Finite Group Databasebeta

← #17195257

Subgroups — #17195257

Order 4352.

2858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763285C211yesmaximal, characteristic
2C2×#11763285C211yesmaximal, characteristic
3C2×C4×Q8×D17C211yesmaximal, characteristic
4#14531703C213yesmaximal, complemented
5#14531702C211yesmaximal, characteristic, complemented
6#14531701C213yesmaximal, complemented
7Q8×C4⋊Dic17C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic34C213yesmaximal, complemented
9#14531698C213yesmaximal, complemented
10Q8×#63225C211yesmaximal, characteristic, complemented
11#14531409C213yesmaximal
12#14531406C213yesmaximal
13#14531265C213yesmaximal
14#14531264C213yesmaximal
15C17×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D17C2211yescharacteristic
17#11763285—28no—
18#11763285C4116yescomplemented
19C2×Q8×Dic17C2211yescharacteristic
20C2×Q8×Dic17C2211yescharacteristic
21C2×Q8×Dic17C2211yescharacteristic
22C2×C4×Dic34C2213yes—
23C2×C4×Dic34C2213yes—
24C2×C4×Dic34C2213yes—
25C2×#10587573C2213yes—
26C2×#10587573C2213yes—
27C2×#10587573C2213yes—
28C4×Q8×D17—28no—
29C2×#10587578C2213yes—
30C2×#10587578C2213yes—
31C2×#10587558C2213yes—
32C2×#10587558C2213yes—
33C2×C4⋊C4×D17C2213yes—
34C2×C42×D17C2213yes—
35#11762708C2213yes—
36#11762708C2213yes—
37#11762707C2213yes—
38#11762707C2213yes—
39#11762706C2216yes—
40#11762706C2216yes—
41#11762706C2213yes—
42#11762706C2213yes—
43#11762705C2213yes—
44#11762705C2213yes—
45#11762704C2213yes—
46#11762704C2213yes—
47#11762683C2216yes—
48#11762683C2213yes—
49#11762683C2211yescharacteristic
50#11762680C2213yes—
51#11762680C2213yes—
52#11762679C2213yes—
53#11762678C2216yes—
54#11762678C2213yes—
55#11762674C2213yes—
56#11762673C2216yes—
57C4⋊C4×Dic17C2216yes—
58C4×#63225C2213yes—
59C4×#63225C2213yes—
60C4×C4⋊Dic17C2213yes—
61#11762660C2216yes—
62C2×C4×Q8×C17C2211yescharacteristic
63C2×C4×Q8×C17C2211yescharacteristic
64C2×C4×Q8×C17C2211yescharacteristic
65C17×#389C2213yes—
66C17×#389C2213yes—
67C17×#382C2213yes—
68C4×C4⋊C4×C17C2213yes—
69C22×Q8×C17C2311yescharacteristic
70C2×Q8×D17C2×C418yes—
71C2×Q8×D17D418yes—
72C2×Q8×D17Q818yes—
73C22×C4×D17C2311yescharacteristic
74C22×C4×D17C2313yes—
75C22×C4×D17C2313yes—
76C22×Dic34C2313yes—
77C4×Q8×C17—24no—
78C4×Q8×C17—24no—
79C4×Q8×C17C2×C418yescomplemented
80C2×C4⋊C4×C17C2313yes—
81C2×C4⋊C4×C17C2313yes—
82C2×C4⋊C4×C17C2313yes—
83C2×C4⋊C4×C17C2313yes—
84C2×C4⋊C4×C17C2313yes—
85C2×C42×C17C2313yes—
86C2×C42×C17C2313yes—
87C2×C42×C17C2313yes—
88Q8×Dic17—24no—
89Q8×Dic17—24no—
90Q8×Dic17C2×C418yescomplemented
91C2×#63227C2311yescharacteristic
92C2×#63227C2313yes—
93C2×#63227C2311yescharacteristic
94C2×#63227C2313yes—
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2313yes—
97C2×C4⋊Dic17C2313yes—
98C2×C4⋊Dic17C2313yes—
99C2×C4⋊Dic17C2316yes—
100C2×C4⋊Dic17C2313yes—
101C2×#63225C2311yescharacteristic
102C2×#63225C2313yes—
103C2×#63225C2313yes—
104C2×#63225C2313yes—
105C2×#63225C2316yes—
106C2×#63225C2313yes—
107C2×#63225C2316yes—
108C2×#63225C2313yes—
109C2×#63225C2313yes—
110C2×#63225C2313yes—
111C2×C4×Dic17C2313yes—
112C2×C4×Dic17C2313yes—
113C2×C4×Dic17C2313yes—
114C2×C4×Dic17C2313yes—
115C2×C4×Dic17C2316yes—
116C2×C4×Dic17C2313yes—
117C2×C4×Dic17C2316yes—
118#10587578—212no—
119#10587578C2×C4124yes—
120C4⋊C4×D17—212no—
121#10587573—212no—
122#10587573—212no—
123#10587573C2×C4124yescomplemented
124#10587558—212no—
125#10587558C2×C4124yes—
126C42×D17—212no—
127C4×Dic34—212no—
128C4×Dic34—212no—
129C4×Dic34C2×C4124yescomplemented
130C17×#95C2311yescharacteristic
131C17×#95C2313yes—
132C17×#95C2316yes—
133#10587502C2311yescharacteristic
134#10587502C2313yes—
135#10587502C2313yes—
136#10587502C2313yes—
137#10587502C2313yes—
138#10587502C2316yes—
139#10587502C2313yes—
140C23×D17C2411yescharacteristic
141C2×Q8×C17C22×C414yes—
142C2×Q8×C17C2×D414yes—
143C2×Q8×C17C2×Q814yes—
144C22×C4×C17C2413yes—
145C22×C4×C17C2413yes—
146C22×C4×C17C2411yescharacteristic
147C22×C4×C17C2413yes—
148C22×C4×C17C2411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×C4×C17C2413yes—
151C22×Dic17C2413yes—
152C22×Dic17C2411yescharacteristic
153C22×Dic17C2413yes—
154C22×Dic17C2411yescharacteristic
155C22×Dic17C2413yes—
156C22×Dic17C2411yescharacteristic
157C22×Dic17C2413yes—
158Q8×D17C4⋊C4164yescomplemented
159C2×C4×D17—22no—
160C2×C4×D17—22no—
161C2×C4×D17—22no—
162C2×C4×D17—26no—
163C2×C4×D17—43no—
164C2×C4×D17—43no—
165C2×C4×D17C22×C4112yes—
166C2×C4×D17C2×D4112yes—
167C2×C4×D17C2×Q8112yes—
168C2×Dic34C22×C4112yes—
169C2×Dic34C2×D4112yes—
170C2×Dic34C2×Q8112yes—
171C4⋊C4×C17—26no—
172C4⋊C4×C17—26no—
173C4⋊C4×C17—26no—
174C4⋊C4×C17—43no—
175C4⋊C4×C17C22×C4112yes—
176C42×C17—26no—
177C42×C17—26no—
178C42×C17C22×C4112yes—
179#63227—22no—
180#63227—26no—
181#63227C22×C414yes—
182#63227C22×C4112yes—
183C4⋊Dic17—26no—
184C4⋊Dic17—26no—
185C4⋊Dic17—22no—
186C4⋊Dic17—43no—
187C4⋊Dic17—46no—
188C4⋊Dic17C22×C4112yes—
189#63225—22no—
190#63225—26no—
191#63225—212no—
192#63225—26no—
193#63225—26no—
194#63225—43no—
195#63225—46no—
196#63225—43no—
197#63225C22×C4112yes—
198#63225C22×C4112yes—
199C4×Dic17—26no—
200C4×Dic17—26no—
201C4×Dic17—26no—
202C4×Dic17—212no—
203C4×Dic17—26no—
204C4×Dic17C22×C4124yes—
205C4×Dic17C22×C4112yes—
206#34249—171nomaximal, Sylow
207C23×C17C2511yescharacteristic
208C22×D17—21no—
209C22×D17—21no—
210C22×D17—21no—
211C22×D17—21no—
212C22×D17C23×C412yes—
213C22×D17C22×D412yes—
214C22×D17C22×Q812yes—
215Q8×C17C2×C4⋊C4116yes—
216C2×C4×C17—21no—
217C2×C4×C17—23no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—23no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—23no—
224C2×C4×C17—23no—
225C2×C4×C17—23no—
226C2×C4×C17—23no—
227C2×C4×C17—21no—
228C2×C4×C17—23no—
229C2×C4×C17—23no—
230C2×C4×C17—21no—
231C2×C4×C17C23×C412yes—
232C2×C4×C17C23×C416yes—
233C2×C4×C17C23×C416yes—
234C2×C4×C17C22×D416yes—
235C2×C4×C17C22×Q816yes—
236C2×C4×C172−1+412yes—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—23no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—23no—
243C2×Dic17—23no—
244C2×Dic17—21no—
245C2×Dic17—23no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—23no—
249C2×Dic17—23no—
250C2×Dic17—23no—
251C2×Dic17—23no—
252C2×Dic17C23×C412yes—
253C2×Dic17C23×C416yes—
254C2×Dic17C23×C416yes—
255C2×Dic17C22×D416yes—
256C2×Dic17C22×Q816yes—
257C2×Dic172−1+412yes—
258C4×D17—28no—
259C4×D17—412no—
260C4×D17C2×C4⋊C4148yes—
261Dic34C2×C4⋊C4148yes—
262C2×#519—171no—
263C2×#519—171no—
264C22×C4×Q8—171no—
265#2351—173no—
266#2351—173no—
267#2351—171no—
268#2351D1711yescharacteristic, complemented
269Q8×C4⋊C4—173no—
270Q8×C4⋊C4—173no—
271Q8×C4⋊C4—171no—
272Q8×C4⋊C4—171no—
273#2317—173no—
274#2317—173no—
275#2296—173no—
276#2296—173no—
277C22×C17C24×C411yescharacteristic
278C22×C17C23×D411yescharacteristic
279C22×C17C23×Q811yescharacteristic
280C22×C17C2×2−1+411yescharacteristic
281C22×C17C2×2−1+411yescharacteristic
282C22×C17#58511yescharacteristic
283C22×C17#58511yescharacteristic
284C2×D17—22no—
285C2×D17—22no—
286C2×D17—22no—
287C2×D17—22no—
288C2×D17—22no—
289C2×D17—22no—
290C2×D17C22×C4⋊C414yes—
291C4×C17—22no—
292C4×C17—26no—
293C4×C17—22no—
294C4×C17—43no—
295C4×C17—43no—
296C4×C17—41no—
297C4×C17C22×C4⋊C4112yes—
298Dic17—26no—
299Dic17—22no—
300Dic17—22no—
301Dic17—41no—
302Dic17—43no—
303Dic17—43no—
304Dic17C22×C4⋊C4112yes—
305C23×Q8—171no—
306#519—1716no—
307#519—348no—
308C2×C4×Q8—171no—
309C2×C4×Q8—171no—
310C2×C4×Q8—171no—
311C2×C4×Q8—173no—
312C2×C4×Q8—173no—
313C2×C4×Q8—173no—
314C2×C4×Q8—173no—
315C2×C4×Q8—173no—
316C2×C4×Q8—173no—
317C2×C4×Q8—348no—
318C2×C4×Q8C2×D1711yescharacteristic
319C2×C4×Q8C2×D1711yescharacteristic
320C2×C4×Q8C2×D1711yescharacteristic
321C2×#117—173no—
322C2×#117—173no—
323C2×#117—173no—
324C2×#117—173no—
325C22×C4⋊C4—173no—
326C22×C42—173no—
327#389—173no—
328#389—173no—
329#389—173no—
330#389—173no—
331#389—173no—
332#389—173no—
333#389—173no—
334#389—173no—
335#389—176no—
336#389—176no—
337#389—173no—
338#389—173no—
339#389C2×D1713yes—
340#389C2×D1713yes—
341#385—173no—
342#385—173no—
343#385—176no—
344#385—173no—
345#385—171no—
346#384—176no—
347#384—173no—
348#384—173no—
349#382—173no—
350#382—176no—
351#382C2×D1713yes—
352C4×C4⋊C4—176no—
353C4×C4⋊C4—173no—
354C4×C4⋊C4—173no—
355C4×C4⋊C4—173no—
356C4×C4⋊C4—176no—
357C4×C4⋊C4C2×D1713yes—
358C2×C17C23×C4⋊C411yescharacteristic
359C2×C17C4×2−1+411yescharacteristic
360C2×C17C4×2−1+411yescharacteristic
361C2×C17#348611yescharacteristic
362C2×C17#348711yescharacteristic
363C2×C17#350211yescharacteristic
364C2×C17#350211yescharacteristic
365D17—24no—
366C22×Q8—178no—
367C22×Q8—178no—
368C22×Q8—178no—
369C22×Q8—173no—
370C22×Q8C22×D1711yescharacteristic
371C23×C4—173no—
372C23×C4—171no—
373C23×C4—173no—
374C4×Q8—24no—
375C4×Q8—24no—
376C4×Q8—178no—
377C4×Q8—1724no—
378C4×Q8—1724no—
379C4×Q8—3412no—
380C4×Q8—3412no—
381C4×Q8—3412no—
382C4×Q8—3412no—
383C4×Q8—344no—
384C4×Q8—344no—
385C4×Q8C4×D1718yescomplemented
386#117—1724no—
387#117—1724no—
388#117—3412no—
389#117—3412no—
390C2×C4⋊C4—173no—
391C2×C4⋊C4—173no—
392C2×C4⋊C4—173no—
393C2×C4⋊C4—176no—
394C2×C4⋊C4—173no—
395C2×C4⋊C4—173no—
396C2×C4⋊C4—173no—
397C2×C4⋊C4—173no—
398C2×C4⋊C4—173no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—173no—
402C2×C4⋊C4—173no—
403C2×C4⋊C4—176no—
404C2×C4⋊C4—176no—
405C2×C4⋊C4—173no—
406C2×C4⋊C4—3412no—
407C2×C4⋊C4C22×D1713yes—
408C2×C4⋊C4C22×D1713yes—
409C2×C4⋊C4C22×D1713yes—
410C2×C4⋊C4C22×D1713yes—
411C2×C4⋊C4C22×D1713yes—
412C2×C22⋊C4—173no—
413C2×C22⋊C4—171no—
414C2×C22⋊C4—171no—
415C2×C22⋊C4—173no—
416C2×C42—176no—
417C2×C42—173no—
418C2×C42—176no—
419C2×C42—173no—
420C2×C42—173no—
421C2×C42—173no—
422C2×C42—173no—
423C2×C42—3412no—
424C2×C42C22×D1713yes—
425C2×C42C22×D1713yes—
426C2×C42C22×D1713yes—
427#95—171no—
428#95—173no—
429#95—173no—
430#95—173no—
431#95—173no—
432#95—176no—
433#95—173no—
434#95C22×D1716yes—
435#95C22×D1711yescharacteristic
436#95C22×D1713yes—
437C17#3424911yescharacteristic, complemented, Sylow
438C24—171no—
439C2×Q8—1764no—
440C2×Q8—1712no—
441C2×Q8—1712no—
442C2×Q8—1712no—
443C2×Q8C2×C4×D1714yes—
444C2×Q8D4×D1714yes—
445C2×Q8Q8×D1714yes—
446C22×C4—173no—
447C22×C4—1712no—
448C22×C4—1712no—
449C22×C4—1712no—
450C22×C4—171no—
451C22×C4—173no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—173no—
455C22×C4—173no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—346no—
459C22×C4—342no—
460C22×C4—683no—
461C22×C4—683no—
462C22×C4C23×D1713yes—
463C22×C4C23×D1713yes—
464C22×C4C23×D1711yescharacteristic
465C22×C4C23×D1713yes—
466C22×C4C23×D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468C22×C4C23×D1713yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—26no—
472C4⋊C4—43no—
473C4⋊C4—1712no—
474C4⋊C4—1712no—
475C4⋊C4—1712no—
476C4⋊C4—3412no—
477C4⋊C4—346no—
478C4⋊C4—346no—
479C4⋊C4—346no—
480C4⋊C4—342no—
481C4⋊C4—346no—
482C4⋊C4—346no—
483C4⋊C4—342no—
484C4⋊C4—683no—
485C4⋊C4—683no—
486C4⋊C4—686no—
487C4⋊C4—686no—
488C4⋊C4—683no—
489C4⋊C4C2×C4×D17112yes—
490C22⋊C4—1712no—
491C22⋊C4—174no—
492C22⋊C4—342no—
493C22⋊C4—346no—
494C42—26no—
495C42—26no—
496C42—1724no—
497C42—1712no—
498C42—346no—
499C42—346no—
500C42—3412no—
501C42—346no—
502C42—346no—
503C42C2×C4×D17112yes—
504C23—172no—
505C23—172no—
506C23—172no—
507C23—341no—
508C23—341no—
509C23—341no—
510C23—341no—
511C23C24×D1711yescharacteristic
512Q8—1748no—
513Q8C4⋊C4×D17116yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—21no—
526C2×C4—23no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—176no—
530C2×C4—176no—
531C2×C4—176no—
532C2×C4—1748no—
533C2×C4—176no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—348no—
537C2×C4—341no—
538C2×C4—343no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—343no—
543C2×C4—341no—
544C2×C4—343no—
545C2×C4—343no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—343no—
549C2×C4—343no—
550C2×C4—343no—
551C2×C4—343no—
552C2×C4—6812no—
553C2×C4C22×C4×D1716yes—
554C2×C4C22×C4×D1712yes—
555C2×C4C22×C4×D1716yes—
556C2×C4C2×D4×D1716yes—
557C2×C4C2×Q8×D1716yes—
558C2×C4#1058769712yes—
559C22—174no—
560C22—342no—
561C22—342no—
562C22—342no—
563C22—342no—
564C22—342no—
565C22—342no—
566C22C23×C4×D1711yescharacteristic
567C22C22×D4×D1711yescharacteristic
568C22C22×Q8×D1711yescharacteristic
569C222−1+4×D1711yescharacteristic
570C22C2×#1058769711yescharacteristic
571C22#1176364511yescharacteristic
572C22#1176364611yescharacteristic
573C4—22no—
574C4—22no—
575C4—26no—
576C4—43no—
577C4—43no—
578C4—41no—
579C4—1712no—
580C4—342no—
581C4—346no—
582C4—342no—
583C4—681no—
584C4—683no—
585C4—683no—
586C4C2×C4⋊C4×D17112yes—
587C2—344no—
588C2C22×C4⋊C4×D1711yescharacteristic
589C2#1454199311yescharacteristic
590C2C4×#1058769711yescharacteristic
591C2#1454243011yescharacteristic
592C2#1454244111yescharacteristic
593C2#1454258811yescharacteristic
594C2#1454259111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2351C2
socle, cosocleC23×C17C25
prime radicalC23×D17C24
hypercenter#2351D17
hypocenterC17#34249
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #14531703C2semidirect
5. #14531702C2semidirect
6. #14531701C2semidirect
7. Q8×C4⋊Dic17C2semidirect
8. C4⋊C4×Dic34C2semidirect
9. #14531698C2semidirect
10. Q8×#63225C2semidirect
15. C17×#2351C2semidirect
18. #11763285C4semidirect
79. C4×Q8×C17C2×C4semidirect
90. Q8×Dic17C2×C4semidirect
123. #10587573C2×C4semidirect
129. C4×Dic34C2×C4semidirect
158. Q8×D17C4⋊C4semidirect
268. #2351D17semidirect
385. C4×Q8C4×D17semidirect
437. C17#34249semidirect

Maximal subgroups

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

subgroupindex
C2×#117632852
C2×C4×Q8×D172
#145317032
#145317022
#145317012
Q8×C4⋊Dic172
C4⋊C4×Dic342
#145316982
Q8×#632252
#145314092
#145314062
#145312652
#145312642
C17×#23512
#3424917