Finite Group Databasebeta

← #17083890

Subgroups — #17083890

Order 4352.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763028C211yesmaximal, characteristic
2C2×Q8×#3622C211yesmaximal, characteristic, complemented
3#14534150C211yesmaximal, characteristic, complemented
4#14530172C211yesmaximal, characteristic
5#14530165C211yesmaximal, characteristic, complemented
6#14530106C211yesmaximal, characteristic, complemented
7#14530101C211yesmaximal, characteristic
8#14529644C211yesmaximal, characteristic, complemented
9#14529637C211yesmaximal, characteristic
10#14529596C211yesmaximal, characteristic
11#14529552C211yesmaximal, characteristic, complemented
12#14529470C211yesmaximal, characteristic
13#14529463C211yesmaximal, characteristic, complemented
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2592C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63229C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19#11763028—24no—
20Q8×#3622—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×C4×#3622C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36C2×C4×#3622C2211yescharacteristic
37#11762747—21no—
38#11762728—21no—
39#11762715C2211yescharacteristic
40#11762715C2211yescharacteristic
41#11762707C2211yescharacteristic
42#11762707C2211yescharacteristic
43#11762690—21no—
44#11762690—21no—
45#11762675C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C17×#124C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C17×#397—21no—
53C17×#393—21no—
54C17×#391C2211yescharacteristic
55C17×#391C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#390C2211yescharacteristic
58C17×#388—21no—
59C17×#388—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17D411yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C22×Dic34C2311yescharacteristic
65C17×#124—22no—
66C17×#124—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C2×C4⋊C4×C17D411yescharacteristic
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
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×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×#63231—22no—
85C2×#63231—22no—
86C2×#63231—41no—
87C2×#63229—22no—
88C2×#63229—22no—
89C2×#63229—41no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C4×Dic34—22no—
102C4×Dic34—22no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108#10587506—22no—
109#10587506—22no—
110#10587506—41no—
111#10587503—22no—
112#10587503—22no—
113#10587503—41no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587481—22no—
123#10587481—22no—
124#10587479—41no—
125#10587478—41no—
126C4⋊#3622—22no—
127C4⋊#3622—22no—
128C4⋊#3622D414yes—
129C4×#3622—22no—
130C4×#3622—22no—
131C4×#3622—22no—
132C24×C17—21no—
133C2×Q8×C17—22no—
134C2×Q8×C17—22no—
135C2×Q8×C17—22no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17—21no—
144C2×D4×C17—21no—
145C2×D4×C17—21no—
146C2×D4×C17—21no—
147C2×D4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
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#4112yes—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C22⋊C4×C17—42no—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—42no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—81no—
180C22⋊C4×C17—81no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—81no—
186C42×C17C2×D412yes—
187C42×C17#4112yes—
188#63231—44no—
189#63229—44no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196#63225—42no—
197#63225—42no—
198#63225—42no—
199C4×Dic17—42no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622C2×D412yes—
210C2×#3622C2×D412yes—
211C2×#3622#4112yes—
212#25297—171nomaximal, Sylow
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17C22×D411yescharacteristic
221Q8×C17—28no—
222Q8×C17—41no—
223Q8×C17—41no—
224Q8×C17—41no—
225Q8×C17—41no—
226D4×C17—41no—
227D4×C17—41no—
228D4×C17—41no—
229D4×C17—41no—
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—21no—
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—41no—
253C2×C4×C17#12112yes—
254C2×C4×C17#12112yes—
255C2×C4×C17#13512yes—
256C2×C4×C17#13512yes—
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C172−1+411yescharacteristic
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—22no—
276Dic34—22no—
277Dic34—41no—
278Dic34—41no—
279#3622—22no—
280#3622—22no—
281#3622—22no—
282#3622—41no—
283C2×#464—171no—
284C2×C8×Q8—171no—
285#2592D1711yescharacteristic
286#2581—171no—
287#2067—171no—
288#2066—171no—
289#2056—171no—
290#2055—171no—
291#1969—171no—
292#1968—171no—
293#1958—171no—
294#1949—171no—
295#1934—171no—
296#1933—171no—
297C4×#103—171no—
298C22×C17—81no—
299C22×C17—81no—
300C22×C17—81no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17C2×#12111yescharacteristic
306C22×C17#54811yescharacteristic
307C22×C17#55011yescharacteristic
308C22×C17C2×#13511yescharacteristic
309C22×C17C2×#13511yescharacteristic
310C22×C17#57711yescharacteristic
311C22×C17#57811yescharacteristic
312C4×C17—22no—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17—81no—
319C4×C17—81no—
320C4×C17—81no—
321C4×C17#52911yescharacteristic
322C4×C17#54211yescharacteristic
323C4×C17#54611yescharacteristic
324C4×C17#54811yescharacteristic
325Dic17—42no—
326Dic17—81no—
327Dic17—81no—
328Dic17—81no—
329C22×Q16—171no—
330C22×SD16—171no—
331C2×#124C2×D1711yescharacteristic
332C2×C4×Q8—171no—
333C2×C4×Q8C2×D1711yescharacteristic
334#464—344no—
335C8×Q8—344no—
336C2×#107—171no—
337C2×#106—171no—
338C2×C4⋊C8—171no—
339C2×C4⋊C8—171no—
340C2×C4⋊C8—171no—
341C2×#103—171no—
342C2×#103—171no—
343C2×#103—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#102—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×C4×C8—171no—
350C2×C4×C8—171no—
351C2×C4×C8—171no—
352#400—341no—
353#397—21no—
354#395—341no—
355#393—21no—
356#391—171no—
357#391—171no—
358#391C2×D1711yescharacteristic
359#391C2×D1711yescharacteristic
360#390C2×D1711yescharacteristic
361#390C2×D1711yescharacteristic
362#389—171no—
363#389—171no—
364#388—21no—
365#388—21no—
366#387—341no—
367#387—341no—
368#382—171no—
369C4×C4⋊C4C2×D1711yescharacteristic
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374C2×C17—161no—
375C2×C17#242611yescharacteristic
376C2×C17#305011yescharacteristic
377C2×C17#306311yescharacteristic
378C2×C17#336411yescharacteristic
379C2×C17#336511yescharacteristic
380C2×C17#339311yescharacteristic
381C2×C17#339411yescharacteristic
382C22×Q8—171no—
383C22×Q8C22⋊D1711yescharacteristic
384C22×Q8C22×D1711yescharacteristic
385C22×D4C22×D1711yescharacteristic
386C2×Q16—342no—
387C2×Q16—342no—
388C2×Q16—681no—
389C2×SD16—342no—
390C2×SD16—342no—
391C2×SD16—681no—
392C22×C8—171no—
393C22×C8—171no—
394C22×C8—171no—
395C22×C8—171no—
396#124—22no—
397#124—22no—
398C4×Q8—22no—
399C4×Q8—22no—
400C4×Q8—342no—
401C4×Q8—342no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4C22⋊D1711yescharacteristic
409C2×C4⋊C4C22⋊D1711yescharacteristic
410C2×C4⋊C4C22⋊D1711yescharacteristic
411C2×C4⋊C4C22×D1711yescharacteristic
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—21no—
420C2×C42—341no—
421C2×C42C22×D1711yescharacteristic
422C2×C42C22×D1711yescharacteristic
423C2×C42C22×D1711yescharacteristic
424#107—681no—
425#106—681no—
426C4⋊C8—174no—
427C4⋊C8—342no—
428C4⋊C8—342no—
429#103—342no—
430#103—342no—
431#103—342no—
432#103—342no—
433#103—681no—
434#102—342no—
435#102—342no—
436#102—681no—
437C4×C8—342no—
438C4×C8—342no—
439C4×C8—342no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—341no—
446#95—341no—
447#95—341no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453C17#2529711yescharacteristic, complemented, Sylow
454C24—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—172no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C2×C22⋊D1711yescharacteristic
486C22×C4C2×C22⋊D1711yescharacteristic
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C2×C22⋊D1711yescharacteristic
489C22×C4C23×D1711yescharacteristic
490Q16—684no—
491SD16—684no—
492C2×C8—172no—
493C2×C8—172no—
494C2×C8—172no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—341no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—682no—
518C4⋊C4—682no—
519C4⋊C4—682no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—81no—
531C42—682no—
532C42D4×D1712yes—
533C42#6325412yes—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D1711yescharacteristic
542Q8—28no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—342no—
548Q8—342no—
549Q8—681no—
550Q8—681no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—342no—
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#1058762512yes—
591C2×C4#1058762512yes—
592C2×C4#1058763612yes—
593C2×C4#1058763612yes—
594C2×C4C2×D4×D1711yescharacteristic
595C2×C4C2×#6325411yescharacteristic
596C2×C4#41×D1711yescharacteristic
597C2×C4#41×D1711yescharacteristic
598C2×C4#1058769611yescharacteristic
599C2×C4#1058769711yescharacteristic
600C8—342no—
601C8—342no—
602C8—342no—
603C8—681no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1058762511yescharacteristic
612C22#1176346611yescharacteristic
613C22#1176347811yescharacteristic
614C22C2×#1058763611yescharacteristic
615C22C2×#1058763611yescharacteristic
616C22#1176362011yescharacteristic
617C22#1176362411yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—682no—
628C4—1361no—
629C4—1361no—
630C4—1361no—
631C4#1176333811yescharacteristic
632C4#1176342811yescharacteristic
633C4#1176345211yescharacteristic
634C4#1176346711yescharacteristic
635C2—161no—
636C2#1453235311yescharacteristic
637C2#1453836211yescharacteristic
638C2#1453847711yescharacteristic
639C2#1454146511yescharacteristic
640C2#1454147611yescharacteristic
641C2#1454176811yescharacteristic
642C2#1454177611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2592C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2592D17
hypocenterC17#25297
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#3622C2semidirect
3. #14534150C2semidirect
5. #14530165C2semidirect
6. #14530106C2semidirect
8. #14529644C2semidirect
11. #14529552C2semidirect
13. #14529463C2semidirect
14. C4×#10587481C2semidirect
453. C17#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#117630282
C2×Q8×#36222
#145341502
#145301722
#145301652
#145301062
#145301012
#145296442
#145296372
#145295962
#145295522
#145294702
#145294632
C4×#105874812
C17×#25922
#2529717