Finite Group Databasebeta

← #17010593

Subgroups — #17010593

Order 4352.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#14539241C211yesmaximal, characteristic, complemented
2#14538875C212yesmaximal, complemented
3#14527929C214yesmaximal, complemented
4#14527867C214yesmaximal, complemented
5#14527510C211yesmaximal, characteristic, complemented
6#14527227C212yesmaximal, complemented
7C2×C17×#555C211yesmaximal, characteristic, complemented
8C2×#10587628—22no—
9#11763359C2211yescharacteristic
10#11763201—22no—
11#11763188C2212yescomplemented
12#11763183C2212yescomplemented
13#11763127—22no—
14#11763099C2214yescomplemented
15#11763004C2214yescomplemented
16#11762943—22no—
17#11762919—24no—
18#11762866C2212yes—
19C4×#63223C2211yescharacteristic
20#11762558C2214yescomplemented
21D4⋊#3622C2218yescomplemented
22C17×#555—24no—
23C17×#555—22no—
24C17×#555C2214yes—
25C2×C17×C4⋊Q8C2211yescharacteristic
26C2×C17×#124—22no—
27C2×C17×#121—22no—
28C2×C4×D4×C17C2212yes—
29C2×C17×#117—22no—
30C22×Q8×C17D412yes—
31C22×D4×C17—22no—
32C22×D4×C17D412yes—
33C23×C4×C17—22no—
34C2×#63252—21no—
35C17×C4⋊Q8—22no—
36C17×C4⋊Q8D412yes—
37C17×C4⋊Q8C2312yes—
38C17×#124—28no—
39C17×#124—28no—
40C17×#121—28no—
41C17×#121—28no—
42C4×D4×C17—22no—
43C4×D4×C17—22no—
44C4×D4×C17D414yes—
45C4×D4×C17C2314yes—
46C17×#117—28no—
47C17×#117—28no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17D412yes—
50C2×C22⋊C4×C17—24no—
51C2×C22⋊C4×C17—22no—
52C2×C42×C17—22no—
53C2×C42×C17C2311yescharacteristic
54C2×#63231—24no—
55#10587628—48no—
56C2×#63230—24no—
57C2×#63228—24no—
58#10587616—22no—
59C2×#63223—21no—
60#10587561—21no—
61#10587560C2311yescharacteristic
62C4×D68—22no—
63#10587555C2311yescharacteristic
64#10587503—24no—
65C22⋊#3622—22no—
66#10587480—24no—
67#10587479—24no—
68C4⋊#3622C2314yes—
69#10587474—21no—
70#10587474—22no—
71C4×#3622C2312yes—
72C24×C17—22no—
73C2×Q8×C17—22no—
74C2×Q8×C17—44no—
75C2×Q8×C17C2×D414yes—
76C2×D4×C17—28no—
77C2×D4×C17—22no—
78C2×D4×C17—44no—
79C2×D4×C17—84no—
80C2×D4×C17C2×D414yes—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—24no—
84C22×C4×C17—24no—
85C22×C4×C17—48no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D412yes—
88C2×C22⋊D17—41no—
89#63252—44no—
90C2×D68—21no—
91C2×D68—22no—
92C2×C4×D17—41no—
93C2×Dic34—21no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—22no—
96C4⋊C4×C17—24no—
97C4⋊C4×C17C2×D414yes—
98C22⋊C4×C17—216no—
99C22⋊C4×C17—24no—
100C22⋊C4×C17—24no—
101C42×C17—28no—
102C42×C17C2×D412yes—
103C42×C17C2×D411yescharacteristic
104C42×C17C2411yescharacteristic
105#63231—48no—
106#63230—48no—
107#63228—48no—
108#63227—42no—
109C4⋊Dic17—22no—
110#63223—24no—
111C2×#3622—22no—
112C2×#3622—24no—
113#18547—171nomaximal, Sylow
114C23×C17—24no—
115C23×C17—44no—
116C23×C17—44no—
117C23×C17—44no—
118C23×C17#12011yescharacteristic
119C22×D17—41no—
120Q8×C17—44no—
121Q8×C17—82no—
122D4×C17—24no—
123D4×C17—24no—
124D4×C17—44no—
125D4×C17—88no—
126C2×C4×C17—22no—
127C2×C4×C17—24no—
128C2×C4×C17—24no—
129C2×C4×C17—24no—
130C2×C4×C17—48no—
131C2×C4×C17—44no—
132C2×C4×C17—44no—
133C2×C4×C17—48no—
134C2×C4×C17#12011yescharacteristic
135C2×C4×C17#12012yes—
136C2×C4×C17C22×D411yescharacteristic
137C2×C4×C17C22×D412yes—
138C22⋊D17—82no—
139C2×Dic17—41no—
140D68—42no—
141D68—44no—
142C4×D17—82no—
143Dic34—42no—
144#3622—24no—
145#3622—44no—
146C2×#555D1711yescharacteristic, complemented
147#3152—171no—
148#3109—172no—
149#1634—174no—
150#1624—174no—
151#1558—171no—
152#1490—172no—
153C22×C17—22no—
154C22×C17—44no—
155C22×C17—44no—
156C22×C17—44no—
157C22×C17—88no—
158C22×C17#45312yes—
159C22×C17C2×#12011yescharacteristic
160C2×D17—82no—
161C2×D17—81no—
162C4×C17—22no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—44no—
166C4×C17—44no—
167C4×C17#57514yes—
168Dic17—81no—
169C2×#137—342no—
170#555—22no—
171#555—24no—
172#555C2×D1714yes—
173#532—171no—
174C2×C4⋊Q8C2×D1711yescharacteristic
175C2×#124—22no—
176C2×#121—22no—
177C2×C4×D4C2×D1712yes—
178C2×#117—22no—
179#497—342no—
180#494—172no—
181#492—172no—
182#481—342no—
183#476—174no—
184#459—174no—
185#447—342no—
186#442—344no—
187#432—172no—
188C4×M16—171no—
189#333—174no—
190D4⋊C8—178no—
191C2×C17—22no—
192C2×C17—84no—
193C2×C17#300512yes—
194C2×C17C2×#45311yescharacteristic
195D17—161no—
196C2×#41—341no—
197C22×Q8C22⋊D1712yes—
198C22×D4—22no—
199C22×D4C22⋊D1712yes—
200C23×C4—22no—
201#137—688no—
202C2×Q16—344no—
203C2×SD16—344no—
204C2×D8—344no—
205C2×M16—341no—
206C4⋊Q8—22no—
207C4⋊Q8—171no—
208C4⋊Q8C22⋊D1712yes—
209C4⋊Q8C22×D1712yes—
210#127—171no—
211#124—28no—
212#124—28no—
213#124—341no—
214#121—28no—
215#121—28no—
216#121—342no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—342no—
220C4×D4C22⋊D1714yes—
221C4×D4C22×D1714yes—
222#117—28no—
223#117—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C22⋊D1712yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C42—22no—
229C2×C42C22×D1711yescharacteristic
230#106—344no—
231C4⋊C8—174no—
232#102—344no—
233#102—344no—
234C22⋊C8—342no—
235#97—341no—
236#97—342no—
237C4×C8—172no—
238C17#1854711yescharacteristic, complemented, Sylow
239C24—22no—
240#41—684no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—341no—
244C2×Q8C2×C22⋊D1714yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—342no—
250C2×D4—341no—
251C2×D4—681no—
252C2×D4C2×C22⋊D1714yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—681no—
259C22×C4C2×C22⋊D1712yes—
260C22×C4C2×C22⋊D1711yescharacteristic
261Q16—688no—
262SD16—688no—
263D8—688no—
264M16—344no—
265C2×C8—342no—
266C2×C8—344no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—342no—
271C4⋊C4C2×C22⋊D1714yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—682no—
276C42—28no—
277C42C2×C22⋊D1711yescharacteristic
278C42C2×C22⋊D1712yes—
279C42C23×D1711yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—681no—
285C23#1058763911yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—682no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—682no—
294D4—684no—
295D4—1362no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—681no—
305C2×C4—1362no—
306C2×C4#1058763911yescharacteristic
307C2×C4#1058763912yes—
308C2×C4C22×C22⋊D1711yescharacteristic
309C2×C4C22×C22⋊D1712yes—
310C8—344no—
311C8—684no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—1362no—
318C22—1361no—
319C22#1176297712yes—
320C22C2×#1058763911yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—44no—
326C4—1361no—
327C4#1176360714yes—
328C2—22no—
329C2—84no—
330C2—2721no—
331C2#1453802612yes—
332C2C2×#1176297711yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#555C2
socleC22×C17C2×#120
hypercenterC2×#555D17
hypocenterC17#18547
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #14539241C2semidirect
2. #14538875C2semidirect
3. #14527929C2semidirect
4. #14527867C2semidirect
5. #14527510C2semidirect
6. #14527227C2semidirect
7. C2×C17×#555C2semidirect
11. #11763188C22semidirect
12. #11763183C22semidirect
14. #11763099C22semidirect
15. #11763004C22semidirect
20. #11762558C22semidirect
21. D4⋊#3622C22semidirect
146. C2×#555D17semidirect
238. C17#18547semidirect

Maximal subgroups

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

subgroupindex
#145392412
#145388752
#145279292
#145278672
#145275102
#145272272
C2×C17×#5552
#1854717