Finite Group Databasebeta

← #17097884

Subgroups — #17097884

Order 4352.

802 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532475C211yesmaximal, characteristic, complemented
2#14530352C211yesmaximal, characteristic, complemented
3#14530285C211yesmaximal, characteristic, complemented
4#14530158C211yesmaximal, characteristic, complemented
5#14529920C211yesmaximal, characteristic, complemented
6#14529736C211yesmaximal, characteristic, complemented
7#14529376C211yesmaximal, characteristic, complemented
8#14529213C211yesmaximal, characteristic
9#14529107C211yesmaximal, characteristic
10#14529031C211yesmaximal, characteristic, complemented
11#14528983C211yesmaximal, characteristic, complemented
12#14528964C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic, complemented
14#14528500C211yesmaximal, characteristic, complemented
15C17×#2363C211yesmaximal, characteristic
16C23×#3622C2211yescharacteristic
17C2×#10587555C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28#11762732—21no—
29#11762721—21no—
30#11762713—21no—
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762696—21no—
34#11762675C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#121C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C2×C17×#117C2211yescharacteristic
51C17×#390—21no—
52C17×#390C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#386—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C22×Dic34—21no—
64C17×#121—22no—
65C17×#121Q814yescomplemented
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C17×#117—22no—
69C17×#117—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87#10587612—22no—
88#10587612—41no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C22×#3622—22no—
98C22×#3622—41no—
99C22×#3622—41no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587555—22no—
105#10587555—41no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95D411yescharacteristic
109C17×#95D411yescharacteristic
110#10587503—22no—
111#10587503—41no—
112#10587503—41no—
113#10587503—41no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119C22⋊#3622—22no—
120#10587479—41no—
121#10587478—22no—
122C4⋊#3622—41no—
123C4⋊#3622—41no—
124C24×C17—21no—
125C24×C17C2×D411yescharacteristic
126C2×D4×C17—24no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—81no—
133C2×D4×C17—81no—
134C2×D4×C17C2×Q812yes—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—81no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×Q812yes—
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C2×Dic34—42no—
158C2×Dic34—42no—
159C2×Dic34—81no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—81no—
165C4⋊C4×C17C2×Q812yes—
166C4⋊C4×C17#4112yes—
167C22⋊C4×C17—24no—
168C22⋊C4×C17—22no—
169C22⋊C4×C17—41no—
170C22⋊C4×C17—42no—
171C22⋊C4×C17—42no—
172C22⋊C4×C17—42no—
173C22⋊C4×C17—81no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—42no—
179C42×C17—81no—
180C22⋊Dic17—42no—
181C22⋊Dic17—81no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—42no—
190C4⋊Dic17#4112yes—
191#63225—42no—
192#63225—81no—
193C4×Dic17—42no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—44no—
207#26409—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—22no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17C22⋊Q812yes—
222C23×C17C22×D411yescharacteristic
223D4×C17—22no—
224D4×C17—22no—
225D4×C17—41no—
226D4×C17—41no—
227D4×C17—84no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—41no—
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×C17—84no—
255C2×C4×C17C22⋊Q812yes—
256C2×C4×C17#13512yes—
257C2×C4×C17C22×Q811yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C17H(5,𝔽2)11yescharacteristic
263C2×Dic17—22no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275Dic34—84no—
276#3622—41no—
277#3622—41no—
278#3622—41no—
279#3622—41no—
280#2439—171no—
281#2363D1711yescharacteristic
282#2101—171no—
283#2088—171no—
284#2065—171no—
285#2023—171no—
286#1988—171no—
287#1921—171no—
288#1893—171no—
289#1875—171no—
290#1862—171no—
291#1854—171no—
292#1850—171no—
293#1804—171no—
294#1735—171no—
295C22×C17—22no—
296C22×C17—22no—
297C22×C17—22no—
298C22×C17—41no—
299C22×C17—41no—
300C22×C17—41no—
301C22×C17—41no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—81no—
306C22×C17—81no—
307C22×C17—81no—
308C22×C17—81no—
309C22×C17C2×C22⋊Q811yescharacteristic
310C22×C17#54611yescharacteristic
311C22×C17#54611yescharacteristic
312C22×C17C2×#13511yescharacteristic
313C22×C17#57511yescharacteristic
314C22×C17#57611yescharacteristic
315C22×C17#57711yescharacteristic
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—42no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—42no—
322C4×C17—41no—
323C4×C17—81no—
324C4×C17#52911yescharacteristic
325C4×C17#53811yescharacteristic
326C4×C17#55411yescharacteristic
327C4×C17#55411yescharacteristic
328Dic17—42no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C23×C8—171no—
333C2×C4⋊Q8—171no—
334C2×#121C2×D1711yescharacteristic
335C2×C4×D4C2×D1711yescharacteristic
336C2×#117—171no—
337C2×#117C2×D1711yescharacteristic
338C2×#107—171no—
339C2×#106—171no—
340C2×C4⋊C8—171no—
341C2×C4⋊C8—171no—
342C2×#102—171no—
343C2×#102—171no—
344C2×#102—171no—
345C2×#102—171no—
346C2×C22⋊C8—171no—
347#396—341no—
348#393—341no—
349#391—341no—
350#390—21no—
351#390C2×D1711yescharacteristic
352#389—171no—
353#389—171no—
354#389C2×D1711yescharacteristic
355#388—21no—
356#387—341no—
357#386—21no—
358#382—171no—
359#380—171no—
360C4×C4⋊C4C2×D1711yescharacteristic
361C4×C22⋊C4—21no—
362C4×C22⋊C4C2×D1711yescharacteristic
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#336—171no—
374#336—171no—
375C2×C17—42no—
376C2×C17—82no—
377C2×C17#245011yescharacteristic
378C2×C17#307611yescharacteristic
379C2×C17#308111yescharacteristic
380C2×C17#329711yescharacteristic
381C2×C17#330411yescharacteristic
382C2×C17#331911yescharacteristic
383C2×C17#332111yescharacteristic
384C22×Q8—341no—
385C22×D4—21no—
386C22×D4C22×D1711yescharacteristic
387C23×C4—21no—
388C23×C4C22×D1711yescharacteristic
389C22×C8—171no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8—171no—
393C22×C8—342no—
394C22×C8—681no—
395C22×C8—681no—
396C4⋊Q8—342no—
397C4⋊Q8—681no—
398#121—22no—
399#121Dic3414yescomplemented
400C4×D4—22no—
401C4×D4—41no—
402#117—22no—
403#117—41no—
404#117—342no—
405#117—681no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4C22×D1711yescharacteristic
415C2×C4⋊C4C22×D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4C22⋊D1711yescharacteristic
425C2×C22⋊C4C22⋊D1711yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—341no—
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431#107—342no—
432#106—681no—
433C4⋊C8—681no—
434C4⋊C8—681no—
435#102—342no—
436#102—681no—
437#102—681no—
438#102—681no—
439C22⋊C8—342no—
440#95—21no—
441#95—21no—
442#95—341no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95C22⋊D1711yescharacteristic
448#95C22⋊D1711yescharacteristic
449C17#2640911yescharacteristic, complemented, Sylow
450C24—21no—
451C24C2×C22⋊D1711yescharacteristic
452C2×Q8—682no—
453C2×Q8—682no—
454C2×Q8—1361no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×Dic3412yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4C2×Dic3412yes—
479C22×C4#6325212yes—
480C22×C4C2×C22⋊D1711yescharacteristic
481C22×C4C2×C22⋊D1711yescharacteristic
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—684no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—81no—
504C4⋊C4—172no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—682no—
513C4⋊C4—682no—
514C4⋊C4—1361no—
515C4⋊C4C2×Dic3412yes—
516C4⋊C4#6325412yes—
517C22⋊C4—24no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—682no—
525C22⋊C4—1361no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—81no—
532C42—682no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23#1058761012yes—
547C23C22×C22⋊D1711yescharacteristic
548Q8—1364no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—342no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4#1058761012yes—
594C2×C4#1058763612yes—
595C2×C4C22×Dic3411yescharacteristic
596C2×C4C2×#6325211yescharacteristic
597C2×C4C2×#6325411yescharacteristic
598C2×C4#1058769111yescharacteristic
599C2×C4#41×D1711yescharacteristic
600C2×C4#1058769611yescharacteristic
601C8—681no—
602C8—681no—
603C8—681no—
604C8—681no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1058761011yescharacteristic
620C22#1176344711yescharacteristic
621C22#1176345311yescharacteristic
622C22C2×#1058763611yescharacteristic
623C22#1176360711yescharacteristic
624C22#1176361211yescharacteristic
625C22#1176362011yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—682no—
635C4—1361no—
636C4—1361no—
637C4—1361no—
638C4#1176333911yescharacteristic
639C4#1176339211yescharacteristic
640C4#1176351611yescharacteristic
641C4#1176351811yescharacteristic
642C2—42no—
643C2—82no—
644C2#1453260211yescharacteristic
645C2#1453857211yescharacteristic
646C2#1453861511yescharacteristic
647C2#1454062811yescharacteristic
648C2#1454072311yescharacteristic
649C2#1454094011yescharacteristic
650C2#1454096411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2363C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2363D17
hypocenterC17#26409
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #14532475C2semidirect
2. #14530352C2semidirect
3. #14530285C2semidirect
4. #14530158C2semidirect
5. #14529920C2semidirect
6. #14529736C2semidirect
7. #14529376C2semidirect
10. #14529031C2semidirect
11. #14528983C2semidirect
12. #14528964C2semidirect
13. #14528794C2semidirect
14. #14528500C2semidirect
65. C17×#121Q8semidirect
399. #121Dic34semidirect
449. C17#26409semidirect

Maximal subgroups

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

subgroupindex
#145324752
#145303522
#145302852
#145301582
#145299202
#145297362
#145293762
#145292132
#145291072
#145290312
#145289832
#145289642
#145287942
#145285002
C17×#23632
#2640917