Finite Group Databasebeta

← #31795910

Subgroups — #31795910

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#22578311C211yesmaximal, characteristic, complemented
2#22578258C211yesmaximal, characteristic, complemented
3#22577581C211yesmaximal, characteristic
4#17984149—21no—
5#17984143—21no—
6#17984118—21no—
7#17984075C2211yescharacteristic
8#17984015—21no—
9#17984005C411yescharacteristic
10H(5,𝔽2)×C137C411yescharacteristic
11#14555519—41no—
12C2×#11764047—41no—
13C2×Dic274D411yescharacteristic
14#14555500—41no—
15#14555498—41no—
16#14555494—21no—
17C22⋊Dic137—41no—
18C4×Dic137D411yescharacteristic
19#41×C137—21no—
20#41×C137—41no—
21C2×D4×C137—41no—
22C2×D4×C137—41no—
23C2×D4×C137C2×C411yescharacteristic
24#11764047—42no—
25#11764047—81no—
26Dic274—41no—
27C2×Dic137—21no—
28C2×Dic137—21no—
29#11764040—41no—
30C23×C137—21no—
31C23×C137—41no—
32Q8×C137—21no—
33D4×C137—21no—
34D4×C137—41no—
35D4×C137—42no—
36D4×C137—41no—
37C2×C4×C137—41no—
38C2×C4×C137—41no—
39C2×C4×C137C22⋊C411yescharacteristic
40C22×C137—41no—
41C22×C137—41no—
42C22×C137—81no—
43C22×C137—161no—
44C22×C137#9911yescharacteristic
45C4×C137—21no—
46C4×C137—41no—
47Dic137—41no—
48Dic137—41no—
49C2×C137—21no—
50C2×C137—81no—
51C2×C137—81no—
52C2×C137AΣL(1,𝔽16)11yescharacteristic
53C137#140311yescharacteristic, complemented, Sylow
54#1403—1371nomaximal, Sylow
55#455—1371no—
56#355—1371no—
57#329—1371no—
58H(5,𝔽2)#1058773911yescharacteristic
59#137—2741no—
60#124—1371no—
61#104—2741no—
62#101—2741no—
63#100—1371no—
64#97—2741no—
65#41—21no—
66#41—41no—
67C2×Q8—1371no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4C2×#1058773911yescharacteristic
71Q16—5481no—
72SD16—5481no—
73M16—2741no—
74M16—5481no—
75C2×C8—5481no—
76C22⋊C4—5481no—
77C42—1371no—
78C23—21no—
79C23—41no—
80Q8—21no—
81Q8—5481no—
82D4—21no—
83D4—41no—
84D4—42no—
85D4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—2741no—
89C2×C4—2741no—
90C2×C4C22⋊#1058773911yescharacteristic
91C8—5481no—
92C8—5482no—
93C8—10961no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22—161no—
98C22#1798414611yescharacteristic
99C4—21no—
100C4—41no—
101C4—5481no—
102C4—5481no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#2257830311yescharacteristic
subgroupG/H
centerC2#22578303
commutatorC2×D4×C137C2×C4
FrattiniC2×D4C2×#10587739
Fitting, prime radicalH(5,𝔽2)×C137C4
socleC2×C137AΣL(1,𝔽16)
cosocle#17984075C22
hypercenterH(5,𝔽2)#10587739
hypocenterC137#1403
LCSG ▷ C2×D4×C137 ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22578311C2semidirect
2. #22578258C2semidirect
53. C137#1403semidirect

Maximal subgroups

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

subgroupindex
#225783112
#225782582
#225775812
#1403137