Finite Group Databasebeta

← #30636730

Subgroups — #30636730

Order 16256.

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

normal subgroups only

This group has 146 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#22549643C211yesmaximal, characteristic
2#22549600C211yesmaximal, characteristic
3#397×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127—21no—
5C22⋊(C127⋊C8)—21no—
6C2×C4⋊C4×C127C411yescharacteristic, complemented
7C2×C22⋊C4×C127—21no—
8C2×C22⋊C4×C127C2211yescharacteristic
9#95×C127—21no—
10#95×C127C411yescharacteristic, complemented
11C22×Dic127—41no—
12C2×C127⋊C8—41no—
13C22⋊Dic127—22no—
14C22⋊Dic127—41no—
15C24×C127D411yescharacteristic
16C22×C4×C127—21no—
17C22×C4×C127—21no—
18C22×C4×C127C2×C411yescharacteristic
19C22×C4×C127D411yescharacteristic
20C4⋊C4×C127—41no—
21C22⋊C4×C127—41no—
22C22⋊C4×C127—41no—
23C22⋊C4×C127—41no—
24C2×Dic127—42no—
25C2×Dic127—81no—
26C2×Dic127—81no—
27C127⋊C8—81no—
28C23×C127—21no—
29C23×C127—21no—
30C23×C127—41no—
31C23×C127—41no—
32C23×C127C22⋊C411yescharacteristic
33C23×C127D811yescharacteristic
34C23×C127SD1611yescharacteristic
35C2×C4×C127—21no—
36C2×C4×C127—21no—
37C2×C4×C127—41no—
38C2×C4×C127—41no—
39C2×C4×C127—41no—
40C2×C4×C127—41no—
41C2×C4×C127—41no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127—41no—
45C22×C127—21no—
46C22×C127—21no—
47C22×C127—21no—
48C22×C127—21no—
49C22×C127—41no—
50C22×C127—41no—
51C22×C127—41no—
52C22×C127—41no—
53C22×C127—81no—
54C22×C127#9911yescharacteristic
55C22×C127#10211yescharacteristic
56C22×C127#10411yescharacteristic
57C4×C127—41no—
58C4×C127—41no—
59C4×C127—81no—
60C4×C127—81no—
61Dic127—82no—
62C2×C127—21no—
63C2×C127—21no—
64C2×C127—41no—
65C2×C127—41no—
66C2×C127#32711yescharacteristic
67C2×C127#35211yescharacteristic
68C2×C127#35411yescharacteristic
69#1344—1271nomaximal, Sylow
70C127#134411yescharacteristic, complemented, Sylow
71#397D12711yescharacteristic
72#342—1271no—
73#323—1271no—
74C2×C4⋊C4Dic12711yescharacteristic, complemented
75C2×C22⋊C4—21no—
76C2×C22⋊C4—2541no—
77C2×C22⋊C4C2×D12711yescharacteristic
78C22⋊C8—2541no—
79#95—21no—
80#95Dic12711yescharacteristic, complemented
81C24C22⋊D12711yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—5081no—
85C22×C4C2×Dic12711yescharacteristic
86C22×C4C22⋊D12711yescharacteristic
87C2×C8—5081no—
88C4⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—5081no—
93C22⋊C4—2542no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C22⋊Dic12711yescharacteristic
99C23#1452323311yescharacteristic
100C23#1452323611yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—10161no—
112C2×C4—10161no—
113C2×C4—5082no—
114C8—10161no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#1686112411yescharacteristic
125C22#1686113011yescharacteristic
126C22#1686113611yescharacteristic
127C4—41no—
128C4—41no—
129C4—81no—
130C4—81no—
131C4—10162no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2254961011yescharacteristic
137C2#2254966611yescharacteristic
138C2#2254967111yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC22×C4×C127C2×C4
FrattiniC2×C22⋊C4C2×D127
Fitting#397×C127C2
socleC22×C127#99
prime radicalC24×C127D4
cosocleC2×C22⋊C4×C127C22
hypercenter#397D127
hypocenterC127#1344
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
6. C2×C4⋊C4×C127C4semidirect
10. #95×C127C4semidirect
70. C127#1344semidirect
74. C2×C4⋊C4Dic127semidirect
80. #95Dic127semidirect

Maximal subgroups

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

subgroupindex
#225496432
#225496002
#397×C1272
#1344127