Finite Group Databasebeta

← #31795908

Subgroups — #31795908

Order 17536.

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

normal subgroups only

This group has 122 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#22578307C211yesmaximal, characteristic, complemented
2#22578261C211yesmaximal, characteristic, complemented
3#22577571C211yesmaximal, characteristic
4#17984155—21no—
5#17984147—21no—
6#17984113—21no—
7#17984084C2211yescharacteristic
8#17984015—21no—
9#17984005C411yescharacteristic, complemented
10H(5,𝔽2)×C137C411yescharacteristic, complemented
11C2×#11764047—41no—
12C22⋊#10587739—41no—
13C2×D548D411yescharacteristic
14C2×C22⋊D137—41no—
15#14555497—41no—
16#14555495—41no—
17#14555494—21no—
18C4×Dic137D411yescharacteristic
19#41×C137—21no—
20#41×C137—41no—
21C2×D4×C137—41no—
22C2×D4×C137—41no—
23C2×D4×C137C2×C411yescharacteristic
24C2×#10587739—81no—
25#11764047—81no—
26C22×D137—21no—
27D548—41no—
28C22⋊D137—81no—
29C22⋊D137—81no—
30C2×Dic137—21no—
31#11764040—41no—
32C23×C137—21no—
33C23×C137—41no—
34Q8×C137—21no—
35D4×C137—21no—
36D4×C137—41no—
37D4×C137—42no—
38D4×C137—41no—
39C2×C4×C137—41no—
40C2×C4×C137—41no—
41C2×C4×C137C22⋊C411yescharacteristic
42C22×C137—41no—
43C22×C137—41no—
44C22×C137—81no—
45C22×C137—161no—
46C22×C137#9911yescharacteristic
47C2×D137—41no—
48C2×D137—81no—
49#10587739—161no—
50C4×C137—21no—
51C4×C137—41no—
52Dic137—21no—
53Dic137—21no—
54C2×C137—21no—
55C2×C137—81no—
56C2×C137—81no—
57C2×C137AΣL(1,𝔽16)11yescharacteristic
58D137—81no—
59C137#140211yescharacteristic, complemented, Sylow
60#1402—1371nomaximal, Sylow
61#453—1371no—
62#353—1371no—
63#331—1371no—
64H(5,𝔽2)#1058773911yescharacteristic, complemented
65AGL(1,ℤ/8)—2741no—
66#127—1371no—
67C4⋊C8—2741no—
68#104—2741no—
69#100—1371no—
70#99—2741no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—5481no—
76C2×D4—1371no—
77C2×D4C2×#1058773911yescharacteristic
78SD16—5481no—
79D8—5481no—
80M16—2741no—
81C2×C8—5481no—
82C22⋊C4—5481no—
83C42—1371no—
84C23—21no—
85C23—41no—
86C23—2741no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—5481no—
93D4—10961no—
94D4—10961no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—2741no—
98C2×C4—10961no—
99C2×C4C22⋊#1058773911yescharacteristic
100C8—5481no—
101C8—10961no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—161no—
106C22—5481no—
107C22—10961no—
108C22#1798414611yescharacteristic
109C4—21no—
110C4—41no—
111C4—2741no—
112C4—2741no—
113C4—21921no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—10961no—
118C2#2257830311yescharacteristic
subgroupG/H
centerC2#22578303
commutatorC2×D4×C137C2×C4
FrattiniC2×D4C2×#10587739
FittingH(5,𝔽2)×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#22577571C2
cosocle#17984084C22
hypercenterH(5,𝔽2)#10587739
hypocenterC137#1402
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. #22578307C2semidirect
2. #22578261C2semidirect
9. #17984005C4semidirect
10. H(5,𝔽2)×C137C4semidirect
59. C137#1402semidirect
64. H(5,𝔽2)#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783072
#225782612
#225775712
#1402137