Finite Group Databasebeta

← #30640899

Subgroups — #30640899

Order 16256.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550273C211yesmaximal, characteristic, complemented
2#22549993C211yesmaximal, characteristic
3#22549699C211yesmaximal, characteristic, complemented
4#22549698C211yesmaximal, characteristic, complemented
5#22549696C211yesmaximal, characteristic, complemented
6#22549686C211yesmaximal, characteristic
7#360×C127C211yesmaximal, characteristic
8#16861216—21no—
9#16861208—21no—
10#16861149C2211yescharacteristic
11#16861145C2211yescharacteristic
12#16861141C2211yescharacteristic
13#16861140—21no—
14#16861136—21no—
15C8×Dic127C2211yescharacteristic
16#135×C127C2211yescharacteristic
17M32×C127C2211yescharacteristic
18#106×C127C2211yescharacteristic
19C2×Dic254—41no—
20#14523232—21no—
21#14523232—41no—
22C2×C127⋊C8—41no—
23C2×C127⋊C8D411yescharacteristic
24C4⋊Dic127—21no—
25Dic127⋊C4—41no—
26C4×Dic127D411yescharacteristic
27C127⋊C16—21no—
28#41×C127—21no—
29Q16×C127Q811yescharacteristic, complemented
30SD16×C127—21no—
31D8×C127Q811yescharacteristic, complemented
32C2×C8×C127C2311yescharacteristic
33C4⋊C4×C127—21no—
34C16×C127—21no—
35Dic254—82no—
36C2×Dic127—21no—
37C2×Dic127—41no—
38C127⋊C8—41no—
39C127⋊C8SD1611yescharacteristic
40C127⋊C8SD1611yescharacteristic
41Q8×C127—21no—
42D4×C127—21no—
43D4×C127—41no—
44C2×C4×C127—41no—
45C2×C4×C127—41no—
46C2×C4×C127C2×D411yescharacteristic
47C8×C127C2×Q811yescharacteristic
48C8×C127#4111yescharacteristic
49C22×C127—41no—
50C22×C127AGL(1,ℤ/8)11yescharacteristic
51C4×C127—41no—
52C4×C127—81no—
53C4×C127C22⋊Q811yescharacteristic
54C4×C127C2×SD1611yescharacteristic
55Dic127—22no—
56Dic127—81no—
57C2×C127—21no—
58C2×C127—81no—
59C2×C127#47611yescharacteristic
60#2230—1271nomaximal, Sylow
61C127#223011yescharacteristic, complemented, Sylow
62#498—1271no—
63#443—1271no—
64#365—1271no—
65#365—1271no—
66#364—1271no—
67#360—1271no—
68#360D12711yescharacteristic
69#135C2×D12711yescharacteristic
70#131—2541no—
71C4⋊Q8—2541no—
72M32—1271no—
73M32C2×D12711yescharacteristic
74#108—1271no—
75#106—1271no—
76#106—2541no—
77#106C2×D12711yescharacteristic
78#104—2541no—
79C4×C8—1271no—
80#41—21no—
81C2×Q8—5081no—
82Q16Dic25411yescharacteristic, complemented
83SD16—21no—
84D8Dic25411yescharacteristic, complemented
85M16—5081no—
86M16—2541no—
87C2×C8—1271no—
88C2×C8—5081no—
89C2×C8C22×D12711yescharacteristic
90C4⋊C4—21no—
91C4⋊C4—5081no—
92C4⋊C4—2541no—
93C42—1271no—
94C16—21no—
95C16—2541no—
96Q8—21no—
97Q8—10162no—
98D4—21no—
99D4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—5081no—
103C2×C4—2541no—
104C2×C4D4×D12711yescharacteristic
105C8—1271no—
106C8—1271no—
107C8—5081no—
108C8C2×Dic25411yescharacteristic
109C8#1452324711yescharacteristic
110C22—41no—
111C22#1686123311yescharacteristic
112C4—41no—
113C4—81no—
114C4—10161no—
115C4—2542no—
116C4#1686118611yescharacteristic
117C4SD16×D12711yescharacteristic
118C2—21no—
119C2—81no—
120C2#2255016411yescharacteristic
subgroupG/H
centerC2#22550164
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127#476
prime radical#135×C127C22
hypercenter#360D127
hypocenterC127#2230
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #22550273C2semidirect
3. #22549699C2semidirect
4. #22549698C2semidirect
5. #22549696C2semidirect
29. Q16×C127Q8semidirect
31. D8×C127Q8semidirect
61. C127#2230semidirect
82. Q16Dic254semidirect
84. D8Dic254semidirect

Maximal subgroups

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

subgroupindex
#225502732
#225499932
#225496992
#225496982
#225496962
#225496862
#360×C1272
#2230127