Finite Group Databasebeta

← #17091441

Subgroups — #17091441

Order 4352.

994 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763089C211yesmaximal, characteristic, complemented
2C2×#11762805C211yesmaximal, characteristic
3#14532475C211yesmaximal, characteristic, complemented
4#14530332C211yesmaximal, characteristic, complemented
5#14530179C211yesmaximal, characteristic, complemented
6#14530179C211yesmaximal, characteristic, complemented
7#14529925C211yesmaximal, characteristic, complemented
8#14529045C211yesmaximal, characteristic, complemented
9#14529031C211yesmaximal, characteristic, complemented
10#14528981C211yesmaximal, characteristic, complemented
11#14528964C211yesmaximal, characteristic, complemented
12#14528811C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic, complemented
14#14528724C211yesmaximal, characteristic
15C17×#2339C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763089—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26#11762805—24no—
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587503C2211yescharacteristic
35C2×C22⋊#3622C2211yescharacteristic
36#11762732—21no—
37#11762721—21no—
38#11762713—21no—
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762696—21no—
42#11762675C2211yescharacteristic
43#11762669C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C22×#41×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C17×#391C2211yescharacteristic
54C17×#391C2211yescharacteristic
55C17×#386—21no—
56C17×#386—21no—
57C17×#381—21no—
58C17×#380C2211yescharacteristic
59C4×C22⋊C4×C17C2211yescharacteristic
60C2×C17×#95—21no—
61C2×#41×C17—22no—
62C2×#41×C17—22no—
63C2×#41×C17—22no—
64C2×#41×C17—41no—
65C2×#41×C17—41no—
66C2×#41×C17Q814yescomplemented
67C22×Q8×C17C2311yescharacteristic
68C22×D4×C17—21no—
69C22×D4×C17C2311yescharacteristic
70C23×C4×C17—21no—
71C23×C4×C17—21no—
72C23×C4×C17—21no—
73C23×C4×C17C2311yescharacteristic
74C22×Dic34—21no—
75C4×D4×C17—22no—
76C4×D4×C17—22no—
77C4×D4×C17—41no—
78C4×D4×C17—41no—
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C22⋊Dic17—21no—
89C2×C22⋊Dic17—21no—
90#10587612—22no—
91#10587612—41no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C22×#3622—22no—
101C22×#3622—22no—
102C22×#3622—41no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587555—22no—
108#10587555—41no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587506—22no—
117#10587506—22no—
118#10587506—41no—
119#10587506—41no—
120#10587503—22no—
121#10587503—22no—
122#10587503—22no—
123#10587503D414yes—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129C22⋊#3622—22no—
130#10587479—41no—
131#10587478—22no—
132#10587478—22no—
133#10587478—41no—
134C4⋊#3622—22no—
135C4⋊#3622—41no—
136C24×C17—21no—
137C24×C17C2×D411yescharacteristic
138#41×C17—28no—
139#41×C17—44no—
140#41×C17—44no—
141#41×C17—44no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17C2×Q812yes—
148C2×D4×C17—24no—
149C2×D4×C17—21no—
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—42no—
153C2×D4×C17—42no—
154C2×D4×C17C2×D412yes—
155C2×D4×C17C2×Q812yes—
156C2×D4×C17#4112yes—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—21no—
160C22×C4×C17—21no—
161C22×C4×C17—21no—
162C22×C4×C17—24no—
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—42no—
166C22×C4×C17—42no—
167C22×C4×C17—42no—
168C22×C4×C17—42no—
169C22×C4×C17—42no—
170C22×C4×C17—42no—
171C22×C4×C17—81no—
172C22×C4×C17—81no—
173C22×C4×C17C2×D411yescharacteristic
174C22×C4×C17C2×D411yescharacteristic
175C22×C4×C17C2×D411yescharacteristic
176C22×C4×C17C2×D411yescharacteristic
177C22×C4×C17C2×D411yescharacteristic
178C22×C4×C17C2×Q812yes—
179C22×C4×C17#4112yes—
180C22×C4×C17C2411yescharacteristic
181C22×Dic17—21no—
182C22×Dic17—21no—
183C22×Dic17—21no—
184C22×Dic17—21no—
185C2×Dic34—42no—
186C2×Dic34—42no—
187C2×Dic34—81no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C22⋊C4×C17—22no—
193C22⋊C4×C17—41no—
194C22⋊C4×C17—42no—
195C22⋊C4×C17—42no—
196C22⋊C4×C17—81no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C22⋊Dic17—42no—
202C22⋊Dic17—81no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—42no—
211C4⋊Dic17C2×D412yes—
212#63225—42no—
213#63225—81no—
214C4×Dic17—42no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—44no—
227C2×#3622C2×D412yes—
228#25954—171nomaximal, Sylow
229C23×C17—22no—
230C23×C17—21no—
231C23×C17—22no—
232C23×C17—22no—
233C23×C17—21no—
234C23×C17—21no—
235C23×C17—21no—
236C23×C17—21no—
237C23×C17—21no—
238C23×C17—41no—
239C23×C17—41no—
240C23×C17—41no—
241C23×C17—41no—
242C23×C17C22⋊Q812yes—
243C23×C17C22×D411yescharacteristic
244Q8×C17—22no—
245Q8×C17—22no—
246Q8×C17—41no—
247Q8×C17—41no—
248D4×C17—22no—
249D4×C17—22no—
250D4×C17—22no—
251D4×C17—48no—
252D4×C17C22⋊Q814yes—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—22no—
258C2×C4×C17—22no—
259C2×C4×C17—22no—
260C2×C4×C17—22no—
261C2×C4×C17—22no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—21no—
267C2×C4×C17—21no—
268C2×C4×C17—21no—
269C2×C4×C17—21no—
270C2×C4×C17—21no—
271C2×C4×C17—48no—
272C2×C4×C17—41no—
273C2×C4×C17—41no—
274C2×C4×C17—41no—
275C2×C4×C17—41no—
276C2×C4×C17—41no—
277C2×C4×C17—41no—
278C2×C4×C17—41no—
279C2×C4×C17—41no—
280C2×C4×C17—84no—
281C2×C4×C17—84no—
282C2×C4×C17C22⋊Q812yes—
283C2×C4×C17#13512yes—
284C2×C4×C17C22×D411yescharacteristic
285C2×C4×C17C22×Q811yescharacteristic
286C2×C4×C17C2×#4111yescharacteristic
287C2×C4×C17C2×#4111yescharacteristic
288C2×C4×C17H(5,𝔽2)11yescharacteristic
289C2×C4×C17H(5,𝔽2)11yescharacteristic
290C2×Dic17—22no—
291C2×Dic17—41no—
292C2×Dic17—41no—
293C2×Dic17—41no—
294C2×Dic17—41no—
295C2×Dic17—41no—
296C2×Dic17—41no—
297C2×Dic17—41no—
298C2×Dic17—41no—
299C2×Dic17—41no—
300C2×Dic17—41no—
301C2×Dic17—41no—
302Dic34—84no—
303#3622—22no—
304#3622—41no—
305#3622—41no—
306#3622—41no—
307C2×#474—171no—
308C2×#416—171no—
309#2439—171no—
310#2339D1711yescharacteristic
311#2097—171no—
312#2068—171no—
313#2068—171no—
314#2024—171no—
315#1864—171no—
316#1862—171no—
317#1854—171no—
318#1850—171no—
319#1808—171no—
320#1804—171no—
321#1788—171no—
322C22×C17—22no—
323C22×C17—22no—
324C22×C17—22no—
325C22×C17—24no—
326C22×C17—42no—
327C22×C17—42no—
328C22×C17—41no—
329C22×C17—42no—
330C22×C17—42no—
331C22×C17—41no—
332C22×C17—41no—
333C22×C17—42no—
334C22×C17—42no—
335C22×C17—41no—
336C22×C17C2×C22⋊Q811yescharacteristic
337C22×C17D4211yescharacteristic
338C22×C17#54611yescharacteristic
339C22×C17C2×#13511yescharacteristic
340C22×C17#57511yescharacteristic
341C22×C17#57611yescharacteristic
342C22×C17#57811yescharacteristic
343C4×C17—22no—
344C4×C17—22no—
345C4×C17—42no—
346C4×C17—41no—
347C4×C17—41no—
348C4×C17—41no—
349C4×C17—81no—
350C4×C17—81no—
351C4×C17C2×C22⋊Q811yescharacteristic
352C4×C17#54311yescharacteristic
353C4×C17#54611yescharacteristic
354C4×C17#55111yescharacteristic
355Dic17—42no—
356Dic17—81no—
357Dic17—81no—
358Dic17—81no—
359C22×#41C2×D1711yescharacteristic
360C23×C8—171no—
361C2×C4⋊Q8—171no—
362C2×C4×D4C2×D1711yescharacteristic
363C2×C4×D4C2×D1711yescharacteristic
364C2×#117—171no—
365#474—344no—
366C2×#107—171no—
367C2×#107—171no—
368C2×#107—171no—
369C2×#106—171no—
370C2×C4⋊C8—171no—
371C2×C4⋊C8—171no—
372#416—344no—
373C2×#103—171no—
374C2×#103—171no—
375C2×#103—171no—
376C2×#103—171no—
377C2×#102—171no—
378C2×#102—171no—
379C2×#102—171no—
380C2×#102—171no—
381C2×C22⋊C8—171no—
382#396—341no—
383#393—341no—
384#391—341no—
385#391C2×D1711yescharacteristic
386#391C2×D1711yescharacteristic
387#389—171no—
388#389—171no—
389#387—341no—
390#386—21no—
391#386—21no—
392#382—171no—
393#381—21no—
394#380—171no—
395#380C2×D1711yescharacteristic
396C4×C22⋊C4C2×D1711yescharacteristic
397C2×#95—21no—
398#340—171no—
399#340—171no—
400#340—171no—
401#340—171no—
402#336—171no—
403#336—171no—
404C2×C17—42no—
405C2×C17—44no—
406C2×C17#243711yescharacteristic
407C2×C17#307611yescharacteristic
408C2×C17#308311yescharacteristic
409C2×C17#326911yescharacteristic
410C2×C17#328011yescharacteristic
411C2×C17#330211yescharacteristic
412C2×C17#331911yescharacteristic
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—41no—
417C2×#41—41no—
418C2×#41Dic3414yescomplemented
419C22×Q8—341no—
420C22×Q8C22×D1711yescharacteristic
421C22×D4—21no—
422C22×D4C22×D1711yescharacteristic
423C23×C4—21no—
424C23×C4—21no—
425C23×C4—21no—
426C23×C4C22×D1711yescharacteristic
427C22×C8—171no—
428C22×C8—171no—
429C22×C8—171no—
430C22×C8—171no—
431C22×C8—342no—
432C22×C8—342no—
433C22×C8—681no—
434C4⋊Q8—342no—
435C4⋊Q8—681no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—41no—
439C4×D4—41no—
440#117—342no—
441#117—681no—
442C2×C4⋊C4—171no—
443C2×C4⋊C4—171no—
444C2×C4⋊C4—171no—
445C2×C4⋊C4—171no—
446C2×C4⋊C4—341no—
447C2×C4⋊C4—341no—
448C2×C4⋊C4—341no—
449C2×C4⋊C4C22×D1711yescharacteristic
450C2×C4⋊C4C22×D1711yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—341no—
455C2×C22⋊C4—341no—
456C2×C22⋊C4C22⋊D1711yescharacteristic
457C2×C22⋊C4C22⋊D1711yescharacteristic
458C2×C42—341no—
459C2×C42C22×D1711yescharacteristic
460C2×C42C22×D1711yescharacteristic
461#107—342no—
462#107—342no—
463#107—681no—
464#106—681no—
465C4⋊C8—342no—
466C4⋊C8—681no—
467#103—342no—
468#103—342no—
469#103—681no—
470#103—681no—
471#102—174no—
472#102—342no—
473#102—342no—
474#102—342no—
475C22⋊C8—342no—
476#95—21no—
477#95—21no—
478#95—21no—
479#95—21no—
480#95—21no—
481#95—341no—
482#95—341no—
483#95—341no—
484#95—341no—
485#95—341no—
486#95C22⋊D1711yescharacteristic
487#95C22⋊D1711yescharacteristic
488C17#2595411yescharacteristic, complemented, Sylow
489C24—21no—
490C24C2×C22⋊D1711yescharacteristic
491#41—28no—
492#41—44no—
493#41—44no—
494#41—44no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—682no—
501C2×Q8—682no—
502C2×Q8—1361no—
503C2×Q8C2×Dic3412yes—
504C2×D4—24no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×Dic3412yes—
511C2×D4D4×D1712yes—
512C2×D4#6325412yes—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—24no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4—42no—
527C22×C4—81no—
528C22×C4—81no—
529C22×C4—341no—
530C22×C4—341no—
531C22×C4—341no—
532C22×C4—341no—
533C22×C4C2×Dic3412yes—
534C22×C4#6325212yes—
535C22×C4C2×C22⋊D1711yescharacteristic
536C22×C4C2×C22⋊D1711yescharacteristic
537C22×C4C2×C22⋊D1711yescharacteristic
538C22×C4C2×C22⋊D1711yescharacteristic
539C22×C4C2×C22⋊D1711yescharacteristic
540C22×C4C23×D1711yescharacteristic
541C2×C8—172no—
542C2×C8—341no—
543C2×C8—341no—
544C2×C8—341no—
545C2×C8—341no—
546C2×C8—341no—
547C2×C8—341no—
548C2×C8—341no—
549C2×C8—341no—
550C2×C8—341no—
551C2×C8—341no—
552C2×C8—341no—
553C2×C8—684no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—172no—
559C4⋊C4—341no—
560C4⋊C4—341no—
561C4⋊C4—341no—
562C4⋊C4—341no—
563C4⋊C4—341no—
564C4⋊C4—341no—
565C4⋊C4—341no—
566C4⋊C4—682no—
567C4⋊C4—682no—
568C4⋊C4—1361no—
569C22⋊C4—22no—
570C22⋊C4—41no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—81no—
574C22⋊C4—682no—
575C22⋊C4—1361no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—682no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—21no—
588C23—22no—
589C23—21no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23#1058761012yes—
595C23C22×C22⋊D1711yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—1364no—
601D4—22no—
602D4—22no—
603D4—22no—
604D4—48no—
605D4#1058756414yes—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—48no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—342no—
636C2×C4—681no—
637C2×C4—681no—
638C2×C4—681no—
639C2×C4—681no—
640C2×C4—681no—
641C2×C4—681no—
642C2×C4—681no—
643C2×C4—681no—
644C2×C4—681no—
645C2×C4—681no—
646C2×C4—681no—
647C2×C4#1058761012yes—
648C2×C4#1058763612yes—
649C2×C4C22×Dic3411yescharacteristic
650C2×C4C2×#6325211yescharacteristic
651C2×C4C2×D4×D1711yescharacteristic
652C2×C4C2×#6325411yescharacteristic
653C2×C4#1058769111yescharacteristic
654C2×C4#1058769111yescharacteristic
655C8—342no—
656C8—681no—
657C8—681no—
658C8—681no—
659C22—22no—
660C22—22no—
661C22—24no—
662C22—22no—
663C22—41no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—41no—
669C22—41no—
670C22—42no—
671C22—42no—
672C22—41no—
673C22C2×#1058761011yescharacteristic
674C22D4×C22⋊D1711yescharacteristic
675C22#1176345311yescharacteristic
676C22C2×#1058763611yescharacteristic
677C22#1176360711yescharacteristic
678C22#1176361211yescharacteristic
679C22#1176362411yescharacteristic
680C4—22no—
681C4—22no—
682C4—41no—
683C4—42no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—682no—
689C4—1361no—
690C4—1361no—
691C4—1361no—
692C4C2×#1058756411yescharacteristic
693C4#1176343511yescharacteristic
694C4#1176344911yescharacteristic
695C4#1176348511yescharacteristic
696C2—42no—
697C2—44no—
698C2#1453244511yescharacteristic
699C2#1453857211yescharacteristic
700C2#1453863511yescharacteristic
701C2#1454033511yescharacteristic
702C2#1454046511yescharacteristic
703C2#1454070111yescharacteristic
704C2#1454094011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2339C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
hypercenter#2339D17
hypocenterC17#25954
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2339

Complements

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

NH ≅ G/Ntype
1. C2×#11763089C2semidirect
3. #14532475C2semidirect
4. #14530332C2semidirect
5. #14530179C2semidirect
6. #14530179C2semidirect
7. #14529925C2semidirect
8. #14529045C2semidirect
9. #14529031C2semidirect
10. #14528981C2semidirect
11. #14528964C2semidirect
12. #14528811C2semidirect
13. #14528794C2semidirect
66. C2×#41×C17Q8semidirect
418. C2×#41Dic34semidirect
488. C17#25954semidirect

Maximal subgroups

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

subgroupindex
C2×#117630892
C2×#117628052
#145324752
#145303322
#145301792
#145299252
#145290452
#145290312
#145289812
#145289642
#145288112
#145287942
#145287242
C17×#23392
#2595417