Finite Group Databasebeta

← #16931762

Subgroups — #16931762

Order 4352.

618 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 107 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#14534164C211yesmaximal, characteristic
2#14531141C211yesmaximal, characteristic
3#14527305C212yesmaximal
4#14526669C212yesmaximal
5C17×#1463C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7C2×#10587612—21no—
8#11762739C412yescomplemented
9#11762729C412yescomplemented
10#11762704—22no—
11#11762682—21no—
12#11762682—21no—
13C4⋊C4×Dic17—24no—
14C4×C22⋊Dic17—21no—
15C4×C22⋊Dic17C2211yescharacteristic
16C4×C22⋊Dic17C2212yes—
17#11762547—24no—
18C22×C4⋊C4×C17C2211yescharacteristic
19C17×#407C2212yes—
20C17×#324—22no—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C2×M16×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C4⋊C4×C17D412yes—
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33#10587612—28no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36#10587610—22no—
37#10587610—22no—
38C2×#63225—22no—
39C2×#63225—22no—
40C2×#63225—21no—
41C2×C4×Dic17—22no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—22no—
50#10587502C2×C412yes—
51#10587502D414yes—
52C24×C17C2×D411yescharacteristic
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—24no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—81no—
59C22×C4×C17—81no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17#4112yes—
66C22×Dic17—22no—
67C22×Dic17—24no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C2×Dic34—22no—
71C2×Dic34—81no—
72M16×C17—82no—
73C2×C8×C17—42no—
74C4⋊C4×C17—24no—
75C4⋊C4×C17—24no—
76C22⋊Dic17—28no—
77C22⋊Dic17—24no—
78C22⋊Dic17—41no—
79C22⋊Dic17—41no—
80C4⋊Dic17—28no—
81C4⋊Dic17—22no—
82#63225—44no—
83#63225—42no—
84#63225—82no—
85C4×Dic17—28no—
86C4×Dic17—82no—
87C4×Dic17—82no—
88C4×Dic17—81no—
89#9408—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—41no—
93C23×C17C2×C22⋊C411yescharacteristic
94C23×C17#12011yescharacteristic
95C23×C17#12311yescharacteristic
96C2×C4×C17—24no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—81no—
109C2×C4×C17#10414yes—
110C2×C4×C17C4×D412yes—
111C2×C4×C17#12112yes—
112C2×Dic17—24no—
113C2×Dic17—28no—
114C2×Dic17—44no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—44no—
118C2×Dic17—42no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—42no—
122Dic34—82no—
123C8×C17—82no—
124#2583—171no—
125#2274—171no—
126#1508—172no—
127#1463D1711yescharacteristic
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—42no—
144Dic17—48no—
145Dic17—44no—
146Dic17—81no—
147Dic17—81no—
148C2×C22⋊Q8—171no—
149C2×#117—341no—
150C22×C4⋊C4C2×D1711yescharacteristic
151#407C2×D1712yes—
152#398—172no—
153#395—172no—
154#389—342no—
155#385—341no—
156#385—341no—
157C4×C4⋊C4—344no—
158C4×C22⋊C4—171no—
159C4×C22⋊C4—172no—
160C4×C22⋊C4—341no—
161#328—344no—
162#324—22no—
163C2×C17—22no—
164C2×C17—22no—
165C2×C17—41no—
166C2×C17#188611yescharacteristic
167C2×C17#189611yescharacteristic
168C2×C17#191311yescharacteristic
169C22×Q8—171no—
170C23×C4D6811yescharacteristic
171C23×C4D6811yescharacteristic
172C23×C4C22×D1711yescharacteristic
173C2×M16—22no—
174C22⋊Q8—342no—
175C22⋊Q8—342no—
176#117—348no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—171no—
180C2×C4⋊C4—342no—
181C2×C4⋊C4—341no—
182C2×C4⋊C4—342no—
183C2×C4⋊C4—342no—
184C2×C4⋊C4C4×D1712yes—
185C2×C4⋊C4C22⋊D1712yes—
186C2×C22⋊C4—342no—
187C2×C22⋊C4—341no—
188C2×C22⋊C4—341no—
189C2×C42—342no—
190C2×C42—342no—
191C2×C42—342no—
192C2×C42—341no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195#95—172no—
196#95—174no—
197#95—342no—
198#95—342no—
199#95—342no—
200C17#940811yescharacteristic, complemented, Sylow
201C24C2×D6811yescharacteristic
202C2×Q8—342no—
203C2×Q8—1361no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—344no—
212C22×C4—342no—
213C22×C4—341no—
214C22×C4—341no—
215C22×C4#6322712yes—
216C22×C4#6322712yes—
217C22×C4C2×C4×D1711yescharacteristic
218C22×C4D4×D1712yes—
219C22×C4#6325412yes—
220C22×C4C2×C22⋊D1711yescharacteristic
221M16—82no—
222C2×C8—42no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—348no—
226C4⋊C4—342no—
227C4⋊C4—684no—
228C4⋊C4—682no—
229C4⋊C4—1362no—
230C22⋊C4—344no—
231C22⋊C4—348no—
232C22⋊C4—681no—
233C22⋊C4—681no—
234C42—348no—
235C42—1362no—
236C42—1361no—
237C42—1362no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23#1058756811yescharacteristic
242C23#1058757211yescharacteristic
243C23C2×#6322711yescharacteristic
244Q8—1362no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—348no—
259C2×C4—344no—
260C2×C4—684no—
261C2×C4—682no—
262C2×C4—684no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4#1058749614yes—
269C2×C4#1058756712yes—
270C2×C4#1058762512yes—
271C8—82no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1176268611yescharacteristic
281C22C2×#1058749612yes—
282C22#1176299412yes—
283C4—22no—
284C4—22no—
285C4—24no—
286C4—42no—
287C4—688no—
288C4—684no—
289C4—1361no—
290C4—1361no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#1452917511yescharacteristic
295C2#1452922511yescharacteristic
296C2#1452932411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1463C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1463D17
hypocenterC17#9408
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
8. #11762739C4semidirect
9. #11762729C4semidirect
200. C17#9408semidirect

Maximal subgroups

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

subgroupindex
#145341642
#145311412
#145273052
#145266692
C17×#14632
#940817