Finite Group Databasebeta

← #30640619

Subgroups — #30640619

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#22550317C211yesmaximal, characteristic
2#22549999C211yesmaximal, characteristic
3#22549698C211yesmaximal, characteristic, complemented
4#22549692C211yesmaximal, characteristic, complemented
5#22549685C211yesmaximal, characteristic
6#22549654C211yesmaximal, characteristic, complemented
7#441×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9#16861215—21no—
10#16861155C412yescomplemented
11#16861152C412yescomplemented
12#16861149C2211yescharacteristic
13#16861149C2211yescharacteristic
14#16861145C2211yescharacteristic
15#16861137—21no—
16C2×Q16×C127C2211yescharacteristic
17C4×Q8×C127—21no—
18#106×C127C2211yescharacteristic
19#103×C127—21no—
20#97×C127C2211yescharacteristic
21#14523248—21no—
22Dic508C2×C411yescharacteristic
23#14523237—21no—
24D1016C2×C411yescharacteristic
25#14523232—21no—
26#14523231—41no—
27C8×D127—41no—
28C127⋊C16—21no—
29C127⋊C16C2×C412yes—
30C2×Q8×C127—21no—
31Q16×C127—21no—
32Q16×C127C2×C412yes—
33C2×C8×C127D411yescharacteristic
34C2×C8×C127C2311yescharacteristic
35C4⋊C4×C127—21no—
36C4⋊C4×C127—41no—
37C42×C127—41no—
38C42×C127D411yescharacteristic
39Dic254—21no—
40D508—21no—
41C22⋊D127—41no—
42C4×D127—41no—
43C127⋊C8—41no—
44Q8×C127—22no—
45Q8×C127—41no—
46C2×C4×C127—21no—
47C2×C4×C127—41no—
48C2×C4×C127—41no—
49C2×C4×C127C2×D411yescharacteristic
50C8×C127—21no—
51C8×C127C22×C411yescharacteristic
52C8×C127#4111yescharacteristic
53C22×C127AGL(1,ℤ/8)11yescharacteristic
54C2×D127—41no—
55C4×C127—41no—
56C4×C127—42no—
57C4×C127—81no—
58C4×C127C4×D411yescharacteristic
59C4×C127AGL(1,ℤ/8)11yescharacteristic
60Dic127—41no—
61C2×C127—21no—
62C2×C127#44211yescharacteristic
63D127—81no—
64#2180—1271nomaximal, Sylow
65C127#218011yescharacteristic, complemented, Sylow
66#510—1271no—
67#444—1271no—
68#441D12711yescharacteristic, complemented
69#365—1271no—
70#362—1271no—
71#360—1271no—
72#347—1271no—
73#135—1271no—
74C2×Q16C2×D12711yescharacteristic
75#131—2541no—
76C4×Q8—21no—
77Q32—1272no—
78SD32—1272no—
79M32—1271no—
80M32—1271no—
81#108—1271no—
82#106C2×D12711yescharacteristic
83#104—2541no—
84#103—21no—
85#97C2×D12711yescharacteristic
86#41—2541no—
87C2×Q8—21no—
88Q16—21no—
89Q16—1271no—
90Q16C4×D12712yes—
91SD16—2541no—
92D8—1271no—
93M16—5081no—
94M16—2541no—
95C2×C8—5081no—
96C2×C8C22⋊D12711yescharacteristic
97C2×C8C22×D12711yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—41no—
100C42—41no—
101C42C22⋊D12711yescharacteristic
102C16—1272no—
103C16—2541no—
104Q8—22no—
105Q8—41no—
106Q8—2541no—
107D4—5081no—
108D4—2541no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—5081no—
113C2×C4C2×C22⋊D12711yescharacteristic
114C8—21no—
115C8—5081no—
116C8C2×C4×D12711yescharacteristic
117C8#1452324811yescharacteristic
118C22—5081no—
119C22#1686123511yescharacteristic
120C4—41no—
121C4—42no—
122C4—81no—
123C4—5081no—
124C4C4×C22⋊D12711yescharacteristic
125C4#1686123411yescharacteristic
126C2—21no—
127C2—10161no—
128C2#2254998611yescharacteristic
subgroupG/H
centerC2#22549986
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#441×C127C2
socleC2×C127#442
prime radical#16861231C22
cosocleC2×C8×C127C23
hypercenter#441D127
hypocenterC127#2180
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #22549698C2semidirect
4. #22549692C2semidirect
6. #22549654C2semidirect
7. #441×C127C2semidirect
10. #16861155C4semidirect
11. #16861152C4semidirect
65. C127#2180semidirect
68. #441D127semidirect

Maximal subgroups

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

subgroupindex
#225503172
#225499992
#225496982
#225496922
#225496852
#225496542
#441×C1272
#2180127