Finite Group Databasebeta

← #30640618

Subgroups — #30640618

Order 16256.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550316C211yesmaximal, characteristic
2#22549998C211yesmaximal, characteristic
3#22549982C211yesmaximal, characteristic, complemented
4#22549698C211yesmaximal, characteristic, complemented
5#22549693C211yesmaximal, characteristic, complemented
6#22549653C211yesmaximal, characteristic, complemented
7#360×C127C211yesmaximal, characteristic
8C2×Dic508C2211yescharacteristic
9#16861216—21no—
10#16861174—21no—
11#16861155C412yescomplemented
12#16861153C412yescomplemented
13#16861149C2211yescharacteristic
14#16861145C2211yescharacteristic
15#16861136—21no—
16#16861132—21no—
17#16861119C2211yescharacteristic
18#135×C127C2211yescharacteristic
19M32×C127C2211yescharacteristic
20#106×C127C2211yescharacteristic
21C2×Dic254—21no—
22Dic508—21no—
23Dic508C2×C412yes—
24#14523232—21no—
25#14523232—41no—
26C2×C127⋊C8—41no—
27C2×C127⋊C8D411yescharacteristic
28Dic127⋊C4—41no—
29C4×Dic127—41no—
30C4×Dic127D411yescharacteristic
31C127⋊C16C2×C412yes—
32#41×C127—21no—
33Q16×C127C2×C411yescharacteristic
34SD16×C127—21no—
35D8×C127C2×C411yescharacteristic
36C2×C8×C127C2311yescharacteristic
37C4⋊C4×C127—21no—
38C16×C127—21no—
39Dic254—22no—
40Dic254—41no—
41C2×Dic127—21no—
42C2×Dic127—41no—
43C127⋊C8—21no—
44C127⋊C8—41no—
45Q8×C127—21no—
46D4×C127—21no—
47D4×C127—41no—
48C2×C4×C127—41no—
49C2×C4×C127—41no—
50C2×C4×C127C2×D411yescharacteristic
51C8×C127C22×C411yescharacteristic
52C8×C127#4111yescharacteristic
53C22×C127—41no—
54C22×C127AGL(1,ℤ/8)11yescharacteristic
55C4×C127—41no—
56C4×C127—81no—
57C4×C127C4×D411yescharacteristic
58C4×C127AGL(1,ℤ/8)11yescharacteristic
59Dic127—41no—
60Dic127—42no—
61C2×C127—21no—
62C2×C127—81no—
63C2×C127#44211yescharacteristic
64#2180—1271nomaximal, Sylow
65C127#218011yescharacteristic, complemented, Sylow
66#510—1271no—
67#444—1271no—
68#441—1271no—
69#365—1271no—
70#362—1271no—
71#360D12711yescharacteristic
72#347—1271no—
73#135C2×D12711yescharacteristic
74C2×Q16—1271no—
75#131—2541no—
76C4×Q8—2541no—
77Q32—1272no—
78SD32—1272no—
79M32—1271no—
80M32C2×D12711yescharacteristic
81#108—1271no—
82#106C2×D12711yescharacteristic
83#104—2541no—
84#103—2541no—
85#97—1271no—
86#41—21no—
87C2×Q8—2541no—
88Q16—1272no—
89Q16—2541no—
90Q16C4×D12711yescharacteristic
91SD16—21no—
92D8C4×D12711yescharacteristic
93M16—5081no—
94M16—2541no—
95C2×C8—1271no—
96C2×C8—5081no—
97C2×C8C22×D12711yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—5081no—
100C42—1271no—
101C42—5081no—
102C16—21no—
103C16—1272no—
104Q8—21no—
105Q8—5081no—
106Q8—2542no—
107D4—21no—
108D4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—5081no—
112C2×C4—2541no—
113C2×C4D4×D12711yescharacteristic
114C8—5081no—
115C8—2541no—
116C8C2×C4×D12711yescharacteristic
117C8#1452324711yescharacteristic
118C22—41no—
119C22#1686123311yescharacteristic
120C4—41no—
121C4—81no—
122C4—5081no—
123C4—5082no—
124C4(C22⋊D127)⋊C411yescharacteristic
125C4#1686123611yescharacteristic
126C2—21no—
127C2—81no—
128C2#2254998511yescharacteristic
subgroupG/H
centerC2#22549985
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#360×C127C2
socleC2×C127#442
prime radical#135×C127C22
cosocleC2×C8×C127C23
hypercenter#360D127
hypocenterC127#2180
LCSG ▷ 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
3. #22549982C2semidirect
4. #22549698C2semidirect
5. #22549693C2semidirect
6. #22549653C2semidirect
11. #16861155C4semidirect
12. #16861153C4semidirect
65. C127#2180semidirect

Maximal subgroups

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

subgroupindex
#225503162
#225499982
#225499822
#225496982
#225496932
#225496532
#360×C1272
#2180127