Finite Group Databasebeta

← #17002049

Subgroups — #17002049

Order 4352.

1038 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542623C211yesmaximal, characteristic
2#14531724C211yesmaximal, characteristic, complemented
3#14527362C212yesmaximal, complemented
4#14527357C211yesmaximal, characteristic, complemented
5#14527353C211yesmaximal, characteristic, complemented
6#14527341C212yesmaximal, complemented
7#14527276C212yesmaximal, complemented
8#14527271C212yesmaximal, complemented
9C2×#11762547C211yesmaximal, characteristic, complemented
10C2×#11762544C211yesmaximal, characteristic, complemented
11C17×#2972C211yesmaximal, 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—
26#11762702—21no—
27#11762695—21no—
28#11762689—21no—
29#11762677C2211yescharacteristic, complemented
30#11762672C2211yescharacteristic, complemented
31#11762669C2211yescharacteristic
32C22⋊C4×Dic17—21no—
33C4×C22⋊Dic17C2211yescharacteristic, complemented
34#11762655C2211yescharacteristic, complemented
35C2×#10587502C2211yescharacteristic
36#11762547C2212yescomplemented
37#11762547C2212yescomplemented
38#11762547C2212yescomplemented
39#11762547C2212yescomplemented
40#11762544C2212yes—
41#11762544C2212yes—
42#11762544C2212yes—
43#11762544C2212yes—
44C2×C4×D4×C17C2211yescharacteristic
45C8×D4×C17—22no—
46C8×D4×C17—22no—
47C17×#431C2212yescomplemented
48C2×C17×C22⋊C8C2211yescharacteristic
49C2×C17×C22⋊C8C2211yescharacteristic
50C2×C17×C22⋊C8C2212yescomplemented
51C22×D4×C17D411yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C23×C4×C17C2311yescharacteristic
54C22×C22⋊D17—21no—
55C23×Dic17—21no—
56C2×#63252C2×C412yescomplemented
57C22×D68C2×C411yescharacteristic
58C22×Dic34C2×C411yescharacteristic
59C22×C8×C17—21no—
60C22×C8×C17—21no—
61C22×C8×C17—22no—
62C4×D4×C17C2×C412yes—
63C4×D4×C17C2×C412yes—
64C4×D4×C17D412yes—
65C4×D4×C17D412yes—
66C2×C4⋊C4×C17D411yescharacteristic
67C2×C22⋊C4×C17D411yescharacteristic
68C2×C22⋊C4×C17D411yescharacteristic
69C2×C42×C17C2311yescharacteristic
70#10587639—22no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75#10587625—22no—
76#10587625—22no—
77#10587623—22no—
78#10587621—22no—
79#10587616—21no—
80#10587616—21no—
81#10587616C2×C412yes—
82#10587616C2×C412yescomplemented
83#10587616C2312yes—
84#10587615—22no—
85#10587615—22no—
86C4×C22⋊D17—22no—
87C4×C22⋊D17—22no—
88C4×C22⋊D17—22no—
89C2×#63227—21no—
90#10587612C2×C412yescomplemented
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17C2×C411yescharacteristic
93C2×C4⋊Dic17C2×C411yescharacteristic
94#10587610—21no—
95#10587610—21no—
96#10587610C2×C412yes—
97#10587610C2×C412yescomplemented
98#10587610C2312yes—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102#10587584—22no—
103#10587582—22no—
104#10587580—22no—
105#10587575—22no—
106#10587571—22no—
107#10587570—22no—
108#10587569—22no—
109#10587567—22no—
110#10587567—22no—
111C22⋊C4×D17—22no—
112#10587565—22no—
113#10587563—22no—
114#10587562—22no—
115#10587561C2×C412yescomplemented
116C4×D68C2×C412yescomplemented
117#10587558—22no—
118#10587556C2×C412yescomplemented
119C4×Dic34C2×C412yescomplemented
120C17×C4⋊C8—22no—
121C17×C22⋊C8—21no—
122C17×C22⋊C8—21no—
123C17×C22⋊C8—22no—
124C17×C22⋊C8C2312yes—
125C17×C22⋊C8C2312yes—
126C4×C8×C17—22no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587502C2311yescharacteristic
133#10587502C2311yescharacteristic
134#10587502C2311yescharacteristic
135#10587502C2311yescharacteristic
136C24×C17C2×D411yescharacteristic
137C24×C17C2×D411yescharacteristic
138C23×D17—21no—
139C2×D4×C17—22no—
140C2×D4×C17—22no—
141C2×D4×C17C22⋊C412yes—
142C2×D4×C17C22⋊C412yes—
143C22×C4×C17—81no—
144C22×C4×C17—81no—
145C22×C4×C17C22×C411yescharacteristic
146C22×C4×C17C22×C411yescharacteristic
147C22×C4×C17C22×C412yes—
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×C17C2×D412yes—
155C22×C4×C17C2411yescharacteristic
156C2×C22⋊D17—22no—
157C2×C22⋊D17—22no—
158C2×C22⋊D17—22no—
159C2×C22⋊D17—22no—
160C2×C22⋊D17—42no—
161C2×C22⋊D17—42no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—42no—
170C22×Dic17—42no—
171C22×Dic17—81no—
172#63252—82no—
173C2×D68—21no—
174C2×D68—81no—
175C2×D68C22×C412yes—
176C2×C4×D17—22no—
177C2×C4×D17—22no—
178C2×Dic34—21no—
179C2×Dic34—81no—
180C2×Dic34C22×C412yes—
181C2×C8×C17—22no—
182C2×C8×C17—22no—
183C2×C8×C17—81no—
184C2×C8×C17—81no—
185C2×C8×C17—82no—
186C4⋊C4×C17C22⋊C412yes—
187C4⋊C4×C17C22⋊C412yes—
188C22⋊C4×C17C22⋊C412yes—
189C22⋊C4×C17C22⋊C412yes—
190C22⋊C4×C17C22⋊C412yes—
191C22⋊C4×C17C22⋊C412yes—
192C42×C17C22×C412yes—
193C42×C17C2×D412yes—
194C22⋊Dic17—22no—
195C22⋊Dic17—22no—
196C22⋊Dic17—22no—
197C22⋊Dic17—22no—
198C22⋊Dic17—42no—
199C22⋊Dic17—42no—
200C22⋊Dic17—81no—
201#63227—22no—
202#63227—22no—
203#63227—22no—
204#63227—22no—
205#63227—22no—
206#63227—22no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—81no—
210C4⋊Dic17C22×C412yes—
211C4⋊Dic17C22×C412yes—
212#63225—22no—
213#63225—22no—
214#63225—22no—
215#63225—22no—
216#63225—22no—
217#63225—22no—
218C4×Dic17—22no—
219C4×Dic17—22no—
220C4×Dic17—42no—
221C4×Dic17—81no—
222#17615—171nomaximal, Sylow
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17C2×C22⋊C412yes—
230C23×C17C2×C22⋊C411yescharacteristic
231C23×C17C2×C22⋊C411yescharacteristic
232C23×C17#13512yes—
233C23×C17#13512yes—
234C23×C17C22×D411yescharacteristic
235C22×D17—22no—
236C22×D17—42no—
237C22×D17—81no—
238C22×D17—81no—
239D4×C17—24no—
240D4×C17—24no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—41no—
244C2×C4×C17—42no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—42no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17C2×C22⋊C412yes—
254C2×C4×C17C2×C22⋊C412yes—
255C2×C4×C17C2×C22⋊C411yescharacteristic
256C2×C4×C17C2×C22⋊C412yes—
257C2×C4×C17C2×C22⋊C411yescharacteristic
258C2×C4×C17C23×C411yescharacteristic
259C2×C4×C17C22×D411yescharacteristic
260C22⋊D17—42no—
261C22⋊D17—42no—
262C22⋊D17—44no—
263C22⋊D17—82no—
264C22⋊D17—82no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—42no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—81no—
282C2×Dic17—81no—
283C2×Dic17—84no—
284D68—42no—
285D68—81no—
286C4×D17—82no—
287C4×D17—82no—
288Dic34—42no—
289Dic34—81no—
290C8×C17—42no—
291C8×C17—42no—
292#3505—171no—
293#2972D1711yescharacteristic, complemented
294#2353—171no—
295#1521—172no—
296#1520—171no—
297#1519—171no—
298#1516—172no—
299#1501—172no—
300#1500—172no—
301C2×#328—171no—
302C2×#327—171no—
303C22×C17—21no—
304C22×C17—21no—
305C22×C17—21no—
306C22×C17—21no—
307C22×C17—21no—
308C22×C17—21no—
309C22×C17—22no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17#41612yes—
317C22×C17#41612yes—
318C22×C17C22×C22⋊C411yescharacteristic
319C22×C17C2×#13511yescharacteristic
320C22×C17C2×#13511yescharacteristic
321C2×D17—42no—
322C2×D17—42no—
323C2×D17—42no—
324C2×D17—162no—
325C4×C17—21no—
326C4×C17—21no—
327C4×C17—21no—
328C4×C17—22no—
329C4×C17—21no—
330C4×C17—22no—
331C4×C17—22no—
332Dic17—42no—
333Dic17—42no—
334Dic17—42no—
335Dic17—42no—
336Dic17—81no—
337Dic17—81no—
338Dic17—81no—
339#553—172no—
340#553—342no—
341#552—172no—
342#551—342no—
343#548—172no—
344#546—172no—
345#539—341no—
346#538—341no—
347#534—341no—
348#529—172no—
349#525—172no—
350C2×C22⋊Q8—171no—
351C2×#121—171no—
352#518—341no—
353C2×C4×D4C2×D1711yescharacteristic
354C8×D4—22no—
355C8×D4—22no—
356#431C2×D1712yescomplemented
357C2×C22⋊C8C2×D1711yescharacteristic
358C2×C22⋊C8C2×D1711yescharacteristic
359C2×C22⋊C8C2×D1712yescomplemented
360#388—341no—
361#387—341no—
362#386—341no—
363#383—171no—
364#381—171no—
365#380—171no—
366C4×C22⋊C4—171no—
367C4×C22⋊C4—341no—
368#376—171no—
369C2×#95—171no—
370#328—172no—
371#328—172no—
372#328—172no—
373#328—172no—
374#327—172no—
375#327—172no—
376#327—172no—
377#327—172no—
378C2×C17—21no—
379C2×C17—21no—
380C2×C17—22no—
381C2×C17—22no—
382C2×C17—42no—
383C2×C17#288311yescharacteristic
384C2×C17C2×#41611yescharacteristic
385C2×C17#290111yescharacteristic
386D17—82no—
387C2×#41—172no—
388C22×Q8—171no—
389C22×D4—171no—
390C22×D4—341no—
391C22×D4D6811yescharacteristic
392C23×C4—341no—
393C23×C4C22×D1711yescharacteristic
394C23×C4C22×D1711yescharacteristic
395C22×C8—21no—
396C22×C8—21no—
397C22×C8—22no—
398#126—342no—
399#126—342no—
400#126—342no—
401#125—172no—
402#125—342no—
403#124—172no—
404#124—342no—
405#123—342no—
406#123—342no—
407#123—342no—
408#123—342no—
409#123—342no—
410C22⋊Q8—172no—
411C22⋊Q8—172no—
412C22⋊Q8—172no—
413C22⋊Q8—341no—
414C22⋊Q8—341no—
415C22⋊Q8—342no—
416#121—172no—
417#121—172no—
418#121—172no—
419#121—342no—
420#121—342no—
421#121—341no—
422#121—341no—
423#121—342no—
424#120—342no—
425#120—342no—
426C4×Q8—172no—
427C4×D4—172no—
428C4×D4—342no—
429C4×D4—342no—
430C4×D4—342no—
431C4×D4—342no—
432C4×D4—342no—
433C4×D4C4×D1712yes—
434C4×D4C4×D1712yes—
435C4×D4C22⋊D1712yes—
436C4×D4C22⋊D1712yes—
437#117—172no—
438#117—342no—
439#117—342no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—171no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4—341no—
444C2×C4⋊C4D6811yescharacteristic
445C2×C22⋊C4—341no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4—342no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C22⋊C4D6811yescharacteristic
452C2×C22⋊C4D6811yescharacteristic
453C2×C42—341no—
454C2×C42—341no—
455C2×C42C22×D1711yescharacteristic
456C4⋊C8—22no—
457C22⋊C8—21no—
458C22⋊C8—21no—
459C22⋊C8—22no—
460C22⋊C8C22×D1712yes—
461C22⋊C8C22×D1712yes—
462C4×C8—22no—
463#95—171no—
464#95—171no—
465#95—171no—
466#95—171no—
467#95—341no—
468#95—341no—
469#95—341no—
470#95—341no—
471#95—341no—
472C17#1761511yescharacteristic, complemented, Sylow
473C24—341no—
474C24C2×D6811yescharacteristic
475C24C2×D6811yescharacteristic
476#41—1362no—
477C2×Q8—172no—
478C2×Q8—341no—
479C2×Q8—1361no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—172no—
483C2×D4—342no—
484C2×D4—342no—
485C2×D4—342no—
486C2×D4—342no—
487C2×D4—341no—
488C2×D4—682no—
489C2×D4—682no—
490C2×D4—1361no—
491C2×D4#6322712yes—
492C2×D4#6322712yes—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—682no—
505C22×C4—682no—
506C22×C4—1361no—
507C22×C4C2×C4×D1711yescharacteristic
508C22×C4C2×C4×D1711yescharacteristic
509C22×C4C2×C4×D1712yes—
510C22×C4C2×D6811yescharacteristic
511C22×C4C2×D6811yescharacteristic
512C22×C4C2×D6811yescharacteristic
513C22×C4C2×D6811yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C2×C22⋊D1711yescharacteristic
516C22×C4C2×C22⋊D1712yes—
517C22×C4C23×D1711yescharacteristic
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—81no—
521C2×C8—81no—
522C2×C8—82no—
523C4⋊C4—172no—
524C4⋊C4—172no—
525C4⋊C4—342no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—342no—
529C4⋊C4—342no—
530C4⋊C4—342no—
531C4⋊C4—342no—
532C4⋊C4—342no—
533C4⋊C4—1361no—
534C4⋊C4#6322712yes—
535C4⋊C4#6322712yes—
536C22⋊C4—342no—
537C22⋊C4—342no—
538C22⋊C4—342no—
539C22⋊C4—342no—
540C22⋊C4—342no—
541C22⋊C4—342no—
542C22⋊C4—342no—
543C22⋊C4—342no—
544C22⋊C4—342no—
545C22⋊C4—342no—
546C22⋊C4—682no—
547C22⋊C4—682no—
548C22⋊C4—1361no—
549C22⋊C4#6322712yes—
550C22⋊C4#6322712yes—
551C22⋊C4#6322712yes—
552C22⋊C4#6322712yes—
553C42—342no—
554C42—342no—
555C42—682no—
556C42—1361no—
557C42C2×C4×D1712yes—
558C42C2×C22⋊D1712yes—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—342no—
566C23—682no—
567C23—1361no—
568C23—1361no—
569C23#1058759012yes—
570C23#1058759012yes—
571C23C2×#6322712yes—
572C23C2×#6322711yescharacteristic
573C23C2×#6322711yescharacteristic
574C23C22×D6811yescharacteristic
575Q8—682no—
576Q8—1361no—
577D4—24no—
578D4—24no—
579D4—682no—
580D4—684no—
581D4—682no—
582D4—682no—
583D4—1361no—
584D4—1362no—
585D4—1362no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—681no—
603C2×C4—682no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—1362no—
615C2×C4—1362no—
616C2×C4—1361no—
617C2×C4—1361no—
618C2×C4—1364no—
619C2×C4C2×#6322712yes—
620C2×C4C2×#6322712yes—
621C2×C4C2×#6322712yes—
622C2×C4C2×#6322711yescharacteristic
623C2×C4C2×#6322711yescharacteristic
624C2×C4C22×C4×D1711yescharacteristic
625C2×C4C22×C22⋊D1711yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—2722no—
642C22—682no—
643C22—682no—
644C22—682no—
645C22#1176280612yes—
646C22#1176280612yes—
647C22C22×#6322711yescharacteristic
648C22C2×#1058759011yescharacteristic
649C22C2×#1058759011yescharacteristic
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—22no—
656C4—22no—
657C4—682no—
658C4—682no—
659C4—682no—
660C4—682no—
661C4—1361no—
662C4—1361no—
663C4—1361no—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—22no—
668C2—42no—
669C2—1362no—
670C2#1453724411yescharacteristic
671C2C2×#1176280611yescharacteristic
672C2#1453732911yescharacteristic
subgroupG/H
centerC22C22×#63227
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2972C2
socleC22×C17C22×C22⋊C4
prime radical#14542623C2
cosocleC22×C4×C17C24
hypercenter#2972D17
hypocenterC17#17615
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #14531724C2semidirect
3. #14527362C2semidirect
4. #14527357C2semidirect
5. #14527353C2semidirect
6. #14527341C2semidirect
7. #14527276C2semidirect
8. #14527271C2semidirect
9. C2×#11762547C2semidirect
10. C2×#11762544C2semidirect
11. C17×#2972C2semidirect
13. #11763505C4semidirect
14. #11763499C4semidirect
16. #11763468C4semidirect
17. #11763455C4semidirect
29. #11762677C22semidirect
30. #11762672C22semidirect
33. C4×C22⋊Dic17C22semidirect
34. #11762655C22semidirect
36. #11762547C22semidirect
37. #11762547C22semidirect
38. #11762547C22semidirect
39. #11762547C22semidirect
47. C17×#431C22semidirect
50. C2×C17×C22⋊C8C22semidirect
56. C2×#63252C2×C4semidirect
82. #10587616C2×C4semidirect
90. #10587612C2×C4semidirect
97. #10587610C2×C4semidirect
115. #10587561C2×C4semidirect
116. C4×D68C2×C4semidirect
118. #10587556C2×C4semidirect
119. C4×Dic34C2×C4semidirect
293. #2972D17semidirect
356. #431C2×D17semidirect
359. C2×C22⋊C8C2×D17semidirect
472. C17#17615semidirect

Maximal subgroups

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

subgroupindex
#145426232
#145317242
#145273622
#145273572
#145273532
#145273412
#145272762
#145272712
C2×#117625472
C2×#117625442
C17×#29722
#1761517