Finite Group Databasebeta

← #31795912

Subgroups — #31795912

Order 17536.

112 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 20 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#22578309C211yesmaximal, characteristic, complemented
2#22578259C211yesmaximal, characteristic, complemented
3#22577576C211yesmaximal, characteristic
4#17984147—21no—
5#17984143—21no—
6#17984113—21no—
7#17984077C2211yescharacteristic
8#17984015—21no—
9#17984007C411yescharacteristic, complemented
102−1+4×C137C411yescharacteristic, complemented
11C2×#11764047—41no—
12C22⋊#10587739—41no—
13C2×D548D411yescharacteristic
14#14555497—41no—
15#14555495—41no—
16#14555494—21no—
17#14555487—41no—
18C4×Dic137D411yescharacteristic
19#41×C137—21no—
20#41×C137—41no—
21#41×C137—41no—
22C2×Q8×C137—41no—
23C2×Q8×C137C2×C411yescharacteristic
24C2×#10587739—81no—
25#11764047—42no—
26C22×D137—21no—
27D548—41no—
28C2×Dic137—21no—
29#11764040—41no—
30Q8×C137—21no—
31Q8×C137—41no—
32Q8×C137—41no—
33D4×C137—21no—
34D4×C137—41no—
35D4×C137—41no—
36C2×C4×C137—21no—
37C2×C4×C137—41no—
38C2×C4×C137—42no—
39C2×C4×C137C22⋊C411yescharacteristic
40C22×C137—41no—
41C22×C137#9911yescharacteristic
42C2×D137—41no—
43C2×D137—81no—
44#10587739—161no—
45C4×C137—21no—
46C4×C137—41no—
47C4×C137—41no—
48Dic137—41no—
49C2×C137—21no—
50C2×C137—81no—
51C2×C137AΣL(1,𝔽16)11yescharacteristic
52D137—81no—
53C137#140411yescharacteristic, complemented, Sylow
54#1404—1371nomaximal, Sylow
55#454—1371no—
56#354—1371no—
57#329—1371no—
582−1+4#1058773911yescharacteristic, complemented
59AGL(1,ℤ/8)—2741no—
60#124—1371no—
61#104—2741no—
62#101—1371no—
63#99—2741no—
64#97—2741no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×#1058773911yescharacteristic
70C2×D4—1371no—
71SD16—5481no—
72D8—5481no—
73M16—2741no—
74C2×C8—5481no—
75C22⋊C4—5481no—
76C22⋊C4—5481no—
77C42—1371no—
78C23—2741no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—5481no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—2741no—
90C2×C4—10961no—
91C2×C4C22⋊#1058773911yescharacteristic
92C8—5481no—
93C8—5482no—
94C22—41no—
95C22—5481no—
96C22—10961no—
97C22#1798414611yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—5481no—
102C4—21921no—
103C2—21no—
104C2—81no—
105C2—10961no—
106C2#2257830311yescharacteristic
subgroupG/H
centerC2#22578303
commutatorC2×Q8×C137C2×C4
FrattiniC2×Q8C2×#10587739
Fitting2−1+4×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#22577576C2
cosocle#17984077C22
hypercenter2−1+4#10587739
hypocenterC137#1404
LCSG ▷ C2×Q8×C137 ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #22578309C2semidirect
2. #22578259C2semidirect
9. #17984007C4semidirect
10. 2−1+4×C137C4semidirect
53. C137#1404semidirect
58. 2−1+4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783092
#225782592
#225775762
#1404137