Finite Group Databasebeta

← #14544696

Subgroups — #14544696

Order 2176.

296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#10587539C212yesmaximal
2#11763686C214yesmaximal, complemented
3#11763283C211yesmaximal, characteristic
4C22×D68C2211yescharacteristic
5C2×C22⋊#595C2214yes—
6C17×#117C411yescharacteristic, complemented
7#10587612C411yescharacteristic, complemented
8#10587579—22no—
9C22⋊C4×D17C412yescomplemented
10C22⋊C4×D17C2212yes—
11C4×D68—22no—
12#10587539—22no—
13#10587539—22no—
14C23×D17C2×C411yescharacteristic
15C23×D17C2311yescharacteristic
16C22×#595—24no—
17C22×C4×C17C2×C411yescharacteristic
18C2×D68—24no—
19C2×D68D412yes—
20C2×D68D412yes—
21C2×C4×D17—22no—
22C2×C4×D17—22no—
23C22⋊#595—24no—
24C22⋊#595—24no—
25C4⋊C4×C17—21no—
26C22⋊C4×C17C2×C412yes—
27C42×C17—21no—
28C22⋊Dic17C2×C412yes—
29#63227—22no—
30#63227—22no—
31C4⋊Dic17—21no—
32C4×Dic17—21no—
33C23×C17C22×C411yescharacteristic
34C22×D17—22no—
35C22×D17—81no—
36C22×D17—81no—
37C22×D17C22⋊C412yes—
38C22×D17C2×D412yes—
39C22×D17#4114yes—
40C2×#595—44no—
41C2×#595—84no—
42C2×C4×C17—22no—
43C2×C4×C17—22no—
44C2×C4×C17C22⋊C411yescharacteristic
45C2×C4×C17C22⋊C411yescharacteristic
46C2×Dic17—22no—
47D68—24no—
48D68—24no—
49C4×D17—82no—
50C4×D17—82no—
51#1828—171nomaximal, Sylow
52C22×C17—41no—
53C22×C17C2×C22⋊C411yescharacteristic
54C22×C17#11712yes—
55C2×D17—22no—
56C2×D17—42no—
57C2×D17—44no—
58C2×D17—42no—
59C2×D17—82no—
60C2×D17#12314yes—
61#595—84no—
62C4×C17—21no—
63C4×C17—21no—
64C4×C17—42no—
65Dic17—42no—
66#518—171no—
67C2×#99—172no—
68#342—174no—
69C2×C17—21no—
70C2×C17—22no—
71C2×C17#38111yescharacteristic
72D17—24no—
73D17—42no—
74C22×D4—171no—
75C4×D4—342no—
76C4×D4—342no—
77#117—171no—
78#117#59511yescharacteristic, complemented
79C2×C22⋊C4—172no—
80C2×C22⋊C4—172no—
81C2×C22⋊C4—174no—
82#99—342no—
83#99—342no—
84C17#182811yescharacteristic, complemented, Sylow
85C24—171no—
86C24—171no—
87C2×D4—172no—
88C2×D4—172no—
89C2×D4—344no—
90C22×C4—342no—
91C22×C4—342no—
92C22×C4—344no—
93C22×C4C2×#59511yescharacteristic
94C4⋊C4—21no—
95C4⋊C4—341no—
96C22⋊C4—172no—
97C22⋊C4—342no—
98C22⋊C4—342no—
99C22⋊C4—344no—
100C22⋊C4—344no—
101C22⋊C4C2×#59512yes—
102C42—21no—
103C42—341no—
104C23—174no—
105C23—172no—
106C23—172no—
107C23—342no—
108C23—1361no—
109C23—1361no—
110C23C22×#59511yescharacteristic
111D4—344no—
112D4—344no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—342no—
116C2×C4—684no—
117C2×C4—1362no—
118C2×C4—1362no—
119C2×C4—1364no—
120C2×C4C22⋊#59511yescharacteristic
121C2×C4C22⋊#59511yescharacteristic
122C22—41no—
123C22—174no—
124C22—342no—
125C22—682no—
126C22—684no—
127C22—682no—
128C22—1362no—
129C22#1058767712yes—
130C22C2×C22⋊#59511yescharacteristic
131C4—21no—
132C4—21no—
133C4—42no—
134C4—682no—
135C4—1364no—
136C2—21no—
137C2—22no—
138C2—344no—
139C2—682no—
140C2#1176375211yescharacteristic
subgroupG/H
centerC2#11763752
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#117C4
socleC2×C17#381
prime radicalC22×D68C22
cosocleC23×D17C23
hypercenter#117#595
hypocenterC17#1828
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11763686C2semidirect
6. C17×#117C4semidirect
7. #10587612C4semidirect
9. C22⋊C4×D17C4semidirect
78. #117#595semidirect
84. C17#1828semidirect

Maximal subgroups

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

subgroupindex
C2×#105875392
#117636862
#117632832
#182817