Finite Group Databasebeta

← #16965678

Subgroups — #16965678

Order 4352.

1770 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587565C211yesmaximal, characteristic
2C2×#11762676C211yesmaximal, characteristic, complemented
3C2×C42×Dic17C211yesmaximal, characteristic, complemented
4#14527021C212yesmaximal
5#14527021C212yesmaximal
6#14527021C211yesmaximal, characteristic
7#14527019C211yesmaximal, characteristic, complemented
8#14527018C212yesmaximal, complemented
9#14527017C211yesmaximal, characteristic, complemented
10#14527015C212yesmaximal, complemented
11C17×#1437C211yesmaximal, characteristic
12C2×#10587565—24no—
13C2×#10587565—24no—
14C2×#10587565C418yescomplemented
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×C22⋊Dic17C2212yes—
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C4×Dic17C2212yes—
22C22×C4×Dic17C2212yes—
23C22×C4×Dic17C2211yescharacteristic
24C22×C4×Dic17C2211yescharacteristic
25#11762676—24no—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C42×Dic17—24no—
35C22×C22⋊C4×C17C2212yes—
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C25×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×Dic17C2312yes—
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2312yes—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17—22no—
53C2×C22⋊C4×C17—24no—
54C2×C22⋊C4×C17—24no—
55C2×C22⋊C4×C17—24no—
56C2×C22⋊C4×C17C2×C414yes—
57C2×C42×C17—22no—
58C2×C42×C17—24no—
59C2×C22⋊Dic17—22no—
60C2×C22⋊Dic17—22no—
61C2×C22⋊Dic17—22no—
62C2×C22⋊Dic17—22no—
63C2×C22⋊Dic17—24no—
64C2×C22⋊Dic17—24no—
65C2×C22⋊Dic17—24no—
66C2×C22⋊Dic17C2×C414yes—
67C2×C22⋊Dic17C2×C414yes—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17—22no—
70C2×C4⋊Dic17C2×C414yes—
71C2×#63225—22no—
72C2×#63225—22no—
73C2×#63225—22no—
74C2×#63225—22no—
75C2×#63225C2×C414yes—
76C2×#63225C2×C414yes—
77C2×C4×Dic17—22no—
78C2×C4×Dic17—24no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17—24no—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17—24no—
84C2×C4×Dic17—24no—
85C2×C4×Dic17C2×C414yes—
86C2×C4×Dic17D414yes—
87C2×C4×Dic17D414yes—
88#10587565—232no—
89C17×#95—22no—
90C17×#95—24no—
91#10587502—22no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—24no—
97#10587502—24no—
98#10587502—24no—
99C24×C17—21no—
100C24×C17—21no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17—22no—
106C24×C17—22no—
107C24×C17C22×C412yes—
108C24×C17C2411yescharacteristic
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—21no—
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—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17#4114yes—
131C22×C4×C17#4114yes—
132C22×C4×C17#4114yes—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17C22×C412yes—
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C412yes—
160C22×Dic17C2×D412yes—
161C22×Dic17C2×D412yes—
162C22×Dic17C2×D412yes—
163C22×Dic17C2×D412yes—
164C22×Dic17#4114yes—
165C22×Dic17#4114yes—
166C22×Dic17#4114yes—
167C22⋊C4×C17—28no—
168C22⋊C4×C17—48no—
169C42×C17—28no—
170C22⋊Dic17—28no—
171C22⋊Dic17—28no—
172C22⋊Dic17—48no—
173C4⋊Dic17—28no—
174#63225—28no—
175#63225—28no—
176C4×Dic17—216no—
177C4×Dic17—216no—
178C4×Dic17—28no—
179C4×Dic17C22⋊C4116yescomplemented
180#14711—171nomaximal, Sylow
181C23×C17—22no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—42no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17C23×C411yescharacteristic
203C23×C17C22×D411yescharacteristic
204C23×C17C22×D411yescharacteristic
205C23×C17C2×#4112yes—
206C23×C17C2×#4112yes—
207C23×C17C2×#4112yes—
208C23×C17C2×#4112yes—
209C23×C17C2×#4112yes—
210C23×C17C2×#4112yes—
211C2×C4×C17—24no—
212C2×C4×C17—22no—
213C2×C4×C17—24no—
214C2×C4×C17—22no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—22no—
222C2×C4×C17—24no—
223C2×C4×C17—42no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17C2×C22⋊C414yes—
234C2×C4×C17#12618yescomplemented
235C2×Dic17—24no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—24no—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—24no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—42no—
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×C22⋊C414yes—
267C2×Dic17C2×C22⋊C414yes—
268C2×Dic17#12618yescomplemented
269C22×#126—171no—
270C2×#383—171no—
271C2×C43—171no—
272#1437—172no—
273#1437—172no—
274#1437—171no—
275#1437D1711yescharacteristic
276#1436—171no—
277#1436—172no—
278#1436—172no—
279#1436—171no—
280C22×C17—44no—
281C22×C17—44no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—82no—
286C22×C17—82no—
287C22×C17—82no—
288C22×C17—82no—
289C22×C17C22×C22⋊C411yescharacteristic
290C22×C17C4×#4112yes—
291C22×C17C4×#4112yes—
292C22×C17C4×#4111yescharacteristic
293C22×C17C4×#4111yescharacteristic
294C22×C17#52511yescharacteristic
295C22×C17#52511yescharacteristic
296C22×C17#52512yes—
297C22×C17#52512yes—
298C22×C17C2×#12612yes—
299C22×C17C2×#12612yes—
300C22×C17#52912yes—
301C22×C17#52912yes—
302C22×C17#52912yes—
303C22×C17#52912yes—
304C22×C17#52912yes—
305C22×C17#52912yes—
306C22×C17#53212yes—
307C22×C17#53212yes—
308C22×C17#53211yescharacteristic
309C22×C17#53211yescharacteristic
310C4×C17—28no—
311C4×C17—24no—
312C4×C17—42no—
313C4×C17—82no—
314Dic17—28no—
315Dic17—24no—
316Dic17—24no—
317Dic17—42no—
318Dic17—42no—
319Dic17—82no—
320C2×#126—178no—
321C2×#126—344no—
322C2×#126—344no—
323C22×C4⋊C4—171no—
324C22×C4⋊C4—171no—
325C22×C4⋊C4—171no—
326C22×C22⋊C4—171no—
327C22×C22⋊C4—171no—
328C22×C22⋊C4—172no—
329C22×C22⋊C4C2×D1712yes—
330C22×C22⋊C4C2×D1711yescharacteristic
331C22×C42—171no—
332C22×C42—172no—
333C22×C42—172no—
334C22×C42—171no—
335C22×C42C2×D1711yescharacteristic
336#383—344no—
337C2×#95—172no—
338C2×#95—172no—
339C2×#95—172no—
340C2×#95—172no—
341C2×#95—172no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—171no—
345C2×#95C2×D1712yes—
346C2×#95C2×D1711yescharacteristic
347C43—344no—
348C2×C17—82no—
349C2×C17#229711yescharacteristic
350C2×C17#229711yescharacteristic
351C2×C17#229711yescharacteristic
352C2×C17C4×#12612yes—
353C2×C17C4×#12612yes—
354C2×C17#285212yes—
355C2×C17#285212yes—
356C2×C17#286012yes—
357C2×C17#286012yes—
358C25C22×D1711yescharacteristic
359C23×C4—171no—
360C23×C4—171no—
361C23×C4—172no—
362C23×C4—171no—
363C23×C4—171no—
364C23×C4—172no—
365C23×C4C22×D1712yes—
366C23×C4C22×D1711yescharacteristic
367C23×C4C22×D1711yescharacteristic
368C23×C4C22×D1712yes—
369#126—3432no—
370C2×C4⋊C4—174no—
371C2×C4⋊C4—174no—
372C2×C4⋊C4—174no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4—342no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—342no—
378C2×C4⋊C4—342no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—174no—
385C2×C22⋊C4—174no—
386C2×C22⋊C4—342no—
387C2×C22⋊C4—342no—
388C2×C22⋊C4—342no—
389C2×C22⋊C4—342no—
390C2×C22⋊C4—344no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4C4×D1714yes—
394C2×C42—22no—
395C2×C42—24no—
396C2×C42—174no—
397C2×C42—174no—
398C2×C42—174no—
399C2×C42—344no—
400C2×C42—344no—
401C2×C42—344no—
402C2×C42—344no—
403C2×C42—344no—
404C2×C42—344no—
405C2×C42—342no—
406C2×C42—344no—
407#95—22no—
408#95—24no—
409#95—344no—
410#95—344no—
411#95—344no—
412#95—344no—
413#95—344no—
414#95—344no—
415#95—342no—
416#95—344no—
417C17#1471111yescharacteristic, complemented, Sylow
418C24—22no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24C2×C4×D1712yes—
427C24C23×D1711yescharacteristic
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—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—172no—
446C22×C4—172no—
447C22×C4—172no—
448C22×C4—172no—
449C22×C4—172no—
450C22×C4—172no—
451C22×C4—174no—
452C22×C4—174no—
453C22×C4—174no—
454C22×C4—172no—
455C22×C4—172no—
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—342no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4C2×C4×D1712yes—
480C22×C4C2×C4×D1712yes—
481C22×C4#6325214yes—
482C22×C4D4×D1712yes—
483C22×C4D4×D1712yes—
484C22×C4#6325414yes—
485C22×C4#6325414yes—
486C4⋊C4—348no—
487C4⋊C4—348no—
488C4⋊C4—348no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—348no—
492C22⋊C4—348no—
493C22⋊C4—688no—
494C42—28no—
495C42—1716no—
496C42—348no—
497C42—3416no—
498C42—3416no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23C22×C4×D1711yescharacteristic
521C23C2×#6325212yes—
522C23C2×D4×D1711yescharacteristic
523C23C2×D4×D1711yescharacteristic
524C23C2×#6325412yes—
525C23C2×#6325412yes—
526C23#41×D1712yes—
527C23#41×D1712yes—
528C23#41×D1712yes—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—174no—
552C2×C4—174no—
553C2×C4—178no—
554C2×C4—344no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—344no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—344no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—344no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—342no—
569C2×C4—342no—
570C2×C4—344no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4#1058756518yescomplemented
586C2×C4C22⋊C4×D1714yes—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×C22⋊C4×D1711yescharacteristic
597C22C4×#6325412yes—
598C22C4×#6325412yes—
599C22#1176327711yescharacteristic
600C22C4×#6325211yescharacteristic
601C22#1176331511yescharacteristic
602C22#1176331711yescharacteristic
603C22#1176331812yes—
604C22#1176331912yes—
605C22#126×D1712yes—
606C22C2×#1058756512yes—
607C22#1176333612yes—
608C22#1176333712yes—
609C22#1176333812yes—
610C22#1176333912yes—
611C22#1176334012yes—
612C22#1176334112yes—
613C22#1176335412yes—
614C22#1176335612yes—
615C22#1176335711yescharacteristic
616C22#1176335811yescharacteristic
617C4—28no—
618C4—24no—
619C4—42no—
620C4—82no—
621C4—348no—
622C4—344no—
623C4—344no—
624C4—682no—
625C4—682no—
626C4—1362no—
627C2—82no—
628C2#1453126911yescharacteristic
629C2#1453127011yescharacteristic
630C2#1453127111yescharacteristic
631C2C4×#1058756512yes—
632C2#1453133812yes—
633C2#1453709712yes—
634C2#1453709812yes—
635C2#1453714812yes—
636C2#1453715012yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1437C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1437D17
hypocenterC17#14711
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#11762676C2semidirect
3. C2×C42×Dic17C2semidirect
7. #14527019C2semidirect
8. #14527018C2semidirect
9. #14527017C2semidirect
10. #14527015C2semidirect
14. C2×#10587565C4semidirect
179. C4×Dic17C22⋊C4semidirect
234. C2×C4×C17#126semidirect
268. C2×Dic17#126semidirect
417. C17#14711semidirect
585. C2×C4#10587565semidirect

Maximal subgroups

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

subgroupindex
C22×#105875652
C2×#117626762
C2×C42×Dic172
#145270212
#145270192
#145270182
#145270172
#145270152
C17×#14372
#1471117