Finite Group Databasebeta

← #17051598

Subgroups — #17051598

Order 4352.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531677C211yesmaximal, characteristic
2Q8×C22⋊Dic17C211yesmaximal, characteristic
3#14529476C211yesmaximal, characteristic, complemented
4#14529462C211yesmaximal, characteristic, complemented
5#14529427C211yesmaximal, characteristic, complemented
6#14529420C211yesmaximal, characteristic, complemented
7#14529054C211yesmaximal, characteristic, complemented
8#14529040C211yesmaximal, characteristic, complemented
9#14528879C211yesmaximal, characteristic
10#14528817C211yesmaximal, characteristic
11#14528719C211yesmaximal, characteristic
12#14528616C211yesmaximal, characteristic, complemented
13#103×Dic17C211yesmaximal, characteristic, complemented
14#14528565C211yesmaximal, characteristic
15C2×C17×#419C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762716—21no—
28#11762716—21no—
29#11762716C2211yescharacteristic
30#11762716C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762702—21no—
34#11762702—21no—
35#11762677—21no—
36#11762669—21no—
37#11762669C2211yescharacteristic
38#11762669C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×C4⋊Dic17—21no—
42C4×C22⋊Dic17—21no—
43C4×C22⋊Dic17C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C23×Q8×C17C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C17×#419C418yescomplemented
56C2×C17×#103C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C22×Q8×C17—21no—
61C22×Q8×C17—22no—
62C22×Q8×C17—22no—
63C22×Q8×C17—22no—
64C22×Q8×C17—41no—
65C22×Q8×C17—41no—
66C22×Q8×C17C2×C414yes—
67C22×Q8×C17C2311yescharacteristic
68C22×Q8×C17C2311yescharacteristic
69C23×C4×C17—21no—
70C23×C4×C17C2311yescharacteristic
71C2×M16×C17—41no—
72C2×M16×C17—41no—
73C2×M16×C17C2×C414yes—
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C17×#117—22no—
77C17×#117C2×C414yes—
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C42×C17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17D411yescharacteristic
85C2×C22⋊Dic17D411yescharacteristic
86Q8×Dic17—22no—
87Q8×Dic17—22no—
88Q8×Dic17—41no—
89Q8×Dic17—41no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C17×#103C2×C414yes—
107C17×#103C2×C414yes—
108C17×#103C2×C414yes—
109C17×#103C2×C414yes—
110#10587506—22no—
111#10587506—22no—
112#10587506—41no—
113#10587506—41no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502D411yescharacteristic
125#10587502D411yescharacteristic
126C22⋊#3622—22no—
127#10587489—41no—
128#10587488—41no—
129#10587486—41no—
130C8×Dic17—41no—
131C24×C17C2×D411yescharacteristic
132C2×Q8×C17—24no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—28no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—42no—
143C2×Q8×C17—42no—
144C2×Q8×C17—44no—
145C2×Q8×C17—44no—
146C2×Q8×C17—44no—
147C2×Q8×C17C22×C412yes—
148C2×Q8×C17C22×C412yes—
149C2×Q8×C17#4112yes—
150C2×Q8×C17#4112yes—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—24no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—42no—
161C22×C4×C17—42no—
162C22×C4×C17C22×C412yes—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17#4112yes—
165C22×C4×C17C2411yescharacteristic
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17C2×D411yescharacteristic
171C22×Dic17C2×D411yescharacteristic
172C22×Dic17C2×D411yescharacteristic
173C22×Dic17C2×D411yescharacteristic
174M16×C17—44no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C2×C8×C17C22×C412yes—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17C22×C412yes—
184C4⋊C4×C17C22×C412yes—
185C22⋊C4×C17—24no—
186C42×C17—24no—
187C22⋊Dic17—22no—
188C22⋊Dic17—41no—
189C22⋊Dic17—42no—
190C22⋊Dic17—81no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17—42no—
201C4⋊Dic17—42no—
202C4⋊Dic17—42no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—42no—
208C4×Dic17—42no—
209C4×Dic17—81no—
210C4×Dic17—81no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217#22581—171nomaximal, Sylow
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17C4×D412yes—
225C23×C17C22×D411yescharacteristic
226Q8×C17—22no—
227Q8×C17—22no—
228Q8×C17—22no—
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17—48no—
232Q8×C17—41no—
233Q8×C17—41no—
234Q8×C17#11714yes—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—48no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17C4×D412yes—
255C2×C4×C17C23×C411yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17H(5,𝔽2)11yescharacteristic
260C2×C4×C17H(5,𝔽2)11yescharacteristic
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17#13712yes—
285C8×C17—22no—
286C8×C17—22no—
287#3622—41no—
288#3622—41no—
289C2×#419D1711yescharacteristic
290#2347—171no—
291Q8×C22⋊C4—171no—
292#1935—171no—
293#1933—171no—
294#1928—171no—
295#1927—171no—
296#1865—171no—
297#1863—171no—
298#1825—171no—
299#1809—171no—
300#1787—171no—
301#1763—171no—
302C4×#103—171no—
303#1753—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17C2×C4×D411yescharacteristic
312C22×C17#54611yescharacteristic
313C22×C17#54611yescharacteristic
314C22×C17C2×#13711yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57811yescharacteristic
318C4×C17—22no—
319C4×C17—24no—
320C4×C17—42no—
321C4×C17—42no—
322C4×C17—42no—
323C4×C17C2×#11711yescharacteristic
324C4×C17#51811yescharacteristic
325C4×C17#55111yescharacteristic
326C4×C17#55111yescharacteristic
327Dic17—22no—
328Dic17—41no—
329Dic17—41no—
330Dic17—41no—
331Dic17—81no—
332Dic17—81no—
333Dic17—81no—
334Dic17—81no—
335C23×Q8C2×D1711yescharacteristic
336C22×M16C2×D1711yescharacteristic
337C2×C4×Q8—171no—
338C2×C4×Q8—171no—
339C2×#117C2×D1711yescharacteristic
340C2×#107—171no—
341C2×#106—171no—
342#419Dic1718yescomplemented
343C2×#103—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#103C2×D1711yescharacteristic
348C2×#103C2×D1711yescharacteristic
349C2×#103C2×D1711yescharacteristic
350C2×#103C2×D1711yescharacteristic
351C2×C22⋊C8—171no—
352C2×#97—171no—
353C2×C4×C8—171no—
354#391—171no—
355#391—171no—
356#391—341no—
357#391—341no—
358#389—171no—
359#389—171no—
360#388—341no—
361#388—341no—
362#383—341no—
363#380—171no—
364#380—171no—
365#380—341no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4—341no—
369C4×C22⋊C4—171no—
370C4×C22⋊C4—341no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#336—171no—
378#336—171no—
379C2×C17—42no—
380C2×C17#234211yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294711yescharacteristic
383C2×C17#331311yescharacteristic
384C2×C17#332011yescharacteristic
385C2×C17#332611yescharacteristic
386C2×C17#332811yescharacteristic
387C22×Q8—21no—
388C22×Q8—22no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—41no—
392C22×Q8—41no—
393C22×Q8C2×Dic1714yes—
394C22×Q8C22×D1711yescharacteristic
395C22×Q8C22×D1711yescharacteristic
396C23×C4—21no—
397C23×C4C22×D1711yescharacteristic
398C2×M16—41no—
399C2×M16—41no—
400C2×M16C2×Dic1714yes—
401C22×C8—171no—
402C22×C8—171no—
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C4×Q8—342no—
406C4×Q8—342no—
407C4×Q8—681no—
408C4×Q8—681no—
409#117—22no—
410#117C2×Dic1714yes—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—171no—
423C2×C22⋊C4—171no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C42—21no—
427C2×C42—171no—
428C2×C42—171no—
429C2×C42—341no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42—341no—
433#107—681no—
434#106—681no—
435#103—342no—
436#103—342no—
437#103—681no—
438#103—681no—
439#103C2×Dic1714yes—
440#103C2×Dic1714yes—
441#103C2×Dic1714yes—
442#103C2×Dic1714yes—
443C22⋊C8—342no—
444#97—681no—
445C4×C8—681no—
446#95—171no—
447#95—171no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458C17#2258111yescharacteristic, complemented, Sylow
459C24D4×D1711yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—24no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8#6325212yes—
476C2×Q8#6325212yes—
477C2×Q8C22×Dic1712yes—
478C2×Q8C22×Dic1712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—171no—
491C22×C4—171no—
492C22×C4—171no—
493C22×C4—171no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4D4×D1711yescharacteristic
499C22×C4#6325412yes—
500C22×C4C22×Dic1712yes—
501C22×C4C23×D1711yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8C22×Dic1712yes—
514C2×C8C22×Dic1712yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4C22×Dic1712yes—
530C4⋊C4C22×Dic1712yes—
531C22⋊C4—24no—
532C22⋊C4—342no—
533C22⋊C4—681no—
534C22⋊C4—682no—
535C22⋊C4—1361no—
536C42—24no—
537C42—341no—
538C42—341no—
539C42—341no—
540C42—341no—
541C42—682no—
542C42—682no—
543C42—1361no—
544C42—1361no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23D4×Dic1712yes—
552C23C2×D4×D1711yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1058761214yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—172no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4D4×Dic1712yes—
606C2×C4C2×#6325211yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4C2×#6325411yescharacteristic
609C2×C4#1058769111yescharacteristic
610C2×C4#1058769111yescharacteristic
611C2×C4C23×Dic1711yescharacteristic
612C8—22no—
613C8—22no—
614C8—681no—
615C8—681no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×D4×Dic1711yescharacteristic
624C22#1176344911yescharacteristic
625C22#1176344911yescharacteristic
626C22#137×D1711yescharacteristic
627C22#1176360811yescharacteristic
628C22#1176361811yescharacteristic
629C22#1176362511yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—342no—
636C4—681no—
637C4—681no—
638C4—681no—
639C4—1361no—
640C4—1361no—
641C4—1361no—
642C4—1361no—
643C4C2×#1058761211yescharacteristic
644C4#1176328411yescharacteristic
645C4#1176348511yescharacteristic
646C4#1176348511yescharacteristic
647C2—42no—
648C2#1453163311yescharacteristic
649C2#137×Dic1711yescharacteristic
650C2#1453762811yescharacteristic
651C2#1454085111yescharacteristic
652C2#1454095611yescharacteristic
653C2#1454103311yescharacteristic
654C2#1454106311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#419C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterC2×#419D17
hypocenterC17#22581
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #14529476C2semidirect
4. #14529462C2semidirect
5. #14529427C2semidirect
6. #14529420C2semidirect
7. #14529054C2semidirect
8. #14529040C2semidirect
12. #14528616C2semidirect
13. #103×Dic17C2semidirect
55. C17×#419C4semidirect
342. #419Dic17semidirect
458. C17#22581semidirect

Maximal subgroups

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

subgroupindex
#145316772
Q8×C22⋊Dic172
#145294762
#145294622
#145294272
#145294202
#145290542
#145290402
#145288792
#145288172
#145287192
#145286162
#103×Dic172
#145285652
C2×C17×#4192
#2258117