Finite Group Databasebeta

← #17035211

Subgroups — #17035211

Order 4352.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14541987C211yesmaximal, characteristic
2C2×#11762849C211yesmaximal, characteristic, complemented
3C2×#11762826C211yesmaximal, characteristic
4#14531408C211yesmaximal, characteristic, complemented
5#14529015C214yesmaximal, complemented
6#14529008C214yesmaximal, complemented
7#14528522C211yesmaximal, characteristic, complemented
8#14528495C211yesmaximal, characteristic, complemented
9C2×C17×#417C211yesmaximal, characteristic
10C22×#63223C2211yescharacteristic
11#11763292—21no—
12#11763292—21no—
13#11763292C2212yes—
14#11763292C2212yes—
15#11763287—21no—
16#11763284—22no—
17#11763284—21no—
18#41×Dic17—22no—
19#41×Dic17—22no—
20C2×#10587612—22no—
21C2×#10587612C2211yescharacteristic
22C2×#10587612C2211yescharacteristic
23C2×#10587612C2211yescharacteristic
24#11762849—22no—
25#11762849C2214yescomplemented
26C2×#10587478—21no—
27C2×#10587479—21no—
28#11762826C418yescomplemented
29C2×#10587508C2214yes—
30#11762705—22no—
31#11762682—22no—
32#11762678C2212yescomplemented
33#11762674C2212yescomplemented
34#11762577C2212yescomplemented
35#11762577C2212yescomplemented
36#11762570C2214yescomplemented
37C22×#41×C17C2211yescharacteristic
38C22×M16×C17C2211yescharacteristic
39C22×C4⋊C4×C17C2211yescharacteristic
40C17×#417—22no—
41C17×#417C2214yes—
42C2×C17×#103—21no—
43C2×C17×#102—21no—
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×C17—21no—
57C23×C4×C17C2311yescharacteristic
58C2×M16×C17—21no—
59C2×M16×C17—22no—
60C2×M16×C17—22no—
61C2×M16×C17C2312yes—
62C22×C8×C17—21no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2312yes—
69C2×C22⋊Dic17—24no—
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17D412yes—
73Q8×Dic17—22no—
74Q8×Dic17—22no—
75Q8×Dic17—22no—
76Q8×Dic17—22no—
77D4×Dic17—22no—
78D4×Dic17—22no—
79D4×Dic17—24no—
80D4×Dic17—24no—
81D4×Dic17—22no—
82D4×Dic17—22no—
83D4×Dic17—24no—
84D4×Dic17—24no—
85#10587612—24no—
86#10587612—24no—
87#10587612—22no—
88#10587612—22no—
89#10587612—21no—
90#10587612—21no—
91#10587612—21no—
92#10587612—21no—
93#10587612—21no—
94#10587612—21no—
95#10587612—22no—
96#10587612C2×C414yescomplemented
97#10587612D412yes—
98#10587612C2312yes—
99C2×C4⋊Dic17—22no—
100C2×C4⋊Dic17—22no—
101C2×C4⋊Dic17—21no—
102C2×C4⋊Dic17D411yescharacteristic
103C2×C4⋊Dic17D411yescharacteristic
104C2×C4⋊Dic17C2311yescharacteristic
105C2×C4⋊Dic17C2311yescharacteristic
106C2×#63225—24no—
107C2×C4×Dic17—24no—
108C2×C4×Dic17—21no—
109C2×C4×Dic17D412yes—
110C2×C4×Dic17C2312yes—
111C2×#63223—21no—
112C2×#63223—22no—
113C2×#63223C2×C414yescomplemented
114C2×#63223C2312yes—
115C22×#3622—21no—
116C17×#103—24no—
117C17×#102—24no—
118#10587508C2×C4116yescomplemented
119#10587502—24no—
120#10587479—24no—
121#10587478—24no—
122C24×C17—21no—
123C24×C17C2×D411yescharacteristic
124#41×C17—24no—
125#41×C17—24no—
126#41×C17—24no—
127#41×C17—24no—
128#41×C17—24no—
129#41×C17—22no—
130#41×C17—22no—
131#41×C17C4⋊C418yescomplemented
132#41×C17C22×C414yes—
133#41×C17C2×Q814yes—
134C2×Q8×C17—21no—
135C2×Q8×C17—22no—
136C2×Q8×C17C4⋊C414yescomplemented
137C2×Q8×C17C2×D412yes—
138C2×D4×C17—24no—
139C2×D4×C17—22no—
140C2×D4×C17—24no—
141C2×D4×C17—22no—
142C2×D4×C17—21no—
143C2×D4×C17—22no—
144C2×D4×C17C4⋊C414yescomplemented
145C2×D4×C17C2×D412yes—
146C22×C4×C17—21no—
147C22×C4×C17—24no—
148C22×C4×C17—22no—
149C22×C4×C17—24no—
150C22×C4×C17—22no—
151C22×C4×C17—22no—
152C22×C4×C17—41no—
153C22×C4×C17—41no—
154C22×C4×C17—82no—
155C22×C4×C17C22×C412yes—
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×Q812yes—
162C22×C4×C17#4112yes—
163C22×C4×C17#4112yes—
164C22×C4×C17C2411yescharacteristic
165C22×Dic17—22no—
166C22×Dic17—24no—
167C22×Dic17—24no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172M16×C17—22no—
173M16×C17—22no—
174M16×C17—42no—
175C2×C8×C17—22no—
176C2×C8×C17—42no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—22no—
179C4⋊C4×C17—41no—
180C4⋊C4×C17—41no—
181C22⋊Dic17—24no—
182C22⋊Dic17—24no—
183C22⋊Dic17—24no—
184C22⋊Dic17—22no—
185C22⋊Dic17—22no—
186C22⋊Dic17—22no—
187C22⋊Dic17—22no—
188C4⋊Dic17—24no—
189C4⋊Dic17—24no—
190C4⋊Dic17—24no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17C2×D412yes—
198C4⋊Dic17C2×D412yes—
199#63225—84no—
200C4×Dic17—24no—
201C4×Dic17—24no—
202C4×Dic17—22no—
203C4×Dic17—22no—
204C4×Dic17—22no—
205C4×Dic17—22no—
206C4×Dic17—22no—
207C4×Dic17C22×C414yes—
208#63223—24no—
209#63223—22no—
210#63223C22×C414yes—
211C2×#3622—24no—
212C2×#3622—22no—
213#20865—171nomaximal, Sylow
214C23×C17—22no—
215C23×C17—24no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—81no—
219C23×C17—81no—
220C23×C17C4×D412yes—
221C23×C17#12011yescharacteristic
222C23×C17#12011yescharacteristic
223C23×C17C22⋊Q812yes—
224C23×C17C22×D411yescharacteristic
225Q8×C17—22no—
226Q8×C17—24no—
227Q8×C17C2×C4⋊C414yes—
228D4×C17—24no—
229D4×C17—24no—
230D4×C17—28no—
231D4×C17—22no—
232D4×C17—24no—
233D4×C17C2×C4⋊C414yes—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—28no—
239C2×C4×C17—44no—
240C2×C4×C17—42no—
241C2×C4×C17—82no—
242C2×C4×C17—82no—
243C2×C4×C17C2×C4⋊C412yes—
244C2×C4×C17C2×C4⋊C412yes—
245C2×C4×C17C4×D412yes—
246C2×C4×C17C4×D412yes—
247C2×C4×C17C4×D412yes—
248C2×C4×C17#12011yescharacteristic
249C2×C4×C17#12011yescharacteristic
250C2×C4×C17C22⋊Q812yes—
251C2×C4×C17C22⋊Q812yes—
252C2×C4×C17C22⋊Q812yes—
253C2×C4×C17#12312yes—
254C2×C4×C17#12312yes—
255C2×C4×C17C23×C411yescharacteristic
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C22×Q811yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—81no—
271C2×Dic17—81no—
272C2×Dic17—84no—
273C2×Dic17—84no—
274C8×C17—42no—
275#3622—24no—
276#3428—171no—
277C2×#428—171no—
278C2×#421—171no—
279C2×#417D1711yescharacteristic
280#2317—171no—
281#1860—174no—
282#1859—174no—
283#1742—171no—
284#1733—171no—
285C22×C17—21no—
286C22×C17—21no—
287C22×C17—21no—
288C22×C17—21no—
289C22×C17—21no—
290C22×C17—21no—
291C22×C17—24no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—84no—
297C22×C17#38512yes—
298C22×C17#38512yes—
299C22×C17C2×C4×D411yescharacteristic
300C22×C17C2×#12011yescharacteristic
301C22×C17C2×C22⋊Q811yescharacteristic
302C4×C17—24no—
303C4×C17—82no—
304C4×C17#38512yes—
305C4×C17#38512yes—
306C4×C17C22×C4⋊C411yescharacteristic
307C4×C17C2×C4×D411yescharacteristic
308C4×C17C2×C22⋊Q811yescharacteristic
309C4×C17C2×#12311yescharacteristic
310Dic17—44no—
311Dic17—42no—
312Dic17—42no—
313Dic17—42no—
314Dic17—42no—
315C22×#41C2×D1711yescharacteristic
316C22×M16—171no—
317C22×M16C2×D1711yescharacteristic
318#520—172no—
319#520—172no—
320#520—341no—
321#520—341no—
322#519—341no—
323#518—342no—
324#518—341no—
325C4×#41—342no—
326C4×#41—342no—
327C2×#117—171no—
328C2×#117—171no—
329C2×#117—171no—
330C2×#117—342no—
331C22×C4⋊C4C2×D1711yescharacteristic
332#428—174no—
333#428—342no—
334C2×#107—341no—
335C2×#106—341no—
336#421—178no—
337C2×#104—174no—
338#417—22no—
339#417C2×D1714yes—
340C2×#103—21no—
341C2×#102—21no—
342#389—342no—
343#385—342no—
344#384—172no—
345#382—172no—
346#343—172no—
347#343—172no—
348#339—174no—
349C2×C17—21no—
350C2×C17—22no—
351C2×C17—21no—
352C2×C17—22no—
353C2×C17—44no—
354C2×C17C2×#38511yescharacteristic
355C2×C17#302111yescharacteristic
356C2×C17#302211yescharacteristic
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41Dic3414yes—
362C2×#41C4×D1714yes—
363C2×#41C22⋊D1712yes—
364C2×#41C22×D1712yes—
365C22×Q8C22⋊D1711yescharacteristic
366C22×D4—21no—
367C22×D4C22⋊D1711yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4C22×D1711yescharacteristic
371C2×M16—21no—
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—174no—
375C2×M16—172no—
376C2×M16—341no—
377C2×M16—342no—
378C2×M16C22×D1712yes—
379C22×C8—21no—
380C22×C8—341no—
381C4×Q8—342no—
382C4×Q8—342no—
383C4×Q8—342no—
384C4×Q8—342no—
385C4×D4—344no—
386C4×D4—344no—
387C4×D4—344no—
388C4×D4—344no—
389C4×D4—342no—
390C4×D4—342no—
391C4×D4—342no—
392C4×D4—342no—
393#117—174no—
394#117—172no—
395#117—172no—
396#117—342no—
397#117—342no—
398#117—344no—
399#117—344no—
400#117—341no—
401#117—341no—
402#117—341no—
403#117—342no—
404#117—341no—
405#117—341no—
406#117—341no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—342no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—344no—
420C2×C4⋊C4C22×D1712yes—
421C2×C22⋊C4—172no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4—341no—
425C2×C42—172no—
426C2×C42—172no—
427C2×C42—344no—
428C2×C42—341no—
429#107—344no—
430#106—344no—
431#104—1716no—
432#103—24no—
433#102—24no—
434#95—344no—
435C17#2086511yescharacteristic, complemented, Sylow
436C24—21no—
437C24D4×D1711yescharacteristic
438#41—22no—
439#41—22no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41#6322518yescomplemented
446#41C2×Dic3414yes—
447#41C2×C4×D1714yes—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8#6322514yescomplemented
451C2×Q8C2×C22⋊D1712yes—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4#6322514yescomplemented
459C2×D4C2×C22⋊D1712yes—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—82no—
469C22×C4—172no—
470C22×C4—172no—
471C22×C4—342no—
472C22×C4—344no—
473C22×C4—344no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4C2×Dic3412yes—
477C22×C4C2×C4×D1712yes—
478C22×C4D4×D1711yescharacteristic
479C22×C4D4×D1711yescharacteristic
480C22×C4D4×D1711yescharacteristic
481C22×C4#6325412yes—
482C22×C4#6325412yes—
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486M16—22no—
487M16—22no—
488M16—42no—
489M16—174no—
490M16—342no—
491M16—344no—
492C2×C8—22no—
493C2×C8—42no—
494C2×C8—342no—
495C2×C8—344no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—172no—
501C4⋊C4—172no—
502C4⋊C4—344no—
503C4⋊C4—344no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—344no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—1364no—
512C22⋊C4—344no—
513C22⋊C4—344no—
514C22⋊C4—342no—
515C22⋊C4—344no—
516C22⋊C4—342no—
517C22⋊C4—342no—
518C22⋊C4—342no—
519C42—174no—
520C42—344no—
521C42—344no—
522C42—342no—
523C42—342no—
524C42—342no—
525C42—342no—
526C42—342no—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23#1058756412yes—
534C23#1058756712yes—
535C23#1058762311yescharacteristic
536C23#1058762311yescharacteristic
537C23C2×D4×D1711yescharacteristic
538Q8—22no—
539Q8—24no—
540Q8C2×#6322514yes—
541D4—24no—
542D4—24no—
543D4—28no—
544D4—22no—
545D4—24no—
546D4C2×#6322514yes—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—344no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—342no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4—1361no—
566C2×C4—1361no—
567C2×C4—1364no—
568C2×C4—1364no—
569C2×C4#1058756412yes—
570C2×C4#1058756412yes—
571C2×C4#1058756412yes—
572C2×C4#1058756712yes—
573C2×C4#1058756712yes—
574C2×C4#1058756712yes—
575C2×C4C2×#6322512yes—
576C2×C4C2×#6322512yes—
577C2×C4#1058762112yes—
578C2×C4#1058762112yes—
579C2×C4#1058762311yescharacteristic
580C2×C4#1058762311yescharacteristic
581C2×C4C22×Dic3411yescharacteristic
582C2×C4C22×C4×D1711yescharacteristic
583C2×C4C2×D4×D1711yescharacteristic
584C2×C4C2×#6325411yescharacteristic
585C2×C4C2×#6325411yescharacteristic
586C2×C4C22×C22⋊D1711yescharacteristic
587C8—42no—
588C8—344no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1176268112yes—
602C22#1176268112yes—
603C22C2×#1058756711yescharacteristic
604C22C2×#1058762311yescharacteristic
605C22C2×#1058756411yescharacteristic
606C4—24no—
607C4—82no—
608C4—684no—
609C4—682no—
610C4—682no—
611C4—682no—
612C4—682no—
613C4#1176268112yes—
614C4#1176268112yes—
615C4C22×#6322511yescharacteristic
616C4C2×#1058756711yescharacteristic
617C4C2×#1058756411yescharacteristic
618C4C2×#1058762111yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—44no—
624C2C2×#1176268111yescharacteristic
625C2#1453815311yescharacteristic
626C2#1453816011yescharacteristic
subgroupG/H
centerC22C2×#10587623
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#417C2
socleC22×C17C2×#120
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenterC2×#417D17
hypocenterC17#20865
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#11762849C2semidirect
4. #14531408C2semidirect
5. #14529015C2semidirect
6. #14529008C2semidirect
7. #14528522C2semidirect
8. #14528495C2semidirect
25. #11762849C22semidirect
28. #11762826C4semidirect
32. #11762678C22semidirect
33. #11762674C22semidirect
34. #11762577C22semidirect
35. #11762577C22semidirect
36. #11762570C22semidirect
96. #10587612C2×C4semidirect
113. C2×#63223C2×C4semidirect
118. #10587508C2×C4semidirect
131. #41×C17C4⋊C4semidirect
136. C2×Q8×C17C4⋊C4semidirect
144. C2×D4×C17C4⋊C4semidirect
435. C17#20865semidirect
445. #41#63225semidirect
450. C2×Q8#63225semidirect
458. C2×D4#63225semidirect

Maximal subgroups

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

subgroupindex
#145419872
C2×#117628492
C2×#117628262
#145314082
#145290152
#145290082
#145285222
#145284952
C2×C17×#4172
#2086517