Finite Group Databasebeta

← #17008519

Subgroups — #17008519

Order 4352.

950 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 183 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#14539605C212yesmaximal
2#14534164C211yesmaximal, characteristic
3#14527792C214yesmaximal, complemented
4#14527758C214yesmaximal, complemented
5#14527304C212yesmaximal
6#14527114C211yesmaximal, characteristic
7C2×C17×#544C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763139—22no—
10#11763137C2214yes—
11#11763065—24no—
12#11762965—22no—
13#11762964C2214yes—
14#11762773C2212yes—
15#11762739C2212yes—
16#11762729C2212yes—
17#11762704—22no—
18#11762682—21no—
19#11762682—21no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762546C2218yes—
22#11762535C2214yes—
23C17×#544—22no—
24C17×#544—24no—
25C17×#544C2214yes—
26C2×C17×C4⋊Q8—22no—
27C2×C17×#125—22no—
28C2×C17×C22⋊Q8—21no—
29C2×C17×C22⋊Q8C2212yes—
30C2×C17×#117—21no—
31C22×C4⋊C4×C17C2211yescharacteristic
32C22×Q8×C17—21no—
33C22×Q8×C17D412yes—
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C22×Dic34C2311yescharacteristic
38C17×C4⋊Q8—28no—
39C17×C4⋊Q8—28no—
40C17×#125—28no—
41C17×#125—28no—
42C17×C22⋊Q8—24no—
43C17×C22⋊Q8—24no—
44C17×C22⋊Q8—22no—
45C17×C22⋊Q8—22no—
46C17×C22⋊Q8D414yes—
47C17×C22⋊Q8C2314yes—
48C17×#117—24no—
49C17×#117—24no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—21no—
53C2×C4⋊C4×C17—22no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17D412yes—
58C2×C4⋊C4×C17D412yes—
59C2×C4⋊C4×C17C2312yes—
60C2×C22⋊C4×C17—22no—
61C2×C42×C17—22no—
62C2×C22⋊Dic17—21no—
63C2×C22⋊Dic17—21no—
64C2×#63231—24no—
65C2×C4⋊Dic17C2311yescharacteristic
66#10587610—22no—
67#10587610—22no—
68C2×#63225—21no—
69C2×#63225—22no—
70C2×#63225—22no—
71C2×C4×Dic17—21no—
72C2×#63223—22no—
73#10587506—24no—
74#10587502—22no—
75#10587502—22no—
76#10587502C2312yes—
77C22⋊#3622—22no—
78C22⋊#3622C2314yes—
79#10587481—24no—
80#10587479—24no—
81C24×C17C2×D411yescharacteristic
82C2×Q8×C17—24no—
83C2×Q8×C17—22no—
84C2×Q8×C17—82no—
85C2×Q8×C17—82no—
86C2×Q8×C17C2×D414yes—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—24no—
90C22×C4×C17—21no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—21no—
94C22×C4×C17—81no—
95C22×C4×C17—81no—
96C22×C4×C17C2×D412yes—
97C22×C4×C17C2×D412yes—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D412yes—
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2411yescharacteristic
102C22×Dic17—22no—
103C22×Dic17—21no—
104C22×Dic17—21no—
105C2×Dic34—22no—
106C2×Dic34—81no—
107C4⋊C4×C17—24no—
108C4⋊C4×C17—28no—
109C4⋊C4×C17—24no—
110C4⋊C4×C17—24no—
111C4⋊C4×C17—22no—
112C4⋊C4×C17—24no—
113C4⋊C4×C17—24no—
114C4⋊C4×C17—24no—
115C4⋊C4×C17—24no—
116C4⋊C4×C17C2×D414yes—
117C22⋊C4×C17—24no—
118C22⋊C4×C17—24no—
119C42×C17—28no—
120C22⋊Dic17—24no—
121C22⋊Dic17—41no—
122C22⋊Dic17—41no—
123#63231—84no—
124C4⋊Dic17—22no—
125#63225—42no—
126#63225—44no—
127#63225—82no—
128C4×Dic17—81no—
129#63223—82no—
130C2×#3622—24no—
131#18327—171nomaximal, Sylow
132C23×C17—22no—
133C23×C17—22no—
134C23×C17—41no—
135C23×C17#12011yescharacteristic
136C23×C17#12011yescharacteristic
137C23×C17C22×D411yescharacteristic
138Q8×C17—44no—
139Q8×C17—82no—
140Q8×C17—84no—
141C2×C4×C17—24no—
142C2×C4×C17—24no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—24no—
146C2×C4×C17—24no—
147C2×C4×C17—22no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—41no—
152C2×C4×C17—42no—
153C2×C4×C17—44no—
154C2×C4×C17—44no—
155C2×C4×C17—42no—
156C2×C4×C17—81no—
157C2×C4×C17#12012yes—
158C2×C4×C17#13714yes—
159C2×C4×C17C22×D412yes—
160C2×Dic17—24no—
161C2×Dic17—42no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—42no—
167Dic34—44no—
168#3622—44no—
169C2×#544D1711yescharacteristic
170#3195—172no—
171#2583—171no—
172#1611—174no—
173#1605—174no—
174#1508—172no—
175#1463—171no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—44no—
184C22×C17#45612yes—
185C22×C17C2×#12011yescharacteristic
186C22×C17C2×#13712yes—
187C4×C17—24no—
188C4×C17—22no—
189C4×C17—22no—
190C4×C17—44no—
191C4×C17—42no—
192C4×C17—44no—
193Dic17—44no—
194Dic17—81no—
195Dic17—81no—
196#544—24no—
197#544—22no—
198#544C2×D1714yes—
199C2×C4⋊Q8—22no—
200C2×#125—22no—
201C2×C22⋊Q8—21no—
202C2×C22⋊Q8—171no—
203C2×C22⋊Q8C2×D1712yes—
204C2×#117—21no—
205C22×C4⋊C4C2×D1711yescharacteristic
206#483—174no—
207#483—342no—
208#470—344no—
209#451—174no—
210#451—342no—
211#407—172no—
212#398—172no—
213#395—172no—
214#389—342no—
215#385—341no—
216#385—341no—
217C4×C22⋊C4—171no—
218#328—178no—
219#324—174no—
220C2×C17—22no—
221C2×C17—22no—
222C2×C17—41no—
223C2×C17#300411yescharacteristic
224C2×C17C2×#45611yescharacteristic
225C2×C17#302611yescharacteristic
226C22×Q8—21no—
227C22×Q8—171no—
228C22×Q8C22⋊D1712yes—
229C23×C4C22⋊D1711yescharacteristic
230C23×C4C22⋊D1711yescharacteristic
231C23×C4C22×D1711yescharacteristic
232C2×Q16—344no—
233C2×M16—342no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236#125—28no—
237#125—28no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—342no—
243C22⋊Q8—342no—
244C22⋊Q8C22⋊D1714yes—
245C22⋊Q8C22×D1714yes—
246#117—24no—
247#117—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—341no—
257C2×C4⋊C4—342no—
258C2×C4⋊C4—342no—
259C2×C4⋊C4C22⋊D1712yes—
260C2×C4⋊C4C22⋊D1712yes—
261C2×C4⋊C4C22×D1712yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—341no—
264C2×C22⋊C4—341no—
265C2×C42—22no—
266C2×C42—341no—
267#106—344no—
268#103—344no—
269#103—344no—
270C22⋊C8—174no—
271C22⋊C8—342no—
272#95—172no—
273#95—342no—
274#95—342no—
275C17#1832711yescharacteristic, complemented, Sylow
276C24C2×C22⋊D1711yescharacteristic
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—342no—
282C2×Q8—1361no—
283C2×Q8C2×C22⋊D1714yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—342no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4C2×C22⋊D1712yes—
297C22×C4C2×C22⋊D1712yes—
298C22×C4C2×C22⋊D1712yes—
299C22×C4C2×C22⋊D1712yes—
300C22×C4C2×C22⋊D1711yescharacteristic
301C22×C4C23×D1711yescharacteristic
302Q16—1364no—
303M16—1362no—
304C2×C8—344no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—342no—
315C4⋊C4—684no—
316C4⋊C4—682no—
317C4⋊C4—1362no—
318C4⋊C4C2×C22⋊D1714yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—344no—
322C22⋊C4—681no—
323C22⋊C4—681no—
324C42—28no—
325C42—1361no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23#1058763911yescharacteristic
330C23#1058763911yescharacteristic
331C23C22×C22⋊D1711yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—84no—
335Q8—684no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—344no—
353C2×C4—682no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4—682no—
359C2×C4#1058762814yes—
360C2×C4#1058763912yes—
361C2×C4C22×C22⋊D1712yes—
362C8—684no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#1176299212yes—
372C22C2×#1058763911yescharacteristic
373C22C2×#1058762812yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—44no—
380C4—684no—
381C4—1361no—
382C4—1361no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2#1453801811yescharacteristic
387C2C2×#1176299211yescharacteristic
388C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#544C2
socleC22×C17C2×#120
prime radicalC24×C17C2×D4
hypercenterC2×#544D17
hypocenterC17#18327
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #14527792C2semidirect
4. #14527758C2semidirect
275. C17#18327semidirect

Maximal subgroups

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

subgroupindex
#145396052
#145341642
#145277922
#145277582
#145273042
#145271142
C2×C17×#5442
#1832717