Finite Group Databasebeta

← #16924448

Subgroups — #16924448

Order 4352.

954 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 159 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
1D17×#407C211yesmaximal, characteristic
2#14531165C212yesmaximal
3#14526670C212yesmaximal, complemented
4#14526669C211yesmaximal, characteristic, complemented
5C17×#1297C211yesmaximal, characteristic, complemented
6C23×C4×D17C2211yescharacteristic
7C2×M16×D17C412yescomplemented
8C2×#10587558—22no—
9#11762773C2211yescharacteristic
10#11762772C412yescomplemented
11#11762771—22no—
12C22⋊C8×D17—22no—
13#11762671C414yescomplemented
14#11762668C414yescomplemented
15#11762667—22no—
16#11762664—22no—
17C4×#63227—22no—
18C4×C22⋊Dic17C2212yes—
19C17×#407C2211yescharacteristic
20C4×C22⋊C4×C17C2212yes—
21C24×D17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17D411yescharacteristic
24C22×C4×D17—44no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17C2×C411yescharacteristic
27C22×C4×D17C2×C412yes—
28C22×C4×D17C2×C412yes—
29C22×C4×D17D412yes—
30C22×C4×D17D412yes—
31C22×C4×D17Q811yescharacteristic
32C22×C4×D17Q811yescharacteristic
33C2×M16×C17C2×C412yes—
34C2×C22⋊C4×C17—22no—
35C2×C42×C17—22no—
36C2×C22⋊Dic17—22no—
37C2×#63227—24no—
38C2×#63227—24no—
39C2×#63225—24no—
40C2×C4×Dic17—22no—
41C2×#63223C2×C412yes—
42M16×D17—44no—
43C2×#63218—22no—
44C2×C8×D17—22no—
45#10587558—48no—
46C17×C22⋊C8—22no—
47C17×#95C2×C414yes—
48#10587502—24no—
49#10587502C2×C414yes—
50C22⋊#3622—22no—
51#10587491—24no—
52C24×C17C2×D411yescharacteristic
53C23×D17—42no—
54C23×D17—44no—
55C23×D17C22⋊C412yes—
56C23×D17C22⋊C412yes—
57C23×D17C4⋊C411yescharacteristic
58C23×D17C4⋊C411yescharacteristic
59C22×C4×C17—24no—
60C22×C4×C17—42no—
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C22×C412yes—
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×Q811yescharacteristic
65C22×Dic17—24no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C412yes—
68C22×Dic17C22⋊C412yes—
69C22×Dic17C4⋊C411yescharacteristic
70C22×Dic17C4⋊C411yescharacteristic
71C2×C4×D17—24no—
72C2×C4×D17—28no—
73C2×C4×D17—44no—
74C2×C4×D17—42no—
75C2×C4×D17—42no—
76C2×C4×D17—44no—
77C2×C4×D17—44no—
78C2×C4×D17—42no—
79C2×C4×D17C4211yescharacteristic
80C2×C4×D17C4211yescharacteristic
81C2×C4×D17C4212yes—
82C2×C4×D17C22⋊C412yes—
83C2×C4×D17C22⋊C411yescharacteristic
84C2×C4×D17C22⋊C411yescharacteristic
85C2×C4×D17C4⋊C412yes—
86C2×C4×D17C4⋊C412yes—
87C2×C4×D17C4⋊C412yes—
88C2×C4×D17C4⋊C412yes—
89M16×C17—42no—
90C2×C8×C17—22no—
91C22⋊C4×C17—44no—
92C42×C17—44no—
93C22⋊Dic17—44no—
94#63227—48no—
95#63227—48no—
96#63225—48no—
97C4×Dic17—44no—
98#63223—42no—
99C2×#3622—22no—
100#63218—44no—
101C8×D17—44no—
102#8049—171nomaximal, Sylow
103C23×C17—24no—
104C23×C17—41no—
105C23×C17C2×C22⋊C412yes—
106C23×C17C2×C4⋊C411yescharacteristic
107C22×D17—24no—
108C22×D17—24no—
109C22×D17—24no—
110C22×D17—22no—
111C22×D17—42no—
112C22×D17—41no—
113C22×D17—42no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—416no—
118C22×D17#9511yescharacteristic
119C22×D17#9512yes—
120C22×D17#9511yescharacteristic
121C2×C4×C17—24no—
122C2×C4×C17—22no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—44no—
126C2×C4×C17—44no—
127C2×C4×C17—44no—
128C2×C4×C17C2×C4211yescharacteristic
129C2×C4×C17C2×C22⋊C411yescharacteristic
130C2×C4×C17C2×C4⋊C412yes—
131C2×Dic17—24no—
132C2×Dic17—22no—
133C2×Dic17—42no—
134C2×Dic17—41no—
135C2×Dic17—44no—
136C2×Dic17—44no—
137C2×Dic17—44no—
138C2×Dic17#9511yescharacteristic
139C2×Dic17#9512yes—
140C2×Dic17#9511yescharacteristic
141C4×D17—22no—
142C4×D17—24no—
143C4×D17—24no—
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—24no—
147C4×D17—42no—
148C4×D17—42no—
149C4×D17—42no—
150C4×D17#9914yes—
151C8×C17—42no—
152#3622—42no—
153C2×#407—171no—
154#2280—172no—
155#1297—171no—
156#1297—172no—
157#1297D1711yescharacteristic, complemented
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—21no—
162C22×C17—44no—
163C22×C17C2×#9511yescharacteristic
164C22×C17#42114yes—
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—22no—
168C2×D17—22no—
169C2×D17—22no—
170C2×D17—21no—
171C2×D17—48no—
172C2×D17—44no—
173C2×D17—48no—
174C2×D17—44no—
175C2×D17#34212yes—
176C4×C17—21no—
177C4×C17—22no—
178C4×C17—84no—
179C4×C17C2×#9912yes—
180Dic17—21no—
181Dic17—22no—
182Dic17—84no—
183Dic17#34212yes—
184C24×C4—171no—
185C22×M16—172no—
186C2×#117—342no—
187#407—171no—
188#407—172no—
189#407—342no—
190#407C2×D1711yescharacteristic
191C2×C22⋊C8—342no—
192#381—174no—
193#380—174no—
194#380—342no—
195#379—342no—
196C4×C22⋊C4—172no—
197C4×C22⋊C4—342no—
198C4×C22⋊C4C2×D1712yes—
199C2×C17—24no—
200C2×C17—41no—
201C2×C17#173312yes—
202C2×C17C2×#34211yescharacteristic
203D17—22no—
204D17—42no—
205D17—41no—
206C25—171no—
207C23×C4—171no—
208C23×C4—171no—
209C23×C4—172no—
210C23×C4—172no—
211C23×C4—172no—
212C23×C4—171no—
213C23×C4—172no—
214C23×C4—171no—
215C23×C4—171no—
216C23×C4—684no—
217C23×C4C22×D1711yescharacteristic
218C2×M16—172no—
219C2×M16—342no—
220C2×M16—684no—
221C2×M16C4×D1712yes—
222C22×C8—342no—
223#117—688no—
224C2×C4⋊C4—344no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—344no—
227C2×C22⋊C4—344no—
228C2×C22⋊C4—342no—
229C2×C42—22no—
230C2×C42—342no—
231C22⋊C8—22no—
232C22⋊C8—344no—
233C22⋊C8—342no—
234#95—174no—
235#95—344no—
236#95C4×D1714yes—
237C17#804911yescharacteristic, complemented, Sylow
238C24—172no—
239C24—172no—
240C24—171no—
241C24—171no—
242C24—684no—
243C24—682no—
244C24D4×D1711yescharacteristic
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—171no—
248C22×C4—172no—
249C22×C4—171no—
250C22×C4—172no—
251C22×C4—171no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—172no—
255C22×C4—171no—
256C22×C4—172no—
257C22×C4—172no—
258C22×C4—172no—
259C22×C4—172no—
260C22×C4—172no—
261C22×C4—348no—
262C22×C4—344no—
263C22×C4—344no—
264C22×C4—684no—
265C22×C4—682no—
266C22×C4—684no—
267C22×C4—682no—
268C22×C4—684no—
269C22×C4—682no—
270C22×C4—682no—
271C22×C4C2×C4×D1711yescharacteristic
272C22×C4C2×C4×D1712yes—
273C22×C4D4×D1712yes—
274C22×C4Q8×D1711yescharacteristic
275M16—42no—
276M16—684no—
277M16—682no—
278C2×C8—22no—
279C2×C8—342no—
280C2×C8—684no—
281C4⋊C4—688no—
282C22⋊C4—44no—
283C22⋊C4—688no—
284C22⋊C4—688no—
285C22⋊C4—684no—
286C42—44no—
287C42—684no—
288C23—24no—
289C23—41no—
290C23—171no—
291C23—171no—
292C23—172no—
293C23—344no—
294C23—342no—
295C23—344no—
296C23—344no—
297C23—682no—
298C23—681no—
299C23—682no—
300C23—682no—
301C23—682no—
302C23—682no—
303C23—6816no—
304C23C22⋊C4×D1712yes—
305C23C4⋊C4×D1711yescharacteristic
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—171no—
314C2×C4—174no—
315C2×C4—172no—
316C2×C4—171no—
317C2×C4—342no—
318C2×C4—344no—
319C2×C4—342no—
320C2×C4—344no—
321C2×C4—344no—
322C2×C4—342no—
323C2×C4—342no—
324C2×C4—344no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—682no—
329C2×C4—681no—
330C2×C4—684no—
331C2×C4—684no—
332C2×C4—684no—
333C2×C4C42×D1711yescharacteristic
334C2×C4C22⋊C4×D1711yescharacteristic
335C2×C4C4⋊C4×D1712yes—
336C8—42no—
337C8—682no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—22no—
342C22—44no—
343C22—172no—
344C22—342no—
345C22—341no—
346C22—342no—
347C22—342no—
348C22—342no—
349C22—342no—
350C22—688no—
351C22—688no—
352C22—684no—
353C22—684no—
354C22#95×D1711yescharacteristic
355C22#1176282414yes—
356C4—21no—
357C4—22no—
358C4—84no—
359C4—172no—
360C4—342no—
361C4—341no—
362C4—1364no—
363C4#99×D1712yes—
364C2—24no—
365C2—41no—
366C2—342no—
367C2—682no—
368C2—681no—
369C2#1452849212yes—
370C2D17×#34211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D17
commutatorC2×C4×C17C2×C42
FrattiniC23×C4C22×D17
FittingC17×#1297C2
socleC22×C17C2×#95
prime radicalC24×D17D4
cosocleC23×C4×C17C23
hypercenter#1297D17
hypocenterC17#8049
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #14526670C2semidirect
4. #14526669C2semidirect
5. C17×#1297C2semidirect
7. C2×M16×D17C4semidirect
10. #11762772C4semidirect
13. #11762671C4semidirect
14. #11762668C4semidirect
157. #1297D17semidirect
237. C17#8049semidirect

Maximal subgroups

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

subgroupindex
D17×#4072
#145311652
#145266702
#145266692
C17×#12972
#804917