Finite Group Databasebeta

← #17007240

Subgroups — #17007240

Order 4352.

1454 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762947C212yesmaximal, complemented
2#14532149C211yesmaximal, characteristic, complemented
3#14527753C214yesmaximal, complemented
4#14527736C214yesmaximal, complemented
5#14527352C212yesmaximal, complemented
6#14527102C211yesmaximal, characteristic, complemented
7C17×#3496C211yesmaximal, characteristic
8C23×Dic34C2211yescharacteristic, complemented
9C22×#63229—22no—
10C2×#10587610—21no—
11#11762947—24no—
12#11762947C2218yescomplemented
13C2×#10587481—22no—
14C2×C22⋊#3622C2212yescomplemented
15#11762729C2212yescomplemented
16#11762721C2212yescomplemented
17#11762715—21no—
18#11762689—21no—
19#11762681—21no—
20#11762669C2211yescharacteristic, complemented
21#11762543C2218yescomplemented
22#11762535C2212yescomplemented
23#11762533C2212yescomplemented
24C2×C17×H(5,𝔽2)—21no—
25C17×#547—22no—
26C17×#546—24no—
27D42×C17—22no—
28C17×#537C2212yes—
29C17×#534—21no—
30C17×#534C2212yes—
31C17×#532—21no—
32C2×C17×#121—21no—
33C2×C17×#121C2212yes—
34C2×C4×D4×C17C2211yescharacteristic
35C17×H(5,𝔽2)—44no—
36C17×H(5,𝔽2)—44no—
37C2×#41×C17—24no—
38C2×#41×C17—22no—
39C22×D4×C17—24no—
40C22×D4×C17—22no—
41C22×D4×C17—22no—
42C22×D4×C17—21no—
43C22×D4×C17D411yescharacteristic
44C22×D4×C17D412yes—
45C23×C4×C17D411yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×Dic17—21no—
48C22×Dic34—22no—
49C22×Dic34—21no—
50C22×Dic34—22no—
51C22×Dic34—44no—
52C22×Dic34C2312yes—
53C17×C4⋊Q8—21no—
54C17×#127—21no—
55C17×#124—22no—
56C17×#123—24no—
57C17×#123—24no—
58C17×C22⋊Q8—24no—
59C17×#121—22no—
60C17×#121—22no—
61C17×#121—24no—
62C17×#121—22no—
63C17×#121—22no—
64C17×#121—22no—
65C17×#121—22no—
66C17×#121D414yes—
67C17×#121C2314yes—
68C17×#120—24no—
69C17×#120—24no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17—22no—
73C4×D4×C17—22no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D412yes—
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—22no—
78C2×C22⋊C4×C17—22no—
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C22⋊C4×C17C2311yescharacteristic
81C2×C42×C17D411yescharacteristic
82C2×C22⋊Dic17—22no—
83C2×C22⋊Dic17—21no—
84C2×#63229—28no—
85C2×#63229—48no—
86C2×C4⋊Dic17—21no—
87#10587610—44no—
88C2×#63225—22no—
89C22×#3622—22no—
90#10587502—22no—
91#10587502—22no—
92#10587502C2312yes—
93C22⋊#3622—22no—
94C22⋊#3622C2314yes—
95#10587481—28no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17C2×D411yescharacteristic
99C24×C17C2×D411yescharacteristic
100#41×C17—44no—
101#41×C17—416no—
102#41×C17—44no—
103C2×Q8×C17—22no—
104C2×D4×C17—22no—
105C2×D4×C17—22no—
106C2×D4×C17—24no—
107C2×D4×C17—22no—
108C2×D4×C17—24no—
109C2×D4×C17—24no—
110C2×D4×C17—22no—
111C2×D4×C17—22no—
112C2×D4×C17—22no—
113C2×D4×C17—44no—
114C2×D4×C17—416no—
115C2×D4×C17—44no—
116C2×D4×C17—44no—
117C2×D4×C17—44no—
118C2×D4×C17—44no—
119C2×D4×C17—82no—
120C2×D4×C17SD1614yescomplemented
121C2×D4×C17C2×D414yes—
122C22×C4×C17—24no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—42no—
128C22×C4×C17—42no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—44no—
139C22×Dic17—42no—
140C2×Dic34—28no—
141C2×Dic34—24no—
142C2×Dic34—24no—
143C2×Dic34—44no—
144C2×Dic34—48no—
145C2×Dic34—44no—
146C2×Dic34—42no—
147C2×Dic34—41no—
148C2×Dic34—41no—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17C2×D414yes—
154C22⋊C4×C17—24no—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—22no—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—22no—
160C42×C17—21no—
161C42×C17—21no—
162C22⋊Dic17—44no—
163C22⋊Dic17—42no—
164#63229—416no—
165C4⋊Dic17—42no—
166#63225—44no—
167C2×#3622—24no—
168C2×#3622—44no—
169#18195—171nomaximal, Sylow
170C23×C17—21no—
171C23×C17—21no—
172C23×C17—24no—
173C23×C17—41no—
174C23×C17—41no—
175C23×C17—41no—
176C23×C17—41no—
177C23×C17—44no—
178C23×C17—44no—
179C23×C17—82no—
180C23×C17—82no—
181C23×C17—82no—
182C23×C17—82no—
183C23×C17#12012yes—
184C23×C17C2×SD1612yes—
185C23×C17C2×SD1612yescomplemented
186C23×C17AGL(1,ℤ/8)12yes—
187C23×C17C22×D411yescharacteristic
188Q8×C17—44no—
189D4×C17—28no—
190D4×C17—44no—
191D4×C17—44no—
192D4×C17—48no—
193D4×C17—44no—
194D4×C17—44no—
195D4×C17—48no—
196D4×C17—44no—
197D4×C17—82no—
198D4×C17—82no—
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—24no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—24no—
206C2×C4×C17—42no—
207C2×C4×C17—41no—
208C2×C4×C17—48no—
209C2×C4×C17—44no—
210C2×C4×C17—44no—
211C2×C4×C17—42no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—81no—
215C2×C4×C17#12011yescharacteristic
216C2×C4×C17#12011yescharacteristic
217C2×C4×C17C2×SD1612yescomplemented
218C2×C4×C17C22×D412yes—
219C2×Dic17—24no—
220C2×Dic17—48no—
221C2×Dic17—42no—
222C2×Dic17—42no—
223C2×Dic17—42no—
224C2×Dic17—42no—
225C2×Dic17—42no—
226Dic34—28no—
227Dic34—44no—
228Dic34—44no—
229Dic34—44no—
230#3622—44no—
231#3496D1711yescharacteristic
232C2×#448—172no—
233#2409—171no—
234#1604—174no—
235#1601—174no—
236#1519—172no—
237#1459—171no—
238C22×C17—21no—
239C22×C17—22no—
240C22×C17—21no—
241C22×C17—22no—
242C22×C17—22no—
243C22×C17—42no—
244C22×C17—42no—
245C22×C17—42no—
246C22×C17—41no—
247C22×C17—41no—
248C22×C17—44no—
249C22×C17—44no—
250C22×C17—44no—
251C22×C17—81no—
252C22×C17—164no—
253C22×C17—164no—
254C22×C17#45014yes—
255C22×C17C2×#12011yescharacteristic
256C22×C17C22×SD1611yescharacteristic
257C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
258C4×C17—22no—
259C4×C17—22no—
260C4×C17—21no—
261C4×C17—44no—
262C4×C17—41no—
263C4×C17—41no—
264C4×C17B(4,𝔽2)12yes—
265Dic17—44no—
266Dic17—82no—
267C2×H(5,𝔽2)—21no—
268C23×Q8—171no—
269C22×SD16—342no—
270#547—22no—
271#546—24no—
272D42—22no—
273#537C2×D1712yes—
274#534—21no—
275#534C2×D1712yes—
276#532—21no—
277C2×C22⋊Q8—341no—
278C2×#121—21no—
279C2×#121C2×D1712yes—
280C2×C4×D4C2×D1711yescharacteristic
281#448—178no—
282#448—344no—
283C2×#103—342no—
284C2×C22⋊C8—172no—
285#395—172no—
286#393—172no—
287#391—341no—
288#386—341no—
289#385—341no—
290#380—171no—
291#327—178no—
292#324—172no—
293#323—172no—
294C2×C17—21no—
295C2×C17—21no—
296C2×C17—24no—
297C2×C17—42no—
298C2×C17—84no—
299C2×C17C2×#45011yescharacteristic
300C2×C17#302211yescharacteristic
301C2×C17C2×B(4,𝔽2)11yescharacteristic
302H(5,𝔽2)—44no—
303H(5,𝔽2)—44no—
304C2×#41—24no—
305C2×#41—22no—
306C22×Q8—172no—
307C22×Q8—341no—
308C22×Q8—342no—
309C22×Q8—342no—
310C22×Q8—684no—
311C22×D4—24no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—21no—
315C22×D4C22⋊D1711yescharacteristic
316C22×D4C22⋊D1712yes—
317C23×C4—341no—
318C23×C4C22⋊D1711yescharacteristic
319C23×C4C22×D1711yescharacteristic
320C2×SD16—348no—
321C2×SD16—688no—
322C22×C8—342no—
323C4⋊Q8—21no—
324#127—21no—
325#124—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8—684no—
330#121—22no—
331#121—22no—
332#121—24no—
333#121—22no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121C22⋊D1714yes—
338#121C22×D1714yes—
339#120—24no—
340#120—24no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—341no—
347C2×C4⋊C4—342no—
348C2×C4⋊C4C22⋊D1712yes—
349C2×C4⋊C4C22×D1711yescharacteristic
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—342no—
353C2×C22⋊C4—341no—
354C2×C22⋊C4C22⋊D1711yescharacteristic
355C2×C22⋊C4C22×D1711yescharacteristic
356C2×C42C22⋊D1711yescharacteristic
357#103—348no—
358C22⋊C8—174no—
359C22⋊C8—342no—
360#95—172no—
361#95—342no—
362#95—342no—
363C17#1819511yescharacteristic, complemented, Sylow
364C24—22no—
365C24—22no—
366C24C2×C22⋊D1711yescharacteristic
367C24C2×C22⋊D1711yescharacteristic
368#41—44no—
369#41—416no—
370#41—44no—
371C2×Q8—22no—
372C2×Q8—344no—
373C2×Q8—348no—
374C2×Q8—344no—
375C2×Q8—682no—
376C2×Q8—684no—
377C2×Q8—684no—
378C2×Q8—688no—
379C2×Q8—681no—
380C2×Q8—681no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—416no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D4#6322914yescomplemented
398C2×D4C2×C22⋊D1714yes—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—342no—
407C22×C4—342no—
408C22×C4—684no—
409C22×C4—682no—
410C22×C4C2×C22⋊D1712yes—
411C22×C4C2×C22⋊D1711yescharacteristic
412C22×C4C2×C22⋊D1711yescharacteristic
413C22×C4C2×C22⋊D1711yescharacteristic
414C22×C4C2×C22⋊D1711yescharacteristic
415C22×C4C2×C22⋊D1712yes—
416C22×C4C23×D1711yescharacteristic
417SD16—6816no—
418C2×C8—344no—
419C2×C8—684no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—682no—
425C4⋊C4—684no—
426C4⋊C4C2×C22⋊D1714yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—24no—
433C22⋊C4—684no—
434C22⋊C4—682no—
435C42—21no—
436C42—21no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—44no—
445C23—44no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23#1058761812yes—
451C23C2×#6322912yes—
452C23C2×#6322912yescomplemented
453C23#1058763912yes—
454C23C22×C22⋊D1711yescharacteristic
455Q8—44no—
456Q8—348no—
457Q8—684no—
458Q8—684no—
459Q8—684no—
460D4—28no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—48no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—344no—
487C2×C4—682no—
488C2×C4—688no—
489C2×C4—682no—
490C2×C4—682no—
491C2×C4—682no—
492C2×C4—682no—
493C2×C4C2×#6322912yescomplemented
494C2×C4#1058763911yescharacteristic
495C2×C4#1058763911yescharacteristic
496C2×C4C22×C22⋊D1712yes—
497C8—684no—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—164no—
513C22—164no—
514C22#1176296014yes—
515C22C2×#1058763911yescharacteristic
516C22C22×#6322911yescharacteristic
517C22C2×#1058761811yescharacteristic
518C4—22no—
519C4—21no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—44no—
524C4—684no—
525C4—1362no—
526C4#1176299612yes—
527C2—21no—
528C2—21no—
529C2—24no—
530C2—42no—
531C2—84no—
532C2C2×#1176296011yescharacteristic
533C2#1453816311yescharacteristic
534C2C2×#1176299611yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC17×#3496C2
socleC22×C17C2×#120
hypercenter#3496D17
hypocenterC17#18195
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#11762947C2semidirect
2. #14532149C2semidirect
3. #14527753C2semidirect
4. #14527736C2semidirect
5. #14527352C2semidirect
6. #14527102C2semidirect
8. C23×Dic34C22semidirect
12. #11762947C22semidirect
14. C2×C22⋊#3622C22semidirect
15. #11762729C22semidirect
16. #11762721C22semidirect
20. #11762669C22semidirect
21. #11762543C22semidirect
22. #11762535C22semidirect
23. #11762533C22semidirect
120. C2×D4×C17SD16semidirect
185. C23×C17C2×SD16semidirect
217. C2×C4×C17C2×SD16semidirect
363. C17#18195semidirect
397. C2×D4#63229semidirect
452. C23C2×#63229semidirect
493. C2×C4C2×#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117629472
#145321492
#145277532
#145277362
#145273522
#145271022
C17×#34962
#1819517