Finite Group Databasebeta

← #30636751

Subgroups — #30636751

Order 16256.

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

normal subgroups only

This group has 162 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#393×C127C211yesmaximal, characteristic
4C2×C22⋊Dic127—21no—
5C22⋊(C127⋊C8)—21no—
6C22×Q8×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
16C2×Q8×C127—42no—
17C2×Q8×C127—41no—
18C22×C4×C127—21no—
19C22×C4×C127—21no—
20C22×C4×C127C2×C411yescharacteristic
21C22×C4×C127D411yescharacteristic
22C22⋊C4×C127—41no—
23C22⋊C4×C127—42no—
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×C127SD1611yescharacteristic
34C23×C127Q1611yescharacteristic
35Q8×C127—42no—
36Q8×C127—42no—
37C2×C4×C127—22no—
38C2×C4×C127—21no—
39C2×C4×C127—21no—
40C2×C4×C127—41no—
41C2×C4×C127—41no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127—41no—
45C2×C4×C127—41no—
46C2×C4×C127—41no—
47C22×C127—21no—
48C22×C127—21no—
49C22×C127—21no—
50C22×C127—21no—
51C22×C127—41no—
52C22×C127—41no—
53C22×C127—41no—
54C22×C127—41no—
55C22×C127—81no—
56C22×C127#9911yescharacteristic
57C22×C127#10311yescharacteristic
58C22×C127#10411yescharacteristic
59C4×C127—42no—
60C4×C127—41no—
61C4×C127—41no—
62C4×C127—81no—
63Dic127—82no—
64C2×C127—21no—
65C2×C127—21no—
66C2×C127—41no—
67C2×C127—41no—
68C2×C127#32811yescharacteristic
69C2×C127AΣL(1,𝔽16)11yescharacteristic
70C2×C127#35411yescharacteristic
71#1351—1271nomaximal, Sylow
72C127#135111yescharacteristic, complemented, Sylow
73#393D12711yescharacteristic
74#342—1271no—
75#323—1271no—
76C22×Q8Dic12711yescharacteristic, complemented
77C2×C22⋊C4—21no—
78C2×C22⋊C4—2541no—
79C2×C22⋊C4C2×D12711yescharacteristic
80C22⋊C8—2541no—
81#95—21no—
82#95Dic12711yescharacteristic, complemented
83C24C22⋊D12711yescharacteristic
84C2×Q8—42no—
85C2×Q8—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—5081no—
89C22×C4C2×Dic12711yescharacteristic
90C22×C4C22⋊D12711yescharacteristic
91C2×C8—5081no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—5081no—
95C22⋊C4—2542no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23C22⋊Dic12711yescharacteristic
101C23#1452323511yescharacteristic
102C23#1452323811yescharacteristic
103Q8—42no—
104Q8—42no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—10161no—
116C2×C4—10161no—
117C2×C4—5082no—
118C8—10161no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—41no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—81no—
128C22#1686112411yescharacteristic
129C22#1686113311yescharacteristic
130C22#1686113611yescharacteristic
131C4—42no—
132C4—41no—
133C4—41no—
134C4—81no—
135C4—10162no—
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2#2254961311yescharacteristic
141C2#2254966311yescharacteristic
142C2#2254967111yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC22×C4×C127C2×C4
FrattiniC2×C22⋊C4C2×D127
Fitting#393×C127C2
socleC22×C127#99
prime radicalC24×C127D4
cosocleC2×C22⋊C4×C127C22
hypercenter#393D127
hypocenterC127#1351
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
6. C22×Q8×C127C4semidirect
10. #95×C127C4semidirect
72. C127#1351semidirect
76. C22×Q8Dic127semidirect
82. #95Dic127semidirect

Maximal subgroups

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

subgroupindex
#225496432
#225496002
#393×C1272
#1351127