Finite Group Databasebeta

← #31795915

Subgroups — #31795915

Order 17536.

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

normal subgroups only

This group has 106 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#22578313C211yesmaximal, characteristic, complemented
2#22578263C211yesmaximal, characteristic, complemented
3#22577588C211yesmaximal, characteristic
4#17984155—21no—
5#17984149—21no—
6#17984136C411yescharacteristic
7#17984121—21no—
8#17984086C2211yescharacteristic
9#17984014—21no—
10#101×C137C411yescharacteristic
11#14555519—41no—
12C2×#11764047—41no—
13#14555511—41no—
14#14555510—21no—
15#14555510—41no—
16C2×Dic274D411yescharacteristic
17Q8×D137—41no—
18Dic548—41no—
19#14555499—41no—
20#14555489—41no—
21C4×Dic137D411yescharacteristic
22C2×Q8×C137C2×C411yescharacteristic
23M16×C137—21no—
24#11764047—81no—
25#11764047—81no—
26Dic274—21no—
27Dic274—41no—
28Dic274—41no—
29D548—21no—
30D548—41no—
31C22⋊D137—41no—
32C2×Dic137—21no—
33C2×Dic137—21no—
34C4×D137—41no—
35C4×D137—42no—
36Q8×C137—41no—
37C2×C4×C137—21no—
38C2×C4×C137C22⋊C411yescharacteristic
39C8×C137—41no—
40C22×C137#9911yescharacteristic
41C2×D137—41no—
42C4×C137—21no—
43C4×C137—41no—
44Dic137—21no—
45Dic137—21no—
46Dic137—41no—
47Dic137—41no—
48C2×C137—21no—
49C2×C137AΣL(1,𝔽16)11yescharacteristic
50D137—81no—
51C137#140511yescharacteristic, complemented, Sylow
52#1405—1371nomaximal, Sylow
53#456—1371no—
54#356—1371no—
55#333—1371no—
562−1+4—1371no—
57#137—2741no—
58C4⋊Q8—1371no—
59C4⋊C8—2741no—
60#104—2741no—
61#101—2741no—
62#101#1058773911yescharacteristic
63#41—2741no—
64#41—5481no—
65#41—5481no—
66C2×Q8—5481no—
67C2×Q8—1371no—
68C2×Q8C2×#1058773911yescharacteristic
69Q16—5481no—
70SD16—5481no—
71M16—21no—
72M16—5481no—
73C2×C8—5481no—
74C4⋊C4—5481no—
75C42—1371no—
76Q8—41no—
77Q8—2741no—
78Q8—5481no—
79Q8—5481no—
80D4—2741no—
81D4—5481no—
82D4—5481no—
83C2×C4—21no—
84C2×C4—2741no—
85C2×C4—2741no—
86C2×C4—5481no—
87C2×C4—5482no—
88C2×C4C22⋊#1058773911yescharacteristic
89C8—41no—
90C8—10961no—
91C8—10961no—
92C22—5481no—
93C22#1798414711yescharacteristic
94C4—21no—
95C4—41no—
96C4—2741no—
97C4—2741no—
98C4—5481no—
99C4—5481no—
100C2—21no—
101C2—10961no—
102C2#2257830411yescharacteristic
subgroupG/H
centerC2#22578304
commutatorC2×Q8×C137C2×C4
FrattiniC2×Q8C2×#10587739
Fitting#101×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#17984136C4
cosocle#17984086C22
hypercenter#101#10587739
hypocenterC137#1405
LCSG ▷ C2×Q8×C137 ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #22578313C2semidirect
2. #22578263C2semidirect
51. C137#1405semidirect

Maximal subgroups

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

subgroupindex
#225783132
#225782632
#225775882
#1405137