Finite Group Databasebeta

← #27164207

Subgroups — #27164207

Order 12736.

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

normal subgroups only

This group has 160 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#20264000C211yesmaximal, characteristic
2#20263938C212yesmaximal, complemented
3#20263870C211yesmaximal, characteristic, complemented
4#20263869C212yesmaximal, complemented
5#123×C199C211yesmaximal, characteristic, complemented
6#15691886—21no—
7#15691885—22no—
8C2×C22⋊D199—21no—
9C2×C22⋊D199C2212yes—
10D4×D199—22no—
11Dic199⋊C4—22no—
12C22⋊Dic199C2212yescomplemented
13(C2×D199)⋊C4—22no—
14C2×D4×C199C2211yescharacteristic
15C22×C4×C199C2211yescharacteristic, complemented
16C4⋊C4×C199—21no—
17C22⋊C4×C199—21no—
18C22⋊C4×C199C2211yescharacteristic, complemented
19C22×D199—21no—
20C22×D199D412yes—
21Dic398—21no—
22D796—21no—
23C22⋊D199—22no—
24C22⋊D199—22no—
25C22⋊D199—22no—
26C2×Dic199—21no—
27C2×Dic199—22no—
28C2×Dic199D412yes—
29C4×D199—22no—
30C23×C199D411yescharacteristic
31C23×C199C2311yescharacteristic
32D4×C199—22no—
33C2×C4×C199—21no—
34C2×C4×C199—21no—
35C2×C4×C199—22no—
36C2×C4×C199D411yescharacteristic
37C22×C199—21no—
38C22×C199—41no—
39C22×C199—41no—
40C22×C199C2×D412yes—
41C22×C199C2×D411yescharacteristic
42C2×D199—22no—
43C2×D199—42no—
44C2×D199—81no—
45C4×C199—21no—
46C4×C199—41no—
47C4×C199—41no—
48Dic199—22no—
49Dic199—42no—
50C2×C199—22no—
51C2×C199—21no—
52C2×C199—41no—
53C2×C199#12011yescharacteristic
54D199—42no—
55C199#45811yescharacteristic, complemented, Sylow
56#458—1991nomaximal, Sylow
57H(5,𝔽2)—1991no—
58#123—1992no—
59#123D19911yescharacteristic, complemented
60#99—1991no—
61#99—1992no—
62#41—3982no—
63#41—3981no—
64C2×D4—3982no—
65C2×D4—3981no—
66C2×D4—1992no—
67C2×D4C2×D19911yescharacteristic
68C22×C4C2×D19911yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—3982no—
71C22⋊C4—21no—
72C22⋊C4—3982no—
73C22⋊C4—1992no—
74C22⋊C4C2×D19911yescharacteristic, complemented
75C23—3981no—
76C23—1992no—
77C23C22⋊D19911yescharacteristic
78C23C22×D19911yescharacteristic
79Q8—3981no—
80D4—22no—
81D4—3981no—
82D4—3982no—
83D4—3982no—
84D4—3982no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—3982no—
89C2×C4—3982no—
90C2×C4—3981no—
91C2×C4—1992no—
92C2×C4C22⋊D19911yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—7962no—
97C22—15921no—
98C22—3982no—
99C22D4×D19912yes—
100C22C2×C22⋊D19911yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—7962no—
105C4—3982no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—7962no—
110C2#2026392311yescharacteristic
subgroupG/H
centerC2#20263923
commutator, cosocleC23×C199C23
FrattiniC23C22×D199
Fitting#123×C199C2
socleC2×C199#120
prime radical#20264000C2
hypercenter#123D199
hypocenterC199#458
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #20263938C2semidirect
3. #20263870C2semidirect
4. #20263869C2semidirect
5. #123×C199C2semidirect
12. C22⋊Dic199C22semidirect
15. C22×C4×C199C22semidirect
18. C22⋊C4×C199C22semidirect
55. C199#458semidirect
59. #123D199semidirect
68. C22×C4C2×D199semidirect
74. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640002
#202639382
#202638702
#202638692
#123×C1992
#458199