Finite Group Databasebeta

← #14544208

Subgroups — #14544208

Order 2176.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#11763832C211yesmaximal, characteristic
2#11763754C211yesmaximal, characteristic
3#11763663C214yesmaximal
4#11763364C211yesmaximal, characteristic
5C17×C4⋊Q8C411yescharacteristic, complemented
6#10587632C2212yes—
7Q8×Dic17C412yescomplemented
8#10587583—21no—
9#10587578—21no—
10#10587576—21no—
11#10587573—21no—
12C42×D17C2211yescharacteristic
13#10587555C411yescharacteristic, complemented
14(C2×#595)⋊C4—21no—
15(C2×#595)⋊C4—21no—
16(C2×#595)⋊C4C2212yes—
17#10587544—21no—
18#10587543C2212yes—
19#10587542—21no—
20C22×#595—22no—
21C2×Q8×C17C2×C412yes—
22C2×C4×D17D411yescharacteristic
23C2×C4×D17D411yescharacteristic
24C2×C4×D17C2311yescharacteristic
25C2×Dic34—21no—
26C2×#3624—22no—
27C4⋊C4×C17—21no—
28C4⋊C4×C17—21no—
29C42×C17C2×C411yescharacteristic
30#63227—22no—
31C4⋊Dic17—21no—
32C4⋊Dic17—22no—
33C4⋊Dic17C2×C412yes—
34#63225—22no—
35C4×Dic17—22no—
36C4×Dic17C2×C411yescharacteristic
37C4×Dic17D411yescharacteristic
38C4×Dic17D411yescharacteristic
39C22×D17C2×D411yescharacteristic
40C2×#595—42no—
41C2×#595—42no—
42C2×#595—42no—
43Q8×C17—42no—
44C2×C4×C17—22no—
45C2×C4×C17C22⋊C411yescharacteristic
46C2×C4×C17C22⋊C411yescharacteristic
47C2×C4×C17C22×C411yescharacteristic
48C2×Dic17—22no—
49C2×Dic17C22⋊C411yescharacteristic
50C2×Dic17C22⋊C411yescharacteristic
51C2×Dic17C2×D411yescharacteristic
52C4×D17—22no—
53C4×D17—41no—
54C4×D17—41no—
55Dic34—42no—
56#3624—42no—
57#1515—171nomaximal, Sylow
58C22×C17C2×C22⋊C411yescharacteristic
59C2×D17—21no—
60C2×D17—21no—
61C2×D17#13712yes—
62#595—82no—
63C4×C17—21no—
64C4×C17—21no—
65C4×C17—21no—
66C4×C17—42no—
67Dic17—21no—
68Dic17—21no—
69Dic17—21no—
70Dic17—42no—
71#533—171no—
72#432—171no—
73#383—171no—
74#328—174no—
75C2×C17#41011yescharacteristic
76C2×C17#41911yescharacteristic
77C2×C17#42111yescharacteristic
78D17—22no—
79C4⋊Q8—171no—
80C4⋊Q8#59511yescharacteristic, complemented
81#125—341no—
82C22⋊Q8—172no—
83C22⋊Q8—341no—
84C4×Q8—172no—
85C4×Q8—341no—
86#117—341no—
87C2×C42—171no—
88C4⋊C8—341no—
89C22⋊C8—172no—
90#97—341no—
91#95—172no—
92#95—341no—
93#95—341no—
94C17#151511yescharacteristic, complemented, Sylow
95C2×Q8—341no—
96C2×Q8C2×#59512yes—
97C22×C4—171no—
98C22×C4—171no—
99C22×C4—171no—
100C22×C4—342no—
101C2×C8—342no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—172no—
105C4⋊C4—342no—
106C4⋊C4—342no—
107C4⋊C4—341no—
108C22⋊C4—342no—
109C42—171no—
110C42—171no—
111C42—171no—
112C42—342no—
113C42C2×#59511yescharacteristic
114C23—171no—
115Q8—42no—
116Q8—682no—
117C2×C4—22no—
118C2×C4—171no—
119C2×C4—171no—
120C2×C4—171no—
121C2×C4—342no—
122C2×C4—342no—
123C2×C4—681no—
124C2×C4—681no—
125C2×C4—682no—
126C2×C4—682no—
127C2×C4—682no—
128C2×C4C22⋊#59511yescharacteristic
129C2×C4C22⋊#59511yescharacteristic
130C2×C4C22×#59511yescharacteristic
131C8—682no—
132C22—172no—
133C22—341no—
134C22—341no—
135C22C2×C22⋊#59511yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—341no—
141C4—341no—
142C4—341no—
143C4—682no—
144C4—1362no—
145C2—342no—
146C2#1176378911yescharacteristic
147C2#1176380211yescharacteristic
148C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C4⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4⋊Q8#595
hypocenterC17#1515
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
5. C17×C4⋊Q8C4semidirect
7. Q8×Dic17C4semidirect
13. #10587555C4semidirect
80. C4⋊Q8#595semidirect
94. C17#1515semidirect

Maximal subgroups

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

subgroupindex
#117638322
#117637542
#117636632
#117633642
#151517