Finite Group Databasebeta

← #14543907

Subgroups — #14543907

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#11762982C211yesmaximal, characteristic
4C17×2−1+4C411yescharacteristic, complemented
5#10587635—21no—
6#10587633C2211yescharacteristic
7#10587542—21no—
8#10587539—21no—
9#10587508—21no—
10#10587507C411yescharacteristic, complemented
11#41×C17—21no—
12#41×C17—41no—
13#41×C17—41no—
14C2×Q8×C17—41no—
15C2×Q8×C17C2×C411yescharacteristic
16C2×D68D411yescharacteristic
17C22⋊#595—41no—
18C2×#3624—41no—
19#63230—41no—
20#63228—41no—
21#63227—41no—
22C4×Dic17D411yescharacteristic
23#63223—21no—
24C22×D17—21no—
25C2×#595—81no—
26Q8×C17—21no—
27Q8×C17—41no—
28Q8×C17—41no—
29D4×C17—21no—
30D4×C17—41no—
31D4×C17—41no—
32C2×C4×C17—21no—
33C2×C4×C17—41no—
34C2×C4×C17—42no—
35C2×C4×C17C22⋊C411yescharacteristic
36C2×Dic17—21no—
37D68—41no—
38#3624—42no—
39#3622—41no—
40#1404—171nomaximal, Sylow
41C22×C17—41no—
42C22×C17#9911yescharacteristic
43C2×D17—41no—
44C2×D17—81no—
45#595—161no—
46C4×C17—21no—
47C4×C17—41no—
48C4×C17—41no—
49Dic17—41no—
50#454—171no—
51#354—171no—
52#329—171no—
53C2×C17—21no—
54C2×C17—81no—
55C2×C17AΣL(1,𝔽16)11yescharacteristic
56D17—81no—
572−1+4#59511yescharacteristic, complemented
58AGL(1,ℤ/8)—341no—
59#124—171no—
60#104—341no—
61#101—171no—
62#99—341no—
63#97—341no—
64C17#140411yescharacteristic, complemented, Sylow
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×#59511yescharacteristic
70C2×D4—171no—
71SD16—681no—
72D8—681no—
73M16—341no—
74C2×C8—681no—
75C22⋊C4—681no—
76C22⋊C4—681no—
77C42—171no—
78C23—341no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—681no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—341no—
90C2×C4—1361no—
91C2×C4C22⋊#59511yescharacteristic
92C8—681no—
93C8—682no—
94C22—41no—
95C22—681no—
96C22—1361no—
97C22#1058755111yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—2721no—
102C4—681no—
103C2—21no—
104C2—81no—
105C2—1361no—
106C2#1176370911yescharacteristic
subgroupG/H
centerC2#11763709
commutatorC2×Q8×C17C2×C4
FrattiniC2×Q8C2×#595
FittingC17×2−1+4C4
socleC2×C17AΣL(1,𝔽16)
prime radical#11762982C2
cosocle#10587633C22
hypercenter2−1+4#595
hypocenterC17#1404
LCSG ▷ C2×Q8×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #11763715C2semidirect
2. #11763665C2semidirect
4. C17×2−1+4C4semidirect
10. #10587507C4semidirect
57. 2−1+4#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
#117629822
#140417