Finite Group Databasebeta

← #30636731

Subgroups — #30636731

Order 16256.

146 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 35 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#22549801C211yesmaximal, characteristic
2#22549643C211yesmaximal, characteristic
3#323×C127C211yesmaximal, characteristic
4C2×C4⋊Dic127C411yescharacteristic, complemented
5C2×C22⋊Dic127—21no—
6C2×C22⋊Dic127—21no—
7C127⋊#95—21no—
8C127⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C127C2211yescharacteristic
10C22⋊C8×C127—21no—
11C22×Dic127—21no—
12C22×Dic127—21no—
13C22×Dic127—41no—
14C4⋊Dic127—41no—
15C22⋊Dic127—22no—
16C22⋊Dic127—41no—
17C22⋊Dic127—41no—
18C22⋊Dic127—41no—
19C24×C127D411yescharacteristic
20C22×C4×C127C2×C411yescharacteristic
21C22×C4×C127D411yescharacteristic
22C2×C8×C127—41no—
23C22⋊C4×C127—41no—
24C2×Dic127—42no—
25C2×Dic127—41no—
26C2×Dic127—41no—
27C2×Dic127—41no—
28C2×Dic127—41no—
29C2×Dic127—41no—
30C2×Dic127—41no—
31C2×Dic127—81no—
32C2×Dic127—81no—
33C23×C127—21no—
34C23×C127—21no—
35C23×C127—41no—
36C23×C127—41no—
37C23×C127C22⋊C411yescharacteristic
38C23×C127D811yescharacteristic
39C23×C127SD1611yescharacteristic
40C2×C4×C127—21no—
41C2×C4×C127—21no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C8×C127—81no—
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—
59Dic127—81no—
60Dic127—82no—
61Dic127—81no—
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#397—1271no—
72#342—1271no—
73#323D12711yescharacteristic
74C2×C4⋊C4—1271no—
75C2×C22⋊C4—2541no—
76C2×C22⋊C4—2541no—
77C2×C22⋊C4C2×D12711yescharacteristic
78C22⋊C8—21no—
79#95—1271no—
80#95—2541no—
81C24D50811yescharacteristic
82C22×C4—5081no—
83C22×C4—2541no—
84C22×C4—2541no—
85C22×C4C4×D12711yescharacteristic
86C22×C4C22⋊D12711yescharacteristic
87C2×C8—41no—
88C4⋊C4—5081no—
89C22⋊C4—41no—
90C22⋊C4—5081no—
91C22⋊C4—5081no—
92C22⋊C4—5081no—
93C22⋊C4—2542no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23(C2×D127)⋊C411yescharacteristic
99C23D101611yescharacteristic
100C23#1452323711yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—10161no—
106C2×C4—10161no—
107C2×C4—5082no—
108C2×C4—5081no—
109C2×C4—5081no—
110C2×C4—5081no—
111C2×C4—5081no—
112C2×C4—5081no—
113C2×C4—5081no—
114C8—81no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#1686112311yescharacteristic
125C22#1686113111yescharacteristic
126C22#1686113511yescharacteristic
127C4—41no—
128C4—41no—
129C4—10161no—
130C4—10162no—
131C4—10161no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2254961111yescharacteristic
137C2#2254966711yescharacteristic
138C2#2254967011yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC22×C4×C127C2×C4
FrattiniC2×C22⋊C4C2×D127
Fitting#323×C127C2
socleC22×C127#99
prime radicalC24×C127D4
cosocleC2×C22⋊C4×C127C22
hypercenter#323D127
hypocenterC127#1344
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic127C4semidirect
8. C127⋊#95C4semidirect
70. C127#1344semidirect

Maximal subgroups

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

subgroupindex
#225498012
#225496432
#323×C1272
#1344127