Finite Group Databasebeta

← #17195270

Subgroups — #17195270

Order 4352.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763291C211yesmaximal, characteristic
2C2×#11763291C211yesmaximal, characteristic
3C2×C4×#63256C211yesmaximal, characteristic
4#14531702C211yesmaximal, characteristic, complemented
5Q8×C4⋊Dic17C211yesmaximal, characteristic, complemented
6Q8×#63225C211yesmaximal, characteristic, complemented
7#14531693C213yesmaximal, complemented
8#14531692C213yesmaximal, complemented
9C4⋊C4×D68C213yesmaximal, complemented
10#14531685C213yesmaximal, complemented
11#14531409C213yesmaximal, complemented
12#14531399C213yesmaximal, complemented
13#14531264C213yesmaximal, complemented
14D17×#389C213yesmaximal, complemented
15C17×#2351C211yesmaximal, characteristic, complemented
16C22×#63256C2211yescharacteristic
17#11763291—28no—
18#11763291C4116yescomplemented
19C4×#63256—28no—
20C2×Q8×Dic17C2211yescharacteristic
21C2×Q8×Dic17C2211yescharacteristic
22C2×Q8×Dic17C2211yescharacteristic
23C2×C4×D68C2213yes—
24C2×C4×D68C2213yes—
25C2×C4×D68C2213yes—
26C2×#10587579C2213yes—
27C2×#10587579C2213yes—
28C2×#10587579C2213yes—
29C2×#10587578C2213yes—
30C2×#10587558C2213yes—
31C2×#10587558C2213yes—
32C2×C4⋊C4×D17C2213yes—
33C2×C4⋊C4×D17C2216yes—
34C2×C4⋊C4×D17C2213yes—
35C2×C42×D17C2213yes—
36#11762708C2213yes—
37#11762708C2213yes—
38#11762707C2213yes—
39#11762707C2213yes—
40#11762706C2216yes—
41#11762706C2213yes—
42#11762706C2211yescharacteristic
43#11762705C2213yes—
44#11762705C2213yes—
45#11762704C2213yes—
46#11762704C2213yes—
47#11762683C2216yes—
48#11762683C2216yes—
49#11762683C2213yes—
50#11762683C2213yes—
51#11762680C2213yes—
52#11762680C2213yes—
53#11762679C2213yes—
54#11762678C2213yes—
55#11762674C2213yes—
56#11762668C2216yes—
57#11762666C2216yes—
58C4×#63225C2213yes—
59C4×#63225C2213yes—
60C4×C4⋊Dic17C2213yes—
61#95×D17C2216yes—
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×#63256C2×C418yes—
71C2×#63256D418yes—
72C2×#63256Q818yes—
73C22×D68C2313yes—
74C22×C4×D17C2313yes—
75C22×C4×D17C2313yes—
76C22×C4×D17C2316yes—
77C22×C4×D17C2316yes—
78C22×C4×D17C2313yes—
79C4×Q8×C17—24no—
80C4×Q8×C17—24no—
81C4×Q8×C17C2×C418yescomplemented
82C2×C4⋊C4×C17C2313yes—
83C2×C4⋊C4×C17C2313yes—
84C2×C4⋊C4×C17C2313yes—
85C2×C4⋊C4×C17C2313yes—
86C2×C4⋊C4×C17C2313yes—
87C2×C42×C17C2313yes—
88C2×C42×C17C2313yes—
89C2×C42×C17C2313yes—
90Q8×Dic17—24no—
91Q8×Dic17—24no—
92Q8×Dic17C2×C418yescomplemented
93C2×#63227C2313yes—
94C2×#63227C2313yes—
95C2×#63227C2316yes—
96C2×#63227C2313yes—
97C2×#63227C2313yes—
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2313yes—
100C2×C4⋊Dic17C2313yes—
101C2×C4⋊Dic17C2316yes—
102C2×C4⋊Dic17C2313yes—
103C2×C4⋊Dic17C2313yes—
104C2×#63225C2311yescharacteristic
105C2×#63225C2313yes—
106C2×#63225C2313yes—
107C2×#63225C2311yescharacteristic
108C2×#63225C2313yes—
109C2×#63225C2316yes—
110C2×#63225C2313yes—
111C2×#63225C2313yes—
112C2×#63225C2311yescharacteristic
113C2×C4×Dic17C2311yescharacteristic
114C2×C4×Dic17C2313yes—
115C2×C4×Dic17C2313yes—
116C2×C4×Dic17C2313yes—
117C2×C4×Dic17C2313yes—
118#10587579—212no—
119#10587579—212no—
120#10587579C2×C4124yescomplemented
121#10587578—212no—
122C4⋊C4×D17—212no—
123C4⋊C4×D17—412no—
124C4⋊C4×D17C2×C4124yescomplemented
125C4×D68—212no—
126C4×D68—212no—
127C4×D68C2×C4124yescomplemented
128#10587558—212no—
129#10587558C2×C4124yescomplemented
130C42×D17—212no—
131C17×#95C2311yescharacteristic
132C17×#95C2313yes—
133C17×#95C2316yes—
134#10587502C2313yes—
135#10587502C2313yes—
136#10587502C2313yes—
137#10587502C2313yes—
138#10587502C2316yes—
139#10587502C2311yescharacteristic
140#10587502C2313yes—
141C23×D17C2413yes—
142C2×Q8×C17C22×C414yes—
143C2×Q8×C17C2×D414yes—
144C2×Q8×C17C2×Q814yes—
145C22×C4×C17C2413yes—
146C22×C4×C17C2413yes—
147C22×C4×C17C2411yescharacteristic
148C22×C4×C17C2413yes—
149C22×C4×C17C2411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×C4×C17C2413yes—
152C22×Dic17C2413yes—
153C22×Dic17C2413yes—
154C22×Dic17C2411yescharacteristic
155C22×Dic17C2413yes—
156C22×Dic17C2411yescharacteristic
157C22×Dic17C2411yescharacteristic
158C22×Dic17C2411yescharacteristic
159#63256C4⋊C4164yescomplemented
160C2×D68C22×C4112yes—
161C2×D68C2×D4112yes—
162C2×D68C2×Q8112yes—
163C2×C4×D17—26no—
164C2×C4×D17—26no—
165C2×C4×D17—26no—
166C2×C4×D17—26no—
167C2×C4×D17—212no—
168C2×C4×D17—46no—
169C2×C4×D17—43no—
170C2×C4×D17—43no—
171C2×C4×D17—46no—
172C2×C4×D17—46no—
173C2×C4×D17—46no—
174C2×C4×D17C22×C4124yes—
175C2×C4×D17C22×C4112yes—
176C2×C4×D17C2×D4112yes—
177C2×C4×D17C2×Q8112yes—
178C4⋊C4×C17—26no—
179C4⋊C4×C17—26no—
180C4⋊C4×C17—26no—
181C4⋊C4×C17—43no—
182C4⋊C4×C17C22×C4112yes—
183C42×C17—26no—
184C42×C17—26no—
185C42×C17C22×C4112yes—
186#63227—26no—
187#63227—26no—
188#63227—212no—
189#63227C22×C4112yes—
190#63227C22×C4112yes—
191C4⋊Dic17—26no—
192C4⋊Dic17—26no—
193C4⋊Dic17—22no—
194C4⋊Dic17—46no—
195C4⋊Dic17—43no—
196C4⋊Dic17C22×C4112yes—
197#63225—26no—
198#63225—22no—
199#63225—26no—
200#63225—22no—
201#63225—46no—
202#63225—43no—
203#63225—43no—
204#63225C22×C414yes—
205#63225C22×C4112yes—
206C4×Dic17—26no—
207C4×Dic17—26no—
208C4×Dic17—22no—
209C4×Dic17—26no—
210C4×Dic17C22×C4112yes—
211#34250—171nomaximal, Sylow
212C23×C17C2511yescharacteristic
213C22×D17—23no—
214C22×D17—23no—
215C22×D17—23no—
216C22×D17—23no—
217C22×D17C23×C416yes—
218C22×D17C22×D416yes—
219C22×D17C22×Q816yes—
220Q8×C17C2×C4⋊C4116yescomplemented
221C2×C4×C17—21no—
222C2×C4×C17—23no—
223C2×C4×C17—21no—
224C2×C4×C17—23no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—23no—
229C2×C4×C17—23no—
230C2×C4×C17—23no—
231C2×C4×C17—23no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—23no—
235C2×C4×C17—23no—
236C2×C4×C17C23×C412yes—
237C2×C4×C17C23×C416yes—
238C2×C4×C17C23×C416yes—
239C2×C4×C17C22×D416yes—
240C2×C4×C17C22×Q816yes—
241C2×C4×C17H(5,𝔽2)12yes—
242C2×Dic17—21no—
243C2×Dic17—23no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—23no—
247C2×Dic17—21no—
248C2×Dic17—23no—
249C2×Dic17—23no—
250C2×Dic17—21no—
251C2×Dic17—23no—
252C2×Dic17—23no—
253C2×Dic17—21no—
254C2×Dic17—23no—
255C2×Dic17—23no—
256C2×Dic17—21no—
257C2×Dic17C23×C416yes—
258C2×Dic17C23×C412yes—
259C2×Dic17C23×C412yes—
260C2×Dic17C22×D412yes—
261C2×Dic17C22×Q812yes—
262C2×Dic172−1+412yes—
263D68C2×C4⋊C4148yescomplemented
264C4×D17—424no—
265C4×D17—412no—
266C4×D17—412no—
267C4×D17—812no—
268C4×D17C2×C4⋊C4148yescomplemented
269C2×#520—171no—
270C2×#520—171no—
271C2×C4×#41—171no—
272#2351—171no—
273#2351D1711yescharacteristic, complemented
274Q8×C4⋊C4—171no—
275Q8×C4⋊C4—171no—
276#2349—173no—
277#2349—173no—
278D4×C4⋊C4—173no—
279D4×C4⋊C4—173no—
280#2317—173no—
281#2316—173no—
282#2296—173no—
283C2×#389—173no—
284C22×C17C24×C411yescharacteristic
285C22×C17C23×D411yescharacteristic
286C22×C17C23×Q811yescharacteristic
287C22×C17C2×H(5,𝔽2)11yescharacteristic
288C22×C17C2×2−1+411yescharacteristic
289C22×C17#58511yescharacteristic
290C22×C17#58511yescharacteristic
291C2×D17—26no—
292C2×D17—26no—
293C2×D17—26no—
294C2×D17—26no—
295C2×D17—26no—
296C2×D17—26no—
297C2×D17C22×C4⋊C4112yes—
298C4×C17—26no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—43no—
302C4×C17—43no—
303C4×C17—41no—
304C4×C17C22×C4⋊C4112yes—
305Dic17—26no—
306Dic17—22no—
307Dic17—22no—
308Dic17—43no—
309Dic17—43no—
310Dic17—41no—
311Dic17C22×C4⋊C414yes—
312C22×#41—171no—
313#520—1716no—
314#520—348no—
315C4×#41—348no—
316C2×C4×Q8—171no—
317C2×C4×Q8—171no—
318C2×C4×Q8—171no—
319C2×C4×Q8C2×D1711yescharacteristic
320C2×C4×Q8C2×D1711yescharacteristic
321C2×C4×Q8C2×D1711yescharacteristic
322C2×C4×D4—173no—
323C2×C4×D4—173no—
324C2×C4×D4—173no—
325C2×C4×D4—173no—
326C2×C4×D4—173no—
327C2×C4×D4—173no—
328C2×#117—173no—
329C2×#117—173no—
330C2×#117—173no—
331C22×C4⋊C4—173no—
332C22×C4⋊C4—173no—
333C22×C4⋊C4—176no—
334C22×C42—173no—
335#389—173no—
336#389—173no—
337#389—173no—
338#389—173no—
339#389—173no—
340#389—173no—
341#389—173no—
342#389—173no—
343#389—176no—
344#389—173no—
345#389—171no—
346#389C2×D1713yes—
347#389C2×D1713yes—
348#385—173no—
349#385—173no—
350#385—176no—
351#385—176no—
352#385—173no—
353#385—173no—
354#384—173no—
355#384—173no—
356#382—173no—
357#382C2×D1713yes—
358#380—176no—
359#380—176no—
360C4×C4⋊C4—173no—
361C4×C4⋊C4—173no—
362C4×C4⋊C4—173no—
363C4×C4⋊C4C2×D1713yes—
364C2×#95—176no—
365C2×C17C23×C4⋊C411yescharacteristic
366C2×C17C4×H(5,𝔽2)11yescharacteristic
367C2×C17C4×2−1+411yescharacteristic
368C2×C17#348811yescharacteristic
369C2×C17#348911yescharacteristic
370C2×C17#350311yescharacteristic
371C2×C17#350411yescharacteristic
372D17—212no—
373C2×#41—178no—
374C2×#41—178no—
375C2×#41—178no—
376C22×Q8C22×D1711yescharacteristic
377C22×D4—173no—
378C23×C4—173no—
379C23×C4—176no—
380C23×C4—173no—
381C23×C4—173no—
382C23×C4—176no—
383C4×Q8—24no—
384C4×Q8—24no—
385C4×Q8—178no—
386C4×Q8—344no—
387C4×Q8—344no—
388C4×Q8C4×D1718yescomplemented
389C4×D4—1724no—
390C4×D4—1724no—
391C4×D4—3412no—
392C4×D4—3412no—
393C4×D4—3412no—
394C4×D4—3412no—
395#117—1724no—
396#117—3412no—
397#117—3412no—
398C2×C4⋊C4—173no—
399C2×C4⋊C4—173no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—1724no—
402C2×C4⋊C4—173no—
403C2×C4⋊C4—173no—
404C2×C4⋊C4—173no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—173no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—176no—
410C2×C4⋊C4—176no—
411C2×C4⋊C4—173no—
412C2×C4⋊C4—173no—
413C2×C4⋊C4—173no—
414C2×C4⋊C4—3412no—
415C2×C4⋊C4—6812no—
416C2×C4⋊C4C22×D1713yes—
417C2×C4⋊C4C22×D1713yes—
418C2×C4⋊C4C22×D1713yes—
419C2×C4⋊C4C22×D1713yes—
420C2×C4⋊C4C22×D1713yes—
421C2×C22⋊C4—173no—
422C2×C22⋊C4—173no—
423C2×C22⋊C4—173no—
424C2×C22⋊C4—173no—
425C2×C22⋊C4—176no—
426C2×C42—173no—
427C2×C42—173no—
428C2×C42—173no—
429C2×C42—173no—
430C2×C42—171no—
431C2×C42—3412no—
432C2×C42C22×D1713yes—
433C2×C42C22×D1713yes—
434C2×C42C22×D1713yes—
435#95—173no—
436#95—173no—
437#95—176no—
438#95—171no—
439#95—173no—
440#95—173no—
441#95—173no—
442#95C22×D1716yes—
443#95C22×D1711yescharacteristic
444#95C22×D1713yes—
445C17#3425011yescharacteristic, complemented, Sylow
446C24—173no—
447#41—1764no—
448C2×Q8C2×C4×D1714yes—
449C2×Q8D4×D1714yes—
450C2×Q8Q8×D1714yes—
451C2×D4—1712no—
452C2×D4—1712no—
453C2×D4—1712no—
454C22×C4—171no—
455C22×C4—1712no—
456C22×C4—1712no—
457C22×C4—1712no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—173no—
461C22×C4—1724no—
462C22×C4—171no—
463C22×C4—173no—
464C22×C4—173no—
465C22×C4—346no—
466C22×C4—346no—
467C22×C4—346no—
468C22×C4—3412no—
469C22×C4—346no—
470C22×C4—686no—
471C22×C4—686no—
472C22×C4—683no—
473C22×C4—683no—
474C22×C4—686no—
475C22×C4—686no—
476C22×C4C23×D1713yes—
477C22×C4C23×D1711yescharacteristic
478C22×C4C23×D1713yes—
479C22×C4C23×D1713yes—
480C22×C4C23×D1711yescharacteristic
481C22×C4C23×D1711yescharacteristic
482C22×C4C23×D1713yes—
483C4⋊C4—26no—
484C4⋊C4—26no—
485C4⋊C4—26no—
486C4⋊C4—43no—
487C4⋊C4—1712no—
488C4⋊C4—1712no—
489C4⋊C4—174no—
490C4⋊C4—346no—
491C4⋊C4—342no—
492C4⋊C4—342no—
493C4⋊C4—342no—
494C4⋊C4—346no—
495C4⋊C4—346no—
496C4⋊C4—346no—
497C4⋊C4—686no—
498C4⋊C4—686no—
499C4⋊C4—683no—
500C4⋊C4—683no—
501C4⋊C4—683no—
502C4⋊C4C2×C4×D17112yes—
503C22⋊C4—1712no—
504C22⋊C4—1712no—
505C22⋊C4—346no—
506C22⋊C4—3412no—
507C22⋊C4—346no—
508C42—26no—
509C42—26no—
510C42—1712no—
511C42—342no—
512C42—346no—
513C42—346no—
514C42—346no—
515C42C2×C4×D17112yes—
516C23—176no—
517C23—176no—
518C23—176no—
519C23—343no—
520C23—343no—
521C23—343no—
522C23—343no—
523C23C24×D1711yescharacteristic
524Q8C4⋊C4×D17116yescomplemented
525D4—1748no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—21no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—23no—
536C2×C4—21no—
537C2×C4—23no—
538C2×C4—21no—
539C2×C4—23no—
540C2×C4—23no—
541C2×C4—172no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—1748no—
545C2×C4—172no—
546C2×C4—176no—
547C2×C4—172no—
548C2×C4—341no—
549C2×C4—343no—
550C2×C4—341no—
551C2×C4—343no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—343no—
556C2×C4—343no—
557C2×C4—343no—
558C2×C4—343no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—343no—
562C2×C4—343no—
563C2×C4—6824no—
564C2×C4—6812no—
565C2×C4—6812no—
566C2×C4—13612no—
567C2×C4C22×C4×D1712yes—
568C2×C4C22×C4×D1716yes—
569C2×C4C22×C4×D1716yes—
570C2×C4C2×D4×D1716yes—
571C2×C4C2×Q8×D1716yes—
572C2×C4#1058769612yes—
573C22—1712no—
574C22—346no—
575C22—346no—
576C22—346no—
577C22—346no—
578C22—346no—
579C22—346no—
580C22C23×C4×D1711yescharacteristic
581C22C22×D4×D1711yescharacteristic
582C22C22×Q8×D1711yescharacteristic
583C22C2×#1058769611yescharacteristic
584C222−1+4×D1711yescharacteristic
585C22#1176364511yescharacteristic
586C22#1176364611yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4—174no—
594C4—342no—
595C4—346no—
596C4—342no—
597C4—683no—
598C4—683no—
599C4—681no—
600C4C2×C4⋊C4×D17112yes—
601C2—3412no—
602C2C22×C4⋊C4×D1711yescharacteristic
603C2C4×#1058769611yescharacteristic
604C2#1454199311yescharacteristic
605C2#1454245011yescharacteristic
606C2#1454246311yescharacteristic
607C2#1454260111yescharacteristic
608C2#1454260911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2351C2
socle, cosocleC23×C17C25
prime radicalC22×#63256C22
hypercenter#2351D17
hypocenterC17#34250
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #14531702C2semidirect
5. Q8×C4⋊Dic17C2semidirect
6. Q8×#63225C2semidirect
7. #14531693C2semidirect
8. #14531692C2semidirect
9. C4⋊C4×D68C2semidirect
10. #14531685C2semidirect
11. #14531409C2semidirect
12. #14531399C2semidirect
13. #14531264C2semidirect
14. D17×#389C2semidirect
15. C17×#2351C2semidirect
18. #11763291C4semidirect
81. C4×Q8×C17C2×C4semidirect
92. Q8×Dic17C2×C4semidirect
120. #10587579C2×C4semidirect
124. C4⋊C4×D17C2×C4semidirect
127. C4×D68C2×C4semidirect
129. #10587558C2×C4semidirect
159. #63256C4⋊C4semidirect
220. Q8×C17C2×C4⋊C4semidirect
263. D68C2×C4⋊C4semidirect
268. C4×D17C2×C4⋊C4semidirect
273. #2351D17semidirect
388. C4×Q8C4×D17semidirect
445. C17#34250semidirect
524. Q8C4⋊C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117632912
C2×C4×#632562
#145317022
Q8×C4⋊Dic172
Q8×#632252
#145316932
#145316922
C4⋊C4×D682
#145316852
#145314092
#145313992
#145312642
D17×#3892
C17×#23512
#3425017