Finite Group Databasebeta

← #30636760

Subgroups — #30636760

Order 16256.

140 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 37 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#22549645C211yesmaximal, characteristic
2#22549602C211yesmaximal, characteristic
3#395×C127C211yesmaximal, characteristic
4C2×#14523232—21no—
5C22⋊(C127⋊C8)—21no—
6C22×Q8×C127C411yescharacteristic
7C2×C4⋊C4×C127—21no—
8C2×C4⋊C4×C127C2211yescharacteristic
9#95×C127—21no—
10#95×C127C411yescharacteristic
11#14523232—22no—
12#14523232—41no—
13C2×C127⋊C8—41no—
14C2×C127⋊C8—41no—
15C2×Q8×C127—42no—
16C2×Q8×C127—41no—
17C22×C4×C127—21no—
18C22×C4×C127—21no—
19C22×C4×C127C2×C411yescharacteristic
20C22×C4×C127D411yescharacteristic
21C22×C4×C127D411yescharacteristic
22C4⋊C4×C127—41no—
23C4⋊C4×C127—42no—
24C127⋊C8—42no—
25C127⋊C8—81no—
26C23×C127C22⋊C411yescharacteristic
27Q8×C127—42no—
28Q8×C127—42no—
29C2×C4×C127—21no—
30C2×C4×C127—21no—
31C2×C4×C127—22no—
32C2×C4×C127—22no—
33C2×C4×C127—21no—
34C2×C4×C127—21no—
35C2×C4×C127—41no—
36C2×C4×C127—41no—
37C2×C4×C127—41no—
38C2×C4×C127—41no—
39C2×C4×C127—41no—
40C2×C4×C127—41no—
41C2×C4×C127SD1611yescharacteristic
42C2×C4×C127Q1611yescharacteristic
43C22×C127—21no—
44C22×C127—21no—
45C22×C127#9911yescharacteristic
46C22×C127#10311yescharacteristic
47C22×C127#10411yescharacteristic
48C4×C127—21no—
49C4×C127—21no—
50C4×C127—42no—
51C4×C127—42no—
52C4×C127—41no—
53C4×C127—41no—
54C2×C127—21no—
55C2×C127—21no—
56C2×C127#32811yescharacteristic
57C2×C127AΣL(1,𝔽16)11yescharacteristic
58C2×C127#35611yescharacteristic
59#1354—1271nomaximal, Sylow
60C127#135411yescharacteristic, complemented, Sylow
61#395D12711yescharacteristic
62#343—1271no—
63#324—1271no—
64C22×Q8Dic12711yescharacteristic
65C2×M16—2541no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×D12711yescharacteristic
68C22⋊C8—2541no—
69#95—21no—
70#95Dic12711yescharacteristic
71C2×Q8—42no—
72C2×Q8—41no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C2×Dic12711yescharacteristic
76C22×C4C22⋊D12711yescharacteristic
77C22×C4C22⋊D12711yescharacteristic
78M16—5081no—
79M16—2542no—
80C2×C8—5081no—
81C2×C8—5081no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C23C22⋊Dic12711yescharacteristic
85Q8—42no—
86Q8—42no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4#1452323511yescharacteristic
100C2×C4#1452323811yescharacteristic
101C8—10161no—
102C8—5082no—
103C22—21no—
104C22—21no—
105C22#1686112411yescharacteristic
106C22#1686113311yescharacteristic
107C22#1686113611yescharacteristic
108C4—21no—
109C4—21no—
110C4—42no—
111C4—41no—
112C4—42no—
113C4—41no—
114C2—21no—
115C2—21no—
116C2#2254961311yescharacteristic
117C2#2254966311yescharacteristic
118C2#2254967511yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC22×C4×C127C2×C4
FrattiniC2×C4⋊C4C2×D127
Fitting#395×C127C2
socleC22×C127#99
prime radicalC23×C127C22⋊C4
cosocleC2×C4⋊C4×C127C22
hypercenter#395D127
hypocenterC127#1354
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
60. C127#1354semidirect

Maximal subgroups

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

subgroupindex
#225496452
#225496022
#395×C1272
#1354127