Finite Group Databasebeta

← #17084414

Subgroups — #17084414

Order 4352.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic
2C2×#11763028C211yesmaximal, characteristic
3#14533999C211yesmaximal, characteristic, complemented
4#14530202C211yesmaximal, characteristic, complemented
5#14530197C211yesmaximal, characteristic
6#14530106C211yesmaximal, characteristic, complemented
7#14530101C211yesmaximal, characteristic
8#14529732C211yesmaximal, characteristic, complemented
9#14529508C211yesmaximal, characteristic, complemented
10#14529125C211yesmaximal, characteristic, complemented
11#14529118C211yesmaximal, characteristic
12#14528799C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic
14#14528617C211yesmaximal, characteristic, complemented
15C17×#2367C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587555C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763168—24no—
21#11763028—24no—
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587474C2211yescharacteristic
34#11762749—21no—
35#11762749—21no—
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762714—21no—
39#11762714—21no—
40#11762675C2211yescharacteristic
41#11762675C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#124C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×D4×C17C2211yescharacteristic
53C17×#391C2211yescharacteristic
54C17×#390C2211yescharacteristic
55C17×#388—21no—
56C17×#388—21no—
57C17×#385—21no—
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C4×C22⋊C4×C17—21no—
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C22×Dic34—21no—
65C22×Dic34C2311yescharacteristic
66C17×#124—22no—
67C17×#124D414yes—
68C4×Q8×C17—22no—
69C4×Q8×C17—41no—
70C4×D4×C17—22no—
71C4×D4×C17—41no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17—21no—
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×#63231—22no—
88C2×#63231—22no—
89C2×#63231—41no—
90C2×#63229—22no—
91C2×#63229—22no—
92C2×#63229—41no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103#10587555—22no—
104#10587555—22no—
105C4×Dic34—22no—
106C4×Dic34—22no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
112#10587506—22no—
113#10587506—22no—
114#10587506—22no—
115#10587503—22no—
116#10587503—22no—
117#10587503—22no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587481—22no—
127#10587481—22no—
128C4⋊#3622—22no—
129C4⋊#3622—41no—
130C4⋊#3622—41no—
131#10587474—22no—
132C24×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17C2×D412yes—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17C2×D412yes—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—42no—
149C22×C4×C17—42no—
150C22×C4×C17—81no—
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2411yescharacteristic
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—42no—
168C2×Dic34—42no—
169C2×Dic34—42no—
170C2×Dic34#4112yes—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—41no—
175C4⋊C4×C17—41no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17#4112yes—
178C22⋊C4×C17—24no—
179C22⋊C4×C17—24no—
180C22⋊C4×C17—42no—
181C22⋊C4×C17—42no—
182C22⋊C4×C17—81no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—42no—
188C42×C17—81no—
189C42×C17C2×D412yes—
190C42×C17#4112yes—
191#63231—44no—
192#63229—44no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200#63225—42no—
201C4×Dic17—42no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214#25337—171nomaximal, Sylow
215C23×C17—22no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17C22×D411yescharacteristic
223Q8×C17—22no—
224Q8×C17—22no—
225Q8×C17—41no—
226Q8×C17—41no—
227D4×C17—22no—
228D4×C17—22no—
229D4×C17—41no—
230D4×C17—41no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—21no—
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—21no—
248C2×C4×C17—21no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—84no—
257C2×C4×C17#12112yes—
258C2×C4×C17#12112yes—
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17H(5,𝔽2)11yescharacteristic
264C2×C4×C172−1+411yescharacteristic
265C2×Dic17—22no—
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—
275C2×Dic17—41no—
276C2×Dic17—41no—
277Dic34—22no—
278Dic34—22no—
279Dic34—41no—
280Dic34—41no—
281Dic34—84no—
282#3622—41no—
283#3622—41no—
284#3622—41no—
285#3622—41no—
286C2×#489—171no—
287C2×#464—171no—
288#2568—171no—
289#2367D1711yescharacteristic
290#2074—171no—
291#2073—171no—
292#2056—171no—
293#2055—171no—
294#1987—171no—
295#1940—171no—
296#1878—171no—
297#1877—171no—
298#1805—171no—
299#1804—171no—
300#1763—171no—
301C22×C17—42no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—81no—
305C22×C17—81no—
306C22×C17—81no—
307C22×C17—81no—
308C22×C17C2×#12111yescharacteristic
309C22×C17#54611yescharacteristic
310C22×C17#54811yescharacteristic
311C22×C17#57511yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57811yescharacteristic
315C4×C17—22no—
316C4×C17—22no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—42no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—81no—
324C4×C17#52911yescharacteristic
325C4×C17#54211yescharacteristic
326C4×C17#54611yescharacteristic
327C4×C17#54711yescharacteristic
328Dic17—42no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C22×Q16—171no—
333C22×SD16—171no—
334C2×C4⋊Q8—171no—
335C2×#124C2×D1711yescharacteristic
336C2×C4×Q8—171no—
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×D4C2×D1711yescharacteristic
339#489—344no—
340#464—344no—
341C2×C4⋊C8—171no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×#97—171no—
353#400—341no—
354#400—341no—
355#391—171no—
356#391—171no—
357#391—341no—
358#391—341no—
359#391C2×D1711yescharacteristic
360#390C2×D1711yescharacteristic
361#388—21no—
362#388—21no—
363#385—21no—
364#382—171no—
365#382—171no—
366C4×C4⋊C4C2×D1711yescharacteristic
367C4×C4⋊C4C2×D1711yescharacteristic
368C4×C22⋊C4—21no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#336—171no—
376#336—171no—
377C2×C17—82no—
378C2×C17#242711yescharacteristic
379C2×C17#304011yescharacteristic
380C2×C17#304611yescharacteristic
381C2×C17#331111yescharacteristic
382C2×C17#331311yescharacteristic
383C2×C17#337111yescharacteristic
384C2×C17#337711yescharacteristic
385C22×Q8—171no—
386C22×Q8—341no—
387C22×Q8C22×D1711yescharacteristic
388C22×D4C22×D1711yescharacteristic
389C23×C4—21no—
390C2×Q16—342no—
391C2×Q16—342no—
392C2×Q16—681no—
393C2×SD16—342no—
394C2×SD16—342no—
395C2×SD16—681no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C4⋊Q8—342no—
401C4⋊Q8—342no—
402#124—22no—
403#124D6814yes—
404C4×Q8—22no—
405C4×Q8—41no—
406C4×Q8—342no—
407C4×Q8—342no—
408C4×D4—22no—
409C4×D4—41no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4C22⋊D1711yescharacteristic
417C2×C4⋊C4C22⋊D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42—341no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431C4⋊C8—342no—
432C4⋊C8—681no—
433C4⋊C8—681no—
434#103—342no—
435#103—342no—
436#103—342no—
437#103—342no—
438#103—342no—
439#102—342no—
440#102—342no—
441#102—342no—
442#97—342no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95C22⋊D1711yescharacteristic
455#95C22⋊D1711yescharacteristic
456C17#2533711yescharacteristic, complemented, Sylow
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—172no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—682no—
470C2×Q8—682no—
471C2×Q8—682no—
472C2×Q8C2×D6812yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D6812yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4C2×C22⋊D1711yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C23×D1711yescharacteristic
496Q16—684no—
497SD16—684no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—682no—
523C4⋊C4—682no—
524C4⋊C4#6325412yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C42—682no—
537C42C2×D6812yes—
538C42#6325212yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D1711yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—342no—
552Q8—342no—
553Q8—681no—
554Q8—681no—
555Q8—1364no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—342no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4#1058761612yes—
599C2×C4#1058761612yes—
600C2×C4C22×D6811yescharacteristic
601C2×C4C2×#6325211yescharacteristic
602C2×C4C2×#6325411yescharacteristic
603C2×C4#1058769111yescharacteristic
604C2×C4#41×D1711yescharacteristic
605C2×C4#1058769711yescharacteristic
606C8—681no—
607C8—681no—
608C8—681no—
609C8—681no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1058761611yescharacteristic
618C22#1176345311yescharacteristic
619C22#1176346611yescharacteristic
620C22#1176360711yescharacteristic
621C22#1176360711yescharacteristic
622C22#1176362011yescharacteristic
623C22#1176362411yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—682no—
634C4—1361no—
635C4—1361no—
636C4—1361no—
637C4#1176334111yescharacteristic
638C4#1176342411yescharacteristic
639C4#1176345411yescharacteristic
640C4#1176346311yescharacteristic
641C2—82no—
642C2#1453235511yescharacteristic
643C2#1453827811yescharacteristic
644C2#1453833411yescharacteristic
645C2#1454082011yescharacteristic
646C2#1454085011yescharacteristic
647C2#1454154611yescharacteristic
648C2#1454162011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2367C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2367D17
hypocenterC17#25337
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #14533999C2semidirect
4. #14530202C2semidirect
6. #14530106C2semidirect
8. #14529732C2semidirect
9. #14529508C2semidirect
10. #14529125C2semidirect
12. #14528799C2semidirect
14. #14528617C2semidirect
456. C17#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
C2×#117630282
#145339992
#145302022
#145301972
#145301062
#145301012
#145297322
#145295082
#145291252
#145291182
#145287992
#145287942
#145286172
C17×#23672
#2533717