Finite Group Databasebeta

← #14544695

Subgroups — #14544695

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×#10587551C212yesmaximal
2#11763686C214yesmaximal, complemented
3#11763282C211yesmaximal, characteristic
4C22×C22⋊D17C2211yescharacteristic
5C2×C22⋊#595C2214yes—
6C2×C22⋊C4×C17C411yescharacteristic, complemented
7C2×C22⋊Dic17C411yescharacteristic, complemented
8C4×C22⋊D17—22no—
9#10587567—22no—
10C22⋊C4×D17C2212yes—
11#10587563C412yescomplemented
12#10587551—22no—
13#10587551—22no—
14C24×C17C2×C411yescharacteristic
15C23×D17C2311yescharacteristic
16C22×#595—24no—
17C22×C4×C17—21no—
18C2×C22⋊D17—24no—
19C2×C22⋊D17D412yes—
20C2×C22⋊D17D412yes—
21C22×Dic17—21no—
22C22×Dic17C2×C411yescharacteristic
23C2×C4×D17—22no—
24C22⋊#595—24no—
25C22⋊#595—24no—
26C22⋊C4×C17—21no—
27C22⋊C4×C17C2×C412yes—
28C22⋊Dic17—21no—
29C22⋊Dic17C2×C412yes—
30#63227—22no—
31#63225—22no—
32C4×Dic17—22no—
33C23×C17—22no—
34C23×C17—81no—
35C23×C17C22⋊C411yescharacteristic
36C23×C17C22⋊C411yescharacteristic
37C23×C17C22×C411yescharacteristic
38C22×D17—81no—
39C22×D17C2×D412yes—
40C22×D17#4114yes—
41C2×#595—44no—
42C2×#595—84no—
43C2×C4×C17—22no—
44C2×C4×C17—81no—
45C22⋊D17—24no—
46C22⋊D17—24no—
47C2×Dic17—22no—
48C2×Dic17—22no—
49C2×Dic17—81no—
50C2×Dic17C22⋊C412yes—
51C4×D17—82no—
52#1828—171nomaximal, Sylow
53C22×C17—21no—
54C22×C17—21no—
55C22×C17—41no—
56C22×C17—41no—
57C22×C17—41no—
58C22×C17—82no—
59C22×C17C2×C22⋊C411yescharacteristic
60C22×C17#11712yes—
61C2×D17—42no—
62C2×D17—44no—
63C2×D17#12314yes—
64#595—84no—
65C4×C17—42no—
66Dic17—22no—
67Dic17—42no—
68#518—171no—
69C2×#99—172no—
70#342—174no—
71C2×C17—21no—
72C2×C17—22no—
73C2×C17—41no—
74C2×C17—41no—
75C2×C17#38111yescharacteristic
76D17—24no—
77C22×D4—171no—
78C4×D4—342no—
79C4×D4—342no—
80#117—172no—
81C2×C22⋊C4—171no—
82C2×C22⋊C4—172no—
83C2×C22⋊C4—174no—
84C2×C22⋊C4#59511yescharacteristic, complemented
85#99—342no—
86#99—342no—
87C17#182811yescharacteristic, complemented, Sylow
88C24—171no—
89C24C2×#59511yescharacteristic
90C2×D4—172no—
91C2×D4—172no—
92C2×D4—344no—
93C22×C4—21no—
94C22×C4—171no—
95C22×C4—342no—
96C22×C4—341no—
97C22×C4—344no—
98C4⋊C4—342no—
99C22⋊C4—21no—
100C22⋊C4—172no—
101C22⋊C4—342no—
102C22⋊C4—341no—
103C22⋊C4—344no—
104C22⋊C4—344no—
105C22⋊C4C2×#59512yes—
106C42—342no—
107C23—22no—
108C23—81no—
109C23—174no—
110C23—172no—
111C23—1361no—
112C23C22⋊#59511yescharacteristic
113C23C22⋊#59511yescharacteristic
114C23C22×#59511yescharacteristic
115D4—344no—
116D4—344no—
117C2×C4—22no—
118C2×C4—81no—
119C2×C4—172no—
120C2×C4—342no—
121C2×C4—342no—
122C2×C4—684no—
123C2×C4—1362no—
124C2×C4—1361no—
125C2×C4—1364no—
126C22—21no—
127C22—21no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—82no—
132C22—174no—
133C22—682no—
134C22—684no—
135C22#1058767712yes—
136C22C2×C22⋊#59511yescharacteristic
137C4—42no—
138C4—342no—
139C4—682no—
140C4—1364no—
141C2—21no—
142C2—22no—
143C2—41no—
144C2—41no—
145C2—344no—
146C2#1176375211yescharacteristic
subgroupG/H
centerC2#11763752
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC2×C17#381
prime radicalC22×C22⋊D17C22
cosocleC23×D17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1828
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11763686C2semidirect
6. C2×C22⋊C4×C17C4semidirect
7. C2×C22⋊Dic17C4semidirect
11. #10587563C4semidirect
84. C2×C22⋊C4#595semidirect
87. C17#1828semidirect

Maximal subgroups

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

subgroupindex
C2×#105875512
#117636862
#117632822
#182817