Finite Group Databasebeta

← #30640853

Subgroups — #30640853

Order 16256.

140 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#16861155C211yesmaximal, characteristic, complemented
2#22550293C211yesmaximal, characteristic
3#22550151C211yesmaximal, characteristic
4#22549705C211yesmaximal, characteristic, complemented
5#22549692C211yesmaximal, characteristic, complemented
6#22549691C211yesmaximal, characteristic, complemented
7#473×C127C211yesmaximal, characteristic
8#16861239—21no—
9C2×Dic508C2211yescharacteristic
10#16861155—21no—
11#16861155—21no—
12#16861149—21no—
13#16861149C2211yescharacteristic
14C2×C127⋊C16—21no—
15C2×C127⋊C16C2211yescharacteristic
16#16861145C2211yescharacteristic
17#16861127—21no—
18#137×C127—21no—
19C2×Q16×C127C2211yescharacteristic
20#131×C127C2211yescharacteristic
21#108×C127C2211yescharacteristic
22#101×C127—21no—
23C2×Dic254—21no—
24Dic508—21no—
25Dic508—21no—
26#14523238—41no—
27#14523236—41no—
28#14523232—21no—
29C127⋊C16—21no—
30C127⋊C16D411yescharacteristic
31C127⋊C16D411yescharacteristic
32#41×C127D411yescharacteristic
33C2×Q8×C127—21no—
34Q16×C127—21no—
35Q16×C127—21no—
36Q16×C127—41no—
37SD16×C127—41no—
38M16×C127—21no—
39M16×C127—21no—
40M16×C127D411yescharacteristic
41C2×C8×C127—21no—
42C2×C8×C127C2311yescharacteristic
43Dic254—41no—
44Dic254—41no—
45C2×Dic127—41no—
46C127⋊C8—41no—
47Q8×C127—41no—
48Q8×C127—41no—
49Q8×C127D811yescharacteristic
50D4×C127—21no—
51D4×C127D811yescharacteristic
52C2×C4×C127—21no—
53C2×C4×C127—41no—
54C2×C4×C127C2×D411yescharacteristic
55C8×C127—21no—
56C8×C127—41no—
57C8×C127C2×D411yescharacteristic
58C8×C127#4111yescharacteristic
59C22×C127—21no—
60C22×C127#13511yescharacteristic
61C4×C127—21no—
62C4×C127—81no—
63C4×C127#12111yescharacteristic
64C4×C127C2×D811yescharacteristic
65Dic127—81no—
66C2×C127—21no—
67C2×C127—41no—
68C2×C127#46611yescharacteristic
69#2222—1271nomaximal, Sylow
70C127#222211yescharacteristic, complemented, Sylow
71C2×Q32—1271no—
72#504—1271no—
73#473—1271no—
74#473D12711yescharacteristic
75#368—1271no—
76#362—1271no—
77#362—1271no—
78#137—21no—
79#137—2541no—
80C2×Q16—1271no—
81C2×Q16C2×D12711yescharacteristic
82#131C2×D12711yescharacteristic
83Q32—2541no—
84Q32—2541no—
85M32—1271no—
86M32—2541no—
87C2×C16—1271no—
88C2×C16—2541no—
89#108—1271no—
90#108C2×D12711yescharacteristic
91#101—21no—
92#101—2541no—
93#41C22⋊D12711yescharacteristic
94C2×Q8—21no—
95C2×Q8—2541no—
96Q16—21no—
97Q16—21no—
98Q16—41no—
99Q16—5081no—
100Q16—2541no—
101Q16—2541no—
102SD16—41no—
103SD16—5081no—
104M16—21no—
105M16—21no—
106M16—2541no—
107M16C22⋊D12711yescharacteristic
108C2×C8—21no—
109C2×C8C22×D12711yescharacteristic
110C16—1271no—
111C16—1271no—
112C16—2541no—
113Q8—41no—
114Q8—41no—
115Q8—5081no—
116Q8—5081no—
117Q8#1452323311yescharacteristic
118D4—21no—
119D4#1452323311yescharacteristic
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—5081no—
123C2×C4C2×C22⋊D12711yescharacteristic
124C8—21no—
125C8—41no—
126C8—5081no—
127C8D4×D12711yescharacteristic
128C8#1452324711yescharacteristic
129C22—21no—
130C22#1686122911yescharacteristic
131C4—21no—
132C4—81no—
133C4—10161no—
134C4#1686118111yescharacteristic
135C4C2×#1452323311yescharacteristic
136C2—21no—
137C2—41no—
138C2#2255011011yescharacteristic
subgroupG/H
centerC2#22550110
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#473×C127C2
socleC2×C127#466
prime radical#41×C127D4
hypercenter#473D127
hypocenterC127#2222
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
1. C2×#16861155C2semidirect
4. #22549705C2semidirect
5. #22549692C2semidirect
6. #22549691C2semidirect
70. C127#2222semidirect

Maximal subgroups

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

subgroupindex
C2×#168611552
#225502932
#225501512
#225497052
#225496922
#225496912
#473×C1272
#2222127