Finite Group Databasebeta

← #17035100

Subgroups — #17035100

Order 4352.

1462 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×#11763289C211yesmaximal, characteristic
2C2×#11762851C211yesmaximal, characteristic, complemented
3C2×#11762845C211yesmaximal, characteristic, complemented
4C2×#11762777C211yesmaximal, characteristic
5#14531377C211yesmaximal, characteristic
6#14528986C212yesmaximal, complemented
7#14528984C212yesmaximal, complemented
8#14528978C212yesmaximal, complemented
9#14528973C212yesmaximal, complemented
10C2×#11762571C211yesmaximal, characteristic, complemented
11C17×#1736C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C23×#3622C2211yescharacteristic
14#11763289—24no—
15#11763289—24no—
16C2×C4×Dic34—21no—
17C2×C4×D68—21no—
18C2×#10587567—22no—
19C2×#10587612C2211yescharacteristic
20C2×#10587578C2212yes—
21C22×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊C4×D17C2212yes—
23#11762851—24no—
24#11762845—24no—
25C2×#10587489C2212yes—
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587488C2212yes—
29C2×#10587479C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31#11762777C418yescomplemented
32C2×#10587491C2212yes—
33C2×#10587491C2212yes—
34#11762714—21no—
35#11762710—21no—
36#11762688—22no—
37#11762666C2212yes—
38#11762666C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C22×M16×C17C2211yescharacteristic
42C2×C17×#117C2211yescharacteristic
43C22×C4⋊C4×C17C2211yescharacteristic
44C17×#340C2212yes—
45C17×#340C2212yes—
46C23×C4×C17—21no—
47C23×C4×C17C2311yescharacteristic
48C22×C22⋊D17D412yes—
49C23×Dic17—21no—
50C2×#63252—22no—
51C2×#63252—22no—
52C2×#63252—22no—
53C2×#63252—22no—
54C2×#63252C2×C414yes—
55C2×#63252Q814yes—
56C22×D68D411yescharacteristic
57C22×C4×D17—22no—
58C22×C4×D17C2312yes—
59C22×Dic34D411yescharacteristic
60C2×M16×C17—22no—
61C2×M16×C17—22no—
62C2×M16×C17C2×C414yescomplemented
63C22×C8×C17C2312yes—
64C17×#117—22no—
65C17×#117D414yes—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17C2312yes—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C22⋊C4×C17—21no—
73C2×C42×C17—21no—
74C2×C22⋊Dic17—21no—
75C2×#63227—22no—
76C2×#63227—22no—
77C2×#63227—22no—
78#10587612—22no—
79#10587612—22no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17D414yes—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2312yes—
86C2×#63225—22no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—22no—
89C22×#3622—22no—
90C22×#3622—22no—
91C22×#3622C2×C414yescomplemented
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587578—24no—
95#10587578—24no—
96C4⋊C4×D17—24no—
97C4⋊C4×D17—24no—
98#10587567—216no—
99C4×D68—28no—
100C4×Dic34—28no—
101#10587502—22no—
102#10587502—22no—
103#10587491C2×C418yescomplemented
104#10587491C2×C418yescomplemented
105#10587489—24no—
106#10587488—24no—
107#10587479—22no—
108#10587479—22no—
109#10587478—22no—
110#10587478—22no—
111C24×C17C2×D411yescharacteristic
112C23×D17C2×D412yes—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—42no—
117C22×C4×C17—81no—
118C22×C4×C17—81no—
119C22×C4×C17C22×C412yes—
120C22×C4×C17C2×D412yes—
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D412yes—
123C22×C4×C17C2×Q812yes—
124C22×C4×C17#4112yes—
125C22×C4×C17#4112yes—
126C22×C4×C17C2411yescharacteristic
127C2×C22⋊D17—24no—
128C2×C22⋊D17—24no—
129C2×C22⋊D17C4⋊C418yes—
130C22×Dic17—22no—
131C22×Dic17—24no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—81no—
135C22×Dic17—81no—
136C22×Dic17C2×D412yes—
137#63252—28no—
138#63252—28no—
139#63252—28no—
140#63252—28no—
141C2×D68—22no—
142C2×D68—22no—
143C2×D68C4⋊C414yes—
144C2×C4×D17—22no—
145C2×C4×D17—22no—
146C2×C4×D17—22no—
147C2×C4×D17—22no—
148C2×C4×D17—28no—
149C2×C4×D17—82no—
150C2×C4×D17—82no—
151C2×C4×D17C22×C414yes—
152C2×C4×D17C2×Q814yes—
153C2×Dic34—22no—
154C2×Dic34—22no—
155C2×Dic34C4⋊C414yes—
156M16×C17—28no—
157C2×C8×C17—22no—
158C2×C8×C17—22no—
159C2×C8×C17C22×C414yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—22no—
163C4⋊C4×C17—22no—
164C4⋊C4×C17—42no—
165C4⋊C4×C17C2×D412yes—
166C4⋊C4×C17C2×D412yes—
167C22⋊C4×C17—24no—
168C42×C17—24no—
169C22⋊Dic17—42no—
170#63227—28no—
171#63227—82no—
172#63227—82no—
173C4⋊Dic17—24no—
174C4⋊Dic17—22no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17C2×D414yes—
180#63225—28no—
181C4×Dic17—28no—
182C4×Dic17—42no—
183C2×#3622—28no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622C22×C412yes—
189C2×#3622C22×C412yes—
190#20849—171nomaximal, Sylow
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17C2×C4⋊C412yes—
196C23×C17#12012yes—
197C23×C17#12312yes—
198C23×C17C22×D411yescharacteristic
199C22×D17—22no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—22no—
205C22×D17C2×C4⋊C414yes—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—84no—
216C2×C4×C17C2×C4⋊C412yes—
217C2×C4×C17C4×D414yes—
218C2×C4×C17C4×D414yes—
219C2×C4×C17#12012yes—
220C2×C4×C17C22⋊Q814yes—
221C2×C4×C17C22⋊Q814yes—
222C2×C4×C17#12312yes—
223C2×C4×C17C23×C411yescharacteristic
224C2×C4×C17C22×D411yescharacteristic
225C2×C4×C17C22×D411yescharacteristic
226C2×C4×C17C22×Q811yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C22⋊D17—216no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—24no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—84no—
242C2×Dic17C2×C4⋊C414yes—
243D68—28no—
244C4×D17—24no—
245C4×D17—24no—
246C4×D17—24no—
247C4×D17—24no—
248C4×D17—88no—
249Dic34—28no—
250C8×C17—24no—
251#3622—22no—
252#3622—22no—
253C2×#520—171no—
254C2×#428—171no—
255C2×#427—171no—
256C2×#408—171no—
257#2311—171no—
258#1854—172no—
259#1854—172no—
260#1853—172no—
261#1852—172no—
262#1736D1711yescharacteristic, complemented
263C2×#340—171no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17#38514yes—
271C22×C17C22×C4⋊C411yescharacteristic
272C22×C17C2×#12011yescharacteristic
273C22×C17C2×#12311yescharacteristic
274C22×C17#57611yescharacteristic
275C22×C17#57711yescharacteristic
276C22×C17#57711yescharacteristic
277C22×C17#57811yescharacteristic
278C2×D17—24no—
279C2×D17—24no—
280C2×D17—24no—
281C2×D17—42no—
282C2×D17—42no—
283C2×D17—42no—
284C2×D17—42no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—82no—
288C4×C17#38514yes—
289C4×C17C2×C4×D411yescharacteristic
290C4×C17C2×C4×D411yescharacteristic
291C4×C17C2×C22⋊Q811yescharacteristic
292C4×C17C2×C22⋊Q811yescharacteristic
293Dic17—24no—
294Dic17—44no—
295Dic17—81no—
296Dic17—81no—
297C22×#41—171no—
298C22×M16C2×D1711yescharacteristic
299C23×C8—171no—
300#520—344no—
301#520—344no—
302C2×C4×Q8—341no—
303C2×C4×D4—341no—
304C2×C4×D4—342no—
305C2×#117—171no—
306C2×#117—172no—
307C2×#117C2×D1711yescharacteristic
308C22×C4⋊C4—171no—
309C22×C4⋊C4—172no—
310C22×C4⋊C4C2×D1711yescharacteristic
311#428—344no—
312#427—344no—
313C2×#107—172no—
314C2×#107—171no—
315C2×#107—171no—
316C2×#106—172no—
317C2×#106—171no—
318C2×#106—171no—
319#408—178no—
320C2×C22⋊C8—172no—
321C2×C22⋊C8—172no—
322#391—341no—
323#390—341no—
324#386—342no—
325#380—172no—
326#380—172no—
327#340—172no—
328#340—172no—
329#340C2×D1712yes—
330#340C2×D1712yes—
331C2×C17—24no—
332C2×C17C2×#38511yescharacteristic
333C2×C17#291211yescharacteristic
334C2×C17#291311yescharacteristic
335C2×C17#301211yescharacteristic
336C2×C17#301311yescharacteristic
337C2×C17#309611yescharacteristic
338C2×C17#309711yescharacteristic
339D17—44no—
340C2×#41—174no—
341C2×#41—174no—
342C2×#41—342no—
343C2×#41—342no—
344C2×#41—342no—
345C2×#41—342no—
346C22×Q8—171no—
347C22×D4—172no—
348C22×D4—171no—
349C23×C4—21no—
350C23×C4—172no—
351C23×C4—342no—
352C23×C4—341no—
353C23×C4C22×D1711yescharacteristic
354C2×M16—22no—
355C2×M16—22no—
356C2×M16C4×D1714yescomplemented
357C22×C8—174no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—342no—
361C22×C8—342no—
362C22×C8C22×D1712yes—
363C4×Q8—348no—
364C4×D4—348no—
365C4×D4—3416no—
366#117—22no—
367#117—344no—
368#117—344no—
369#117—342no—
370#117—342no—
371#117D6814yes—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—174no—
376C2×C4⋊C4—172no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—344no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—344no—
384C2×C4⋊C4C22×D1712yes—
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—342no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—342no—
392C2×C42—21no—
393C2×C42—342no—
394C2×C42—341no—
395#107—344no—
396#107—342no—
397#107—342no—
398#106—344no—
399#106—342no—
400#106—342no—
401C22⋊C8—178no—
402C22⋊C8—178no—
403#95—342no—
404#95—342no—
405C17#2084911yescharacteristic, complemented, Sylow
406C24—172no—
407C24D4×D1711yescharacteristic
408#41—348no—
409#41—348no—
410#41—348no—
411#41—348no—
412C2×Q8—174no—
413C2×Q8—342no—
414C2×Q8—342no—
415C2×D4—178no—
416C2×D4—174no—
417C2×D4—344no—
418C2×D4—344no—
419C2×D4—342no—
420C2×D4—342no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—172no—
428C22×C4—174no—
429C22×C4—174no—
430C22×C4—348no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—344no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—1362no—
440C22×C4—1362no—
441C22×C4—1361no—
442C22×C4—1361no—
443C22×C4C2×C4×D1712yes—
444C22×C4C2×D6812yes—
445C22×C4#6325212yes—
446C22×C4D4×D1712yes—
447C22×C4D4×D1711yescharacteristic
448C22×C4#6325412yes—
449C22×C4Q8×D1712yes—
450C22×C4C23×D1711yescharacteristic
451M16—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—172no—
455C2×C8—172no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—348no—
461C2×C8C2×C4×D1714yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—174no—
468C4⋊C4—348no—
469C4⋊C4—342no—
470C4⋊C4—344no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4C2×D6812yes—
476C4⋊C4C2×D6812yes—
477C22⋊C4—24no—
478C22⋊C4—348no—
479C22⋊C4—682no—
480C22⋊C4—1362no—
481C22⋊C4—1362no—
482C42—24no—
483C42—348no—
484C42—682no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—174no—
490C23—342no—
491C23—342no—
492C23—342no—
493C23—342no—
494C23—342no—
495C23—342no—
496C23#1058756812yes—
497C23#1058756912yes—
498C23C4⋊C4×D1712yes—
499C23C2×D4×D1711yescharacteristic
500Q8—348no—
501D4—3416no—
502D4—348no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—174no—
514C2×C4—342no—
515C2×C4—342no—
516C2×C4—344no—
517C2×C4—344no—
518C2×C4—344no—
519C2×C4—344no—
520C2×C4—344no—
521C2×C4—682no—
522C2×C4—682no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—1368no—
530C2×C4—1364no—
531C2×C4C4×D6814yes—
532C2×C4#1058756714yes—
533C2×C4#1058756812yes—
534C2×C4#1058756912yes—
535C2×C4C4⋊C4×D1712yes—
536C2×C4#1058758214yes—
537C2×C4#1058758314yes—
538C2×C4C22×C4×D1711yescharacteristic
539C2×C4C22×D6811yescharacteristic
540C2×C4C2×#6325211yescharacteristic
541C2×C4C2×D4×D1711yescharacteristic
542C2×C4C2×#6325411yescharacteristic
543C2×C4C2×Q8×D1711yescharacteristic
544C8—24no—
545C8—342no—
546C8—342no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—344no—
554C22—344no—
555C22—344no—
556C22—682no—
557C22—682no—
558C22—682no—
559C22—682no—
560C22#1176268314yes—
561C22C2×C4⋊C4×D1711yescharacteristic
562C22C2×#1058756811yescharacteristic
563C22C2×#1058756911yescharacteristic
564C22#1176361311yescharacteristic
565C22#1176361811yescharacteristic
566C22#1176361911yescharacteristic
567C22#1176362511yescharacteristic
568C4—42no—
569C4—42no—
570C4—82no—
571C4—344no—
572C4—684no—
573C4—1361no—
574C4—1361no—
575C4#1176268314yes—
576C4C2×#1058756711yescharacteristic
577C4C2×C4×D6811yescharacteristic
578C4C2×#1058758211yescharacteristic
579C4C2×#1058758311yescharacteristic
580C2—24no—
581C2—684no—
582C2C2×#1176268311yescharacteristic
583C2#1453737811yescharacteristic
584C2#1453738311yescharacteristic
585C2#1453809011yescharacteristic
586C2#1453810111yescharacteristic
587C2#1453874011yescharacteristic
588C2#1453875111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1736C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#1736D17
hypocenterC17#20849
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#11762851C2semidirect
3. C2×#11762845C2semidirect
6. #14528986C2semidirect
7. #14528984C2semidirect
8. #14528978C2semidirect
9. #14528973C2semidirect
10. C2×#11762571C2semidirect
11. C17×#1736C2semidirect
31. #11762777C4semidirect
62. C2×M16×C17C2×C4semidirect
91. C22×#3622C2×C4semidirect
103. #10587491C2×C4semidirect
104. #10587491C2×C4semidirect
262. #1736D17semidirect
356. C2×M16C4×D17semidirect
405. C17#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#117632892
C2×#117628512
C2×#117628452
C2×#117627772
#145313772
#145289862
#145289842
#145289782
#145289732
C2×#117625712
C17×#17362
#2084917