Finite Group Databasebeta

← #17043780

Subgroups — #17043780

Order 4352.

990 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763437C211yesmaximal, characteristic
2#14531385C212yesmaximal
3#14529767C218yesmaximal
4#14528861C211yesmaximal, characteristic
5#14528859C212yesmaximal
6C17×#1820C211yesmaximal, characteristic
7#11763437—24no—
8#11763437—24no—
9C2×#10587555C412yescomplemented
10C2×#10587574C412yescomplemented
11C2×#10587556C412yescomplemented
12C2×#10587576C412yescomplemented
13C2×#10587564C2214yes—
14C2×#10587612C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16#11762694—24no—
17#11762692—24no—
18#11762672C2212yes—
19#11762667C2214yes—
20C22⋊C4×Dic17C2214yes—
21#11762576C2214yes—
22#11762575C2218yes—
23C2×C17×#117C2211yescharacteristic
24C17×#381C2212yes—
25C17×#342C2214yes—
26C23×C4×C17C2311yescharacteristic
27C23×Dic17C2312yes—
28C22×Dic34C2×C414yes—
29C17×#117—22no—
30C17×#117—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17C2314yes—
34C2×C22⋊C4×C17C2312yes—
35C2×C42×C17C2×C412yes—
36C2×C22⋊Dic17C2314yes—
37C2×C22⋊Dic17C2312yes—
38#10587612—22no—
39#10587612—22no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C414yes—
43C2×C4⋊Dic17C2×C412yes—
44C2×C4⋊Dic17C2×C412yes—
45C2×C4⋊Dic17D414yes—
46C2×C4⋊Dic17D414yes—
47C2×#63225—24no—
48C2×#63225C2×C414yes—
49C2×#63225C2×C414yes—
50C2×C4×Dic17—24no—
51C2×C4×Dic17C2×C412yes—
52#10587576—28no—
53#10587574—28no—
54#10587564—28no—
55#10587564—28no—
56#10587556—24no—
57#10587556—24no—
58#10587555—24no—
59#10587555—24no—
60C17×#95—22no—
61C17×#95—22no—
62#10587502—22no—
63#10587502—22no—
64#10587502—24no—
65#10587502—24no—
66C24×C17C2411yescharacteristic
67C22×C4×C17—24no—
68C22×C4×C17—24no—
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C22×C412yes—
72C22×C4×C17C22×C412yes—
73C22×C4×C17C2×D412yes—
74C22×C4×C17C2×D412yes—
75C22×Dic17—24no—
76C22×Dic17—82no—
77C22×Dic17C22×C412yes—
78C22×Dic17C22×C412yes—
79C22×Dic17C22×C414yes—
80C22×Dic17C2×D414yes—
81C22×Dic17C2×D414yes—
82C2×Dic34—28no—
83C2×Dic34—84no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C22⋊C4×C17—24no—
87C22⋊C4×C17—28no—
88C42×C17—22no—
89C42×C17—22no—
90C22⋊Dic17—24no—
91C22⋊Dic17—28no—
92C4⋊Dic17—21no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—28no—
97C4⋊Dic17C22⋊C414yes—
98C4⋊Dic17C22⋊C414yes—
99C4⋊Dic17C22⋊C418yes—
100#63225—28no—
101#63225—28no—
102#63225—84no—
103C4×Dic17—24no—
104C4×Dic17—84no—
105#21828—171nomaximal, Sylow
106C23×C17—41no—
107C23×C17—41no—
108C23×C17C23×C411yescharacteristic
109C23×C17C22×D412yes—
110C23×C172−1+414yes—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—24no—
114C2×C4×C17—24no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—44no—
120C2×C4×C17—44no—
121C2×C4×C17—44no—
122C2×C4×C17C2×C22⋊C411yescharacteristic
123C2×C4×C17C2×C22⋊C411yescharacteristic
124C2×C4×C17C2×C22⋊C412yes—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—44no—
128C2×Dic17—44no—
129C2×Dic17—44no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—44no—
133C2×Dic17—44no—
134C2×Dic17—44no—
135C2×Dic17C2×C22⋊C418yes—
136Dic34—88no—
137C2×#544—171no—
138#2313—172no—
139#1996—178no—
140#1820—172no—
141#1820—171no—
142#1820D1711yescharacteristic
143C22×C17—21no—
144C22×C17—22no—
145C22×C17—21no—
146C22×C17—22no—
147C22×C17—44no—
148C22×C17C22×C22⋊C411yescharacteristic
149C22×C17#51912yes—
150C22×C17#53614yes—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—42no—
155C4×C17—42no—
156C4×C17—84no—
157Dic17—28no—
158Dic17—42no—
159Dic17—42no—
160Dic17—84no—
161#544—344no—
162#544—344no—
163C2×C4⋊Q8—172no—
164C2×C4⋊Q8—172no—
165C2×#125—172no—
166C2×#125—172no—
167C2×C22⋊Q8—174no—
168C2×#117—171no—
169C2×#117C2×D1711yescharacteristic
170C22×C4⋊C4—171no—
171#387—344no—
172#387—344no—
173#381—172no—
174#381C2×D1712yes—
175#380—174no—
176C4×C22⋊C4—174no—
177#342—178no—
178#342—174no—
179#342C2×D1714yes—
180C2×C17—24no—
181C2×C17—22no—
182C2×C17#231011yescharacteristic
183C2×C17#288311yescharacteristic
184C2×C17#288411yescharacteristic
185C22×Q8—174no—
186C23×C4—172no—
187C23×C4C22×D1711yescharacteristic
188C4⋊Q8—344no—
189C4⋊Q8—344no—
190C4⋊Q8—348no—
191#125—344no—
192#125—344no—
193#125—348no—
194C22⋊Q8—348no—
195C22⋊Q8—348no—
196#117—22no—
197#117—22no—
198#117—342no—
199#117—342no—
200C2×C4⋊C4—172no—
201C2×C4⋊C4—172no—
202C2×C4⋊C4—174no—
203C2×C4⋊C4—174no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—174no—
207C2×C4⋊C4—174no—
208C2×C4⋊C4—174no—
209C2×C4⋊C4—344no—
210C2×C4⋊C4C4×D1711yescharacteristic
211C2×C4⋊C4C4×D1711yescharacteristic
212C2×C22⋊C4—172no—
213C2×C22⋊C4—174no—
214C2×C22⋊C4C22×D1714yes—
215C2×C22⋊C4C22×D1712yes—
216C2×C42—172no—
217C2×C42—344no—
218C2×C42C4×D1712yes—
219#95—22no—
220#95—22no—
221#95—344no—
222#95—344no—
223#95—342no—
224#95—342no—
225C17#2182811yescharacteristic, complemented, Sylow
226C24C23×D1711yescharacteristic
227C2×Q8—348no—
228C2×Q8—1364no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—174no—
232C22×C4—174no—
233C22×C4—174no—
234C22×C4—172no—
235C22×C4—172no—
236C22×C4—344no—
237C22×C4—1362no—
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×C4×D1712yes—
241C22×C4C2×C4×D1712yes—
242C22×C4D4×D1712yes—
243C22×C4D4×D1712yes—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—174no—
247C4⋊C4—174no—
248C4⋊C4—178no—
249C4⋊C4—348no—
250C4⋊C4—348no—
251C4⋊C4—348no—
252C4⋊C4—341no—
253C4⋊C4—341no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—1364no—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—344no—
260C22⋊C4—348no—
261C42—22no—
262C42—22no—
263C42—344no—
264C42—1364no—
265C23—41no—
266C23—41no—
267C23C22×C4×D1711yescharacteristic
268C23C2×D4×D1712yes—
269C23#1058769714yes—
270Q8—1368no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—178no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4—684no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—684no—
291C2×C4—684no—
292C2×C4—684no—
293C2×C4C22⋊C4×D1711yescharacteristic
294C2×C4C22⋊C4×D1711yescharacteristic
295C2×C4C22⋊C4×D1712yes—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—22no—
300C22—44no—
301C22C2×C22⋊C4×D1711yescharacteristic
302C22#1176328612yes—
303C22#1176338114yes—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—84no—
310C4—348no—
311C4—682no—
312C4—682no—
313C4—1364no—
314C2—24no—
315C2—22no—
316C2#1453136911yescharacteristic
317C2#1453724311yescharacteristic
318C2#1453724611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC17×#1820C2
socleC22×C17C22×C22⋊C4
prime radical, cosocleC24×C17C24
hypercenter#1820D17
hypocenterC17#21828
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
9. C2×#10587555C4semidirect
10. C2×#10587574C4semidirect
11. C2×#10587556C4semidirect
12. C2×#10587576C4semidirect
225. C17#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#117634372
#145313852
#145297672
#145288612
#145288592
C17×#18202
#2182817