Finite Group Databasebeta

← #17034872

Subgroups — #17034872

Order 4352.

886 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×Dic17C211yesmaximal, characteristic
2#14531186C211yesmaximal, characteristic
3#14531179C211yesmaximal, characteristic
4#14528951C211yesmaximal, characteristic, complemented
5#14528949C211yesmaximal, characteristic, complemented
6#14528916C211yesmaximal, characteristic, complemented
7#14528916C211yesmaximal, characteristic, complemented
8#14528817C211yesmaximal, characteristic
9#14528815C211yesmaximal, characteristic
10#14528815C211yesmaximal, characteristic
11#107×Dic17C211yesmaximal, characteristic, complemented
12#14528676C211yesmaximal, characteristic, complemented
13#106×Dic17C211yesmaximal, characteristic, complemented
14#14528672C211yesmaximal, characteristic, complemented
15C17×#1809C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587563—21no—
18C22×C4×Dic17C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25M16×Dic17C418yescomplemented
26C2×#10587486C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762678C2211yescharacteristic
31#11762678C2211yescharacteristic
32#11762676—21no—
33#11762674C2211yescharacteristic
34#11762674C2211yescharacteristic
35#11762673—21no—
36#11762670—21no—
37#11762669C2211yescharacteristic
38#11762669C2211yescharacteristic
39#11762667—21no—
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44#11762660—21no—
45C22⋊C4×Dic17—21no—
46#11762654—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17D411yescharacteristic
62C23×Dic17D411yescharacteristic
63C2×M16×C17—22no—
64C2×M16×C17—22no—
65C2×M16×C17C2×C414yes—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2×C414yes—
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17Q814yes—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C2×#63223—22no—
95C2×#63223—22no—
96C2×#63223C2×C414yes—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587563—44no—
100C17×#107—22no—
101C17×#106—22no—
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#10587502—21no—
114#10587489—22no—
115#10587488—22no—
116#10587486C2×C414yes—
117#10587486C2×C414yes—
118C8×Dic17C2×C414yes—
119C8×Dic17C2×C414yes—
120#10587479—22no—
121#10587479—22no—
122#10587478—22no—
123#10587478—22no—
124C24×C17C2×D411yescharacteristic
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×Q812yes—
136C22×C4×C17C2411yescharacteristic
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—41no—
144C22×Dic17—41no—
145C22×Dic17C4⋊C414yes—
146C22×Dic17C4⋊C414yes—
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151M16×C17—28no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17C22×C412yes—
155C2×C8×C17C22×C412yes—
156C2×C8×C17#4112yes—
157C2×C8×C17#4112yes—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C22⋊C4×C17—42no—
167C22⋊C4×C17—42no—
168C22⋊Dic17—42no—
169C22⋊Dic17—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178#63225—42no—
179#63225—42no—
180#63225—42no—
181#63225—42no—
182C4×Dic17—42no—
183C4×Dic17—42no—
184C4×Dic17—42no—
185C4×Dic17—42no—
186C4×Dic17C4⋊C414yes—
187C4×Dic17C4⋊C414yes—
188C4×Dic17C22×C412yes—
189C4×Dic17C22×C412yes—
190C4×Dic17C2×Q812yes—
191C4×Dic17C2×Q812yes—
192#63223—28no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622C22×C412yes—
196C2×#3622C22×C412yes—
197C2×#3622#4112yes—
198C2×#3622#4112yes—
199#20819—171nomaximal, Sylow
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—21no—
205C23×C17—21no—
206C23×C17C2×C4⋊C412yes—
207C23×C17C22×D411yescharacteristic
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
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×C17C23×C411yescharacteristic
228C2×C4×C17C22×Q811yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—28no—
240C2×Dic17—44no—
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×Dic17—41no—
253C2×Dic17C2×C4⋊C412yes—
254C2×Dic17C2×C4⋊C412yes—
255C2×Dic17C2×C4⋊C412yes—
256C2×Dic17C2×C4⋊C412yes—
257C2×Dic17AGL(1,ℤ/8)12yes—
258C2×Dic17#13712yes—
259C8×C17—22no—
260C8×C17#11714yescomplemented
261#3622—22no—
262#3622#11714yescomplemented
263C2×C4×M16—171no—
264#2285—171no—
265#2283—171no—
266#1847—171no—
267#1847—171no—
268#1836—171no—
269#1836—171no—
270#1809—171no—
271#1809—171no—
272#1809—171no—
273#1809D1711yescharacteristic
274C4×#107—171no—
275C4×#107—171no—
276C4×#106—171no—
277C4×#106—171no—
278C22×C17—22no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—41no—
285C22×C17C22×C4⋊C411yescharacteristic
286C22×C17#53211yescharacteristic
287C22×C17#53211yescharacteristic
288C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C17C2×#13711yescharacteristic
290C22×C17#57511yescharacteristic
291C22×C17#57511yescharacteristic
292C4×C17—22no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17—81no—
296C4×C17—81no—
297C4×C17C2×#11711yescharacteristic
298C4×C17C2×#11711yescharacteristic
299C4×C17#53311yescharacteristic
300C4×C17#53311yescharacteristic
301Dic17—22no—
302Dic17—22no—
303Dic17—22no—
304Dic17—81no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308Dic17#42814yes—
309C22×M16—171no—
310C22×M16C2×D1711yescharacteristic
311C2×#117—341no—
312C22×C42—171no—
313C2×#107—171no—
314C2×#107—171no—
315C2×#107—171no—
316C2×#107C2×D1711yescharacteristic
317C2×#106—171no—
318C2×#106—171no—
319C2×#106—171no—
320C2×#106C2×D1711yescharacteristic
321C4×M16—178no—
322C2×#97—171no—
323C2×#97—171no—
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326#384—171no—
327#384—171no—
328#383—341no—
329#382—171no—
330#382—171no—
331#382—341no—
332#381—341no—
333#380—171no—
334#380—171no—
335#380—341no—
336#380C2×D1711yescharacteristic
337#380C2×D1711yescharacteristic
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—341no—
343C4×C22⋊C4—341no—
344#376—341no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340C2×D1711yescharacteristic
352#340C2×D1711yescharacteristic
353C2×C17—42no—
354C2×C17#228511yescharacteristic
355C2×C17C2×#42811yescharacteristic
356C2×C17#291111yescharacteristic
357C2×C17#317111yescharacteristic
358C2×C17#317111yescharacteristic
359C2×C17#317211yescharacteristic
360C2×C17#317211yescharacteristic
361C23×C4—171no—
362C23×C4—171no—
363C23×C4C22×D1711yescharacteristic
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—174no—
367C2×M16—342no—
368C2×M16—342no—
369C2×M16C4×D1714yes—
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⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
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—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—341no—
401#107—22no—
402#107—342no—
403#107—342no—
404#107—342no—
405#106—22no—
406#106—342no—
407#106—342no—
408#106—342no—
409#97—174no—
410#97—174no—
411C4×C8—174no—
412C4×C8—174no—
413#95—21no—
414#95—21no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425C17#2081911yescharacteristic, complemented, Sylow
426C24D4×D1711yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—171no—
436C22×C4—171no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—174no—
440C22×C4—174no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—681no—
448C22×C4—681no—
449C22×C4C2×C4×D1712yes—
450C22×C4D4×D1711yescharacteristic
451C22×C4Q8×D1712yes—
452C22×C4C23×D1711yescharacteristic
453M16—28no—
454M16—348no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—172no—
458C2×C8—172no—
459C2×C8—172no—
460C2×C8—172no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8C2×C4×D1712yes—
464C2×C8C2×C4×D1712yes—
465C2×C8#6325412yes—
466C2×C8#6325412yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—341no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—682no—
484C4⋊C4—682no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—682no—
490C22⋊C4—682no—
491C42—172no—
492C42—172no—
493C42—172no—
494C42—172no—
495C42—174no—
496C42—174no—
497C42—682no—
498C42—682no—
499C42—682no—
500C42—682no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23C4⋊C4×D1712yes—
508C23C2×D4×D1711yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
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—172no—
528C2×C4—172no—
529C2×C4—172no—
530C2×C4—172no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—348no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—684no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4C4⋊C4×D1712yes—
554C2×C4C22×C4×D1711yescharacteristic
555C2×C4C2×#6325411yescharacteristic
556C2×C4C2×#6325411yescharacteristic
557C2×C4C2×Q8×D1711yescharacteristic
558C2×C4#41×D1711yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C8—22no—
561C8—174no—
562C8—342no—
563C8#1058756314yescomplemented
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C2×C4⋊C4×D1711yescharacteristic
572C22#1176335411yescharacteristic
573C22#1176335411yescharacteristic
574C22AGL(1,ℤ/8)×D1711yescharacteristic
575C22#137×D1711yescharacteristic
576C22#1176360511yescharacteristic
577C22#1176360811yescharacteristic
578C4—22no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—174no—
584C4—342no—
585C4—342no—
586C4—342no—
587C4—1361no—
588C4—1361no—
589C4—1361no—
590C4—1361no—
591C4#117×D1711yescharacteristic
592C4C2×#1058756311yescharacteristic
593C4#1176336111yescharacteristic
594C4#1176336111yescharacteristic
595C2—42no—
596C2#1453118611yescharacteristic
597C2D17×#42811yescharacteristic
598C2#1453737111yescharacteristic
599C2#1453937211yescharacteristic
600C2#1453937411yescharacteristic
601C2#1453938311yescharacteristic
602C2#1453938511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1809C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1809D17
hypocenterC17#20819
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #14528951C2semidirect
5. #14528949C2semidirect
6. #14528916C2semidirect
7. #14528916C2semidirect
11. #107×Dic17C2semidirect
12. #14528676C2semidirect
13. #106×Dic17C2semidirect
14. #14528672C2semidirect
25. M16×Dic17C4semidirect
260. C8×C17#117semidirect
262. #3622#117semidirect
425. C17#20819semidirect
563. C8#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×M16×Dic172
#145311862
#145311792
#145289512
#145289492
#145289162
#145288172
#145288152
#107×Dic172
#145286762
#106×Dic172
#145286722
C17×#18092
#2081917