Finite Group Databasebeta

← #31795911

Subgroups — #31795911

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#22577583C211yesmaximal, characteristic
4#17984149—21no—
5#17984143—21no—
6#17984133C411yescharacteristic
7#17984121—21no—
8#17984075C2211yescharacteristic
9#17984014—21no—
10#100×C137C411yescharacteristic
11#14555519—41no—
12C2×#11764047—41no—
13#14555510—21no—
14#14555509—41no—
15C2×Dic274D411yescharacteristic
16C2×C22⋊D137—41no—
17D4×D137—41no—
18Dic548—41no—
19#14555499—41no—
20C22⋊Dic137—41no—
21C4×Dic137D411yescharacteristic
22C2×D4×C137C2×C411yescharacteristic
23M16×C137—21no—
24#11764047—42no—
25#11764047—81no—
26C22×D137—41no—
27Dic274—21no—
28Dic274—41no—
29D548—21no—
30C22⋊D137—42no—
31C22⋊D137—41no—
32C2×Dic137—21no—
33C2×Dic137—21no—
34C2×Dic137—41no—
35C4×D137—41no—
36C23×C137—21no—
37D4×C137—41no—
38C2×C4×C137C22⋊C411yescharacteristic
39C8×C137—41no—
40C22×C137—41no—
41C22×C137—81no—
42C22×C137#9911yescharacteristic
43C2×D137—41no—
44C2×D137—161no—
45C4×C137—21no—
46Dic137—41no—
47Dic137—41no—
48Dic137—41no—
49C2×C137—21no—
50C2×C137—81no—
51C2×C137AΣL(1,𝔽16)11yescharacteristic
52D137—81no—
53C137#140311yescharacteristic, complemented, Sylow
54#1403—1371nomaximal, Sylow
55#455—1371no—
56#355—1371no—
57#329—1371no—
58H(5,𝔽2)—1371no—
59#137—2741no—
60#124—1371no—
61#104—2741no—
62#101—2741no—
63#100#1058773911yescharacteristic
64#97—2741no—
65#41—2741no—
66#41—5481no—
67C2×Q8—1371no—
68C2×D4—5481no—
69C2×D4—5481no—
70C2×D4C2×#1058773911yescharacteristic
71Q16—5481no—
72SD16—5481no—
73M16—21no—
74M16—5481no—
75C2×C8—5481no—
76C22⋊C4—5481no—
77C42—1371no—
78C23—21no—
79C23—5481no—
80Q8—2741no—
81Q8—5481no—
82D4—41no—
83D4—2741no—
84D4—5482no—
85D4—5481no—
86C2×C4—2741no—
87C2×C4—2741no—
88C2×C4—5481no—
89C2×C4—5481no—
90C2×C4C22⋊#1058773911yescharacteristic
91C8—41no—
92C8—5482no—
93C8—10961no—
94C22—41no—
95C22—81no—
96C22—5481no—
97C22—21921no—
98C22#1798414711yescharacteristic
99C4—21no—
100C4—5481no—
101C4—5481no—
102C4—5481no—
103C2—21no—
104C2—81no—
105C2—10961no—
106C2#2257830411yescharacteristic
subgroupG/H
centerC2#22578304
commutatorC2×D4×C137C2×C4
FrattiniC2×D4C2×#10587739
Fitting#100×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#17984133C4
cosocle#17984075C22
hypercenter#100#10587739
hypocenterC137#1403
LCSG ▷ C2×D4×C137 ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ #100

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
#225775832
#1403137