Finite Group Databasebeta

← #17240750

Subgroups — #17240750

Order 4352.

1778 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537048C211yesmaximal, characteristic
2#14537048C212yesmaximal
3#14537041C214yesmaximal, complemented
4#14537039C218yesmaximal, complemented
5#14537037C211yesmaximal, characteristic
6#14532058C218yesmaximal, complemented
7#14532052C214yesmaximal, complemented
8#14531167C212yesmaximal
9C17×#2280C211yesmaximal, characteristic
10C2×C4×Dic34C2214yescomplemented
11C2×#10587612C2214yes—
12#11762751C2218yescomplemented
13#11762748C2214yescomplemented
14#11762745C2218yes—
15#11762745C2214yes—
16#11762743C2218yescomplemented
17#11762742C2218yescomplemented
18#11762741C2218yescomplemented
19#11762732C2218yes—
20#11762732C2218yes—
21#11762732C2218yes—
22#11762730C2218yescomplemented
23#11762729C2214yescomplemented
24#11762721C2218yes—
25#11762721C2214yes—
26#11762672C2214yes—
27#11762669C2214yes—
28#11762669C2214yes—
29#11762665C2212yes—
30#11762665C2212yes—
31C4×#63225C2218yescomplemented
32C4×C4⋊Dic17C2214yescomplemented
33C4×C22⋊Dic17C2214yes—
34C4×C22⋊Dic17C2214yes—
35C24×C4×C17C2211yescharacteristic
36C2×C17×#117C2212yes—
37C17×#381C2212yes—
38C17×#380C2214yes—
39C17×#379C2211yescharacteristic
40C17×#379C2211yescharacteristic
41C4×C22⋊C4×C17C2214yes—
42C25×C17C2311yescharacteristic
43C23×C4×C17—44no—
44C23×C4×C17Q818yescomplemented
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2312yes—
47C23×C4×C17C2312yes—
48C23×C4×C17C2311yescharacteristic
49C22×Dic34C2314yes—
50C17×#117—44no—
51C2×C4⋊C4×C17C2314yes—
52C2×C22⋊C4×C17C2312yes—
53C2×C22⋊C4×C17C2312yes—
54C2×C22⋊C4×C17C2314yes—
55C2×C22⋊C4×C17C2314yes—
56C2×C42×C17C2314yes—
57C2×C22⋊Dic17C2314yes—
58C2×C22⋊Dic17C2314yes—
59C2×C22⋊Dic17C2314yes—
60C2×C22⋊Dic17C2314yes—
61C2×C22⋊Dic17C2314yes—
62C2×C22⋊Dic17C2314yes—
63#10587612—48no—
64C2×C4⋊Dic17C2318yes—
65C2×C4⋊Dic17C2318yes—
66C2×C4⋊Dic17C2314yes—
67C2×#63225C2318yes—
68C2×#63225C2318yes—
69C2×#63225C2318yes—
70C2×C4×Dic17C2318yes—
71C4×Dic34—48no—
72C17×#95C2318yes—
73#10587502C2318yes—
74#10587502C2314yes—
75#10587502C2314yes—
76#10587502C2318yes—
77#10587502C2318yes—
78#10587502C2318yes—
79C24×C17—42no—
80C24×C17—44no—
81C24×C17C2×Q814yes—
82C24×C17C2412yes—
83C24×C17C2411yescharacteristic
84C22×C4×C17—28no—
85C22×C4×C17—24no—
86C22×C4×C17—416no—
87C22×C4×C17—44no—
88C22×C4×C17—42no—
89C22×C4×C17C2×Q814yes—
90C22×C4×C17C2×Q818yes—
91C22×C4×C17C2×Q818yes—
92C22×C4×C17C2×Q814yes—
93C22×C4×C17C2414yes—
94C22×C4×C17C2414yes—
95C22×C4×C17C2411yescharacteristic
96C22×C4×C17C2412yes—
97C22×C4×C17C2411yescharacteristic
98C22×Dic17C2418yes—
99C22×Dic17C2418yes—
100C2×Dic34—48no—
101C2×Dic34—44no—
102C4⋊C4×C17—44no—
103C22⋊C4×C17—44no—
104C22⋊C4×C17—44no—
105C22⋊C4×C17—42no—
106C22⋊C4×C17—42no—
107C42×C17—28no—
108C22⋊Dic17—44no—
109C22⋊Dic17—44no—
110C22⋊Dic17—44no—
111C22⋊Dic17—44no—
112C22⋊Dic17—44no—
113C22⋊Dic17—44no—
114C4⋊Dic17—44no—
115C4⋊Dic17—48no—
116C4⋊Dic17—48no—
117#63225—48no—
118#63225—48no—
119#63225—48no—
120C4×Dic17—216no—
121#36674—171nomaximal, Sylow
122C23×C17—24no—
123C23×C17—22no—
124C23×C17—24no—
125C23×C17—24no—
126C23×C17—24no—
127C23×C17—44no—
128C23×C17—48no—
129C23×C17—416no—
130C23×C17C22×Q814yes—
131C23×C17C22×Q812yes—
132C23×C17C2511yescharacteristic
133C2×C4×C17—22no—
134C2×C4×C17—24no—
135C2×C4×C17—216no—
136C2×C4×C17—28no—
137C2×C4×C17—24no—
138C2×C4×C17—24no—
139C2×C4×C17—28no—
140C2×C4×C17—28no—
141C2×C4×C17—48no—
142C2×C4×C17—44no—
143C2×C4×C17C22×Q812yes—
144C2×C4×C17C22×Q814yes—
145C2×C4×C17C22×Q812yes—
146C2×C4×C17H(5,𝔽2)14yes—
147C2×Dic17—28no—
148C2×Dic17—28no—
149C2×Dic17—28no—
150C2×Dic17—28no—
151C2×Dic17—28no—
152C2×Dic17—28no—
153Dic34—416no—
154#2842—172no—
155#2842—171no—
156#2841—174no—
157#2841—178no—
158#2840—171no—
159#2396—178no—
160#2396—174no—
161#2280—172no—
162#2280D1711yescharacteristic
163C22×C17—24no—
164C22×C17—22no—
165C22×C17—24no—
166C22×C17—24no—
167C22×C17—24no—
168C22×C17—416no—
169C22×C17—48no—
170C22×C17—44no—
171C22×C17C23×Q811yescharacteristic
172C22×C17C2×H(5,𝔽2)12yes—
173C22×C17#58514yes—
174C4×C17—22no—
175C4×C17—24no—
176C4×C17—44no—
177C4×C17—44no—
178C4×C17#54314yes—
179Dic17—48no—
180Dic17—48no—
181C24×C4C2×D1711yescharacteristic
182C2×C4×Q8—174no—
183C2×#117—174no—
184C2×#117C2×D1712yes—
185#400—174no—
186#400—178no—
187#399—178no—
188#399—174no—
189#398—178no—
190#398—178no—
191#398—178no—
192#396—178no—
193#396—178no—
194#396—178no—
195#395—174no—
196#395—178no—
197#393—178no—
198#393—174no—
199#381—174no—
200#381C2×D1712yes—
201#380—174no—
202#380—174no—
203#380C2×D1714yes—
204#379—172no—
205#379—172no—
206#379C2×D1711yescharacteristic
207#379C2×D1711yescharacteristic
208C4×C4⋊C4—174no—
209C4×C4⋊C4—178no—
210C4×C22⋊C4—174no—
211C4×C22⋊C4—174no—
212C4×C22⋊C4C2×D1714yes—
213C2×C17—44no—
214C2×C17—42no—
215C2×C17C2×#54311yescharacteristic
216C2×C17#345812yes—
217C2×C17#352914yes—
218C25C22×D1711yescharacteristic
219C22×Q8—174no—
220C23×C4—44no—
221C23×C4Dic3418yescomplemented
222C23×C4C22×D1712yes—
223C23×C4C22×D1711yescharacteristic
224C23×C4C22×D1711yescharacteristic
225C23×C4C22×D1712yes—
226C4×Q8—688no—
227#117—44no—
228#117—688no—
229C2×C4⋊C4—178no—
230C2×C4⋊C4—178no—
231C2×C4⋊C4—174no—
232C2×C4⋊C4—178no—
233C2×C4⋊C4—178no—
234C2×C4⋊C4—178no—
235C2×C4⋊C4C22×D1714yes—
236C2×C22⋊C4—174no—
237C2×C22⋊C4—174no—
238C2×C22⋊C4—174no—
239C2×C22⋊C4—174no—
240C2×C22⋊C4—174no—
241C2×C22⋊C4—174no—
242C2×C22⋊C4C22×D1712yes—
243C2×C22⋊C4C22×D1712yes—
244C2×C22⋊C4C22×D1714yes—
245C2×C22⋊C4C22×D1714yes—
246C2×C42—178no—
247C2×C42C22×D1714yes—
248#95—178no—
249#95—178no—
250#95—178no—
251#95—174no—
252#95—178no—
253#95—174no—
254#95C22×D1718yes—
255C17#3667411yescharacteristic, complemented, Sylow
256C24—42no—
257C24—44no—
258C24C2×Dic3414yes—
259C24C23×D1711yescharacteristic
260C24C23×D1712yes—
261C2×Q8—684no—
262C2×Q8—688no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—416no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—178no—
269C22×C4—178no—
270C22×C4C2×Dic3418yes—
271C22×C4C2×Dic3414yes—
272C22×C4C2×Dic3414yes—
273C22×C4C2×Dic3418yes—
274C22×C4C23×D1714yes—
275C22×C4C23×D1714yes—
276C22×C4C23×D1711yescharacteristic
277C22×C4C23×D1712yes—
278C22×C4C23×D1711yescharacteristic
279C4⋊C4—44no—
280C4⋊C4—688no—
281C4⋊C4—688no—
282C4⋊C4—684no—
283C4⋊C4—688no—
284C4⋊C4—688no—
285C4⋊C4—688no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—684no—
291C22⋊C4—684no—
292C22⋊C4—684no—
293C22⋊C4—684no—
294C22⋊C4—684no—
295C22⋊C4—684no—
296C42—28no—
297C42—3416no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—44no—
304C23—48no—
305C23—416no—
306C23C22×Dic3412yes—
307C23C22×Dic3414yes—
308C23C24×D1711yescharacteristic
309Q8—6816no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—216no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—348no—
321C2×C4—348no—
322C2×C4—348no—
323C2×C4—348no—
324C2×C4—348no—
325C2×C4—348no—
326C2×C4C22×Dic3412yes—
327C2×C4C22×Dic3414yes—
328C2×C4C22×Dic3412yes—
329C2×C4#1058769114yes—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—416no—
336C22—44no—
337C22—48no—
338C22C23×Dic3411yescharacteristic
339C22C2×#1058769112yes—
340C22#1176364514yes—
341C4—22no—
342C4—24no—
343C4—44no—
344C4—44no—
345C4—688no—
346C4—688no—
347C4#1176343514yes—
348C2—42no—
349C2—44no—
350C2C2×#1176343511yescharacteristic
351C2#1454218912yes—
352C2#1454292214yes—
subgroupG/H
centerC22×C4C23×D17
commutator, socle, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC17×#2280C2
prime radicalC25×C17C23
hypercenter#2280D17
hypocenterC17#36674
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
3. #14537041C2semidirect
4. #14537039C2semidirect
6. #14532058C2semidirect
7. #14532052C2semidirect
10. C2×C4×Dic34C22semidirect
12. #11762751C22semidirect
13. #11762748C22semidirect
16. #11762743C22semidirect
17. #11762742C22semidirect
18. #11762741C22semidirect
22. #11762730C22semidirect
23. #11762729C22semidirect
31. C4×#63225C22semidirect
32. C4×C4⋊Dic17C22semidirect
44. C23×C4×C17Q8semidirect
221. C23×C4Dic34semidirect
255. C17#36674semidirect

Maximal subgroups

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

subgroupindex
#145370482
#145370412
#145370392
#145370372
#145320582
#145320522
#145311672
C17×#22802
#3667417