Finite Group Databasebeta

← #30638022

Subgroups — #30638022

Order 16256.

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

normal subgroups only

This group has 230 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#22550624C211yesmaximal, characteristic, complemented
2#22550263C212yesmaximal, complemented
3#22549618C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#490×C127C211yesmaximal, characteristic, complemented
6C2×#14523236—22no—
7C2×#14523233—22no—
8#16861197C2212yes—
9#16861194—22no—
10#16861187—22no—
11#16861184—22no—
12#16861120C2212yes—
13#127×C127C2211yescharacteristic
14#125×C127C2211yescharacteristic
15#102×C127—22no—
16#97×C127C2211yescharacteristic
17C2×Dic254D412yes—
18C2×D508—22no—
19C2×D508D412yes—
20C2×C4×D127—22no—
21#14523236—44no—
22#14523233—44no—
23C2×C127⋊C8—22no—
24Dic127⋊C4—22no—
25(C2×D127)⋊C4—22no—
26(C2×D127)⋊C4—22no—
27(C2×D127)⋊C4—22no—
28C2×D4×C127—22no—
29C2×D4×C127—21no—
30C2×C8×C127—21no—
31C4⋊C4×C127—22no—
32C4⋊C4×C127D412yes—
33C42×C127C2311yescharacteristic
34C22×D127—22no—
35Dic254—42no—
36D508—42no—
37D508—82no—
38C2×Dic127—22no—
39C4×D127—82no—
40C127⋊C8—24no—
41C23×C127—41no—
42D4×C127—42no—
43D4×C127—42no—
44D4×C127—42no—
45C2×C4×C127—22no—
46C2×C4×C127C2×D412yes—
47C2×C4×C127C2×D411yescharacteristic
48C8×C127—41no—
49C22×C127—82no—
50C22×C127—81no—
51C22×C127#12011yescharacteristic
52C2×D127—42no—
53C2×D127—42no—
54C2×D127—42no—
55C4×C127—22no—
56C4×C127—21no—
57C4×C127—42no—
58Dic127—42no—
59C2×C127—161no—
60C2×C127#45312yes—
61C2×C127#45411yescharacteristic
62D127—82no—
63#1656—1271nomaximal, Sylow
64C127#165611yescharacteristic, complemented, Sylow
65#562—1271no—
66#496—1272no—
67#490D12711yescharacteristic, complemented
68#330—1271no—
69#329—1272no—
70C2×SD16—2542no—
71C2×D8—2542no—
72#127C2×D12711yescharacteristic
73#125C2×D12711yescharacteristic
74#124—1272no—
75#123—2542no—
76C22⋊Q8—2542no—
77#121—2542no—
78#102—22no—
79#97—1272no—
80#97C2×D12711yescharacteristic
81C2×Q8—1272no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—1272no—
85C2×D4—2542no—
86C22×C4—2542no—
87SD16—5084no—
88D8—5084no—
89C2×C8—21no—
90C2×C8—2542no—
91C4⋊C4—22no—
92C4⋊C4—2542no—
93C4⋊C4C22⋊D12712yes—
94C22⋊C4—2542no—
95C22⋊C4—2542no—
96C22⋊C4—2542no—
97C42C22×D12711yescharacteristic
98C23—41no—
99C23—2542no—
100Q8—5082no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—10162no—
105D4—5082no—
106C2×C4—22no—
107C2×C4—10162no—
108C2×C4—2542no—
109C2×C4D4×D12712yes—
110C2×C4C2×C22⋊D12711yescharacteristic
111C8—41no—
112C8—2544no—
113C22—82no—
114C22—81no—
115C22—5082no—
116C22—5082no—
117C22—5082no—
118C22#1686117711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—5082no—
123C2—161no—
124C2—10162no—
125C2#2255004212yes—
126C2#2255004811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#490×C127C2
socleC22×C127#120
hypercenter#490D127
hypocenterC127#1656
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #22550624C2semidirect
2. #22550263C2semidirect
3. #22549618C2semidirect
4. #22549615C2semidirect
5. #490×C127C2semidirect
64. C127#1656semidirect
67. #490D127semidirect

Maximal subgroups

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

subgroupindex
#225506242
#225502632
#225496182
#225496152
#490×C1272
#1656127