Finite Group Databasebeta

← #16965465

Subgroups — #16965465

Order 4352.

1486 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762671C211yesmaximal, characteristic
2C2×#11762668C211yesmaximal, characteristic
3C2×#11762666C211yesmaximal, characteristic
4#14527049C211yesmaximal, characteristic, complemented
5#14527047C211yesmaximal, characteristic, complemented
6#14527040C211yesmaximal, characteristic, complemented
7#14527038C212yesmaximal, complemented
8#14527035C212yesmaximal, complemented
9#14527030C212yesmaximal
10#14527028C212yesmaximal
11C17×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C4⋊Dic17C2212yes—
15C22×#63225C2212yes—
16C22×#63225C2212yes—
17C22×#63227C2211yescharacteristic
18C22×#63227C2212yes—
19C22×#63227C2211yescharacteristic
20C22×#63227C2212yes—
21#11762671—24no—
22#11762668C418yescomplemented
23#11762666—24no—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C2×C17×#95C2211yescharacteristic
35C2×C17×#95C2212yes—
36C2×C17×#95C2212yes—
37C2×C17×#95C2211yescharacteristic
38C24×D17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×Dic17C2311yescharacteristic
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C22×C4×D17—42no—
50C22×C4×D17—42no—
51C22×C4×D17C2×C414yes—
52C22×C4×D17D414yes—
53C22×C4×D17Q814yes—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17—42no—
56C2×#63227—24no—
57C2×#63227—22no—
58C2×#63227—22no—
59C2×#63227—42no—
60C2×#63227—42no—
61C2×#63227—42no—
62C2×#63227—42no—
63C2×#63227—42no—
64C2×#63227—42no—
65C2×#63227C2×C418yes—
66C2×C4⋊Dic17—22no—
67C2×C4⋊Dic17—42no—
68C2×C4⋊Dic17—42no—
69C2×C4⋊Dic17—42no—
70C2×C4⋊Dic17—42no—
71C2×#63225—42no—
72C2×#63225—42no—
73C2×#63225—42no—
74C2×#63225—42no—
75C2×#63225—42no—
76C2×#63225C2×C418yes—
77C17×#95—22no—
78C17×#95—42no—
79C17×#95—42no—
80C17×#95C2×C414yes—
81#10587502—22no—
82#10587502—24no—
83#10587502—22no—
84#10587502—22no—
85#10587502—42no—
86#10587502—42no—
87#10587502—42no—
88#10587502—42no—
89#10587502C2×C414yes—
90C24×C17C2411yescharacteristic
91C23×D17—22no—
92C23×D17—22no—
93C23×D17—21no—
94C23×D17—21no—
95C23×D17—22no—
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—22no—
99C23×D17C22×C412yes—
100C23×D17C2×D412yes—
101C23×D17C2×Q812yes—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17C22×C414yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2×Q812yes—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—21no—
154C22×Dic17—22no—
155C22×Dic17—21no—
156C22×Dic17C22×C414yes—
157C22×Dic17C22×C412yes—
158C22×Dic17C2×D412yes—
159C22×Dic17C2×Q812yes—
160C2×C4×D17—48no—
161C2×C4×D17—44no—
162C2×C4×D17—44no—
163C2×C4×D17—48no—
164C2×C4×D17C4⋊C4116yescomplemented
165C4⋊C4×C17—44no—
166#63227—48no—
167#63227—44no—
168#63227—44no—
169#63227—48no—
170C4⋊Dic17—44no—
171C4⋊Dic17—48no—
172#63225—48no—
173#63225—48no—
174#14675—171nomaximal, Sylow
175C23×C17C23×C411yescharacteristic
176C23×C17C22×D411yescharacteristic
177C23×C17C22×Q811yescharacteristic
178C23×C17H(5,𝔽2)12yes—
179C23×C17H(5,𝔽2)11yescharacteristic
180C23×C17H(5,𝔽2)12yes—
181C23×C17H(5,𝔽2)11yescharacteristic
182C23×C17H(5,𝔽2)12yes—
183C23×C172−1+411yescharacteristic
184C23×C172−1+411yescharacteristic
185C23×C172−1+412yes—
186C22×D17—24no—
187C22×D17—22no—
188C22×D17—22no—
189C22×D17—24no—
190C22×D17—22no—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—24no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—24no—
200C22×D17—42no—
201C22×D17—42no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17C2×C4⋊C414yes—
209C2×C4×C17—24no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—42no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17C2×C4⋊C414yes—
238C2×Dic17—24no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—42no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17C2×C4⋊C414yes—
267C4×D17—416no—
268C2×#381—171no—
269C2×#380—171no—
270C2×#380—171no—
271#1444—171no—
272#1444—171no—
273#1442—171no—
274#1442D1711yescharacteristic, complemented
275#1441—172no—
276#1441—172no—
277#1439—172no—
278#1439—172no—
279C22×C17C22×C4⋊C411yescharacteristic
280C22×C17#51812yes—
281C22×C17#51811yescharacteristic
282C22×C17#51911yescharacteristic
283C22×C17#52012yes—
284C22×C17#53411yescharacteristic
285C22×C17#53411yescharacteristic
286C22×C17#53511yescharacteristic
287C22×C17#53611yescharacteristic
288C22×C17#53711yescharacteristic
289C22×C17#53711yescharacteristic
290C22×C17#54312yes—
291C22×C17#54411yescharacteristic
292C22×C17#54411yescharacteristic
293C22×C17#54412yes—
294C22×C17#56012yes—
295C22×C17#56012yes—
296C22×C17#56212yes—
297C22×C17#56212yes—
298C22×C17#56212yes—
299C22×C17#56212yes—
300C22×C17#56312yes—
301C22×C17#56312yes—
302C2×D17—24no—
303C2×D17—24no—
304C2×D17—24no—
305C2×D17—44no—
306C2×D17—44no—
307C2×D17—44no—
308C2×D17—44no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C4×C17—24no—
314C4×C17—44no—
315C4×C17—82no—
316C4×C17—81no—
317C4×C17—81no—
318Dic17—24no—
319Dic17—44no—
320Dic17—81no—
321Dic17—82no—
322Dic17—81no—
323C24×C4—171no—
324C22×C4⋊C4—172no—
325C22×C4⋊C4—171no—
326C22×C4⋊C4—172no—
327C22×C4⋊C4—172no—
328C22×C4⋊C4C2×D1711yescharacteristic
329C22×C22⋊C4—172no—
330C22×C22⋊C4—172no—
331C22×C22⋊C4—171no—
332C22×C22⋊C4—171no—
333#381—344no—
334#380—178no—
335#380—344no—
336C2×#95—171no—
337C2×#95—172no—
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—171no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95C2×D1711yescharacteristic
346C2×#95C2×D1712yes—
347C2×#95C2×D1712yes—
348C2×#95C2×D1711yescharacteristic
349C2×C17#231511yescharacteristic
350C2×C17#231611yescharacteristic
351C2×C17#231711yescharacteristic
352C2×C17#237712yes—
353C2×C17#237812yes—
354C2×C17#281311yescharacteristic
355C2×C17#281311yescharacteristic
356C2×C17#281411yescharacteristic
357C2×C17#281411yescharacteristic
358C2×C17#284112yes—
359C2×C17#284112yes—
360D17—44no—
361C25—171no—
362C23×C4—171no—
363C23×C4—174no—
364C23×C4—174no—
365C23×C4—174no—
366C23×C4—172no—
367C23×C4—171no—
368C23×C4—171no—
369C23×C4—172no—
370C23×C4—682no—
371C23×C4—682no—
372C23×C4C22×D1712yes—
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—178no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4—682no—
383C2×C4⋊C4—682no—
384C2×C4⋊C4—682no—
385C2×C4⋊C4—682no—
386C2×C4⋊C4—682no—
387C2×C4⋊C4—682no—
388C2×C4⋊C4—682no—
389C2×C4⋊C4—682no—
390C2×C22⋊C4—178no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—342no—
393C2×C22⋊C4—342no—
394C2×C22⋊C4—682no—
395C2×C22⋊C4—682no—
396C2×C22⋊C4—682no—
397C2×C22⋊C4—682no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400#95—22no—
401#95—42no—
402#95—42no—
403#95—174no—
404#95—344no—
405#95—342no—
406#95—342no—
407#95—342no—
408#95—682no—
409#95—682no—
410#95—682no—
411#95—682no—
412#95C4×D1714yes—
413C17#1467511yescharacteristic, complemented, Sylow
414C24—172no—
415C24—172no—
416C24—172no—
417C24—341no—
418C24—342no—
419C24—341no—
420C24—341no—
421C24—342no—
422C24—342no—
423C24—341no—
424C24—342no—
425C24C23×D1711yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—172no—
452C22×C4—172no—
453C22×C4—172no—
454C22×C4—1716no—
455C22×C4—174no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—341no—
469C22×C4—342no—
470C22×C4—341no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—341no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4—688no—
482C22×C4—684no—
483C22×C4—684no—
484C22×C4—688no—
485C22×C4C2×C4×D1714yes—
486C22×C4C2×C4×D1712yes—
487C22×C4D4×D1712yes—
488C22×C4Q8×D1712yes—
489C4⋊C4—44no—
490C4⋊C4—688no—
491C4⋊C4—684no—
492C4⋊C4—688no—
493C4⋊C4—688no—
494C22⋊C4—688no—
495C22⋊C4—684no—
496C22⋊C4—684no—
497C22⋊C4—688no—
498C23—174no—
499C23—342no—
500C23—344no—
501C23—342no—
502C23—344no—
503C23—342no—
504C23—342no—
505C23—342no—
506C23—342no—
507C23—342no—
508C23—342no—
509C23—344no—
510C23—342no—
511C23—342no—
512C23—344no—
513C23—682no—
514C23—682no—
515C23—682no—
516C23—682no—
517C23—682no—
518C23—682no—
519C23—682no—
520C23—682no—
521C23C22×C4×D1711yescharacteristic
522C23C2×D4×D1711yescharacteristic
523C23#1058769111yescharacteristic
524C23#1058769112yes—
525C23C2×Q8×D1711yescharacteristic
526C23#1058769411yescharacteristic
527C23#1058769611yescharacteristic
528C23#1058769612yes—
529C23#1058769612yes—
530C23#1058769711yescharacteristic
531C23#1058769712yes—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—174no—
561C2×C4—344no—
562C2×C4—342no—
563C2×C4—342no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—344no—
568C2×C4—6816no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—682no—
590C2×C4C4⋊C4×D1714yes—
591C22—344no—
592C22—344no—
593C22—344no—
594C22—684no—
595C22—682no—
596C22—684no—
597C22—682no—
598C22—682no—
599C22—682no—
600C22—684no—
601C22—684no—
602C22C2×C4⋊C4×D1711yescharacteristic
603C22#1176328112yes—
604C22#1176328111yescharacteristic
605C22#1176328511yescharacteristic
606C22#1176328912yes—
607C22#1176336911yescharacteristic
608C22#1176337011yescharacteristic
609C22#1176337711yescharacteristic
610C22#1176337911yescharacteristic
611C22#1176338811yescharacteristic
612C22#1176338911yescharacteristic
613C22#1176343412yes—
614C22#1176343611yescharacteristic
615C22#1176343911yescharacteristic
616C22#1176343912yes—
617C22#1176354712yes—
618C22#1176355012yes—
619C22#1176355312yes—
620C22#1176355612yes—
621C22#1176355612yes—
622C22#1176355712yes—
623C22#1176356012yes—
624C22#1176356112yes—
625C4—24no—
626C4—44no—
627C4—82no—
628C4—81no—
629C4—81no—
630C4—344no—
631C4—684no—
632C4—1361no—
633C4—1361no—
634C4—1362no—
635C2—684no—
636C2#1453139511yescharacteristic
637C2#1453139911yescharacteristic
638C2#1453140911yescharacteristic
639C2#1453193712yes—
640C2#1453194912yes—
641C2#1453678511yescharacteristic
642C2#1453678911yescharacteristic
643C2#1453679411yescharacteristic
644C2#1453679811yescharacteristic
645C2#1453704012yes—
646C2#1453704312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1442C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1442D17
hypocenterC17#14675
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #14527049C2semidirect
5. #14527047C2semidirect
6. #14527040C2semidirect
7. #14527038C2semidirect
8. #14527035C2semidirect
11. C17×#1442C2semidirect
22. #11762668C4semidirect
164. C2×C4×D17C4⋊C4semidirect
274. #1442D17semidirect
413. C17#14675semidirect

Maximal subgroups

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

subgroupindex
C2×#117626712
C2×#117626682
C2×#117626662
#145270492
#145270472
#145270402
#145270382
#145270352
#145270302
#145270282
C17×#14422
#1467517