Finite Group Databasebeta

← #17980373

Subgroups — #17980373

Order 4352.

1864 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545737C214yesmaximal, complemented
2#14545721C214yesmaximal, complemented
3#14545709C214yesmaximal, complemented
4#14545703C214yesmaximal, complemented
5#14545577C212yesmaximal, complemented
6#14545515C214yesmaximal, complemented
7#14545513C212yesmaximal, complemented
8#14545480C212yesmaximal, complemented
9D17⋊#432C211yesmaximal, characteristic, complemented
10#14545429C212yesmaximal, complemented
11D17⋊#423C211yesmaximal, characteristic, complemented
12#14542079C211yesmaximal, characteristic
13C2×#10587678C2214yes—
14C2×D17⋊M16C2212yes—
15#11763861C2218yescomplemented
16#11763860C2218yescomplemented
17#11763844C2218yes—
18#11763841C2218yes—
19#11763840C2218yes—
20#11763840C2218yescomplemented
21#11763840C2218yescomplemented
22#11763836C2218yes—
23#11763833C2218yescomplemented
24#11763832C2212yes—
25#11763831C2212yes—
26#11763830C2212yes—
27#11763830C2214yescomplemented
28#11763829C2212yes—
29#11763825C2214yescomplemented
30#11763816C2214yes—
31#11763815C2218yescomplemented
32#11763814C2212yes—
33#11763813C2212yes—
34#11763812C2212yes—
35#11763812C2212yes—
36#11763812C2214yescomplemented
37#11763810C2212yes—
38#11763810C2212yes—
39#11763810C2211yescharacteristic
40#11763810C2211yescharacteristic
41C2×C4⋊#3624C2214yescomplemented
42#11763783C2214yes—
43#11763777C2212yes—
44#11763774C2214yes—
45#11763768C2212yes—
46#11763366C411yescharacteristic
47#11763364C411yescharacteristic
48#11763364C2211yescharacteristic
49#11763357C411yescharacteristic
50#11763357C2211yescharacteristic
51#11763341C412yes—
52#11763341C412yes—
53#11763340C412yes—
54#11763340C2212yes—
55#11763338C412yes—
56#11763338C2212yes—
57#11763319C412yes—
58#11763319C2212yes—
59#11763316C412yes—
60#41×Dic17C2212yes—
61C4×#63254C2212yes—
62C4×#63254C2212yes—
63C2×C42×D17C2211yescharacteristic
64C17×#532C411yescharacteristic
65C2×#41×C17C2×C412yes—
66C2×#63254C2×C412yes—
67C2×#63254C2312yes—
68C22×C4×D17C2×C411yescharacteristic
69C22×C4×D17C2×C411yescharacteristic
70C22×C4×D17C2311yescharacteristic
71C2×#63247C2314yes—
72C2×#63247C2318yes—
73C22×#3624C2318yes—
74#10587678—48no—
75D17⋊M16—44no—
76C2×#63243C2314yes—
77C2×#63242C2312yes—
78C2×#63242C2312yes—
79C2×#63242C2314yes—
80C17×C4⋊Q8C2×C411yescharacteristic
81C17×#127C2×C411yescharacteristic
82C17×#124C2×C412yes—
83C17×#121C2×C412yes—
84C17×#121C2×C412yes—
85C4×D4×C17C2×C412yes—
86C4×D4×C17C2×C412yes—
87C2×C42×C17C2×C411yescharacteristic
88#10587632C2×C412yes—
89#10587632C2×C412yes—
90Q8×Dic17C2×C412yes—
91Q8×Dic17C2312yes—
92#10587624C2×C412yes—
93#10587624C2×C412yes—
94#10587624C2×C412yes—
95#10587624C2×C412yes—
96#10587623C2×C414yes—
97#10587622C2×C412yes—
98#10587622C2×C412yes—
99#10587621C2×C414yes—
100D4×Dic17C2×C412yes—
101D4×Dic17C2×C414yes—
102D4×Dic17C2×C412yes—
103D4×Dic17C2312yes—
104D4×Dic17C2314yes—
105D4×Dic17C2312yes—
106C4×C22⋊D17C2×C414yes—
107C4×C22⋊D17C2×C414yes—
108#10587612C2×C412yes—
109#10587612C2×C414yes—
110#10587610C2×C412yes—
111#10587610C2×C412yes—
112C2×C4×Dic17C2×C411yescharacteristic
113C2×C4×Dic17C2311yescharacteristic
114C2×C4×Dic17C2311yescharacteristic
115C2×C4×Dic17C2314yes—
116#10587583C2×C412yes—
117#10587583C2×C412yes—
118#10587578C2×C412yes—
119#10587578C2312yes—
120#10587576C2×C412yes—
121#10587576C2×C412yes—
122#10587573C2×C412yes—
123#10587573C2312yes—
124#10587569C2×C414yes—
125#10587569C2×C414yes—
126#10587567C2×C414yes—
127#10587567C2314yes—
128#10587565C2×C414yes—
129#10587565C2×C414yes—
130#10587563C2×C414yes—
131#10587563C2314yes—
132C42×D17C2×C411yescharacteristic
133C42×D17C2×C411yescharacteristic
134C42×D17C2×C411yescharacteristic
135C42×D17C2×C411yescharacteristic
136C42×D17C2311yescharacteristic
137C42×D17C2311yescharacteristic
138C42×D17C2311yescharacteristic
139C42×D17C2311yescharacteristic
140#10587556C2×C412yes—
141#10587555C2×C411yescharacteristic
142#10587555C2×C411yescharacteristic
143C4×Dic34C2×C412yes—
144C4×Dic34C2×C412yes—
145C22⋊#3624C2314yes—
146C22⋊#3624C2318yes—
147C22⋊#3624C2318yes—
148#10587544C2314yes—
149#10587544C2314yes—
150#10587544C2312yes—
151#10587544C2312yes—
152#10587544C2318yes—
153#10587543C2314yes—
154#10587542C2314yes—
155#10587542C2314yes—
156#10587542C2318yes—
157C4⋊#3624C2312yes—
158C4⋊#3624C2314yes—
159C4⋊#3624C2318yes—
160C4⋊#3624C2314yes—
161C4⋊#3624C2312yes—
162C4×#3624C2318yes—
163C23×D17C22×C411yescharacteristic
164#41×C17—44no—
165C2×Q8×C17C22×C412yes—
166C2×D4×C17C22×C412yes—
167C2×D4×C17C22×C412yes—
168C2×D4×C17C22×C412yes—
169C2×D4×C17C22×C414yes—
170C22×C4×C17C22×C411yescharacteristic
171C22×C4×C17C22×C411yescharacteristic
172C22×C4×C17C22×C414yes—
173C22×C4×C17C22×C411yescharacteristic
174C2×C22⋊D17C22×C414yes—
175C2×C22⋊D17C22×C414yes—
176C22×Dic17C22×C411yescharacteristic
177C22×Dic17C22×C411yescharacteristic
178C22×Dic17C22×C414yes—
179C22×Dic17C2414yes—
180C22×Dic17C2411yescharacteristic
181#63254—24no—
182#63254—24no—
183#63254—44no—
184C2×C4×D17—22no—
185C2×C4×D17—22no—
186C2×C4×D17—42no—
187C2×C4×D17—42no—
188C2×C4×D17C22×C411yescharacteristic
189C2×C4×D17C22×C411yescharacteristic
190C2×C4×D17C22×C411yescharacteristic
191C2×C4×D17C22×C411yescharacteristic
192C2×C4×D17C22×C411yescharacteristic
193C2×C4×D17C22×C411yescharacteristic
194C2×C4×D17C22×C411yescharacteristic
195C2×C4×D17C22×C411yescharacteristic
196C2×C4×D17C22×C411yescharacteristic
197C2×C4×D17C22×C411yescharacteristic
198C2×C4×D17C2411yescharacteristic
199C2×C4×D17C2411yescharacteristic
200C2×Dic34C22×C412yes—
201C2×Dic34C22×C412yes—
202#63247—44no—
203#63247—48no—
204C2×#3624—48no—
205C2×#3624C2418yes—
206C2×#3624C2414yes—
207C2×#3624C2414yes—
208#63243—28no—
209#63242—24no—
210#63242—24no—
211#63242—44no—
212C4⋊C4×C17C22×C412yes—
213C4⋊C4×C17C22×C412yes—
214C22⋊C4×C17C22×C414yes—
215C22⋊C4×C17C22×C414yes—
216C42×C17C22×C411yescharacteristic
217C42×C17C22×C411yescharacteristic
218C42×C17C22×C411yescharacteristic
219C42×C17C22×C411yescharacteristic
220C22⋊Dic17C22×C414yes—
221C22⋊Dic17C22×C414yes—
222C22⋊Dic17C22×C414yes—
223C22⋊Dic17C22×C414yes—
224C22⋊Dic17C22×C414yes—
225#63227C22×C414yes—
226#63227C22×C414yes—
227C4⋊Dic17C22×C412yes—
228C4⋊Dic17C22×C412yes—
229C4⋊Dic17C22×C412yes—
230C4⋊Dic17C22×C412yes—
231C4⋊Dic17C22×C414yes—
232C4⋊Dic17C22×C414yes—
233#63225C22×C414yes—
234#63225C22×C414yes—
235#63225C22×C414yes—
236#63225C22×C414yes—
237C4×Dic17C22×C414yes—
238C4×Dic17C22×C411yescharacteristic
239C4×Dic17C22×C411yescharacteristic
240C4×Dic17C22×C411yescharacteristic
241C4×Dic17C22×C411yescharacteristic
242C4×Dic17C22×C411yescharacteristic
243C4×Dic17C22×C411yescharacteristic
244C4×Dic17C22×C411yescharacteristic
245C4×Dic17C22×C411yescharacteristic
246C4×Dic17C22×C414yes—
247C4×Dic17C2414yes—
248C4×Dic17C2411yescharacteristic
249C4×Dic17C2411yescharacteristic
250C4×Dic17C2411yescharacteristic
251C4×Dic17C2411yescharacteristic
252#60269—171nomaximal, Sylow
253C23×C17C23×C414yes—
254C23×C17C23×C411yescharacteristic
255C22×D17—22no—
256C22×D17—22no—
257C22×D17—41no—
258C22×D17C23×C411yescharacteristic
259C22×D17C23×C411yescharacteristic
260Q8×C17—42no—
261D4×C17—24no—
262D4×C17—42no—
263D4×C17—42no—
264D4×C17—44no—
265C2×C4×C17—22no—
266C2×C4×C17—22no—
267C2×C4×C17—28no—
268C2×C4×C17—41no—
269C2×C4×C17C23×C411yescharacteristic
270C2×C4×C17C23×C411yescharacteristic
271C2×C4×C17C23×C411yescharacteristic
272C2×C4×C17C23×C411yescharacteristic
273C2×C4×C17C23×C414yes—
274C2×C4×C17C23×C411yescharacteristic
275C2×C4×C17C23×C411yescharacteristic
276C22⋊D17—28no—
277C22⋊D17—28no—
278C2×Dic17—24no—
279C2×Dic17—21no—
280C2×Dic17—21no—
281C2×Dic17—24no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—44no—
285C2×Dic17C23×C411yescharacteristic
286C2×Dic17C23×C411yescharacteristic
287C2×Dic17C23×C411yescharacteristic
288C2×Dic17C23×C411yescharacteristic
289C2×Dic17C23×C414yes—
290C2×Dic17C23×C414yes—
291C2×Dic17C23×C411yescharacteristic
292C2×Dic17C2511yescharacteristic
293C4×D17—22no—
294C4×D17—22no—
295C4×D17—22no—
296C4×D17—22no—
297C4×D17—22no—
298C4×D17—22no—
299C4×D17—21no—
300C4×D17—21no—
301C4×D17—41no—
302C4×D17—41no—
303C4×D17—41no—
304C4×D17—41no—
305C4×D17H(5,𝔽2)12yes—
306C4×D172−1+412yes—
307Dic34—24no—
308Dic34—42no—
309#3624—24no—
310#3624—28no—
311#3624—24no—
312#3444—171no—
313#2995—174no—
314#2992—174no—
315#2990—174no—
316#2989—174no—
317#2961—172no—
318#2933—174no—
319#2932—172no—
320#2925—172no—
321C2×#432—171no—
322#2906—172no—
323C2×#423—171no—
324C22×C17—24no—
325C22×C17—21no—
326C22×C17—24no—
327C22×C17—21no—
328C22×C17—21no—
329C22×C17—24no—
330C22×C17C24×C411yescharacteristic
331C2×D17—21no—
332C2×D17—21no—
333C2×D17—21no—
334C2×D17—21no—
335C2×D17—44no—
336C2×D17#52012yes—
337C2×D17#56812yes—
338C4×C17—21no—
339C4×C17—21no—
340C4×C17—21no—
341C4×C17—21no—
342C4×C17—24no—
343C4×C17#52012yes—
344C4×C17#56812yes—
345Dic17—24no—
346Dic17—21no—
347Dic17—24no—
348Dic17—21no—
349Dic17—21no—
350Dic17—21no—
351Dic17—21no—
352Dic17C2×H(5,𝔽2)11yescharacteristic
353Dic17C2×2−1+411yescharacteristic
354C2×#131—174no—
355C22×M16—172no—
356#533—171no—
357#533—171no—
358#533—171no—
359#532—171no—
360#532—171no—
361#532#59511yescharacteristic
362#529—172no—
363#529—172no—
364#529—172no—
365#529—172no—
366#529—172no—
367#529—172no—
368#525—172no—
369#525—172no—
370#525—172no—
371C4×#41—172no—
372C4×#41—172no—
373C4×#41—172no—
374C22×C42—171no—
375#446—178no—
376#446—178no—
377#436—178no—
378#435—178no—
379#435—178no—
380#435—178no—
381#435—178no—
382C8×D4—178no—
383#433—178no—
384#432—172no—
385#432—172no—
386#432—172no—
387#432—172no—
388#432—174no—
389#431—174no—
390#424—178no—
391#424—174no—
392#423—172no—
393#423—172no—
394#423—172no—
395#423—172no—
396#423—174no—
397C2×C4⋊C8—172no—
398C2×C4⋊C8—172no—
399C2×C4⋊C8—171no—
400C2×C4⋊C8—174no—
401C2×C4⋊C8—171no—
402#408—174no—
403C2×C22⋊C8—172no—
404#405—174no—
405C2×#97—172no—
406C2×C17—41no—
407C2×C17—44no—
408C2×C17C2×#52011yescharacteristic
409C2×C17C2×#56811yescharacteristic
410C2×C17C2×#56811yescharacteristic
411D17—22no—
412D17—22no—
413C2×#41—172no—
414C2×#41—172no—
415C2×#41C2×#59512yes—
416C23×C4—171no—
417C23×C4—171no—
418C23×C4—171no—
419#131—688no—
420C2×M16—174no—
421C2×M16—174no—
422C2×M16—178no—
423C2×M16—684no—
424C22×C8—172no—
425C22×C8—172no—
426C22×C8—174no—
427C22×C8—178no—
428C4⋊Q8—171no—
429C4⋊Q8—171no—
430C4⋊Q8C2×#59511yescharacteristic
431#127C2×#59511yescharacteristic
432#126—174no—
433#126—174no—
434#125—172no—
435#125—172no—
436#125—172no—
437#124—172no—
438#124—172no—
439#124C2×#59512yes—
440#123—174no—
441#123—174no—
442#123—174no—
443C22⋊Q8—172no—
444C22⋊Q8—172no—
445C22⋊Q8—172no—
446C22⋊Q8—172no—
447C22⋊Q8—172no—
448C22⋊Q8—172no—
449#121—172no—
450#121—172no—
451#121—172no—
452#121—172no—
453#121C2×#59512yes—
454#121C2×#59512yes—
455#120—174no—
456C4×Q8—172no—
457C4×Q8—172no—
458C4×Q8—172no—
459C4×Q8—172no—
460C4×Q8—172no—
461C4×Q8—172no—
462C4×D4—174no—
463C4×D4—174no—
464C4×D4—172no—
465C4×D4—172no—
466C4×D4—172no—
467C4×D4—172no—
468C4×D4—174no—
469C4×D4—174no—
470C4×D4—174no—
471C4×D4—174no—
472C4×D4C2×#59512yes—
473C4×D4C2×#59512yes—
474#117—172no—
475#117—174no—
476#117—172no—
477#117—172no—
478#117—174no—
479#117—174no—
480C2×C42—174no—
481C2×C42—171no—
482C2×C42—171no—
483C2×C42—171no—
484C2×C42—171no—
485C2×C42—171no—
486C2×C42—171no—
487C2×C42—171no—
488C2×C42—171no—
489C2×C42—171no—
490C2×C42—171no—
491C2×C42—171no—
492C2×C42C2×#59511yescharacteristic
493C4⋊C8—172no—
494C4⋊C8—174no—
495C4⋊C8—174no—
496C4⋊C8—178no—
497C4⋊C8—172no—
498C4⋊C8—174no—
499C4⋊C8—174no—
500C4⋊C8—172no—
501C4⋊C8—172no—
502C4⋊C8—178no—
503C22⋊C8—174no—
504C22⋊C8—174no—
505C22⋊C8—178no—
506C22⋊C8—178no—
507#97—174no—
508#97—174no—
509#97—178no—
510C4×C8—178no—
511C17#6026911yescharacteristic, complemented, Sylow
512C24—171no—
513#41—44no—
514#41—344no—
515#41—344no—
516#41—684no—
517C2×Q8—172no—
518C2×Q8—172no—
519C2×Q8C22×#59512yes—
520C2×D4—174no—
521C2×D4—174no—
522C2×D4C22×#59512yes—
523C2×D4C22×#59514yes—
524C2×D4C22×#59512yes—
525C2×D4C22×#59512yes—
526C22×C4—171no—
527C22×C4—171no—
528C22×C4—174no—
529C22×C4—171no—
530C22×C4—171no—
531C22×C4—174no—
532C22×C4—171no—
533C22×C4—171no—
534C22×C4—171no—
535C22×C4—171no—
536C22×C4—171no—
537C22×C4—171no—
538C22×C4—171no—
539C22×C4—171no—
540C22×C4—171no—
541C22×C4—171no—
542C22×C4—171no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—682no—
546C22×C4—682no—
547C22×C4C22×#59511yescharacteristic
548C22×C4C22×#59511yescharacteristic
549C22×C4C22×#59514yes—
550C22×C4C22×#59511yescharacteristic
551M16—348no—
552M16—684no—
553M16—688no—
554C2×C8—174no—
555C2×C8—178no—
556C2×C8—174no—
557C2×C8—344no—
558C2×C8—344no—
559C2×C8—684no—
560C2×C8—688no—
561C4⋊C4—172no—
562C4⋊C4—172no—
563C4⋊C4—172no—
564C4⋊C4—172no—
565C4⋊C4—174no—
566C4⋊C4—174no—
567C4⋊C4—174no—
568C4⋊C4—174no—
569C4⋊C4—174no—
570C4⋊C4—174no—
571C4⋊C4C22×#59512yes—
572C4⋊C4C22×#59512yes—
573C22⋊C4—174no—
574C22⋊C4—174no—
575C22⋊C4—174no—
576C22⋊C4—174no—
577C22⋊C4—174no—
578C22⋊C4—174no—
579C22⋊C4—174no—
580C22⋊C4C22×#59514yes—
581C22⋊C4C22×#59514yes—
582C42—174no—
583C42—171no—
584C42—171no—
585C42—171no—
586C42—171no—
587C42—171no—
588C42—171no—
589C42—171no—
590C42—171no—
591C42—171no—
592C42—171no—
593C42—171no—
594C42—171no—
595C42—174no—
596C42—174no—
597C42C22×#59511yescharacteristic
598C42C22×#59511yescharacteristic
599C42C22×#59511yescharacteristic
600C42C22×#59511yescharacteristic
601C23—171no—
602C23—171no—
603C23—342no—
604C23—342no—
605C23—681no—
606C23C23×#59514yes—
607C23C23×#59511yescharacteristic
608Q8—42no—
609Q8—344no—
610Q8—682no—
611D4—24no—
612D4—42no—
613D4—44no—
614D4—42no—
615D4—348no—
616D4—348no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—28no—
620C2×C4—41no—
621C2×C4—171no—
622C2×C4—172no—
623C2×C4—172no—
624C2×C4—171no—
625C2×C4—171no—
626C2×C4—171no—
627C2×C4—171no—
628C2×C4—171no—
629C2×C4—174no—
630C2×C4—174no—
631C2×C4—341no—
632C2×C4—341no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—342no—
636C2×C4—344no—
637C2×C4—341no—
638C2×C4—341no—
639C2×C4—344no—
640C2×C4—342no—
641C2×C4—342no—
642C2×C4—342no—
643C2×C4—681no—
644C2×C4—681no—
645C2×C4—681no—
646C2×C4—681no—
647C2×C4—681no—
648C2×C4—681no—
649C2×C4—684no—
650C2×C4C23×#59511yescharacteristic
651C2×C4C23×#59511yescharacteristic
652C2×C4C23×#59511yescharacteristic
653C2×C4C23×#59511yescharacteristic
654C2×C4C23×#59514yes—
655C2×C4C23×#59511yescharacteristic
656C2×C4C23×#59511yescharacteristic
657C8—344no—
658C8—344no—
659C8—348no—
660C22—21no—
661C22—24no—
662C22—21no—
663C22—24no—
664C22—21no—
665C22—24no—
666C22—172no—
667C22—172no—
668C22—341no—
669C22—341no—
670C22—341no—
671C22—341no—
672C22—684no—
673C22C24×#59511yescharacteristic
674C4—21no—
675C4—21no—
676C4—24no—
677C4—21no—
678C4—21no—
679C4—171no—
680C4—171no—
681C4—341no—
682C4—344no—
683C4—341no—
684C4—341no—
685C4—341no—
686C4—341no—
687C4—344no—
688C4#1176387812yes—
689C4#1176388712yes—
690C2—41no—
691C2—44no—
692C2—342no—
693C2—342no—
694C2C2×#1176387811yescharacteristic
695C2C2×#1176388711yescharacteristic
696C2D17⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#532C4
prime radical#14542079C2
cosocleC2×Dic17C25
hypercenter#532#595
hypocenterC17#60269
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #14545737C2semidirect
2. #14545721C2semidirect
3. #14545709C2semidirect
4. #14545703C2semidirect
5. #14545577C2semidirect
6. #14545515C2semidirect
7. #14545513C2semidirect
8. #14545480C2semidirect
9. D17⋊#432C2semidirect
10. #14545429C2semidirect
11. D17⋊#423C2semidirect
15. #11763861C22semidirect
16. #11763860C22semidirect
20. #11763840C22semidirect
21. #11763840C22semidirect
23. #11763833C22semidirect
27. #11763830C22semidirect
29. #11763825C22semidirect
31. #11763815C22semidirect
36. #11763812C22semidirect
41. C2×C4⋊#3624C22semidirect
511. C17#60269semidirect

Maximal subgroups

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

subgroupindex
#145457372
#145457212
#145457092
#145457032
#145455772
#145455152
#145455132
#145454802
D17⋊#4322
#145454292
D17⋊#4232
#145420792
#6026917