Finite Group Databasebeta

← #17951544

Subgroups — #17951544

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×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D412yes—
28C22×C4×D17Q811yescharacteristic
29C22×C4×D17Q811yescharacteristic
30C2×C22⋊#595—24no—
31C2×C22⋊#595—24no—
32C2×C22⋊#595C2×C418yes—
33#10587677—48no—
34C2×C4⋊#595—24no—
35C2×C4×#595—24no—
36C2×C22⋊C4×C17C2×C412yes—
37C2×C22⋊Dic17C2×C412yes—
38C2×#63227—24no—
39C2×#63225—22no—
40C2×C4×Dic17—22no—
41#10587563—44no—
42(C2×#595)⋊C4—24no—
43(C2×#595)⋊C4—24no—
44C17×#95—22no—
45#10587502—22no—
46C24×C17C22×C411yescharacteristic
47C23×D17—44no—
48C23×D17—42no—
49C23×D17C22×C412yes—
50C23×D17C2×D411yescharacteristic
51C23×D17C2×D412yes—
52C23×D17C2×Q811yescharacteristic
53C22×#595—28no—
54C22×C4×C17—22no—
55C22×C4×C17—42no—
56C22×C4×C17C4212yes—
57C22×C4×C17C22⋊C412yes—
58C22×C4×C17C4⋊C411yescharacteristic
59C22×C4×C17C4⋊C411yescharacteristic
60C22×Dic17—24no—
61C22×Dic17—22no—
62C22×Dic17C4211yescharacteristic
63C22×Dic17C4211yescharacteristic
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C4⋊C411yescharacteristic
67C22×Dic17C4⋊C411yescharacteristic
68C2×C4×D17—28no—
69C2×C4×D17—44no—
70C2×C4×D17—44no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×C4×D17—42no—
74C2×C4×D17—42no—
75C2×C4×D17—44no—
76C2×C4×D17C22⋊C412yes—
77C2×C4×D17C22⋊C412yes—
78C2×C4×D17C22⋊C412yes—
79C2×C4×D17C22⋊C412yes—
80C2×C4×D17C4⋊C412yes—
81C2×C4×D17C4⋊C412yes—
82C2×C4×D17C4⋊C412yes—
83C2×C4×D17C4⋊C412yes—
84C22⋊#595—48no—
85C22⋊#595—48no—
86C22⋊#595—48no—
87C4⋊#595—48no—
88C4×#595—48no—
89C22⋊C4×C17—42no—
90C22⋊Dic17—42no—
91#63227—48no—
92#63225—44no—
93C4×Dic17—44no—
94#8100—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—42no—
97C23×C17C2×C4211yescharacteristic
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17C2×C4⋊C411yescharacteristic
100C22×D17—24no—
101C22×D17—24no—
102C22×D17—24no—
103C22×D17—24no—
104C22×D17—42no—
105C22×D17—41no—
106C22×D17—41no—
107C22×D17—42no—
108C22×D17—42no—
109C22×D17—42no—
110C22×D17—416no—
111C22×D17C2×C22⋊C412yes—
112C22×D17C2×C4⋊C412yes—
113C2×#595—48no—
114C2×#595—48no—
115C2×#595—48no—
116C2×C4×C17—24no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—44no—
121C2×C4×C17—42no—
122C2×C4×C17#9512yes—
123C2×C4×C17#9512yes—
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—44no—
130C2×Dic17—42no—
131C2×Dic17#9511yescharacteristic
132C2×Dic17#9511yescharacteristic
133C2×Dic17#9511yescharacteristic
134C2×Dic17#9511yescharacteristic
135C2×Dic17#9914yes—
136C4×D17—24no—
137C4×D17—24no—
138C4×D17—24no—
139C4×D17—24no—
140C4×D17—42no—
141C4×D17—42no—
142C4×D17—42no—
143C4×D17—42no—
144#2280—171no—
145#2280—172no—
146#1304—174no—
147C22×C17—22no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—44no—
153C22×C17C2×#9511yescharacteristic
154C22×C17C2×#9912yes—
155C2×D17—22no—
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—44no—
161C2×D17—48no—
162C2×D17—48no—
163C2×D17—44no—
164C2×D17#41014yes—
165#595—88no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—82no—
169Dic17—21no—
170Dic17—21no—
171Dic17—82no—
172Dic17#34212yes—
173Dic17#34212yes—
174C24×C4—171no—
175C2×#117—172no—
176C2×#117—342no—
177#381—171no—
178#381—342no—
179#381#59511yescharacteristic, complemented
180#380—174no—
181#380—342no—
182#380—342no—
183#379—174no—
184#379—172no—
185C4×C22⋊C4—174no—
186C4×C22⋊C4—342no—
187C4×C22⋊C4—342no—
188C2×C17—22no—
189C2×C17—41no—
190C2×C17—41no—
191C2×C17C2×#34211yescharacteristic
192C2×C17#174012yes—
193D17—24no—
194D17—42no—
195C25—171no—
196C23×C4—171no—
197C23×C4—172no—
198C23×C4—172no—
199C23×C4—172no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—171no—
203C23×C4—171no—
204C23×C4—172no—
205C23×C4—684no—
206C23×C4C2×#59511yescharacteristic
207#117—688no—
208#117—684no—
209C2×C4⋊C4—342no—
210C2×C4⋊C4—344no—
211C2×C22⋊C4—178no—
212C2×C22⋊C4—172no—
213C2×C22⋊C4—344no—
214C2×C22⋊C4—344no—
215C2×C22⋊C4—344no—
216C2×C22⋊C4C2×#59512yes—
217C2×C42—342no—
218C2×C42—344no—
219#95—22no—
220#95—344no—
221#95—344no—
222#95—342no—
223C17#810011yescharacteristic, complemented, Sylow
224C24—171no—
225C24—172no—
226C24—172no—
227C24—171no—
228C24—682no—
229C24—684no—
230C24C22×#59511yescharacteristic
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4—172no—
247C22×C4—348no—
248C22×C4—344no—
249C22×C4—348no—
250C22×C4—342no—
251C22×C4—684no—
252C22×C4—684no—
253C22×C4—684no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4—682no—
257C22×C4—682no—
258C22×C4C4×#59512yes—
259C22×C4C4⋊#59511yescharacteristic
260C22×C4C4⋊#59511yescharacteristic
261C22×C4C22⋊#59512yes—
262C4⋊C4—684no—
263C4⋊C4—688no—
264C22⋊C4—42no—
265C22⋊C4—688no—
266C22⋊C4—688no—
267C22⋊C4—688no—
268C22⋊C4—682no—
269C22⋊C4—688no—
270C42—684no—
271C42—688no—
272C23—22no—
273C23—42no—
274C23—172no—
275C23—172no—
276C23—344no—
277C23—344no—
278C23—344no—
279C23—344no—
280C23—682no—
281C23—681no—
282C23—681no—
283C23—682no—
284C23—682no—
285C23—682no—
286C23—6816no—
287C23C2×C4×#59511yescharacteristic
288C23C2×C4⋊#59511yescharacteristic
289C23C2×C22⋊#59511yescharacteristic
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—171no—
297C2×C4—171no—
298C2×C4—174no—
299C2×C4—171no—
300C2×C4—171no—
301C2×C4—344no—
302C2×C4—344no—
303C2×C4—344no—
304C2×C4—344no—
305C2×C4—342no—
306C2×C4—342no—
307C2×C4—342no—
308C2×C4—342no—
309C2×C4—342no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—682no—
313C2×C4—682no—
314C2×C4—684no—
315C2×C4—688no—
316C2×C4—688no—
317C2×C4—688no—
318C2×C4—682no—
319C2×C4(C2×#595)⋊C412yes—
320C2×C4(C2×#595)⋊C412yes—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—44no—
327C22—174no—
328C22—342no—
329C22—342no—
330C22—342no—
331C22—342no—
332C22—342no—
333C22—688no—
334C22—684no—
335C22—688no—
336C22—684no—
337C22C2×(C2×#595)⋊C411yescharacteristic
338C22D17⋊#9912yes—
339C4—22no—
340C4—22no—
341C4—82no—
342C4—172no—
343C4—172no—
344C4—341no—
345C4—341no—
346C4—1362no—
347C4—1368no—
348C2—22no—
349C2—41no—
350C2—41no—
351C2—344no—
352C2—682no—
353C2#1454448111yescharacteristic
354C2#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#8100
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
179. #381#595semidirect
223. C17#8100semidirect

Maximal subgroups

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

subgroupindex
#145452282
#145436832
#145311642
#810017