Finite Group Databasebeta

← #17979389

Subgroups — #17979389

Order 4352.

2216 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545330C211yesmaximal, characteristic, complemented
2#14545327C212yesmaximal, complemented
3#14545316C214yesmaximal, complemented
4#14545311C214yesmaximal, complemented
5#14545307C214yesmaximal, complemented
6#14545283C214yesmaximal, complemented
7#14545273C214yesmaximal, complemented
8#14545271C214yesmaximal, complemented
9#125×#595C211yesmaximal, characteristic, complemented
10#124×#595C212yesmaximal, complemented
11D17×#562C211yesmaximal, characteristic
12C2×Q8×#595C2212yes—
13C2×D4×#595C2212yes—
14C2×D4×#595C2214yes—
15C2×#10587677C2214yes—
16(C2×#595)⋊Q8C2212yes—
17(C2×#595)⋊Q8C2214yes—
18#11763761C2212yes—
19#11763761C2214yes—
20#11763760C2212yes—
21#11763760C2214yes—
22#11763760C2214yes—
23#11763759C2214yes—
24#11763759C2214yes—
25#11763759C2214yes—
26#11763759C2214yes—
27#11763759C2214yes—
28#11763759C2214yes—
29#11763758C2214yes—
30#11763758C2214yes—
31#11763758C2214yes—
32#11763758C2214yes—
33#11763758C2214yes—
34#11763757C2214yes—
35#11763757C2214yes—
36#11763757C2214yes—
37#11763756C2214yes—
38#11763756C2214yes—
39#11763756C2214yes—
40#11763753C2212yes—
41#11763753C2211yescharacteristic
42#11763751C2214yes—
43C4⋊C4×#595C2212yes—
44C4⋊C4×#595C2214yes—
45C4⋊C4×#595C2214yes—
46C22⋊C4×#595C2214yes—
47C22⋊C4×#595C2214yes—
48C22⋊C4×#595C2214yes—
49C4×C22⋊#595C2214yes—
50C2×(C2×#595)⋊C4C2214yes—
51C42×#595C2211yescharacteristic
52#11763557C411yescharacteristic, complemented
53#11763556C414yescomplemented
54#11763555C414yescomplemented
55#11763554C414yescomplemented
56#11763553C412yescomplemented
57#125×D17C2211yescharacteristic
58#124×D17C2212yes—
59#123×D17C2214yes—
60C22⋊Q8×D17C2214yes—
61#121×D17C2214yes—
62C17×#562C411yescharacteristic, complemented
63C23×#595C2314yes—
64C23×#595C2314yes—
65C2×Q8×D17C2312yes—
66C2×D4×D17C2314yes—
67C2×D4×D17C2312yes—
68C22×C4×D17C2314yes—
69C2×C22⋊#595C2314yes—
70C2×C22⋊#595C2314yes—
71C2×C22⋊#595C2314yes—
72C2×C22⋊#595C2314yes—
73C2×C22⋊#595C2314yes—
74C2×C22⋊#595C2314yes—
75Q8×#595—44no—
76D4×#595—44no—
77D4×#595—48no—
78#10587677—48no—
79C2×C4⋊#595C2314yes—
80C2×C4⋊#595C2314yes—
81C2×C4⋊#595C2312yes—
82C2×C4⋊#595C2314yes—
83C2×C4⋊#595C2312yes—
84C2×C4⋊#595C2312yes—
85C2×C4⋊#595C2314yes—
86C2×C4×#595C2311yescharacteristic
87C2×C4×#595C2312yes—
88C2×C4×#595C2314yes—
89C2×C4×#595C2311yescharacteristic
90C2×C4×#595C2312yes—
91C2×C4×#595C2314yes—
92C2×C4×#595C2314yes—
93C2×C4×#595C2314yes—
94C17×#125C2×C411yescharacteristic, complemented
95C17×#124C2×C412yescomplemented
96C17×#123C2×C414yescomplemented
97C17×C22⋊Q8C2×C414yescomplemented
98C17×#121C2×C414yescomplemented
99#10587633C2×C412yescomplemented
100#10587632C2×C414yescomplemented
101#10587625C2×C414yescomplemented
102#10587625C2×C414yescomplemented
103#10587624C2×C414yescomplemented
104#10587622C2×C412yescomplemented
105#10587621C2×C414yescomplemented
106#10587616C2×C414yescomplemented
107#10587615C2×C414yescomplemented
108#10587610C2×C414yescomplemented
109#10587583C2×C414yescomplemented
110#10587582C2×C414yescomplemented
111#10587582C2×C414yescomplemented
112#10587581C2×C414yescomplemented
113#10587580C2×C414yescomplemented
114#10587580C2×C414yescomplemented
115C4⋊C4×D17C2314yes—
116C4⋊C4×D17C2314yes—
117C4⋊C4×D17C2312yes—
118#10587576C2×C412yescomplemented
119#10587575C2×C414yescomplemented
120#10587572C2×C414yescomplemented
121#10587571C2×C414yescomplemented
122#10587571C2×C414yescomplemented
123#10587570C2×C414yescomplemented
124#10587570C2×C414yescomplemented
125#10587569C2×C414yescomplemented
126#10587569C2×C414yescomplemented
127C22⋊C4×D17C2314yes—
128C22⋊C4×D17C2314yes—
129C22⋊C4×D17C2314yes—
130#10587564C2×C414yescomplemented
131#10587564C2×C414yescomplemented
132#10587561C2×C412yescomplemented
133C42×D17C2311yescharacteristic
134#10587556C2×C411yescharacteristic, complemented
135(C2×#595)⋊C4C2314yes—
136(C2×#595)⋊C4C2314yes—
137(C2×#595)⋊C4C2314yes—
138(C2×#595)⋊C4C2314yes—
139(C2×#595)⋊C4C2314yes—
140(C2×#595)⋊C4C2314yes—
141(C2×#595)⋊C4C2314yes—
142(C2×#595)⋊C4C2314yes—
143(C2×#595)⋊C4C2314yes—
144(C2×#595)⋊C4C2314yes—
145(C2×#595)⋊C4C2314yes—
146C23×D17C2414yes—
147C22×#595—28no—
148C22×#595—44no—
149C22×#595—44no—
150C22×#595—44no—
151C22×#595—44no—
152C22×#595—44no—
153C22×#595C2414yes—
154C22×#595C2411yescharacteristic
155C22×#595C2414yes—
156C22×#595C2414yes—
157C22×#595C2412yes—
158C22×#595C2411yescharacteristic
159C2×Q8×C17C22×C412yes—
160C2×D4×C17C22×C414yes—
161C2×D4×C17C22×C412yes—
162C22×C4×C17C22×C414yes—
163C2×C22⋊D17C22×C414yes—
164C2×C22⋊D17C22×C414yes—
165C2×C22⋊D17C22×C414yes—
166C22×Dic17C22×C414yes—
167Q8×D17—42no—
168Q8×D17—42no—
169D4×D17—42no—
170D4×D17—44no—
171D4×D17—44no—
172D4×D17—42no—
173C2×D68C22×C412yes—
174C2×D68C22×C414yes—
175C2×C4×D17—44no—
176C2×C4×D17—44no—
177C2×C4×D17C22×C414yes—
178C2×C4×D17C22×C414yes—
179C2×C4×D17C2414yes—
180C2×C4×D17C2414yes—
181C2×C4×D17C2412yes—
182C2×C4×D17C2411yescharacteristic
183C2×Dic34C22×C412yes—
184C2×Dic34C22×C414yes—
185C22⋊#595—28no—
186C22⋊#595—28no—
187C22⋊#595—44no—
188C22⋊#595—44no—
189C22⋊#595—44no—
190C22⋊#595—44no—
191C4⋊#595—28no—
192C4⋊#595—42no—
193C4⋊#595—44no—
194C4⋊#595—44no—
195C4⋊#595—42no—
196C4⋊#595—44no—
197C4⋊#595—42no—
198C4×#595—28no—
199C4×#595—24no—
200C4×#595—22no—
201C4×#595—41no—
202C4×#595—44no—
203C4×#595—42no—
204C4×#595—44no—
205C4×#595—44no—
206C4⋊C4×C17C22×C414yes—
207C4⋊C4×C17C22×C414yes—
208C4⋊C4×C17C22×C412yes—
209C22⋊C4×C17C22×C414yes—
210C22⋊C4×C17C22×C414yes—
211C22⋊C4×C17C22×C414yes—
212C42×C17C22×C411yescharacteristic
213C22⋊Dic17C22×C414yes—
214C22⋊Dic17C22×C414yes—
215C22⋊Dic17C22×C414yes—
216#63227C22×C414yes—
217#63227C22×C414yes—
218#63227C22×C414yes—
219#63227C22×C414yes—
220#63227C22×C414yes—
221#63227C22×C414yes—
222C4⋊Dic17C22×C412yes—
223C4⋊Dic17C22×C414yes—
224C4⋊Dic17C22×C414yes—
225#63225C22×C414yes—
226#63225C22×C414yes—
227#63225C22×C414yes—
228#63225C22×C414yes—
229#63225C22×C414yes—
230C4×Dic17C22×C411yescharacteristic
231C4×Dic17C22×C412yes—
232#34522—171nomaximal, Sylow
233C23×C17C23×C414yes—
234C22×D17—24no—
235C22×D17—24no—
236C22×D17—24no—
237C22×D17—24no—
238C22×D17—24no—
239C22×D17—24no—
240C22×D17C23×C414yes—
241C22×D17C2511yescharacteristic
242C2×#595—22no—
243C2×#595—22no—
244C2×#595—22no—
245C2×#595—24no—
246C2×#595—22no—
247C2×#595—24no—
248C2×#595—24no—
249C2×#595—24no—
250C2×#595—21no—
251C2×#595—24no—
252C2×#595—24no—
253C2×#595—24no—
254C2×#595—24no—
255C2×#595—24no—
256C2×#595—88no—
257C2×#595—88no—
258C2×#595H(5,𝔽2)14yes—
259C2×#5952−1+412yes—
260Q8×C17—42no—
261D4×C17—42no—
262D4×C17—44no—
263C2×C4×C17—44no—
264C2×C4×C17C23×C411yescharacteristic
265C2×C4×C17C23×C412yes—
266C2×C4×C17C23×C414yes—
267C2×C4×C17C23×C414yes—
268C22⋊D17—44no—
269C22⋊D17—44no—
270C22⋊D17—44no—
271C2×Dic17—44no—
272C2×Dic17C23×C411yescharacteristic
273C2×Dic17C23×C414yes—
274C2×Dic17C23×C412yes—
275C2×Dic17C23×C414yes—
276D68—42no—
277D68—44no—
278C4×D17—24no—
279C4×D17—22no—
280C4×D17—24no—
281C4×D17—21no—
282C4×D17—21no—
283C4×D17—24no—
284C4×D17—22no—
285C4×D17—24no—
286C4×D17—44no—
287C4×D17—44no—
288Dic34—42no—
289Dic34—44no—
290C2×#562—171no—
291#2382—171no—
292#2379—172no—
293#2368—174no—
294#2363—174no—
295#2359—174no—
296#2335—174no—
297#2325—174no—
298#2323—174no—
299C4×#125—171no—
300C4×#124—172no—
301C22×C17—24no—
302C22×C17—24no—
303C22×C17—24no—
304C22×C17C24×C411yescharacteristic
305C2×D17—24no—
306C2×D17—24no—
307C2×D17—24no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—44no—
312C2×D17C2×H(5,𝔽2)12yes—
313C2×D17C2×2−1+411yescharacteristic
314C2×D17#58511yescharacteristic
315C2×D17#58512yes—
316#595—24no—
317#595—22no—
318#595—44no—
319#595—44no—
320#595—44no—
321#595—41no—
322C4×C17—24no—
323C4×C17—22no—
324C4×C17—24no—
325C4×C17—21no—
326Dic17—24no—
327Dic17—22no—
328Dic17—24no—
329Dic17—21no—
330#562—171no—
331#562—172no—
332#562—174no—
333#562—174no—
334#562—174no—
335#562#59511yescharacteristic, complemented
336C2×#125—171no—
337C2×#124—172no—
338C2×#123—174no—
339C2×C22⋊Q8—174no—
340C2×#121—174no—
341C2×C4×Q8—172no—
342C2×C4×D4—172no—
343C2×C4×D4—174no—
344C2×#117—174no—
345#391—174no—
346#391—172no—
347#390—174no—
348#390—172no—
349#389—172no—
350#389—174no—
351#389—174no—
352#388—174no—
353#388—174no—
354#388—174no—
355#388—174no—
356#388—174no—
357#388—174no—
358#387—174no—
359#387—174no—
360#387—174no—
361#387—174no—
362#387—174no—
363#386—174no—
364#386—174no—
365#386—174no—
366#385—174no—
367#385—174no—
368#385—174no—
369#382—171no—
370#382—172no—
371#380—174no—
372C4×C4⋊C4—172no—
373C4×C4⋊C4—174no—
374C4×C4⋊C4—174no—
375C4×C22⋊C4—174no—
376C4×C22⋊C4—174no—
377C4×C22⋊C4—174no—
378C4×C22⋊C4—174no—
379C2×#95—174no—
380C43—171no—
381C2×C17—44no—
382C2×C17C4×H(5,𝔽2)12yes—
383C2×C17C4×2−1+411yescharacteristic
384D17—44no—
385D17#354512yes—
386D17#354811yescharacteristic
387D17#355311yescharacteristic
388C22×Q8—172no—
389C22×D4—174no—
390C22×D4—172no—
391C23×C4—174no—
392C23×C4—174no—
393C23×C4—174no—
394#125—174no—
395#125—172no—
396#125—171no—
397#125C2×#59511yescharacteristic, complemented
398#124—172no—
399#124—172no—
400#124—174no—
401#124—174no—
402#124—172no—
403#124C2×#59512yescomplemented
404#123—174no—
405#123—174no—
406#123—174no—
407#123—174no—
408#123—174no—
409#123—174no—
410#123—174no—
411#123C2×#59514yescomplemented
412C22⋊Q8—174no—
413C22⋊Q8—174no—
414C22⋊Q8—174no—
415C22⋊Q8—174no—
416C22⋊Q8—174no—
417C22⋊Q8—174no—
418C22⋊Q8—174no—
419C22⋊Q8C2×#59514yescomplemented
420#121—174no—
421#121—174no—
422#121—174no—
423#121—174no—
424#121—174no—
425#121—174no—
426#121—174no—
427#121C2×#59514yescomplemented
428C4×Q8—684no—
429C4×D4—684no—
430C4×D4—688no—
431#117—688no—
432C2×C4⋊C4—174no—
433C2×C4⋊C4—174no—
434C2×C4⋊C4—172no—
435C2×C4⋊C4—174no—
436C2×C4⋊C4—174no—
437C2×C4⋊C4—172no—
438C2×C4⋊C4—174no—
439C2×C4⋊C4—172no—
440C2×C4⋊C4—174no—
441C2×C4⋊C4—172no—
442C2×C22⋊C4—174no—
443C2×C22⋊C4—174no—
444C2×C22⋊C4—174no—
445C2×C22⋊C4—174no—
446C2×C22⋊C4—174no—
447C2×C22⋊C4—174no—
448C2×C22⋊C4—174no—
449C2×C22⋊C4—174no—
450C2×C22⋊C4—174no—
451C2×C42—171no—
452C2×C42—174no—
453C2×C42—172no—
454C2×C42—171no—
455C2×C42—174no—
456C2×C42—174no—
457C2×C42—171no—
458C2×C42—172no—
459C2×C42—174no—
460#95—174no—
461#95—174no—
462#95—174no—
463#95—174no—
464#95—174no—
465#95—174no—
466#95—174no—
467#95—174no—
468#95—174no—
469#95—174no—
470#95—174no—
471C17#3452211yescharacteristic, complemented, Sylow
472C24—174no—
473C2×Q8—174no—
474C2×Q8—172no—
475C2×Q8—682no—
476C2×Q8—682no—
477C2×Q8C22×#59512yes—
478C2×D4—174no—
479C2×D4—174no—
480C2×D4—174no—
481C2×D4—174no—
482C2×D4—172no—
483C2×D4—682no—
484C2×D4—684no—
485C2×D4—684no—
486C2×D4—682no—
487C2×D4C22×#59514yes—
488C2×D4C22×#59512yes—
489C22×C4—171no—
490C22×C4—172no—
491C22×C4—174no—
492C22×C4—171no—
493C22×C4—172no—
494C22×C4—174no—
495C22×C4—174no—
496C22×C4—174no—
497C22×C4—174no—
498C22×C4—174no—
499C22×C4—174no—
500C22×C4—171no—
501C22×C4—174no—
502C22×C4—348no—
503C22×C4—684no—
504C22×C4—684no—
505C22×C4—684no—
506C22×C4—684no—
507C22×C4—684no—
508C22×C4—684no—
509C22×C4—684no—
510C22×C4C22×#59514yes—
511C4⋊C4—174no—
512C4⋊C4—174no—
513C4⋊C4—174no—
514C4⋊C4—174no—
515C4⋊C4—174no—
516C4⋊C4—174no—
517C4⋊C4—174no—
518C4⋊C4—172no—
519C4⋊C4—348no—
520C4⋊C4—682no—
521C4⋊C4—684no—
522C4⋊C4—684no—
523C4⋊C4—682no—
524C4⋊C4—684no—
525C4⋊C4—682no—
526C4⋊C4C22×#59514yes—
527C4⋊C4C22×#59514yes—
528C4⋊C4C22×#59512yes—
529C22⋊C4—174no—
530C22⋊C4—174no—
531C22⋊C4—174no—
532C22⋊C4—174no—
533C22⋊C4—174no—
534C22⋊C4—174no—
535C22⋊C4—174no—
536C22⋊C4—174no—
537C22⋊C4—174no—
538C22⋊C4—348no—
539C22⋊C4—348no—
540C22⋊C4—684no—
541C22⋊C4—684no—
542C22⋊C4—684no—
543C22⋊C4—684no—
544C22⋊C4C22×#59514yes—
545C22⋊C4C22×#59514yes—
546C22⋊C4C22×#59514yes—
547C42—172no—
548C42—171no—
549C42—342no—
550C42—348no—
551C42—344no—
552C42—681no—
553C42—684no—
554C42—682no—
555C42—684no—
556C42—684no—
557C42C22×#59511yescharacteristic
558C23—171no—
559C23—174no—
560C23—344no—
561C23—344no—
562C23—344no—
563C23—344no—
564C23—344no—
565C23—344no—
566C23C23×#59514yes—
567Q8—42no—
568Q8—684no—
569Q8—682no—
570D4—44no—
571D4—42no—
572D4—684no—
573D4—682no—
574D4—684no—
575D4—684no—
576D4—684no—
577C2×C4—44no—
578C2×C4—174no—
579C2×C4—172no—
580C2×C4—171no—
581C2×C4—172no—
582C2×C4—174no—
583C2×C4—174no—
584C2×C4—344no—
585C2×C4—342no—
586C2×C4—341no—
587C2×C4—344no—
588C2×C4—341no—
589C2×C4—344no—
590C2×C4—342no—
591C2×C4—344no—
592C2×C4—342no—
593C2×C4—342no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—344no—
597C2×C4—341no—
598C2×C4—344no—
599C2×C4—344no—
600C2×C4—344no—
601C2×C4—344no—
602C2×C4—344no—
603C2×C4—344no—
604C2×C4—344no—
605C2×C4—344no—
606C2×C4—684no—
607C2×C4—684no—
608C2×C4—684no—
609C2×C4—1368no—
610C2×C4—1368no—
611C2×C4C23×#59511yescharacteristic
612C2×C4C23×#59512yes—
613C2×C4C23×#59514yes—
614C2×C4C23×#59514yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—172no—
619C22—172no—
620C22—171no—
621C22—171no—
622C22—344no—
623C22—344no—
624C22—344no—
625C22—684no—
626C22—684no—
627C22—684no—
628C22—684no—
629C22C24×#59511yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—21no—
634C4—344no—
635C4—342no—
636C4—344no—
637C4—344no—
638C4—342no—
639C4—341no—
640C4—684no—
641C4—684no—
642C4—681no—
643C4—684no—
644C2—44no—
645C2—172no—
646C2—171no—
647C2—171no—
648C2—684no—
649C2H(5,𝔽2)×#59512yes—
650C22−1+4×#59511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#562C4
prime radicalD17×#562C2
cosocleC22×D17C25
hypercenter#562#595
hypocenterC17#34522
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #14545330C2semidirect
2. #14545327C2semidirect
3. #14545316C2semidirect
4. #14545311C2semidirect
5. #14545307C2semidirect
6. #14545283C2semidirect
7. #14545273C2semidirect
8. #14545271C2semidirect
9. #125×#595C2semidirect
10. #124×#595C2semidirect
52. #11763557C4semidirect
53. #11763556C4semidirect
54. #11763555C4semidirect
55. #11763554C4semidirect
56. #11763553C4semidirect
62. C17×#562C4semidirect
94. C17×#125C2×C4semidirect
95. C17×#124C2×C4semidirect
96. C17×#123C2×C4semidirect
97. C17×C22⋊Q8C2×C4semidirect
98. C17×#121C2×C4semidirect
99. #10587633C2×C4semidirect
100. #10587632C2×C4semidirect
101. #10587625C2×C4semidirect
102. #10587625C2×C4semidirect
103. #10587624C2×C4semidirect
104. #10587622C2×C4semidirect
105. #10587621C2×C4semidirect
106. #10587616C2×C4semidirect
107. #10587615C2×C4semidirect
108. #10587610C2×C4semidirect
109. #10587583C2×C4semidirect
110. #10587582C2×C4semidirect
111. #10587582C2×C4semidirect
112. #10587581C2×C4semidirect
113. #10587580C2×C4semidirect
114. #10587580C2×C4semidirect
118. #10587576C2×C4semidirect
119. #10587575C2×C4semidirect
120. #10587572C2×C4semidirect
121. #10587571C2×C4semidirect
122. #10587571C2×C4semidirect
123. #10587570C2×C4semidirect
124. #10587570C2×C4semidirect
125. #10587569C2×C4semidirect
126. #10587569C2×C4semidirect
130. #10587564C2×C4semidirect
131. #10587564C2×C4semidirect
132. #10587561C2×C4semidirect
134. #10587556C2×C4semidirect
335. #562#595semidirect
397. #125C2×#595semidirect
403. #124C2×#595semidirect
411. #123C2×#595semidirect
419. C22⋊Q8C2×#595semidirect
427. #121C2×#595semidirect
471. C17#34522semidirect

Maximal subgroups

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

subgroupindex
#145453302
#145453272
#145453162
#145453112
#145453072
#145452832
#145452732
#145452712
#125×#5952
#124×#5952
D17×#5622
#3452217