Finite Group Databasebeta

← #17979494

Subgroups — #17979494

Order 4352.

4812 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17⋊#568C212yesmaximal, complemented
2#14545397C212yesmaximal, complemented
3#14545396C216yesmaximal, complemented
4#14545395C212yesmaximal, complemented
5#14545394C216yesmaximal, complemented
6D17⋊#413C216yesmaximal, complemented
7C2×#11763795C216yesmaximal, complemented
8C2×#11763645C211yesmaximal, characteristic
9#11763889C2214yescomplemented
10#11763888—28no—
11#11763887C22112yescomplemented
12D17⋊#131—28no—
13C2×D17⋊M16C22112yescomplemented
14#11763796C22124yescomplemented
15#11763795C2214yescomplemented
16#11763795C22112yescomplemented
17#11763795C22112yescomplemented
18#11763795C22112yescomplemented
19D17⋊#101C22112yescomplemented
20C2×#10587545C22112yescomplemented
21D17⋊#100C22112yescomplemented
22C2×#10587553C22112yescomplemented
23#11763645—28no—
24#11763645—28no—
25C2×#10587694C411yescharacteristic
26C2×#10587694C2211yescharacteristic
27C2×#10587697C411yescharacteristic
28C2×#10587697C2213yes—
29C2×#10587696C413yes—
30C2×#10587696C2211yescharacteristic
31C2×#10587691C413yes—
32C2×#10587691C2213yes—
33C22×#63252C413yes—
34C22×#63252C2213yes—
35C2×#41×D17C411yescharacteristic
36C2×#41×D17C413yes—
37C2×#41×D17C2213yes—
38C2×#41×D17C2211yescharacteristic
39C22×#41×C17C411yescharacteristic
40C2×#41×C17—24no—
41C2×#41×C17—24no—
42C2×#41×C17C2×C413yes—
43C2×#41×C17C2×C413yes—
44C2×#41×C17C2×C411yescharacteristic
45C2×#41×C17C2×C411yescharacteristic
46C22×Q8×C17C2×C411yescharacteristic
47C22×D4×C17C2×C413yes—
48C23×C4×C17C2×C413yes—
49C22×C22⋊D17C2×C416yes—
50C22×C22⋊D17C2×C413yes—
51C22×C22⋊D17C2313yes—
52#10587697—28no—
53#10587697—224no—
54#10587696—28no—
55#10587696—224no—
56#41×D17—24no—
57#41×D17—24no—
58#41×D17—212no—
59#41×D17—212no—
60#41×D17—212no—
61#41×D17—212no—
62#41×D17D418yes—
63#41×D17D418yes—
64#10587694—28no—
65#10587694—28no—
66C2×#63256C2×C413yes—
67C2×#63256C2×C411yescharacteristic
68C2×#63256C2×C413yes—
69C2×#63256C2311yescharacteristic
70C2×Q8×D17C2×C411yescharacteristic
71C2×Q8×D17C2×C413yes—
72C2×Q8×D17C2×C411yescharacteristic
73C2×Q8×D17C2313yes—
74#10587691—224no—
75#10587691—224no—
76C2×#63254C2×C413yes—
77C2×#63254C2×C416yes—
78C2×#63254C2×C413yes—
79C2×#63254C2×C413yes—
80C2×#63254C2313yes—
81C2×#63254C2316yes—
82C2×D4×D17C2×C413yes—
83C2×D4×D17C2×C416yes—
84C2×D4×D17C2×C413yes—
85C2×D4×D17C2×C416yes—
86C2×D4×D17C2×C413yes—
87C2×D4×D17C2313yes—
88C2×#63252—212no—
89C2×#63252—212no—
90C2×#63252—212no—
91C2×#63252—212no—
92C2×#63252C2×C413yes—
93C2×#63252C2×C413yes—
94C2×#63252C2×C413yes—
95C2×#63252C2×C413yes—
96C2×#63252C2×C413yes—
97C2×#63252C2×C416yes—
98C2×#63252C2×C413yes—
99C2×#63252C2×C413yes—
100C2×#63252C2×C413yes—
101C2×#63252C2313yes—
102C2×#63252C2311yescharacteristic
103C2×#63252C2313yes—
104C2×#63252C2313yes—
105C22×D68C2×C413yes—
106C22×D68C2×C413yes—
107C22×C4×D17C2×C413yes—
108C22×C4×D17C2×C416yes—
109C22×C4×D17C2313yes—
110C22×Dic34C2×C413yes—
111C22×Dic34C2313yes—
112C2×#63247C23124yes—
113#10587680—216no—
114#10587678—248no—
115D17⋊M16—212no—
116D17⋊M16—212no—
117D17⋊M16C23124yes—
118D17⋊M16C2318yes—
119C2×#63243—212no—
120C2×#63242—212no—
121#10587553C2318yes—
122#10587553C23124yes—
123#10587545C2318yes—
124#10587545C23124yes—
125C24×C17C22×C413yes—
126C23×D17C22×C413yes—
127C23×D17C22×C416yes—
128C23×D17C22×C413yes—
129#41×C17—24no—
130#41×C17—212no—
131#41×C17—26no—
132#41×C17—26no—
133#41×C17C22⋊C414yes—
134#41×C17C22⋊C414yes—
135C2×Q8×C17—24no—
136C2×Q8×C17C22×C413yes—
137C2×Q8×C17C22×C411yescharacteristic
138C2×D4×C17—212no—
139C2×D4×C17C22×C416yes—
140C2×D4×C17C22×C413yes—
141C2×D4×C17C22×C413yes—
142C22×C4×C17—26no—
143C22×C4×C17—26no—
144C22×C4×C17C22×C413yes—
145C22×C4×C17C22×C413yes—
146C22×C4×C17C22×C413yes—
147C22×C4×C17C22×C411yescharacteristic
148C22×C4×C17C22×C413yes—
149C22×C4×C17C22×C413yes—
150C2×C22⋊D17—212no—
151C2×C22⋊D17—224no—
152C2×C22⋊D17—212no—
153C2×C22⋊D17C22×C413yes—
154C2×C22⋊D17C22×C413yes—
155C2×C22⋊D17C22×C411yescharacteristic
156C2×C22⋊D17C22×C413yes—
157C2×C22⋊D17C22×C416yes—
158C2×C22⋊D17C22×C413yes—
159C2×C22⋊D17C22×C413yes—
160C2×C22⋊D17C22×C416yes—
161C2×C22⋊D17C2411yescharacteristic
162C2×C22⋊D17C2413yes—
163C22×Dic17C22×C416yes—
164C22×Dic17C22×C413yes—
165C22×Dic17C2413yes—
166#63256—24no—
167#63256—212no—
168#63256—212no—
169#63256C2×D418yes—
170Q8×D17—24no—
171Q8×D17—212no—
172Q8×D17—212no—
173Q8×D17C22⋊C418yes—
174#63254—212no—
175#63254—224no—
176#63254—224no—
177#63254—212no—
178#63254—212no—
179#63254C2×D4124yes—
180D4×D17—224no—
181D4×D17—212no—
182D4×D17—224no—
183D4×D17—212no—
184D4×D17—212no—
185D4×D17C22⋊C4124yes—
186#63252—26no—
187#63252—26no—
188#63252—224no—
189#63252—212no—
190#63252—26no—
191#63252—26no—
192#63252—212no—
193#63252—26no—
194#63252—26no—
195#63252—212no—
196#63252—22no—
197#63252—22no—
198#63252—212no—
199#63252—26no—
200#63252—26no—
201#63252—212no—
202#63252—212no—
203#63252C22⋊C4112yes—
204#63252C22⋊C4112yes—
205C2×D68—212no—
206C2×D68—212no—
207C2×D68C22×C413yes—
208C2×D68C22×C411yescharacteristic
209C2×D68C22×C413yes—
210C2×D68C22×C413yes—
211C2×D68C22×C413yes—
212C2×D68C22×C413yes—
213C2×C4×D17—26no—
214C2×C4×D17—26no—
215C2×C4×D17—212no—
216C2×C4×D17—212no—
217C2×C4×D17C22×C411yescharacteristic
218C2×C4×D17C22×C413yes—
219C2×C4×D17C22×C416yes—
220C2×C4×D17C22×C416yes—
221C2×C4×D17C22×C413yes—
222C2×C4×D17C22×C413yes—
223C2×C4×D17C22×C413yes—
224C2×C4×D17C22×C413yes—
225C2×C4×D17C2×D4112yes—
226C2×C4×D17C2×D4112yes—
227C2×C4×D17C2411yescharacteristic
228C2×C4×D17C2413yes—
229C2×Dic34—212no—
230C2×Dic34—212no—
231C2×Dic34C22×C413yes—
232C2×Dic34C22×C413yes—
233C2×Dic34C22×C413yes—
234C2×Dic34C22×C413yes—
235C2×Dic34C2411yescharacteristic
236C2×Dic34C2413yes—
237#63247—224no—
238#63247C24116yes—
239C2×#3624—224no—
240#63243—28no—
241#63243—224no—
242#63242—28no—
243#63242—224no—
244#60076—171nomaximal, Sylow
245C23×C17—26no—
246C23×C17—83no—
247C23×C17C23×C413yes—
248C23×C17C23×C411yescharacteristic
249C23×C17C23×C413yes—
250C22×D17—26no—
251C22×D17—212no—
252C22×D17—83no—
253C22×D17—86no—
254C22×D17—83no—
255C22×D17C2×C22⋊C4112yes—
256C22×D17C23×C413yes—
257C22×D17C23×C411yescharacteristic
258C22×D17C23×C413yes—
259C22×D17C23×C411yescharacteristic
260Q8×C17—26no—
261Q8×C17C2×C22⋊C414yes—
262D4×C17—212no—
263D4×C17—26no—
264D4×C17C2×C22⋊C4112yes—
265C2×C4×C17—26no—
266C2×C4×C17—26no—
267C2×C4×C17—23no—
268C2×C4×C17—23no—
269C2×C4×C17—21no—
270C2×C4×C17—21no—
271C2×C4×C17—26no—
272C2×C4×C17C2×C22⋊C416yes—
273C2×C4×C17C2×C22⋊C416yes—
274C2×C4×C17C23×C411yescharacteristic
275C2×C4×C17C23×C413yes—
276C2×C4×C17C23×C413yes—
277C2×C4×C17C23×C411yescharacteristic
278C22⋊D17—212no—
279C22⋊D17—26no—
280C22⋊D17—22no—
281C22⋊D17—26no—
282C22⋊D17—26no—
283C22⋊D17—26no—
284C22⋊D17—26no—
285C22⋊D17—22no—
286C22⋊D17—26no—
287C22⋊D17C2×C22⋊C4124yes—
288C2×Dic17—26no—
289C2×Dic17—212no—
290C2×Dic17C23×C413yes—
291C2×Dic17C23×C411yescharacteristic
292C2×Dic17C23×C413yes—
293C2×Dic17C22×D4112yes—
294C2×Dic17C2511yescharacteristic
295D68—26no—
296D68—22no—
297D68—26no—
298D68—26no—
299D68—26no—
300D68C2×C22⋊C4112yes—
301C4×D17—212no—
302C4×D17—21no—
303C4×D17—23no—
304C4×D17—21no—
305C4×D17—23no—
306C4×D17—26no—
307C4×D17—26no—
308C4×D17—212no—
309C4×D17C2×C22⋊C416yes—
310C4×D17C2×C22⋊C416yes—
311C4×D17C2×C22⋊C4112yes—
312C4×D17C22×D412yes—
313C4×D17C22×D4112yes—
314C4×D17C22×D412yes—
315Dic34—26no—
316Dic34—26no—
317Dic34—26no—
318Dic34—22no—
319Dic34—26no—
320Dic34C2×C22⋊C4112yes—
321#3624—216no—
322C2×#585—171no—
323C2×#568—172no—
324#2889—176no—
325#2889—172no—
326#2888—176no—
327#2888—172no—
328C2×#413—176no—
329C2×#413—176no—
330C22×C17—23no—
331C22×C17—21no—
332C22×C17—43no—
333C22×C17—43no—
334C22×C17—41no—
335C22×C17—86no—
336C22×C17C22×C22⋊C416yes—
337C22×C17C24×C411yescharacteristic
338C2×D17—21no—
339C2×D17—23no—
340C2×D17—41no—
341C2×D17—43no—
342C2×D17—412no—
343C2×D17—43no—
344C2×D17—41no—
345C2×D17—812no—
346C2×D17—86no—
347C2×D17C22×C22⋊C416yes—
348C2×D17C22×C22⋊C412yes—
349C4×C17—21no—
350C4×C17—23no—
351C4×C17C22×C22⋊C416yes—
352C4×C17C22×C22⋊C411yescharacteristic
353C4×C17C22×C22⋊C411yescharacteristic
354Dic17—23no—
355Dic17—21no—
356Dic17C22×C22⋊C416yes—
357Dic17C23×D412yes—
358#585—348no—
359#585—348no—
360C2×2−1+4—173no—
361C2×2−1+4—171no—
362C2×2−1+4—171no—
363C2×2−1+4—171no—
364C2×H(5,𝔽2)—171no—
365C2×H(5,𝔽2)—173no—
366C2×H(5,𝔽2)—173no—
367C2×H(5,𝔽2)—173no—
368C22×#41—171no—
369C22×#41—173no—
370C22×#41—173no—
371C22×#41—171no—
372C22×#41—173no—
373C22×#41—173no—
374C22×#41#59511yescharacteristic
375#568—1712no—
376#568—174no—
377#568—348no—
378C2×#131—348no—
379C22×M16—1712no—
380#413—174no—
381#413—1712no—
382#413—1712no—
383#413—1712no—
384#413—1724no—
385C2×#101—1712no—
386C2×#101—1712no—
387C2×#100—1712no—
388C2×#100—1712no—
389C2×C17—21no—
390C2×C17—26no—
391C2×C17—43no—
392C2×C17—41no—
393C2×C17C23×C22⋊C411yescharacteristic
394D17—22no—
395D17—26no—
396D17—43no—
397D17—41no—
3982−1+4—348no—
3992−1+4—348no—
4002−1+4—348no—
4012−1+4—3424no—
402H(5,𝔽2)—3424no—
403H(5,𝔽2)—3424no—
404H(5,𝔽2)—3424no—
405H(5,𝔽2)—348no—
406C2×#41—24no—
407C2×#41—24no—
408C2×#41—178no—
409C2×#41—178no—
410C2×#41—171no—
411C2×#41—173no—
412C2×#41—173no—
413C2×#41—171no—
414C2×#41—173no—
415C2×#41—173no—
416C2×#41—176no—
417C2×#41—173no—
418C2×#41—173no—
419C2×#41—176no—
420C2×#41—173no—
421C2×#41—173no—
422C2×#41—173no—
423C2×#41—173no—
424C2×#41—173no—
425C2×#41—173no—
426C2×#41—176no—
427C2×#41—173no—
428C2×#41—173no—
429C2×#41—173no—
430C2×#41—171no—
431C2×#41—173no—
432C2×#41—173no—
433C2×#41—3412no—
434C2×#41—3412no—
435C2×#41—3412no—
436C2×#41—3412no—
437C2×#41—3412no—
438C2×#41—3412no—
439C2×#41—3412no—
440C2×#41—3412no—
441C2×#41—344no—
442C2×#41—344no—
443C2×#41C2×#59513yes—
444C2×#41C2×#59511yescharacteristic
445C2×#41C2×#59513yes—
446C2×#41C2×#59511yescharacteristic
447C22×Q8—171no—
448C22×Q8—173no—
449C22×Q8—171no—
450C22×Q8—173no—
451C22×Q8—173no—
452C22×Q8—173no—
453C22×Q8C2×#59511yescharacteristic
454C22×D4—173no—
455C22×D4—176no—
456C22×D4—173no—
457C22×D4—173no—
458C22×D4—176no—
459C22×D4—173no—
460C22×D4—176no—
461C22×D4—173no—
462C22×D4—173no—
463C22×D4—173no—
464C22×D4—173no—
465C22×D4C2×#59513yes—
466C23×C4—173no—
467C23×C4—176no—
468C23×C4—173no—
469C23×C4C2×#59513yes—
470#131—3448no—
471#131—3416no—
472C2×M16—178no—
473C2×M16—1724no—
474C2×M16—1724no—
475C2×M16—3412no—
476C2×M16—3412no—
477C2×M16—3412no—
478C22×C8—3412no—
479#101—178no—
480#101—1724no—
481#100—178no—
482#100—1724no—
483C17#6007611yescharacteristic, complemented, Sylow
484C24—173no—
485C24—176no—
486C24—173no—
487C24C22×#59513yes—
488#41—24no—
489#41—212no—
490#41—26no—
491#41—26no—
492#41—178no—
493#41—1724no—
494#41—1712no—
495#41—1712no—
496#41—342no—
497#41—346no—
498#41—346no—
499#41—346no—
500#41—342no—
501#41—346no—
502#41—346no—
503#41—346no—
504#41—3412no—
505#41—3412no—
506#41—3412no—
507#41—3412no—
508#41—3424no—
509#41—3412no—
510#41—3412no—
511#41—3424no—
512#41—3412no—
513#41—3412no—
514#41—3424no—
515#41—3412no—
516#41—3412no—
517#41—344no—
518#41—3412no—
519#41—346no—
520#41—346no—
521#41C22⋊#59514yes—
522#41C22⋊#59514yes—
523C2×Q8—24no—
524C2×Q8—178no—
525C2×Q8—173no—
526C2×Q8—173no—
527C2×Q8—173no—
528C2×Q8—173no—
529C2×Q8—173no—
530C2×Q8—171no—
531C2×Q8—3412no—
532C2×Q8—3412no—
533C2×Q8—3412no—
534C2×Q8—3412no—
535C2×Q8—344no—
536C2×Q8C22×#59513yes—
537C2×Q8C22×#59511yescharacteristic
538C2×D4—212no—
539C2×D4—176no—
540C2×D4—1724no—
541C2×D4—173no—
542C2×D4—173no—
543C2×D4—173no—
544C2×D4—173no—
545C2×D4—173no—
546C2×D4—171no—
547C2×D4—176no—
548C2×D4—173no—
549C2×D4—171no—
550C2×D4—173no—
551C2×D4—173no—
552C2×D4—173no—
553C2×D4—173no—
554C2×D4—171no—
555C2×D4—173no—
556C2×D4—3412no—
557C2×D4—3412no—
558C2×D4—3412no—
559C2×D4—3412no—
560C2×D4—3424no—
561C2×D4—3412no—
562C2×D4—3424no—
563C2×D4—3424no—
564C2×D4—3412no—
565C2×D4—3412no—
566C2×D4C22×#59516yes—
567C2×D4C22×#59513yes—
568C2×D4C22×#59513yes—
569C22×C4—26no—
570C22×C4—26no—
571C22×C4—173no—
572C22×C4—1712no—
573C22×C4—1712no—
574C22×C4—173no—
575C22×C4—173no—
576C22×C4—171no—
577C22×C4—173no—
578C22×C4—176no—
579C22×C4—173no—
580C22×C4—173no—
581C22×C4—171no—
582C22×C4—176no—
583C22×C4—176no—
584C22×C4—173no—
585C22×C4—173no—
586C22×C4—3412no—
587C22×C4—3412no—
588C22×C4—346no—
589C22×C4—346no—
590C22×C4C22×#59513yes—
591C22×C4C22×#59511yescharacteristic
592C22×C4C22×#59513yes—
593C22×C4C22×#59513yes—
594C22×C4C22×#59513yes—
595C22×C4C22×#59513yes—
596M16—1716no—
597M16—348no—
598M16—3424no—
599M16—3424no—
600C2×C8—348no—
601C2×C8—3424no—
602C2×C8—3424no—
603C23—26no—
604C23—83no—
605C23—1712no—
606C23—171no—
607C23—173no—
608C23—173no—
609C23—171no—
610C23—3412no—
611C23—346no—
612C23—1363no—
613C23—1363no—
614C23—1366no—
615C23C23×#59513yes—
616C23C23×#59511yescharacteristic
617C23C23×#59513yes—
618Q8—26no—
619Q8—1712no—
620Q8—342no—
621Q8—346no—
622Q8—346no—
623Q8—346no—
624Q8—346no—
625Q8C2×C22⋊#59514yes—
626D4—212no—
627D4—26no—
628D4—1724no—
629D4—1712no—
630D4—342no—
631D4—346no—
632D4—346no—
633D4—346no—
634D4—342no—
635D4—346no—
636D4—346no—
637D4—346no—
638D4—342no—
639D4—346no—
640D4—346no—
641D4—346no—
642D4—3412no—
643D4—346no—
644D4C2×C22⋊#595112yes—
645C2×C4—26no—
646C2×C4—21no—
647C2×C4—23no—
648C2×C4—26no—
649C2×C4—26no—
650C2×C4—21no—
651C2×C4—23no—
652C2×C4—1712no—
653C2×C4—171no—
654C2×C4—173no—
655C2×C4—1712no—
656C2×C4—172no—
657C2×C4—176no—
658C2×C4—1712no—
659C2×C4—172no—
660C2×C4—176no—
661C2×C4—173no—
662C2×C4—171no—
663C2×C4—3412no—
664C2×C4—3412no—
665C2×C4—3412no—
666C2×C4—346no—
667C2×C4—343no—
668C2×C4—341no—
669C2×C4—346no—
670C2×C4—346no—
671C2×C4—343no—
672C2×C4—341no—
673C2×C4C2×C22⋊#59516yes—
674C2×C4C2×C22⋊#59516yes—
675C2×C4C23×#59513yes—
676C2×C4C23×#59511yescharacteristic
677C2×C4C23×#59513yes—
678C2×C4C23×#59511yescharacteristic
679C8—3416no—
680C22—21no—
681C22—23no—
682C22—43no—
683C22—41no—
684C22—43no—
685C22—86no—
686C22—172no—
687C22—176no—
688C22—343no—
689C22—341no—
690C22—681no—
691C22—683no—
692C22—6812no—
693C22—681no—
694C22—683no—
695C22—1366no—
696C22—13612no—
697C22C22×C22⋊#59516yes—
698C22C24×#59511yescharacteristic
699C4—21no—
700C4—23no—
701C4—172no—
702C4—176no—
703C4—343no—
704C4—341no—
705C4C22×C22⋊#59511yescharacteristic
706C4C22×C22⋊#59516yes—
707C4C22×C22⋊#59511yescharacteristic
708C2—21no—
709C2—26no—
710C2—41no—
711C2—43no—
712C2—342no—
713C2—346no—
714C2—681no—
715C2—683no—
716C2C23×C22⋊#59511yescharacteristic
subgroupG/H
centerC4C22×C22⋊#595
commutatorC22×C17C24×C4
FrattiniC22C24×#595
FittingC22×#41×C17C4
socleC2×C17C23×C22⋊C4
prime radicalC2×#11763645C2
cosocleC2×Dic17C25
hypercenterC22×#41#595
hypocenterC17#60076
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×#41 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. D17⋊#568C2semidirect
2. #14545397C2semidirect
3. #14545396C2semidirect
4. #14545395C2semidirect
5. #14545394C2semidirect
6. D17⋊#413C2semidirect
7. C2×#11763795C2semidirect
9. #11763889C22semidirect
11. #11763887C22semidirect
13. C2×D17⋊M16C22semidirect
14. #11763796C22semidirect
15. #11763795C22semidirect
16. #11763795C22semidirect
17. #11763795C22semidirect
18. #11763795C22semidirect
19. D17⋊#101C22semidirect
20. C2×#10587545C22semidirect
21. D17⋊#100C22semidirect
22. C2×#10587553C22semidirect
483. C17#60076semidirect

Maximal subgroups

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

subgroupindex
D17⋊#5682
#145453972
#145453962
#145453952
#145453942
D17⋊#4132
C2×#117637952
C2×#117636452
#6007617