Finite Group Databasebeta

← #30640621

Subgroups — #30640621

Order 16256.

142 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 53 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#22549984C211yesmaximal, characteristic, complemented
3#22549700C211yesmaximal, characteristic, complemented
4#22549691C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic
6#22549654C211yesmaximal, characteristic, complemented
7#444×C127C211yesmaximal, characteristic
8C2×Dic508C2211yescharacteristic
9C4×Dic254—21no—
10#16861155C412yescomplemented
11#16861153C412yescomplemented
12#16861149C2211yescharacteristic
13#16861149C2211yescharacteristic
14#16861141C2211yescharacteristic
15#16861134—21no—
16#135×C127C2211yescharacteristic
17#131×C127—21no—
18#108×C127C2211yescharacteristic
19#104×C127—21no—
20#97×C127C2211yescharacteristic
21C2×Dic254—21no—
22Dic508—21no—
23Dic508C2×C412yes—
24C4⋊Dic127—21no—
25Dic127⋊C4—41no—
26C4×Dic127—41no—
27C127⋊C16—21no—
28C127⋊C16C2×C412yes—
29#41×C127—21no—
30Q16×C127C2×C411yescharacteristic
31SD16×C127—21no—
32D8×C127C2×C411yescharacteristic
33M16×C127—21no—
34M16×C127—41no—
35C2×C8×C127—41no—
36C2×C8×C127D411yescharacteristic
37C2×C8×C127C2311yescharacteristic
38C42×C127D411yescharacteristic
39Dic254—22no—
40Dic254—41no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43Q8×C127—21no—
44D4×C127—21no—
45D4×C127—41no—
46C2×C4×C127—21no—
47C2×C4×C127—41no—
48C2×C4×C127C2×D411yescharacteristic
49C8×C127—21no—
50C8×C127—41no—
51C8×C127C22×C411yescharacteristic
52C8×C127#4111yescharacteristic
53C22×C127—41no—
54C22×C127AGL(1,ℤ/8)11yescharacteristic
55C4×C127—41no—
56C4×C127—41no—
57C4×C127C4×D411yescharacteristic
58C4×C127AGL(1,ℤ/8)11yescharacteristic
59Dic127—42no—
60Dic127—81no—
61C2×C127—21no—
62C2×C127—81no—
63C2×C127#44211yescharacteristic
64#2180—1271nomaximal, Sylow
65C127#218011yescharacteristic, complemented, Sylow
66#510—1271no—
67#444D12711yescharacteristic
68#441—1271no—
69#365—1271no—
70#362—1271no—
71#360—1271no—
72#347—1271no—
73#135C2×D12711yescharacteristic
74C2×Q16—1271no—
75#131—21no—
76C4×Q8—2541no—
77Q32—1272no—
78SD32—1272no—
79M32—1271no—
80M32—1271no—
81#108C2×D12711yescharacteristic
82#106—1271no—
83#104—21no—
84#103—2541no—
85#97C2×D12711yescharacteristic
86#41—21no—
87C2×Q8—2541no—
88Q16—1272no—
89Q16—2541no—
90Q16C4×D12711yescharacteristic
91SD16—21no—
92D8C4×D12711yescharacteristic
93M16—21no—
94M16—41no—
95C2×C8—41no—
96C2×C8C22⋊D12711yescharacteristic
97C2×C8C22×D12711yescharacteristic
98C4⋊C4—5081no—
99C4⋊C4—2541no—
100C42—5081no—
101C42C22⋊D12711yescharacteristic
102C16—1272no—
103C16—2541no—
104Q8—21no—
105Q8—5081no—
106Q8—2542no—
107D4—21no—
108D4—41no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—5081no—
112C2×C4—5081no—
113C2×C4C2×C22⋊D12711yescharacteristic
114C8—21no—
115C8—41no—
116C8C2×C4×D12711yescharacteristic
117C8#1452324811yescharacteristic
118C22—41no—
119C22#1686123411yescharacteristic
120C4—41no—
121C4—41no—
122C4—10161no—
123C4—5082no—
124C4C4×C22⋊D12711yescharacteristic
125C4#1686123511yescharacteristic
126C2—21no—
127C2—81no—
128C2#2254998611yescharacteristic
subgroupG/H
centerC2#22549986
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#444×C127C2
socleC2×C127#442
prime radical#135×C127C22
cosocleC2×C8×C127C23
hypercenter#444D127
hypocenterC127#2180
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #22549984C2semidirect
3. #22549700C2semidirect
4. #22549691C2semidirect
6. #22549654C2semidirect
10. #16861155C4semidirect
11. #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
#225499842
#225497002
#225496912
#225496862
#225496542
#444×C1272
#2180127