Finite Group Databasebeta

← #17063277

Subgroups — #17063277

Order 4352.

888 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762858C211yesmaximal, characteristic, complemented
2#14537304C211yesmaximal, characteristic
3#14537301C211yesmaximal, characteristic
4C2×#11762791C211yesmaximal, characteristic
5#14530250C214yesmaximal, complemented
6#14530243C214yesmaximal, complemented
7C2×#11762580C211yesmaximal, characteristic, complemented
8C2×#11762565C211yesmaximal, characteristic, complemented
9C2×C17×#344C211yesmaximal, characteristic, complemented
10C22×#63252C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C22×#63223C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14#11762858—22no—
15#11762858C2214yes—
16C2×#10587490—22no—
17#11762819C2212yes—
18#11762817C2212yes—
19#11762815C2212yes—
20#11762813C2212yes—
21#11762811—22no—
22#11762808—22no—
23#11762806—22no—
24#11762803—22no—
25#11762791—22no—
26#11762791—22no—
27C2×#10587495C2212yes—
28C2×#10587494C2212yes—
29#11762580C2214yes—
30#11762565C2214yes—
31C22×M16×C17C2211yescharacteristic
32C22×M16×C17C2211yescharacteristic
33C2×C17×#117C2211yescharacteristic
34C17×#344C2214yes—
35C23×C4×C17C2311yescharacteristic
36C22×C22⋊D17—21no—
37C2×#63252—21no—
38C2×#63252—21no—
39C2×#63252—21no—
40C2×#63252—21no—
41C2×#63252—21no—
42C2×#63252—21no—
43C2×#63252—21no—
44C2×#63252—21no—
45C2×#63252—21no—
46C2×#63252—21no—
47C2×#63252D412yes—
48C2×#63252D412yes—
49C22×D68C2311yescharacteristic
50C22×C4×D17—21no—
51C22×Dic34C2311yescharacteristic
52C2×M16×C17—21no—
53C2×M16×C17—21no—
54C2×M16×C17—21no—
55C2×M16×C17—21no—
56C2×M16×C17—21no—
57C2×M16×C17—21no—
58C2×M16×C17—21no—
59C2×M16×C17—21no—
60C2×M16×C17—21no—
61C2×M16×C17D412yes—
62C2×M16×C17C2312yes—
63C2×M16×C17C2312yes—
64C22×C8×C17—21no—
65C22×C8×C17D412yes—
66C17×#117—21no—
67C17×#117—21no—
68C17×#117—21no—
69C17×#117C2312yes—
70C2×C4⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C42×C17—21no—
73C2×C22⋊Dic17—21no—
74#10587612—21no—
75#10587612—21no—
76#10587612—21no—
77#10587612C2312yes—
78C2×C4⋊Dic17—21no—
79C2×C4×Dic17—21no—
80C2×#63223—21no—
81C2×#63223—21no—
82C2×#63223—21no—
83C2×#63223—21no—
84C2×#63223—21no—
85C2×#63223—21no—
86C2×#63223—21no—
87C2×#63223—21no—
88C2×#63223—21no—
89C2×#63223—21no—
90C2×#63223C2312yes—
91C2×#63223C2312yes—
92C22×#3622—21no—
93C22×#3622—21no—
94#10587495—24no—
95#10587494—24no—
96#10587492—24no—
97#10587490—28no—
98#10587487—24no—
99#10587481—24no—
100#10587480—24no—
101C24×C17#4111yescharacteristic
102C23×D17—21no—
103C22×C4×C17—41no—
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17#4111yescharacteristic
109C22×C4×C17#4111yescharacteristic
110C22×C4×C17#4111yescharacteristic
111C22×C4×C17#4111yescharacteristic
112C22×C4×C17#4111yescharacteristic
113C22×C4×C17#4111yescharacteristic
114C22×C4×C17#4111yescharacteristic
115C22×C4×C17#4111yescharacteristic
116C22×C4×C17#4111yescharacteristic
117C22×C4×C17C2411yescharacteristic
118C2×C22⋊D17—22no—
119C2×C22⋊D17—22no—
120C2×C22⋊D17—41no—
121C2×C22⋊D17—41no—
122C2×C22⋊D17—41no—
123C2×C22⋊D17—41no—
124C22×Dic17—21no—
125C22×Dic17—41no—
126#63252—22no—
127#63252—22no—
128#63252—22no—
129#63252—22no—
130#63252—22no—
131#63252—22no—
132#63252—22no—
133#63252—22no—
134#63252—41no—
135#63252—41no—
136#63252—41no—
137#63252—41no—
138#63252—41no—
139#63252—41no—
140#63252—41no—
141#63252—41no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68C2×D412yes—
148C2×C4×D17—22no—
149C2×C4×D17—22no—
150C2×C4×D17—41no—
151C2×C4×D17—41no—
152C2×C4×D17—41no—
153C2×C4×D17—41no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C2×D412yes—
160M16×C17—22no—
161M16×C17—22no—
162M16×C17—22no—
163M16×C17—22no—
164M16×C17—22no—
165M16×C17—22no—
166M16×C17—41no—
167M16×C17—41no—
168C2×C8×C17—22no—
169C2×C8×C17—22no—
170C2×C8×C17—22no—
171C2×C8×C17—41no—
172C2×C8×C17—41no—
173C2×C8×C17C2×D414yes—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—41no—
176C22⋊C4×C17—22no—
177C22⋊C4×C17—41no—
178C42×C17—22no—
179C42×C17—41no—
180C22⋊Dic17—22no—
181C22⋊Dic17—41no—
182C4⋊Dic17—22no—
183C4⋊Dic17—41no—
184C4×Dic17—22no—
185C4×Dic17—41no—
186#63223—22no—
187#63223—22no—
188#63223—22no—
189#63223—22no—
190#63223—41no—
191#63223—41no—
192#63223—41no—
193#63223—41no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—41no—
197C2×#3622—41no—
198C2×#3622—41no—
199C2×#3622—41no—
200#23652—171nomaximal, Sylow
201C23×C17—81no—
202C23×C17#12311yescharacteristic
203C23×C17#12311yescharacteristic
204C23×C17#12411yescharacteristic
205C23×C17#12411yescharacteristic
206C23×C17#12611yescharacteristic
207C23×C17#12611yescharacteristic
208C23×C17C2×#4111yescharacteristic
209C22×D17—22no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—82no—
213C2×C4×C17—42no—
214C2×C4×C17—82no—
215C2×C4×C17#12111yescharacteristic
216C2×C4×C17#12111yescharacteristic
217C2×C4×C17#12111yescharacteristic
218C2×C4×C17#12111yescharacteristic
219C2×C4×C17#12311yescharacteristic
220C2×C4×C17#12311yescharacteristic
221C2×C4×C17#12311yescharacteristic
222C2×C4×C17#12311yescharacteristic
223C2×C4×C17#12311yescharacteristic
224C2×C4×C17#12311yescharacteristic
225C2×C4×C17#12311yescharacteristic
226C2×C4×C17#12311yescharacteristic
227C2×C4×C17#12311yescharacteristic
228C2×C4×C17#12311yescharacteristic
229C2×C4×C17#12411yescharacteristic
230C2×C4×C17#12411yescharacteristic
231C2×C4×C17#12611yescharacteristic
232C2×C4×C17#12611yescharacteristic
233C2×C4×C17#12611yescharacteristic
234C2×C4×C17#12611yescharacteristic
235C2×C4×C17#12611yescharacteristic
236C2×C4×C17#12611yescharacteristic
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C22×D411yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C22⋊D17—42no—
244C22⋊D17—42no—
245C22⋊D17—42no—
246C22⋊D17—42no—
247C2×Dic17—22no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—42no—
251C2×Dic17—82no—
252D68—22no—
253D68—22no—
254D68—41no—
255D68—41no—
256C4×D17—42no—
257C4×D17—42no—
258C4×D17—42no—
259C4×D17—42no—
260Dic34—22no—
261Dic34—22no—
262Dic34—41no—
263Dic34—41no—
264C8×C17—24no—
265C8×C17—42no—
266#3622—42no—
267#3622—42no—
268C2×#430—171no—
269#2896—171no—
270#2896—171no—
271C2×#413—171no—
272#2082—174no—
273#2081—174no—
274C2×#344—171no—
275C2×#344D1711yescharacteristic, complemented
276C2×H(3,ℤ/4)—171no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17#39711yescharacteristic
285C22×C17#39711yescharacteristic
286C22×C17#39711yescharacteristic
287C22×C17#39711yescharacteristic
288C22×C17C2×#12311yescharacteristic
289C22×C17C2×#12411yescharacteristic
290C22×C17C2×#12611yescharacteristic
291C2×D17—42no—
292C2×D17—82no—
293C2×D17—82no—
294C2×D17—82no—
295C4×C17—82no—
296C4×C17#39711yescharacteristic
297C4×C17#39711yescharacteristic
298C4×C17#39711yescharacteristic
299C4×C17#39711yescharacteristic
300C4×C17C2×#12111yescharacteristic
301C4×C17C2×#12311yescharacteristic
302C4×C17C2×#12311yescharacteristic
303C4×C17C2×#12611yescharacteristic
304Dic17—42no—
305Dic17—82no—
306C22×#41—171no—
307C22×M16—171no—
308C22×M16—171no—
309C22×M16C2×D1711yescharacteristic
310C22×M16C2×D1711yescharacteristic
311C2×#117—171no—
312C2×#117C2×D1711yescharacteristic
313#430—174no—
314#430—342no—
315C2×#108—342no—
316#419—172no—
317#419—172no—
318#418—172no—
319#418—172no—
320#417—342no—
321#417—342no—
322#416—342no—
323#416—342no—
324#413—342no—
325#413—342no—
326C2×#101—172no—
327C2×#100—172no—
328#344—174no—
329#344C2×D1714yes—
330H(3,ℤ/4)—174no—
331C2×C17—21no—
332C2×C17—21no—
333C2×C17—21no—
334C2×C17—21no—
335C2×C17—21no—
336C2×C17—21no—
337C2×C17—21no—
338C2×C17C2×#39711yescharacteristic
339D17—82no—
340C2×#41—172no—
341C2×#41—172no—
342C2×#41—341no—
343C2×#41—341no—
344C2×#41—341no—
345C2×#41—341no—
346C2×#41—341no—
347C2×#41—341no—
348C2×#41—341no—
349C2×#41—341no—
350C2×#41—341no—
351C2×#41—341no—
352C22×Q8—171no—
353C22×D4—171no—
354C22×D4—341no—
355C23×C4—341no—
356C23×C4C22×D1711yescharacteristic
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—172no—
367C2×M16—172no—
368C2×M16—341no—
369C2×M16—341no—
370C2×M16—341no—
371C2×M16—341no—
372C2×M16—341no—
373C2×M16—341no—
374C2×M16—341no—
375C2×M16—341no—
376C2×M16—341no—
377C2×M16—341no—
378C2×M16D6812yes—
379C2×M16C22×D1712yes—
380C2×M16C22×D1712yes—
381C22×C8—21no—
382C22×C8—341no—
383C22×C8—341no—
384C22×C8D6812yes—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—172no—
389#117—341no—
390#117—341no—
391#117—341no—
392#117C22×D1712yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—341no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—341no—
397C2×C42—21no—
398C2×C42—341no—
399#108—348no—
400#103—344no—
401#103—344no—
402#102—344no—
403#102—344no—
404#101—344no—
405#100—344no—
406C17#2365211yescharacteristic, complemented, Sylow
407C24—341no—
408C24#6325211yescharacteristic
409#41—342no—
410#41—342no—
411#41—342no—
412#41—342no—
413#41—342no—
414#41—342no—
415#41—342no—
416#41—342no—
417#41—681no—
418#41—681no—
419#41—681no—
420#41—681no—
421#41—681no—
422#41—681no—
423#41—681no—
424#41—681no—
425C2×Q8—172no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×Q8—341no—
430C2×Q8—341no—
431C2×D4—172no—
432C2×D4—342no—
433C2×D4—342no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C2×D4—341no—
438C2×D4—341no—
439C2×D4—681no—
440C2×D4—681no—
441C2×D4—681no—
442C2×D4—681no—
443C22×C4—41no—
444C22×C4—341no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—681no—
448C22×C4—681no—
449C22×C4—681no—
450C22×C4—681no—
451C22×C4—681no—
452C22×C4C2×D6811yescharacteristic
453C22×C4C2×D6811yescharacteristic
454C22×C4#6325211yescharacteristic
455C22×C4#6325211yescharacteristic
456C22×C4#6325211yescharacteristic
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4#6325411yescharacteristic
460C22×C4#6325411yescharacteristic
461C22×C4#6325411yescharacteristic
462C22×C4#6325411yescharacteristic
463C22×C4#6325611yescharacteristic
464C22×C4#6325611yescharacteristic
465C22×C4C23×D1711yescharacteristic
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—22no—
471M16—22no—
472M16—41no—
473M16—41no—
474M16—342no—
475M16—342no—
476M16—342no—
477M16—342no—
478M16—681no—
479M16—681no—
480M16—681no—
481M16—681no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—342no—
488C2×C8—342no—
489C2×C8—681no—
490C2×C8—681no—
491C2×C8—681no—
492C2×C8—681no—
493C2×C8C2×D6814yes—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—342no—
497C4⋊C4—681no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—342no—
501C22⋊C4—681no—
502C42—22no—
503C42—41no—
504C42—342no—
505C42—681no—
506C23—81no—
507C23—342no—
508C23—681no—
509C23—681no—
510C23—1362no—
511C23#1058756111yescharacteristic
512C23#1058756111yescharacteristic
513C23#1058756911yescharacteristic
514C23#1058756911yescharacteristic
515C23#1058758411yescharacteristic
516C23#1058758411yescharacteristic
517C23C2×#6325211yescharacteristic
518Q8—342no—
519Q8—342no—
520Q8—681no—
521Q8—681no—
522D4—342no—
523D4—342no—
524D4—682no—
525D4—682no—
526D4—682no—
527D4—682no—
528D4—681no—
529D4—681no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—342no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—682no—
536C2×C4—682no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—1362no—
541C2×C4#1058756111yescharacteristic
542C2×C4#1058756111yescharacteristic
543C2×C4#1058756911yescharacteristic
544C2×C4#1058756911yescharacteristic
545C2×C4#1058756911yescharacteristic
546C2×C4#1058756911yescharacteristic
547C2×C4#1058756911yescharacteristic
548C2×C4#1058756911yescharacteristic
549C2×C4#1058757211yescharacteristic
550C2×C4#1058757211yescharacteristic
551C2×C4#1058757211yescharacteristic
552C2×C4#1058757211yescharacteristic
553C2×C4#1058758111yescharacteristic
554C2×C4#1058758111yescharacteristic
555C2×C4#1058758111yescharacteristic
556C2×C4#1058758111yescharacteristic
557C2×C4#1058758411yescharacteristic
558C2×C4#1058758411yescharacteristic
559C2×C4#1058758411yescharacteristic
560C2×C4#1058758411yescharacteristic
561C2×C4#1058758411yescharacteristic
562C2×C4#1058758411yescharacteristic
563C2×C4C22×D6811yescharacteristic
564C2×C4C2×#6325211yescharacteristic
565C2×C4C2×D4×D1711yescharacteristic
566C2×C4C2×#6325411yescharacteristic
567C2×C4C2×#6325411yescharacteristic
568C2×C4C2×#6325611yescharacteristic
569C8—24no—
570C8—42no—
571C8—682no—
572C8—682no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—682no—
581C22—1362no—
582C22—1362no—
583C22—1362no—
584C22#1176273611yescharacteristic
585C22#1176273611yescharacteristic
586C22#1176273611yescharacteristic
587C22#1176273611yescharacteristic
588C22C2×#1058756911yescharacteristic
589C22C2×#1058756111yescharacteristic
590C22C2×#1058758411yescharacteristic
591C4—82no—
592C4—682no—
593C4—1362no—
594C4#1176273611yescharacteristic
595C4#1176273611yescharacteristic
596C4#1176273611yescharacteristic
597C4#1176273611yescharacteristic
598C4C2×#1058758111yescharacteristic
599C4C2×#1058756911yescharacteristic
600C4C2×#1058757211yescharacteristic
601C4C2×#1058758411yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—1362no—
610C2C2×#1176273611yescharacteristic
subgroupG/H
centerC4#11762736
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#344C2
socleC2×C17C2×#397
prime radicalC22×#63252C22
hypercenterC2×#344D17
hypocenterC17#23652
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
1. C2×#11762858C2semidirect
5. #14530250C2semidirect
6. #14530243C2semidirect
7. C2×#11762580C2semidirect
8. C2×#11762565C2semidirect
9. C2×C17×#344C2semidirect
275. C2×#344D17semidirect
406. C17#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#117628582
#145373042
#145373012
C2×#117627912
#145302502
#145302432
C2×#117625802
C2×#117625652
C2×C17×#3442
#2365217