Finite Group Databasebeta

← #17044005

Subgroups — #17044005

Order 4352.

1222 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763363C211yesmaximal, characteristic
2C2×#11762861C211yesmaximal, characteristic
3D17×#384C211yesmaximal, characteristic
4#14529743C212yesmaximal, complemented
5#14529584C212yesmaximal, complemented
6#14529090C212yesmaximal
7#14529088C212yesmaximal
8#14528790C211yesmaximal, characteristic, complemented
9#14528789C211yesmaximal, characteristic, complemented
10#14528786C211yesmaximal, characteristic, complemented
11C17×#1803C211yesmaximal, characteristic, complemented
12#11763363—24no—
13#11763363C418yescomplemented
14C2×#10587574C2211yescharacteristic
15C2×#10587630C2211yescharacteristic
16C2×#10587575—21no—
17C2×#10587583C2211yescharacteristic
18C2×#10587582—21no—
19C2×#10587632C2211yescharacteristic
20C2×C4×Dic34—21no—
21C2×C4×Dic34C2211yescharacteristic
22C2×#10587558—21no—
23C2×C4⋊C4×D17—21no—
24C2×C4⋊C4×D17—21no—
25C2×C4⋊C4×D17C2212yes—
26C2×C42×D17C2211yescharacteristic
27#11762861—24no—
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587487C2212yes—
30C2×#10587506C2212yes—
31C2×#10587481C2212yes—
32C2×#10587491C2212yes—
33C2×C8×Dic17C2212yes—
34#11762679C2211yescharacteristic
35#11762678—21no—
36#11762678—21no—
37#11762678C2212yes—
38#11762572C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C2×C4×Q8×C17C2211yescharacteristic
42C2×C17×C4⋊C8C2211yescharacteristic
43C2×C17×#103C2212yes—
44C17×#384C2211yescharacteristic
45C17×#340C2212yes—
46C22×Q8×C17C2311yescharacteristic
47C22×C4×D17—22no—
48C22×C4×D17D411yescharacteristic
49C22×C4×D17D411yescharacteristic
50C22×C4×D17C2311yescharacteristic
51C22×Dic34—21no—
52C22×Dic34C2311yescharacteristic
53C22×C8×C17C2312yes—
54C4×Q8×C17—22no—
55C4×Q8×C17C2×C414yescomplemented
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17C2312yes—
60C2×C4⋊C4×C17C2311yescharacteristic
61C2×C42×C17—21no—
62C2×C42×C17C2311yescharacteristic
63#10587632—22no—
64#10587632C2×C414yes—
65#10587630—22no—
66#10587630C2×C414yescomplemented
67C2×#63227—21no—
68C2×#63227—21no—
69C2×C4⋊Dic17—21no—
70C2×C4⋊Dic17—21no—
71C2×C4⋊Dic17—21no—
72C2×C4⋊Dic17C2312yes—
73C2×C4⋊Dic17C2311yescharacteristic
74C2×#63225—21no—
75C2×#63225—21no—
76C2×#63225—21no—
77C2×#63225—21no—
78C2×#63225—21no—
79C2×#63225—22no—
80C2×#63225—22no—
81C2×#63225—22no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17D411yescharacteristic
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C22×#3622C2312yes—
87#10587583—22no—
88#10587583C2×C414yes—
89#10587582—28no—
90C4⋊C4×D17—42no—
91C4⋊C4×D17—42no—
92C4⋊C4×D17—44no—
93C4⋊C4×D17—44no—
94#10587575—28no—
95#10587574—22no—
96#10587574C2×C414yescomplemented
97#10587558—28no—
98C42×D17C2×C414yes—
99C42×D17D414yes—
100C4×Dic34—28no—
101C4×Dic34—22no—
102C4×Dic34C2×C414yescomplemented
103C17×C4⋊C8—22no—
104C17×#103—42no—
105#10587506—42no—
106#10587491—24no—
107#10587487—42no—
108C8×Dic17—24no—
109#10587481—42no—
110C4⋊#3622—22no—
111C23×D17C2×D411yescharacteristic
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17—21no—
117C2×Q8×C17C22×C412yes—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2411yescharacteristic
124C22×Dic17—21no—
125C22×Dic17—21no—
126C22×Dic17—22no—
127C22×Dic17C2×D411yescharacteristic
128C22×Dic17C2×D411yescharacteristic
129C22×Dic17C2×D411yescharacteristic
130C2×C4×D17—21no—
131C2×C4×D17—21no—
132C2×C4×D17—21no—
133C2×C4×D17—21no—
134C2×C4×D17—22no—
135C2×C4×D17—22no—
136C2×C4×D17—44no—
137C2×C4×D17—44no—
138C2×C4×D17—44no—
139C2×C4×D17C22⋊C414yes—
140C2×C4×D17C22⋊C414yes—
141C2×C4×D17SD1614yes—
142C2×C4×D17Q1614yes—
143C2×C4×D17C22×C412yes—
144C2×C4×D17C2×D412yes—
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—24no—
151C2×Dic34—42no—
152C2×Dic34—42no—
153C2×Dic34C22×C412yes—
154C2×C8×C17—22no—
155C2×C8×C17—22no—
156C2×C8×C17—22no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—22no—
160C4⋊C4×C17—22no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17C22×C412yes—
164C42×C17—24no—
165C42×C17C22×C412yes—
166C42×C17C2×D412yes—
167#63227—24no—
168#63227—24no—
169C4⋊Dic17—21no—
170C4⋊Dic17—24no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17—42no—
174C4⋊Dic17—42no—
175C4⋊Dic17C22×C412yes—
176#63225—24no—
177#63225—24no—
178#63225—24no—
179#63225—24no—
180#63225—24no—
181#63225—44no—
182#63225—44no—
183#63225—44no—
184C4×Dic17—24no—
185C4×Dic17C22⋊C414yes—
186C4×Dic17C22⋊C414yes—
187C4×Dic17C22×C412yes—
188C4×Dic17C2×D412yes—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—22no—
192#21860—171nomaximal, Sylow
193C23×C17C22×D411yescharacteristic
194C22×D17—21no—
195C22×D17—21no—
196C22×D17—21no—
197C22×D17—21no—
198C22×D17C2×C22⋊C412yes—
199C22×D17C2×SD1612yes—
200C22×D17C2×Q1612yes—
201Q8×C17—22no—
202Q8×C17—22no—
203Q8×C17—41no—
204Q8×C17—41no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17C2×C22⋊C412yes—
217C2×C4×C17C2×C22⋊C412yes—
218C2×C4×C17C2×SD1612yes—
219C2×C4×C17C2×Q1612yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C22×D411yescharacteristic
222C2×C4×C17H(5,𝔽2)11yescharacteristic
223C2×C4×C17H(5,𝔽2)11yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17C2×C22⋊C412yes—
238C2×Dic17C2×C22⋊C412yes—
239C2×Dic17C2×C22⋊C412yes—
240C2×Dic17C2×SD1612yes—
241C2×Dic17C2×Q1612yes—
242C2×Dic17AGL(1,ℤ/8)12yes—
243C2×Dic17#13712yes—
244C4×D17—22no—
245C4×D17—22no—
246C4×D17—22no—
247C4×D17—22no—
248C4×D17—28no—
249C4×D17—88no—
250C4×D17#103116yescomplemented
251Dic34—22no—
252Dic34—22no—
253Dic34—41no—
254Dic34—41no—
255Dic34—48no—
256C8×C17—42no—
257#3622—42no—
258C2×#533—171no—
259C2×#431—171no—
260C2×#384—171no—
261#1990—172no—
262#1956—172no—
263#1872—172no—
264#1872—172no—
265#1803—171no—
266#1803—171no—
267#1803—171no—
268#1803D1711yescharacteristic, complemented
269C22×C17C22×C22⋊C411yescharacteristic
270C22×C17#53511yescharacteristic
271C22×C17#53611yescharacteristic
272C22×C17C22×SD1611yescharacteristic
273C22×C17C22×Q1611yescharacteristic
274C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C17C2×#13711yescharacteristic
276C2×D17—22no—
277C2×D17—22no—
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—22no—
281C2×D17—22no—
282C2×D17C2×#10314yes—
283C4×C17—22no—
284C4×C17—42no—
285C4×C17—42no—
286C4×C17—82no—
287C4×C17C2×#10314yes—
288C4×C17#52011yescharacteristic
289C4×C17#52011yescharacteristic
290C4×C17#53711yescharacteristic
291C4×C17#53711yescharacteristic
292Dic17—22no—
293Dic17—42no—
294Dic17—42no—
295Dic17—82no—
296Dic17C2×#10314yes—
297Dic17#41714yes—
298#533—178no—
299#533—344no—
300C2×C4⋊Q8—171no—
301C2×C4⋊Q8—171no—
302C2×#125—341no—
303C2×C22⋊Q8—171no—
304C2×C22⋊Q8—171no—
305C2×C22⋊Q8—341no—
306C2×C4×Q8—171no—
307C2×C4×Q8—341no—
308C2×C4×Q8C2×D1711yescharacteristic
309C2×#117—341no—
310C22×C4⋊C4—172no—
311C22×C4⋊C4—341no—
312C22×C4⋊C4—341no—
313C22×C42—171no—
314#431—344no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8C2×D1711yescharacteristic
317C2×#103—172no—
318C2×#103—172no—
319C2×#103—172no—
320C2×#103C2×D1712yes—
321C2×C22⋊C8—172no—
322C2×C4×C8—172no—
323#384—172no—
324#384—171no—
325#384—341no—
326#384—341no—
327#384C2×D1711yescharacteristic
328#340—172no—
329#340—172no—
330#340—172no—
331#340C2×D1712yes—
332C2×C17#231111yescharacteristic
333C2×C17C22×#10311yescharacteristic
334C2×C17C2×#41711yescharacteristic
335C2×C17#320811yescharacteristic
336C2×C17#320911yescharacteristic
337C2×C17#322911yescharacteristic
338C2×C17#323011yescharacteristic
339D17—24no—
340C22×Q8—171no—
341C22×Q8—341no—
342C22×Q8C22×D1711yescharacteristic
343C23×C4—171no—
344C23×C4—171no—
345C23×C4—171no—
346C23×C4—342no—
347C22×C8—172no—
348C22×C8C22×D1712yes—
349C4⋊Q8—174no—
350C4⋊Q8—174no—
351C4⋊Q8—342no—
352C4⋊Q8—342no—
353#125—348no—
354C22⋊Q8—174no—
355C22⋊Q8—174no—
356C22⋊Q8—342no—
357C22⋊Q8—348no—
358C22⋊Q8—342no—
359C4×Q8—22no—
360C4×Q8—174no—
361C4×Q8—348no—
362C4×Q8—342no—
363C4×Q8C4×D1714yescomplemented
364#117—348no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—172no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
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—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4—682no—
383C2×C4⋊C4—684no—
384C2×C4⋊C4—684no—
385C2×C4⋊C4C22×D1712yes—
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C42—21no—
390C2×C42—171no—
391C2×C42—171no—
392C2×C42—171no—
393C2×C42—174no—
394C2×C42—174no—
395C2×C42—341no—
396C2×C42C22×D1711yescharacteristic
397C4⋊C8—22no—
398C4⋊C8—342no—
399#103—42no—
400#103—682no—
401#103—682no—
402#103—682no—
403C22⋊C8—344no—
404C4×C8—344no—
405C17#2186011yescharacteristic, complemented, Sylow
406C24—171no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—172no—
413C2×Q8—344no—
414C2×Q8—341no—
415C2×Q8—341no—
416C2×Q8—341no—
417C2×Q8—341no—
418C2×Q8—341no—
419C2×Q8—682no—
420C2×Q8—682no—
421C2×Q8C2×C4×D1712yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—171no—
426C22×C4—171no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—174no—
430C22×C4—171no—
431C22×C4—172no—
432C22×C4—172no—
433C22×C4—174no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—342no—
443C22×C4—684no—
444C22×C4—684no—
445C22×C4—684no—
446C22×C4D4×D1711yescharacteristic
447C22×C4D4×D1711yescharacteristic
448C22×C4C23×D1711yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C4⋊C4—21no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—172no—
462C4⋊C4—344no—
463C4⋊C4—344no—
464C4⋊C4—344no—
465C4⋊C4—344no—
466C4⋊C4—344no—
467C4⋊C4—341no—
468C4⋊C4—344no—
469C4⋊C4—342no—
470C4⋊C4—342no—
471C4⋊C4—682no—
472C4⋊C4—684no—
473C4⋊C4—684no—
474C4⋊C4—684no—
475C4⋊C4—682no—
476C4⋊C4C2×C4×D1712yes—
477C22⋊C4—344no—
478C22⋊C4—344no—
479C42—24no—
480C42—172no—
481C42—172no—
482C42—174no—
483C42—174no—
484C42—344no—
485C42C2×C4×D1712yes—
486C42D4×D1712yes—
487C23—172no—
488C23—172no—
489C23—172no—
490C23—341no—
491C23—341no—
492C23—341no—
493C23—341no—
494C23C2×D4×D1711yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—342no—
500Q8—342no—
501Q8—688no—
502Q8—681no—
503Q8—681no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—172no—
516C2×C4—172no—
517C2×C4—172no—
518C2×C4—172no—
519C2×C4—172no—
520C2×C4—172no—
521C2×C4—1716no—
522C2×C4—172no—
523C2×C4—348no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—342no—
527C2×C4—342no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—1368no—
540C2×C4C22⋊C4×D1712yes—
541C2×C4C22⋊C4×D1712yes—
542C2×C4SD16×D1712yes—
543C2×C4Q16×D1712yes—
544C2×C4C22×C4×D1711yescharacteristic
545C2×C4C2×D4×D1711yescharacteristic
546C2×C4#1058769111yescharacteristic
547C2×C4#1058769411yescharacteristic
548C2×C4#1058769611yescharacteristic
549C2×C4#1058769711yescharacteristic
550C8—42no—
551C8—682no—
552C22—174no—
553C22—342no—
554C22—342no—
555C22—342no—
556C22—342no—
557C22—342no—
558C22—342no—
559C22C2×C22⋊C4×D1711yescharacteristic
560C22#1176337511yescharacteristic
561C22#1176338211yescharacteristic
562C22C2×SD16×D1711yescharacteristic
563C22C2×Q16×D1711yescharacteristic
564C22AGL(1,ℤ/8)×D1711yescharacteristic
565C22#137×D1711yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—174no—
571C4—174no—
572C4—342no—
573C4—682no—
574C4—682no—
575C4—1362no—
576C4#103×D1714yes—
577C4#1176328911yescharacteristic
578C4#1176329011yescharacteristic
579C4#1176338811yescharacteristic
580C4#1176338911yescharacteristic
581C2—344no—
582C2#1453137311yescharacteristic
583C2C2×#103×D1711yescharacteristic
584C2D17×#41711yescharacteristic
585C2#1453972411yescharacteristic
586C2#1453973111yescharacteristic
587C2#1453992711yescharacteristic
588C2#1453993411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1803C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1803D17
hypocenterC17#21860
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #14529743C2semidirect
5. #14529584C2semidirect
8. #14528790C2semidirect
9. #14528789C2semidirect
10. #14528786C2semidirect
11. C17×#1803C2semidirect
13. #11763363C4semidirect
55. C4×Q8×C17C2×C4semidirect
66. #10587630C2×C4semidirect
96. #10587574C2×C4semidirect
102. C4×Dic34C2×C4semidirect
250. C4×D17#103semidirect
268. #1803D17semidirect
363. C4×Q8C4×D17semidirect
405. C17#21860semidirect

Maximal subgroups

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

subgroupindex
C2×#117633632
C2×#117628612
D17×#3842
#145297432
#145295842
#145290902
#145290882
#145287902
#145287892
#145287862
C17×#18032
#2186017