Finite Group Databasebeta

← #30637332

Subgroups — #30637332

Order 16256.

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

normal subgroups only

This group has 212 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
2C4×#14523232C211yesmaximal, characteristic
3#22549618C214yesmaximal, complemented
4C2×#125×C127C211yesmaximal, characteristic
5C2×#14523232—21no—
6C4⋊(C127⋊C8)—21no—
7C22⋊(C127⋊C8)—21no—
8#16861120C2212yes—
9#16861120C2212yes—
10C4×C127⋊C8—21no—
11#125×C127—21no—
12#125×C127—21no—
13#125×C127C412yes—
14#125×C127C2212yes—
15C2×C4⋊C4×C127—21no—
16C2×C4⋊C4×C127—21no—
17C2×C4⋊C4×C127C411yescharacteristic
18C2×C4⋊C4×C127C411yescharacteristic
19C2×C42×C127C2211yescharacteristic
20#14523232—24no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C22×C4×C127—21no—
24C22×C4×C127—21no—
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127D411yescharacteristic
27C22×C4×C127D411yescharacteristic
28C4⋊C4×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—21no—
31C4⋊C4×C127—21no—
32C4⋊C4×C127—22no—
33C4⋊C4×C127—22no—
34C4⋊C4×C127C2×C412yes—
35C4⋊C4×C127C2×C412yes—
36C42×C127C2×C411yescharacteristic
37C42×C127D411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127C2311yescharacteristic
40C127⋊C8—24no—
41C127⋊C8—42no—
42C23×C127C22⋊C411yescharacteristic
43C2×C4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—41no—
48C2×C4×C127—42no—
49C2×C4×C127—42no—
50C2×C4×C127C22⋊C411yescharacteristic
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C22⋊C411yescharacteristic
53C2×C4×C127C22×C411yescharacteristic
54C2×C4×C127C2×D411yescharacteristic
55C2×C4×C127C2×D411yescharacteristic
56C22×C127—21no—
57C22×C127—21no—
58C22×C127#10412yes—
59C22×C127C2×C22⋊C411yescharacteristic
60C4×C127—21no—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—21no—
64C4×C127—21no—
65C4×C127—42no—
66C4×C127—42no—
67C4×C127#10112yes—
68C2×C127—22no—
69C2×C127C2×#10111yescharacteristic
70C2×C127C2×#10411yescharacteristic
71C2×C127#42111yescharacteristic
72#1528—1271nomaximal, Sylow
73C127#152811yescharacteristic, complemented, Sylow
74C2×#125D12711yescharacteristic
75#432—1271no—
76C4×M16—1271no—
77#330—1274no—
78C2×M16—2541no—
79#125—21no—
80#125—21no—
81#125Dic12712yes—
82#125C2×D12712yes—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4Dic12711yescharacteristic
86C2×C4⋊C4Dic12711yescharacteristic
87C2×C42C2×D12711yescharacteristic
88C4⋊C8—2541no—
89C22⋊C8—2541no—
90#97—1272no—
91#97—1272no—
92C4×C8—2541no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4C2×Dic12711yescharacteristic
96C22×C4C22⋊D12711yescharacteristic
97C22×C4C22⋊D12711yescharacteristic
98M16—2544no—
99C2×C8—2542no—
100C2×C8—2542no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4C2×Dic12712yes—
108C4⋊C4C2×Dic12712yes—
109C42C2×Dic12711yescharacteristic
110C42C22⋊D12711yescharacteristic
111C42C22⋊D12711yescharacteristic
112C42C22×D12711yescharacteristic
113C23C22⋊Dic12711yescharacteristic
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4C22⋊Dic12711yescharacteristic
122C2×C4C22⋊Dic12711yescharacteristic
123C2×C4C22⋊Dic12711yescharacteristic
124C2×C4C22×Dic12711yescharacteristic
125C2×C4C2×C22⋊D12711yescharacteristic
126C2×C4C2×C22⋊D12711yescharacteristic
127C8—5082no—
128C8—2544no—
129C22—21no—
130C22—21no—
131C22#1686113612yes—
132C22C2×C22⋊Dic12711yescharacteristic
133C4—21no—
134C4—21no—
135C4—21no—
136C4—21no—
137C4—21no—
138C4—42no—
139C4—42no—
140C4#1686112812yes—
141C2—22no—
142C2C2×#1686112811yescharacteristic
143C2C2×#1686113611yescharacteristic
144C2#2254989311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×#125×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C22⋊C4
cosocleC42×C127C23
hypercenterC2×#125D127
hypocenterC127#1528
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #22549618C2semidirect
73. C127#1528semidirect

Maximal subgroups

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

subgroupindex
#225499332
C4×#145232322
#225496182
C2×#125×C1272
#1528127