Finite Group Databasebeta

← #27164199

Subgroups — #27164199

Order 12736.

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

normal subgroups only

This group has 142 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#20264002C211yesmaximal, characteristic
2#20263986C211yesmaximal, characteristic, complemented
3#20263953C211yesmaximal, characteristic, complemented
4#20263882C211yesmaximal, characteristic, complemented
5#20263881C211yesmaximal, characteristic, complemented
6#20263873C211yesmaximal, characteristic, complemented
7#137×C199C211yesmaximal, characteristic, complemented
8#15691886—21no—
9#15691886C2211yescharacteristic
10#15691885—22no—
11#15691885—21no—
12C2×Dic398—21no—
13C2×Dic398C2211yescharacteristic
14Q8×D199—21no—
15#15691876—21no—
16#15691873—21no—
17#15691870C2211yescharacteristic
18Dic199⋊C4—22no—
19C4×Dic199C2211yescharacteristic
20#41×C199C2211yescharacteristic
21C2×Q8×C199C2211yescharacteristic
22Q16×C199—21no—
23SD16×C199—21no—
24M16×C199C2211yescharacteristic
25Dic398—22no—
26Dic398—21no—
27Dic398—21no—
28Dic398D411yescharacteristic
29D796D411yescharacteristic
30C22⋊D199—22no—
31C22⋊D199—21no—
32C2×Dic199—22no—
33C2×Dic199—21no—
34C2×Dic199D412yes—
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—21no—
50C4×C199—21no—
51C4×C199—41no—
52C4×C199C2×D411yescharacteristic
53C4×C199C2×D411yescharacteristic
54Dic199—22no—
55Dic199—21no—
56Dic199—22no—
57C2×C199—21no—
58C2×C199—41no—
59C2×C199#12011yescharacteristic
60D199—41no—
61C199#45611yescharacteristic, complemented, Sylow
62#456—1991nomaximal, Sylow
632−1+4—1991no—
64#137—1991no—
65#137D19911yescharacteristic, complemented
66C4⋊Q8—1991no—
67#104—1991no—
68#104—1991no—
69#101—1991no—
70#41—3981no—
71#41—3981no—
72#41—3982no—
73#41—1991no—
74#41C2×D19911yescharacteristic
75C2×Q8—3981no—
76C2×Q8—3981no—
77C2×Q8—1991no—
78C2×Q8C2×D19911yescharacteristic
79Q16—21no—
80Q16—3981no—
81SD16—21no—
82SD16—3981no—
83M16—1991no—
84M16C2×D19911yescharacteristic
85C4⋊C4—3982no—
86C42—1991no—
87Q8—21no—
88Q8—21no—
89Q8—3982no—
90Q8—3981no—
91Q8—3981no—
92Q8—1991no—
93Q8C22⋊D19911yescharacteristic
94D4—21no—
95D4—3982no—
96D4—3981no—
97D4—1991no—
98D4C22⋊D19911yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—3982no—
102C2×C4—3982no—
103C2×C4—3981no—
104C2×C4—3981no—
105C2×C4—1992no—
106C2×C4C22×D19911yescharacteristic
107C8—21no—
108C8—3981no—
109C22—21no—
110C22—3981no—
111C22D4×D19911yescharacteristic
112C4—21no—
113C4—41no—
114C4—3982no—
115C4—3981no—
116C4—3982no—
117C4D4×D19911yescharacteristic
118C4C2×C22⋊D19911yescharacteristic
119C2—21no—
120C2—41no—
121C2—7961no—
122C2#2026392311yescharacteristic
subgroupG/H
centerC2#20263923
commutator, cosocleC2×C4×C199C23
FrattiniC2×C4C22×D199
Fitting#137×C199C2
socleC2×C199#120
prime radical#20264002C2
hypercenter#137D199
hypocenterC199#456
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. #20263953C2semidirect
4. #20263882C2semidirect
5. #20263881C2semidirect
6. #20263873C2semidirect
7. #137×C199C2semidirect
61. C199#456semidirect
65. #137D199semidirect

Maximal subgroups

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

subgroupindex
#202640022
#202639862
#202639532
#202638822
#202638812
#202638732
#137×C1992
#456199