Finite Group Databasebeta

← #17070405

Subgroups — #17070405

Order 4352.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763366C211yesmaximal, characteristic, complemented
2C2×#11763099C211yesmaximal, characteristic, complemented
3C2×#11763089C211yesmaximal, characteristic, complemented
4#14530122C212yesmaximal, complemented
5#14529930C212yesmaximal, complemented
6#14529736C211yesmaximal, characteristic, complemented
7#14529732C211yesmaximal, characteristic, complemented
8#14529107C212yesmaximal
9#14528981C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C4×C17×#121C211yesmaximal, characteristic
12#11763366—24no—
13#11763366—24no—
14C2×#10587555C2211yescharacteristic
15C2×#10587555C2211yescharacteristic
16C2×#10587556C2212yes—
17C2×#10587610—21no—
18C2×#10587610—21no—
19C2×C4×Dic34—21no—
20C2×C4×Dic34—21no—
21C2×#10587612C2212yes—
22#11763099—24no—
23#11763089—24no—
24C2×#10587478C2212yes—
25C2×#10587479C2212yes—
26C2×C4⋊#3622C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587503C2212yes—
29C2×#10587503C2212yes—
30C2×C22⋊#3622C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C2×C17×#121C2211yescharacteristic
37C2×C4×D4×C17—21no—
38C2×C4×D4×C17C2211yescharacteristic
39C22×C42×C17C2211yescharacteristic
40C17×#390C2211yescharacteristic
41C17×#389C2211yescharacteristic
42C17×#388—21no—
43C17×#386—21no—
44C17×#380C2211yescharacteristic
45C4×C4⋊C4×C17C2211yescharacteristic
46C4×C22⋊C4×C17—21no—
47C22×D4×C17—21no—
48C22×D4×C17C2311yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17D411yescharacteristic
51C23×C4×C17D411yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C22×Dic34—21no—
54C22×Dic34—21no—
55C17×#121—22no—
56C17×#121Q814yescomplemented
57C4×D4×C17—22no—
58C4×D4×C17—22no—
59C4×D4×C17—44no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C4⋊C4×C17C2311yescharacteristic
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C22⋊C4×C17—21no—
65C2×C22⋊C4×C17—21no—
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C42×C17—22no—
70C2×C42×C17—22no—
71C2×C42×C17—21no—
72C2×C42×C17D411yescharacteristic
73C2×C42×C17D411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—22no—
77#10587612—24no—
78#10587612—24no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2312yes—
82C2×C4⋊Dic17C2312yes—
83#10587610—28no—
84#10587610—28no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×C4×Dic17—22no—
88C22×#3622C2312yes—
89C22×#3622C2312yes—
90#10587556—24no—
91#10587556—24no—
92#10587555—22no—
93#10587555—22no—
94#10587555D414yescomplemented
95#10587555D414yescomplemented
96C4×Dic34—28no—
97C4×Dic34—28no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101#10587503—24no—
102#10587503—24no—
103C22⋊#3622—24no—
104#10587479—24no—
105#10587478—24no—
106C4⋊#3622—22no—
107C4⋊#3622—22no—
108C24×C17—21no—
109C24×C17C2×D411yescharacteristic
110C2×D4×C17—24no—
111C2×D4×C17—21no—
112C2×D4×C17—21no—
113C2×D4×C17—21no—
114C2×D4×C17—21no—
115C2×D4×C17—21no—
116C2×D4×C17—42no—
117C2×D4×C17—42no—
118C2×D4×C17C2×Q812yes—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—41no—
131C22×C4×C17—41no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17—42no—
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×Q812yes—
141C22×C4×C17#4112yes—
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—22no—
144C22×Dic17—22no—
145C2×Dic34—24no—
146C2×Dic34—24no—
147C2×Dic34—42no—
148C2×Dic34—42no—
149C2×Dic34—42no—
150C2×Dic34—42no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—42no—
157C4⋊C4×C17C2×Q812yes—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—42no—
161C22⋊C4×C17—42no—
162C22⋊C4×C17—42no—
163C42×C17—22no—
164C42×C17—22no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—42no—
168C42×C17C2×D412yes—
169C42×C17C2×D412yes—
170C22⋊Dic17—28no—
171C4⋊Dic17—24no—
172C4⋊Dic17—24no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17C2×D414yes—
176C4⋊Dic17C2×D414yes—
177#63225—28no—
178#63225—28no—
179C4×Dic17—28no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186#24300—171nomaximal, Sylow
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—21no—
191C23×C17—21no—
192C23×C17—21no—
193C23×C17—22no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C22⋊Q812yes—
201C23×C17C22×D411yescharacteristic
202D4×C17—22no—
203D4×C17—22no—
204D4×C17—48no—
205D4×C17—41no—
206D4×C17—41no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—44no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—44no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—84no—
233C2×C4×C17#12012yes—
234C2×C4×C17#12012yes—
235C2×C4×C17C22⋊Q812yes—
236C2×C4×C17#13512yes—
237C2×C4×C17AGL(1,ℤ/8)12yes—
238C2×C4×C17C22×D411yescharacteristic
239C2×C4×C17C22×D411yescharacteristic
240C2×C4×C17C22×Q811yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C172−1+411yescharacteristic
243C2×C4×C172−1+411yescharacteristic
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250Dic34—48no—
251Dic34—48no—
252#3622—42no—
253#3622—42no—
254C2×#533—171no—
255C2×#476—171no—
256C2×#474—171no—
257C4×#121D1711yescharacteristic
258#2059—172no—
259#2025—172no—
260#1988—171no—
261#1987—171no—
262#1875—172no—
263#1854—172no—
264#1797—171no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—41no—
269C22×C17—41no—
270C22×C17—41no—
271C22×C17—41no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17—81no—
279C22×C17C2×#12011yescharacteristic
280C22×C17C2×C22⋊Q811yescharacteristic
281C22×C17#53611yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C17#57511yescharacteristic
285C22×C17#57511yescharacteristic
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—42no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17#44914yes—
294C4×C17#54811yescharacteristic
295C4×C17#54811yescharacteristic
296C4×C17D4×Q811yescharacteristic
297C4×C17D4×Q811yescharacteristic
298Dic17—44no—
299Dic17—44no—
300#533—344no—
301#533—344no—
302C2×C4⋊Q8—171no—
303C2×C4⋊Q8—171no—
304C2×#125—172no—
305C2×C22⋊Q8—341no—
306C2×C22⋊Q8—341no—
307C2×#121C2×D1711yescharacteristic
308C2×C4×Q8—341no—
309C2×C4×Q8—341no—
310C2×C4×D4—21no—
311C2×C4×D4C2×D1711yescharacteristic
312C2×#117—172no—
313C22×C42C2×D1711yescharacteristic
314#476—344no—
315#474—344no—
316C2×#107—172no—
317C2×#106—172no—
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×#102—172no—
321C2×#102—172no—
322C2×C22⋊C8—172no—
323#390C2×D1711yescharacteristic
324#389C2×D1711yescharacteristic
325#388—21no—
326#386—21no—
327#380C2×D1711yescharacteristic
328C4×C4⋊C4C2×D1711yescharacteristic
329C4×C22⋊C4—21no—
330#340—172no—
331#340—172no—
332#340—172no—
333#340—172no—
334#336—172no—
335C2×C17—42no—
336C2×C17—82no—
337C2×C17#240911yescharacteristic
338C2×C17C2×#44911yescharacteristic
339C2×C17#300511yescharacteristic
340C2×C17#307611yescharacteristic
341C2×C17#307711yescharacteristic
342C2×C17#323311yescharacteristic
343C2×C17#323511yescharacteristic
344C22×Q8—341no—
345C22×Q8—341no—
346C22×D4—21no—
347C22×D4C22×D1711yescharacteristic
348C23×C4—21no—
349C23×C4C22⋊D1711yescharacteristic
350C23×C4C22⋊D1711yescharacteristic
351C23×C4C22×D1711yescharacteristic
352C22×C8—172no—
353C22×C8—172no—
354C4⋊Q8—174no—
355C4⋊Q8—174no—
356C4⋊Q8—342no—
357C4⋊Q8—342no—
358#125—344no—
359#125—344no—
360C22⋊Q8—348no—
361C22⋊Q8—348no—
362#121—22no—
363#121Dic3414yescomplemented
364C4×Q8—348no—
365C4×Q8—348no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—44no—
369#117—344no—
370#117—344no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—172no—
373C2×C4⋊C4—172no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4—342no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—342no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—342no—
391C2×C42C22⋊D1711yescharacteristic
392C2×C42C22⋊D1711yescharacteristic
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395#107—344no—
396#106—344no—
397C4⋊C8—342no—
398C4⋊C8—342no—
399#102—344no—
400#102—344no—
401C22⋊C8—344no—
402#95—21no—
403#95—21no—
404#95—21no—
405C17#2430011yescharacteristic, complemented, Sylow
406C24—21no—
407C24C2×C22⋊D1711yescharacteristic
408C2×Q8—344no—
409C2×Q8—344no—
410C2×Q8—682no—
411C2×Q8—682no—
412C2×Q8—682no—
413C2×Q8—682no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×Dic3412yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4C2×Dic3412yes—
442C22×C4#6325212yes—
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C23×D1711yescharacteristic
449C2×C8—342no—
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—174no—
462C4⋊C4—174no—
463C4⋊C4—348no—
464C4⋊C4—348no—
465C4⋊C4—342no—
466C4⋊C4—342no—
467C4⋊C4—344no—
468C4⋊C4—344no—
469C4⋊C4C2×Dic3412yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—348no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—348no—
482C42D4×D1712yes—
483C42D4×D1712yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1058761012yes—
498C23C22×C22⋊D1711yescharacteristic
499Q8—688no—
500Q8—688no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4#1058761012yes—
539C2×C4#1058761812yes—
540C2×C4#1058762312yes—
541C2×C4#1058762312yes—
542C2×C4#1058763612yes—
543C2×C4C22×Dic3411yescharacteristic
544C2×C4C2×#6325211yescharacteristic
545C2×C4C2×D4×D1711yescharacteristic
546C2×C4C2×D4×D1711yescharacteristic
547C2×C4#1058769711yescharacteristic
548C2×C4#1058769711yescharacteristic
549C8—682no—
550C8—682no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#1058762311yescharacteristic
566C22C2×#1058761011yescharacteristic
567C22#1176338011yescharacteristic
568C22C2×#1058763611yescharacteristic
569C22C2×#1058761811yescharacteristic
570C22#1176360711yescharacteristic
571C22#1176360711yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—684no—
580C4—684no—
581C4#1176295214yes—
582C4#1176346811yescharacteristic
583C4#1176346811yescharacteristic
584C4D4×Dic3411yescharacteristic
585C4D4×Dic3411yescharacteristic
586C2—42no—
587C2—82no—
588C2#1453214911yescharacteristic
589C2C2×#1176295211yescharacteristic
590C2#1453802411yescharacteristic
591C2#1453857211yescharacteristic
592C2#1453858111yescharacteristic
593C2#1453995911yescharacteristic
594C2#1453997811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×#121C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC4×#121D17
hypocenterC17#24300
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#11763366C2semidirect
2. C2×#11763099C2semidirect
3. C2×#11763089C2semidirect
4. #14530122C2semidirect
5. #14529930C2semidirect
6. #14529736C2semidirect
7. #14529732C2semidirect
9. #14528981C2semidirect
10. #14528764C2semidirect
56. C17×#121Q8semidirect
94. #10587555D4semidirect
95. #10587555D4semidirect
363. #121Dic34semidirect
405. C17#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#117633662
C2×#117630992
C2×#117630892
#145301222
#145299302
#145297362
#145297322
#145291072
#145289812
#145287642
C4×C17×#1212
#2430017