Finite Group Databasebeta

← #17035201

Subgroups — #17035201

Order 4352.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763292C211yesmaximal, characteristic
2#14537373C211yesmaximal, characteristic, complemented
3C2×#11762826C211yesmaximal, characteristic
4#14531404C211yesmaximal, characteristic, complemented
5#14529015C214yesmaximal, complemented
6#14529008C214yesmaximal, complemented
7C2×#11762577C211yesmaximal, characteristic, complemented
8#14528495C211yesmaximal, characteristic, complemented
9C17×#2896C211yesmaximal, characteristic
10C22×#63223C2211yescharacteristic
11#11763292—22no—
12#11763292—22no—
13#11763292—22no—
14#11763292C2214yes—
15C2×Q8×Dic17—21no—
16C2×D4×Dic17—21no—
17C2×D4×Dic17—22no—
18C2×#10587612—21no—
19C2×#10587612C2211yescharacteristic
20C2×#10587612C2211yescharacteristic
21C22×C4⋊Dic17—21no—
22C22×C4⋊Dic17C2211yescharacteristic
23#11762852—21no—
24#11762852—21no—
25#11762849C2212yescomplemented
26#11762849C2212yescomplemented
27#11762845—22no—
28#11762826C418yescomplemented
29C2×#10587508C2214yes—
30#11762707—22no—
31#11762681—22no—
32#11762678C2212yescomplemented
33#11762674C2212yescomplemented
34#11762577C2214yescomplemented
35#11762570C2214yescomplemented
36C22×#41×C17C2211yescharacteristic
37C22×M16×C17C2211yescharacteristic
38C2×C17×#117C2211yescharacteristic
39C17×#419—21no—
40C17×#418—21no—
41C17×#417C2212yes—
42C17×#417C2212yes—
43C17×#416—22no—
44C2×#41×C17—21no—
45C2×#41×C17—21no—
46C2×#41×C17—22no—
47C2×#41×C17—22no—
48C2×#41×C17C2×C414yes—
49C2×#41×C17D412yes—
50C2×#41×C17Q814yes—
51C2×#41×C17C2312yes—
52C22×Q8×C17D411yescharacteristic
53C22×D4×C17—21no—
54C22×D4×C17D411yescharacteristic
55C23×C4×C17—21no—
56C23×C4×C17C2311yescharacteristic
57C23×Dic17—22no—
58C23×Dic17D412yes—
59C2×M16×C17—21no—
60C2×M16×C17—22no—
61C2×M16×C17—22no—
62C2×M16×C17C2312yes—
63C22×C8×C17—21no—
64C17×#117—21no—
65C17×#117—21no—
66C17×#117—21no—
67C17×#117—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75Q8×Dic17—24no—
76Q8×Dic17—24no—
77D4×Dic17—24no—
78D4×Dic17—28no—
79D4×Dic17—24no—
80D4×Dic17—28no—
81#10587612—24no—
82#10587612—24no—
83#10587612—21no—
84#10587612—21no—
85#10587612—22no—
86#10587612C2×C414yescomplemented
87#10587612D412yes—
88#10587612C2312yes—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—24no—
91C2×C4⋊Dic17—24no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17D411yescharacteristic
98C2×C4⋊Dic17D411yescharacteristic
99C2×C4⋊Dic17D412yes—
100C2×C4⋊Dic17C2312yes—
101C2×#63225—24no—
102C2×C4×Dic17—22no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2312yes—
105C2×#63223—21no—
106C2×#63223—22no—
107C2×#63223C2×C414yescomplemented
108C2×#63223C2312yes—
109C22×#3622—21no—
110C17×#103—22no—
111C17×#103—22no—
112C17×#102—22no—
113C17×#102—22no—
114#10587508C2×C4116yescomplemented
115#10587502—24no—
116#10587479—22no—
117#10587479—22no—
118#10587478—22no—
119#10587478—22no—
120C24×C17—21no—
121C24×C17C2×D411yescharacteristic
122#41×C17—24no—
123#41×C17—24no—
124#41×C17—24no—
125#41×C17—24no—
126#41×C17—24no—
127#41×C17—22no—
128#41×C17—22no—
129#41×C17C4⋊C418yescomplemented
130#41×C17C22×C414yes—
131#41×C17C2×Q814yes—
132C2×Q8×C17—21no—
133C2×Q8×C17—22no—
134C2×Q8×C17C4⋊C414yescomplemented
135C2×Q8×C17C2×D412yes—
136C2×D4×C17—24no—
137C2×D4×C17—22no—
138C2×D4×C17—24no—
139C2×D4×C17—22no—
140C2×D4×C17—21no—
141C2×D4×C17—22no—
142C2×D4×C17C4⋊C414yescomplemented
143C2×D4×C17C2×D412yes—
144C22×C4×C17—21no—
145C22×C4×C17—24no—
146C22×C4×C17—22no—
147C22×C4×C17—24no—
148C22×C4×C17—22no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17C22×C412yes—
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×Q812yes—
158C22×C4×C17#4112yes—
159C22×C4×C17#4112yes—
160C22×C4×C17C2411yescharacteristic
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—21no—
164C22×Dic17—22no—
165C22×Dic17—28no—
166C22×Dic17—22no—
167C22×Dic17—82no—
168C22×Dic17—84no—
169C22×Dic17C2×D414yes—
170M16×C17—22no—
171M16×C17—22no—
172M16×C17—42no—
173C2×C8×C17—22no—
174C2×C8×C17—42no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C22⋊C4×C17—41no—
180C22⋊C4×C17—41no—
181C42×C17—41no—
182C42×C17—41no—
183C22⋊Dic17—28no—
184C22⋊Dic17—24no—
185C22⋊Dic17—22no—
186C22⋊Dic17—22no—
187C4⋊Dic17—28no—
188C4⋊Dic17—24no—
189C4⋊Dic17—21no—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—21no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195C4⋊Dic17C2×D412yes—
196C4⋊Dic17C2×D412yes—
197#63225—84no—
198C4×Dic17—28no—
199C4×Dic17—22no—
200C4×Dic17—22no—
201C4×Dic17—22no—
202C4×Dic17C22×C414yescomplemented
203#63223—24no—
204#63223—22no—
205#63223C22×C414yescomplemented
206C2×#3622—24no—
207C2×#3622—22no—
208#20864—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—24no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—42no—
214C23×C17—42no—
215C23×C17C4×D412yes—
216C23×C17#12011yescharacteristic
217C23×C17#12011yescharacteristic
218C23×C17C22⋊Q812yes—
219C23×C17C22×D411yescharacteristic
220Q8×C17—22no—
221Q8×C17—24no—
222Q8×C17C2×C4⋊C414yes—
223D4×C17—24no—
224D4×C17—24no—
225D4×C17—28no—
226D4×C17—22no—
227D4×C17—24no—
228D4×C17C2×C4⋊C414yes—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—28no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17C2×C4⋊C412yes—
239C2×C4×C17C2×C4⋊C412yes—
240C2×C4×C17C4×D412yes—
241C2×C4×C17C4×D412yes—
242C2×C4×C17C4×D412yes—
243C2×C4×C17#12011yescharacteristic
244C2×C4×C17#12011yescharacteristic
245C2×C4×C17C22⋊Q812yes—
246C2×C4×C17C22⋊Q812yes—
247C2×C4×C17C22⋊Q812yes—
248C2×C4×C17#12312yes—
249C2×C4×C17#12312yes—
250C2×C4×C17C23×C411yescharacteristic
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×Q811yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×Dic17—24no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—24no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—44no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—44no—
266C2×Dic17—82no—
267C2×Dic17—88no—
268C8×C17—42no—
269#3622—24no—
270C2×#520—171no—
271#2911—171no—
272C2×#421—171no—
273#2896D1711yescharacteristic
274#2317—171no—
275#1860—174no—
276#1859—174no—
277C2×#343—171no—
278#1733—171no—
279C22×C17—21no—
280C22×C17—21no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17—21no—
284C22×C17—21no—
285C22×C17—24no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—48no—
289C22×C17—42no—
290C22×C17—42no—
291C22×C17#38512yes—
292C22×C17#38512yes—
293C22×C17C2×C4×D411yescharacteristic
294C22×C17C2×#12011yescharacteristic
295C22×C17C2×C22⋊Q811yescharacteristic
296C4×C17—24no—
297C4×C17—81no—
298C4×C17—81no—
299C4×C17#38512yes—
300C4×C17#38512yes—
301C4×C17C22×C4⋊C411yescharacteristic
302C4×C17C2×C4×D411yescharacteristic
303C4×C17C2×C22⋊Q811yescharacteristic
304C4×C17C2×#12311yescharacteristic
305Dic17—44no—
306Dic17—44no—
307Dic17—42no—
308Dic17—42no—
309C22×#41C2×D1711yescharacteristic
310C22×M16—171no—
311C22×M16C2×D1711yescharacteristic
312#520—174no—
313#520—342no—
314#520—342no—
315#520—342no—
316C2×C4×Q8—341no—
317C2×C4×D4—342no—
318C2×C4×D4—341no—
319C2×#117—171no—
320C2×#117—171no—
321C2×#117—341no—
322C2×#117C2×D1711yescharacteristic
323C22×C4⋊C4—171no—
324C22×C4⋊C4—341no—
325#428—172no—
326#428—172no—
327#428—341no—
328#428—341no—
329#427—342no—
330#421—178no—
331C2×#104—174no—
332#419—21no—
333#418—21no—
334#417C2×D1712yes—
335#417C2×D1712yes—
336#416—22no—
337#389—342no—
338#385—342no—
339#384—172no—
340#382—172no—
341#343—174no—
342#339—174no—
343C2×C17—21no—
344C2×C17—22no—
345C2×C17—21no—
346C2×C17—22no—
347C2×C17—44no—
348C2×C17C2×#38511yescharacteristic
349C2×C17#301911yescharacteristic
350C2×C17#302011yescharacteristic
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41Dic3414yes—
356C2×#41C4×D1714yes—
357C2×#41C22⋊D1712yes—
358C2×#41C22×D1712yes—
359C22×Q8C22⋊D1711yescharacteristic
360C22×D4—21no—
361C22×D4C22⋊D1711yescharacteristic
362C23×C4—21no—
363C23×C4—172no—
364C23×C4—342no—
365C23×C4C22×D1711yescharacteristic
366C2×M16—21no—
367C2×M16—22no—
368C2×M16—22no—
369C2×M16—174no—
370C2×M16—172no—
371C2×M16—341no—
372C2×M16—342no—
373C2×M16C22×D1712yes—
374C22×C8—21no—
375C22×C8—341no—
376C4×Q8—344no—
377C4×Q8—344no—
378C4×D4—348no—
379C4×D4—348no—
380C4×D4—344no—
381C4×D4—344no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—174no—
387#117—172no—
388#117—172no—
389#117—344no—
390#117—344no—
391#117—341no—
392#117—342no—
393#117—341no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—172no—
397C2×C4⋊C4—172no—
398C2×C4⋊C4—344no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—344no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—344no—
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—342no—
411C2×C22⋊C4—341no—
412C2×C22⋊C4—341no—
413C2×C42—21no—
414C2×C42—172no—
415C2×C42—342no—
416C2×C42—341no—
417#107—342no—
418#107—342no—
419#106—342no—
420#106—342no—
421#104—1716no—
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426#95—344no—
427C17#2086411yescharacteristic, complemented, Sylow
428C24—21no—
429C24D4×D1711yescharacteristic
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41#6322518yescomplemented
438#41C2×Dic3414yes—
439#41C2×C4×D1714yes—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8#6322514yescomplemented
443C2×Q8C2×C22⋊D1712yes—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4#6322514yescomplemented
451C2×D4C2×C22⋊D1712yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—174no—
460C22×C4—342no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—342no—
464C22×C4—348no—
465C22×C4—342no—
466C22×C4—1362no—
467C22×C4—1364no—
468C22×C4C2×Dic3412yes—
469C22×C4C2×C4×D1712yes—
470C22×C4D4×D1711yescharacteristic
471C22×C4D4×D1711yescharacteristic
472C22×C4D4×D1711yescharacteristic
473C22×C4#6325412yes—
474C22×C4#6325412yes—
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478M16—22no—
479M16—22no—
480M16—42no—
481M16—174no—
482M16—342no—
483M16—344no—
484C2×C8—22no—
485C2×C8—42no—
486C2×C8—342no—
487C2×C8—344no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—172no—
493C4⋊C4—172no—
494C4⋊C4—348no—
495C4⋊C4—342no—
496C4⋊C4—342no—
497C4⋊C4—344no—
498C4⋊C4—341no—
499C4⋊C4—342no—
500C4⋊C4—342no—
501C4⋊C4—341no—
502C4⋊C4—1364no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—348no—
506C22⋊C4—342no—
507C22⋊C4—344no—
508C22⋊C4—342no—
509C42—41no—
510C42—41no—
511C42—174no—
512C42—348no—
513C42—342no—
514C42—342no—
515C42—342no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23#1058756412yes—
523C23#1058756712yes—
524C23#1058762311yescharacteristic
525C23#1058762311yescharacteristic
526C23C2×D4×D1711yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8C2×#6322514yes—
530D4—24no—
531D4—24no—
532D4—28no—
533D4—22no—
534D4—24no—
535D4C2×#6322514yes—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—344no—
546C2×C4—342no—
547C2×C4—342no—
548C2×C4—344no—
549C2×C4—682no—
550C2×C4—684no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—682no—
554C2×C4—684no—
555C2×C4—1362no—
556C2×C4—1368no—
557C2×C4#1058756412yes—
558C2×C4#1058756412yes—
559C2×C4#1058756412yes—
560C2×C4#1058756712yes—
561C2×C4#1058756712yes—
562C2×C4#1058756712yes—
563C2×C4C2×#6322512yes—
564C2×C4C2×#6322512yes—
565C2×C4#1058762112yes—
566C2×C4#1058762112yes—
567C2×C4#1058762311yescharacteristic
568C2×C4#1058762311yescharacteristic
569C2×C4C22×Dic3411yescharacteristic
570C2×C4C22×C4×D1711yescharacteristic
571C2×C4C2×D4×D1711yescharacteristic
572C2×C4C2×#6325411yescharacteristic
573C2×C4C2×#6325411yescharacteristic
574C2×C4C22×C22⋊D1711yescharacteristic
575C8—42no—
576C8—344no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1176268112yes—
590C22#1176268112yes—
591C22C2×#1058756711yescharacteristic
592C22C2×#1058762311yescharacteristic
593C22C2×#1058756411yescharacteristic
594C4—24no—
595C4—81no—
596C4—81no—
597C4—684no—
598C4—684no—
599C4—682no—
600C4—682no—
601C4#1176268112yes—
602C4#1176268112yes—
603C4C22×#6322511yescharacteristic
604C4C2×#1058756711yescharacteristic
605C4C2×#1058756411yescharacteristic
606C4C2×#1058762111yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—44no—
612C2C2×#1176268111yescharacteristic
613C2#1453813911yescharacteristic
614C2#1453814611yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2896C2
socleC22×C17C2×#120
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenter#2896D17
hypocenterC17#20864
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #14537373C2semidirect
4. #14531404C2semidirect
5. #14529015C2semidirect
6. #14529008C2semidirect
7. C2×#11762577C2semidirect
8. #14528495C2semidirect
25. #11762849C22semidirect
26. #11762849C22semidirect
28. #11762826C4semidirect
32. #11762678C22semidirect
33. #11762674C22semidirect
34. #11762577C22semidirect
35. #11762570C22semidirect
86. #10587612C2×C4semidirect
107. C2×#63223C2×C4semidirect
114. #10587508C2×C4semidirect
129. #41×C17C4⋊C4semidirect
134. C2×Q8×C17C4⋊C4semidirect
142. C2×D4×C17C4⋊C4semidirect
202. C4×Dic17C22×C4semidirect
205. #63223C22×C4semidirect
427. C17#20864semidirect
437. #41#63225semidirect
442. C2×Q8#63225semidirect
450. C2×D4#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117632922
#145373732
C2×#117628262
#145314042
#145290152
#145290082
C2×#117625772
#145284952
C17×#28962
#2086417