Finite Group Databasebeta

← #16931765

Subgroups — #16931765

Order 4352.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 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#14531141C211yesmaximal, characteristic
2#14527304C212yesmaximal
3#14527114C211yesmaximal, characteristic
4#14526670C212yesmaximal
5C17×#2583C211yesmaximal, characteristic
6C2×#10587612—21no—
7#11762773C2212yes—
8C4⋊C4×Dic17—24no—
9C4×C22⋊Dic17—21no—
10C4×C22⋊Dic17C2212yes—
11#11762546—24no—
12#11762535—22no—
13C2×C17×C22⋊Q8C2211yescharacteristic
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#398C412yescomplemented
16C17×#395C412yescomplemented
17C17×#389—22no—
18C17×#385—21no—
19C17×#385—21no—
20C4×C22⋊C4×C17C2211yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C17×C22⋊Q8—22no—
26C17×C22⋊Q8—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17D412yes—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C42×C17—21no—
38C2×C22⋊Dic17—22no—
39#10587612—28no—
40C2×C4⋊Dic17—22no—
41C2×C4×Dic17—22no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—22no—
44C2×#63223—22no—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95C2×C412yes—
48#10587502—22no—
49#10587502D414yes—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C24×C17C2×D411yescharacteristic
53C2×Q8×C17—22no—
54C2×Q8×C17—81no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—24no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—81no—
64C22×C4×C17—81no—
65C22×C4×C17C22⋊C412yes—
66C22×C4×C17C22⋊C412yes—
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D412yes—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17#4112yes—
71C22×Dic17—24no—
72C4⋊C4×C17—24no—
73C4⋊C4×C17—24no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—44no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—82no—
78C22⋊C4×C17—24no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C42×C17—81no—
82C22⋊Dic17—28no—
83C4⋊Dic17—28no—
84C4×Dic17—28no—
85C4×Dic17—82no—
86C4×Dic17—82no—
87#63223—82no—
88C2×#3622—42no—
89#9408—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—41no—
93C23×C17C2×C22⋊C411yescharacteristic
94C23×C17#12011yescharacteristic
95C23×C17#12311yescharacteristic
96Q8×C17—82no—
97C2×C4×C17—24no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—24no—
103C2×C4×C17—22no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—81no—
117C2×C4×C17#10414yes—
118C2×C4×C17C4×D412yes—
119C2×C4×C17#12112yes—
120C2×Dic17—28no—
121C2×Dic17—44no—
122C2×Dic17—44no—
123#3622—82no—
124#2583D1711yescharacteristic
125#2274—171no—
126#1508—172no—
127#1463—171no—
128#1297—172no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—44no—
137C22×C17#38611yescharacteristic
138C22×C17C2×#10412yes—
139C22×C17#45612yes—
140C4×C17—24no—
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—44no—
144C4×C17—42no—
145C4×C17—81no—
146C4×C17—81no—
147Dic17—48no—
148C2×C22⋊Q8C2×D1711yescharacteristic
149C2×#117—341no—
150C22×C4⋊C4C2×D1711yescharacteristic
151#407—172no—
152#398Dic1712yescomplemented
153#395Dic1712yescomplemented
154#389—22no—
155#385—21no—
156#385—21no—
157C4×C4⋊C4—344no—
158C4×C22⋊C4—172no—
159C4×C22⋊C4—341no—
160C4×C22⋊C4C2×D1711yescharacteristic
161#328—344no—
162#324—342no—
163C2×C17—22no—
164C2×C17—22no—
165C2×C17—41no—
166C2×C17#188611yescharacteristic
167C2×C17#189611yescharacteristic
168C2×C17#191311yescharacteristic
169C22×Q8C2×Dic1711yescharacteristic
170C23×C4C22⋊D1711yescharacteristic
171C23×C4C22⋊D1711yescharacteristic
172C23×C4C22×D1711yescharacteristic
173C2×M16—342no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176#117—348no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—342no—
183C2×C4⋊C4C2×Dic1711yescharacteristic
184C2×C4⋊C4C2×Dic1712yes—
185C2×C4⋊C4C22⋊D1712yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—342no—
189C2×C42—21no—
190C2×C42—342no—
191C2×C42—342no—
192C2×C42—342no—
193C22⋊C8—342no—
194C22⋊C8—342no—
195#95—22no—
196#95—22no—
197#95—174no—
198#95—342no—
199#95C2×Dic1712yes—
200C17#940811yescharacteristic, complemented, Sylow
201C24C2×C22⋊D1711yescharacteristic
202C2×Q8—22no—
203C2×Q8—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—344no—
215C22×C4C22⋊Dic1712yes—
216C22×C4C22⋊Dic1712yes—
217C22×C4D4×D1712yes—
218C22×C4#6325412yes—
219C22×C4C22×Dic1711yescharacteristic
220C22×C4C2×C22⋊D1711yescharacteristic
221M16—1362no—
222C2×C8—682no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—348no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—348no—
234C42—81no—
235C42—348no—
236C42—1362no—
237C42—1362no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23#1058762111yescharacteristic
242C23#1058762311yescharacteristic
243C23C2×C22⋊Dic1711yescharacteristic
244Q8—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—348no—
266C2×C4—684no—
267C2×C4—684no—
268C2×C4#1058750814yes—
269C2×C4D4×Dic1712yes—
270C2×C4#1058762512yes—
271C8—1362no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1176268911yescharacteristic
281C22C2×#1058750812yes—
282C22#1176299112yes—
283C4—22no—
284C4—22no—
285C4—24no—
286C4—44no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—688no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#1452917811yescharacteristic
295C2#1452922811yescharacteristic
296C2#1452932711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2583C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2583D17
hypocenterC17#9408
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
15. C17×#398C4semidirect
16. C17×#395C4semidirect
152. #398Dic17semidirect
153. #395Dic17semidirect
200. C17#9408semidirect

Maximal subgroups

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

subgroupindex
#145311412
#145273042
#145271142
#145266702
C17×#25832
#940817