367C2316yes—
41C2×C4×C367C2316yes—
42C22×C367—43no—
43C22×C367C2411yescharacteristic
44C22×C367C2416yes—
45C2×D367—412no—
46C2×D367C2418yes—
47C4×C367C2411yescharacteristic
48C4×C367C2416yes—
49C4×C367C2411yescharacteristic
50Dic367C2418yes—
51C2×C367—26no—
52C2×C367—21no—
53C2×C367C2511yescharacteristic
54D367—28no—
55C367#58511yescharacteristic, complemented, Sylow
56#585—3671nomaximal, Sylow
572−1+4—3674no—
582−1+4—3672no—
59H(5,𝔽2)—3674no—
60H(5,𝔽2)—3676no—
61C2×#41—3678no—
62C2×#41—3676no—
63C2×#41D36711yescharacteristic, complemented
64#41—3678no—
65#41—36724no—
66#41—36712no—
67#41—3674no—
68#41—36712no—
69#41—36712no—
70#41C2×D36714yescomplemented
71#41C2×D36714yescomplemented
72C2×Q8—3678no—
73C2×Q8—3676no—
74C2×Q8C2×D36711yescharacteristic, complemented
75C2×D4—36712no—
76C2×D4—36724no—
77C2×D4—3676no—
78C2×D4C2×D36713yescomplemented
79C22×C4—36712no—
80C22×C4C2×D36713yescomplemented
81C23—36712no—
82C23C22×D36713yes—
83Q8—36712no—
84Q8—3674no—
85Q8C22×D36714yes—
86D4—36724no—
87D4—3678no—
88D4—36712no—
89D4—3674no—
90D4C22×D367112yes—
91C2×C4—36712no—
92C2×C4—36724no—
93C2×C4—3678no—
94C2×C4C22×D36716yes—
95C2×C4C22×D36713yes—
96C2×C4C22×D36716yes—
97C2×C4C22×D36711yescharacteristic
98C22—43no—
99C22—3678no—
100C22—146812no—
101C22C23×D36716yes—
102C22C23×D36711yescharacteristic
103C4—3678no—
104C4C23×D36716yes—
105C4C23×D36711yescharacteristic
106C4C23×D36711yescharacteristic
107C2—21no—
108C2—26no—
109C2—7348no—
110C2C24×D36711yescharacteristic
subgroupG/H
centerC4C23×D367
commutator, socle, cosocleC2×C367C25
FrattiniC2C24×D367
FittingC2×#41×C367C2
hypercenterC2×#41D367
hypocenterC367#585
LCSG ▷ C2×C367 ▷ C367
UCSC1 ◁ C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26044309C2semidirect
2. #26044308C2semidirect
3. #26044307C2semidirect
4. #26044306C2semidirect
5. C2×#19154577C2semidirect
6. #41×D367C2semidirect
7. C2×#41×C367C2semidirect
8. #19154578C22semidirect
9. #19154577C22semidirect
10. #19154577C22semidirect
11. #19154577C22semidirect
12. #19154577C22semidirect
13. #19154576C22semidirect
14. C2×Dic734C22semidirect
15. Q8×D367C22semidirect
16. C2×D1468C22semidirect
17. C2×C22⋊D367C22semidirect
18. D4×D367C22semidirect
19. C2×C4×D367C22semidirect
20. #41×C367C22semidirect
21. #41×C367C22semidirect
22. C2×Q8×C367C22semidirect
23. C2×D4×C367C22semidirect
24. C22×C4×C367C22semidirect
55. C367#585semidirect
63. C2×#41D367semidirect
70. #41C2×D367semidirect
71. #41C2×D367semidirect
74. C2×Q8C2×D367semidirect
78. C2×D4C2×D367semidirect
80. C22×C4C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260443092
#260443082
#260443072
#260443062
C2×#191545772
#41×D3672
C2×#41×C3672
#585367