Finite Group Databasebeta

← D17⋊#1503 #17965225

Subgroups — D17⋊#1503 #17965225

Order 4352.

1194 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545448C211yesmaximal, characteristic
2C2×C4×C4⋊#595C211yesmaximal, characteristic
3#14544168C212yesmaximal, complemented
4#14544167C212yesmaximal, complemented
5#14544166C214yesmaximal, complemented
6D17⋊#328C214yesmaximal
7D17×#533C211yesmaximal, characteristic
8C22×C4⋊#595—21no—
9C22×C4×#595—21no—
10#11763828C2212yes—
11#11763827C2212yes—
12#11763826—21no—
13#11763825—21no—
14#11763810—21no—
15#11763777C2212yes—
16C4×#63242—21no—
17C4×C4⋊#595—21no—
18C4×C4⋊#595—21no—
19C4×C4⋊#595C2212yes—
20C4×C4⋊#595C2212yes—
21C2×(C2×#595)⋊C4C2212yes—
22#11763663C2218yes—
23#11763662C2218yes—
24#11763366C411yescharacteristic, complemented
25#11763365—21no—
26#11763364C2211yescharacteristic
27#11763364C2211yescharacteristic
28#11763363—21no—
29#11763363C2212yes—
30#11763362—21no—
31#11763361—21no—
32#11763361C412yescomplemented
33C4⋊Q8×D17C411yescharacteristic, complemented
34C4⋊Q8×D17C411yescharacteristic, complemented
35#125×D17—21no—
36C22⋊Q8×D17—21no—
37C22⋊Q8×D17C2212yes—
38C4×Q8×D17—21no—
39C4×Q8×D17C412yescomplemented
40#117×D17—21no—
41C2×C42×D17C2211yescharacteristic
42C17×#533C411yescharacteristic, complemented
43C23×#595—22no—
44C2×Q8×D17—21no—
45C2×Q8×D17C2×C412yes—
46C22×C4×D17D411yescharacteristic
47C22×C4×D17D411yescharacteristic
48C22×C4×D17C2311yescharacteristic
49C2×C4⋊#595—28no—
50C2×C4⋊#595—22no—
51C2×C4⋊#595—22no—
52C2×C4×#595—22no—
53C2×C4×#595—22no—
54C2×C4×#595—44no—
55C2×#63242—22no—
56C17×C4⋊Q8C2×C411yescharacteristic, complemented
57C17×C4⋊Q8C2×C411yescharacteristic, complemented
58C17×#125—21no—
59C17×C22⋊Q8—21no—
60C17×C22⋊Q8C2×C412yes—
61C4×Q8×C17—21no—
62C4×Q8×C17C2×C412yescomplemented
63C17×#117—21no—
64C2×C42×C17C2×C411yescharacteristic
65#10587632—21no—
66#10587632C2312yes—
67Q8×Dic17—21no—
68Q8×Dic17C2×C412yescomplemented
69#10587630—21no—
70#10587630C2×C412yescomplemented
71#10587612—21no—
72#10587610—21no—
73#10587610C2×C412yes—
74C2×C4×Dic17C2×C411yescharacteristic
75C2×C4×Dic17D411yescharacteristic
76C2×C4×Dic17D411yescharacteristic
77#10587583—21no—
78#10587583C2312yes—
79#10587582—22no—
80#10587582—22no—
81#10587578—21no—
82#10587578—21no—
83C4⋊C4×D17—21no—
84C4⋊C4×D17—21no—
85C4⋊C4×D17—21no—
86C4⋊C4×D17—22no—
87C4⋊C4×D17—22no—
88C4⋊C4×D17C2×C412yes—
89#10587576—21no—
90#10587576—21no—
91#10587575—22no—
92#10587575—22no—
93#10587574—21no—
94#10587574—21no—
95#10587574—21no—
96#10587574C2×C412yescomplemented
97#10587573—22no—
98#10587573—22no—
99#10587573—21no—
100#10587573C2×C412yescomplemented
101C22⋊C4×D17—22no—
102#10587564—22no—
103#10587564—22no—
104#10587563—22no—
105#10587558—22no—
106C42×D17—22no—
107C42×D17C2×C411yescharacteristic
108C42×D17C2×C411yescharacteristic
109C42×D17D411yescharacteristic
110C42×D17D411yescharacteristic
111C42×D17D411yescharacteristic
112C42×D17D411yescharacteristic
113C42×D17C2311yescharacteristic
114C42×D17C2311yescharacteristic
115#10587556—21no—
116#10587555C2×C411yescharacteristic, complemented
117#10587555C2×C411yescharacteristic, complemented
118C4×Dic34—22no—
119C4×Dic34—22no—
120C4×Dic34—21no—
121C4×Dic34C2×C412yescomplemented
122C22⋊#3624—22no—
123(C2×#595)⋊C4—22no—
124(C2×#595)⋊C4C2314yes—
125#10587544—22no—
126#10587543C2314yes—
127C4⋊#3624—22no—
128C4×#3624—22no—
129C4×#3624—22no—
130C23×D17C2×D411yescharacteristic
131C22×#595—28no—
132C22×#595—24no—
133C22×#595—44no—
134C22×#595—44no—
135C2×Q8×C17—21no—
136C2×Q8×C17C22×C412yes—
137C22×C4×C17C22⋊C411yescharacteristic
138C22×C4×C17C22⋊C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×Dic17C22⋊C411yescharacteristic
141C22×Dic17C22⋊C411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143Q8×D17—44no—
144Q8×D17—44no—
145C2×C4×D17—24no—
146C2×C4×D17—22no—
147C2×C4×D17—22no—
148C2×C4×D17—42no—
149C2×C4×D17C22⋊C411yescharacteristic
150C2×C4×D17C22⋊C411yescharacteristic
151C2×C4×D17C22⋊C411yescharacteristic
152C2×C4×D17C22⋊C411yescharacteristic
153C2×C4×D17SD1614yes—
154C2×C4×D17Q1614yes—
155C2×C4×D17C22×C411yescharacteristic
156C2×C4×D17C2×D411yescharacteristic
157C2×C4×D17C2×D411yescharacteristic
158C2×C4×D17C2×D411yescharacteristic
159C2×C4×D17C2×D411yescharacteristic
160C2×C4×D17C2×D411yescharacteristic
161C2×C4×D17C2×D411yescharacteristic
162C2×C4×D17C2411yescharacteristic
163C2×Dic34—21no—
164C2×Dic34—22no—
165C2×Dic34—22no—
166C2×Dic34C22×C412yes—
167C2×#3624—24no—
168C4⋊#595—28no—
169C4⋊#595—28no—
170C4×#595—44no—
171C4×#595—44no—
172#63242—44no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C22×C412yes—
179C22⋊C4×C17—22no—
180C42×C17—22no—
181C42×C17C22⋊C411yescharacteristic
182C42×C17C22⋊C411yescharacteristic
183C42×C17C22×C411yescharacteristic
184C42×C17C22×C411yescharacteristic
185C22⋊Dic17—22no—
186#63227—22no—
187#63227—22no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—22no—
192C4⋊Dic17—22no—
193C4⋊Dic17C22×C412yes—
194#63225—22no—
195#63225—22no—
196#63225—22no—
197#63225—22no—
198#63225—22no—
199#63225—22no—
200#63225—22no—
201#63225—22no—
202C4×Dic17—22no—
203C4×Dic17—22no—
204C4×Dic17—22no—
205C4×Dic17C22⋊C411yescharacteristic
206C4×Dic17C22⋊C411yescharacteristic
207C4×Dic17C22⋊C411yescharacteristic
208C4×Dic17C22⋊C411yescharacteristic
209C4×Dic17C22⋊C411yescharacteristic
210C4×Dic17C22⋊C411yescharacteristic
211C4×Dic17C22×C411yescharacteristic
212C4×Dic17C22×C411yescharacteristic
213C4×Dic17C2×D411yescharacteristic
214C4×Dic17C2×D411yescharacteristic
215C4×Dic17C2×D411yescharacteristic
216C4×Dic17C2×D411yescharacteristic
217C2×#1503—171nomaximal, Sylow
218C23×C17C2×C22⋊C411yescharacteristic
219C22×D17—22no—
220C22×D17—22no—
221C22×D17C2×C22⋊C411yescharacteristic
222C22×D17C2×SD1612yes—
223C22×D17C2×Q1612yes—
224C22×D17C22×D411yescharacteristic
225C2×#595—28no—
226C2×#595—44no—
227C2×#595—416no—
228C2×#595—44no—
229Q8×C17—42no—
230Q8×C17—42no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—41no—
235C2×C4×C17#10314yes—
236C2×C4×C17C2×C22⋊C411yescharacteristic
237C2×C4×C17C2×C22⋊C411yescharacteristic
238C2×C4×C17C2×C22⋊C411yescharacteristic
239C2×C4×C17C2×C22⋊C411yescharacteristic
240C2×C4×C17C2×C22⋊C411yescharacteristic
241C2×C4×C17C23×C411yescharacteristic
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—41no—
246C2×Dic17#10314yes—
247C2×Dic17C2×C22⋊C411yescharacteristic
248C2×Dic17C2×C22⋊C411yescharacteristic
249C2×Dic17C2×C22⋊C411yescharacteristic
250C2×Dic17C2×C22⋊C411yescharacteristic
251C2×Dic17C2×C22⋊C411yescharacteristic
252C2×Dic17C22×D411yescharacteristic
253C4×D17—22no—
254C4×D17—22no—
255C4×D17—22no—
256C4×D17—22no—
257C4×D17—22no—
258C4×D17—41no—
259C4×D17—41no—
260C4×D17—44no—
261C4×D17—44no—
262C4×D17#10314yes—
263C4×D17#10314yes—
264C4×D17#10414yes—
265C4×D17C2×SD1612yes—
266C4×D17C2×SD1612yes—
267C4×D17C2×Q1612yes—
268C4×D17C2×Q1612yes—
269Dic34—44no—
270Dic34—42no—
271Dic34—44no—
272Dic34—42no—
273#3624—44no—
274C2×#533—171no—
275C2×#431—171no—
276C2×C4×C4⋊C4—171no—
277#1503—174no—
278#1503—172no—
279#1503—172no—
280C2×#328—174no—
281C22×C17—21no—
282C22×C17—21no—
283C22×C17C2×#10312yes—
284C22×C17C22×C22⋊C411yescharacteristic
285C2×D17—21no—
286C2×D17—21no—
287C2×D17—24no—
288C2×D17—24no—
289C2×D17#41012yes—
290C2×D17C2×#10312yes—
291C2×D17C2×#10412yes—
292C2×D17C22×SD1611yescharacteristic
293C2×D17C22×Q1611yescharacteristic
294#595—48no—
295C4×C17—21no—
296C4×C17—21no—
297C4×C17—21no—
298C4×C17—42no—
299C4×C17—42no—
300C4×C17C2×#10312yes—
301C4×C17C2×#10312yes—
302C4×C17C2×#10412yes—
303Dic17—21no—
304Dic17—21no—
305Dic17—21no—
306Dic17—42no—
307Dic17—42no—
308Dic17C2×#10312yes—
309Dic17C2×#10312yes—
310Dic17C2×#10412yes—
311#533—171no—
312#533—171no—
313#533—172no—
314#533—172no—
315#533—171no—
316#533—341no—
317#533—341no—
318#533—341no—
319#533—341no—
320#533#59511yescharacteristic, complemented
321C2×C4⋊Q8—171no—
322C2×C4⋊Q8—171no—
323C2×#125—341no—
324C2×C22⋊Q8—172no—
325C2×C22⋊Q8—341no—
326C2×C4×Q8—172no—
327C2×C4×Q8—341no—
328C2×#117—341no—
329C22×C4⋊C4—341no—
330C22×C42—171no—
331C22×C42—341no—
332#431—172no—
333#431—172no—
334#431—341no—
335#431—341no—
336C2×C4⋊C8—341no—
337C2×C22⋊C8—172no—
338C2×C4×C8—341no—
339C4×C4⋊C4—172no—
340C4×C4⋊C4—172no—
341C4×C4⋊C4—341no—
342C4×C4⋊C4—341no—
343C2×#95—172no—
344#328—178no—
345#328—178no—
346C2×C17—22no—
347C2×C17C2×#41011yescharacteristic
348C2×C17C22×#10311yescharacteristic
349C2×C17C22×#10411yescharacteristic
350D17—21no—
351D17—21no—
352D17#150314yescomplemented
353C22×Q8—172no—
354C22×Q8—341no—
355C23×C4—171no—
356C23×C4—171no—
357C23×C4—171no—
358C23×C4—342no—
359C22×C8—342no—
360C4⋊Q8—172no—
361C4⋊Q8—172no—
362C4⋊Q8—171no—
363C4⋊Q8—171no—
364C4⋊Q8—341no—
365C4⋊Q8—341no—
366C4⋊Q8—341no—
367C4⋊Q8—341no—
368C4⋊Q8C2×#59511yescharacteristic, complemented
369C4⋊Q8C2×#59511yescharacteristic, complemented
370#125—21no—
371#125—342no—
372#125—342no—
373#125—341no—
374#125—341no—
375#125—341no—
376C22⋊Q8—21no—
377C22⋊Q8—172no—
378C22⋊Q8—172no—
379C22⋊Q8—172no—
380C22⋊Q8—342no—
381C22⋊Q8—342no—
382C22⋊Q8—342no—
383C22⋊Q8—342no—
384C22⋊Q8—341no—
385C22⋊Q8—341no—
386C22⋊Q8—341no—
387C22⋊Q8C2×#59512yes—
388C4×Q8—21no—
389C4×Q8—172no—
390C4×Q8—172no—
391C4×Q8—172no—
392C4×Q8—342no—
393C4×Q8—342no—
394C4×Q8—342no—
395C4×Q8—341no—
396C4×Q8—342no—
397C4×Q8—341no—
398C4×Q8—341no—
399C4×Q8C2×#59512yescomplemented
400#117—21no—
401#117—342no—
402#117—342no—
403#117—341no—
404#117—341no—
405#117—341no—
406C2×C4⋊C4—172no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—342no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4—348no—
415C2×C22⋊C4—342no—
416C2×C42—171no—
417C2×C42—171no—
418C2×C42—171no—
419C2×C42—171no—
420C2×C42—171no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—171no—
424C2×C42—171no—
425C2×C42—171no—
426C2×C42—171no—
427C2×C42—342no—
428C2×C42—342no—
429C2×C42—342no—
430C2×C42—684no—
431C2×C42C2×#59511yescharacteristic
432C4⋊C8—342no—
433C4⋊C8—342no—
434C22⋊C8—174no—
435C22⋊C8—342no—
436C4×C8—342no—
437C4×C8—342no—
438#95—174no—
439#95—342no—
440C17C2×#150311yescharacteristic, complemented, Sylow
441C24—171no—
442C2×Q8—21no—
443C2×Q8—172no—
444C2×Q8—342no—
445C2×Q8—342no—
446C2×Q8—341no—
447C2×Q8—684no—
448C2×Q8—684no—
449C2×Q8C22×#59512yes—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—171no—
456C22×C4—174no—
457C22×C4—174no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—171no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—344no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—344no—
471C22×C4—348no—
472C22×C4—682no—
473C22×C4—684no—
474C22×C4—684no—
475C22×C4C22⋊#59511yescharacteristic
476C22×C4C22⋊#59511yescharacteristic
477C22×C4C22×#59511yescharacteristic
478C2×C8—344no—
479C2×C8—684no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—172no—
486C4⋊C4—342no—
487C4⋊C4—342no—
488C4⋊C4—342no—
489C4⋊C4—342no—
490C4⋊C4—342no—
491C4⋊C4—342no—
492C4⋊C4—342no—
493C4⋊C4—342no—
494C4⋊C4—348no—
495C4⋊C4—348no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—342no—
499C4⋊C4—342no—
500C4⋊C4—341no—
501C4⋊C4C22×#59512yes—
502C22⋊C4—22no—
503C22⋊C4—342no—
504C22⋊C4—342no—
505C22⋊C4—342no—
506C42—22no—
507C42—171no—
508C42—171no—
509C42—171no—
510C42—171no—
511C42—171no—
512C42—171no—
513C42—171no—
514C42—171no—
515C42—171no—
516C42—171no—
517C42—171no—
518C42—171no—
519C42—342no—
520C42—342no—
521C42—342no—
522C42—684no—
523C42—684no—
524C42C22⋊#59511yescharacteristic
525C42C22⋊#59511yescharacteristic
526C42C22×#59511yescharacteristic
527C42C22×#59511yescharacteristic
528C23—171no—
529C23—172no—
530C23—172no—
531C23—171no—
532C23—342no—
533C23—342no—
534C23C2×C22⋊#59511yescharacteristic
535Q8—42no—
536Q8—42no—
537Q8—682no—
538Q8—684no—
539Q8—684no—
540Q8—682no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—174no—
546C2×C4—171no—
547C2×C4—171no—
548C2×C4—171no—
549C2×C4—172no—
550C2×C4—172no—
551C2×C4—174no—
552C2×C4—174no—
553C2×C4—172no—
554C2×C4—172no—
555C2×C4—174no—
556C2×C4—171no—
557C2×C4—171no—
558C2×C4—171no—
559C2×C4—342no—
560C2×C4—342no—
561C2×C4—342no—
562C2×C4—342no—
563C2×C4—342no—
564C2×C4—342no—
565C2×C4—348no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—684no—
571C2×C4—684no—
572C2×C4—681no—
573C2×C4—684no—
574C2×C4—6816no—
575C2×C4—684no—
576C2×C4#1058754914yes—
577C2×C4C2×C22⋊#59511yescharacteristic
578C2×C4C2×C22⋊#59511yescharacteristic
579C2×C4C2×C22⋊#59511yescharacteristic
580C2×C4C2×C22⋊#59511yescharacteristic
581C2×C4C2×C22⋊#59511yescharacteristic
582C2×C4C23×#59511yescharacteristic
583C8—684no—
584C22—21no—
585C22—21no—
586C22—172no—
587C22—172no—
588C22—172no—
589C22—171no—
590C22—171no—
591C22—344no—
592C22—344no—
593C22—341no—
594C22—341no—
595C22C2×#1058754912yes—
596C22C22×C22⋊#59511yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—172no—
603C4—172no—
604C4—172no—
605C4—341no—
606C4—341no—
607C4—341no—
608C4—688no—
609C4—682no—
610C4—682no—
611C4C2×#1058754912yes—
612C4C2×#1058754912yes—
613C4D17⋊#10412yes—
614C2—22no—
615C2—174no—
616C2—341no—
617C2—341no—
618C2D17⋊#41011yescharacteristic
619C2C22×#1058754911yescharacteristic
620C2C2×D17⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC2×C4×C17C23×C4
FrattiniC2×C4C23×#595
FittingC17×#533C4
socleC22×C17C22×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC2×C4×D17C24
hypercenter#533#595
hypocenterC17C2×#1503
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #14544168C2semidirect
4. #14544167C2semidirect
5. #14544166C2semidirect
24. #11763366C4semidirect
32. #11763361C4semidirect
33. C4⋊Q8×D17C4semidirect
34. C4⋊Q8×D17C4semidirect
39. C4×Q8×D17C4semidirect
42. C17×#533C4semidirect
56. C17×C4⋊Q8C2×C4semidirect
57. C17×C4⋊Q8C2×C4semidirect
62. C4×Q8×C17C2×C4semidirect
68. Q8×Dic17C2×C4semidirect
70. #10587630C2×C4semidirect
96. #10587574C2×C4semidirect
100. #10587573C2×C4semidirect
116. #10587555C2×C4semidirect
117. #10587555C2×C4semidirect
121. C4×Dic34C2×C4semidirect
320. #533#595semidirect
352. D17#1503semidirect
368. C4⋊Q8C2×#595semidirect
369. C4⋊Q8C2×#595semidirect
399. C4×Q8C2×#595semidirect
440. C17C2×#1503semidirect

Maximal subgroups

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

subgroupindex
#145454482
C2×C4×C4⋊#5952
#145441682
#145441672
#145441662
D17⋊#3282
D17×#5332
C2×#150317