Finite Group Databasebeta

← #37588527

Subgroups — #37588527

Order 24512.

152 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 39 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#27133887C211yesmaximal, characteristic
2#27133872C211yesmaximal, characteristic, complemented
3#27133830C211yesmaximal, characteristic, complemented
4#27133768C211yesmaximal, characteristic, complemented
5#27133767C211yesmaximal, characteristic, complemented
6#27133757C211yesmaximal, characteristic, complemented
7#137×C383C211yesmaximal, characteristic, complemented
8#20259458—21no—
9#20259457—21no—
10#20259457C2211yescharacteristic
11C2×D1532—21no—
12C2×D1532C2211yescharacteristic
13D4×D383—22no—
14D4×D383—21no—
15#20259447—21no—
16#20259444—21no—
17#20259441C2211yescharacteristic, complemented
18(C2×D383)⋊C4—22no—
19C4×Dic383C2211yescharacteristic, complemented
20#41×C383C2211yescharacteristic
21C2×Q8×C383C2211yescharacteristic, complemented
22Q16×C383—21no—
23SD16×C383—21no—
24M16×C383C2211yescharacteristic, complemented
25C22×D383—22no—
26C22×D383D412yes—
27Dic766D411yescharacteristic
28D1532—22no—
29D1532—21no—
30D1532—21no—
31D1532D411yescharacteristic
32C22⋊D383—22no—
33C22⋊D383—21no—
34C2×Dic383—21no—
35C4×D383—22no—
36C4×D383—21no—
37C383⋊C8—21no—
38Q8×C383—21no—
39Q8×C383—21no—
40Q8×C383D411yescharacteristic
41D4×C383—21no—
42D4×C383D411yescharacteristic
43C2×C4×C383—21no—
44C2×C4×C383—21no—
45C2×C4×C383C2311yescharacteristic
46C8×C383—21no—
47C22×C383—21no—
48C22×C383C2×D411yescharacteristic
49C2×D383—22no—
50C2×D383—21no—
51C2×D383—42no—
52C2×D383—82no—
53C4×C383—21no—
54C4×C383—41no—
55C4×C383C2×D411yescharacteristic
56C4×C383C2×D411yescharacteristic
57Dic383—21no—
58Dic383—41no—
59C2×C383—21no—
60C2×C383—41no—
61C2×C383#12011yescharacteristic
62D383—42no—
63D383—41no—
64C383#45511yescharacteristic, complemented, Sylow
65#455—3831nomaximal, Sylow
66H(5,𝔽2)—3831no—
67#137—3831no—
68#137D38311yescharacteristic, complemented
69#124—3831no—
70#104—3831no—
71#104—3831no—
72#100—3831no—
73#41—3831no—
74#41—7661no—
75#41—7661no—
76#41C2×D38311yescharacteristic
77C2×Q8C2×D38311yescharacteristic, complemented
78C2×D4—3831no—
79C2×D4—7661no—
80C2×D4—7662no—
81C2×D4—7661no—
82Q16—21no—
83Q16—7661no—
84SD16—21no—
85SD16—7661no—
86M16—3831no—
87M16C2×D38311yescharacteristic, complemented
88C22⋊C4—7662no—
89C42—3831no—
90C23—3832no—
91C23—7662no—
92Q8—21no—
93Q8—21no—
94Q8—3831no—
95Q8C22⋊D38311yescharacteristic
96D4—21no—
97D4—3831no—
98D4—7662no—
99D4—7662no—
100D4—7661no—
101D4—7661no—
102D4—7661no—
103D4C22⋊D38311yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—7662no—
107C2×C4—7661no—
108C2×C4—7661no—
109C2×C4C22×D38311yescharacteristic
110C8—21no—
111C8—7661no—
112C22—21no—
113C22—30642no—
114C22—15322no—
115C22—7661no—
116C22—7662no—
117C22D4×D38311yescharacteristic
118C4—21no—
119C4—41no—
120C4—15321no—
121C4—7661no—
122C4D4×D38311yescharacteristic
123C4C2×C22⋊D38311yescharacteristic
124C2—21no—
125C2—41no—
126C2—15321no—
127C2—15322no—
128C2#2713380911yescharacteristic
subgroupG/H
centerC2#27133809
commutator, cosocleC2×C4×C383C23
FrattiniC2×C4C22×D383
Fitting#137×C383C2
socleC2×C383#120
prime radical#27133887C2
hypercenter#137D383
hypocenterC383#455
LCSG ▷ C2×C4×C383 ▷ C2×C383 ▷ C383
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #27133872C2semidirect
3. #27133830C2semidirect
4. #27133768C2semidirect
5. #27133767C2semidirect
6. #27133757C2semidirect
7. #137×C383C2semidirect
17. #20259441C22semidirect
19. C4×Dic383C22semidirect
21. C2×Q8×C383C22semidirect
24. M16×C383C22semidirect
64. C383#455semidirect
68. #137D383semidirect
77. C2×Q8C2×D383semidirect
87. M16C2×D383semidirect

Maximal subgroups

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

subgroupindex
#271338872
#271338722
#271338302
#271337682
#271337672
#271337572
#137×C3832
#455383