Finite Group Databasebeta

← #17044739

Subgroups — #17044739

Order 4352.

754 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×#11762834C211yesmaximal, characteristic
2#14531392C211yesmaximal, characteristic
3#14531389C211yesmaximal, characteristic
4#14529555C211yesmaximal, characteristic, complemented
5#14529554C211yesmaximal, characteristic, complemented
6#14529552C211yesmaximal, characteristic, complemented
7#14529551C211yesmaximal, characteristic, complemented
8#14529545C211yesmaximal, characteristic, complemented
9#14529544C211yesmaximal, characteristic, complemented
10#14529544C211yesmaximal, characteristic, complemented
11#14528816C211yesmaximal, characteristic
12#14528814C211yesmaximal, characteristic
13#14528808C211yesmaximal, characteristic
14#14528806C211yesmaximal, characteristic
15C17×#1948C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×#10587558C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25#11762834C418yescomplemented
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587485C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29#11762700—21no—
30#11762699—21no—
31#11762699—21no—
32#11762698—21no—
33#11762693—21no—
34#11762691—21no—
35#11762690—21no—
36#11762690—21no—
37#11762678C2211yescharacteristic
38#11762675C2211yescharacteristic
39#11762674C2211yescharacteristic
40#11762666C2211yescharacteristic
41#11762666C2211yescharacteristic
42#11762666C2211yescharacteristic
43#11762666C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C17×#384C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227D411yescharacteristic
76C2×#63227D411yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225D411yescharacteristic
84C2×#63225D411yescharacteristic
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C2×#63218—41no—
92C2×#63218—41no—
93C2×#63218C2×C414yes—
94C2×C8×D17—41no—
95C2×C8×D17—41no—
96C2×C8×D17C2×C414yes—
97#10587558C2×C414yes—
98#10587558Q814yes—
99C17×#106—41no—
100C17×#106—41no—
101C17×C4⋊C8C2×C414yes—
102C17×#95—21no—
103#10587502—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—41no—
115#10587488—41no—
116#10587485C2×C414yes—
117#10587485C2×C414yes—
118#10587479—41no—
119#10587479—41no—
120#10587479—41no—
121#10587478—41no—
122C4⋊#3622C2×C414yes—
123C23×D17C2×D411yescharacteristic
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17C2×D411yescharacteristic
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17C22×C412yes—
143C2×C4×D17C2×Q812yes—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17C22×C412yes—
149C2×C8×C17C22×C412yes—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—81no—
159C4⋊C4×C17—81no—
160C42×C17—81no—
161C42×C17C22×C412yes—
162C42×C17C2×Q812yes—
163#63227—81no—
164#63227—81no—
165#63227—81no—
166#63227—81no—
167#63227C4⋊C414yes—
168#63227C4⋊C414yes—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177#63225—81no—
178#63225—81no—
179#63225C4⋊C414yes—
180#63225C4⋊C414yes—
181C4×Dic17—81no—
182C4×Dic17—81no—
183C4×Dic17—81no—
184C4×Dic17C22×C412yes—
185C4×Dic17C2×Q812yes—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622C22×C412yes—
191C2×#3622C22×C412yes—
192#63218—44no—
193C8×D17—44no—
194#21967—171nomaximal, Sylow
195C23×C17C22×D411yescharacteristic
196C22×D17—21no—
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17C2×C4⋊C412yes—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—41no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17C2×C4⋊C412yes—
220C2×C4×C17C2×C4⋊C412yes—
221C2×C4×C17C23×C411yescharacteristic
222C2×C4×C17C22×Q811yescharacteristic
223C2×C4×C17H(5,𝔽2)11yescharacteristic
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17C2×C4⋊C412yes—
246C2×Dic17C2×C4⋊C412yes—
247C2×Dic17C2×C4⋊C412yes—
248C4×D17—22no—
249C4×D17—22no—
250C4×D17—22no—
251C4×D17—22no—
252C8×C17—22no—
253C8×C17—22no—
254#3622—22no—
255#3622—22no—
256C2×#424—171no—
257#2314—171no—
258#2314—171no—
259#1949—171no—
260#1949—171no—
261#1949—171no—
262#1949—171no—
263#1948—171no—
264#1948—171no—
265#1948—171no—
266#1948D1711yescharacteristic, complemented
267#1809—171no—
268#1809—171no—
269#1807—171no—
270#1807—171no—
271C22×C17C22×C4⋊C411yescharacteristic
272C22×C17#53711yescharacteristic
273C22×C17#53711yescharacteristic
274C22×C17#57711yescharacteristic
275C22×C17#57711yescharacteristic
276C22×C17#57711yescharacteristic
277C22×C17#57711yescharacteristic
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—22no—
281C2×D17—41no—
282C2×D17—41no—
283C2×D17—41no—
284C2×D17—41no—
285C4×C17—22no—
286C4×C17—22no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17—81no—
290C4×C17—81no—
291C4×C17#51811yescharacteristic
292C4×C17#51911yescharacteristic
293C4×C17#54411yescharacteristic
294C4×C17#54411yescharacteristic
295Dic17—22no—
296Dic17—22no—
297Dic17—22no—
298Dic17—81no—
299Dic17—81no—
300Dic17—81no—
301Dic17—81no—
302C22×M16—171no—
303C23×C8—171no—
304C2×#117—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×#106—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×#106C2×D1711yescharacteristic
312C2×#106C2×D1711yescharacteristic
313#424—178no—
314C2×C4⋊C8—171no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8C2×D1711yescharacteristic
318#388—341no—
319#388—341no—
320#388—341no—
321#388—341no—
322#387—341no—
323#387—341no—
324#387—341no—
325#387—341no—
326#384—171no—
327#384C2×D1711yescharacteristic
328#382—171no—
329#382—171no—
330#380—171no—
331#380—171no—
332#380—171no—
333#380—171no—
334C4×C4⋊C4—171no—
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4C2×D1711yescharacteristic
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340C2×D1711yescharacteristic
345#340C2×D1711yescharacteristic
346C2×C17#231611yescharacteristic
347C2×C17#291211yescharacteristic
348C2×C17#291211yescharacteristic
349C2×C17#326011yescharacteristic
350C2×C17#326011yescharacteristic
351C2×C17#326311yescharacteristic
352C2×C17#326311yescharacteristic
353D17—42no—
354C23×C4—171no—
355C2×M16—174no—
356C2×M16—681no—
357C2×M16—681no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—174no—
361C22×C8—681no—
362C22×C8—681no—
363C22×C8C22×D1711yescharacteristic
364C22×C8C22×D1711yescharacteristic
365#117—174no—
366#117—174no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—171no—
382C2×C22⋊C4—171no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—21no—
388C2×C42—171no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42C22×D1711yescharacteristic
393#107—681no—
394#107—681no—
395#106—41no—
396#106—41no—
397#106—681no—
398#106—681no—
399#106—681no—
400#106—681no—
401C4⋊C8—174no—
402C4⋊C8—174no—
403C4⋊C8—174no—
404C4⋊C8C4×D1714yes—
405#95—21no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417C17#2196711yescharacteristic, complemented, Sylow
418C24—171no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—172no—
427C22×C4—172no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4D4×D1711yescharacteristic
437C22×C4D4×D1711yescharacteristic
438C22×C4C23×D1711yescharacteristic
439M16—684no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—172no—
445C2×C8—172no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—684no—
451C2×C8C2×C4×D1712yes—
452C2×C8C2×C4×D1712yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—174no—
464C4⋊C4—174no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—1361no—
474C4⋊C4—1361no—
475C22⋊C4—174no—
476C22⋊C4—174no—
477C22⋊C4—1361no—
478C22⋊C4—1361no—
479C22⋊C4—1361no—
480C22⋊C4—1361no—
481C42—81no—
482C42—172no—
483C42—172no—
484C42—1361no—
485C42—1361no—
486C42—1361no—
487C42C2×C4×D1712yes—
488C42Q8×D1712yes—
489C23—172no—
490C23—341no—
491C23—341no—
492C23—341no—
493C23—341no—
494C23—341no—
495C23—341no—
496C23C2×D4×D1711yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—172no—
514C2×C4—172no—
515C2×C4—172no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—342no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4C4⋊C4×D1712yes—
539C2×C4C4⋊C4×D1712yes—
540C2×C4C22×C4×D1711yescharacteristic
541C2×C4#1058769111yescharacteristic
542C2×C4C2×Q8×D1711yescharacteristic
543C2×C4#1058769411yescharacteristic
544C2×C4#1058769611yescharacteristic
545C2×C4#1058769711yescharacteristic
546C8—22no—
547C8—22no—
548C8—342no—
549C8—342no—
550C22—342no—
551C22—342no—
552C22—342no—
553C22—681no—
554C22—681no—
555C22—681no—
556C22—681no—
557C22C2×C4⋊C4×D1711yescharacteristic
558C22#1176338811yescharacteristic
559C22#1176338911yescharacteristic
560C22#1176361511yescharacteristic
561C22#1176361511yescharacteristic
562C22#1176361811yescharacteristic
563C22#1176361911yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—342no—
571C4—342no—
572C4—342no—
573C4—1361no—
574C4—1361no—
575C4—1361no—
576C4—1361no—
577C4#1176328111yescharacteristic
578C4#1176328511yescharacteristic
579C4#1176343611yescharacteristic
580C4#1176343911yescharacteristic
581C2—682no—
582C2#1453139911yescharacteristic
583C2#1453737511yescharacteristic
584C2#1453737811yescharacteristic
585C2#1454022711yescharacteristic
586C2#1454022811yescharacteristic
587C2#1454025311yescharacteristic
588C2#1454025611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1948C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1948D17
hypocenterC17#21967
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. #14529555C2semidirect
5. #14529554C2semidirect
6. #14529552C2semidirect
7. #14529551C2semidirect
8. #14529545C2semidirect
9. #14529544C2semidirect
10. #14529544C2semidirect
15. C17×#1948C2semidirect
25. #11762834C4semidirect
266. #1948D17semidirect
417. C17#21967semidirect

Maximal subgroups

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

subgroupindex
C2×#117628342
#145313922
#145313892
#145295552
#145295542
#145295522
#145295512
#145295452
#145295442
#145288162
#145288142
#145288082
#145288062
C17×#19482
#2196717