Finite Group Databasebeta

← #17951550

Subgroups — #17951550

Order 4352.

994 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 148 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#14545228C212yesmaximal
2#14543683C214yesmaximal
3#14531164C211yesmaximal, characteristic
4C2×#10587677—22no—
5#11763752—22no—
6#11763751—22no—
7#11763751C414yescomplemented
8#11763750C2214yes—
9C4×C22⋊#595—22no—
10C4×C22⋊#595C414yescomplemented
11C23×C4×D17C2211yescharacteristic
12C2×#10587563C412yescomplemented
13#11762672C411yescharacteristic, complemented
14#11762668—22no—
15#11762664C2212yes—
16#11762656—22no—
17C17×#381C411yescharacteristic, complemented
18C24×D17C2311yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C23×Dic17C2×C411yescharacteristic
21C22×C4×D17—44no—
22C22×C4×D17C2×C412yes—
23C22×C4×D17C2×C412yes—
24C22×C4×D17D412yes—
25C22×C4×D17D412yes—
26C22×C4×D17D412yes—
27C22×C4×D17Q812yes—
28C2×C22⋊#595—24no—
29C2×C22⋊#595—24no—
30C2×C22⋊#595C2×C418yes—
31#10587677—48no—
32C2×C4⋊#595—24no—
33C2×C4×#595—24no—
34C2×C22⋊C4×C17C2×C412yes—
35C2×C22⋊Dic17C2×C412yes—
36C2×#63227—24no—
37C2×#63225—22no—
38C2×C4×Dic17—22no—
39#10587563—44no—
40(C2×#595)⋊C4—24no—
41(C2×#595)⋊C4—24no—
42C17×#95—22no—
43#10587502—22no—
44C24×C17C22×C411yescharacteristic
45C23×D17—44no—
46C23×D17—42no—
47C23×D17C22×C412yes—
48C23×D17C2×D411yescharacteristic
49C23×D17C2×D412yes—
50C23×D17C2×Q811yescharacteristic
51C22×#595—28no—
52C22×C4×C17—22no—
53C22×C4×C17—42no—
54C22×C4×C17C4211yescharacteristic
55C22×C4×C17C4211yescharacteristic
56C22×C4×C17C22⋊C412yes—
57C22×C4×C17C4⋊C412yes—
58C22×Dic17—24no—
59C22×Dic17—22no—
60C22×Dic17C4212yes—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C4⋊C412yes—
64C2×C4×D17—28no—
65C2×C4×D17—44no—
66C2×C4×D17—44no—
67C2×C4×D17—44no—
68C2×C4×D17—44no—
69C2×C4×D17—44no—
70C2×C4×D17C22⋊C412yes—
71C2×C4×D17C22⋊C412yes—
72C2×C4×D17C22⋊C412yes—
73C2×C4×D17C22⋊C412yes—
74C2×C4×D17C4⋊C412yes—
75C2×C4×D17C4⋊C412yes—
76C2×C4×D17C4⋊C412yes—
77C2×C4×D17C4⋊C412yes—
78C22⋊#595—48no—
79C22⋊#595—48no—
80C22⋊#595—48no—
81C4⋊#595—48no—
82C4×#595—48no—
83C22⋊C4×C17—42no—
84C22⋊Dic17—42no—
85#63227—48no—
86#63225—44no—
87C4×Dic17—44no—
88#8102—171nomaximal, Sylow
89C23×C17—22no—
90C23×C17—42no—
91C23×C17C2×C4211yescharacteristic
92C23×C17C2×C22⋊C411yescharacteristic
93C23×C17C2×C4⋊C411yescharacteristic
94C22×D17—24no—
95C22×D17—24no—
96C22×D17—24no—
97C22×D17—24no—
98C22×D17—42no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—42no—
102C22×D17—42no—
103C22×D17—42no—
104C22×D17—416no—
105C22×D17C2×C22⋊C412yes—
106C22×D17C2×C4⋊C412yes—
107C2×#595—48no—
108C2×#595—48no—
109C2×#595—48no—
110C2×C4×C17—24no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—44no—
115C2×C4×C17—42no—
116C2×C4×C17#9512yes—
117C2×C4×C17#9512yes—
118C2×Dic17—22no—
119C2×Dic17—24no—
120C2×Dic17—24no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17#9512yes—
124C2×Dic17#9512yes—
125C2×Dic17#9914yes—
126C4×D17—24no—
127C4×D17—24no—
128C4×D17—24no—
129C4×D17—24no—
130C4×D17—42no—
131C4×D17—42no—
132C4×D17—42no—
133C4×D17—42no—
134#2280—171no—
135#2280—172no—
136#1304—174no—
137C22×C17—22no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—44no—
143C22×C17C2×#9511yescharacteristic
144C22×C17C2×#9912yes—
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17—44no—
151C2×D17—48no—
152C2×D17—48no—
153C2×D17—44no—
154C2×D17#41014yes—
155#595—88no—
156C4×C17—22no—
157C4×C17—22no—
158C4×C17—82no—
159Dic17—22no—
160Dic17—82no—
161Dic17#34214yes—
162C24×C4—171no—
163C2×#117—172no—
164C2×#117—342no—
165#381—171no—
166#381—342no—
167#381#59511yescharacteristic, complemented
168#380—174no—
169#380—342no—
170#380—342no—
171#379—174no—
172#379—172no—
173C4×C22⋊C4—174no—
174C4×C22⋊C4—342no—
175C4×C22⋊C4—342no—
176C2×C17—22no—
177C2×C17—41no—
178C2×C17—41no—
179C2×C17C2×#34211yescharacteristic
180C2×C17#174012yes—
181D17—24no—
182D17—42no—
183C25—171no—
184C23×C4—171no—
185C23×C4—172no—
186C23×C4—172no—
187C23×C4—172no—
188C23×C4—172no—
189C23×C4—172no—
190C23×C4—172no—
191C23×C4—684no—
192C23×C4C2×#59511yescharacteristic
193#117—684no—
194#117—688no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—344no—
197C2×C22⋊C4—178no—
198C2×C22⋊C4—172no—
199C2×C22⋊C4—344no—
200C2×C22⋊C4—344no—
201C2×C22⋊C4—344no—
202C2×C22⋊C4C2×#59512yes—
203C2×C42—342no—
204C2×C42—344no—
205#95—22no—
206#95—344no—
207#95—344no—
208#95—342no—
209C17#810211yescharacteristic, complemented, Sylow
210C24—171no—
211C24—172no—
212C24—172no—
213C24—171no—
214C24—682no—
215C24—684no—
216C24C22×#59511yescharacteristic
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—171no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—171no—
223C22×C4—172no—
224C22×C4—172no—
225C22×C4—172no—
226C22×C4—172no—
227C22×C4—172no—
228C22×C4—172no—
229C22×C4—172no—
230C22×C4—172no—
231C22×C4—348no—
232C22×C4—344no—
233C22×C4—348no—
234C22×C4—342no—
235C22×C4—684no—
236C22×C4—684no—
237C22×C4—684no—
238C22×C4—684no—
239C22×C4—684no—
240C22×C4C4×#59511yescharacteristic
241C22×C4C4×#59511yescharacteristic
242C22×C4C4⋊#59512yes—
243C22×C4C22⋊#59512yes—
244C4⋊C4—684no—
245C4⋊C4—688no—
246C22⋊C4—42no—
247C22⋊C4—688no—
248C22⋊C4—688no—
249C22⋊C4—688no—
250C22⋊C4—682no—
251C22⋊C4—688no—
252C42—684no—
253C42—688no—
254C23—22no—
255C23—42no—
256C23—172no—
257C23—172no—
258C23—344no—
259C23—344no—
260C23—344no—
261C23—344no—
262C23—682no—
263C23—681no—
264C23—681no—
265C23—682no—
266C23—682no—
267C23—682no—
268C23—6816no—
269C23C2×C4×#59511yescharacteristic
270C23C2×C4⋊#59511yescharacteristic
271C23C2×C22⋊#59511yescharacteristic
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—172no—
279C2×C4—174no—
280C2×C4—172no—
281C2×C4—344no—
282C2×C4—344no—
283C2×C4—344no—
284C2×C4—344no—
285C2×C4—344no—
286C2×C4—344no—
287C2×C4—342no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—684no—
293C2×C4—688no—
294C2×C4—688no—
295C2×C4—688no—
296C2×C4—682no—
297C2×C4(C2×#595)⋊C412yes—
298C2×C4(C2×#595)⋊C412yes—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—22no—
304C22—44no—
305C22—174no—
306C22—342no—
307C22—342no—
308C22—342no—
309C22—342no—
310C22—342no—
311C22—688no—
312C22—684no—
313C22—688no—
314C22—684no—
315C22C2×(C2×#595)⋊C411yescharacteristic
316C22D17⋊#9912yes—
317C4—22no—
318C4—22no—
319C4—82no—
320C4—174no—
321C4—342no—
322C4—1362no—
323C4—1368no—
324C2—22no—
325C2—41no—
326C2—41no—
327C2—344no—
328C2—682no—
329C2#1454448111yescharacteristic
330C2#1454448212yes—
subgroupG/H
centerC22C2×(C2×#595)⋊C4
commutatorC23×C17C2×C42
FrattiniC24C22×#595
FittingC17×#381C4
socleC22×C17C2×#95
prime radical, cosocleC24×D17C23
hypercenter#381#595
hypocenterC17#8102
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
7. #11763751C4semidirect
10. C4×C22⋊#595C4semidirect
12. C2×#10587563C4semidirect
13. #11762672C4semidirect
17. C17×#381C4semidirect
167. #381#595semidirect
209. C17#8102semidirect

Maximal subgroups

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

subgroupindex
#145452282
#145436832
#145311642
#810217