Finite Group Databasebeta

← #17010591

Subgroups — #17010591

Order 4352.

1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1068 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, complemented
2#14538871C212yesmaximal, complemented
3#14527927C214yesmaximal, complemented
4#14527865C214yesmaximal, complemented
5#14527510C211yesmaximal, characteristic, complemented
6#14527228C212yesmaximal, complemented
7C17×#3152C211yesmaximal, characteristic, complemented
8#11763525—24no—
9#11763525—22no—
10#11763525C2214yes—
11C2×#10587633—22no—
12C2×#10587581—22no—
13C2×C4×D68C2212yes—
14C2×#10587578—22no—
15#11763125—22no—
16#11763096C2214yescomplemented
17#11763001C2214yescomplemented
18#11762942—22no—
19#11762919—24no—
20#11762866C2212yes—
21#11762558C2214yescomplemented
22#11762538C2218yescomplemented
23C2×C17×#137—22no—
24C17×#532C2211yescharacteristic
25C2×C17×C4⋊Q8C2211yescharacteristic
26C17×#497—22no—
27C17×#494C2212yescomplemented
28C17×#492C2212yescomplemented
29C4×M16×C17C2211yescharacteristic
30C2×#41×C17—21no—
31C22×Q8×C17D412yes—
32C22×D68—22no—
33C22×D68D412yes—
34C22×C4×D17—22no—
35C17×#137—48no—
36C2×Q16×C17—24no—
37C2×SD16×C17—24no—
38C2×M16×C17—21no—
39C17×C4⋊Q8—22no—
40C17×C4⋊Q8D412yes—
41C17×C4⋊Q8C2311yescharacteristic
42C17×C4⋊Q8C2312yes—
43C17×#127C2311yescharacteristic
44C17×#124—21no—
45C17×#121—22no—
46C4×D4×C17—22no—
47C2×C4⋊C4×C17—22no—
48C2×C42×C17C2311yescharacteristic
49#10587633—28no—
50#10587633—28no—
51C2×#63228—24no—
52C2×#63227—24no—
53C2×#63227—22no—
54C2×C4⋊Dic17D412yes—
55C2×C4×Dic17—22no—
56#10587581—28no—
57#10587581—28no—
58#10587578—28no—
59#10587578—28no—
60C4×D68—22no—
61C4×D68—22no—
62C4×D68D414yes—
63C4×D68C2314yes—
64C17×#97—21no—
65C4×C8×C17C2312yes—
66#10587503—24no—
67C22⋊#3622—22no—
68#10587480—24no—
69#10587479—24no—
70C4⋊#3622C2314yes—
71#10587474—22no—
72C23×D17—22no—
73#41×C17—44no—
74C2×Q8×C17—22no—
75C2×Q8×C17—21no—
76C2×Q8×C17—44no—
77C2×Q8×C17C2×D414yes—
78C2×D4×C17—21no—
79C2×D4×C17—22no—
80C2×D4×C17—41no—
81C22×C4×C17—22no—
82C22×C4×C17—41no—
83C22×C4×C17C2×D412yes—
84C22×C4×C17C2×D411yescharacteristic
85C22×Dic17—22no—
86C2×D68—22no—
87C2×D68—28no—
88C2×D68—44no—
89C2×D68—84no—
90C2×D68C2×D414yes—
91C2×C4×D17—24no—
92C2×C4×D17—24no—
93C2×C4×D17—48no—
94Q16×C17—48no—
95SD16×C17—48no—
96M16×C17—24no—
97C2×C8×C17—22no—
98C4⋊C4×C17—24no—
99C4⋊C4×C17—24no—
100C4⋊C4×C17—22no—
101C22⋊C4×C17—42no—
102C42×C17C2×D412yes—
103C42×C17C2×D411yescharacteristic
104C42×C17C2411yescharacteristic
105#63228—48no—
106#63227—216no—
107#63227—24no—
108#63227—24no—
109C4⋊Dic17—22no—
110C4⋊Dic17C2×D414yes—
111C4×Dic17—28no—
112C2×#3622—24no—
113#18547—171nomaximal, Sylow
114C23×C17—41no—
115C23×C17#12011yescharacteristic
116C22×D17—24no—
117C22×D17—44no—
118C22×D17—44no—
119C22×D17—44no—
120Q8×C17—44no—
121Q8×C17—42no—
122Q8×C17—82no—
123D4×C17—42no—
124D4×C17—44no—
125D4×C17—82no—
126C2×C4×C17—22no—
127C2×C4×C17—24no—
128C2×C4×C17—24no—
129C2×C4×C17—44no—
130C2×C4×C17—41no—
131C2×C4×C17—82no—
132C2×C4×C17#12012yes—
133C2×C4×C17#12011yescharacteristic
134C2×C4×C17C22×D412yes—
135C2×C4×C17C22×D411yescharacteristic
136C2×Dic17—24no—
137C2×Dic17—44no—
138D68—24no—
139D68—24no—
140D68—44no—
141D68—88no—
142C4×D17—48no—
143C4×D17—48no—
144C8×C17—24no—
145#3622—44no—
146C2×#555—171no—
147#3152D1711yescharacteristic, complemented
148#3109—172no—
149#1634—174no—
150#1624—174no—
151#1558—171no—
152#1490—172no—
153C22×C17—22no—
154C22×C17—82no—
155C22×C17—81no—
156C22×C17#45312yes—
157C22×C17C2×#12011yescharacteristic
158C2×D17—44no—
159C2×D17—44no—
160C2×D17—44no—
161C2×D17—88no—
162C4×C17—22no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—44no—
166C4×C17—81no—
167C4×C17#57514yes—
168Dic17—44no—
169C2×#137—22no—
170#555—174no—
171#555—344no—
172#555—342no—
173#532C2×D1711yescharacteristic
174C2×C4⋊Q8C2×D1711yescharacteristic
175C2×#124—342no—
176C2×#121—342no—
177C2×C4×D4—172no—
178C2×#117—342no—
179#497—22no—
180#494C2×D1712yescomplemented
181#492C2×D1712yescomplemented
182#481—342no—
183#476—174no—
184#459—174no—
185#447—342no—
186#442—344no—
187#432—172no—
188C4×M16C2×D1711yescharacteristic
189#333—174no—
190D4⋊C8—178no—
191C2×C17—22no—
192C2×C17—161no—
193C2×C17#300512yes—
194C2×C17C2×#45311yescharacteristic
195D17—84no—
196C2×#41—21no—
197C22×Q8C22⋊D1712yes—
198C22×D4—172no—
199C22×D4—342no—
200C23×C4—342no—
201#137—48no—
202C2×Q16—24no—
203C2×SD16—24no—
204C2×D8—344no—
205C2×M16—21no—
206C4⋊Q8—22no—
207C4⋊Q8C22⋊D1712yes—
208C4⋊Q8C22×D1711yescharacteristic
209C4⋊Q8C22×D1712yes—
210#127C22×D1711yescharacteristic
211#124—21no—
212#124—348no—
213#124—348no—
214#121—22no—
215#121—348no—
216#121—348no—
217C4×D4—22no—
218C4×D4—174no—
219C4×D4—174no—
220C4×D4—342no—
221C4×D4—342no—
222#117—348no—
223#117—348no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—172no—
226C2×C22⋊C4—344no—
227C2×C22⋊C4—342no—
228C2×C42—342no—
229C2×C42C22×D1711yescharacteristic
230#106—344no—
231C4⋊C8—174no—
232#102—344no—
233#102—344no—
234C22⋊C8—342no—
235#97—21no—
236#97—342no—
237C4×C8C22×D1712yes—
238C17#1854711yescharacteristic, complemented, Sylow
239C24—342no—
240#41—44no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—44no—
244C2×Q8C2×C22⋊D1714yes—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—174no—
249C2×D4—348no—
250C2×D4—342no—
251C2×D4—684no—
252C2×D4—1364no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—342no—
256C22×C4—344no—
257C22×C4—344no—
258C22×C4—688no—
259C22×C4D4×D1712yes—
260C22×C4C2×C22⋊D1711yescharacteristic
261Q16—48no—
262SD16—48no—
263D8—688no—
264M16—24no—
265C2×C8—22no—
266C2×C8—344no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—174no—
271C4⋊C4—342no—
272C22⋊C4—42no—
273C22⋊C4—3416no—
274C22⋊C4—344no—
275C22⋊C4—344no—
276C42—348no—
277C42D4×D1712yes—
278C42C2×C22⋊D1711yescharacteristic
279C42C23×D1711yescharacteristic
280C23—41no—
281C23—344no—
282C23—684no—
283C23—684no—
284C23—684no—
285C23#1058762311yescharacteristic
286Q8—44no—
287Q8—42no—
288Q8—82no—
289D4—42no—
290D4—44no—
291D4—82no—
292D4—344no—
293D4—344no—
294D4—684no—
295D4—1368no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—344no—
303C2×C4—688no—
304C2×C4—684no—
305C2×C4—688no—
306C2×C4#1058762311yescharacteristic
307C2×C4#1058762312yes—
308C2×C4C2×D4×D1712yes—
309C2×C4C22×C22⋊D1711yescharacteristic
310C8—24no—
311C8—684no—
312C22—22no—
313C22—82no—
314C22—81no—
315C22—684no—
316C22—684no—
317C22—684no—
318C22—1368no—
319C22#1176297612yes—
320C22C2×#1058762311yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—81no—
326C4—684no—
327C4#1176360814yes—
328C2—22no—
329C2—161no—
330C2—1364no—
331C2#1453802512yes—
332C2C2×#1176297611yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3152C2
socleC22×C17C2×#120
hypercenter#3152D17
hypocenterC17#18547
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#11763525C2semidirect
2. #14538871C2semidirect
3. #14527927C2semidirect
4. #14527865C2semidirect
5. #14527510C2semidirect
6. #14527228C2semidirect
7. C17×#3152C2semidirect
16. #11763096C22semidirect
17. #11763001C22semidirect
21. #11762558C22semidirect
22. #11762538C22semidirect
27. C17×#494C22semidirect
28. C17×#492C22semidirect
147. #3152D17semidirect
180. #494C2×D17semidirect
181. #492C2×D17semidirect
238. C17#18547semidirect

Maximal subgroups

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

subgroupindex
C2×#117635252
#145388712
#145279272
#145278652
#145275102
#145272282
C17×#31522
#1854717