Finite Group Databasebeta

← #14543903

Subgroups — #14543903

Order 2176.

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#11763713C211yesmaximal, characteristic, complemented
2#11763667C211yesmaximal, characteristic, complemented
3#11762977C211yesmaximal, characteristic
4C17×H(5,𝔽2)C411yescharacteristic, complemented
5#10587635—21no—
6#10587626C2211yescharacteristic
7#10587544—21no—
8#10587539—21no—
9#10587508—21no—
10#10587504C411yescharacteristic, complemented
11#41×C17—21no—
12#41×C17—41no—
13C2×D4×C17—41no—
14C2×D4×C17—41no—
15C2×D4×C17C2×C411yescharacteristic
16C2×C22⋊D17—41no—
17C2×D68D411yescharacteristic
18C22⋊#595—41no—
19C2×#3624—41no—
20#63230—41no—
21#63228—41no—
22C4×Dic17D411yescharacteristic
23#63223—21no—
24C23×C17—21no—
25C23×C17—41no—
26C22×D17—21no—
27C2×#595—81no—
28Q8×C17—21no—
29D4×C17—21no—
30D4×C17—41no—
31D4×C17—42no—
32D4×C17—41no—
33C2×C4×C17—41no—
34C2×C4×C17—41no—
35C2×C4×C17C22⋊C411yescharacteristic
36C22⋊D17—81no—
37C22⋊D17—81no—
38C2×Dic17—21no—
39D68—41no—
40#3624—81no—
41#3622—41no—
42#1402—171nomaximal, Sylow
43C22×C17—41no—
44C22×C17—41no—
45C22×C17—81no—
46C22×C17—161no—
47C22×C17#9911yescharacteristic
48C2×D17—41no—
49C2×D17—81no—
50#595—161no—
51C4×C17—21no—
52C4×C17—41no—
53Dic17—21no—
54Dic17—21no—
55#453—171no—
56#353—171no—
57#331—171no—
58C2×C17—21no—
59C2×C17—81no—
60C2×C17—81no—
61C2×C17AΣL(1,𝔽16)11yescharacteristic
62D17—81no—
63H(5,𝔽2)#59511yescharacteristic, complemented
64AGL(1,ℤ/8)—341no—
65#127—171no—
66C4⋊C8—341no—
67#104—341no—
68#100—171no—
69#99—341no—
70C17#140211yescharacteristic, complemented, Sylow
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—171no—
76C2×D4—681no—
77C2×D4C2×#59511yescharacteristic
78SD16—681no—
79D8—681no—
80M16—341no—
81C2×C8—681no—
82C22⋊C4—681no—
83C42—171no—
84C23—21no—
85C23—41no—
86C23—341no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—681no—
93D4—1361no—
94D4—1361no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—341no—
98C2×C4—1361no—
99C2×C4C22⋊#59511yescharacteristic
100C8—681no—
101C8—1361no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—161no—
106C22—681no—
107C22—1361no—
108C22#1058755111yescharacteristic
109C4—21no—
110C4—41no—
111C4—2721no—
112C4—341no—
113C4—341no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—1361no—
118C2#1176370911yescharacteristic
subgroupG/H
centerC2#11763709
commutatorC2×D4×C17C2×C4
FrattiniC2×D4C2×#595
FittingC17×H(5,𝔽2)C4
socleC2×C17AΣL(1,𝔽16)
prime radical#11762977C2
cosocle#10587626C22
hypercenterH(5,𝔽2)#595
hypocenterC17#1402
LCSG ▷ C2×D4×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
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. #11763713C2semidirect
2. #11763667C2semidirect
4. C17×H(5,𝔽2)C4semidirect
10. #10587504C4semidirect
63. H(5,𝔽2)#595semidirect
70. C17#1402semidirect

Maximal subgroups

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

subgroupindex
#117637132
#117636672
#117629772
#140217