Finite Group Databasebeta

← #17008232

Subgroups — #17008232

Order 4352.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#11763390C211yesmaximal, characteristic
2#14539537C212yesmaximal, complemented
3#14527820C214yesmaximal, complemented
4#14527752C214yesmaximal, complemented
5#14527351C212yesmaximal, complemented
6C2×#11762535C211yesmaximal, characteristic, complemented
7C17×#2627C211yesmaximal, characteristic, complemented
8C22×#63252—21no—
9#11763390—22no—
10#11763390—24no—
11#11763390C2214yes—
12C2×#10587583—22no—
13C2×#10587581—22no—
14C2×#10587616—21no—
15C2×#10587616C2212yes—
16#11763137—22no—
17#11763127C2214yescomplemented
18#11763040—24no—
19#11762959—22no—
20#11762948C2214yescomplemented
21C2×C22⋊#3622C2212yes—
22#11762542C2218yescomplemented
23#11762535C2214yescomplemented
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#400C2212yescomplemented
27C17×#398—22no—
28C17×#395C2212yescomplemented
29C17×#385—21no—
30C17×#381—21no—
31C17×#380C2211yescharacteristic
32C22×Q8×C17C2311yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C22×C22⋊D17—22no—
37C2×#63252—24no—
38C2×#63252—24no—
39C2×#63252—44no—
40C2×#63252—44no—
41C22×D68—21no—
42C22×D68—21no—
43C22×D68D412yes—
44C22×C4×D17—22no—
45C22×Dic34—21no—
46C17×C22⋊Q8—22no—
47C17×C22⋊Q8—22no—
48C2×C4⋊C4×C17—22no—
49C2×C4⋊C4×C17—22no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—21no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17D412yes—
54C2×C4⋊C4×C17C2311yescharacteristic
55C2×C4⋊C4×C17C2312yes—
56C2×C22⋊C4×C17—21no—
57C2×C22⋊C4×C17—21no—
58C2×#63230—24no—
59#10587616—24no—
60#10587616—24no—
61#10587616—22no—
62#10587616—22no—
63#10587616D414yes—
64#10587616C2314yes—
65C2×#63227—22no—
66C2×#63227—22no—
67C2×C4⋊Dic17—21no—
68C2×C4⋊Dic17D412yes—
69C22×#3622—22no—
70#10587583—28no—
71#10587583—28no—
72#10587581—28no—
73#10587581—28no—
74C17×#95—22no—
75C17×#95—22no—
76C17×#95—22no—
77C17×#95C2312yes—
78#10587506—24no—
79C22⋊#3622—22no—
80C22⋊#3622C2314yes—
81#10587480—24no—
82#10587479—24no—
83C24×C17C2×D411yescharacteristic
84C23×D17—22no—
85C2×Q8×C17—22no—
86C2×Q8×C17—81no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—22no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—22no—
95C22×C4×C17—81no—
96C22×C4×C17—81no—
97C22×C4×C17SD1614yescomplemented
98C22×C4×C17C2×D412yes—
99C22×C4×C17C2×D412yes—
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2×D412yes—
102C22×C4×C17C2×D411yescharacteristic
103C22×C4×C17C2411yescharacteristic
104C2×C22⋊D17—24no—
105C2×C22⋊D17—24no—
106C2×C22⋊D17—44no—
107C2×C22⋊D17—44no—
108C22×Dic17—22no—
109#63252—48no—
110#63252—416no—
111#63252—48no—
112C2×D68—24no—
113C2×D68—24no—
114C2×D68—22no—
115C2×D68—44no—
116C2×D68—82no—
117C2×D68—82no—
118C2×D68C2×D414yes—
119C2×C4×D17—28no—
120C2×C4×D17—48no—
121C2×Dic34—24no—
122C2×Dic34—44no—
123C4⋊C4×C17—24no—
124C4⋊C4×C17—24no—
125C4⋊C4×C17—22no—
126C4⋊C4×C17—42no—
127C4⋊C4×C17—81no—
128C4⋊C4×C17—82no—
129C22⋊C4×C17—41no—
130C22⋊C4×C17—41no—
131C22⋊C4×C17—42no—
132#63230—84no—
133#63227—28no—
134#63227—24no—
135#63227—24no—
136C4⋊Dic17—24no—
137C4⋊Dic17—22no—
138C4⋊Dic17C2×D414yes—
139C2×#3622—24no—
140C2×#3622—82no—
141#18293—171nomaximal, Sylow
142C23×C17—41no—
143C23×C17—41no—
144C23×C17#12011yescharacteristic
145C23×C17#12011yescharacteristic
146C23×C17C2×SD1612yes—
147C23×C17AGL(1,ℤ/8)12yes—
148C23×C17C22×D411yescharacteristic
149C22×D17—24no—
150C22×D17—44no—
151C22×D17—44no—
152C22×D17—44no—
153Q8×C17—44no—
154C2×C4×C17—22no—
155C2×C4×C17—22no—
156C2×C4×C17—24no—
157C2×C4×C17—24no—
158C2×C4×C17—22no—
159C2×C4×C17—41no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17—41no—
163C2×C4×C17—42no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—42no—
168C2×C4×C17—42no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17—42no—
172C2×C4×C17—81no—
173C2×C4×C17#12012yes—
174C2×C4×C17C2×SD1614yes—
175C2×C4×C17C22×D412yes—
176C22⋊D17—44no—
177C22⋊D17—44no—
178C22⋊D17—48no—
179C2×Dic17—24no—
180C2×Dic17—44no—
181D68—44no—
182D68—48no—
183D68—82no—
184D68—84no—
185C4×D17—416no—
186Dic34—48no—
187#3622—44no—
188C2×#537—171no—
189#3187—172no—
190#2627D1711yescharacteristic, complemented
191#1616—174no—
192#1604—174no—
193#1519—172no—
194C2×#324—171no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—42no—
202C22×C17—42no—
203C22×C17#45012yes—
204C22×C17#45012yes—
205C22×C17C2×#12011yescharacteristic
206C22×C17C22×SD1611yescharacteristic
207C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
208C2×D17—44no—
209C2×D17—44no—
210C2×D17—44no—
211C2×D17—88no—
212C4×C17—24no—
213C4×C17—22no—
214C4×C17—22no—
215C4×C17—42no—
216C4×C17—82no—
217C4×C17—81no—
218C4×C17—81no—
219Dic17—44no—
220C22×#41—341no—
221#537—174no—
222#537—344no—
223#537—342no—
224C2×C22⋊Q8—342no—
225C2×C22⋊Q8C2×D1711yescharacteristic
226C2×#121—172no—
227C2×#121—342no—
228C2×#121—341no—
229C22×C4⋊C4C2×D1711yescharacteristic
230#483—342no—
231#481—174no—
232#465—344no—
233#450—342no—
234#448—174no—
235C2×C22⋊C8—172no—
236#400C2×D1712yescomplemented
237#398—22no—
238#395C2×D1712yescomplemented
239#385—21no—
240#381—21no—
241#380C2×D1711yescharacteristic
242#327—178no—
243#324—174no—
244C2×C17—21no—
245C2×C17—22no—
246C2×C17—21no—
247C2×C17—22no—
248C2×C17C2×#45011yescharacteristic
249C2×C17#302011yescharacteristic
250C2×C17#302611yescharacteristic
251D17—84no—
252C2×#41—344no—
253C2×#41—344no—
254C2×#41—684no—
255C2×#41—684no—
256C22×Q8—341no—
257C22×Q8C22×D1711yescharacteristic
258C22×D4—172no—
259C22×D4—341no—
260C22×D4—341no—
261C22×D4—342no—
262C23×C4—342no—
263C23×C4C22⋊D1711yescharacteristic
264C23×C4C22⋊D1711yescharacteristic
265C23×C4C22×D1711yescharacteristic
266C2×SD16—344no—
267C22×C8—342no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—348no—
271C22⋊Q8—348no—
272#121—174no—
273#121—174no—
274#121—348no—
275#121—348no—
276#121—344no—
277#121—342no—
278#121—342no—
279#121—344no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—172no—
286C2×C4⋊C4—341no—
287C2×C4⋊C4C22⋊D1712yes—
288C2×C4⋊C4C22×D1711yescharacteristic
289C2×C4⋊C4C22×D1712yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—342no—
293C2×C22⋊C4—342no—
294#106—344no—
295#103—344no—
296#102—344no—
297C22⋊C8—174no—
298C22⋊C8—342no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95C22×D1712yes—
303C17#1829311yescharacteristic, complemented, Sylow
304C24—342no—
305C24C2×C22⋊D1711yescharacteristic
306#41—6816no—
307#41—688no—
308#41—688no—
309C2×Q8—22no—
310C2×Q8—81no—
311C2×Q8—344no—
312C2×Q8—684no—
313C2×D4—174no—
314C2×D4—344no—
315C2×D4—344no—
316C2×D4—344no—
317C2×D4—344no—
318C2×D4—342no—
319C2×D4—684no—
320C2×D4—684no—
321C2×D4—684no—
322C2×D4—1362no—
323C2×D4—1362no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—348no—
335C22×C4—342no—
336C22×C4—688no—
337C22×C4#6323014yescomplemented
338C22×C4D4×D1712yes—
339C22×C4D4×D1712yes—
340C22×C4C2×C22⋊D1712yes—
341C22×C4C2×C22⋊D1712yes—
342C22×C4C2×C22⋊D1711yescharacteristic
343C22×C4C23×D1711yescharacteristic
344SD16—1364no—
345C2×C8—344no—
346C2×C8—1362no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—174no—
354C4⋊C4—344no—
355C4⋊C4—342no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—348no—
360C22⋊C4—344no—
361C22⋊C4—344no—
362C23—41no—
363C23—41no—
364C23—344no—
365C23—684no—
366C23—684no—
367C23—684no—
368C23#1058762311yescharacteristic
369C23#1058762311yescharacteristic
370C23C2×#6323012yes—
371C23#1058763512yes—
372C23C22×C22⋊D1711yescharacteristic
373Q8—44no—
374Q8—688no—
375D4—684no—
376D4—688no—
377D4—688no—
378D4—684no—
379D4—684no—
380D4—1362no—
381D4—1364no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—344no—
402C2×C4—6816no—
403C2×C4—684no—
404C2×C4#1058762312yes—
405C2×C4C2×#6323014yes—
406C2×C4C2×D4×D1712yes—
407C8—684no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—684no—
417C22—684no—
418C22—684no—
419C22—1368no—
420C22#1176295912yes—
421C22#1176295912yes—
422C22C2×#1058762311yescharacteristic
423C22C22×#6323011yescharacteristic
424C22C2×#1058763511yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—684no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—1364no—
438C2C2×#1176295911yescharacteristic
439C2#1453814711yescharacteristic
440C2#1453818011yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2627C2
socleC22×C17C2×#120
prime radicalC2×#11763390C2
hypercenter#2627D17
hypocenterC17#18293
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
2. #14539537C2semidirect
3. #14527820C2semidirect
4. #14527752C2semidirect
5. #14527351C2semidirect
6. C2×#11762535C2semidirect
7. C17×#2627C2semidirect
17. #11763127C22semidirect
20. #11762948C22semidirect
22. #11762542C22semidirect
23. #11762535C22semidirect
26. C17×#400C22semidirect
28. C17×#395C22semidirect
97. C22×C4×C17SD16semidirect
190. #2627D17semidirect
236. #400C2×D17semidirect
238. #395C2×D17semidirect
303. C17#18293semidirect
337. C22×C4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117633902
#145395372
#145278202
#145277522
#145273512
C2×#117625352
C17×#26272
#1829317