Finite Group Databasebeta

← #27164194

Subgroups — #27164194

Order 12736.

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#20264001C211yesmaximal, characteristic
2#20263986C211yesmaximal, characteristic, complemented
3#20263944C211yesmaximal, characteristic, complemented
4#20263882C211yesmaximal, characteristic, complemented
5#20263881C211yesmaximal, characteristic, complemented
6#20263871C211yesmaximal, characteristic, complemented
7#137×C199C211yesmaximal, characteristic, complemented
8#15691887—21no—
9#15691886—21no—
10#15691886C2211yescharacteristic
11C2×D796—21no—
12C2×D796C2211yescharacteristic
13D4×D199—22no—
14D4×D199—21no—
15#15691876—21no—
16#15691873—21no—
17#15691870C2211yescharacteristic, complemented
18(C2×D199)⋊C4—22no—
19C4×Dic199C2211yescharacteristic, complemented
20#41×C199C2211yescharacteristic
21C2×Q8×C199C2211yescharacteristic, complemented
22Q16×C199—21no—
23SD16×C199—21no—
24M16×C199C2211yescharacteristic, complemented
25C22×D199—22no—
26C22×D199D412yes—
27Dic398D411yescharacteristic
28D796—22no—
29D796—21no—
30D796—21no—
31D796D411yescharacteristic
32C22⋊D199—22no—
33C22⋊D199—21no—
34C2×Dic199—21no—
35C4×D199—22no—
36C4×D199—21no—
37C199⋊C8—21no—
38Q8×C199—21no—
39Q8×C199—21no—
40Q8×C199D411yescharacteristic
41D4×C199—21no—
42D4×C199D411yescharacteristic
43C2×C4×C199—21no—
44C2×C4×C199—21no—
45C2×C4×C199C2311yescharacteristic
46C8×C199—21no—
47C22×C199—21no—
48C22×C199C2×D411yescharacteristic
49C2×D199—22no—
50C2×D199—21no—
51C2×D199—42no—
52C2×D199—82no—
53C4×C199—21no—
54C4×C199—41no—
55C4×C199C2×D411yescharacteristic
56C4×C199C2×D411yescharacteristic
57Dic199—21no—
58Dic199—41no—
59C2×C199—21no—
60C2×C199—41no—
61C2×C199#12011yescharacteristic
62D199—42no—
63D199—41no—
64C199#45511yescharacteristic, complemented, Sylow
65#455—1991nomaximal, Sylow
66H(5,𝔽2)—1991no—
67#137—1991no—
68#137D19911yescharacteristic, complemented
69#124—1991no—
70#104—1991no—
71#104—1991no—
72#100—1991no—
73#41—3981no—
74#41—3981no—
75#41—1991no—
76#41C2×D19911yescharacteristic
77C2×Q8C2×D19911yescharacteristic, complemented
78C2×D4—3981no—
79C2×D4—3982no—
80C2×D4—3981no—
81C2×D4—1991no—
82Q16—21no—
83Q16—3981no—
84SD16—21no—
85SD16—3981no—
86M16—1991no—
87M16C2×D19911yescharacteristic, complemented
88C22⋊C4—3982no—
89C42—1991no—
90C23—3982no—
91C23—1992no—
92Q8—21no—
93Q8—21no—
94Q8—1991no—
95Q8C22⋊D19911yescharacteristic
96D4—21no—
97D4—3982no—
98D4—3982no—
99D4—3981no—
100D4—3981no—
101D4—3981no—
102D4—1991no—
103D4C22⋊D19911yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—3982no—
107C2×C4—3981no—
108C2×C4—3981no—
109C2×C4C22×D19911yescharacteristic
110C8—21no—
111C8—3981no—
112C22—21no—
113C22—7962no—
114C22—15922no—
115C22—3982no—
116C22—3981no—
117C22D4×D19911yescharacteristic
118C4—21no—
119C4—41no—
120C4—7961no—
121C4—3981no—
122C4D4×D19911yescharacteristic
123C4C2×C22⋊D19911yescharacteristic
124C2—21no—
125C2—41no—
126C2—7962no—
127C2—7961no—
128C2#2026392311yescharacteristic
subgroupG/H
centerC2#20263923
commutator, cosocleC2×C4×C199C23
FrattiniC2×C4C22×D199
Fitting#137×C199C2
socleC2×C199#120
prime radical#20264001C2
hypercenter#137D199
hypocenterC199#455
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20263986C2semidirect
3. #20263944C2semidirect
4. #20263882C2semidirect
5. #20263881C2semidirect
6. #20263871C2semidirect
7. #137×C199C2semidirect
17. #15691870C22semidirect
19. C4×Dic199C22semidirect
21. C2×Q8×C199C22semidirect
24. M16×C199C22semidirect
64. C199#455semidirect
68. #137D199semidirect
77. C2×Q8C2×D199semidirect
87. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640012
#202639862
#202639442
#202638822
#202638812
#202638712
#137×C1992
#455199