Finite Group Databasebeta

← #17044056

Subgroups — #17044056

Order 4352.

1234 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763340C211yesmaximal, characteristic
2C2×#11762863C211yesmaximal, characteristic, complemented
3C2×#11762678C211yesmaximal, characteristic, complemented
4#14529751C212yesmaximal, complemented
5#14529599C212yesmaximal, complemented
6#14529102C211yesmaximal, characteristic
7#14529091C211yesmaximal, characteristic, complemented
8#14529089C211yesmaximal, characteristic, complemented
9#14528788C212yesmaximal, complemented
10#14528782C212yesmaximal, complemented
11C17×#1874C211yesmaximal, characteristic
12#11763340—24no—
13#11763340C418yescomplemented
14C2×#10587565—21no—
15C2×#10587576C2211yescharacteristic
16C2×#10587622C2211yescharacteristic
17C2×#10587621—21no—
18C2×#10587610C2211yescharacteristic
19C2×#10587573C2211yescharacteristic
20C2×D4×Dic17—21no—
21C2×D4×Dic17C2211yescharacteristic
22C2×#10587563—21no—
23C22×C4⋊Dic17C2211yescharacteristic
24C22×#63225—21no—
25C22×#63225—21no—
26C22×C4×Dic17C2211yescharacteristic
27#11762863—24no—
28C2×#10587485C2212yes—
29C2×#10587487C2212yes—
30C2×#10587481C2212yes—
31C2×#10587503C2212yes—
32C2×C22⋊#3622C2211yescharacteristic
33C2×#10587486C2212yes—
34#11762678—21no—
35#11762678—21no—
36#11762678C2212yes—
37#11762678C2212yes—
38#11762572C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C2×C17×#121C2211yescharacteristic
42C22×C4⋊C4×C17C2211yescharacteristic
43C2×C17×#102C2212yes—
44C2×C17×C22⋊C8C2211yescharacteristic
45C17×#340C2212yes—
46C22×D4×C17—21no—
47C22×D4×C17C2311yescharacteristic
48C23×C4×C17—21no—
49C23×C4×C17C2311yescharacteristic
50C23×Dic17—21no—
51C23×Dic17—21no—
52C23×Dic17D411yescharacteristic
53C23×Dic17D411yescharacteristic
54C22×Dic34C2311yescharacteristic
55C22×C8×C17C2312yes—
56C17×#121—22no—
57C17×#121C2×C414yes—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—42no—
60C2×C4⋊C4×C17C2312yes—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C22⋊C4×C17—21no—
63C2×C22⋊Dic17—21no—
64C2×C22⋊Dic17—21no—
65C2×C22⋊Dic17—21no—
66C2×C22⋊Dic17—21no—
67#10587622—22no—
68#10587622C2×C414yescomplemented
69#10587621—28no—
70D4×Dic17—22no—
71D4×Dic17—28no—
72D4×Dic17C2×C414yescomplemented
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—42no—
76C2×C4⋊Dic17C2312yes—
77C2×C4⋊Dic17C2311yescharacteristic
78#10587610—22no—
79#10587610C2×C414yes—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—44no—
88C2×#63225—44no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17C2×C414yes—
91C2×C4×Dic17D414yes—
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2312yes—
97#10587576—22no—
98#10587576C2×C414yescomplemented
99#10587573—22no—
100#10587573C2×C414yescomplemented
101#10587565—28no—
102#10587563—28no—
103C17×#102—42no—
104C17×C22⋊C8—22no—
105#10587503—42no—
106C22⋊#3622—22no—
107#10587487—42no—
108#10587486—24no—
109#10587485—24no—
110#10587481—42no—
111C24×C17—21no—
112C24×C17C2×D411yescharacteristic
113C2×D4×C17—24no—
114C2×D4×C17—21no—
115C2×D4×C17—22no—
116C2×D4×C17—22no—
117C2×D4×C17—42no—
118C2×D4×C17—42no—
119C2×D4×C17C22×C412yes—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—42no—
125C22×C4×C17—42no—
126C22×C4×C17—42no—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—24no—
138C22×Dic17—22no—
139C22×Dic17—42no—
140C22×Dic17—42no—
141C22×Dic17—42no—
142C22×Dic17—42no—
143C22×Dic17—42no—
144C22×Dic17C22⋊C414yes—
145C22×Dic17C22⋊C414yes—
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—21no—
153C2×Dic34C22×C412yes—
154C2×C8×C17—22no—
155C2×C8×C17—22no—
156C2×C8×C17—22no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—22no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17C22×C412yes—
162C22⋊C4×C17—24no—
163C22⋊Dic17—24no—
164C22⋊Dic17—24no—
165C22⋊Dic17—24no—
166C22⋊Dic17—24no—
167C4⋊Dic17—21no—
168C4⋊Dic17—24no—
169C4⋊Dic17—22no—
170C4⋊Dic17—22no—
171C4⋊Dic17—42no—
172C4⋊Dic17C22×C412yes—
173#63225—24no—
174#63225—24no—
175#63225—24no—
176#63225—44no—
177#63225—44no—
178#63225—44no—
179#63225—44no—
180C4×Dic17—24no—
181C4×Dic17C22⋊C414yes—
182C4×Dic17C22⋊C414yes—
183C4×Dic17C22×C412yes—
184C4×Dic17C22×C412yes—
185C4×Dic17C2×D412yes—
186C4×Dic17C2×D412yes—
187C2×#3622—22no—
188C2×#3622—22no—
189C2×#3622—22no—
190#21867—171nomaximal, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—22no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17C2×C22⋊C412yes—
203C23×C17AGL(1,ℤ/8)12yes—
204C23×C17#13712yes—
205C23×C17C22×D411yescharacteristic
206D4×C17—24no—
207D4×C17—48no—
208D4×C17—42no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—22no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—84no—
220C2×C4×C17C2×C22⋊C412yes—
221C2×C4×C17C23×C411yescharacteristic
222C2×C4×C17C22×D411yescharacteristic
223C2×C4×C17H(5,𝔽2)12yes—
224C2×C4×C172−1+412yes—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—28no—
232C2×Dic17—28no—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—48no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—84no—
244C2×Dic17C2×C22⋊C412yes—
245C2×Dic17C2×C22⋊C412yes—
246C2×Dic17C2×C22⋊C412yes—
247C2×Dic17C2×C22⋊C412yes—
248C2×Dic17AGL(1,ℤ/8)12yes—
249C2×Dic17#13712yes—
250Dic34—24no—
251Dic34—42no—
252C8×C17—42no—
253#3622—42no—
254C2×#529—171no—
255C2×#432—171no—
256C2×#384—171no—
257#1992—172no—
258#1959—172no—
259#1874—171no—
260#1874D1711yescharacteristic
261#1872—171no—
262#1872—171no—
263#1803—172no—
264#1802—172no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17#41714yes—
279C22×C17C22×C22⋊C411yescharacteristic
280C22×C17#53511yescharacteristic
281C22×C17#53611yescharacteristic
282C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C22×C17C2×#13711yescharacteristic
286C4×C17—22no—
287C4×C17—42no—
288C4×C17—82no—
289C4×C17#52012yes—
290C4×C17#53712yes—
291Dic17—22no—
292Dic17—22no—
293Dic17—22no—
294Dic17—42no—
295Dic17—42no—
296Dic17—82no—
297Dic17#41714yes—
298#529—178no—
299#529—344no—
300C2×#126—341no—
301C2×#125—171no—
302C2×#124—171no—
303C2×#123—341no—
304C2×C22⋊Q8—171no—
305C2×#121C2×D1711yescharacteristic
306C2×C4×Q8—171no—
307C2×C4×D4—171no—
308C2×C4×D4—341no—
309C2×#117—341no—
310C22×C4⋊C4—171no—
311C22×C4⋊C4—341no—
312C22×C4⋊C4—341no—
313C22×C4⋊C4C2×D1711yescharacteristic
314C22×C42—171no—
315#432—344no—
316C2×C4⋊C8—172no—
317C2×#103—172no—
318C2×#103—172no—
319C2×#102—172no—
320C2×#102C2×D1712yes—
321C2×C22⋊C8—171no—
322C2×C22⋊C8C2×D1711yescharacteristic
323C2×#97—172no—
324#384—172no—
325#384—172no—
326#384—341no—
327#384—341no—
328#340—172no—
329#340—172no—
330#340—172no—
331#340C2×D1712yes—
332C2×C17—24no—
333C2×C17—82no—
334C2×C17#231111yescharacteristic
335C2×C17C2×#41711yescharacteristic
336C2×C17C2×#41711yescharacteristic
337C2×C17#321711yescharacteristic
338C2×C17#321811yescharacteristic
339C2×C17#323811yescharacteristic
340C2×C17#323911yescharacteristic
341C22×Q8—171no—
342C22×D4—21no—
343C22×D4C22×D1711yescharacteristic
344C23×C4—21no—
345C23×C4—171no—
346C23×C4—171no—
347C23×C4—341no—
348C23×C4—341no—
349C23×C4C22×D1711yescharacteristic
350C22×C8—172no—
351C22×C8C22×D1712yes—
352#126—348no—
353#125—174no—
354#125—342no—
355#124—174no—
356#124—342no—
357#123—348no—
358C22⋊Q8—174no—
359C22⋊Q8—342no—
360#121—22no—
361#121C4×D1714yes—
362C4×Q8—174no—
363C4×Q8—342no—
364C4×D4—174no—
365C4×D4—342no—
366C4×D4—348no—
367#117—348no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—172no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—342no—
378C2×C4⋊C4—342no—
379C2×C4⋊C4—342no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4—684no—
383C2×C4⋊C4—684no—
384C2×C4⋊C4C22×D1712yes—
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—174no—
392C2×C42—171no—
393C2×C42—171no—
394C2×C42—174no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—341no—
398C4⋊C8—344no—
399#103—682no—
400#103—682no—
401#102—42no—
402#102—682no—
403C22⋊C8—22no—
404C22⋊C8—342no—
405#97—344no—
406C17#2186711yescharacteristic, complemented, Sylow
407C24—21no—
408C24D4×D1711yescharacteristic
409C2×Q8—172no—
410C2×Q8—341no—
411C2×Q8—342no—
412C2×Q8—342no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C4×D1712yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—174no—
432C22×C4—174no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—344no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—342no—
441C22×C4—682no—
442C22×C4—682no—
443C22×C4—682no—
444C22×C4—682no—
445C22×C4—682no—
446C22×C4C2×C4×D1712yes—
447C22×C4D4×D1712yes—
448C22×C4D4×D1711yescharacteristic
449C22×C4C23×D1711yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—342no—
454C2×C8—342no—
455C2×C8—342no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—172no—
461C4⋊C4—344no—
462C4⋊C4—344no—
463C4⋊C4—344no—
464C4⋊C4—341no—
465C4⋊C4—344no—
466C4⋊C4—342no—
467C4⋊C4—342no—
468C4⋊C4—682no—
469C4⋊C4—684no—
470C4⋊C4—684no—
471C4⋊C4—684no—
472C4⋊C4—684no—
473C4⋊C4C2×C4×D1712yes—
474C22⋊C4—24no—
475C22⋊C4—344no—
476C22⋊C4—344no—
477C22⋊C4—344no—
478C22⋊C4—344no—
479C42—172no—
480C42—172no—
481C42—172no—
482C42—172no—
483C42—174no—
484C42—174no—
485C42—344no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22⋊C4×D1712yes—
498C23#1058759712yes—
499C23#1058760112yes—
500C23C2×D4×D1711yescharacteristic
501Q8—344no—
502Q8—682no—
503D4—24no—
504D4—48no—
505D4—42no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—172no—
518C2×C4—172no—
519C2×C4—172no—
520C2×C4—172no—
521C2×C4—172no—
522C2×C4—172no—
523C2×C4—348no—
524C2×C4—348no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—342no—
532C2×C4—342no—
533C2×C4—688no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4—1364no—
542C2×C4C22⋊C4×D1712yes—
543C2×C4C22×C4×D1711yescharacteristic
544C2×C4C2×D4×D1711yescharacteristic
545C2×C4#1058769112yes—
546C2×C4#1058769712yes—
547C8—42no—
548C8—682no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22#1176280714yes—
563C22C2×C22⋊C4×D1711yescharacteristic
564C22#1176337411yescharacteristic
565C22#1176338111yescharacteristic
566C22AGL(1,ℤ/8)×D1711yescharacteristic
567C22C2×#1058759711yescharacteristic
568C22#137×D1711yescharacteristic
569C22C2×#1058760111yescharacteristic
570C4—22no—
571C4—42no—
572C4—82no—
573C4—174no—
574C4—342no—
575C4—342no—
576C4—342no—
577C4—682no—
578C4—682no—
579C4—1362no—
580C4#1176328812yes—
581C4#1176338612yes—
582C2—24no—
583C2—82no—
584C2#1453137111yescharacteristic
585C2D17×#41711yescharacteristic
586C2C2×#1176280711yescharacteristic
587C2#1453980011yescharacteristic
588C2#1453980911yescharacteristic
589C2#1454000311yescharacteristic
590C2#1454001211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1874C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
cosocleC22×C4×C17C24
hypercenter#1874D17
hypocenterC17#21867
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#11762863C2semidirect
3. C2×#11762678C2semidirect
4. #14529751C2semidirect
5. #14529599C2semidirect
7. #14529091C2semidirect
8. #14529089C2semidirect
9. #14528788C2semidirect
10. #14528782C2semidirect
13. #11763340C4semidirect
68. #10587622C2×C4semidirect
72. D4×Dic17C2×C4semidirect
98. #10587576C2×C4semidirect
100. #10587573C2×C4semidirect
406. C17#21867semidirect

Maximal subgroups

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

subgroupindex
C2×#117633402
C2×#117628632
C2×#117626782
#145297512
#145295992
#145291022
#145290912
#145290892
#145287882
#145287822
C17×#18742
#2186717