Finite Group Databasebeta

← #30640082

Subgroups — #30640082

Order 16256.

314 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 75 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#22549748C213yesmaximal
2#22549643C213yesmaximal
3#543×C127C211yesmaximal, characteristic
4C23×Dic127C2211yescharacteristic
5C2×Dic127⋊C4—23no—
6C2×C22⋊Dic127C2213yes—
7C127⋊#95—23no—
8C22⋊Q8×C127—23no—
9C22⋊Q8×C127—23no—
10C2×C22⋊C4×C127C2213yes—
11C22×Dic127—23no—
12C22×Dic127—23no—
13C22×Dic127—42no—
14Dic127⋊C4—46no—
15C22⋊Dic127—23no—
16C22⋊Dic127Q816yescomplemented
17C24×C127C2311yescharacteristic
18C2×Q8×C127—21no—
19C2×Q8×C127—21no—
20C22×C4×C127D413yes—
21C22×C4×C127D413yes—
22C4⋊C4×C127—23no—
23C4⋊C4×C127—23no—
24C22⋊C4×C127—26no—
25C2×Dic127—26no—
26C2×Dic127—26no—
27C2×Dic127—43no—
28C2×Dic127—43no—
29C2×Dic127—43no—
30C2×Dic127—41no—
31C23×C127—41no—
32C23×C127—41no—
33C23×C127C2×D413yes—
34C23×C127C2×Q813yes—
35C23×C127#4111yescharacteristic
36Q8×C127—81no—
37Q8×C127—81no—
38C2×C4×C127—23no—
39C2×C4×C127—23no—
40C2×C4×C127—43no—
41C2×C4×C127—43no—
42C22×C127—23no—
43C22×C127—23no—
44C22×C127—43no—
45C22×C127—41no—
46C22×C127#12011yescharacteristic
47C22×C127C22⋊Q813yes—
48C22×C127C4⋊Q813yes—
49C4×C127—43no—
50C4×C127—43no—
51Dic127—46no—
52Dic127—42no—
53C2×C127—23no—
54C2×C127—23no—
55C2×C127#39511yescharacteristic
56C2×C127B(4,𝔽2)11yescharacteristic
57C2×C127#45811yescharacteristic
58#2062—1271nomaximal, Sylow
59C127#206211yescharacteristic, complemented, Sylow
60#543D12711yescharacteristic
61#385—1273no—
62#342—1273no—
63C23×C4—1271no—
64C22⋊Q8—23no—
65C22⋊Q8—23no—
66C2×C4⋊C4—2543no—
67C2×C22⋊C4—1273no—
68C2×C22⋊C4C2×D12713yes—
69#95—2543no—
70C24C22×D12711yescharacteristic
71C2×Q8—21no—
72C2×Q8—21no—
73C22×C4—5082no—
74C22×C4—2543no—
75C22×C4—2543no—
76C22×C4C22⋊D12713yes—
77C22×C4C22⋊D12713yes—
78C4⋊C4—23no—
79C4⋊C4—23no—
80C4⋊C4—5086no—
81C22⋊C4—26no—
82C22⋊C4—1276no—
83C22⋊C4—2543no—
84C23—41no—
85C23—41no—
86C23C2×C22⋊D12713yes—
87C23Q8×D12713yes—
88C23#1452324911yescharacteristic
89Q8—81no—
90Q8—81no—
91C2×C4—23no—
92C2×C4—23no—
93C2×C4—43no—
94C2×C4—43no—
95C2×C4—5083no—
96C2×C4—5083no—
97C2×C4—5083no—
98C2×C4—5081no—
99C2×C4—2546no—
100C2×C4—2546no—
101C22—23no—
102C22—23no—
103C22—43no—
104C22—41no—
105C22#1686117811yescharacteristic
106C22#1686118513yes—
107C22#1686120713yes—
108C4—43no—
109C4—43no—
110C4—5086no—
111C4—5082no—
112C2—23no—
113C2—23no—
114C2#2254979511yescharacteristic
115C2#2255006311yescharacteristic
116C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, prime radical, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#543×C127C2
socleC22×C127#120
hypercenter#543D127
hypocenterC127#2062
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #543

Complements

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

NH ≅ G/Ntype
16. C22⋊Dic127Q8semidirect
59. C127#2062semidirect

Maximal subgroups

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

subgroupindex
#225497482
#225496432
#543×C1272
#2062127