Finite Group Databasebeta

← #17008498

Subgroups — #17008498

Order 4352.

1256 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1256 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#14539581C212yesmaximal, complemented
2#14534264C211yesmaximal, characteristic, complemented
3#14527821C214yesmaximal, complemented
4#14527719C214yesmaximal, complemented
5#14527300C212yesmaximal, complemented
6#14527114C211yesmaximal, characteristic, complemented
7C2×C17×#537C211yesmaximal, characteristic, complemented
8C2×#10587616C2211yescharacteristic
9#11763127—22no—
10#11763125C2214yescomplemented
11#11763060—24no—
12#11762943—22no—
13#11762942C2214yescomplemented
14#11762773C2212yes—
15#11762748C2212yescomplemented
16#11762726C2212yescomplemented
17#11762699—22no—
18#11762688—21no—
19#11762688—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×C22⋊D17—21no—
45C22×D68C2311yescharacteristic
46C17×C22⋊Q8—28no—
47C17×C22⋊Q8—28no—
48C17×#121—24no—
49C17×#121—28no—
50C17×#121—28no—
51C17×#121—24no—
52C17×#121—22no—
53C17×#121—22no—
54C17×#121D414yes—
55C17×#121C2314yes—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—22no—
59C2×C4⋊C4×C17D412yes—
60C2×C4⋊C4×C17D412yes—
61C2×C4⋊C4×C17C2312yes—
62C2×C22⋊C4×C17—22no—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊Dic17—21no—
65C2×#63228—24no—
66#10587616—22no—
67#10587616—22no—
68C2×#63227—22no—
69C2×#63227—21no—
70C2×#63227—22no—
71C2×C4⋊Dic17C2311yescharacteristic
72C2×C4×Dic17—21no—
73C2×#63223—22no—
74#10587503—24no—
75#10587502—22no—
76#10587502—22no—
77#10587502C2312yes—
78C22⋊#3622—22no—
79C22⋊#3622C2314yes—
80#10587480—24no—
81#10587479—24no—
82C24×C17—22no—
83C24×C17C2×D411yescharacteristic
84C23×D17—21no—
85#41×C17—416no—
86#41×C17—48no—
87#41×C17—48no—
88C2×Q8×C17—24no—
89C2×Q8×C17—44no—
90C2×D4×C17—24no—
91C2×D4×C17—24no—
92C2×D4×C17—24no—
93C2×D4×C17—24no—
94C2×D4×C17—22no—
95C2×D4×C17—44no—
96C2×D4×C17—48no—
97C2×D4×C17—82no—
98C2×D4×C17—82no—
99C2×D4×C17C2×D414yes—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—24no—
103C22×C4×C17—28no—
104C22×C4×C17—22no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—48no—
108C22×C4×C17—81no—
109C22×C4×C17—81no—
110C22×C4×C17C2×D412yes—
111C22×C4×C17C2×D412yes—
112C22×C4×C17C2×D411yescharacteristic
113C22×C4×C17C2×D412yes—
114C22×C4×C17C2×D412yes—
115C22×C4×C17C2411yescharacteristic
116C2×C22⋊D17—44no—
117C2×C22⋊D17—41no—
118C2×C22⋊D17—41no—
119C22×Dic17—21no—
120C22×Dic17—22no—
121C2×D68—22no—
122C2×D68—81no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17—22no—
125C4⋊C4×C17—24no—
126C4⋊C4×C17—24no—
127C4⋊C4×C17C2×D414yes—
128C22⋊C4×C17—28no—
129C22⋊C4×C17—24no—
130C22⋊C4×C17—24no—
131C22⋊Dic17—42no—
132#63228—84no—
133#63227—42no—
134#63227—44no—
135#63227—82no—
136C4⋊Dic17—22no—
137C4×Dic17—81no—
138#63223—82no—
139C2×#3622—24no—
140#18324—171nomaximal, Sylow
141C23×C17—22no—
142C23×C17—22no—
143C23×C17—24no—
144C23×C17—41no—
145C23×C17—44no—
146C23×C17—44no—
147C23×C17—44no—
148C23×C17#12011yescharacteristic
149C23×C17#12011yescharacteristic
150C23×C17C22×D411yescharacteristic
151C22×D17—42no—
152C22×D17—42no—
153C22×D17—41no—
154C22×D17—42no—
155Q8×C17—48no—
156D4×C17—44no—
157D4×C17—48no—
158D4×C17—48no—
159D4×C17—48no—
160D4×C17—82no—
161D4×C17—84no—
162C2×C4×C17—24no—
163C2×C4×C17—24no—
164C2×C4×C17—22no—
165C2×C4×C17—22no—
166C2×C4×C17—22no—
167C2×C4×C17—24no—
168C2×C4×C17—41no—
169C2×C4×C17—42no—
170C2×C4×C17—42no—
171C2×C4×C17—41no—
172C2×C4×C17—416no—
173C2×C4×C17—44no—
174C2×C4×C17—42no—
175C2×C4×C17—42no—
176C2×C4×C17—81no—
177C2×C4×C17#12012yes—
178C2×C4×C17AGL(1,ℤ/8)14yes—
179C2×C4×C17C22×D412yes—
180C22⋊D17—82no—
181C22⋊D17—82no—
182C2×Dic17—42no—
183C2×Dic17—42no—
184C2×Dic17—42no—
185C2×Dic17—41no—
186C2×Dic17—42no—
187D68—44no—
188#3622—44no—
189C2×#537D1711yescharacteristic, complemented
190#3192—172no—
191#2591—171no—
192#1616—174no—
193#1598—174no—
194#1507—172no—
195#1463—171no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—44no—
204C22×C17—44no—
205C22×C17—44no—
206C22×C17—44no—
207C22×C17—88no—
208C22×C17#45412yes—
209C22×C17C2×#12011yescharacteristic
210C22×C17C2×AGL(1,ℤ/8)12yes—
211C2×D17—84no—
212C2×D17—81no—
213C2×D17—81no—
214C2×D17—81no—
215C4×C17—24no—
216C4×C17—22no—
217C4×C17—22no—
218C4×C17—44no—
219C4×C17—42no—
220Dic17—82no—
221Dic17—81no—
222C22×#41—21no—
223#537—24no—
224#537—22no—
225#537C2×D1714yes—
226C2×C22⋊Q8—22no—
227C2×#121—22no—
228C2×#121—21no—
229C2×#121—171no—
230C2×#121C2×D1712yes—
231C22×C4⋊C4C2×D1711yescharacteristic
232#481—174no—
233#481—342no—
234#469—344no—
235#447—174no—
236#447—342no—
237#407—172no—
238#400—172no—
239#394—172no—
240#388—342no—
241#386—341no—
242#386—341no—
243C4×C22⋊C4—171no—
244#327—178no—
245#324—174no—
246C2×C17—22no—
247C2×C17—22no—
248C2×C17—41no—
249C2×C17—84no—
250C2×C17#300311yescharacteristic
251C2×C17C2×#45411yescharacteristic
252C2×C17#302611yescharacteristic
253D17—161no—
254C2×#41—24no—
255C2×#41—24no—
256C2×#41—48no—
257C22×Q8—21no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—171no—
262C22×D4—341no—
263C22×D4C22⋊D1712yes—
264C23×C4—22no—
265C23×C4C22⋊D1711yescharacteristic
266C23×C4C22⋊D1711yescharacteristic
267C23×C4C22×D1711yescharacteristic
268C2×D8—344no—
269C2×M16—342no—
270C22⋊Q8—28no—
271C22⋊Q8—28no—
272#121—24no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—22no—
277#121—22no—
278#121—342no—
279#121—342no—
280#121C22⋊D1714yes—
281#121C22×D1714yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—171no—
286C2×C4⋊C4C22⋊D1712yes—
287C2×C4⋊C4C22⋊D1712yes—
288C2×C4⋊C4C22×D1712yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—341no—
292C2×C22⋊C4—341no—
293C2×C22⋊C4—342no—
294C2×C22⋊C4—342no—
295C2×C42—341no—
296#106—344no—
297#102—344no—
298#102—344no—
299C22⋊C8—174no—
300C22⋊C8—342no—
301#95—172no—
302#95—342no—
303#95—342no—
304C17#1832411yescharacteristic, complemented, Sylow
305C24—22no—
306C24—341no—
307C24C2×C22⋊D1711yescharacteristic
308#41—416no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—342no—
323C2×D4—684no—
324C2×D4—681no—
325C2×D4—681no—
326C2×D4—1361no—
327C2×D4C2×C22⋊D1714yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—48no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—342no—
339C22×C4—341no—
340C22×C4C2×C22⋊D1712yes—
341C22×C4C2×C22⋊D1712yes—
342C22×C4C2×C22⋊D1712yes—
343C22×C4C2×C22⋊D1712yes—
344C22×C4C2×C22⋊D1711yescharacteristic
345C22×C4C23×D1711yescharacteristic
346D8—1364no—
347M16—1362no—
348C2×C8—344no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—342no—
354C4⋊C4C2×C22⋊D1714yes—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—682no—
359C22⋊C4—684no—
360C22⋊C4—682no—
361C22⋊C4—1362no—
362C42—1361no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—682no—
371C23—682no—
372C23—681no—
373C23—682no—
374C23#1058763911yescharacteristic
375C23#1058763911yescharacteristic
376C23C22×C22⋊D1711yescharacteristic
377Q8—48no—
378D4—44no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—684no—
385D4—1362no—
386D4—1362no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—682no—
403C2×C4—682no—
404C2×C4—682no—
405C2×C4—681no—
406C2×C4—682no—
407C2×C4#1058761814yes—
408C2×C4#1058763912yes—
409C2×C4C22×C22⋊D1712yes—
410C8—684no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—1364no—
424C22—1361no—
425C22—1361no—
426C22—1361no—
427C22#1176298212yes—
428C22C2×#1058763911yescharacteristic
429C22C2×#1058761812yes—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—1362no—
436C4—1361no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—2721no—
442C2#1453801111yescharacteristic
443C2C2×#1176298211yescharacteristic
444C2#1453818111yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#537C2
socleC22×C17C2×#120
hypercenterC2×#537D17
hypocenterC17#18324
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. #14539581C2semidirect
2. #14534264C2semidirect
3. #14527821C2semidirect
4. #14527719C2semidirect
5. #14527300C2semidirect
6. #14527114C2semidirect
7. C2×C17×#537C2semidirect
10. #11763125C22semidirect
13. #11762942C22semidirect
15. #11762748C22semidirect
16. #11762726C22semidirect
21. #11762543C22semidirect
22. #11762535C22semidirect
189. C2×#537D17semidirect
304. C17#18324semidirect

Maximal subgroups

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

subgroupindex
#145395812
#145342642
#145278212
#145277192
#145273002
#145271142
C2×C17×#5372
#1832417