Finite Group Databasebeta

← #17063015

Subgroups — #17063015

Order 4352.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762833C211yesmaximal, characteristic, complemented
3C2×#11762808C211yesmaximal, characteristic
4C2×#11762808C211yesmaximal, characteristic
5#14531274C211yesmaximal, characteristic
6#14530176C212yesmaximal, complemented
7#14530169C212yesmaximal, complemented
8#14530162C212yesmaximal, complemented
9#14530155C212yesmaximal, complemented
10#14528503C211yesmaximal, characteristic, complemented
11C2×C17×#389C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C22×#63223C2211yescharacteristic
15C2×#10587612C2211yescharacteristic
16#11762849—24no—
17C2×#10587478C2212yes—
18C2×#10587479C2212yes—
19#11762833—24no—
20C2×C4⋊#3622C2212yes—
21C2×C4⋊#3622C2212yes—
22#11762808—24no—
23#11762808—24no—
24C2×#10587481C2212yes—
25C2×#10587481C2212yes—
26C2×#10587480C2212yes—
27C2×#10587480C2212yes—
28#11762715C2211yescharacteristic
29#11762715C2211yescharacteristic
30#11762711C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762688—21no—
33#11762688—21no—
34C4×#63227—21no—
35C4×#63227—21no—
36#11762571C2212yes—
37#11762571C2212yes—
38C22×C4⋊C4×C17—21no—
39C22×C4⋊C4×C17C2211yescharacteristic
40C22×C4⋊C4×C17C2211yescharacteristic
41C22×C42×C17C2211yescharacteristic
42C17×#389—21no—
43C17×#389—21no—
44C17×#389C2212yes—
45C17×#389C2212yes—
46C2×C17×#95—21no—
47C23×C4×C17—21no—
48C23×C4×C17—21no—
49C23×C4×C17D411yescharacteristic
50C23×C4×C17D411yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C22×C22⋊D17—21no—
53C2×#63252—22no—
54C2×#63252—22no—
55C2×#63252—22no—
56C2×#63252—22no—
57C2×#63252—22no—
58C2×#63252Q814yes—
59C22×D68C2311yescharacteristic
60C22×C4×D17—21no—
61C22×Dic34C2311yescharacteristic
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17—22no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—44no—
70C2×C4⋊C4×C17D414yes—
71C2×C4⋊C4×C17C2312yes—
72C2×C4⋊C4×C17C2312yes—
73C2×C42×C17—22no—
74C2×C42×C17—22no—
75C2×C42×C17D411yescharacteristic
76C2×C42×C17D411yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—21no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84#10587612—22no—
85#10587612—22no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4×Dic17—21no—
89C2×#63223—22no—
90C2×#63223—22no—
91C2×#63223—22no—
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223D414yes—
95C22×#3622C2312yes—
96C22×#3622C2312yes—
97C17×#95—22no—
98C17×#95—22no—
99#10587502—21no—
100#10587502—21no—
101#10587502—21no—
102#10587502—21no—
103#10587481—24no—
104#10587481—24no—
105#10587480—24no—
106#10587480—24no—
107#10587479—24no—
108#10587478—24no—
109C4⋊#3622—24no—
110C4⋊#3622—24no—
111C24×C17C2×D411yescharacteristic
112C23×D17—21no—
113C22×C4×C17—24no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—41no—
121C22×C4×C17—41no—
122C22×C4×C17—42no—
123C22×C4×C17—42no—
124C22×C4×C17—42no—
125C22×C4×C17—42no—
126C22×C4×C17—42no—
127C22×C4×C17C2×D412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×Q812yes—
135C22×C4×C17#4112yes—
136C22×C4×C17#4112yes—
137C22×C4×C17#4112yes—
138C22×C4×C17C2411yescharacteristic
139C2×C22⋊D17—24no—
140C2×C22⋊D17—42no—
141C2×C22⋊D17—42no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145#63252—28no—
146#63252—28no—
147#63252—44no—
148#63252—44no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68C2×Q812yes—
155C2×C4×D17—24no—
156C2×C4×D17—42no—
157C2×C4×D17—42no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34C2×Q812yes—
164C4⋊C4×C17—24no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—22no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—44no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17—44no—
171C4⋊C4×C17C2×D414yes—
172C42×C17—22no—
173C42×C17—22no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C22⋊Dic17—42no—
179#63227—42no—
180#63227—42no—
181#63227—42no—
182#63227—42no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4×Dic17—42no—
188#63223—28no—
189#63223—44no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622C2×D414yes—
196#23617—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17#12112yes—
202C23×C17#12112yes—
203C23×C17C22⋊Q812yes—
204C23×C17C22×D411yescharacteristic
205C22×D17—22no—
206C22×D17—41no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C22×D17—41no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—24no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—44no—
224C2×C4×C17—44no—
225C2×C4×C17—48no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—84no—
232C2×C4×C17#12114yes—
233C2×C4×C17#12112yes—
234C2×C4×C17#12112yes—
235C2×C4×C17C22⋊Q814yes—
236C2×C4×C17C22⋊Q814yes—
237C2×C4×C17C22⋊Q812yes—
238C2×C4×C17#12614yes—
239C2×C4×C17AGL(1,ℤ/8)12yes—
240C2×C4×C17#13712yes—
241C2×C4×C17C22×D411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C22×Q811yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C22⋊D17—48no—
248C2×Dic17—22no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257D68—22no—
258D68—22no—
259D68—41no—
260D68—41no—
261C4×D17—48no—
262Dic34—22no—
263Dic34—22no—
264Dic34—41no—
265Dic34—41no—
266#3622—24no—
267#3622—42no—
268C2×#428—171no—
269C2×#423—171no—
270C2×#417—171no—
271C2×#417—171no—
272#2297—171no—
273C2×#389D1711yescharacteristic, complemented
274#2068—172no—
275#2067—172no—
276#2066—172no—
277#2065—172no—
278#1736—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—22no—
285C22×C17#39614yes—
286C22×C17C2×#12111yescharacteristic
287C22×C17C2×#12111yescharacteristic
288C22×C17C2×C22⋊Q811yescharacteristic
289C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C17C2×#13711yescharacteristic
291C22×C17#57511yescharacteristic
292C22×C17#57511yescharacteristic
293C2×D17—42no—
294C2×D17—42no—
295C2×D17—42no—
296C2×D17—81no—
297C2×D17—81no—
298C2×D17—81no—
299C2×D17—81no—
300C4×C17—22no—
301C4×C17—22no—
302C4×C17—44no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—82no—
306C4×C17#39614yes—
307C4×C17C2×#12111yescharacteristic
308C4×C17C2×C22⋊Q811yescharacteristic
309C4×C17C2×C22⋊Q811yescharacteristic
310C4×C17C2×#12611yescharacteristic
311Dic17—42no—
312Dic17—81no—
313Dic17—81no—
314C22×#41—171no—
315C22×M16—171no—
316C22×M16—171no—
317C2×#117—171no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4C2×D1711yescharacteristic
320C22×C4⋊C4C2×D1711yescharacteristic
321C22×C42C2×D1711yescharacteristic
322#428—344no—
323C2×#107—172no—
324C2×#106—172no—
325#423—344no—
326C2×C4⋊C8—172no—
327C2×C4⋊C8—172no—
328#417—344no—
329#417—344no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#102—172no—
333C2×#102—172no—
334#391—171no—
335#391—171no—
336#390—171no—
337#390—171no—
338#389—21no—
339#389—21no—
340#389C2×D1712yes—
341#389C2×D1712yes—
342#386—341no—
343#386—341no—
344C4×C22⋊C4—341no—
345C4×C22⋊C4—341no—
346C2×#95—21no—
347#340—172no—
348#340—172no—
349C2×C17—24no—
350C2×C17C2×#39611yescharacteristic
351C2×C17#303611yescharacteristic
352C2×C17#303711yescharacteristic
353C2×C17#305511yescharacteristic
354C2×C17#305611yescharacteristic
355C2×C17#307911yescharacteristic
356C2×C17#308011yescharacteristic
357D17—82no—
358C2×#41—174no—
359C2×#41—342no—
360C2×#41—342no—
361C2×#41—342no—
362C2×#41—342no—
363C2×#41—342no—
364C22×Q8—171no—
365C22×D4—171no—
366C22×D4—341no—
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—341no—
370C23×C4C22⋊D1711yescharacteristic
371C23×C4C22⋊D1711yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C2×M16—174no—
374C2×M16—342no—
375C2×M16—342no—
376C2×M16—342no—
377C2×M16—342no—
378C2×M16—342no—
379C22×C8—172no—
380C22×C8—172no—
381#117—342no—
382#117—342no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—44no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4D6814yes—
394C2×C4⋊C4C22×D1712yes—
395C2×C4⋊C4C22×D1712yes—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—341no—
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22⋊D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—344no—
409#106—344no—
410C4⋊C8—344no—
411C4⋊C8—344no—
412#103—344no—
413#103—344no—
414#102—344no—
415#102—344no—
416#95—22no—
417#95—22no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2361711yescharacteristic, complemented, Sylow
423C24—341no—
424C24C2×C22⋊D1711yescharacteristic
425#41—348no—
426#41—348no—
427#41—684no—
428#41—684no—
429C2×Q8—172no—
430C2×Q8—341no—
431C2×Q8—341no—
432C2×Q8—341no—
433C2×Q8—341no—
434C2×Q8—341no—
435C2×D4—172no—
436C2×D4—344no—
437C2×D4—341no—
438C2×D4—341no—
439C2×D4—341no—
440C2×D4—341no—
441C2×D4—341no—
442C2×D4—682no—
443C2×D4—682no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—341no—
459C22×C4—344no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4C2×D6812yes—
465C22×C4#6325212yes—
466C22×C4D4×D1712yes—
467C22×C4#6325412yes—
468C22×C4Q8×D1712yes—
469C22×C4#6325612yes—
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C23×D1711yescharacteristic
476M16—348no—
477M16—684no—
478C2×C8—174no—
479C2×C8—342no—
480C2×C8—342no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4C2×D6814yes—
496C22⋊C4—682no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C22⋊C4—682no—
500C22⋊C4—682no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—682no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—342no—
513C23—681no—
514C23—681no—
515C23—681no—
516C23—681no—
517C23—681no—
518C23—681no—
519C23#1058761612yes—
520C23#1058762512yes—
521C23#1058763212yes—
522C23C22×C22⋊D1711yescharacteristic
523Q8—342no—
524Q8—342no—
525Q8—681no—
526Q8—681no—
527D4—342no—
528D4—342no—
529D4—688no—
530D4—681no—
531D4—681no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—342no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—688no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4#1058758114yes—
563C2×C4#1058758214yes—
564C2×C4#1058758314yes—
565C2×C4#1058758414yes—
566C2×C4#1058761612yes—
567C2×C4#1058761812yes—
568C2×C4#1058762512yes—
569C2×C4#1058762812yes—
570C2×C4#1058763212yes—
571C2×C4C22×D6811yescharacteristic
572C2×C4C2×#6325211yescharacteristic
573C2×C4C2×D4×D1711yescharacteristic
574C2×C4C2×#6325411yescharacteristic
575C2×C4C2×Q8×D1711yescharacteristic
576C2×C4C2×#6325611yescharacteristic
577C8—344no—
578C8—682no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—682no—
586C22—682no—
587C22—682no—
588C22—1361no—
589C22—1361no—
590C22—1361no—
591C22—1361no—
592C22#1176273114yes—
593C22C2×#1058762511yescharacteristic
594C22C2×#1058761611yescharacteristic
595C22C2×#1058763211yescharacteristic
596C22C2×#1058761811yescharacteristic
597C22C2×#1058762811yescharacteristic
598C22#1176360711yescharacteristic
599C22#1176360711yescharacteristic
600C4—22no—
601C4—22no—
602C4—44no—
603C4—41no—
604C4—41no—
605C4—82no—
606C4—682no—
607C4—1361no—
608C4—1361no—
609C4#1176273114yes—
610C4C2×#1058758111yescharacteristic
611C4C2×#1058758211yescharacteristic
612C4C2×#1058758311yescharacteristic
613C4C2×#1058758411yescharacteristic
614C2—24no—
615C2—1362no—
616C2C2×#1176273111yescharacteristic
617C2#1453824811yescharacteristic
618C2#1453825611yescharacteristic
619C2#1453839911yescharacteristic
620C2#1453841011yescharacteristic
621C2#1453859311yescharacteristic
622C2#1453860211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#389C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
hypercenterC2×#389D17
hypocenterC17#23617
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
2. C2×#11762833C2semidirect
6. #14530176C2semidirect
7. #14530169C2semidirect
8. #14530162C2semidirect
9. #14530155C2semidirect
10. #14528503C2semidirect
11. C2×C17×#389C2semidirect
273. C2×#389D17semidirect
422. C17#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628332
C2×#117628082
#145312742
#145301762
#145301692
#145301622
#145301552
#145285032
C2×C17×#3892
#2361717