Finite Group Databasebeta

← #17010629

Subgroups — #17010629

Order 4352.

982 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 163 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×#11763525C211yesmaximal, characteristic
2#14538879C212yesmaximal, complemented
3#14528009C218yesmaximal, complemented
4#14527510C211yesmaximal, characteristic, complemented
5#14527228C212yesmaximal, complemented
6C17×#3139C211yesmaximal, characteristic, complemented
7#11763525—24no—
8#11763525—22no—
9#11763525C2214yes—
10C2×#10587633—22no—
11C2×#10587581—22no—
12C2×C4×D68C2212yes—
13C2×#10587578—22no—
14#11763142—22no—
15#11763086C2214yescomplemented
16#11763011C2214yescomplemented
17#11762959—22no—
18#11762907—24no—
19#11762866C2212yes—
20#11762558C2214yescomplemented
21#11762538C2218yescomplemented
22C17×#533C2211yescharacteristic
23C2×C17×C4⋊Q8C2211yescharacteristic
24C17×#491—22no—
25C17×#487C2214yescomplemented
26C17×#419—22no—
27C4×M16×C17C2211yescharacteristic
28C22×Q8×C17D412yes—
29C22×D68—22no—
30C22×D68D412yes—
31C22×C4×D17—22no—
32C2×M16×C17—21no—
33C17×C4⋊Q8—22no—
34C17×C4⋊Q8D412yes—
35C17×C4⋊Q8C2312yes—
36C17×C4⋊Q8C2312yes—
37C17×#125—21no—
38C17×C22⋊Q8—22no—
39C4×Q8×C17—22no—
40C17×#117—21no—
41C2×C4⋊C4×C17—22no—
42C2×C42×C17C2311yescharacteristic
43#10587633—28no—
44#10587633—28no—
45C2×#63230—24no—
46C2×#63227—24no—
47C2×#63227—22no—
48C2×C4⋊Dic17D412yes—
49C2×C4×Dic17—22no—
50#10587581—28no—
51#10587581—28no—
52#10587578—28no—
53#10587578—28no—
54C4×D68—22no—
55C4×D68—22no—
56C4×D68D414yes—
57C4×D68C2314yes—
58C17×#103—28no—
59C17×#97—21no—
60C4×C8×C17C2312yes—
61#10587506—24no—
62C22⋊#3622—22no—
63#10587480—24no—
64#10587478—24no—
65C4⋊#3622C2314yes—
66#10587474—22no—
67C23×D17—22no—
68C2×Q8×C17—22no—
69C2×Q8×C17—22no—
70C2×Q8×C17—44no—
71C2×Q8×C17C2×D414yes—
72C22×C4×C17—22no—
73C22×C4×C17C2×D412yes—
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—22no—
76C2×D68—22no—
77C2×D68—28no—
78C2×D68—44no—
79C2×D68—84no—
80C2×D68C2×D414yes—
81C2×C4×D17—24no—
82C2×C4×D17—24no—
83C2×C4×D17—48no—
84M16×C17—42no—
85C2×C8×C17—22no—
86C4⋊C4×C17—24no—
87C4⋊C4×C17—24no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—42no—
91C22⋊C4×C17—41no—
92C42×C17—41no—
93C42×C17C2×D412yes—
94C42×C17C2×D411yescharacteristic
95C42×C17C2411yescharacteristic
96#63230—48no—
97#63227—216no—
98#63227—24no—
99#63227—24no—
100C4⋊Dic17—22no—
101C4⋊Dic17C2×D414yes—
102C4×Dic17—28no—
103C2×#3622—24no—
104#18552—171nomaximal, Sylow
105C23×C17#12011yescharacteristic
106C22×D17—24no—
107C22×D17—44no—
108C22×D17—44no—
109C22×D17—44no—
110Q8×C17—44no—
111Q8×C17—44no—
112Q8×C17—82no—
113C2×C4×C17—22no—
114C2×C4×C17—24no—
115C2×C4×C17—24no—
116C2×C4×C17—44no—
117C2×C4×C17—42no—
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12012yes—
120C2×C4×C17C22×D412yes—
121C2×C4×C17C22×D411yescharacteristic
122C2×Dic17—24no—
123C2×Dic17—44no—
124D68—24no—
125D68—24no—
126D68—44no—
127D68—88no—
128C4×D17—48no—
129C4×D17—48no—
130C8×C17—42no—
131#3622—44no—
132C2×#555—171no—
133#3139D1711yescharacteristic, complemented
134#3110—172no—
135#1646—178no—
136#1558—171no—
137#1490—172no—
138C22×C17—22no—
139C22×C17#45512yes—
140C22×C17C2×#12011yescharacteristic
141C2×D17—44no—
142C2×D17—44no—
143C2×D17—44no—
144C2×D17—88no—
145C4×C17—22no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—44no—
149C4×C17—82no—
150C4×C17#57514yes—
151Dic17—44no—
152#555—174no—
153#555—344no—
154#555—342no—
155#533C2×D1711yescharacteristic
156C2×C4⋊Q8C2×D1711yescharacteristic
157C2×#124—342no—
158C2×#121—342no—
159C2×C4×D4—172no—
160C2×#117—342no—
161#491—22no—
162#487C2×D1714yescomplemented
163#484—342no—
164#474—174no—
165#461—174no—
166#450—342no—
167#440—344no—
168#432—172no—
169#419—22no—
170C4×M16C2×D1711yescharacteristic
171#333—174no—
172D4⋊C8—178no—
173C2×C17—22no—
174C2×C17#300512yes—
175C2×C17C2×#45511yescharacteristic
176D17—84no—
177C22×Q8C22⋊D1712yes—
178C22×D4—172no—
179C22×D4—342no—
180C23×C4—342no—
181C2×SD16—344no—
182C2×M16—21no—
183C4⋊Q8—22no—
184C4⋊Q8C22⋊D1712yes—
185C4⋊Q8C22×D1712yes—
186C4⋊Q8C22×D1712yes—
187#125—21no—
188#124—348no—
189#124—348no—
190C22⋊Q8—22no—
191#121—348no—
192#121—348no—
193C4×Q8—22no—
194C4×D4—174no—
195C4×D4—174no—
196C4×D4—342no—
197C4×D4—342no—
198#117—21no—
199#117—348no—
200#117—348no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—172no—
203C2×C22⋊C4—344no—
204C2×C22⋊C4—342no—
205C2×C42—342no—
206C2×C42C22×D1711yescharacteristic
207#107—344no—
208C4⋊C8—174no—
209#103—28no—
210#103—344no—
211#102—344no—
212C22⋊C8—342no—
213#97—21no—
214#97—342no—
215C4×C8C22×D1712yes—
216C17#1855211yescharacteristic, complemented, Sylow
217C24—342no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8C2×C22⋊D1714yes—
222C2×D4—174no—
223C2×D4—348no—
224C2×D4—342no—
225C2×D4—684no—
226C2×D4—1364no—
227C22×C4—22no—
228C22×C4—342no—
229C22×C4—344no—
230C22×C4—344no—
231C22×C4—688no—
232C22×C4D4×D1712yes—
233C22×C4C2×C22⋊D1711yescharacteristic
234SD16—688no—
235M16—42no—
236C2×C8—22no—
237C2×C8—344no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—174no—
244C4⋊C4—342no—
245C22⋊C4—41no—
246C22⋊C4—3416no—
247C22⋊C4—344no—
248C22⋊C4—344no—
249C42—41no—
250C42—348no—
251C42D4×D1712yes—
252C42C2×C22⋊D1711yescharacteristic
253C42C23×D1711yescharacteristic
254C23—344no—
255C23—684no—
256C23—684no—
257C23—684no—
258C23#1058762311yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—82no—
262D4—344no—
263D4—344no—
264D4—684no—
265D4—1368no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—344no—
272C2×C4—688no—
273C2×C4—684no—
274C2×C4—688no—
275C2×C4#1058762311yescharacteristic
276C2×C4#1058762312yes—
277C2×C4C2×D4×D1712yes—
278C2×C4C22×C22⋊D1711yescharacteristic
279C8—42no—
280C8—684no—
281C22—22no—
282C22—684no—
283C22—684no—
284C22—684no—
285C22—1368no—
286C22#1176298612yes—
287C22C2×#1058762311yescharacteristic
288C4—22no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—82no—
293C4—684no—
294C4#1176360814yes—
295C2—22no—
296C2—1364no—
297C2#1453802512yes—
298C2C2×#1176298611yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3139C2
socleC22×C17C2×#120
prime radicalC2×#11763525C2
hypercenter#3139D17
hypocenterC17#18552
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #14538879C2semidirect
3. #14528009C2semidirect
4. #14527510C2semidirect
5. #14527228C2semidirect
6. C17×#3139C2semidirect
15. #11763086C22semidirect
16. #11763011C22semidirect
20. #11762558C22semidirect
21. #11762538C22semidirect
25. C17×#487C22semidirect
133. #3139D17semidirect
162. #487C2×D17semidirect
216. C17#18552semidirect

Maximal subgroups

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

subgroupindex
C2×#117635252
#145388792
#145280092
#145275102
#145272282
C17×#31392
#1855217