Finite Group Databasebeta

← #17097888

Subgroups — #17097888

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#14531820C211yesmaximal, characteristic
3#14530353C211yesmaximal, characteristic, complemented
4#14530282C211yesmaximal, characteristic, complemented
5#14530157C211yesmaximal, characteristic, complemented
6#14529921C211yesmaximal, characteristic, complemented
7#14529735C211yesmaximal, characteristic, complemented
8#14529377C211yesmaximal, characteristic, complemented
9#14529212C211yesmaximal, characteristic
10#14529104C211yesmaximal, characteristic
11#14529034C211yesmaximal, characteristic, complemented
12#14528985C211yesmaximal, characteristic, complemented
13#14528795C211yesmaximal, characteristic, complemented
14#14528501C211yesmaximal, characteristic, complemented
15C17×#1850C211yesmaximal, characteristic, complemented
16C2×#10587555C2211yescharacteristic
17C2×#10587616C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C2×#10587612C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587492C2211yescharacteristic
24C2×#10587492C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27#11762732—21no—
28#11762721—21no—
29#11762713—21no—
30#11762711C2211yescharacteristic
31#11762709—21no—
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762698—21no—
36#11762696—21no—
37#11762685—21no—
38#11762675C2211yescharacteristic
39#11762669C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41#11762656—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762572C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52C23×C8×C17C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17—21no—
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C22×Dic34—21no—
64C22×C8×C17—22no—
65C22×C8×C17—41no—
66C22×C8×C17—41no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78#10587616—22no—
79#10587616Q814yescomplemented
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84#10587612—22no—
85#10587612—22no—
86#10587612—41no—
87#10587612—41no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C4×D68—22no—
103C4×D68—41no—
104#10587555—22no—
105#10587555—41no—
106C17×C4⋊C8—41no—
107C17×C4⋊C8—41no—
108C17×C22⋊C8—22no—
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587492—22no—
119#10587492—41no—
120#10587492—41no—
121#10587492—41no—
122#10587489—22no—
123#10587488—41no—
124C24×C17C2×D411yescharacteristic
125C23×D17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×Q812yes—
136C22×C4×C17#4112yes—
137C22×C4×C17C2411yescharacteristic
138C2×C22⋊D17—24no—
139C2×C22⋊D17—81no—
140C2×C22⋊D17—81no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C2×D68—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68C2×Q812yes—
154C2×C4×D17—42no—
155C2×C4×D17—42no—
156C2×C4×D17—81no—
157C2×Dic34—42no—
158C2×Dic34—42no—
159C2×Dic34—81no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—44no—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—41no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C22⋊Dic17—42no—
180C22⋊Dic17—42no—
181C22⋊Dic17—81no—
182#63227—24no—
183#63227—42no—
184#63227—42no—
185#63227—81no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—42no—
198C4⋊Dic17C2×Q812yes—
199C4⋊Dic17#4112yes—
200C4⋊Dic17#4112yes—
201#63225—42no—
202#63225—81no—
203#63225—81no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206C4×Dic17—81no—
207#26409—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17C22⋊Q812yes—
215C23×C17C22×D411yescharacteristic
216C22×D17—22no—
217C22×D17—41no—
218C22×D17—41no—
219C22×D17—41no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17C22⋊Q812yes—
241C2×C4×C17#13512yes—
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C17H(5,𝔽2)11yescharacteristic
248C22⋊D17—84no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270D68—22no—
271D68—22no—
272D68—41no—
273D68—41no—
274C4×D17—84no—
275Dic34—84no—
276C8×C17—41no—
277C8×C17—41no—
278C8×C17—41no—
279C8×C17—41no—
280#2439—171no—
281#2363—171no—
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#1850D1711yescharacteristic, complemented
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×C17C2×C22⋊Q811yescharacteristic
303C22×C17#54611yescharacteristic
304C22×C17#54611yescharacteristic
305C22×C17C2×#13511yescharacteristic
306C22×C17#57511yescharacteristic
307C22×C17#57611yescharacteristic
308C22×C17#57711yescharacteristic
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—81no—
313C2×D17—81no—
314C2×D17—81no—
315C2×D17—81no—
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17#52911yescharacteristic
322C4×C17#53811yescharacteristic
323C4×C17#55411yescharacteristic
324C4×C17#55411yescharacteristic
325Dic17—42no—
326Dic17—42no—
327Dic17—42no—
328Dic17—81no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C23×C8C2×D1711yescharacteristic
333C2×C4⋊Q8—171no—
334C2×#121—171no—
335C2×C4×D4—171no—
336C2×#117—171no—
337C2×#117—171no—
338C2×#107—171no—
339C2×#106—171no—
340C2×C4⋊C8C2×D1711yescharacteristic
341C2×C4⋊C8C2×D1711yescharacteristic
342C2×#102—171no—
343C2×#102—171no—
344C2×#102—171no—
345C2×#102—171no—
346C2×C22⋊C8C2×D1711yescharacteristic
347#396—341no—
348#393—341no—
349#391—341no—
350#390—171no—
351#390—341no—
352#389—171no—
353#389—171no—
354#389—171no—
355#388—341no—
356#387—341no—
357#386—341no—
358#382—171no—
359#380—171no—
360C4×C4⋊C4—171no—
361C4×C22⋊C4—341no—
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#336C2×D1711yescharacteristic
374#336C2×D1711yescharacteristic
375C2×C17—42no—
376C2×C17#245011yescharacteristic
377C2×C17#307611yescharacteristic
378C2×C17#308111yescharacteristic
379C2×C17#329711yescharacteristic
380C2×C17#330411yescharacteristic
381C2×C17#331911yescharacteristic
382C2×C17#332111yescharacteristic
383D17—82no—
384C22×Q8—341no—
385C22×D4—171no—
386C22×D4—341no—
387C23×C4—341no—
388C23×C4C22×D1711yescharacteristic
389C22×C8—22no—
390C22×C8—41no—
391C22×C8—41no—
392C22×C8C22×D1711yescharacteristic
393C22×C8C22×D1711yescharacteristic
394C22×C8C22×D1711yescharacteristic
395C22×C8C22×D1711yescharacteristic
396C4⋊Q8—342no—
397C4⋊Q8—681no—
398#121—174no—
399#121—342no—
400C4×D4—342no—
401C4×D4—681no—
402#117—342no—
403#117—342no—
404#117—681no—
405#117—681no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—341no—
419C2×C22⋊C4—341no—
420C2×C22⋊C4—341no—
421C2×C22⋊C4—341no—
422C2×C22⋊C4—341no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4D6811yescharacteristic
425C2×C22⋊C4D6811yescharacteristic
426C2×C42—341no—
427C2×C42—341no—
428C2×C42—341no—
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431#107—342no—
432#106—681no—
433C4⋊C8—41no—
434C4⋊C8—41no—
435#102—342no—
436#102—681no—
437#102—681no—
438#102—681no—
439C22⋊C8—22no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447#95D6811yescharacteristic
448#95D6811yescharacteristic
449C17#2640911yescharacteristic, complemented, Sylow
450C24—341no—
451C24C2×D6811yescharacteristic
452C2×Q8—682no—
453C2×Q8—682no—
454C2×Q8—1361no—
455C2×D4—172no—
456C2×D4—344no—
457C2×D4—341no—
458C2×D4—341no—
459C2×D4—341no—
460C2×D4—341no—
461C2×D4—341no—
462C2×D4—1361no—
463C2×D4—1361no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—682no—
476C22×C4—682no—
477C22×C4—1361no—
478C22×C4C2×D6811yescharacteristic
479C22×C4C2×D6811yescharacteristic
480C22×C4C2×D6811yescharacteristic
481C22×C4C2×D6811yescharacteristic
482C22×C4C2×D6811yescharacteristic
483C22×C4#6325412yes—
484C22×C4Q8×D1712yes—
485C22×C4C23×D1711yescharacteristic
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—44no—
499C4⋊C4—172no—
500C4⋊C4—172no—
501C4⋊C4—172no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—682no—
514C4⋊C4—682no—
515C4⋊C4—1361no—
516C4⋊C4—1361no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—344no—
520C22⋊C4—682no—
521C22⋊C4—682no—
522C22⋊C4—682no—
523C22⋊C4—682no—
524C22⋊C4—1361no—
525C22⋊C4—1361no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—682no—
531C42—682no—
532C42—1361no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—342no—
540C23—681no—
541C23—681no—
542C23—681no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23#1058758312yes—
547C23C22×D6811yescharacteristic
548Q8—1364no—
549D4—342no—
550D4—342no—
551D4—681no—
552D4—681no—
553D4—1364no—
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—21no—
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—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
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—1364no—
593C2×C4#1058758312yes—
594C2×C4#1058759012yes—
595C2×C4C2×#6325411yescharacteristic
596C2×C4#1058769111yescharacteristic
597C2×C4#1058769111yescharacteristic
598C2×C4C2×Q8×D1711yescharacteristic
599C2×C4#41×D1711yescharacteristic
600C2×C4#41×D1711yescharacteristic
601C8—41no—
602C8—41no—
603C8—41no—
604C8—41no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—682no—
613C22—682no—
614C22—682no—
615C22—1361no—
616C22—1361no—
617C22—1361no—
618C22—1361no—
619C22C2×#1058758311yescharacteristic
620C22#1176345411yescharacteristic
621C22#1176345411yescharacteristic
622C22C2×#1058759011yescharacteristic
623C22#1176360911yescharacteristic
624C22#1176361411yescharacteristic
625C22#1176362111yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—682no—
632C4—682no—
633C4—682no—
634C4—1361no—
635C4—1361no—
636C4—1361no—
637C4—1361no—
638C4#1176333611yescharacteristic
639C4#1176339411yescharacteristic
640C4#1176351311yescharacteristic
641C4#1176351311yescharacteristic
642C2—42no—
643C2—1362no—
644C2#1453260611yescharacteristic
645C2#1453857411yescharacteristic
646C2#1453861611yescharacteristic
647C2#1454063611yescharacteristic
648C2#1454072411yescharacteristic
649C2#1454094211yescharacteristic
650C2#1454097211yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1850C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#1850D17
hypocenterC17#26409
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #14532475C2semidirect
3. #14530353C2semidirect
4. #14530282C2semidirect
5. #14530157C2semidirect
6. #14529921C2semidirect
7. #14529735C2semidirect
8. #14529377C2semidirect
11. #14529034C2semidirect
12. #14528985C2semidirect
13. #14528795C2semidirect
14. #14528501C2semidirect
15. C17×#1850C2semidirect
79. #10587616Q8semidirect
292. #1850D17semidirect
449. C17#26409semidirect

Maximal subgroups

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

subgroupindex
#145324752
#145318202
#145303532
#145302822
#145301572
#145299212
#145297352
#145293772
#145292122
#145291042
#145290342
#145289852
#145287952
#145285012
C17×#18502
#2640917