Finite Group Databasebeta

← #31795913

Subgroups — #31795913

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#22577578C211yesmaximal, characteristic
4#17984147—21no—
5#17984143—21no—
6#17984136C411yescharacteristic, complemented
7#17984116—21no—
8#17984077C2211yescharacteristic
9#17984014—21no—
10#101×C137C411yescharacteristic, complemented
11C2×#11764047—41no—
12C22⋊#10587739—41no—
13#14555511—41no—
14#14555510—21no—
15#14555510—41no—
16Q8×D137—41no—
17C2×D548D411yescharacteristic
18#14555499—41no—
19D1096—41no—
20#14555487—41no—
21C4×Dic137D411yescharacteristic
22C2×Q8×C137C2×C411yescharacteristic
23M16×C137—21no—
24C2×#10587739—81no—
25#11764047—42no—
26C22×D137—21no—
27Dic274—21no—
28Dic274—41no—
29D548—21no—
30D548—41no—
31D548—41no—
32C22⋊D137—41no—
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—
42C2×D137—41no—
43C2×D137—81no—
44#10587739—161no—
45C4×C137—21no—
46C4×C137—41no—
47Dic137—41no—
48Dic137—41no—
49C2×C137—21no—
50C2×C137AΣL(1,𝔽16)11yescharacteristic
51D137—81no—
52D137—81no—
53C137#140411yescharacteristic, complemented, Sylow
54#1404—1371nomaximal, Sylow
55#454—1371no—
56#354—1371no—
57#329—1371no—
582−1+4—1371no—
59AGL(1,ℤ/8)—2741no—
60#124—1371no—
61#104—2741no—
62#101#1058773911yescharacteristic, complemented
63#99—2741no—
64#97—2741no—
65#41—2741no—
66#41—5481no—
67#41—5481no—
68C2×Q8—5481no—
69C2×Q8C2×#1058773911yescharacteristic
70C2×D4—1371no—
71SD16—5481no—
72D8—5481no—
73M16—21no—
74C2×C8—5481no—
75C22⋊C4—5481no—
76C22⋊C4—5481no—
77C42—1371no—
78C23—2741no—
79Q8—41no—
80Q8—2741no—
81Q8—5481no—
82D4—2741no—
83D4—5481no—
84D4—5481no—
85D4—5481no—
86C2×C4—21no—
87C2×C4—2741no—
88C2×C4—5481no—
89C2×C4—5482no—
90C2×C4—10961no—
91C2×C4C22⋊#1058773911yescharacteristic
92C8—41no—
93C8—5482no—
94C22—5481no—
95C22—5481no—
96C22—10961no—
97C22#1798414711yescharacteristic
98C4—21no—
99C4—41no—
100C4—5481no—
101C4—5481no—
102C4—21921no—
103C2—21no—
104C2—10961no—
105C2—10961no—
106C2#2257830411yescharacteristic
subgroupG/H
centerC2#22578304
commutatorC2×Q8×C137C2×C4
FrattiniC2×Q8C2×#10587739
Fitting#101×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#22577578C2
cosocle#17984077C22
hypercenter#101#10587739
hypocenterC137#1404
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. #22578309C2semidirect
2. #22578259C2semidirect
6. #17984136C4semidirect
10. #101×C137C4semidirect
53. C137#1404semidirect
62. #101#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783092
#225782592
#225775782
#1404137