Finite Group Databasebeta

← #14543908

Subgroups — #14543908

Order 2176.

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#11763715C211yesmaximal, characteristic, complemented
2#11763665C211yesmaximal, characteristic, complemented
3#11762984C211yesmaximal, characteristic
4#10587694C411yescharacteristic, complemented
5#10587633C2211yescharacteristic
6#10587594—21no—
7#10587542—21no—
8#10587539—21no—
9C17×#101C411yescharacteristic, complemented
10#10587496—21no—
11C2×Q8×C17C2×C411yescharacteristic
12#63256—41no—
13Q8×D17—41no—
14#63252—21no—
15#63252—41no—
16C2×D68D411yescharacteristic
17C22⋊#595—41no—
18C2×#3624—41no—
19M16×C17—21no—
20#63227—41no—
21C4×Dic17D411yescharacteristic
22D136—41no—
23#63219—41no—
24C22×D17—21no—
25C2×#595—81no—
26Q8×C17—41no—
27C2×C4×C17—21no—
28C2×C4×C17C22⋊C411yescharacteristic
29C22⋊D17—41no—
30C2×Dic17—21no—
31D68—21no—
32D68—41no—
33D68—41no—
34C4×D17—41no—
35C4×D17—42no—
36Dic34—21no—
37Dic34—41no—
38#3624—42no—
39C8×C17—41no—
40#1404—171nomaximal, Sylow
41C22×C17#9911yescharacteristic
42C2×D17—41no—
43C2×D17—41no—
44C2×D17—81no—
45#595—161no—
46C4×C17—21no—
47C4×C17—41no—
48Dic17—41no—
49Dic17—41no—
50#454—171no—
51#354—171no—
52#329—171no—
53C2×C17—21no—
54C2×C17AΣL(1,𝔽16)11yescharacteristic
55D17—81no—
56D17—81no—
572−1+4—171no—
58AGL(1,ℤ/8)—341no—
59#124—171no—
60#104—341no—
61#101#59511yescharacteristic, complemented
62#99—341no—
63#97—341no—
64C17#140411yescharacteristic, complemented, Sylow
65#41—341no—
66#41—681no—
67#41—681no—
68C2×Q8—681no—
69C2×Q8C2×#59511yescharacteristic
70C2×D4—171no—
71SD16—681no—
72D8—681no—
73M16—21no—
74C2×C8—681no—
75C22⋊C4—681no—
76C22⋊C4—681no—
77C42—171no—
78C23—341no—
79Q8—41no—
80Q8—341no—
81Q8—681no—
82D4—341no—
83D4—681no—
84D4—681no—
85D4—681no—
86C2×C4—21no—
87C2×C4—341no—
88C2×C4—681no—
89C2×C4—682no—
90C2×C4—1361no—
91C2×C4C22⋊#59511yescharacteristic
92C8—41no—
93C8—682no—
94C22—681no—
95C22—681no—
96C22—1361no—
97C22#1058753911yescharacteristic
98C4—21no—
99C4—41no—
100C4—2721no—
101C4—681no—
102C4—681no—
103C2—21no—
104C2—1361no—
105C2—1361no—
106C2#1176371011yescharacteristic
subgroupG/H
centerC2#11763710
commutatorC2×Q8×C17C2×C4
FrattiniC2×Q8C2×#595
FittingC17×#101C4
socleC2×C17AΣL(1,𝔽16)
prime radical#11762984C2
cosocle#10587633C22
hypercenter#101#595
hypocenterC17#1404
LCSG ▷ C2×Q8×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #11763715C2semidirect
2. #11763665C2semidirect
4. #10587694C4semidirect
9. C17×#101C4semidirect
61. #101#595semidirect
64. C17#1404semidirect

Maximal subgroups

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

subgroupindex
#117637152
#117636652
#117629842
#140417