Finite Group Databasebeta

← #27164187

Subgroups — #27164187

Order 12736.

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

normal subgroups only

This group has 168 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, complemented
2#20263983C211yesmaximal, characteristic, complemented
3#20263980C211yesmaximal, characteristic, complemented
4#20263952C211yesmaximal, characteristic, complemented
5#20263883C211yesmaximal, characteristic, complemented
6#20263871C211yesmaximal, characteristic, complemented
7#104×C199C211yesmaximal, characteristic, complemented
8#15691887—21no—
9#15691886—21no—
10#15691886C2211yescharacteristic
11C2×D796—21no—
12C2×D796—22no—
13C2×D796C2211yescharacteristic
14C2×D796C2211yescharacteristic, complemented
15D4×D199—22no—
16D4×D199—21no—
17#15691875—21no—
18#15691873—21no—
19D1592—21no—
20#15691871—21no—
21#15691870C2211yescharacteristic, complemented
22#41×C199C2211yescharacteristic
23M16×C199C2211yescharacteristic, complemented
24C42×C199C2211yescharacteristic, complemented
25C22×D199—22no—
26C22×D199—21no—
27C22×D199D412yes—
28Dic398D411yescharacteristic, complemented
29D796—22no—
30D796—21no—
31D796—21no—
32D796—21no—
33D796—21no—
34D796—44no—
35D796D411yescharacteristic, complemented
36C22⋊D199—22no—
37C22⋊D199—21no—
38C4×D199—22no—
39C4×D199—21no—
40C199⋊C8—21no—
41Q8×C199D411yescharacteristic, complemented
42D4×C199—21no—
43D4×C199D411yescharacteristic, complemented
44C2×C4×C199—21no—
45C2×C4×C199—21no—
46C2×C4×C199C2311yescharacteristic
47C8×C199—21no—
48C22×C199—21no—
49C22×C199C2×D411yescharacteristic
50C2×D199—22no—
51C2×D199—21no—
52C2×D199—42no—
53C2×D199—42no—
54C2×D199—41no—
55C2×D199—82no—
56C4×C199—22no—
57C4×C199—21no—
58C4×C199C2×D411yescharacteristic
59C4×C199C2×D411yescharacteristic
60Dic199—21no—
61C2×C199—21no—
62C2×C199—41no—
63C2×C199#12011yescharacteristic
64D199—42no—
65D199—41no—
66D199—81no—
67C199#45311yescharacteristic, complemented, Sylow
68#453—1991nomaximal, Sylow
69H(5,𝔽2)—1991no—
70AGL(1,ℤ/8)—1991no—
71AGL(1,ℤ/8)—1991no—
72#127—1991no—
73#104—1991no—
74#104D19911yescharacteristic, complemented
75#100—1991no—
76#41—3981no—
77#41—3981no—
78#41—1991no—
79#41C2×D19911yescharacteristic
80C2×D4—3982no—
81C2×D4—3981no—
82C2×D4—3982no—
83C2×D4—3981no—
84C2×D4—1991no—
85C2×D4—1991no—
86SD16—3981no—
87SD16—3981no—
88D8—3981no—
89D8—3981no—
90M16—1991no—
91M16C2×D19911yescharacteristic, complemented
92C42C2×D19911yescharacteristic, complemented
93C23—3982no—
94C23—3981no—
95C23—1992no—
96Q8—1991no—
97Q8D79611yescharacteristic, complemented
98D4—21no—
99D4—7964no—
100D4—3982no—
101D4—3982no—
102D4—3981no—
103D4—3981no—
104D4—3981no—
105D4—3981no—
106D4—3981no—
107D4—1991no—
108D4D79611yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—3982no—
112C2×C4—3981no—
113C2×C4C22×D19911yescharacteristic
114C8—21no—
115C8—3981no—
116C22—21no—
117C22—7962no—
118C22—7962no—
119C22—7961no—
120C22—15922no—
121C22—3982no—
122C22—3981no—
123C22D4×D19911yescharacteristic
124C4—22no—
125C4—21no—
126C4—3981no—
127C4D4×D19911yescharacteristic
128C4C2×D79611yescharacteristic
129C2—21no—
130C2—41no—
131C2—7962no—
132C2—7961no—
133C2—15921no—
134C2#2026392511yescharacteristic
subgroupG/H
centerC2#20263925
commutator, cosocleC2×C4×C199C23
FrattiniC2×C4C22×D199
Fitting#104×C199C2
socleC2×C199#120
hypercenter#104D199
hypocenterC199#453
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #20264001C2semidirect
2. #20263983C2semidirect
3. #20263980C2semidirect
4. #20263952C2semidirect
5. #20263883C2semidirect
6. #20263871C2semidirect
7. #104×C199C2semidirect
14. C2×D796C22semidirect
21. #15691870C22semidirect
23. M16×C199C22semidirect
24. C42×C199C22semidirect
28. Dic398D4semidirect
35. D796D4semidirect
41. Q8×C199D4semidirect
43. D4×C199D4semidirect
67. C199#453semidirect
74. #104D199semidirect
91. M16C2×D199semidirect
92. C42C2×D199semidirect
97. Q8D796semidirect
108. D4D796semidirect

Maximal subgroups

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

subgroupindex
#202640012
#202639832
#202639802
#202639522
#202638832
#202638712
#104×C1992
#453199