Finite Group Databasebeta

← #17063018

Subgroups — #17063018

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×#11762851C211yesmaximal, characteristic, complemented
2C2×#11762811C211yesmaximal, characteristic
3C2×#11762811C211yesmaximal, characteristic
4#14531274C211yesmaximal, characteristic
5C2×#11762707C211yesmaximal, characteristic, complemented
6#14530178C212yesmaximal, complemented
7#14530171C212yesmaximal, complemented
8#14530164C212yesmaximal, complemented
9#14530157C212yesmaximal, complemented
10#14528504C211yesmaximal, characteristic, complemented
11C2×C17×#423C211yesmaximal, characteristic, complemented
12C22×#63252C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225—21no—
17#11762851—24no—
18C2×#10587489C2212yes—
19C2×#10587488C2212yes—
20#11762811—24no—
21#11762811—24no—
22C2×#10587487C2212yes—
23C2×#10587487C2212yes—
24C2×#10587492C2212yes—
25C2×#10587492C2212yes—
26#11762715C2211yescharacteristic
27#11762715C2211yescharacteristic
28#11762711C2211yescharacteristic
29#11762711C2211yescharacteristic
30#11762707—21no—
31#11762707—21no—
32#11762707C2212yes—
33#11762707C2212yes—
34#11762688—21no—
35#11762688—21no—
36C4×#63227—21no—
37C4×#63227—21no—
38C2×#10587502—21no—
39#11762572C2212yes—
40#11762572C2212yes—
41C22×M16×C17C2211yescharacteristic
42C22×M16×C17C2211yescharacteristic
43C22×C42×C17C2211yescharacteristic
44C17×#423—24no—
45C2×C17×C4⋊C8C2212yes—
46C2×C17×C4⋊C8C2212yes—
47C23×C4×C17D411yescharacteristic
48C23×C4×C17D411yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C22×C22⋊D17—21no—
51C23×Dic17—21no—
52C23×Dic17—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×M16×C17—22no—
63C2×M16×C17—22no—
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C2×M16×C17D414yes—
68C22×C8×C17C2312yes—
69C22×C8×C17C2312yes—
70C2×C42×C17—22no—
71C2×C42×C17—22no—
72C2×C42×C17D411yescharacteristic
73C2×C42×C17D411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81#10587612—22no—
82#10587612—22no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—22no—
86C2×C4⋊Dic17—22no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17D414yes—
89C2×C4⋊Dic17C2312yes—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2312yes—
93C2×#63225—22no—
94C2×#63225—22no—
95C2×#63225—44no—
96C2×C4×Dic17—21no—
97C17×C4⋊C8—24no—
98C17×C4⋊C8—24no—
99#10587502—21no—
100#10587502—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—22no—
104#10587502—22no—
105#10587492—24no—
106#10587492—24no—
107#10587489—24no—
108#10587488—24no—
109#10587487—24no—
110#10587487—24no—
111C24×C17C2×D411yescharacteristic
112C23×D17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—41no—
118C22×C4×C17—41no—
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17C2×D411yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D412yes—
125C22×C4×C17C2×D412yes—
126C22×C4×C17C2×Q812yes—
127C22×C4×C17#4112yes—
128C22×C4×C17#4112yes—
129C22×C4×C17#4112yes—
130C22×C4×C17C2411yescharacteristic
131C2×C22⋊D17—24no—
132C2×C22⋊D17—42no—
133C2×C22⋊D17—42no—
134C22×Dic17—24no—
135C22×Dic17—21no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—42no—
141C22×Dic17—42no—
142C22×Dic17—42no—
143C22×Dic17—42no—
144C22×Dic17—42no—
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—
164M16×C17—28no—
165M16×C17—44no—
166C2×C8×C17—22no—
167C2×C8×C17—22no—
168C2×C8×C17—22no—
169C2×C8×C17—22no—
170C2×C8×C17—22no—
171C2×C8×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—24no—
184C4⋊Dic17—22no—
185C4⋊Dic17—22no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—22no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—42no—
192C4⋊Dic17C2×D414yes—
193#63225—44no—
194#63225—44no—
195C4×Dic17—42no—
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—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—44no—
223C2×C4×C17—44no—
224C2×C4×C17#12112yes—
225C2×C4×C17#12112yes—
226C2×C4×C17#12114yes—
227C2×C4×C17C22⋊Q814yes—
228C2×C4×C17C22⋊Q812yes—
229C2×C4×C17C22⋊Q814yes—
230C2×C4×C17#12614yes—
231C2×C4×C17AGL(1,ℤ/8)12yes—
232C2×C4×C17#13712yes—
233C2×C4×C17C22×D411yescharacteristic
234C2×C4×C17C22×D411yescharacteristic
235C2×C4×C17C22×Q811yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C22⋊D17—48no—
240C2×Dic17—24no—
241C2×Dic17—22no—
242C2×Dic17—48no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—42no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—84no—
257D68—22no—
258D68—22no—
259D68—41no—
260D68—41no—
261C4×D17—48no—
262Dic34—22no—
263Dic34—22no—
264Dic34—41no—
265Dic34—41no—
266C8×C17—24no—
267C8×C17—42no—
268C2×#428—171no—
269C2×#423D1711yescharacteristic, complemented
270C2×#417—171no—
271C2×#417—171no—
272#2297—171no—
273C2×#389—171no—
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—41no—
303C4×C17—41no—
304C4×C17#39614yes—
305C4×C17C2×#12111yescharacteristic
306C4×C17C2×C22⋊Q811yescharacteristic
307C4×C17C2×C22⋊Q811yescharacteristic
308C4×C17C2×#12611yescharacteristic
309Dic17—42no—
310Dic17—44no—
311Dic17—81no—
312Dic17—81no—
313Dic17—82no—
314C22×#41—171no—
315C22×M16C2×D1711yescharacteristic
316C22×M16C2×D1711yescharacteristic
317C2×#117—171no—
318C22×C4⋊C4—171no—
319C22×C4⋊C4—171no—
320C22×C4⋊C4—341no—
321C22×C42C2×D1711yescharacteristic
322#428—344no—
323C2×#107—172no—
324C2×#106—172no—
325#423—24no—
326C2×C4⋊C8C2×D1712yes—
327C2×C4⋊C8C2×D1712yes—
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—172no—
339#389—172no—
340#389—341no—
341#389—341no—
342#386—341no—
343#386—341no—
344C4×C22⋊C4—341no—
345C4×C22⋊C4—341no—
346C2×#95—341no—
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—341no—
368C23×C4—341no—
369C23×C4—341no—
370C23×C4D6811yescharacteristic
371C23×C4D6811yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—22no—
378C2×M16C22⋊D1714yes—
379C22×C8C22×D1712yes—
380C22×C8C22×D1712yes—
381#117—342no—
382#117—342no—
383C2×C4⋊C4—174no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—172no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4—342no—
395C2×C4⋊C4—684no—
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×C42D6811yescharacteristic
405C2×C42D6811yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—344no—
409#106—344no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412#103—344no—
413#103—344no—
414#102—344no—
415#102—344no—
416#95—342no—
417#95—342no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2361711yescharacteristic, complemented, Sylow
423C24—341no—
424C24C2×D6811yescharacteristic
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—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—341no—
451C22×C4—342no—
452C22×C4—344no—
453C22×C4—344no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—342no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4C2×D6811yescharacteristic
465C22×C4C2×D6811yescharacteristic
466C22×C4C2×D6811yescharacteristic
467C22×C4C2×D6811yescharacteristic
468C22×C4C2×D6811yescharacteristic
469C22×C4#6325212yes—
470C22×C4D4×D1712yes—
471C22×C4#6325412yes—
472C22×C4Q8×D1712yes—
473C22×C4#6325612yes—
474C22×C4C2×C22⋊D1712yes—
475C22×C4C23×D1711yescharacteristic
476M16—28no—
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8C2×C22⋊D1714yes—
484C4⋊C4—174no—
485C4⋊C4—342no—
486C4⋊C4—344no—
487C4⋊C4—342no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—342no—
493C4⋊C4—684no—
494C4⋊C4—684no—
495C4⋊C4—682no—
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#1058758112yes—
520C23#1058758312yes—
521C23#1058761612yes—
522C23C22×D6811yescharacteristic
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—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—344no—
545C2×C4—342no—
546C2×C4—688no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—688no—
552C2×C4—682no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—682no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—682no—
561C2×C4—1364no—
562C2×C4#1058758112yes—
563C2×C4#1058758214yes—
564C2×C4#1058758312yes—
565C2×C4#1058758414yes—
566C2×C4#1058759412yes—
567C2×C4#1058759512yes—
568C2×C4#1058761612yes—
569C2×C4#1058762514yes—
570C2×C4#1058763214yes—
571C2×C4C2×#6325211yescharacteristic
572C2×C4C2×D4×D1711yescharacteristic
573C2×C4C2×#6325411yescharacteristic
574C2×C4C2×Q8×D1711yescharacteristic
575C2×C4C2×#6325611yescharacteristic
576C2×C4C22×C22⋊D1711yescharacteristic
577C8—24no—
578C8—42no—
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×#1058761611yescharacteristic
594C22C2×#1058758111yescharacteristic
595C22C2×#1058758311yescharacteristic
596C22C2×#1058759411yescharacteristic
597C22C2×#1058759511yescharacteristic
598C22#1176360911yescharacteristic
599C22#1176360911yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—684no—
605C4—682no—
606C4—1361no—
607C4—1361no—
608C4—1362no—
609C4#1176273114yes—
610C4C2×#1058762511yescharacteristic
611C4C2×#1058763211yescharacteristic
612C4C2×#1058758211yescharacteristic
613C4C2×#1058758411yescharacteristic
614C2—24no—
615C2—1362no—
616C2C2×#1176273111yescharacteristic
617C2#1453825011yescharacteristic
618C2#1453825811yescharacteristic
619C2#1453840311yescharacteristic
620C2#1453841411yescharacteristic
621C2#1453860011yescharacteristic
622C2#1453860711yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#423C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
hypercenterC2×#423D17
hypocenterC17#23617
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
1. C2×#11762851C2semidirect
5. C2×#11762707C2semidirect
6. #14530178C2semidirect
7. #14530171C2semidirect
8. #14530164C2semidirect
9. #14530157C2semidirect
10. #14528504C2semidirect
11. C2×C17×#423C2semidirect
269. C2×#423D17semidirect
422. C17#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#117628512
C2×#117628112
#145312742
C2×#117627072
#145301782
#145301712
#145301642
#145301572
#145285042
C2×C17×#4232
#2361717