Finite Group Databasebeta

← #31795909

Subgroups — #31795909

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#22577573C211yesmaximal, characteristic
4#17984155—21no—
5#17984147—21no—
6#17984133C411yescharacteristic, complemented
7#17984116—21no—
8#17984084C2211yescharacteristic
9#17984014—21no—
10#100×C137C411yescharacteristic, complemented
11C2×#11764047—41no—
12C22⋊#10587739—41no—
13#14555510—21no—
14#14555509—41no—
15C2×D548D411yescharacteristic
16C2×C22⋊D137—41no—
17C2×C22⋊D137—41no—
18D4×D137—41no—
19#14555499—41no—
20D1096—41no—
21C4×Dic137D411yescharacteristic
22C2×D4×C137C2×C411yescharacteristic
23M16×C137—21no—
24C2×#10587739—81no—
25#11764047—81no—
26C22×D137—21no—
27C22×D137—41no—
28Dic274—21no—
29D548—21no—
30D548—41no—
31C22⋊D137—42no—
32C22⋊D137—41no—
33C22⋊D137—81no—
34C22⋊D137—81no—
35C2×Dic137—21no—
36C2×Dic137—41no—
37C4×D137—41no—
38C23×C137—21no—
39D4×C137—41no—
40C2×C4×C137C22⋊C411yescharacteristic
41C8×C137—41no—
42C22×C137—41no—
43C22×C137—81no—
44C22×C137#9911yescharacteristic
45C2×D137—41no—
46C2×D137—41no—
47C2×D137—81no—
48C2×D137—161no—
49#10587739—161no—
50C4×C137—21no—
51Dic137—21no—
52Dic137—21no—
53Dic137—41no—
54C2×C137—21no—
55C2×C137—81no—
56C2×C137AΣL(1,𝔽16)11yescharacteristic
57D137—81no—
58D137—81no—
59C137#140211yescharacteristic, complemented, Sylow
60#1402—1371nomaximal, Sylow
61#453—1371no—
62#353—1371no—
63#331—1371no—
64H(5,𝔽2)—1371no—
65AGL(1,ℤ/8)—2741no—
66#127—1371no—
67C4⋊C8—2741no—
68#104—2741no—
69#100#1058773911yescharacteristic, complemented
70#99—2741no—
71#41—2741no—
72#41—5481no—
73C2×D4—5481no—
74C2×D4—5481no—
75C2×D4—5481no—
76C2×D4—1371no—
77C2×D4C2×#1058773911yescharacteristic
78SD16—5481no—
79D8—5481no—
80M16—21no—
81C2×C8—5481no—
82C22⋊C4—5481no—
83C42—1371no—
84C23—21no—
85C23—2741no—
86C23—5481no—
87Q8—2741no—
88D4—41no—
89D4—2741no—
90D4—5481no—
91D4—5482no—
92D4—5481no—
93D4—10961no—
94D4—10961no—
95C2×C4—2741no—
96C2×C4—5481no—
97C2×C4—5481no—
98C2×C4—10961no—
99C2×C4C22⋊#1058773911yescharacteristic
100C8—41no—
101C8—10961no—
102C22—41no—
103C22—81no—
104C22—5481no—
105C22—5481no—
106C22—10961no—
107C22—21921no—
108C22#1798414711yescharacteristic
109C4—21no—
110C4—2741no—
111C4—2741no—
112C4—5481no—
113C4—21921no—
114C2—21no—
115C2—81no—
116C2—10961no—
117C2—10961no—
118C2#2257830411yescharacteristic
subgroupG/H
centerC2#22578304
commutatorC2×D4×C137C2×C4
FrattiniC2×D4C2×#10587739
Fitting#100×C137C4
socleC2×C137AΣL(1,𝔽16)
prime radical#22577573C2
cosocle#17984084C22
hypercenter#100#10587739
hypocenterC137#1402
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. #22578307C2semidirect
2. #22578261C2semidirect
6. #17984133C4semidirect
10. #100×C137C4semidirect
59. C137#1402semidirect
69. #100#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783072
#225782612
#225775732
#1402137