Finite Group Databasebeta

← #14528043

Subgroups — #14528043

Order 2176.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763532C211yesmaximal, characteristic
2#11763214C211yesmaximal, characteristic, complemented
3#11763094C211yesmaximal, characteristic, complemented
4#11763015C211yesmaximal, characteristic
5#11762556C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C4×Q8×C17C2211yescharacteristic
9#10587632—21no—
10C2×#63219—21no—
11#10587584—22no—
12#10587578—21no—
13#10587561—21no—
14C4×D68C2211yescharacteristic
15#10587555C2211yescharacteristic
16#10587555C2211yescharacteristic
17C4×Dic34—21no—
18C17×C4⋊C8C2211yescharacteristic
19C4×C8×C17C2211yescharacteristic
20#10587506—21no—
21#10587489—22no—
22#10587487—21no—
23#10587479—21no—
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17D411yescharacteristic
26C2×D68D411yescharacteristic
27C2×C4×D17—21no—
28C2×Dic34—21no—
29C2×Dic34—21no—
30C2×C8×C17—21no—
31C2×C8×C17—21no—
32C4⋊C4×C17—21no—
33C4⋊C4×C17D411yescharacteristic
34C42×C17—21no—
35C42×C17C2311yescharacteristic
36#63227—22no—
37#63227—22no—
38#63227—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17D411yescharacteristic
43C4⋊Dic17D412yes—
44#63225—22no—
45C4×Dic17—22no—
46C2×#3622—21no—
47#63219—42no—
48C22×D17—21no—
49Q8×C17—21no—
50Q8×C17—21no—
51C2×C4×C17—21no—
52C2×C4×C17—21no—
53C2×C4×C17C2×D411yescharacteristic
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×Dic17—22no—
57C2×Dic17—21no—
58C2×Dic17—41no—
59D68—21no—
60D68—21no—
61C4×D17—41no—
62C4×D17—41no—
63Dic34—41no—
64Dic34—41no—
65Dic34—41no—
66Dic34—41no—
67C8×C17—22no—
68C8×C17—41no—
69#3622—41no—
70#1650—171nomaximal, Sylow
71C22×C17#12011yescharacteristic
72C2×D17—41no—
73C2×D17—41no—
74C2×D17—41no—
75C4×C17—21no—
76C4×C17—41no—
77C4×C17—41no—
78C4×C17#13511yescharacteristic
79C4×C17#13511yescharacteristic
80C4×C17AGL(1,ℤ/8)11yescharacteristic
81C4×C17#13711yescharacteristic
82Dic17—42no—
83Dic17—41no—
84Dic17—81no—
85#556—171no—
86#500—171no—
87#475—171no—
88#461—171no—
89#332—171no—
90Q8⋊C8D1711yescharacteristic, complemented
91D4⋊C8—171no—
92C2×C17#44911yescharacteristic
93C2×C17#45211yescharacteristic
94C2×C17#45611yescharacteristic
95D17—81no—
96C2×SD16—341no—
97C4⋊Q8—171no—
98C4⋊Q8—171no—
99#126—342no—
100#124—341no—
101C22⋊Q8—341no—
102C4×Q8—341no—
103C4×Q8C2×D1711yescharacteristic
104C4×D4—171no—
105#117—341no—
106#107—342no—
107#106—341no—
108C4⋊C8—171no—
109C4⋊C8C2×D1711yescharacteristic
110#103—341no—
111#103—341no—
112C4×C8C2×D1711yescharacteristic
113C17#165011yescharacteristic, complemented, Sylow
114C2×Q8—341no—
115C2×Q8—341no—
116C2×Q8D6811yescharacteristic
117C2×D4—171no—
118C22×C4—341no—
119SD16—682no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—341no—
123C4⋊C4—21no—
124C4⋊C4—171no—
125C4⋊C4—172no—
126C4⋊C4—342no—
127C4⋊C4—341no—
128C4⋊C4—341no—
129C4⋊C4—341no—
130C4⋊C4D6811yescharacteristic
131C22⋊C4—342no—
132C22⋊C4—342no—
133C22⋊C4—341no—
134C42—21no—
135C42—342no—
136C42C22×D1711yescharacteristic
137C23—341no—
138Q8—21no—
139Q8—21no—
140Q8—681no—
141Q8—681no—
142Q8—681no—
143Q8—681no—
144D4—341no—
145D4—341no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—341no—
149C2×C4—342no—
150C2×C4—681no—
151C2×C4—681no—
152C2×C4—681no—
153C2×C4C2×D6811yescharacteristic
154C2×C4D4×D1711yescharacteristic
155C2×C4D4×D1711yescharacteristic
156C8—22no—
157C8—41no—
158C8—681no—
159C22—681no—
160C22—681no—
161C22—681no—
162C22#1058756811yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—682no—
167C4—681no—
168C4—1361no—
169C4#1058759011yescharacteristic
170C4#1058759511yescharacteristic
171C4#1058760011yescharacteristic
172C4#1058760511yescharacteristic
173C2—1361no—
174C2#1176295711yescharacteristic
175C2#1176297311yescharacteristic
176C2#1176299311yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17#120
prime radicalC2×D68D4
hypercenterQ8⋊C8D17
hypocenterC17#1650
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11763214C2semidirect
3. #11763094C2semidirect
5. #11762556C2semidirect
7. C17×Q8⋊C8C2semidirect
90. Q8⋊C8D17semidirect
113. C17#1650semidirect

Maximal subgroups

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

subgroupindex
#117635322
#117632142
#117630942
#117630152
#117625562
#117625382
C17×Q8⋊C82
#165017