Finite Group Databasebeta

← #14528107

Subgroups — #14528107

Order 2176.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#11763179C211yesmaximal, characteristic, complemented
2#11763168C212yesmaximal, complemented
3#11762551C211yesmaximal, characteristic, complemented
4#11762549C212yesmaximal, complemented
5C17×#562C211yesmaximal, characteristic
6C17×#125C2211yescharacteristic
7C17×#124C2212yes—
8C17×#123—22no—
9C17×C22⋊Q8—22no—
10C17×#121—22no—
11#10587555C2211yescharacteristic
12#10587506—22no—
13#10587503—22no—
14#10587481—22no—
15#10587474C2211yescharacteristic
16#10587474C2212yes—
17C2×Q8×C17D412yes—
18C2×D4×C17—22no—
19C2×D4×C17D412yes—
20C22×C4×C17—22no—
21C2×Dic34—21no—
22C4⋊C4×C17—22no—
23C4⋊C4×C17—22no—
24C4⋊C4×C17D412yes—
25C22⋊C4×C17—22no—
26C22⋊C4×C17—22no—
27C22⋊C4×C17—22no—
28C42×C17C2311yescharacteristic
29C4⋊Dic17—22no—
30C2×#3622—21no—
31C2×#3622—22no—
32C23×C17—22no—
33Q8×C17—42no—
34D4×C17—42no—
35D4×C17—82no—
36C2×C4×C17—22no—
37C2×C4×C17—22no—
38C2×C4×C17—82no—
39C2×C4×C17C2×D411yescharacteristic
40C2×C4×C17C2×D412yes—
41C2×Dic17—41no—
42Dic34—42no—
43#3622—41no—
44#3622—42no—
45#1662—171nomaximal, Sylow
46C22×C17—42no—
47C22×C17—42no—
48C22×C17—42no—
49C22×C17#12011yescharacteristic
50C4×C17—21no—
51C4×C17—22no—
52C4×C17—42no—
53C4×C17—42no—
54Dic17—81no—
55#562D1711yescharacteristic
56#491—171no—
57#489—172no—
58#330—171no—
59#329—172no—
60C2×C17—82no—
61C2×C17#45411yescharacteristic
62C2×C17#45512yes—
63C4⋊Q8—171no—
64#125C2×D1711yescharacteristic
65#124C2×D1712yes—
66#123—22no—
67C22⋊Q8—22no—
68#121—22no—
69#103—342no—
70#103—342no—
71#102—342no—
72#97—171no—
73#97—172no—
74C17#166211yescharacteristic, complemented, Sylow
75C2×Q8—341no—
76C2×Q8C22⋊D1712yes—
77C2×D4—22no—
78C2×D4C22⋊D1712yes—
79C22×C4—22no—
80C2×C8—342no—
81C2×C8—341no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—342no—
85C4⋊C4C22⋊D1712yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D1711yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—682no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—681no—
99C2×C4C2×C22⋊D1711yescharacteristic
100C2×C4C2×C22⋊D1712yes—
101C8—681no—
102C8—682no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1058763911yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—1361no—
112C2—82no—
113C2#1176298211yescharacteristic
114C2#1176298712yes—
subgroupG/H
centerC22#10587639
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
Fitting, prime radicalC17×#562C2
socleC22×C17#120
hypercenter#562D17
hypocenterC17#1662
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #11763179C2semidirect
2. #11763168C2semidirect
3. #11762551C2semidirect
4. #11762549C2semidirect
74. C17#1662semidirect

Maximal subgroups

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

subgroupindex
#117631792
#117631682
#117625512
#117625492
C17×#5622
#166217