Finite Group Databasebeta

← #17008505

Subgroups — #17008505

Order 4352.

1230 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1230 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#14539590C212yesmaximal, complemented
2#14534164C211yesmaximal, characteristic, complemented
3#14527787C214yesmaximal, complemented
4#14527753C214yesmaximal, complemented
5#14527300C212yesmaximal, complemented
6#14527114C211yesmaximal, characteristic, complemented
7C2×C17×#537C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763144—22no—
10#11763120C2214yescomplemented
11#11763053—24no—
12#11762960—22no—
13#11762947C2214yescomplemented
14#11762773C2212yes—
15#11762739C2212yescomplemented
16#11762729C2212yescomplemented
17#11762704—22no—
18#11762682—21no—
19#11762682—21no—
20C4×C22⋊Dic17C2211yescharacteristic
21#11762543C2218yescomplemented
22#11762535C2214yescomplemented
23C22×#41×C17—21no—
24C17×#537—24no—
25C17×#537—22no—
26C17×#537C2214yes—
27C2×C17×C22⋊Q8—22no—
28C2×C17×#121—22no—
29C2×C17×#121—21no—
30C2×C17×#121C2212yes—
31C22×C4⋊C4×C17C2211yescharacteristic
32C2×#41×C17—24no—
33C2×#41×C17—24no—
34C2×#41×C17—48no—
35C22×Q8×C17—21no—
36C22×D4×C17—21no—
37C22×D4×C17—22no—
38C22×D4×C17—21no—
39C22×D4×C17D412yes—
40C23×C4×C17—22no—
41C23×C4×C17D411yescharacteristic
42C23×C4×C17D411yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C22×Dic34C2311yescharacteristic
45C17×C22⋊Q8—28no—
46C17×C22⋊Q8—28no—
47C17×#121—24no—
48C17×#121—28no—
49C17×#121—28no—
50C17×#121—24no—
51C17×#121—22no—
52C17×#121—22no—
53C17×#121D414yes—
54C17×#121C2314yes—
55C2×C4⋊C4×C17—21no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17D412yes—
59C2×C4⋊C4×C17D412yes—
60C2×C4⋊C4×C17C2312yes—
61C2×C22⋊C4×C17—22no—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊Dic17—21no—
64C2×C22⋊Dic17—21no—
65C2×#63229—24no—
66C2×C4⋊Dic17C2311yescharacteristic
67#10587610—22no—
68#10587610—22no—
69C2×#63225—21no—
70C2×#63225—22no—
71C2×#63225—22no—
72C2×C4×Dic17—21no—
73C2×#63223—22no—
74#10587503—24no—
75#10587502—22no—
76#10587502—22no—
77#10587502C2312yes—
78C22⋊#3622—22no—
79C22⋊#3622C2314yes—
80#10587481—24no—
81#10587478—24no—
82C24×C17—22no—
83C24×C17C2×D411yescharacteristic
84#41×C17—416no—
85#41×C17—48no—
86#41×C17—48no—
87C2×Q8×C17—24no—
88C2×Q8×C17—44no—
89C2×D4×C17—24no—
90C2×D4×C17—24no—
91C2×D4×C17—24no—
92C2×D4×C17—24no—
93C2×D4×C17—22no—
94C2×D4×C17—44no—
95C2×D4×C17—48no—
96C2×D4×C17—82no—
97C2×D4×C17—82no—
98C2×D4×C17C2×D414yes—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—24no—
102C22×C4×C17—28no—
103C22×C4×C17—22no—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—48no—
107C22×C4×C17—81no—
108C22×C4×C17—81no—
109C22×C4×C17C2×D412yes—
110C22×C4×C17C2×D412yes—
111C22×C4×C17C2×D411yescharacteristic
112C22×C4×C17C2×D412yes—
113C22×C4×C17C2×D412yes—
114C22×C4×C17C2411yescharacteristic
115C22×Dic17—22no—
116C22×Dic17—21no—
117C22×Dic17—21no—
118C2×Dic34—22no—
119C2×Dic34—81no—
120C4⋊C4×C17—24no—
121C4⋊C4×C17—22no—
122C4⋊C4×C17—24no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17C2×D414yes—
125C22⋊C4×C17—28no—
126C22⋊C4×C17—24no—
127C22⋊C4×C17—24no—
128C22⋊Dic17—24no—
129C22⋊Dic17—41no—
130C22⋊Dic17—41no—
131#63229—84no—
132C4⋊Dic17—22no—
133#63225—42no—
134#63225—44no—
135#63225—82no—
136C4×Dic17—81no—
137#63223—82no—
138C2×#3622—24no—
139#18325—171nomaximal, Sylow
140C23×C17—22no—
141C23×C17—22no—
142C23×C17—24no—
143C23×C17—41no—
144C23×C17—44no—
145C23×C17—44no—
146C23×C17—44no—
147C23×C17#12011yescharacteristic
148C23×C17#12011yescharacteristic
149C23×C17C22×D411yescharacteristic
150Q8×C17—48no—
151D4×C17—44no—
152D4×C17—48no—
153D4×C17—48no—
154D4×C17—48no—
155D4×C17—82no—
156D4×C17—84no—
157C2×C4×C17—24no—
158C2×C4×C17—24no—
159C2×C4×C17—22no—
160C2×C4×C17—22no—
161C2×C4×C17—22no—
162C2×C4×C17—24no—
163C2×C4×C17—41no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—41no—
167C2×C4×C17—416no—
168C2×C4×C17—44no—
169C2×C4×C17—42no—
170C2×C4×C17—42no—
171C2×C4×C17—81no—
172C2×C4×C17#12012yes—
173C2×C4×C17AGL(1,ℤ/8)14yes—
174C2×C4×C17C22×D412yes—
175C2×Dic17—24no—
176C2×Dic17—42no—
177C2×Dic17—42no—
178C2×Dic17—42no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181C2×Dic17—42no—
182Dic34—44no—
183#3622—44no—
184C2×#537D1711yescharacteristic
185#3193—172no—
186#2583—171no—
187#1610—174no—
188#1604—174no—
189#1507—172no—
190#1463—171no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—44no—
199C22×C17—44no—
200C22×C17—44no—
201C22×C17—44no—
202C22×C17—88no—
203C22×C17#45612yes—
204C22×C17C2×#12011yescharacteristic
205C22×C17C2×AGL(1,ℤ/8)12yes—
206C4×C17—24no—
207C4×C17—22no—
208C4×C17—22no—
209C4×C17—44no—
210C4×C17—42no—
211Dic17—44no—
212Dic17—81no—
213Dic17—81no—
214C22×#41—21no—
215#537—24no—
216#537—22no—
217#537C2×D1714yes—
218C2×C22⋊Q8—22no—
219C2×C22⋊Q8—171no—
220C2×#121—22no—
221C2×#121—21no—
222C2×#121C2×D1712yes—
223C22×C4⋊C4C2×D1711yescharacteristic
224#484—342no—
225#480—174no—
226#468—344no—
227#450—342no—
228#448—174no—
229#407—172no—
230#398—172no—
231#395—172no—
232#389—342no—
233#385—341no—
234#385—341no—
235C4×C22⋊C4—171no—
236#327—178no—
237#324—174no—
238C2×C17—22no—
239C2×C17—22no—
240C2×C17—41no—
241C2×C17—84no—
242C2×C17#300311yescharacteristic
243C2×C17C2×#45611yescharacteristic
244C2×C17#302611yescharacteristic
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—48no—
248C22×Q8—21no—
249C22×Q8—171no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4C22⋊D1712yes—
254C23×C4—22no—
255C23×C4C22⋊D1711yescharacteristic
256C23×C4C22⋊D1711yescharacteristic
257C23×C4C22×D1711yescharacteristic
258C2×SD16—344no—
259C2×M16—342no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—342no—
263C22⋊Q8—342no—
264#121—24no—
265#121—28no—
266#121—28no—
267#121—24no—
268#121—22no—
269#121—22no—
270#121C22⋊D1714yes—
271#121C22×D1714yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—171no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4—342no—
278C2×C4⋊C4—342no—
279C2×C4⋊C4C22⋊D1712yes—
280C2×C4⋊C4C22⋊D1712yes—
281C2×C4⋊C4C22×D1712yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—341no—
285C2×C22⋊C4—341no—
286C2×C42—341no—
287#107—344no—
288#103—344no—
289#102—344no—
290C22⋊C8—174no—
291C22⋊C8—342no—
292#95—172no—
293#95—342no—
294#95—342no—
295C17#1832511yescharacteristic, complemented, Sylow
296C24—22no—
297C24C2×C22⋊D1711yescharacteristic
298#41—416no—
299#41—48no—
300#41—48no—
301C2×Q8—24no—
302C2×Q8—44no—
303C2×Q8—342no—
304C2×Q8—1361no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C22⋊D1714yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—48no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—342no—
326C22×C4—341no—
327C22×C4—341no—
328C22×C4C2×C22⋊D1712yes—
329C22×C4C2×C22⋊D1712yes—
330C22×C4C2×C22⋊D1712yes—
331C22×C4C2×C22⋊D1712yes—
332C22×C4C2×C22⋊D1711yescharacteristic
333C22×C4C23×D1711yescharacteristic
334SD16—1364no—
335M16—1362no—
336C2×C8—344no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—342no—
342C4⋊C4—684no—
343C4⋊C4—682no—
344C4⋊C4—1362no—
345C4⋊C4C2×C22⋊D1714yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—344no—
350C22⋊C4—681no—
351C22⋊C4—681no—
352C42—1361no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—41no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23#1058763911yescharacteristic
361C23#1058763911yescharacteristic
362C23C22×C22⋊D1711yescharacteristic
363Q8—48no—
364Q8—684no—
365D4—44no—
366D4—48no—
367D4—48no—
368D4—48no—
369D4—82no—
370D4—84no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—344no—
387C2×C4—682no—
388C2×C4—682no—
389C2×C4—682no—
390C2×C4—681no—
391C2×C4—681no—
392C2×C4—682no—
393C2×C4#1058761814yes—
394C2×C4#1058763912yes—
395C2×C4C22×C22⋊D1712yes—
396C8—684no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—88no—
409C22#1176299212yes—
410C22C2×#1058763911yescharacteristic
411C22C2×#1058761812yes—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—42no—
417C4—684no—
418C4—1361no—
419C4—1361no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—84no—
424C2#1453801111yescharacteristic
425C2C2×#1176299211yescharacteristic
426C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#537C2
socleC22×C17C2×#120
hypercenterC2×#537D17
hypocenterC17#18325
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #14539590C2semidirect
2. #14534164C2semidirect
3. #14527787C2semidirect
4. #14527753C2semidirect
5. #14527300C2semidirect
6. #14527114C2semidirect
10. #11763120C22semidirect
13. #11762947C22semidirect
15. #11762739C22semidirect
16. #11762729C22semidirect
21. #11762543C22semidirect
22. #11762535C22semidirect
295. C17#18325semidirect

Maximal subgroups

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

subgroupindex
#145395902
#145341642
#145277872
#145277532
#145273002
#145271142
C2×C17×#5372
#1832517