Finite Group Databasebeta

← #31782687

Subgroups — #31782687

Order 17536.

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

normal subgroups only

This group has 154 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#22577838C211yesmaximal, characteristic
2#22577516C211yesmaximal, characteristic, complemented
3#22577227C211yesmaximal, characteristic, complemented
4#22577215C211yesmaximal, characteristic, complemented
5#22577213C211yesmaximal, characteristic, complemented
6#22577181C211yesmaximal, characteristic, complemented
7#444×C137C211yesmaximal, characteristic, complemented
8C2×D1096C2211yescharacteristic
9C4×D548—21no—
10#17984031C412yescomplemented
11#17984029C412yescomplemented
12#17984028C2211yescharacteristic
13#17984028C2211yescharacteristic
14#17984020C2211yescharacteristic
15#17984010—21no—
16#135×C137C2211yescharacteristic
17#131×C137—21no—
18#108×C137C2211yescharacteristic
19#104×C137—21no—
20#97×C137C2211yescharacteristic
21C2×D548—21no—
22C2×C4×D137—41no—
23D1096—21no—
24D1096C2×C412yes—
25C4⋊Dic137—21no—
26#14555487—41no—
27#14555485—21no—
28#14555485C2×C412yescomplemented
29#41×C137—21no—
30Q16×C137C2×C411yescharacteristic, complemented
31SD16×C137—21no—
32D8×C137C2×C411yescharacteristic, complemented
33M16×C137—21no—
34M16×C137—41no—
35C2×C8×C137—41no—
36C2×C8×C137D411yescharacteristic
37C2×C8×C137C2311yescharacteristic
38C42×C137D411yescharacteristic
39C22×D137—41no—
40D548—22no—
41D548—41no—
42C2×Dic137—41no—
43C4×D137—82no—
44Q8×C137—21no—
45D4×C137—21no—
46D4×C137—41no—
47C2×C4×C137—21no—
48C2×C4×C137—41no—
49C2×C4×C137C2×D411yescharacteristic
50C8×C137—21no—
51C8×C137—41no—
52C8×C137C22×C411yescharacteristic
53C8×C137#4111yescharacteristic
54C22×C137—41no—
55C22×C137AGL(1,ℤ/8)11yescharacteristic
56C2×D137—42no—
57C2×D137—81no—
58C2×D137—81no—
59C4×C137—41no—
60C4×C137—41no—
61C4×C137C4×D411yescharacteristic
62C4×C137AGL(1,ℤ/8)11yescharacteristic
63Dic137—81no—
64C2×C137—21no—
65C2×C137—81no—
66C2×C137#44211yescharacteristic
67D137—82no—
68C137AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
69AGL(1,ℤ/16)—1371nomaximal, Sylow
70#509—1371no—
71#444D13711yescharacteristic, complemented
72#442—1371no—
73#365—1371no—
74#361—1371no—
75#360—1371no—
76#347—1371no—
77#135C2×D13711yescharacteristic
78C2×D8—1371no—
79#131—21no—
80C4×D4—2741no—
81SD32—1372no—
82D16—1372no—
83M32—1371no—
84M32—1371no—
85#108C2×D13711yescharacteristic
86#106—1371no—
87#104—21no—
88#102—2741no—
89#97C2×D13711yescharacteristic
90#41—21no—
91C2×D4—2741no—
92C22×C4—5481no—
93Q16C4×D13711yescharacteristic, complemented
94SD16—21no—
95D8—2741no—
96D8—1372no—
97D8C4×D13711yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D13711yescharacteristic
102C2×C8C22×D13711yescharacteristic
103C4⋊C4—2741no—
104C22⋊C4—5481no—
105C42C22⋊D13711yescharacteristic
106C16—2741no—
107C16—1372no—
108C23—5481no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—2742no—
113D4—5481no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—5481no—
117C2×C4—10962no—
118C2×C4C2×C22⋊D13711yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D13711yescharacteristic
122C8#1455551011yescharacteristic
123C22—41no—
124C22—5482no—
125C22—10961no—
126C22—10961no—
127C22#1798411311yescharacteristic
128C4—41no—
129C4—41no—
130C4—10961no—
131C4C4×C22⋊D13711yescharacteristic
132C4#1798411411yescharacteristic
133C2—21no—
134C2—81no—
135C2—10962no—
136C2#2257751311yescharacteristic
subgroupG/H
centerC2#22577513
commutatorC8×C137C22×C4
FrattiniC2×C8C22×D137
Fitting#444×C137C2
socleC2×C137#442
prime radical#22577838C2
cosocleC2×C8×C137C23
hypercenter#444D137
hypocenterC137AGL(1,ℤ/16)
LCSG ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #22577516C2semidirect
3. #22577227C2semidirect
4. #22577215C2semidirect
5. #22577213C2semidirect
6. #22577181C2semidirect
7. #444×C137C2semidirect
10. #17984031C4semidirect
11. #17984029C4semidirect
28. #14555485C2×C4semidirect
30. Q16×C137C2×C4semidirect
32. D8×C137C2×C4semidirect
68. C137AGL(1,ℤ/16)semidirect
71. #444D137semidirect
93. Q16C4×D137semidirect
97. D8C4×D137semidirect

Maximal subgroups

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

subgroupindex
#225778382
#225775162
#225772272
#225772152
#225772132
#225771812
#444×C1372
AGL(1,ℤ/16)137