Finite Group Databasebeta

← #17044878

Subgroups — #17044878

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762865C211yesmaximal, characteristic
2#14531188C211yesmaximal, characteristic
3D17×#382C211yesmaximal, characteristic
4#14529554C211yesmaximal, characteristic, complemented
5#14529552C211yesmaximal, characteristic, complemented
6#14529550C211yesmaximal, characteristic, complemented
7#14528978C211yesmaximal, characteristic
8#14528975C211yesmaximal, characteristic
9#14528973C211yesmaximal, characteristic
10#14528970C211yesmaximal, characteristic
11#14528951C211yesmaximal, characteristic, complemented
12#14528949C211yesmaximal, characteristic, complemented
13#14528916C211yesmaximal, characteristic, complemented
14#14528916C211yesmaximal, characteristic, complemented
15C17×#1949C211yesmaximal, characteristic, complemented
16C2×#10587558—21no—
17C2×C4⋊C4×D17C2211yescharacteristic
18C2×C4⋊C4×D17C2211yescharacteristic
19C2×C42×D17C2211yescharacteristic
20#11762865C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31C2×#10587486C2211yescharacteristic
32#11762678C2211yescharacteristic
33#11762678C2211yescharacteristic
34#11762675C2211yescharacteristic
35#11762674C2211yescharacteristic
36#11762674C2211yescharacteristic
37#11762673—21no—
38#11762673—21no—
39#11762673—21no—
40#11762668—21no—
41#11762666C2211yescharacteristic
42#11762666C2211yescharacteristic
43C4×#63225—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45#95×D17—21no—
46#95×D17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C17×#382C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17—21no—
61C22×C4×D17—21no—
62C22×C4×D17D411yescharacteristic
63C22×C4×D17D411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×#63227—21no—
74C2×#63227—21no—
75C2×C4⋊Dic17C2311yescharacteristic
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C4⋊C4×D17—41no—
92C4⋊C4×D17—41no—
93C4⋊C4×D17—41no—
94C4⋊C4×D17—41no—
95#10587558—44no—
96C42×D17C2×C414yes—
97C42×D17Q814yes—
98C17×#107—41no—
99C17×#106—41no—
100C17×C4⋊C8C2×C414yescomplemented
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587491C2×C414yes—
114#10587491C2×C414yes—
115#10587489—41no—
116#10587488—41no—
117#10587486C2×C414yescomplemented
118#10587486C2×C414yescomplemented
119#10587479—41no—
120#10587479—41no—
121#10587478—41no—
122#10587478—41no—
123C4⋊#3622C2×C414yescomplemented
124C23×D17C2×D411yescharacteristic
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C2×C4×D17—22no—
140C2×C4×D17—22no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×C4×D17—21no—
144C2×C4×D17—21no—
145C2×C4×D17—41no—
146C2×C4×D17—41no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C4×D17C4⋊C414yes—
154C2×C4×D17C4⋊C414yes—
155C2×C4×D17C22×C412yes—
156C2×C4×D17C2×Q812yes—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17C22×C412yes—
162C2×C8×C17C22×C412yes—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C42×C17—42no—
172C42×C17C22×C412yes—
173C42×C17C2×Q812yes—
174#63227—42no—
175#63227—42no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184#63225—42no—
185#63225—42no—
186#63225—42no—
187#63225—42no—
188#63225—42no—
189#63225—42no—
190C4×Dic17—42no—
191C4×Dic17C4⋊C414yes—
192C4×Dic17C4⋊C414yes—
193C4×Dic17C22×C412yes—
194C4×Dic17C2×Q812yes—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622C22×C412yes—
200C2×#3622C22×C412yes—
201#21986—171nomaximal, Sylow
202C23×C17C22×D411yescharacteristic
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17C2×C4⋊C412yes—
208C22×D17AGL(1,ℤ/8)12yes—
209C22×D17#13712yes—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17C2×C4⋊C412yes—
227C2×C4×C17C2×C4⋊C412yes—
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17H(5,𝔽2)11yescharacteristic
231C2×C4×C17H(5,𝔽2)11yescharacteristic
232C2×C4×C172−1+411yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17C2×C4⋊C412yes—
253C2×Dic17C2×C4⋊C412yes—
254C2×Dic17C2×C4⋊C412yes—
255C4×D17—28no—
256C4×D17—22no—
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—22no—
260C4×D17—44no—
261C4×D17—84no—
262C4×D17—84no—
263C8×C17—22no—
264C8×C17—22no—
265#3622—22no—
266#3622—22no—
267C2×#432—171no—
268#2285—171no—
269C2×#382—171no—
270#1949—171no—
271#1949—171no—
272#1949—171no—
273#1949D1711yescharacteristic, complemented
274#1853—171no—
275#1853—171no—
276#1852—171no—
277#1852—171no—
278#1847—171no—
279#1847—171no—
280#1836—171no—
281#1836—171no—
282C22×C17C22×C4⋊C411yescharacteristic
283C22×C17#53511yescharacteristic
284C22×C17#53611yescharacteristic
285C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C17C2×#13711yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57511yescharacteristic
289C2×D17—22no—
290C2×D17—22no—
291C2×D17—22no—
292C2×D17—22no—
293C2×D17—22no—
294C2×D17—22no—
295C2×D17#42814yes—
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—81no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17#52011yescharacteristic
303C4×C17#52011yescharacteristic
304C4×C17#54411yescharacteristic
305C4×C17#54411yescharacteristic
306Dic17—22no—
307Dic17—22no—
308Dic17—22no—
309Dic17—81no—
310Dic17—81no—
311Dic17—81no—
312Dic17—81no—
313C2×#117—341no—
314C22×C4⋊C4—171no—
315C22×C4⋊C4—171no—
316C22×C42—171no—
317#432—178no—
318C2×#107—171no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#107C2×D1711yescharacteristic
322C2×#106—171no—
323C2×#106—171no—
324C2×#106—171no—
325C2×#106C2×D1711yescharacteristic
326C2×C4⋊C8—171no—
327C2×C4⋊C8C2×D1711yescharacteristic
328C2×C22⋊C8—171no—
329C2×C22⋊C8—171no—
330C2×#97—171no—
331C2×#97—171no—
332#384—171no—
333#384—171no—
334#382—171no—
335#382—171no—
336#382—171no—
337#382—341no—
338#382—341no—
339#382—341no—
340#382C2×D1711yescharacteristic
341#380—171no—
342#380—171no—
343#380—341no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4C2×D1711yescharacteristic
347C2×#95—341no—
348C2×#95—341no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340C2×D1711yescharacteristic
356#340C2×D1711yescharacteristic
357C2×C17#231711yescharacteristic
358C2×C17C2×#42811yescharacteristic
359C2×C17#291111yescharacteristic
360C2×C17#321311yescharacteristic
361C2×C17#321511yescharacteristic
362C2×C17#323411yescharacteristic
363C2×C17#323611yescharacteristic
364D17—24no—
365C23×C4—171no—
366C23×C4—171no—
367C23×C4—171no—
368C23×C4—341no—
369C23×C4—341no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8C22×D1711yescharacteristic
373C22×C8C22×D1711yescharacteristic
374#117—684no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—681no—
386C2×C4⋊C4—681no—
387C2×C4⋊C4—681no—
388C2×C4⋊C4—681no—
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C42—21no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—174no—
400C2×C42—174no—
401C2×C42—341no—
402C2×C42C22×D1711yescharacteristic
403#107—41no—
404#107—681no—
405#107—681no—
406#107—681no—
407#106—41no—
408#106—681no—
409#106—681no—
410#106—681no—
411C4⋊C8—174no—
412C4⋊C8C4×D1714yescomplemented
413C22⋊C8—174no—
414C22⋊C8—174no—
415#97—174no—
416#97—174no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429C17#2198611yescharacteristic, complemented, Sylow
430C24—171no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—171no—
436C22×C4—171no—
437C22×C4—171no—
438C22×C4—172no—
439C22×C4—172no—
440C22×C4—174no—
441C22×C4—174no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—681no—
453C22×C4—681no—
454C22×C4—682no—
455C22×C4—682no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4—682no—
460C22×C4D4×D1711yescharacteristic
461C22×C4D4×D1711yescharacteristic
462C22×C4C23×D1711yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8C2×C4×D1712yes—
474C2×C8C2×C4×D1712yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—682no—
492C4⋊C4—682no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4—682no—
496C4⋊C4—682no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C42—42no—
500C42—172no—
501C42—172no—
502C42—174no—
503C42—174no—
504C42—682no—
505C42C2×C4×D1712yes—
506C42Q8×D1712yes—
507C23—172no—
508C23—172no—
509C23—172no—
510C23—341no—
511C23—341no—
512C23—341no—
513C23—341no—
514C23C2×D4×D1711yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—348no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—342no—
544C2×C4—342no—
545C2×C4—684no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—1364no—
559C2×C4—1364no—
560C2×C4C4⋊C4×D1712yes—
561C2×C4C4⋊C4×D1712yes—
562C2×C4C22×C4×D1711yescharacteristic
563C2×C4#1058769111yescharacteristic
564C2×C4C2×Q8×D1711yescharacteristic
565C2×C4#1058769411yescharacteristic
566C2×C4#1058769611yescharacteristic
567C2×C4#1058769711yescharacteristic
568C8—22no—
569C8—22no—
570C8—342no—
571C8—342no—
572C22—174no—
573C22—342no—
574C22—342no—
575C22—342no—
576C22—342no—
577C22—342no—
578C22—342no—
579C22C2×C4⋊C4×D1711yescharacteristic
580C22#1176337511yescharacteristic
581C22#1176338211yescharacteristic
582C22AGL(1,ℤ/8)×D1711yescharacteristic
583C22#137×D1711yescharacteristic
584C22#1176360611yescharacteristic
585C22#1176360611yescharacteristic
586C4—22no—
587C4—22no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—342no—
593C4—342no—
594C4—342no—
595C4—1361no—
596C4—1361no—
597C4—1361no—
598C4—1361no—
599C4#1176328811yescharacteristic
600C4#1176329111yescharacteristic
601C4#1176343611yescharacteristic
602C4#1176343911yescharacteristic
603C2—344no—
604C2#1453140611yescharacteristic
605C2D17×#42811yescharacteristic
606C2#1453737211yescharacteristic
607C2#1453976911yescharacteristic
608C2#1453978511yescharacteristic
609C2#1453997211yescharacteristic
610C2#1453998811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1949C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1949D17
hypocenterC17#21986
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #14529554C2semidirect
5. #14529552C2semidirect
6. #14529550C2semidirect
11. #14528951C2semidirect
12. #14528949C2semidirect
13. #14528916C2semidirect
14. #14528916C2semidirect
15. C17×#1949C2semidirect
20. #11762865C4semidirect
100. C17×C4⋊C8C2×C4semidirect
117. #10587486C2×C4semidirect
118. #10587486C2×C4semidirect
123. C4⋊#3622C2×C4semidirect
273. #1949D17semidirect
412. C4⋊C8C4×D17semidirect
429. C17#21986semidirect

Maximal subgroups

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

subgroupindex
C2×#117628652
#145311882
D17×#3822
#145295542
#145295522
#145295502
#145289782
#145289752
#145289732
#145289702
#145289512
#145289492
#145289162
C17×#19492
#2198617