Finite Group Databasebeta

← #27164137

Subgroups — #27164137

Order 12736.

104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 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#20263975C211yesmaximal, characteristic
2#20263963C211yesmaximal, characteristic, complemented
3#20263962C211yesmaximal, characteristic, complemented
4#20263890C211yesmaximal, characteristic, complemented
5#20263883C211yesmaximal, characteristic, complemented
6#20263866C211yesmaximal, characteristic, complemented
7#104×C199C211yesmaximal, characteristic, complemented
8#15691886C2211yescharacteristic
9#15691876C411yescharacteristic, complemented
10#15691874C411yescharacteristic, complemented
11#15691873C411yescharacteristic, complemented
12#15691871C411yescharacteristic, complemented
13#15691870—21no—
14#15691870C2211yescharacteristic
15#15691869—21no—
16C2×C199⋊C8—21no—
17C2×C199⋊C8C2211yescharacteristic
18C2×C199⋊C8C2211yescharacteristic
19C8×D199—21no—
20#41×C199C2211yescharacteristic
21M16×C199C2211yescharacteristic
22C42×C199C2211yescharacteristic
23Dic398C2×C411yescharacteristic
24D796C2×C411yescharacteristic
25C22⋊D199—21no—
26C4×D199—21no—
27C199⋊C8—21no—
28C199⋊C8C2×C411yescharacteristic
29C199⋊C8C2×C411yescharacteristic
30C199⋊C8D411yescharacteristic
31C199⋊C8D411yescharacteristic
32Q8×C199C2×C411yescharacteristic
33D4×C199—21no—
34D4×C199C2×C411yescharacteristic
35C2×C4×C199—21no—
36C2×C4×C199—21no—
37C2×C4×C199C2311yescharacteristic
38C8×C199—21no—
39C22×C199—21no—
40C22×C199#4111yescharacteristic
41C2×D199—21no—
42C4×C199—21no—
43C4×C199—41no—
44C4×C199C22×C411yescharacteristic
45C4×C199C2×D411yescharacteristic
46Dic199—21no—
47C2×C199—21no—
48C2×C199—41no—
49C2×C199C4×D411yescharacteristic
50D199—41no—
51C199#44411yescharacteristic, complemented, Sylow
52#444—1991nomaximal, Sylow
53#135—1991no—
54#131—1991no—
55#131—1991no—
56#108—1991no—
57#104—1991no—
58#104D19911yescharacteristic, complemented
59#97—1991no—
60#41—1991no—
61#41C2×D19911yescharacteristic
62Q16—1991no—
63SD16—1991no—
64SD16—1991no—
65D8—1991no—
66M16—3981no—
67M16—3981no—
68M16—1991no—
69M16C2×D19911yescharacteristic
70C2×C8—3981no—
71C2×C8—3981no—
72C2×C8—1991no—
73C2×C8—1991no—
74C42C2×D19911yescharacteristic
75Q8—1991no—
76Q8C4×D19911yescharacteristic
77D4—21no—
78D4—3981no—
79D4—1991no—
80D4C4×D19911yescharacteristic
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—3981no—
84C2×C4C22×D19911yescharacteristic
85C8—21no—
86C8—3981no—
87C8—1991no—
88C8—1991no—
89C8—1991no—
90C8—1991no—
91C22—21no—
92C22—3981no—
93C22#1569188511yescharacteristic
94C4—21no—
95C4—41no—
96C4—3981no—
97C4C2×C4×D19911yescharacteristic
98C4D4×D19911yescharacteristic
99C2—21no—
100C2—41no—
101C2—7961no—
102C2(C22⋊D199)⋊C411yescharacteristic
subgroupG/H
centerC4D4×D199
commutatorC4×C199C22×C4
FrattiniC2×C4C22×D199
Fitting#104×C199C2
socleC2×C199C4×D4
prime radical#20263975C2
cosocleC2×C4×C199C23
hypercenter#104D199
hypocenterC199#444
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #20263963C2semidirect
3. #20263962C2semidirect
4. #20263890C2semidirect
5. #20263883C2semidirect
6. #20263866C2semidirect
7. #104×C199C2semidirect
9. #15691876C4semidirect
10. #15691874C4semidirect
11. #15691873C4semidirect
12. #15691871C4semidirect
51. C199#444semidirect
58. #104D199semidirect

Maximal subgroups

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

subgroupindex
#202639752
#202639632
#202639622
#202638902
#202638832
#202638662
#104×C1992
#444199