Finite Group Databasebeta

← #17001529

Subgroups — #17001529

Order 4352.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542833C211yesmaximal, characteristic
2#14537819C211yesmaximal, characteristic, complemented
3#14527307C211yesmaximal, characteristic, complemented
4#14527303C212yesmaximal, complemented
5#14527295C211yesmaximal, characteristic, complemented
6#14527286C212yesmaximal, complemented
7#14527282C211yesmaximal, characteristic, complemented
8#14527273C212yesmaximal, complemented
9#14527268C211yesmaximal, characteristic, complemented
10C2×#11762545C212yesmaximal, complemented
11C17×#2275C211yesmaximal, characteristic, complemented
12#11763529C411yescharacteristic, complemented
13#11763514—21no—
14#11763499C412yescomplemented
15#11763485—22no—
16D4×Dic34C411yescharacteristic, complemented
17#11763468C412yescomplemented
18#11763464C411yescharacteristic, complemented
19#11763463C411yescharacteristic, complemented
20#11763453—21no—
21#11763424—21no—
22#11763401—22no—
23#11763366C2211yescharacteristic
24#11763343C2211yescharacteristic
25#11763320C2212yes—
26C2×#10587610C2212yes—
27#11763288—21no—
28#11762887—21no—
29#11762881—22no—
30D4×#3622—21no—
31#11762866C2211yescharacteristic
32#11762862C2211yescharacteristic
33#11762773C2212yes—
34C2×C22⋊#3622C2212yes—
35#11762545C2211yescharacteristic
36#11762545C2212yes—
37#11762545C2212yes—
38#11762545C2212yes—
39#11762545C2211yescharacteristic
40#11762545C2212yes—
41#11762545C2212yes—
42#11762542C2211yescharacteristic
43#11762542C2211yescharacteristic
44#11762542C2212yes—
45C2×C4×D4×C17—21no—
46C2×C4×D4×C17C2211yescharacteristic
47C4×C4⋊C4×C17—21no—
48C4×C4⋊C4×C17C2211yescharacteristic
49C4×C22⋊C4×C17—22no—
50C4×C22⋊C4×C17C2212yes—
51C17×#376C2211yescharacteristic
52C2×C17×#95C2212yes—
53C22×D4×C17D411yescharacteristic
54C23×C4×C17—22no—
55C23×C4×C17C2312yes—
56C2×#63254—21no—
57C2×#63252C2×C412yescomplemented
58C22×Dic34C2×C412yescomplemented
59C4×D4×C17—28no—
60C4×D4×C17C2×C411yescharacteristic
61C4×D4×C17C2×C411yescharacteristic
62C4×D4×C17C2×C412yes—
63C4×D4×C17D411yescharacteristic
64C4×D4×C17D412yes—
65C4×D4×C17D411yescharacteristic
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C22⋊C4×C17—22no—
69C2×C22⋊C4×C17D412yes—
70C2×C42×C17—21no—
71C2×C42×C17—21no—
72C2×C42×C17—22no—
73C2×C42×C17C2311yescharacteristic
74#10587639—22no—
75C2×C22⋊Dic17—22no—
76#10587625—22no—
77#10587624—21no—
78#10587622—21no—
79#10587621—22no—
80D4×Dic17—21no—
81#10587616C2×C412yescomplemented
82#10587616C2311yescharacteristic
83#10587615—21no—
84#10587615—22no—
85C4×C22⋊D17—22no—
86C4×C22⋊D17—21no—
87C4×C22⋊D17—22no—
88#10587612—21no—
89#10587612C2×C412yescomplemented
90C2×C4⋊Dic17C2×C412yescomplemented
91#10587610—22no—
92#10587610—21no—
93#10587610—22no—
94#10587610C2×C412yescomplemented
95#10587610C2×C412yescomplemented
96#10587610C2×C412yescomplemented
97#10587610C2311yescharacteristic
98#10587610C2312yes—
99C2×#63225—22no—
100C2×#63223—22no—
101C22×#3622—22no—
102#10587583—21no—
103#10587582—22no—
104C4⋊C4×D17—21no—
105#10587575—22no—
106#10587574—21no—
107#10587573—21no—
108#10587572—22no—
109#10587571—22no—
110#10587569—22no—
111#10587567—22no—
112#10587565—22no—
113#10587565—22no—
114#10587564—22no—
115#10587563—22no—
116#10587562—21no—
117#10587561C2×C411yescharacteristic, complemented
118C4×D68C2×C411yescharacteristic, complemented
119#10587558—21no—
120#10587556—21no—
121#10587556C2×C411yescharacteristic, complemented
122#10587555C2×C411yescharacteristic, complemented
123#10587555C2×C411yescharacteristic, complemented
124C4×Dic34—21no—
125C4×Dic34C2×C412yescomplemented
126C4×Dic34C2×C411yescharacteristic, complemented
127C17×#95—22no—
128C17×#95—21no—
129C17×#95C2311yescharacteristic
130C17×#95C2312yes—
131C17×#95C2311yescharacteristic
132C22⋊#3622—22no—
133C22⋊#3622—22no—
134C22⋊#3622C2311yescharacteristic
135C22⋊#3622C2312yes—
136C22⋊#3622C2311yescharacteristic
137C4⋊#3622—21no—
138C4⋊#3622—21no—
139#10587474—21no—
140C4×#3622—21no—
141C24×C17C2×D412yes—
142C2×D4×C17—24no—
143C2×D4×C17C22⋊C412yes—
144C2×D4×C17C22⋊C411yescharacteristic
145C2×D4×C17C22⋊C411yescharacteristic
146C22×C4×C17—21no—
147C22×C4×C17—22no—
148C22×C4×C17—21no—
149C22×C4×C17—28no—
150C22×C4×C17—82no—
151C22×C4×C17—82no—
152C22×C4×C17—82no—
153C22×C4×C17C22×C412yes—
154C22×C4×C17C22×C412yes—
155C22×C4×C17C2×D412yes—
156C22×C4×C17C2×D412yes—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D412yes—
160C22×C4×C17C2411yescharacteristic
161C2×C22⋊D17—22no—
162C2×C22⋊D17—21no—
163C2×C22⋊D17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166#63254—44no—
167#63252—82no—
168C2×D68C22×C411yescharacteristic
169C2×C4×D17—21no—
170C2×C4×D17—22no—
171C2×Dic34—21no—
172C2×Dic34—22no—
173C2×Dic34—82no—
174C2×Dic34C22×C411yescharacteristic
175C2×Dic34C22×C412yes—
176C4⋊C4×C17—24no—
177C4⋊C4×C17C22⋊C411yescharacteristic
178C4⋊C4×C17C22⋊C411yescharacteristic
179C4⋊C4×C17C22⋊C412yes—
180C22⋊C4×C17—28no—
181C22⋊C4×C17C22⋊C412yes—
182C22⋊C4×C17C22⋊C412yes—
183C22⋊C4×C17C22⋊C412yes—
184C22⋊C4×C17C22⋊C412yes—
185C42×C17—24no—
186C42×C17—81no—
187C42×C17—82no—
188C42×C17C22×C411yescharacteristic
189C42×C17C22×C411yescharacteristic
190C42×C17C2×D411yescharacteristic
191C42×C17C2×D411yescharacteristic
192C22⋊Dic17—22no—
193C22⋊Dic17—22no—
194C22⋊Dic17—22no—
195C22⋊Dic17—22no—
196C22⋊Dic17—22no—
197C22⋊Dic17—21no—
198#63227—21no—
199#63227—21no—
200#63227—21no—
201#63227—22no—
202#63227—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—22no—
207C4⋊Dic17C22×C412yes—
208C4⋊Dic17C22×C411yescharacteristic
209C4⋊Dic17C22×C411yescharacteristic
210#63225—22no—
211#63225—21no—
212#63225—21no—
213#63225—21no—
214#63225—22no—
215#63225—22no—
216#63225—22no—
217#63225—22no—
218#63225—21no—
219C4×Dic17—22no—
220C4×Dic17—21no—
221C4×Dic17—22no—
222#63223—82no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—22no—
226C2×#3622—82no—
227#17548—171nomaximal, Sylow
228C23×C17—22no—
229C23×C17—22no—
230C23×C17—24no—
231C23×C17—42no—
232C23×C17C2×C22⋊C412yes—
233C23×C17C2×C22⋊C412yes—
234C23×C17C22×D411yescharacteristic
235C22×D17—21no—
236D4×C17—24no—
237D4×C17—22no—
238D4×C17#10414yes—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—24no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—41no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17—41no—
254C2×C4×C17—42no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—88no—
258C2×C4×C17C2×C22⋊C411yescharacteristic
259C2×C4×C17C2×C22⋊C411yescharacteristic
260C2×C4×C17C2×C22⋊C411yescharacteristic
261C2×C4×C17C2×C22⋊C412yes—
262C2×C4×C17C2×C22⋊C412yes—
263C2×C4×C17C2×C22⋊C411yescharacteristic
264C2×C4×C17C23×C411yescharacteristic
265C2×C4×C17C22×D411yescharacteristic
266C22⋊D17—41no—
267C22⋊D17—41no—
268C22⋊D17—44no—
269C22⋊D17—82no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—21no—
275C2×Dic17—44no—
276C2×Dic17—82no—
277D68—41no—
278C4×D17—42no—
279C4×D17—82no—
280Dic34—42no—
281Dic34—41no—
282Dic34—42no—
283Dic34—82no—
284#3622—41no—
285#3622—42no—
286#3622—41no—
287#3520—171no—
288#2974—171no—
289#2275D1711yescharacteristic, complemented
290#1509—171no—
291#1508—172no—
292#1506—171no—
293#1504—172no—
294#1503—171no—
295#1501—172no—
296#1500—171no—
297C2×#328—172no—
298C22×C17—22no—
299C22×C17—21no—
300C22×C17—22no—
301C22×C17—21no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17—44no—
306C22×C17—44no—
307C22×C17C2×#10414yes—
308C22×C17C22×C22⋊C411yescharacteristic
309C22×C17#57711yescharacteristic
310C22×C17#57811yescharacteristic
311C2×D17—41no—
312C2×D17—41no—
313C2×D17—41no—
314C4×C17—21no—
315C4×C17—21no—
316C4×C17—22no—
317C4×C17—22no—
318C4×C17—21no—
319C4×C17—21no—
320C4×C17—21no—
321C4×C17—42no—
322C4×C17—44no—
323C4×C17—42no—
324C4×C17C2×#10412yes—
325Dic17—41no—
326Dic17—42no—
327Dic17—41no—
328Dic17—42no—
329Dic17—41no—
330#556—171no—
331#554—341no—
332#552—172no—
333#551—342no—
334D4×Q8—171no—
335#548—172no—
336#547—171no—
337#547—171no—
338#546—341no—
339#542—341no—
340#539—342no—
341#533—171no—
342#529—171no—
343#525—172no—
344C2×C22⋊Q8—172no—
345#520—341no—
346C2×C4×D4—21no—
347C2×C4×D4C2×D1711yescharacteristic
348#436—341no—
349#435—342no—
350C8×D4—341no—
351#432—171no—
352#431—171no—
353#407—172no—
354C2×C22⋊C8—172no—
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D1711yescharacteristic
357C4×C22⋊C4—22no—
358C4×C22⋊C4C2×D1712yes—
359#376C2×D1711yescharacteristic
360C2×#95C2×D1712yes—
361#328—171no—
362#328—172no—
363#328—172no—
364#328—172no—
365#328—171no—
366#328—172no—
367#328—172no—
368#327—171no—
369#327—171no—
370#327—172no—
371C2×C17—21no—
372C2×C17—21no—
373C2×C17—24no—
374C2×C17—42no—
375C2×C17#288311yescharacteristic
376C2×C17#289811yescharacteristic
377C2×C17C22×#10411yescharacteristic
378D17—81no—
379C2×#41—172no—
380C2×#41—341no—
381C22×Q8—172no—
382C22×D4C22⋊D1711yescharacteristic
383C23×C4—22no—
384C23×C4C22×D1712yes—
385C2×M16—342no—
386C22×C8—342no—
387C4⋊Q8—171no—
388C4⋊Q8—171no—
389C4⋊Q8—341no—
390#126—342no—
391#126—342no—
392#126—341no—
393#125—171no—
394#125—342no—
395#125—341no—
396#124—171no—
397#124—341no—
398#124—342no—
399#123—342no—
400#123—342no—
401#123—341no—
402#123—342no—
403#123—342no—
404C22⋊Q8—172no—
405C22⋊Q8—172no—
406C22⋊Q8—172no—
407C22⋊Q8—171no—
408C22⋊Q8—172no—
409C22⋊Q8—342no—
410C22⋊Q8—342no—
411C22⋊Q8—341no—
412C22⋊Q8—342no—
413C22⋊Q8—341no—
414C22⋊Q8—342no—
415#121—172no—
416#121—171no—
417#121—342no—
418#121—341no—
419#120—342no—
420C4×Q8—171no—
421C4×Q8—172no—
422C4×Q8—341no—
423C4×Q8—341no—
424C4×D4—28no—
425C4×D4—171no—
426C4×D4—341no—
427C4×D4—342no—
428C4×D4—342no—
429C4×D4—341no—
430C4×D4—342no—
431C4×D4C4×D1711yescharacteristic
432C4×D4C4×D1711yescharacteristic
433C4×D4C4×D1712yes—
434C4×D4D6811yescharacteristic
435C4×D4D6812yes—
436C4×D4D6811yescharacteristic
437#117—172no—
438#117—341no—
439#117—342no—
440#117—341no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—172no—
443C2×C4⋊C4—342no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4C22⋊D1711yescharacteristic
446C2×C22⋊C4—22no—
447C2×C22⋊C4—342no—
448C2×C22⋊C4C22⋊D1712yes—
449C2×C42—21no—
450C2×C42—21no—
451C2×C42—22no—
452C2×C42C22×D1711yescharacteristic
453C4⋊C8—341no—
454C4⋊C8—341no—
455C22⋊C8—171no—
456C22⋊C8—172no—
457C22⋊C8—171no—
458C22⋊C8—342no—
459C22⋊C8—342no—
460#97—341no—
461C4×C8—341no—
462#95—22no—
463#95—21no—
464#95C22×D1711yescharacteristic
465#95C22×D1712yes—
466#95C22×D1711yescharacteristic
467C17#1754811yescharacteristic, complemented, Sylow
468C24C2×C22⋊D1712yes—
469#41—684no—
470#41—1362no—
471C2×Q8—171no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—342no—
475C2×Q8—1362no—
476C2×D4—24no—
477C2×D4—171no—
478C2×D4—341no—
479C2×D4—342no—
480C2×D4—342no—
481C2×D4#6322712yes—
482C2×D4#6322711yescharacteristic
483C2×D4#6322711yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—341no—
494C22×C4—342no—
495C22×C4C2×C4×D1712yes—
496C22×C4C2×C4×D1712yes—
497C22×C4C2×D6812yes—
498C22×C4C2×D6812yes—
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1712yes—
502C22×C4C23×D1711yescharacteristic
503M16—1362no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—342no—
507C2×C8—1362no—
508C4⋊C4—24no—
509C4⋊C4—171no—
510C4⋊C4—172no—
511C4⋊C4—171no—
512C4⋊C4—341no—
513C4⋊C4—342no—
514C4⋊C4—342no—
515C4⋊C4—342no—
516C4⋊C4—341no—
517C4⋊C4—342no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—342no—
522C4⋊C4—341no—
523C4⋊C4—342no—
524C4⋊C4—341no—
525C4⋊C4#6322711yescharacteristic
526C4⋊C4#6322712yes—
527C4⋊C4#6322711yescharacteristic
528C22⋊C4—28no—
529C22⋊C4—341no—
530C22⋊C4—342no—
531C22⋊C4—342no—
532C22⋊C4—342no—
533C22⋊C4—342no—
534C22⋊C4—342no—
535C22⋊C4—341no—
536C22⋊C4—341no—
537C22⋊C4—341no—
538C22⋊C4—342no—
539C22⋊C4—341no—
540C22⋊C4#6322712yes—
541C22⋊C4#6322712yes—
542C22⋊C4#6322712yes—
543C22⋊C4#6322712yes—
544C42—24no—
545C42—81no—
546C42—82no—
547C42—342no—
548C42—342no—
549C42—341no—
550C42C2×C4×D1711yescharacteristic
551C42C2×C4×D1711yescharacteristic
552C42C2×D6811yescharacteristic
553C42C2×D6811yescharacteristic
554C23—24no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—341no—
559C23C2×#6322712yes—
560C23C2×#6322712yes—
561C23C22×C22⋊D1711yescharacteristic
562Q8—681no—
563Q8—682no—
564Q8—682no—
565Q8—1362no—
566D4—24no—
567D4—22no—
568D4—681no—
569D4—684no—
570D4—681no—
571D4—681no—
572D4—1362no—
573D4#1058747614yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—341no—
594C2×C4—342no—
595C2×C4—341no—
596C2×C4—342no—
597C2×C4—341no—
598C2×C4—684no—
599C2×C4—682no—
600C2×C4—1362no—
601C2×C4—1362no—
602C2×C4C2×#6322711yescharacteristic
603C2×C4C2×#6322712yes—
604C2×C4C2×#6322711yescharacteristic
605C2×C4C2×#6322711yescharacteristic
606C2×C4C2×#6322712yes—
607C2×C4C2×#6322711yescharacteristic
608C2×C4C22×C4×D1711yescharacteristic
609C2×C4C22×D6811yescharacteristic
610C8—681no—
611C8—682no—
612C8—681no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—21no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—681no—
623C22—681no—
624C22—681no—
625C22C2×#1058747614yes—
626C22C22×#6322711yescharacteristic
627C22#1176362011yescharacteristic
628C22#1176362411yescharacteristic
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—682no—
640C4—681no—
641C4—681no—
642C4—681no—
643C4—682no—
644C4C2×#1058747612yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—1361no—
650C2#1453724411yescharacteristic
651C2#1453731411yescharacteristic
652C2C22×#1058747611yescharacteristic
subgroupG/H
centerC22C22×#63227
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2275C2
socleC22×C17C22×C22⋊C4
prime radical#14542833C2
cosocleC22×C4×C17C24
hypercenter#2275D17
hypocenterC17#17548
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #14537819C2semidirect
3. #14527307C2semidirect
4. #14527303C2semidirect
5. #14527295C2semidirect
6. #14527286C2semidirect
7. #14527282C2semidirect
8. #14527273C2semidirect
9. #14527268C2semidirect
10. C2×#11762545C2semidirect
11. C17×#2275C2semidirect
12. #11763529C4semidirect
14. #11763499C4semidirect
16. D4×Dic34C4semidirect
17. #11763468C4semidirect
18. #11763464C4semidirect
19. #11763463C4semidirect
57. C2×#63252C2×C4semidirect
58. C22×Dic34C2×C4semidirect
81. #10587616C2×C4semidirect
89. #10587612C2×C4semidirect
90. C2×C4⋊Dic17C2×C4semidirect
94. #10587610C2×C4semidirect
95. #10587610C2×C4semidirect
96. #10587610C2×C4semidirect
117. #10587561C2×C4semidirect
118. C4×D68C2×C4semidirect
121. #10587556C2×C4semidirect
122. #10587555C2×C4semidirect
123. #10587555C2×C4semidirect
125. C4×Dic34C2×C4semidirect
126. C4×Dic34C2×C4semidirect
289. #2275D17semidirect
467. C17#17548semidirect

Maximal subgroups

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

subgroupindex
#145428332
#145378192
#145273072
#145273032
#145272952
#145272862
#145272822
#145272732
#145272682
C2×#117625452
C17×#22752
#1754817