Finite Group Databasebeta

← #30637086

Subgroups — #30637086

Order 16256.

204 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 81 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#22549933C211yesmaximal, characteristic
2#22549929C211yesmaximal, characteristic
3#22549602C214yesmaximal
4C2×#125×C127C211yesmaximal, characteristic
5C4⋊(C127⋊C8)—21no—
6C4⋊(C127⋊C8)—21no—
7C22⋊(C127⋊C8)C2212yes—
8C22⋊(C127⋊C8)C2212yes—
9#16861120—21no—
10C4×C127⋊C8—21no—
11#125×C127—21no—
12#125×C127—21no—
13#125×C127—21no—
14#125×C127—21no—
15C2×C4⋊C4×C127C411yescharacteristic
16C2×C4⋊C4×C127C412yes—
17C2×C4⋊C4×C127C411yescharacteristic
18C2×C4⋊C4×C127C2212yes—
19C2×C42×C127C2211yescharacteristic
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C22×C4×C127C2×C412yes—
23C22×C4×C127C2×C411yescharacteristic
24C22×C4×C127C2×C411yescharacteristic
25C22×C4×C127C2×C412yes—
26C22×C4×C127C2311yescharacteristic
27C4⋊C4×C127—21no—
28C4⋊C4×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—21no—
31C4⋊C4×C127—21no—
32C4⋊C4×C127—21no—
33C4⋊C4×C127—22no—
34C4⋊C4×C127—22no—
35C42×C127D411yescharacteristic
36C42×C127D411yescharacteristic
37C42×C127D411yescharacteristic
38C42×C127D411yescharacteristic
39C127⋊C8—42no—
40C127⋊C8—42no—
41C23×C127C22×C411yescharacteristic
42C2×C4×C127—22no—
43C2×C4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—24no—
47C2×C4×C127—41no—
48C2×C4×C127—42no—
49C2×C4×C127C22⋊C411yescharacteristic
50C2×C4×C127C22⋊C411yescharacteristic
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C2×D411yescharacteristic
54C2×C4×C127C2×D411yescharacteristic
55C22×C127—21no—
56C22×C127—21no—
57C22×C127C2×C22⋊C411yescharacteristic
58C22×C127#13112yes—
59C4×C127—21no—
60C4×C127—21no—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—21no—
64C4×C127—42no—
65C4×C127—42no—
66C4×C127#10112yes—
67C2×C127—22no—
68C2×C127#40811yescharacteristic
69C2×C127#41011yescharacteristic
70C2×C127C2×#10111yescharacteristic
71#1470—1271nomaximal, Sylow
72C127#147011yescharacteristic, complemented, Sylow
73C2×#125D12711yescharacteristic
74#432—1271no—
75#431—1271no—
76#324—1274no—
77#125—21no—
78#125—21no—
79#125—21no—
80#125—21no—
81C2×C4⋊C4Dic12711yescharacteristic
82C2×C4⋊C4Dic12712yes—
83C2×C4⋊C4Dic12711yescharacteristic
84C2×C4⋊C4C2×D12712yes—
85C2×C42C2×D12711yescharacteristic
86C4⋊C8—2541no—
87C4⋊C8—2541no—
88C22⋊C8—1272no—
89C22⋊C8—1272no—
90#97—2541no—
91C4×C8—2541no—
92C22×C4C2×Dic12712yes—
93C22×C4C2×Dic12711yescharacteristic
94C22×C4C2×Dic12711yescharacteristic
95C22×C4C2×Dic12712yes—
96C22×C4C22×D12711yescharacteristic
97C2×C8—2542no—
98C2×C8—2542no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C42C22⋊D12711yescharacteristic
108C42C22⋊D12711yescharacteristic
109C42C22⋊D12711yescharacteristic
110C42C22⋊D12711yescharacteristic
111C23C22×Dic12711yescharacteristic
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4C22⋊Dic12711yescharacteristic
120C2×C4C22⋊Dic12711yescharacteristic
121C2×C4C22⋊Dic12711yescharacteristic
122C2×C4C22⋊Dic12711yescharacteristic
123C2×C4C2×C22⋊D12711yescharacteristic
124C2×C4C2×C22⋊D12711yescharacteristic
125C8—5082no—
126C8—5082no—
127C22—21no—
128C22—21no—
129C22C2×C22⋊Dic12711yescharacteristic
130C22#1686121612yes—
131C4—21no—
132C4—21no—
133C4—21no—
134C4—21no—
135C4—21no—
136C4—42no—
137C4—42no—
138C4#1686112812yes—
139C2—22no—
140C2#2254984311yescharacteristic
141C2#2254985011yescharacteristic
142C2C2×#1686112811yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutator, prime radicalC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#125×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenterC2×#125D127
hypocenterC127#1470
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
72. C127#1470semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225499292
#225496022
C2×#125×C1272
#1470127