Finite Group Databasebeta

← #17107005

Subgroups — #17107005

Order 4352.

792 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
1#14533217C211yesmaximal, characteristic, complemented
2#14530187C211yesmaximal, characteristic, complemented
3#14530158C211yesmaximal, characteristic, complemented
4#14529970C211yesmaximal, characteristic
5#14529970C211yesmaximal, characteristic
6#14529547C211yesmaximal, characteristic, complemented
7#14529509C211yesmaximal, characteristic, complemented
8#14529392C211yesmaximal, characteristic, complemented
9#14529369C211yesmaximal, characteristic, complemented
10#14529359C211yesmaximal, characteristic, complemented
11#14529045C211yesmaximal, characteristic, complemented
12#14528977C211yesmaximal, characteristic, complemented
13#14528870C211yesmaximal, characteristic, complemented
14C4×#10587503C211yesmaximal, characteristic, complemented
15C17×#2361C211yesmaximal, characteristic
16C2×#10587479C2211yescharacteristic
17C2×#10587479C2211yescharacteristic
18C2×#10587479C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762751—21no—
29#11762732—21no—
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762702—21no—
33#11762696—21no—
34#11762679C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
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×C17—21no—
50C2×C4×D4×C17C2211yescharacteristic
51C22×C4⋊C4×C17C2211yescharacteristic
52C17×#390C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#386—21no—
56C17×#385—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×D4×C17—21no—
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C17×#121—22no—
65C17×#121—22no—
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C4×D4×C17—44no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—41no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17D411yescharacteristic
81C2×C22⋊C4×C17D411yescharacteristic
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587503—22no—
105#10587503—22no—
106#10587503—22no—
107#10587503—41no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—41no—
115#10587479—22no—
116#10587479—22no—
117#10587479—22no—
118#10587479—41no—
119C4⋊#3622—22no—
120C4×#3622—41no—
121C24×C17—21no—
122C24×C17C2×D411yescharacteristic
123C2×D4×C17—24no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—42no—
130C2×D4×C17—42no—
131C2×D4×C17#4112yes—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
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—42no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17—81no—
146C22×C4×C17—81no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17#4112yes—
164C4⋊C4×C17#4112yes—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—41no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—81no—
170C22⋊C4×C17SD1614yes—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—42no—
176C22⋊Dic17—81no—
177C22⋊Dic17—81no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—81no—
187C4⋊Dic17—81no—
188#63225—42no—
189#63225—81no—
190C4×Dic17—81no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
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#4112yes—
203#27032—171nomaximal, Sylow
204C23×C17—21no—
205C23×C17—21no—
206C23×C17—21no—
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—22no—
211C23×C17—41no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17C2×SD1612yes—
218C23×C17C22×D411yescharacteristic
219D4×C17—22no—
220D4×C17—22no—
221D4×C17—48no—
222D4×C17—41no—
223D4×C17—41no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
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—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
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—84no—
251C2×C4×C17—84no—
252C2×C4×C17C2×SD1612yes—
253C2×C4×C17C2×SD1612yes—
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17H(5,𝔽2)11yescharacteristic
259C2×C4×C172−1+411yescharacteristic
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—
270C2×Dic17—41no—
271C2×Dic17—41no—
272#3622—22no—
273#3622—41no—
274#3622—41no—
275#3622—41no—
276#2497—171no—
277#2361D1711yescharacteristic
278#2071—171no—
279#2065—171no—
280#2034—171no—
281#2034—171no—
282#1948—171no—
283#1941—171no—
284#1923—171no—
285#1920—171no—
286#1919—171no—
287#1864—171no—
288#1853—171no—
289#1823—171no—
290C4×#102—171no—
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17#52511yescharacteristic
306C22×C17#54611yescharacteristic
307C22×C17#54711yescharacteristic
308C22×C17C22×SD1611yescharacteristic
309C22×C17#57511yescharacteristic
310C22×C17#57611yescharacteristic
311C22×C17#57711yescharacteristic
312C4×C17—22no—
313C4×C17—22no—
314C4×C17—41no—
315C4×C17—42no—
316C4×C17—41no—
317C4×C17—42no—
318C4×C17—41no—
319C4×C17—81no—
320C4×C17#52911yescharacteristic
321C4×C17#53911yescharacteristic
322C4×C17#55311yescharacteristic
323C4×C17#55611yescharacteristic
324Dic17—81no—
325Dic17—81no—
326Dic17—81no—
327Dic17—81no—
328C2×#121C2×D1711yescharacteristic
329C2×C4×D4—21no—
330C2×C4×D4C2×D1711yescharacteristic
331C22×C4⋊C4C2×D1711yescharacteristic
332C2×#106—171no—
333C2×#106—171no—
334C2×#106—171no—
335C2×#106—171no—
336C2×C4⋊C8—171no—
337C2×#102—171no—
338C2×#102—171no—
339C2×#102—171no—
340C2×#102—171no—
341C2×C22⋊C8—171no—
342C2×C22⋊C8—171no—
343C2×C4×C8—171no—
344#400—341no—
345#396—341no—
346#390C2×D1711yescharacteristic
347#389—171no—
348#389—171no—
349#389C2×D1711yescharacteristic
350#388—21no—
351#388—341no—
352#387—341no—
353#386—21no—
354#385—21no—
355#384—171no—
356#380—171no—
357#380—171no—
358C4×C4⋊C4—171no—
359C4×C4⋊C4C2×D1711yescharacteristic
360C4×C22⋊C4C2×D1711yescharacteristic
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#336—171no—
370#336—171no—
371C2×C17—42no—
372C2×C17—82no—
373C2×C17#246611yescharacteristic
374C2×C17#310111yescharacteristic
375C2×C17#311911yescharacteristic
376C2×C17#329511yescharacteristic
377C2×C17#332111yescharacteristic
378C2×C17#333511yescharacteristic
379C2×C17#335511yescharacteristic
380C22×D4—21no—
381C22×D4C22×D1711yescharacteristic
382C23×C4—21no—
383C23×C4—21no—
384C23×C4C22×D1711yescharacteristic
385C22×C8—171no—
386C22×C8—171no—
387C22×C8—171no—
388C22×C8—171no—
389#121—22no—
390#121—22no—
391C4×D4—22no—
392C4×D4—41no—
393C4×D4—44no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—341no—
414C2×C22⋊C4—341no—
415C2×C22⋊C4C22⋊D1711yescharacteristic
416C2×C22⋊C4C22⋊D1711yescharacteristic
417C2×C42—21no—
418C2×C42—341no—
419C2×C42C22×D1711yescharacteristic
420C2×C42C22×D1711yescharacteristic
421#106—342no—
422#106—342no—
423#106—342no—
424#106—681no—
425C4⋊C8—342no—
426#102—342no—
427#102—342no—
428#102—342no—
429#102—681no—
430C22⋊C8—342no—
431C22⋊C8—681no—
432C4×C8—681no—
433#95—21no—
434#95—21no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95C22⋊D1711yescharacteristic
441#95C22⋊D1711yescharacteristic
442C17#2703211yescharacteristic, complemented, Sylow
443C24—21no—
444C24C2×C22⋊D1711yescharacteristic
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4#6325212yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C23×D1711yescharacteristic
479C2×C8—172no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—682no—
506C4⋊C4—1361no—
507C4⋊C4—1361no—
508C4⋊C4—1361no—
509C4⋊C4#6325212yes—
510C4⋊C4#6325412yes—
511C22⋊C4—24no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—1361no—
517C22⋊C4—1361no—
518C22⋊C4#6322914yes—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—1361no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×#6322912yes—
539C23C22×C22⋊D1711yescharacteristic
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—681no—
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×C4C2×#6322912yes—
586C2×C4C2×#6322912yes—
587C2×C4C2×#6325211yescharacteristic
588C2×C4C2×#6325211yescharacteristic
589C2×C4C2×#6325411yescharacteristic
590C2×C4#1058769111yescharacteristic
591C2×C4#41×D1711yescharacteristic
592C2×C4#1058769711yescharacteristic
593C8—342no—
594C8—681no—
595C8—681no—
596C8—681no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1176332011yescharacteristic
612C22#1176345311yescharacteristic
613C22#1176345811yescharacteristic
614C22C22×#6322911yescharacteristic
615C22#1176360711yescharacteristic
616C22#1176361211yescharacteristic
617C22#1176361711yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—1361no—
627C4—1361no—
628C4—1361no—
629C4—1361no—
630C4#1176333911yescharacteristic
631C4#1176340111yescharacteristic
632C4#1176350911yescharacteristic
633C4#1176352911yescharacteristic
634C2—42no—
635C2—82no—
636C2#1453281411yescharacteristic
637C2#1453879211yescharacteristic
638C2#1453898311yescharacteristic
639C2#1454060811yescharacteristic
640C2#1454097011yescharacteristic
641C2#1454114111yescharacteristic
642C2#1454138411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2361C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2361D17
hypocenterC17#27032
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #14533217C2semidirect
2. #14530187C2semidirect
3. #14530158C2semidirect
6. #14529547C2semidirect
7. #14529509C2semidirect
8. #14529392C2semidirect
9. #14529369C2semidirect
10. #14529359C2semidirect
11. #14529045C2semidirect
12. #14528977C2semidirect
13. #14528870C2semidirect
14. C4×#10587503C2semidirect
442. C17#27032semidirect

Maximal subgroups

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

subgroupindex
#145332172
#145301872
#145301582
#145299702
#145295472
#145295092
#145293922
#145293692
#145293592
#145290452
#145289772
#145288702
C4×#105875032
C17×#23612
#2703217