Finite Group Databasebeta

← #17002106

Subgroups — #17002106

Order 4352.

1074 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542623C211yesmaximal, characteristic
2#14531694C211yesmaximal, characteristic, complemented
3#14527357C211yesmaximal, characteristic, complemented
4#14527353C211yesmaximal, characteristic, complemented
5#14527328C212yesmaximal, complemented
6#14527323C212yesmaximal, complemented
7#14527310C212yesmaximal, complemented
8#14527289C212yesmaximal, complemented
9C2×#11762547C211yesmaximal, characteristic, complemented
10C2×#11762544C211yesmaximal, characteristic, complemented
11C17×#2973C211yesmaximal, characteristic, complemented
12#11763508—22no—
13#11763505C412yescomplemented
14#11763499C412yescomplemented
15#11763488—22no—
16#11763468C412yescomplemented
17#11763455C412yescomplemented
18#11763406—21no—
19#11763397—21no—
20#11763368—21no—
21#11763343C2212yes—
22#11763320C2212yes—
23C2×#10587610C2211yescharacteristic
24C2×#10587616C2211yescharacteristic
25#11763282—21no—
26C2×D4×Dic17—21no—
27#11762708—21no—
28#11762684—21no—
29#11762684—21no—
30#11762675C2211yescharacteristic, complemented
31#11762669C2211yescharacteristic
32#11762669C2211yescharacteristic, complemented
33C4×C4⋊Dic17C2211yescharacteristic, complemented
34C2×#10587502C2211yescharacteristic
35C2×#10587502C2211yescharacteristic, complemented
36#11762547C2212yescomplemented
37#11762547C2212yescomplemented
38#11762547C2212yescomplemented
39#11762547C2212yescomplemented
40#11762544C2212yes—
41#11762544C2212yes—
42#11762544C2212yes—
43#11762544C2212yes—
44C2×C4×D4×C17C2211yescharacteristic
45C17×#435—22no—
46C17×#435—22no—
47C17×#432C2212yescomplemented
48C17×#407C2212yescomplemented
49C2×C17×C22⋊C8C2211yescharacteristic
50C2×C17×C22⋊C8C2211yescharacteristic
51C22×D4×C17D411yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C22×C22⋊D17—21no—
55C23×Dic17—21no—
56C23×Dic17—21no—
57C2×#63252C2×C412yescomplemented
58C22×D68C2×C411yescharacteristic, complemented
59C22×Dic34C2×C411yescharacteristic, complemented
60C2×M16×C17—22no—
61C22×C8×C17—21no—
62C22×C8×C17—21no—
63C4×D4×C17C2×C412yes—
64C4×D4×C17C2×C412yes—
65C4×D4×C17D412yes—
66C4×D4×C17D412yes—
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C22⋊C4×C17D411yescharacteristic
69C2×C22⋊C4×C17D411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71#10587639—22no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75#10587625—22no—
76#10587625—22no—
77#10587623—22no—
78#10587621—22no—
79D4×Dic17—44no—
80#10587616—21no—
81#10587616—21no—
82#10587616C2×C412yescomplemented
83#10587616C2×C412yescomplemented
84#10587616C2312yes—
85#10587615—22no—
86#10587615—22no—
87C4×C22⋊D17—22no—
88C4×C22⋊D17—22no—
89C4×C22⋊D17—22no—
90C2×#63227—21no—
91#10587612C2×C412yescomplemented
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17C2×C411yescharacteristic, complemented
97C2×C4⋊Dic17C2×C411yescharacteristic, complemented
98#10587610—21no—
99#10587610—21no—
100#10587610C2×C412yescomplemented
101#10587610C2×C412yescomplemented
102#10587610C2312yes—
103C2×#63225—21no—
104C2×C4×Dic17—21no—
105#10587584—22no—
106#10587582—22no—
107#10587580—22no—
108#10587575—22no—
109#10587571—22no—
110#10587570—22no—
111#10587569—22no—
112#10587567—22no—
113#10587567—22no—
114C22⋊C4×D17—22no—
115#10587565—22no—
116#10587563—22no—
117#10587562—22no—
118#10587561C2×C412yescomplemented
119C4×D68C2×C412yescomplemented
120#10587558—22no—
121#10587556C2×C412yescomplemented
122C4×Dic34C2×C412yescomplemented
123C17×C4⋊C8—22no—
124C17×C22⋊C8—21no—
125C17×C22⋊C8—21no—
126C17×C22⋊C8—22no—
127C17×C22⋊C8C2312yes—
128C17×C22⋊C8C2312yes—
129C17×#97—22no—
130#10587502—21no—
131#10587502—21no—
132#10587502—21no—
133#10587502C2311yescharacteristic
134#10587502C2311yescharacteristic
135#10587502C2311yescharacteristic
136#10587502C2311yescharacteristic
137C24×C17C2×D411yescharacteristic
138C24×C17C2×D411yescharacteristic
139C23×D17—21no—
140C2×D4×C17—22no—
141C2×D4×C17—22no—
142C2×D4×C17C22⋊C412yes—
143C2×D4×C17C22⋊C412yes—
144C22×C4×C17—81no—
145C22×C4×C17—81no—
146C22×C4×C17C22×C411yescharacteristic
147C22×C4×C17C22×C411yescharacteristic
148C22×C4×C17C22×C412yes—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D412yes—
156C22×C4×C17C2411yescharacteristic
157C2×C22⋊D17—22no—
158C2×C22⋊D17—22no—
159C2×C22⋊D17—22no—
160C2×C22⋊D17—22no—
161C2×C22⋊D17—42no—
162C2×C22⋊D17—42no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—42no—
171C22×Dic17—42no—
172C22×Dic17—42no—
173C22×Dic17—42no—
174C22×Dic17—81no—
175C22×Dic17—81no—
176#63252—82no—
177C2×D68—21no—
178C2×D68—81no—
179C2×D68C22×C412yes—
180C2×C4×D17—22no—
181C2×C4×D17—22no—
182C2×Dic34—21no—
183C2×Dic34—81no—
184C2×Dic34C22×C412yes—
185M16×C17—82no—
186C2×C8×C17—22no—
187C2×C8×C17—22no—
188C2×C8×C17—81no—
189C2×C8×C17—81no—
190C4⋊C4×C17C22⋊C412yes—
191C4⋊C4×C17C22⋊C412yes—
192C22⋊C4×C17C22⋊C412yes—
193C22⋊C4×C17C22⋊C412yes—
194C22⋊C4×C17C22⋊C412yes—
195C22⋊C4×C17C22⋊C412yes—
196C42×C17C22×C412yes—
197C42×C17C2×D412yes—
198C22⋊Dic17—22no—
199C22⋊Dic17—22no—
200C22⋊Dic17—22no—
201C22⋊Dic17—22no—
202C22⋊Dic17—42no—
203C22⋊Dic17—42no—
204#63227—22no—
205#63227—22no—
206#63227—22no—
207#63227—22no—
208#63227—22no—
209#63227—22no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—42no—
213C4⋊Dic17—42no—
214C4⋊Dic17—81no—
215C4⋊Dic17—81no—
216C4⋊Dic17C22×C412yes—
217C4⋊Dic17C22×C412yes—
218#63225—22no—
219#63225—22no—
220#63225—22no—
221#63225—22no—
222#63225—22no—
223#63225—22no—
224C4×Dic17—22no—
225C4×Dic17—22no—
226C4×Dic17—42no—
227#17622—171nomaximal, Sylow
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—41no—
231C23×C17—41no—
232C23×C17—41no—
233C23×C17—41no—
234C23×C17C2×C22⋊C412yes—
235C23×C17C2×C22⋊C411yescharacteristic
236C23×C17C2×C22⋊C411yescharacteristic
237C23×C17AGL(1,ℤ/8)12yes—
238C23×C17#13712yes—
239C23×C17C22×D411yescharacteristic
240C22×D17—22no—
241C22×D17—42no—
242C22×D17—81no—
243C22×D17—81no—
244D4×C17—24no—
245D4×C17—24no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—42no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—42no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17C2×C22⋊C412yes—
259C2×C4×C17C2×C22⋊C412yes—
260C2×C4×C17C2×C22⋊C411yescharacteristic
261C2×C4×C17C2×C22⋊C412yes—
262C2×C4×C17C2×C22⋊C411yescharacteristic
263C2×C4×C17C23×C411yescharacteristic
264C2×C4×C17C22×D411yescharacteristic
265C22⋊D17—42no—
266C22⋊D17—42no—
267C22⋊D17—44no—
268C22⋊D17—82no—
269C22⋊D17—82no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—42no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—81no—
287C2×Dic17—81no—
288C2×Dic17—84no—
289C2×Dic17—84no—
290D68—42no—
291D68—81no—
292C4×D17—82no—
293C4×D17—82no—
294Dic34—42no—
295Dic34—81no—
296C8×C17—42no—
297C8×C17—42no—
298#3505—171no—
299#2973D1711yescharacteristic, complemented
300#2349—171no—
301#1520—171no—
302#1519—171no—
303#1513—172no—
304#1512—172no—
305#1509—172no—
306#1504—172no—
307C2×#328—171no—
308C2×#327—171no—
309C22×C17—21no—
310C22×C17—21no—
311C22×C17—21no—
312C22×C17—21no—
313C22×C17—21no—
314C22×C17—21no—
315C22×C17—22no—
316C22×C17—22no—
317C22×C17—22no—
318C22×C17—42no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17#41712yes—
323C22×C17#41712yes—
324C22×C17C22×C22⋊C411yescharacteristic
325C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
326C22×C17C2×#13711yescharacteristic
327C2×D17—42no—
328C2×D17—42no—
329C2×D17—42no—
330C2×D17—162no—
331C4×C17—21no—
332C4×C17—21no—
333C4×C17—21no—
334C4×C17—22no—
335C4×C17—21no—
336C4×C17—22no—
337C4×C17—22no—
338Dic17—42no—
339Dic17—42no—
340Dic17—42no—
341Dic17—42no—
342Dic17—81no—
343Dic17—81no—
344Dic17—81no—
345#553—172no—
346#553—342no—
347#552—172no—
348#551—342no—
349#548—172no—
350#546—172no—
351#539—341no—
352#538—341no—
353#534—341no—
354#529—172no—
355#525—172no—
356C2×C22⋊Q8—171no—
357C2×#121—171no—
358#518—341no—
359C2×C4×D4—341no—
360C2×C4×D4C2×D1711yescharacteristic
361#435—22no—
362#435—22no—
363#432C2×D1712yescomplemented
364#407C2×D1712yescomplemented
365C2×C22⋊C8C2×D1711yescharacteristic
366C2×C22⋊C8C2×D1711yescharacteristic
367#389—341no—
368#385—341no—
369#385—341no—
370#382—171no—
371#380—171no—
372#380—171no—
373C4×C4⋊C4—171no—
374C2×#95—171no—
375C2×#95—171no—
376#328—172no—
377#328—172no—
378#328—172no—
379#328—172no—
380#327—172no—
381#327—172no—
382#327—172no—
383#327—172no—
384C2×C17—21no—
385C2×C17—22no—
386C2×C17—21no—
387C2×C17—22no—
388C2×C17—42no—
389C2×C17#288311yescharacteristic
390C2×C17C2×#41711yescharacteristic
391C2×C17#290111yescharacteristic
392D17—82no—
393C2×#41—172no—
394C22×Q8—171no—
395C22×D4—171no—
396C22×D4—341no—
397C22×D4D6811yescharacteristic
398C23×C4—341no—
399C23×C4—341no—
400C23×C4C22×D1711yescharacteristic
401C23×C4C22×D1711yescharacteristic
402C2×M16—22no—
403C22×C8—21no—
404C22×C8—21no—
405#126—342no—
406#126—342no—
407#126—342no—
408#125—172no—
409#125—342no—
410#124—172no—
411#124—342no—
412#123—342no—
413#123—342no—
414#123—342no—
415#123—342no—
416#123—342no—
417C22⋊Q8—172no—
418C22⋊Q8—172no—
419C22⋊Q8—172no—
420C22⋊Q8—341no—
421C22⋊Q8—341no—
422C22⋊Q8—342no—
423#121—172no—
424#121—172no—
425#121—172no—
426#121—342no—
427#121—342no—
428#121—341no—
429#121—341no—
430#121—342no—
431#120—342no—
432#120—342no—
433C4×Q8—172no—
434C4×D4—172no—
435C4×D4—342no—
436C4×D4—342no—
437C4×D4—342no—
438C4×D4—342no—
439C4×D4—342no—
440C4×D4—684no—
441C4×D4C4×D1712yes—
442C4×D4C4×D1712yes—
443C4×D4C22⋊D1712yes—
444C4×D4C22⋊D1712yes—
445#117—172no—
446#117—342no—
447#117—342no—
448C2×C4⋊C4—171no—
449C2×C4⋊C4—171no—
450C2×C4⋊C4—341no—
451C2×C4⋊C4—341no—
452C2×C4⋊C4—341no—
453C2×C4⋊C4—341no—
454C2×C4⋊C4—341no—
455C2×C4⋊C4D6811yescharacteristic
456C2×C22⋊C4—341no—
457C2×C22⋊C4—341no—
458C2×C22⋊C4—342no—
459C2×C22⋊C4—341no—
460C2×C22⋊C4—341no—
461C2×C22⋊C4D6811yescharacteristic
462C2×C22⋊C4D6811yescharacteristic
463C2×C42—341no—
464C2×C42C22×D1711yescharacteristic
465C4⋊C8—22no—
466C22⋊C8—21no—
467C22⋊C8—21no—
468C22⋊C8—22no—
469C22⋊C8C22×D1712yes—
470C22⋊C8C22×D1712yes—
471#97—22no—
472#95—171no—
473#95—171no—
474#95—171no—
475#95—171no—
476#95—341no—
477#95—341no—
478#95—341no—
479C17#1762211yescharacteristic, complemented, Sylow
480C24—341no—
481C24C2×D6811yescharacteristic
482C24C2×D6811yescharacteristic
483#41—1362no—
484C2×Q8—172no—
485C2×Q8—341no—
486C2×Q8—1361no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—172no—
490C2×D4—342no—
491C2×D4—342no—
492C2×D4—342no—
493C2×D4—342no—
494C2×D4—341no—
495C2×D4—682no—
496C2×D4—682no—
497C2×D4—1361no—
498C2×D4#6322712yes—
499C2×D4#6322712yes—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4—341no—
511C22×C4—682no—
512C22×C4—682no—
513C22×C4—682no—
514C22×C4—682no—
515C22×C4—1361no—
516C22×C4—1361no—
517C22×C4C2×C4×D1711yescharacteristic
518C22×C4C2×C4×D1711yescharacteristic
519C22×C4C2×C4×D1712yes—
520C22×C4C2×D6811yescharacteristic
521C22×C4C2×D6811yescharacteristic
522C22×C4C2×D6811yescharacteristic
523C22×C4C2×D6811yescharacteristic
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1712yes—
527C22×C4C23×D1711yescharacteristic
528M16—82no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—81no—
532C2×C8—81no—
533C4⋊C4—172no—
534C4⋊C4—172no—
535C4⋊C4—342no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—342no—
539C4⋊C4—342no—
540C4⋊C4—342no—
541C4⋊C4—342no—
542C4⋊C4—342no—
543C4⋊C4—682no—
544C4⋊C4—682no—
545C4⋊C4—1361no—
546C4⋊C4—1361no—
547C4⋊C4#6322712yes—
548C4⋊C4#6322712yes—
549C22⋊C4—342no—
550C22⋊C4—342no—
551C22⋊C4—342no—
552C22⋊C4—342no—
553C22⋊C4—342no—
554C22⋊C4—342no—
555C22⋊C4—342no—
556C22⋊C4—342no—
557C22⋊C4—342no—
558C22⋊C4—342no—
559C22⋊C4—682no—
560C22⋊C4—682no—
561C22⋊C4#6322712yes—
562C22⋊C4#6322712yes—
563C22⋊C4#6322712yes—
564C22⋊C4#6322712yes—
565C42—342no—
566C42—342no—
567C42—682no—
568C42C2×C4×D1712yes—
569C42C2×C22⋊D1712yes—
570C23—21no—
571C23—21no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—342no—
577C23—682no—
578C23—1361no—
579C23—1361no—
580C23#1058759412yes—
581C23#1058759512yes—
582C23C2×#6322712yes—
583C23C2×#6322711yescharacteristic
584C23C2×#6322711yescharacteristic
585C23C22×D6811yescharacteristic
586Q8—682no—
587Q8—1361no—
588D4—24no—
589D4—24no—
590D4—682no—
591D4—684no—
592D4—682no—
593D4—682no—
594D4—1361no—
595D4—1362no—
596D4—1362no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—681no—
614C2×C4—682no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—1362no—
626C2×C4—1362no—
627C2×C4—1361no—
628C2×C4—1361no—
629C2×C4—1364no—
630C2×C4—1364no—
631C2×C4C2×#6322712yes—
632C2×C4C2×#6322712yes—
633C2×C4C2×#6322712yes—
634C2×C4C2×#6322711yescharacteristic
635C2×C4C2×#6322711yescharacteristic
636C2×C4C22×C4×D1711yescharacteristic
637C2×C4C22×C22⋊D1711yescharacteristic
638C8—42no—
639C8—42no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—2722no—
654C22—682no—
655C22—682no—
656C22—682no—
657C22#1176281112yes—
658C22#1176281112yes—
659C22C22×#6322711yescharacteristic
660C22C2×#1058759411yescharacteristic
661C22C2×#1058759511yescharacteristic
662C4—21no—
663C4—21no—
664C4—22no—
665C4—21no—
666C4—21no—
667C4—22no—
668C4—22no—
669C4—682no—
670C4—682no—
671C4—682no—
672C4—682no—
673C4—1361no—
674C4—1361no—
675C4—1361no—
676C2—21no—
677C2—21no—
678C2—22no—
679C2—22no—
680C2—42no—
681C2—1362no—
682C2#1453724411yescharacteristic
683C2C2×#1176281111yescharacteristic
684C2#1453732911yescharacteristic
subgroupG/H
centerC22C22×#63227
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2973C2
socleC22×C17C22×C22⋊C4
prime radical#14542623C2
cosocleC22×C4×C17C24
hypercenter#2973D17
hypocenterC17#17622
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #14531694C2semidirect
3. #14527357C2semidirect
4. #14527353C2semidirect
5. #14527328C2semidirect
6. #14527323C2semidirect
7. #14527310C2semidirect
8. #14527289C2semidirect
9. C2×#11762547C2semidirect
10. C2×#11762544C2semidirect
11. C17×#2973C2semidirect
13. #11763505C4semidirect
14. #11763499C4semidirect
16. #11763468C4semidirect
17. #11763455C4semidirect
30. #11762675C22semidirect
32. #11762669C22semidirect
33. C4×C4⋊Dic17C22semidirect
35. C2×#10587502C22semidirect
36. #11762547C22semidirect
37. #11762547C22semidirect
38. #11762547C22semidirect
39. #11762547C22semidirect
47. C17×#432C22semidirect
48. C17×#407C22semidirect
57. C2×#63252C2×C4semidirect
58. C22×D68C2×C4semidirect
59. C22×Dic34C2×C4semidirect
82. #10587616C2×C4semidirect
83. #10587616C2×C4semidirect
91. #10587612C2×C4semidirect
96. C2×C4⋊Dic17C2×C4semidirect
97. C2×C4⋊Dic17C2×C4semidirect
100. #10587610C2×C4semidirect
101. #10587610C2×C4semidirect
118. #10587561C2×C4semidirect
119. C4×D68C2×C4semidirect
121. #10587556C2×C4semidirect
122. C4×Dic34C2×C4semidirect
299. #2973D17semidirect
363. #432C2×D17semidirect
364. #407C2×D17semidirect
479. C17#17622semidirect

Maximal subgroups

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

subgroupindex
#145426232
#145316942
#145273572
#145273532
#145273282
#145273232
#145273102
#145272892
C2×#117625472
C2×#117625442
C17×#29732
#1762217