Finite Group Databasebeta

← #17084037

Subgroups — #17084037

Order 4352.

802 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×#11763014C211yesmaximal, characteristic
2C2×#11762940C211yesmaximal, characteristic, complemented
3#14533996C211yesmaximal, characteristic, complemented
4#14530176C211yesmaximal, characteristic, complemented
5#14530162C211yesmaximal, characteristic, complemented
6#14530101C211yesmaximal, characteristic
7#14530101C211yesmaximal, characteristic
8#14529664C211yesmaximal, characteristic
9#14529637C211yesmaximal, characteristic
10#14529603C211yesmaximal, characteristic, complemented
11#14529550C211yesmaximal, characteristic, complemented
12#14529473C211yesmaximal, characteristic, complemented
13#14529459C211yesmaximal, characteristic, complemented
14#14528609C211yesmaximal, characteristic
15C17×#2580C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×#10587555C2211yescharacteristic
19#11763014—24no—
20#11762940—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35C2×#10587474C2211yescharacteristic
36C2×C4×#3622C2211yescharacteristic
37#11762748—21no—
38#11762748—21no—
39#11762716—21no—
40#11762715C2211yescharacteristic
41#11762715C2211yescharacteristic
42#11762715C2211yescharacteristic
43#11762715C2211yescharacteristic
44#11762674—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#399—21no—
53C17×#393—21no—
54C17×#390C2211yescharacteristic
55C17×#390C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#385—21no—
59C17×#385—21no—
60C17×#382C2211yescharacteristic
61C22×Q8×C17D411yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C22×Dic34C2311yescharacteristic
65C22×Dic34C2311yescharacteristic
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×#63229—22no—
87C2×#63229—22no—
88C2×#63229—22no—
89C2×#63229—22no—
90C2×#63229—41no—
91C2×#63229—41no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587555—22no—
102#10587555—22no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107#10587503—22no—
108#10587503—22no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587481—22no—
119#10587481—22no—
120#10587481—22no—
121#10587481—22no—
122#10587481—41no—
123#10587481—41no—
124#10587479—41no—
125#10587478—41no—
126C4⋊#3622—22no—
127C4⋊#3622—22no—
128C4⋊#3622D414yes—
129#10587474—22no—
130#10587474—22no—
131C4×#3622—22no—
132C24×C17—21no—
133C2×Q8×C17—22no—
134C2×Q8×C17—22no—
135C2×Q8×C17—41no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17#4112yes—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—42no—
146C22×C4×C17—81no—
147C22×C4×C17—81no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2411yescharacteristic
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—81no—
181C4⋊C4×C17—81no—
182C22⋊C4×C17—42no—
183C22⋊C4×C17—42no—
184C22⋊C4×C17—42no—
185C22⋊C4×C17—81no—
186C22⋊C4×C17—81no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—21no—
191C42×C17C2×D412yes—
192C42×C17#4112yes—
193#63229—44no—
194#63229—44no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17—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×#3622C2×D412yes—
212C2×#3622C2×D412yes—
213C2×#3622#4112yes—
214#25308—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—44no—
224D4×C17—22no—
225D4×C17—22no—
226D4×C17—41no—
227D4×C17—41no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
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×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—84no—
253C2×C4×C17#12112yes—
254C2×C4×C17#12112yes—
255C2×C4×C17#13712yes—
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17H(5,𝔽2)11yescharacteristic
261C2×C4×C172−1+411yescharacteristic
262C2×Dic17—41no—
263C2×Dic17—41no—
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—
274Dic34—41no—
275Dic34—41no—
276Dic34—41no—
277Dic34—41no—
278Dic34—41no—
279Dic34—41no—
280Dic34—41no—
281Dic34—41no—
282#3622—22no—
283#3622—22no—
284#3622—22no—
285#3622—41no—
286C2×#461—171no—
287C2×#446—171no—
288#2580D1711yescharacteristic
289#2568—171no—
290#2068—171no—
291#2066—171no—
292#2055—171no—
293#2055—171no—
294#1973—171no—
295#1968—171no—
296#1960—171no—
297#1949—171no—
298#1935—171no—
299#1933—171no—
300#1762—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#54811yescharacteristic
310C22×C17#55011yescharacteristic
311C22×C17C2×#13711yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57711yescharacteristic
315C4×C17—22no—
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17—81no—
324C4×C17#52911yescharacteristic
325C4×C17#54211yescharacteristic
326C4×C17#54611yescharacteristic
327C4×C17#54811yescharacteristic
328Dic17—81no—
329Dic17—81no—
330Dic17—81no—
331Dic17—81no—
332C22×SD16—171no—
333C22×SD16—171no—
334C2×C4⋊Q8—171no—
335C2×C4×Q8C2×D1711yescharacteristic
336C2×C4×D4C2×D1711yescharacteristic
337#461—344no—
338#446—344no—
339C2×#107—171no—
340C2×#106—171no—
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×#103—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×#97—171no—
353C2×#97—171no—
354C2×C4×C8—171no—
355#400—341no—
356#400—341no—
357#399—21no—
358#393—21no—
359#391—171no—
360#391—171no—
361#391—171no—
362#391—171no—
363#391—341no—
364#390C2×D1711yescharacteristic
365#390C2×D1711yescharacteristic
366#389C2×D1711yescharacteristic
367#389C2×D1711yescharacteristic
368#385—21no—
369#385—21no—
370#382—341no—
371#382C2×D1711yescharacteristic
372C4×C4⋊C4—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377C2×C17—82no—
378C2×C17#242611yescharacteristic
379C2×C17#305611yescharacteristic
380C2×C17#305711yescharacteristic
381C2×C17#337211yescharacteristic
382C2×C17#337611yescharacteristic
383C2×C17#340011yescharacteristic
384C2×C17#340111yescharacteristic
385C22×Q8—171no—
386C22×Q8—171no—
387C22×Q8C22⋊D1711yescharacteristic
388C22×D4C22×D1711yescharacteristic
389C23×C4—21no—
390C2×SD16—342no—
391C2×SD16—342no—
392C2×SD16—342no—
393C2×SD16—342no—
394C2×SD16—681no—
395C2×SD16—681no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C4⋊Q8—342no—
401C4⋊Q8—342no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×D4—22no—
405C4×D4—22no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4C22⋊D1711yescharacteristic
413C2×C4⋊C4C22⋊D1711yescharacteristic
414C2×C4⋊C4C22⋊D1711yescharacteristic
415C2×C4⋊C4C22×D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C4⋊C4C22×D1711yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C42—341no—
424C2×C42C22×D1711yescharacteristic
425C2×C42C22×D1711yescharacteristic
426C2×C42C22×D1711yescharacteristic
427#107—681no—
428#106—681no—
429C4⋊C8—174no—
430C4⋊C8—342no—
431C4⋊C8—342no—
432#103—342no—
433#103—342no—
434#103—342no—
435#103—342no—
436#103—681no—
437#103—681no—
438#102—342no—
439#102—342no—
440#97—342no—
441#97—342no—
442C4×C8—342no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456C17#2530811yescharacteristic, complemented, Sylow
457C24—21no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—41no—
461C2×Q8—341no—
462C2×Q8—341no—
463C2×Q8—341no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×Q8—341no—
471C2×Q8—341no—
472C2×Q8—341no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4#6325612yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—81no—
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
496SD16—684no—
497SD16—684no—
498C2×C8—172no—
499C2×C8—172no—
500C2×C8—172no—
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—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—81no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—682no—
537C42D4×D1712yes—
538C42#6325412yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D1711yescharacteristic
547Q8—44no—
548Q8—681no—
549Q8—681no—
550Q8—681no—
551Q8—681no—
552Q8—681no—
553Q8—681no—
554Q8—681no—
555Q8—681no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
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—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
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—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4#1058762512yes—
598C2×C4#1058762512yes—
599C2×C4#1058763712yes—
600C2×C4C2×D4×D1711yescharacteristic
601C2×C4C2×#6325411yescharacteristic
602C2×C4C2×#6325611yescharacteristic
603C2×C4#1058769411yescharacteristic
604C2×C4#41×D1711yescharacteristic
605C2×C4#1058769611yescharacteristic
606C8—342no—
607C8—342no—
608C8—342no—
609C8—681no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1058762511yescharacteristic
618C22#1176347011yescharacteristic
619C22#1176347811yescharacteristic
620C22C2×#1058763711yescharacteristic
621C22#1176360711yescharacteristic
622C22#1176362011yescharacteristic
623C22#1176362011yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—1361no—
634C4—1361no—
635C4—1361no—
636C4—1361no—
637C4#1176334511yescharacteristic
638C4#1176343211yescharacteristic
639C4#1176345011yescharacteristic
640C4#1176346911yescharacteristic
641C2—82no—
642C2#1453235211yescharacteristic
643C2#1453841611yescharacteristic
644C2#1453842011yescharacteristic
645C2#1454156511yescharacteristic
646C2#1454161111yescharacteristic
647C2#1454183711yescharacteristic
648C2#1454184411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2580C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2580D17
hypocenterC17#25308
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×#11762940C2semidirect
3. #14533996C2semidirect
4. #14530176C2semidirect
5. #14530162C2semidirect
10. #14529603C2semidirect
11. #14529550C2semidirect
12. #14529473C2semidirect
13. #14529459C2semidirect
456. C17#25308semidirect

Maximal subgroups

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

subgroupindex
C2×#117630142
C2×#117629402
#145339962
#145301762
#145301622
#145301012
#145296642
#145296372
#145296032
#145295502
#145294732
#145294592
#145286092
C17×#25802
#2530817