Finite Group Databasebeta

← #17091396

Subgroups — #17091396

Order 4352.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763094C211yesmaximal, characteristic, complemented
2C2×#11762805C211yesmaximal, characteristic
3#14532475C211yesmaximal, characteristic, complemented
4#14530305C211yesmaximal, characteristic, complemented
5#14530158C211yesmaximal, characteristic, complemented
6#14530158C211yesmaximal, characteristic, complemented
7#14529920C211yesmaximal, characteristic, complemented
8#14529052C211yesmaximal, characteristic, complemented
9#14529038C211yesmaximal, characteristic, complemented
10#14528981C211yesmaximal, characteristic, complemented
11#14528964C211yesmaximal, characteristic, complemented
12#14528807C211yesmaximal, characteristic, complemented
13#14528799C211yesmaximal, characteristic, complemented
14#14528724C211yesmaximal, characteristic
15C17×#2341C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763094—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
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×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C17×#390—21no—
54C17×#390—21no—
55C17×#390C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#380—21no—
58C17×#380C2211yescharacteristic
59C4×C22⋊C4×C17—21no—
60C4×C22⋊C4×C17C2211yescharacteristic
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×C17C2311yescharacteristic
72C22×Dic34—21no—
73C4×Q8×C17—22no—
74C4×Q8×C17—22no—
75C4×Q8×C17—41no—
76C4×Q8×C17—41no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
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×C17—21no—
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17D411yescharacteristic
87C2×C22⋊C4×C17D411yescharacteristic
88C2×C42×C17—21no—
89C2×C42×C17—21no—
90C2×C42×C17C2311yescharacteristic
91C2×C42×C17C2311yescharacteristic
92C2×C22⋊Dic17—21no—
93C2×C22⋊Dic17—21no—
94#10587612—22no—
95#10587612—41no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×#63225—21no—
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C22×#3622—22no—
105C22×#3622—22no—
106C22×#3622—41no—
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111#10587555—22no—
112#10587555—41no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587506—22no—
117#10587506—22no—
118#10587506—22no—
119#10587506D414yes—
120#10587503—22no—
121#10587503—22no—
122#10587503—41no—
123#10587503—41no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129C22⋊#3622—22no—
130#10587479—22no—
131#10587479—22no—
132#10587479—41no—
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×C17C2×D412yes—
146C2×Q8×C17C2×Q812yes—
147C2×Q8×C17#4112yes—
148C2×D4×C17—24no—
149C2×D4×C17—21no—
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—21no—
153C2×D4×C17—21no—
154C2×D4×C17—42no—
155C2×D4×C17—42no—
156C2×D4×C17C2×Q812yes—
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×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2×D411yescharacteristic
172C22×C4×C17C2×Q812yes—
173C22×C4×C17#4112yes—
174C22×C4×C17C2411yescharacteristic
175C22×Dic17—21no—
176C22×Dic17—21no—
177C22×Dic17—21no—
178C22×Dic17—21no—
179C2×Dic34—42no—
180C2×Dic34—42no—
181C2×Dic34—81no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—42no—
187C4⋊C4×C17—42no—
188C22⋊C4×C17—22no—
189C22⋊C4×C17—41no—
190C22⋊C4×C17—42no—
191C22⋊C4×C17—42no—
192C22⋊C4×C17—81no—
193C22⋊C4×C17—81no—
194C22⋊C4×C17—81no—
195C42×C17—21no—
196C42×C17—21no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—42no—
200C42×C17—42no—
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#25951—171nomaximal, Sylow
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17—21no—
235C23×C17—22no—
236C23×C17—41no—
237C23×C17—41no—
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—22no—
247Q8×C17C22⋊Q814yes—
248D4×C17—22no—
249D4×C17—22no—
250D4×C17—48no—
251D4×C17—41no—
252D4×C17—41no—
253C2×C4×C17—22no—
254C2×C4×C17—21no—
255C2×C4×C17—22no—
256C2×C4×C17—21no—
257C2×C4×C17—22no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—22no—
261C2×C4×C17—22no—
262C2×C4×C17—22no—
263C2×C4×C17—22no—
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—21no—
272C2×C4×C17—21no—
273C2×C4×C17—48no—
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×C17C22⋊Q812yes—
281C2×C4×C17#13512yes—
282C2×C4×C17C22×D411yescharacteristic
283C2×C4×C17C22×Q811yescharacteristic
284C2×C4×C17C2×#4111yescharacteristic
285C2×C4×C17C2×#4111yescharacteristic
286C2×C4×C17H(5,𝔽2)11yescharacteristic
287C2×C4×C17H(5,𝔽2)11yescharacteristic
288C2×Dic17—22no—
289C2×Dic17—41no—
290C2×Dic17—41no—
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—
300Dic34—84no—
301#3622—22no—
302#3622—41no—
303#3622—41no—
304#3622—41no—
305C2×#475—171no—
306C2×#416—171no—
307#2439—171no—
308#2341D1711yescharacteristic
309#2092—171no—
310#2065—171no—
311#2065—171no—
312#2023—171no—
313#1865—171no—
314#1863—171no—
315#1854—171no—
316#1850—171no—
317#1807—171no—
318#1805—171no—
319#1788—171no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—22no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17—41no—
326C22×C17—41no—
327C22×C17—42no—
328C22×C17—42no—
329C22×C17—42no—
330C22×C17—81no—
331C22×C17—81no—
332C22×C17—81no—
333C22×C17—81no—
334C22×C17C2×C22⋊Q811yescharacteristic
335C22×C17D4211yescharacteristic
336C22×C17#54611yescharacteristic
337C22×C17C2×#13511yescharacteristic
338C22×C17#57511yescharacteristic
339C22×C17#57711yescharacteristic
340C22×C17#57711yescharacteristic
341C4×C17—22no—
342C4×C17—24no—
343C4×C17—22no—
344C4×C17—41no—
345C4×C17—41no—
346C4×C17—41no—
347C4×C17—81no—
348C4×C17—81no—
349C4×C17C2×C22⋊Q811yescharacteristic
350C4×C17#54311yescharacteristic
351C4×C17#54611yescharacteristic
352C4×C17#55111yescharacteristic
353Dic17—42no—
354Dic17—81no—
355Dic17—81no—
356Dic17—81no—
357C22×#41C2×D1711yescharacteristic
358C23×C8—171no—
359C2×C4⋊Q8—171no—
360C2×C4×Q8C2×D1711yescharacteristic
361C2×C4×Q8C2×D1711yescharacteristic
362C2×#117—171no—
363#475—344no—
364C2×#107—171no—
365C2×#106—171no—
366C2×#106—171no—
367C2×#106—171no—
368C2×C4⋊C8—171no—
369C2×C4⋊C8—171no—
370#416—344no—
371C2×#103—171no—
372C2×#103—171no—
373C2×#103—171no—
374C2×#103—171no—
375C2×#102—171no—
376C2×#102—171no—
377C2×#102—171no—
378C2×#102—171no—
379C2×C22⋊C8—171no—
380#396—341no—
381#393—341no—
382#391—341no—
383#390—21no—
384#390—21no—
385#390C2×D1711yescharacteristic
386#390C2×D1711yescharacteristic
387#389—171no—
388#389—171no—
389#387—341no—
390#382—171no—
391#380—21no—
392#380—171no—
393#380C2×D1711yescharacteristic
394C4×C22⋊C4—21no—
395C4×C22⋊C4C2×D1711yescharacteristic
396#340—171no—
397#340—171no—
398#340—171no—
399#340—171no—
400#336—171no—
401#336—171no—
402C2×C17—42no—
403C2×C17—82no—
404C2×C17#243711yescharacteristic
405C2×C17#307611yescharacteristic
406C2×C17#308411yescharacteristic
407C2×C17#327111yescharacteristic
408C2×C17#328211yescharacteristic
409C2×C17#330411yescharacteristic
410C2×C17#332111yescharacteristic
411C2×#41—22no—
412C2×#41—22no—
413C2×#41—22no—
414C2×#41—41no—
415C2×#41—41no—
416C2×#41Dic3414yescomplemented
417C22×Q8—341no—
418C22×Q8C22×D1711yescharacteristic
419C22×D4—21no—
420C22×D4C22×D1711yescharacteristic
421C23×C4—21no—
422C23×C4C22×D1711yescharacteristic
423C22×C8—171no—
424C22×C8—171no—
425C22×C8—171no—
426C22×C8—171no—
427C22×C8—342no—
428C22×C8—342no—
429C22×C8—681no—
430C4⋊Q8—342no—
431C4⋊Q8—681no—
432C4×Q8—22no—
433C4×Q8—22no—
434C4×Q8—41no—
435C4×Q8—41no—
436#117—342no—
437#117—681no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—171no—
442C2×C4⋊C4—171no—
443C2×C4⋊C4—171no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4—341no—
446C2×C4⋊C4—341no—
447C2×C4⋊C4C22×D1711yescharacteristic
448C2×C4⋊C4C22×D1711yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
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—21no—
459C2×C42—21no—
460C2×C42—341no—
461C2×C42C22×D1711yescharacteristic
462C2×C42C22×D1711yescharacteristic
463#107—681no—
464#106—342no—
465#106—342no—
466#106—681no—
467C4⋊C8—342no—
468C4⋊C8—681no—
469#103—174no—
470#103—342no—
471#103—342no—
472#103—342no—
473#102—342no—
474#102—342no—
475#102—681no—
476#102—681no—
477C22⋊C8—342no—
478#95—21no—
479#95—341no—
480#95—341no—
481#95—341no—
482#95—341no—
483#95—341no—
484#95C22⋊D1711yescharacteristic
485#95C22⋊D1711yescharacteristic
486C17#2595111yescharacteristic, complemented, Sylow
487C24—21no—
488C24C2×C22⋊D1711yescharacteristic
489#41—28no—
490#41—44no—
491#41—44no—
492#41—44no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—682no—
497C2×Q8—682no—
498C2×Q8—1361no—
499C2×Q8C2×Dic3412yes—
500C2×Q8D4×D1712yes—
501C2×Q8#6325412yes—
502C2×D4—24no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×Dic3412yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—24no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4—341no—
525C22×C4C2×Dic3412yes—
526C22×C4#6325212yes—
527C22×C4C2×C22⋊D1711yescharacteristic
528C22×C4C2×C22⋊D1711yescharacteristic
529C22×C4C2×C22⋊D1711yescharacteristic
530C22×C4C2×C22⋊D1711yescharacteristic
531C22×C4C2×C22⋊D1711yescharacteristic
532C22×C4C23×D1711yescharacteristic
533C2×C8—172no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C2×C8—341no—
543C2×C8—341no—
544C2×C8—341no—
545C2×C8—684no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—172no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C4⋊C4—341no—
558C4⋊C4—341no—
559C4⋊C4—341no—
560C4⋊C4—682no—
561C4⋊C4—682no—
562C4⋊C4—1361no—
563C22⋊C4—22no—
564C22⋊C4—41no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—81no—
568C22⋊C4—81no—
569C22⋊C4—81no—
570C22⋊C4—682no—
571C22⋊C4—1361no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—42no—
577C42—42no—
578C42—682no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#1058761012yes—
593C23C22×C22⋊D1711yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—1364no—
598Q8#1058756414yes—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
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—342no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—681no—
636C2×C4—681no—
637C2×C4—681no—
638C2×C4—681no—
639C2×C4—681no—
640C2×C4—681no—
641C2×C4—681no—
642C2×C4—681no—
643C2×C4#1058761012yes—
644C2×C4#1058763612yes—
645C2×C4C22×Dic3411yescharacteristic
646C2×C4C2×#6325211yescharacteristic
647C2×C4C2×D4×D1711yescharacteristic
648C2×C4C2×#6325411yescharacteristic
649C2×C4#1058769111yescharacteristic
650C2×C4#1058769111yescharacteristic
651C8—342no—
652C8—681no—
653C8—681no—
654C8—681no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C2×#1058761011yescharacteristic
670C22D4×C22⋊D1711yescharacteristic
671C22#1176345311yescharacteristic
672C22C2×#1058763611yescharacteristic
673C22#1176360711yescharacteristic
674C22#1176361711yescharacteristic
675C22#1176362011yescharacteristic
676C4—22no—
677C4—24no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—81no—
683C4—81no—
684C4—682no—
685C4—1361no—
686C4—1361no—
687C4—1361no—
688C4C2×#1058756411yescharacteristic
689C4#1176343511yescharacteristic
690C4#1176344911yescharacteristic
691C4#1176348511yescharacteristic
692C2—42no—
693C2—82no—
694C2#1453244511yescharacteristic
695C2#1453857211yescharacteristic
696C2#1453864611yescharacteristic
697C2#1454036511yescharacteristic
698C2#1454049211yescharacteristic
699C2#1454072311yescharacteristic
700C2#1454097011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2341C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
hypercenter#2341D17
hypocenterC17#25951
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
1. C2×#11763094C2semidirect
3. #14532475C2semidirect
4. #14530305C2semidirect
5. #14530158C2semidirect
6. #14530158C2semidirect
7. #14529920C2semidirect
8. #14529052C2semidirect
9. #14529038C2semidirect
10. #14528981C2semidirect
11. #14528964C2semidirect
12. #14528807C2semidirect
13. #14528799C2semidirect
66. C2×#41×C17Q8semidirect
416. C2×#41Dic34semidirect
486. C17#25951semidirect

Maximal subgroups

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

subgroupindex
C2×#117630942
C2×#117628052
#145324752
#145303052
#145301582
#145299202
#145290522
#145290382
#145289812
#145289642
#145288072
#145287992
#145287242
C17×#23412
#2595117