Finite Group Databasebeta

← #30638681

Subgroups — #30638681

Order 16256.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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#22549860C213yesmaximal, complemented
2#22549843C213yesmaximal, complemented
3#585×C127C211yesmaximal, characteristic
4C2×#14523232C2213yescomplemented
5C22×C127⋊C8C2211yescharacteristic, complemented
6#16861128—23no—
7#16861126—23no—
8C22⋊(C127⋊C8)—26no—
92−1+4×C127—23no—
10H(5,𝔽2)×C127—23no—
11H(5,𝔽2)×C127C414yes—
12C2×#41×C127—26no—
13C2×#41×C127C2213yes—
14#14523232—26no—
15C2×C127⋊C8—23no—
16C2×C127⋊C8—23no—
17#41×C127—24no—
18#41×C127—212no—
19#41×C127—212no—
20#41×C127—26no—
21#41×C127—26no—
22C2×Q8×C127—26no—
23C2×Q8×C127D413yes—
24C2×D4×C127—26no—
25C2×D4×C127—212no—
26C2×D4×C127C2×C416yes—
27C2×D4×C127D413yes—
28C22×C4×C127—24no—
29C22×C4×C127D416yes—
30C22×C4×C127C2311yescharacteristic
31C127⋊C8—41no—
32C127⋊C8—43no—
33C23×C127—24no—
34C23×C127C22⋊C416yes—
35C23×C127C22×C411yescharacteristic
36Q8×C127—24no—
37Q8×C127—26no—
38D4×C127—212no—
39D4×C127—212no—
40D4×C127—26no—
41C2×C4×C127—212no—
42C2×C4×C127—26no—
43C2×C4×C127—26no—
44C2×C4×C127C22⋊C416yes—
45C2×C4×C127C2×D413yes—
46C2×C4×C127C2×D413yes—
47C22×C127—26no—
48C22×C127—41no—
49C22×C127—46no—
50C22×C127—84no—
51C22×C127C2×C22⋊C413yes—
52C4×C127—26no—
53C4×C127#12013yes—
54C4×C127#12011yescharacteristic
55C2×C127—23no—
56C2×C127—46no—
57C2×C127#37911yescharacteristic
58#1791—1271nomaximal, Sylow
59C127#179111yescharacteristic, complemented, Sylow
60#585D12711yescharacteristic
61#413—1273no—
62#408—1273no—
632−1+4—23no—
64H(5,𝔽2)—23no—
65H(5,𝔽2)Dic12714yes—
66C2×#41—26no—
67C2×#41C2×D12713yes—
68C2×M16—1273no—
69C22×C8—1271no—
70#101—2543no—
71#100—2543no—
72C22⋊C8—2546no—
73#41—24no—
74#41—212no—
75#41—212no—
76#41—26no—
77#41—26no—
78C2×Q8—26no—
79C2×Q8C22⋊D12713yes—
80C2×D4—26no—
81C2×D4—212no—
82C2×D4C2×Dic12716yes—
83C2×D4C22⋊D12713yes—
84C22×C4—24no—
85C22×C4C22⋊D12716yes—
86C22×C4C22×D12711yescharacteristic
87M16—2546no—
88C2×C8—2543no—
89C2×C8—2543no—
90C23—24no—
91C23C22⋊Dic12716yes—
92C23C22×Dic12711yescharacteristic
93Q8—24no—
94Q8—26no—
95D4—212no—
96D4—212no—
97D4—26no—
98C2×C4—212no—
99C2×C4—26no—
100C2×C4—26no—
101C2×C4C22⋊Dic12716yes—
102C2×C4C2×C22⋊D12713yes—
103C2×C4C2×C22⋊D12713yes—
104C8—5083no—
105C8—5081no—
106C22—26no—
107C22—41no—
108C22—46no—
109C22—84no—
110C22C2×C22⋊Dic12713yes—
111C4—26no—
112C4#1686117813yes—
113C4#1686117811yescharacteristic
114C2—23no—
115C2—46no—
116C2#2254973211yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#585×C127C2
socleC2×C127#379
cosocleC22×C4×C127C23
hypercenter#585D127
hypocenterC127#1791
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22549860C2semidirect
2. #22549843C2semidirect
4. C2×#14523232C22semidirect
5. C22×C127⋊C8C22semidirect
59. C127#1791semidirect

Maximal subgroups

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

subgroupindex
#225498602
#225498432
#585×C1272
#1791127