Finite Group Databasebeta

← #17068642

Subgroups — #17068642

Order 4352.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763366C211yesmaximal, characteristic, complemented
2C2×#11763099C211yesmaximal, characteristic, complemented
3C2×#11763014C211yesmaximal, characteristic, complemented
4#14529980C212yesmaximal, complemented
5#14529787C212yesmaximal
6#14529732C211yesmaximal, characteristic, complemented
7#14529707C211yesmaximal, characteristic, complemented
8#14529142C212yesmaximal, complemented
9#14529051C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C4×C17×#121C211yesmaximal, characteristic
12C22×#63229C2212yes—
13#11763366—24no—
14#11763366—24no—
15C2×#10587555C2211yescharacteristic
16C2×#10587555C2211yescharacteristic
17C2×#10587556—21no—
18C2×#10587610—21no—
19C2×#10587610C2212yes—
20C2×C4×Dic34—21no—
21C2×C4×Dic34C2212yes—
22C2×#10587612—21no—
23#11763099—24no—
24#11763014—24no—
25C2×#10587479C2212yes—
26C2×C4⋊#3622C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587481C2212yes—
29C2×#10587481C2212yes—
30C2×#10587481C2212yes—
31C2×#10587503C2212yes—
32C2×C22⋊#3622C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C2×C17×#121C2211yescharacteristic
37C2×C4×D4×C17—21no—
38C2×C4×D4×C17C2211yescharacteristic
39C22×C42×C17C2211yescharacteristic
40C17×#390C2211yescharacteristic
41C17×#389C2211yescharacteristic
42C17×#388—21no—
43C17×#386—21no—
44C17×#380C2211yescharacteristic
45C4×C4⋊C4×C17C2211yescharacteristic
46C4×C22⋊C4×C17—21no—
47C22×D4×C17—21no—
48C22×D4×C17C2311yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17D411yescharacteristic
51C23×C4×C17D411yescharacteristic
52C23×C4×C17C2311yescharacteristic
53C22×Dic34—21no—
54C22×Dic34C2312yes—
55C17×#121—22no—
56C17×#121—22no—
57C4×D4×C17—22no—
58C4×D4×C17—22no—
59C4×D4×C17—44no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C4⋊C4×C17C2311yescharacteristic
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C22⋊C4×C17—21no—
65C2×C22⋊C4×C17—21no—
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C42×C17—22no—
70C2×C42×C17—22no—
71C2×C42×C17—21no—
72C2×C42×C17D411yescharacteristic
73C2×C42×C17D411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—22no—
77C2×#63229—24no—
78C2×#63229—24no—
79C2×#63229—42no—
80#10587612—28no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17C2312yes—
85#10587610—28no—
86#10587610—24no—
87#10587610—24no—
88C2×#63225—22no—
89C2×#63225—22no—
90C2×C4×Dic17—22no—
91C22×#3622C2312yes—
92C22×#3622C2312yes—
93#10587556—28no—
94#10587555—22no—
95#10587555—22no—
96#10587555D414yescomplemented
97#10587555D414yescomplemented
98C4×Dic34—28no—
99C4×Dic34—24no—
100C4×Dic34—24no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104#10587503—24no—
105C22⋊#3622—42no—
106#10587481—24no—
107#10587481—24no—
108#10587481—42no—
109#10587479—24no—
110C4⋊#3622—22no—
111C4⋊#3622—22no—
112C24×C17—21no—
113C24×C17C2×D411yescharacteristic
114C2×D4×C17—24no—
115C2×D4×C17—21no—
116C2×D4×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—42no—
121C2×D4×C17—42no—
122C2×D4×C17#4112yes—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—41no—
136C22×C4×C17—41no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17—42no—
140C22×C4×C17SD1614yescomplemented
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—22no—
148C22×Dic17—22no—
149C2×Dic34—22no—
150C2×Dic34—22no—
151C2×Dic34—22no—
152C2×Dic34—22no—
153C2×Dic34—24no—
154C2×Dic34—22no—
155C2×Dic34—42no—
156C2×Dic34—42no—
157C2×Dic34C2×D414yes—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17#4112yes—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—42no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—42no—
170C42×C17—22no—
171C42×C17—22no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—42no—
175C42×C17C2×D412yes—
176C42×C17C2×D412yes—
177C22⋊Dic17—28no—
178#63229—48no—
179C4⋊Dic17—24no—
180C4⋊Dic17—24no—
181C4⋊Dic17—24no—
182C4⋊Dic17—22no—
183C4⋊Dic17C2×D414yes—
184#63225—28no—
185#63225—28no—
186C4×Dic17—28no—
187C2×#3622—22no—
188C2×#3622—22no—
189C2×#3622—22no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193#24151—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—22no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17C2×SD1612yes—
208C23×C17C22×D411yescharacteristic
209D4×C17—22no—
210D4×C17—22no—
211D4×C17—48no—
212D4×C17—41no—
213D4×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—28no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—44no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—84no—
240C2×C4×C17#12012yes—
241C2×C4×C17#12012yes—
242C2×C4×C17C2×SD1612yes—
243C2×C4×C17C2×SD1612yes—
244C2×C4×C17#13712yes—
245C2×C4×C17C22×D411yescharacteristic
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C172−1+411yescharacteristic
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257Dic34—24no—
258Dic34—24no—
259Dic34—48no—
260Dic34—42no—
261Dic34—42no—
262#3622—42no—
263#3622—42no—
264C2×#533—171no—
265C2×#476—171no—
266C2×#461—171no—
267C4×#121D1711yescharacteristic
268#2035—172no—
269#2000—172no—
270#1987—171no—
271#1981—171no—
272#1880—172no—
273#1865—172no—
274#1797—171no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—81no—
286C22×C17—81no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17C2×#12011yescharacteristic
290C22×C17#52511yescharacteristic
291C22×C17#53711yescharacteristic
292C22×C17C22×SD1611yescharacteristic
293C22×C17C2×#13711yescharacteristic
294C22×C17#57511yescharacteristic
295C22×C17#57511yescharacteristic
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—42no—
299C4×C17—41no—
300C4×C17—41no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17#44814yes—
304C4×C17#54611yescharacteristic
305C4×C17#54711yescharacteristic
306C4×C17#54811yescharacteristic
307C4×C17#55011yescharacteristic
308Dic17—44no—
309Dic17—44no—
310C22×SD16—172no—
311#533—344no—
312#533—344no—
313C2×C4⋊Q8—171no—
314C2×C4⋊Q8—171no—
315C2×#125—341no—
316C2×C22⋊Q8—172no—
317C2×C22⋊Q8—341no—
318C2×#121C2×D1711yescharacteristic
319C2×C4×Q8—172no—
320C2×C4×Q8—341no—
321C2×C4×D4—21no—
322C2×C4×D4C2×D1711yescharacteristic
323C2×#117—341no—
324C22×C42C2×D1711yescharacteristic
325#476—344no—
326#461—344no—
327C2×#106—172no—
328C2×C4⋊C8—171no—
329C2×C4⋊C8—171no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#103—172no—
333C2×#102—172no—
334C2×C22⋊C8—172no—
335#390C2×D1711yescharacteristic
336#389C2×D1711yescharacteristic
337#388—21no—
338#386—21no—
339#380C2×D1711yescharacteristic
340C4×C4⋊C4C2×D1711yescharacteristic
341C4×C22⋊C4—21no—
342#340—172no—
343#340—172no—
344#336—172no—
345C2×C17—42no—
346C2×C17—82no—
347C2×C17#240711yescharacteristic
348C2×C17C2×#44811yescharacteristic
349C2×C17#300511yescharacteristic
350C2×C17#310111yescharacteristic
351C2×C17#311611yescharacteristic
352C2×C17#324611yescharacteristic
353C2×C17#325211yescharacteristic
354C22×Q8—172no—
355C22×Q8—341no—
356C22×D4—21no—
357C22×D4C22×D1711yescharacteristic
358C23×C4—21no—
359C23×C4C22⋊D1711yescharacteristic
360C23×C4C22⋊D1711yescharacteristic
361C23×C4C22×D1711yescharacteristic
362C2×SD16—344no—
363C2×SD16—344no—
364C2×SD16—682no—
365C22×C8—172no—
366C22×C8—172no—
367C4⋊Q8—174no—
368C4⋊Q8—174no—
369C4⋊Q8—342no—
370C4⋊Q8—342no—
371#125—348no—
372C22⋊Q8—344no—
373C22⋊Q8—344no—
374C22⋊Q8—348no—
375#121—22no—
376#121—22no—
377C4×Q8—344no—
378C4×Q8—344no—
379C4×Q8—348no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—44no—
383#117—348no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—172no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—342no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—342no—
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22⋊D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#106—344no—
409C4⋊C8—342no—
410C4⋊C8—342no—
411#103—344no—
412#103—344no—
413#103—682no—
414#102—344no—
415C22⋊C8—682no—
416#95—21no—
417#95—21no—
418#95—21no—
419C17#2415111yescharacteristic, complemented, Sylow
420C24—21no—
421C24C2×C22⋊D1711yescharacteristic
422C2×Q8—174no—
423C2×Q8—342no—
424C2×Q8—344no—
425C2×Q8—342no—
426C2×Q8—342no—
427C2×Q8—342no—
428C2×Q8—342no—
429C2×Q8—682no—
430C2×Q8—682no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#6325212yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4#6322914yescomplemented
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C23×D1711yescharacteristic
466SD16—688no—
467C2×C8—342no—
468C2×C8—342no—
469C2×C8—342no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—174no—
480C4⋊C4—348no—
481C4⋊C4—348no—
482C4⋊C4—342no—
483C4⋊C4—344no—
484C4⋊C4—344no—
485C4⋊C4—344no—
486C4⋊C4#6325212yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—348no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—348no—
499C42D4×D1712yes—
500C42D4×D1712yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#6322912yes—
515C23C22×C22⋊D1711yescharacteristic
516Q8—344no—
517Q8—344no—
518Q8—688no—
519Q8—682no—
520Q8—682no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—344no—
553C2×C4—344no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4C2×#6322912yes—
559C2×C4C2×#6322912yes—
560C2×C4#1058762312yes—
561C2×C4#1058762312yes—
562C2×C4#1058763712yes—
563C2×C4C2×#6325211yescharacteristic
564C2×C4C2×#6325211yescharacteristic
565C2×C4C2×D4×D1711yescharacteristic
566C2×C4C2×D4×D1711yescharacteristic
567C2×C4#1058769611yescharacteristic
568C2×C4#1058769711yescharacteristic
569C8—682no—
570C8—682no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1058762311yescharacteristic
586C22#1176332011yescharacteristic
587C22#1176338711yescharacteristic
588C22C22×#6322911yescharacteristic
589C22C2×#1058763711yescharacteristic
590C22#1176360711yescharacteristic
591C22#1176360711yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—684no—
600C4—684no—
601C4#1176294714yes—
602C4#1176345511yescharacteristic
603C4#1176346411yescharacteristic
604C4#1176346811yescharacteristic
605C4#1176348011yescharacteristic
606C2—42no—
607C2—82no—
608C2#1453212911yescharacteristic
609C2C2×#1176294711yescharacteristic
610C2#1453802411yescharacteristic
611C2#1453879211yescharacteristic
612C2#1453894611yescharacteristic
613C2#1454010011yescharacteristic
614C2#1454015711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×#121C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC4×#121D17
hypocenterC17#24151
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#11763366C2semidirect
2. C2×#11763099C2semidirect
3. C2×#11763014C2semidirect
4. #14529980C2semidirect
6. #14529732C2semidirect
7. #14529707C2semidirect
8. #14529142C2semidirect
9. #14529051C2semidirect
10. #14528764C2semidirect
96. #10587555D4semidirect
97. #10587555D4semidirect
140. C22×C4×C17SD16semidirect
419. C17#24151semidirect
459. C22×C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117633662
C2×#117630992
C2×#117630142
#145299802
#145297872
#145297322
#145297072
#145291422
#145290512
#145287642
C4×C17×#1212
#2415117