Finite Group Databasebeta

← #17063280

Subgroups — #17063280

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×#11762857C211yesmaximal, characteristic, complemented
2#14537301C211yesmaximal, characteristic
3#14537301C211yesmaximal, characteristic
4C2×#11762791C211yesmaximal, characteristic
5#14530253C214yesmaximal, complemented
6#14530246C214yesmaximal, complemented
7C2×#11762579C211yesmaximal, characteristic, complemented
8C2×#11762579C211yesmaximal, characteristic, complemented
9C2×C17×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
10C22×#63252C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C22×#63223C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14#11762857—22no—
15#11762857C2214yes—
16C2×#10587493—22no—
17#11762817C2212yes—
18#11762817C2212yes—
19#11762813C2212yes—
20#11762813C2212yes—
21#11762808—22no—
22#11762808—22no—
23#11762803—22no—
24#11762803—22no—
25#11762791—22no—
26#11762791—22no—
27C2×#10587495C2212yes—
28C2×#10587494C2212yes—
29#11762579C2214yes—
30#11762579C2214yes—
31C22×M16×C17C2211yescharacteristic
32C2×C17×#117C2211yescharacteristic
33C2×C17×#117C2211yescharacteristic
34C17×H(3,ℤ/4)C2214yes—
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×C17C2312yes—
58C22×C8×C17—21no—
59C17×#117—21no—
60C17×#117—21no—
61C17×#117—21no—
62C17×#117—21no—
63C17×#117—21no—
64C17×#117—21no—
65C17×#117C2312yes—
66C17×#117C2312yes—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×C42×C17—21no—
73C2×#63223—21no—
74C2×#63223—21no—
75C2×#63223—21no—
76C2×#63223—21no—
77C2×#63223—21no—
78C2×#63223—21no—
79C2×#63223—21no—
80C2×#63223—21no—
81C2×#63223—21no—
82C2×#63223—21no—
83C2×#63223—21no—
84C2×#63223—21no—
85C2×#63223—21no—
86C2×#63223—21no—
87C2×#63223D412yes—
88C2×#63223C2312yes—
89C2×#63223C2312yes—
90C2×#63223C2312yes—
91C22×#3622—21no—
92C22×#3622—21no—
93C22×#3622D412yes—
94#10587495—24no—
95#10587494—24no—
96#10587493—28no—
97#10587481—24no—
98#10587481—24no—
99#10587480—24no—
100#10587480—24no—
101C24×C17#4111yescharacteristic
102C23×D17—21no—
103C22×C4×C17—41no—
104C22×C4×C17—41no—
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17C2×D411yescharacteristic
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×C17#4111yescharacteristic
118C22×C4×C17C2411yescharacteristic
119C2×C22⋊D17—22no—
120C2×C22⋊D17—22no—
121C2×C22⋊D17—41no—
122C2×C22⋊D17—41no—
123C2×C22⋊D17—41no—
124C2×C22⋊D17—41no—
125C22×Dic17—21no—
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—41no—
163M16×C17—41no—
164C2×C8×C17—22no—
165C2×C8×C17—41no—
166C2×C8×C17—41no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—41no—
170C4⋊C4×C17—41no—
171C22⋊C4×C17—22no—
172C22⋊C4×C17—22no—
173C22⋊C4×C17—41no—
174C22⋊C4×C17—41no—
175C42×C17—22no—
176C42×C17—22no—
177C42×C17—41no—
178C42×C17—41no—
179#63223—22no—
180#63223—22no—
181#63223—22no—
182#63223—22no—
183#63223—22no—
184#63223—22no—
185#63223—22no—
186#63223—22no—
187#63223—41no—
188#63223—41no—
189#63223—41no—
190#63223—41no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—41no—
196C2×#3622—41no—
197C2×#3622—41no—
198C2×#3622—41no—
199C2×#3622C2×D414yes—
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—42no—
215C2×C4×C17—82no—
216C2×C4×C17—82no—
217C2×C4×C17#12111yescharacteristic
218C2×C4×C17#12111yescharacteristic
219C2×C4×C17#12111yescharacteristic
220C2×C4×C17#12111yescharacteristic
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#12311yescharacteristic
230C2×C4×C17#12311yescharacteristic
231C2×C4×C17#12411yescharacteristic
232C2×C4×C17#12411yescharacteristic
233C2×C4×C17#12611yescharacteristic
234C2×C4×C17#12611yescharacteristic
235C2×C4×C17#12611yescharacteristic
236C2×C4×C17#12611yescharacteristic
237C2×C4×C17#12611yescharacteristic
238C2×C4×C17#12611yescharacteristic
239C2×C4×C17C22×D411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C22⋊D17—42no—
246C22⋊D17—42no—
247C22⋊D17—42no—
248C22⋊D17—42no—
249C2×Dic17—22no—
250C2×Dic17—41no—
251C2×Dic17—41no—
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—42no—
265#3622—24no—
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×#344—171no—
276C2×H(3,ℤ/4)D1711yescharacteristic, complemented
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—82no—
297C4×C17#39711yescharacteristic
298C4×C17#39711yescharacteristic
299C4×C17#39711yescharacteristic
300C4×C17#39711yescharacteristic
301C4×C17C2×#12111yescharacteristic
302C4×C17C2×#12311yescharacteristic
303C4×C17C2×#12311yescharacteristic
304C4×C17C2×#12611yescharacteristic
305Dic17—42no—
306C22×#41—171no—
307C22×M16—171no—
308C22×M16—171no—
309C22×M16—171no—
310C22×M16C2×D1711yescharacteristic
311C2×#117C2×D1711yescharacteristic
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#344—174no—
330H(3,ℤ/4)C2×D1714yes—
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—172no—
363C2×M16—172no—
364C2×M16—172no—
365C2×M16—172no—
366C2×M16—341no—
367C2×M16—341no—
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×M16—341no—
379C2×M16—341no—
380C2×M16C22×D1712yes—
381C22×C8—21no—
382C22×C8—172no—
383C22×C8—341no—
384C22×C8—341no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—21no—
391#117C22×D1712yes—
392#117C22×D1712yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—21no—
398C2×C42—21no—
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#6325411yescharacteristic
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—41no—
445C22×C4—341no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—681no—
449C22×C4—681no—
450C22×C4—681no—
451C22×C4—681no—
452C22×C4#6325211yescharacteristic
453C22×C4#6325211yescharacteristic
454C22×C4#6325211yescharacteristic
455C22×C4#6325211yescharacteristic
456C22×C4#6325211yescharacteristic
457C22×C4#6325211yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4D4×D1711yescharacteristic
460C22×C4D4×D1711yescharacteristic
461C22×C4D4×D1711yescharacteristic
462C22×C4#6325411yescharacteristic
463C22×C4#6325611yescharacteristic
464C22×C4#6325611yescharacteristic
465C22×C4C23×D1711yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—342no—
471M16—342no—
472M16—342no—
473M16—342no—
474M16—342no—
475M16—342no—
476M16—342no—
477M16—342no—
478M16—681no—
479M16—681no—
480M16—681no—
481M16—681no—
482C2×C8—22no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—174no—
486C2×C8—342no—
487C2×C8—342no—
488C2×C8—342no—
489C2×C8—342no—
490C2×C8—681no—
491C2×C8—681no—
492C2×C8—681no—
493C2×C8—681no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C23—81no—
507C23—342no—
508C23—681no—
509C23—681no—
510C23—1362no—
511C23#1058756911yescharacteristic
512C23#1058756911yescharacteristic
513C23#1058757111yescharacteristic
514C23#1058757111yescharacteristic
515C23#1058758411yescharacteristic
516C23#1058758411yescharacteristic
517C23C2×#6325411yescharacteristic
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—42no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—342no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—682no—
538C2×C4—682no—
539C2×C4—682no—
540C2×C4—682no—
541C2×C4#1058756211yescharacteristic
542C2×C4#1058756211yescharacteristic
543C2×C4#1058756211yescharacteristic
544C2×C4#1058756211yescharacteristic
545C2×C4#1058756911yescharacteristic
546C2×C4#1058756911yescharacteristic
547C2×C4#1058757011yescharacteristic
548C2×C4#1058757011yescharacteristic
549C2×C4#1058757011yescharacteristic
550C2×C4#1058757011yescharacteristic
551C2×C4#1058757111yescharacteristic
552C2×C4#1058757111yescharacteristic
553C2×C4#1058758011yescharacteristic
554C2×C4#1058758011yescharacteristic
555C2×C4#1058758011yescharacteristic
556C2×C4#1058758011yescharacteristic
557C2×C4#1058758011yescharacteristic
558C2×C4#1058758011yescharacteristic
559C2×C4#1058758011yescharacteristic
560C2×C4#1058758011yescharacteristic
561C2×C4#1058758411yescharacteristic
562C2×C4#1058758411yescharacteristic
563C2×C4C2×#6325211yescharacteristic
564C2×C4C2×#6325211yescharacteristic
565C2×C4C2×#6325211yescharacteristic
566C2×C4C2×D4×D1711yescharacteristic
567C2×C4C2×D4×D1711yescharacteristic
568C2×C4C2×#6325611yescharacteristic
569C8—42no—
570C8—344no—
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#1176273811yescharacteristic
585C22#1176273811yescharacteristic
586C22#1176273811yescharacteristic
587C22#1176273811yescharacteristic
588C22C2×#1058756911yescharacteristic
589C22C2×#1058757111yescharacteristic
590C22C2×#1058758411yescharacteristic
591C4—82no—
592C4—82no—
593C4—682no—
594C4#1176273811yescharacteristic
595C4#1176273811yescharacteristic
596C4#1176273811yescharacteristic
597C4#1176273811yescharacteristic
598C4C2×#1058757011yescharacteristic
599C4C2×#1058758011yescharacteristic
600C4C2×#1058758011yescharacteristic
601C4C2×#1058756211yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—1362no—
610C2C2×#1176273811yescharacteristic
subgroupG/H
centerC4#11762738
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×H(3,ℤ/4)C2
socleC2×C17C2×#397
prime radicalC22×#63252C22
hypercenterC2×H(3,ℤ/4)D17
hypocenterC17#23652
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. C2×#11762857C2semidirect
5. #14530253C2semidirect
6. #14530246C2semidirect
7. C2×#11762579C2semidirect
8. C2×#11762579C2semidirect
9. C2×C17×H(3,ℤ/4)C2semidirect
276. C2×H(3,ℤ/4)D17semidirect
406. C17#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#117628572
#145373012
C2×#117627912
#145302532
#145302462
C2×#117625792
C2×C17×H(3,ℤ/4)2
#2365217