Finite Group Databasebeta

← #17022999

Subgroups — #17022999

Order 4352.

744 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539928C212yesmaximal, complemented
2#14539905C212yesmaximal
3#14539107C211yesmaximal, characteristic
4#14528459C214yesmaximal, complemented
5#14528447C214yesmaximal, complemented
6#14527076C211yesmaximal, characteristic, complemented
7C2×C17×#494C211yesmaximal, characteristic, complemented
8#11763366C2211yescharacteristic
9#11763359C2211yescharacteristic
10#11763212C2214yes—
11#11763205C2214yes—
12#11763164—22no—
13#11763161C2214yes—
14#11763160C2212yes—
15#11763157C2214yes—
16#11763156C2212yes—
17#11763139—24no—
18#11763122—24no—
19#11763091—24no—
20#11763086—24no—
21#11762862C2212yes—
22#11762806—22no—
23#11762531C2214yes—
24C22×Q16×C17—22no—
25C2×C17×C4⋊Q8C2212yes—
26C17×#494—22no—
27C17×#494C2214yes—
28C2×C4×C8×C17C2211yescharacteristic
29C22×Q8×C17—22no—
30C2×#63252—21no—
31C2×Q16×C17—28no—
32C2×Q16×C17—48no—
33C22×C8×C17D412yes—
34C17×C4⋊Q8—22no—
35C17×C4⋊Q8—22no—
36C17×C4⋊Q8—22no—
37C17×C4⋊Q8C2314yes—
38C2×C4⋊C4×C17—22no—
39C2×C4⋊C4×C17—22no—
40C2×C42×C17C2311yescharacteristic
41#10587616—22no—
42#10587612—21no—
43#10587610—22no—
44#10587561—21no—
45#10587560C2311yescharacteristic
46C4×D68—22no—
47#10587556—21no—
48#10587555C2312yes—
49#10587555C2311yescharacteristic
50C4×Dic34—22no—
51C4×C8×C17D412yes—
52C4×C8×C17C2312yes—
53C22⋊#3622D414yes—
54#10587492—24no—
55#10587487—24no—
56#10587481—28no—
57#10587480—28no—
58#10587479—28no—
59#10587478—28no—
60C4⋊#3622D414yes—
61C4×#3622C2314yes—
62C2×Q8×C17—24no—
63C2×Q8×C17—42no—
64C2×Q8×C17—48no—
65C22×C4×C17—42no—
66C22×C4×C17C2×D412yes—
67C22×C4×C17C2×D411yescharacteristic
68C2×C22⋊D17—41no—
69#63252—44no—
70C2×D68—21no—
71C2×D68—22no—
72C2×C4×D17—41no—
73C2×Dic34—21no—
74C2×Dic34—22no—
75Q16×C17—416no—
76C2×C8×C17—24no—
77C2×C8×C17C2×D414yes—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—24no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C42×C17C2×D412yes—
83C42×C17C2×D411yescharacteristic
84C42×C17C2411yescharacteristic
85C22⋊Dic17—41no—
86#63227—42no—
87C4⋊Dic17—22no—
88C4⋊Dic17—22no—
89C4⋊Dic17—22no—
90#63225—42no—
91C4×Dic17—41no—
92C2×#3622C2×D418yes—
93#19699—171nomaximal, Sylow
94C23×C17#12711yescharacteristic
95C22×D17—41no—
96Q8×C17—48no—
97Q8×C17—84no—
98C2×C4×C17—24no—
99C2×C4×C17—44no—
100C2×C4×C17—84no—
101C2×C4×C17#12712yes—
102C2×C4×C17#12711yescharacteristic
103C2×C4×C17AGL(1,ℤ/8)14yes—
104C2×C4×C17C22×D412yes—
105C2×C4×C17C22×D411yescharacteristic
106C22⋊D17—82no—
107C2×Dic17—42no—
108C2×Dic17—41no—
109D68—42no—
110D68—44no—
111C4×D17—82no—
112Dic34—42no—
113Dic34—44no—
114C8×C17—24no—
115#3622—28no—
116#3229—172no—
117#3226—172no—
118C2×#494D1711yescharacteristic, complemented
119#3131—171no—
120#1723—174no—
121#1720—174no—
122#1452—171no—
123C22×C17—22no—
124C22×C17#49612yes—
125C22×C17C2×#12711yescharacteristic
126C2×D17—81no—
127C2×D17—82no—
128C4×C17—22no—
129C4×C17—84no—
130C4×C17#49612yes—
131C4×C17C2×AGL(1,ℤ/8)12yes—
132C4×C17#57814yes—
133Dic17—81no—
134Dic17—82no—
135C22×Q16—22no—
136#533—171no—
137#532—171no—
138C2×C4⋊Q8C2×D1712yes—
139#500—174no—
140#498—174no—
141#494—22no—
142#494C2×D1714yes—
143#488—342no—
144#487—174no—
145#487—172no—
146#486—174no—
147#486—172no—
148#483—344no—
149#480—344no—
150#475—344no—
151#474—344no—
152#431—172no—
153#416—342no—
154C2×C4×C8C2×D1711yescharacteristic
155#322—174no—
156C2×C17—22no—
157C2×C17C2×#49611yescharacteristic
158C2×C17#315612yes—
159D17—161no—
160C2×#41—341no—
161C22×Q8—22no—
162C2×Q16—28no—
163C2×Q16—48no—
164C22×C8C22⋊D1712yes—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—171no—
169C4⋊Q8—172no—
170C4⋊Q8C22×D1714yes—
171#127—171no—
172#125—341no—
173#124—341no—
174C22⋊Q8—342no—
175#121—342no—
176C4×Q8—342no—
177C4×D4—342no—
178#117—341no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C42C22×D1711yescharacteristic
182#107—348no—
183#106—348no—
184C4⋊C8—174no—
185#103—348no—
186#103—344no—
187#102—348no—
188#102—344no—
189C22⋊C8—174no—
190C4×C8—174no—
191C4×C8C22⋊D1712yes—
192C4×C8C22×D1712yes—
193C17#1969911yescharacteristic, complemented, Sylow
194#41—684no—
195C2×Q8—24no—
196C2×Q8—42no—
197C2×Q8—48no—
198C2×Q8—342no—
199C2×Q8—341no—
200C2×D4—342no—
201C2×D4—341no—
202C2×D4—681no—
203C22×C4—42no—
204C22×C4—681no—
205C22×C4D4×D1712yes—
206C22×C4C2×C22⋊D1711yescharacteristic
207Q16—416no—
208C2×C8—24no—
209C2×C8—178no—
210C2×C8C2×C22⋊D1714yes—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—342no—
216C4⋊C4—342no—
217C4⋊C4—342no—
218C4⋊C4—682no—
219C22⋊C4—681no—
220C22⋊C4—682no—
221C42—681no—
222C42D4×D1712yes—
223C42C2×C22⋊D1711yescharacteristic
224C42C23×D1711yescharacteristic
225C23—681no—
226C23#1058762611yescharacteristic
227Q8—48no—
228Q8—84no—
229Q8—682no—
230Q8—684no—
231D4—682no—
232D4—684no—
233D4—1362no—
234C2×C4—24no—
235C2×C4—44no—
236C2×C4—84no—
237C2×C4—682no—
238C2×C4—681no—
239C2×C4—1362no—
240C2×C4#1058761814yes—
241C2×C4#1058762612yes—
242C2×C4#1058762611yescharacteristic
243C2×C4C2×D4×D1712yes—
244C2×C4C22×C22⋊D1711yescharacteristic
245C8—24no—
246C8—348no—
247C22—22no—
248C22—1362no—
249C22—1361no—
250C22#1176319612yes—
251C22C2×#1058762611yescharacteristic
252C4—22no—
253C4—84no—
254C4—1362no—
255C4—1361no—
256C4#1176319612yes—
257C4C2×#1058761812yes—
258C4#1176362214yes—
259C2—22no—
260C2—2721no—
261C2C2×#1176319611yescharacteristic
262C2#1453927312yes—
subgroupG/H
centerC22C2×#10587626
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#494C2
socleC22×C17C2×#127
prime radical#11763359C22
hypercenterC2×#494D17
hypocenterC17#19699
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. #14539928C2semidirect
4. #14528459C2semidirect
5. #14528447C2semidirect
6. #14527076C2semidirect
7. C2×C17×#494C2semidirect
118. C2×#494D17semidirect
193. C17#19699semidirect

Maximal subgroups

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

subgroupindex
#145399282
#145399052
#145391072
#145284592
#145284472
#145270762
C2×C17×#4942
#1969917