Finite Group Databasebeta

← #14543905

Subgroups — #14543905

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#11763717C211yesmaximal, characteristic, complemented
2#11763664C211yesmaximal, characteristic, complemented
3#11762987C211yesmaximal, characteristic
4C17×H(5,𝔽2)C411yescharacteristic
5#10587637—21no—
6#10587622C2211yescharacteristic
7#10587545—21no—
8#10587542—21no—
9#10587508—21no—
10#10587504C411yescharacteristic
11#41×C17—21no—
12#41×C17—41no—
13C2×D4×C17—41no—
14C2×D4×C17—41no—
15C2×D4×C17C2×C411yescharacteristic
16C2×Dic34D411yescharacteristic
17#63247—41no—
18C2×#3624—41no—
19C22⋊Dic17—41no—
20#63231—41no—
21#63229—41no—
22C4×Dic17D411yescharacteristic
23#63223—21no—
24C23×C17—21no—
25C23×C17—41no—
26Q8×C17—21no—
27D4×C17—21no—
28D4×C17—41no—
29D4×C17—42no—
30D4×C17—41no—
31C2×C4×C17—41no—
32C2×C4×C17—41no—
33C2×C4×C17C22⋊C411yescharacteristic
34C2×Dic17—21no—
35C2×Dic17—21no—
36Dic34—41no—
37#3624—42no—
38#3624—81no—
39#3622—41no—
40#1403—171nomaximal, Sylow
41C22×C17—41no—
42C22×C17—41no—
43C22×C17—81no—
44C22×C17—161no—
45C22×C17#9911yescharacteristic
46C4×C17—21no—
47C4×C17—41no—
48Dic17—41no—
49Dic17—41no—
50#455—171no—
51#355—171no—
52#329—171no—
53C2×C17—21no—
54C2×C17—81no—
55C2×C17—81no—
56C2×C17AΣL(1,𝔽16)11yescharacteristic
57H(5,𝔽2)#59511yescharacteristic
58#137—341no—
59#124—171no—
60#104—341no—
61#101—341no—
62#100—171no—
63#97—341no—
64C17#140311yescharacteristic, complemented, Sylow
65#41—21no—
66#41—41no—
67C2×Q8—171no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4C2×#59511yescharacteristic
71Q16—681no—
72SD16—681no—
73M16—341no—
74M16—681no—
75C2×C8—681no—
76C22⋊C4—681no—
77C42—171no—
78C23—21no—
79C23—41no—
80Q8—21no—
81Q8—681no—
82D4—21no—
83D4—41no—
84D4—42no—
85D4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—341no—
89C2×C4—341no—
90C2×C4C22⋊#59511yescharacteristic
91C8—681no—
92C8—682no—
93C8—1361no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22—161no—
98C22#1058755111yescharacteristic
99C4—21no—
100C4—41no—
101C4—681no—
102C4—681no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#1176370911yescharacteristic
subgroupG/H
centerC2#11763709
commutatorC2×D4×C17C2×C4
FrattiniC2×D4C2×#595
Fitting, prime radicalC17×H(5,𝔽2)C4
socleC2×C17AΣL(1,𝔽16)
cosocle#10587622C22
hypercenterH(5,𝔽2)#595
hypocenterC17#1403
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. #11763717C2semidirect
2. #11763664C2semidirect
64. C17#1403semidirect

Maximal subgroups

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

subgroupindex
#117637172
#117636642
#117629872
#140317