Finite Group Databasebeta

← #17076163

Subgroups — #17076163

Order 4352.

1076 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763040C212yesmaximal
2C2×#11763011C211yesmaximal, characteristic, complemented
3C22×C4⋊#3622C211yesmaximal, characteristic, complemented
4C4×#10587616C211yesmaximal, characteristic
5#14530165C212yesmaximal, complemented
6#14530099C212yesmaximal, complemented
7#14529708C211yesmaximal, characteristic, complemented
8#14529545C211yesmaximal, characteristic, complemented
9#14529208C212yesmaximal, complemented
10#14528781C211yesmaximal, characteristic, complemented
11C17×#2857C211yesmaximal, characteristic, complemented
12C22×#63230C2212yes—
13C23×#3622C2212yes—
14C2×#10587616C2211yescharacteristic
15C2×C4×D68C2211yescharacteristic
16C2×C4×C22⋊D17—21no—
17#11763040—28no—
18#11763011—24no—
19C2×#10587479C2212yes—
20C2×C4⋊#3622—24no—
21C2×C4⋊#3622C2211yescharacteristic
22C2×C4⋊#3622C2212yes—
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587506C2212yes—
25C2×#10587506C2212yes—
26C2×#10587506C2212yes—
27C2×#10587480C2212yes—
28#11762711C2211yescharacteristic
29#11762707C2211yescharacteristic
30#11762700—21no—
31#11762688—21no—
32#11762669C2211yescharacteristic
33C4×C4⋊Dic17C2211yescharacteristic
34C4×#63227—21no—
35#11762571C2212yes—
36#11762571C2212yes—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×C22⋊Q8C2212yes—
39C22×C42×C17C2211yescharacteristic
40C17×#391C2212yes—
41C17×#389—21no—
42C17×#389—21no—
43C17×#384C2211yescharacteristic
44C17×#380—21no—
45C17×#380—21no—
46C22×Q8×C17C2312yes—
47C23×C4×C17D411yescharacteristic
48C23×C4×C17D411yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C22×C22⋊D17—21no—
51C22×D68C2311yescharacteristic
52C22×C4×D17—21no—
53C17×C4⋊Q8—22no—
54C17×C4⋊Q8—22no—
55C17×C22⋊Q8—24no—
56C17×C22⋊Q8—24no—
57C2×C4⋊C4×C17—22no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—21no—
60C2×C4⋊C4×C17—21no—
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17C2312yes—
63C2×C22⋊C4×C17—22no—
64C2×C42×C17—22no—
65C2×C42×C17—22no—
66C2×C42×C17D411yescharacteristic
67C2×C42×C17D411yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C22⋊Dic17—21no—
71C2×#63230—24no—
72C2×#63230—24no—
73C2×#63230—42no—
74#10587616—22no—
75#10587616D414yes—
76C4×C22⋊D17—44no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×#63227—21no—
80C2×#63227—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C22×#3622—24no—
87C22×#3622—24no—
88C22×#3622—42no—
89C22×#3622C2312yes—
90C22×#3622C2312yes—
91C4×D68—22no—
92C4×D68—22no—
93C17×#95—22no—
94C17×#95—22no—
95#10587506—24no—
96#10587506—24no—
97#10587506—42no—
98#10587502—21no—
99#10587502—21no—
100#10587502—21no—
101#10587480—24no—
102#10587479—24no—
103C4⋊#3622—24no—
104C4⋊#3622—22no—
105C4⋊#3622D414yescomplemented
106C24×C17C2×D411yescharacteristic
107C23×D17—21no—
108C2×Q8×C17—22no—
109C2×Q8×C17—22no—
110C2×Q8×C17—22no—
111C2×Q8×C17—22no—
112C2×Q8×C17—22no—
113C2×Q8×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—41no—
125C22×C4×C17SD1614yescomplemented
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17#4112yes—
133C22×C4×C17C2411yescharacteristic
134C2×C22⋊D17—24no—
135C2×C22⋊D17—42no—
136C2×C22⋊D17—42no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68C2×D412yes—
146C2×C4×D17—42no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—42no—
152C4⋊C4×C17—44no—
153C4⋊C4×C17—42no—
154C4⋊C4×C17—42no—
155C4⋊C4×C17—42no—
156C22⋊C4×C17—44no—
157C42×C17—22no—
158C42×C17—22no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17C2×D412yes—
162C42×C17#4112yes—
163C22⋊Dic17—42no—
164#63230—48no—
165#63227—24no—
166#63227—42no—
167#63227—42no—
168#63227—42no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17C2×D412yes—
175#63225—42no—
176C4×Dic17—42no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—48no—
183C2×#3622C2×D414yes—
184#24751—171nomaximal, Sylow
185C23×C17—21no—
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17#12112yes—
190C23×C17C2×SD1612yes—
191C23×C17#13712yes—
192C23×C17C22×D411yescharacteristic
193C22×D17—22no—
194C22×D17—41no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—41no—
200Q8×C17—42no—
201Q8×C17—42no—
202Q8×C17—42no—
203Q8×C17—42no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—28no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—44no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17#12112yes—
226C2×C4×C17#12112yes—
227C2×C4×C17#12112yes—
228C2×C4×C17C2×SD1612yes—
229C2×C4×C17C2×SD1612yes—
230C2×C4×C17#13512yes—
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17H(5,𝔽2)11yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C22⋊D17—48no—
238C2×Dic17—22no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247D68—22no—
248D68—22no—
249D68—41no—
250D68—41no—
251C4×D17—84no—
252#3622—24no—
253#3622—42no—
254C2×#465—172no—
255C2×#461—171no—
256C22×C4⋊C8—171no—
257#2857D1711yescharacteristic, complemented
258C4×#121—171no—
259#2066—172no—
260#2055—172no—
261#1981—171no—
262#1948—171no—
263#1892—172no—
264#1802—171no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—22no—
270C22×C17—22no—
271C22×C17#46114yescomplemented
272C22×C17C2×#12111yescharacteristic
273C22×C17C2×#12111yescharacteristic
274C22×C17#53711yescharacteristic
275C22×C17C22×SD1611yescharacteristic
276C22×C17C2×#13511yescharacteristic
277C22×C17C2×#13711yescharacteristic
278C22×C17#57511yescharacteristic
279C2×D17—42no—
280C2×D17—42no—
281C2×D17—42no—
282C2×D17—81no—
283C2×D17—81no—
284C2×D17—81no—
285C2×D17—81no—
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—22no—
289C4×C17—41no—
290C4×C17—82no—
291C4×C17—82no—
292C4×C17#46514yescomplemented
293C4×C17D4211yescharacteristic
294C4×C17#54711yescharacteristic
295C4×C17#54811yescharacteristic
296C4×C17#55311yescharacteristic
297Dic17—42no—
298Dic17—81no—
299Dic17—81no—
300C22×SD16—172no—
301C23×C8—172no—
302C2×C4⋊Q8C2×D1711yescharacteristic
303C2×C22⋊Q8C2×D1712yes—
304C2×#121—171no—
305C2×C4×D4—171no—
306C2×C4×D4—341no—
307C22×C42C2×D1711yescharacteristic
308#465—348no—
309#461—344no—
310C2×#106—172no—
311C2×C4⋊C8—171no—
312C2×C4⋊C8—172no—
313C2×C4⋊C8—171no—
314C2×C4⋊C8—344no—
315C2×#103—172no—
316C2×#103—172no—
317C2×#103—172no—
318C2×#102—172no—
319#391C2×D1712yes—
320#390—171no—
321#389—21no—
322#389—21no—
323#389—171no—
324#388—341no—
325#386—341no—
326#384C2×D1711yescharacteristic
327#380—21no—
328#380—21no—
329#380—171no—
330C4×C4⋊C4—171no—
331C4×C22⋊C4—341no—
332#340—172no—
333#340—172no—
334C2×C17—24no—
335C2×C17#241611yescharacteristic
336C2×C17C2×#46111yescharacteristic
337C2×C17#303511yescharacteristic
338C2×C17C2×#46511yescharacteristic
339C2×C17#305611yescharacteristic
340C2×C17#324611yescharacteristic
341C2×C17#324811yescharacteristic
342D17—82no—
343C22×Q8C22×D1712yes—
344C22×D4—171no—
345C22×D4—341no—
346C23×C4—341no—
347C23×C4C22⋊D1711yescharacteristic
348C23×C4C22⋊D1711yescharacteristic
349C23×C4C22×D1711yescharacteristic
350C2×SD16—344no—
351C2×SD16—344no—
352C2×SD16—682no—
353C22×C8—172no—
354C22×C8—172no—
355C22×C8—344no—
356C22×C8—344no—
357C22×C8—682no—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362#121—174no—
363#121—342no—
364C4×D4—342no—
365C4×D4—342no—
366C4×D4—684no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1712yes—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—341no—
386C2×C42C22⋊D1711yescharacteristic
387C2×C42C22⋊D1711yescharacteristic
388C2×C42C22×D1711yescharacteristic
389C2×C42C22×D1711yescharacteristic
390#106—344no—
391C4⋊C8—174no—
392C4⋊C8—344no—
393C4⋊C8—342no—
394#103—344no—
395#103—344no—
396#103—682no—
397#102—344no—
398#95—22no—
399#95—22no—
400#95—341no—
401#95—341no—
402#95—341no—
403C17#2475111yescharacteristic, complemented, Sylow
404C24—341no—
405C24C2×C22⋊D1711yescharacteristic
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×D4—172no—
413C2×D4—344no—
414C2×D4—341no—
415C2×D4—341no—
416C2×D4—341no—
417C2×D4—341no—
418C2×D4—341no—
419C2×D4—682no—
420C2×D4—682no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—682no—
436C22×C4—682no—
437C22×C4—682no—
438C22×C4#6323014yescomplemented
439C22×C4D4×D1712yes—
440C22×C4#6325412yes—
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C23×D1711yescharacteristic
447SD16—688no—
448C2×C8—174no—
449C2×C8—342no—
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—688no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—172no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—682no—
469C22⋊C4—44no—
470C22⋊C4—344no—
471C22⋊C4—682no—
472C22⋊C4—682no—
473C22⋊C4—682no—
474C22⋊C4—682no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—682no—
480C42D4×D1712yes—
481C42#6325412yes—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—342no—
487C23—681no—
488C23—681no—
489C23—681no—
490C23—681no—
491C23—681no—
492C23—681no—
493C23#1058762412yes—
494C23C2×#6323012yes—
495C23#1058762812yes—
496C23C22×C22⋊D1711yescharacteristic
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501D4—342no—
502D4—342no—
503D4—688no—
504D4—681no—
505D4—681no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—342no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—1364no—
537C2×C4#1058762412yes—
538C2×C4#1058762512yes—
539C2×C4#1058762512yes—
540C2×C4C2×#6323012yes—
541C2×C4C2×#6323012yes—
542C2×C4#1058763612yes—
543C2×C4C2×D4×D1711yescharacteristic
544C2×C4C2×D4×D1711yescharacteristic
545C2×C4C2×#6325411yescharacteristic
546C2×C4C2×#6325411yescharacteristic
547C2×C4#1058769611yescharacteristic
548C2×C4#1058769711yescharacteristic
549C8—344no—
550C8—682no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—682no—
558C22—682no—
559C22—682no—
560C22—1361no—
561C22—1361no—
562C22—1361no—
563C22—1361no—
564C22#1176301114yescomplemented
565C22C2×#1058762411yescharacteristic
566C22C2×#1058762511yescharacteristic
567C22#1176338711yescharacteristic
568C22C22×#6323011yescharacteristic
569C22C2×#1058763611yescharacteristic
570C22C2×#1058762811yescharacteristic
571C22#1176360711yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—82no—
577C4—82no—
578C4—682no—
579C4—1361no—
580C4—1361no—
581C4#1176304014yescomplemented
582C4#1176344311yescharacteristic
583C4#1176345911yescharacteristic
584C4#1176346511yescharacteristic
585C4#1176351111yescharacteristic
586C2—24no—
587C2—1362no—
588C2#1453222511yescharacteristic
589C2C2×#1176301111yescharacteristic
590C2#1453823311yescharacteristic
591C2C2×#1176304011yescharacteristic
592C2#1453841011yescharacteristic
593C2#1454009611yescharacteristic
594C2#1454011511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2857C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#2857D17
hypocenterC17#24751
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C2×#11763011C2semidirect
3. C22×C4⋊#3622C2semidirect
5. #14530165C2semidirect
6. #14530099C2semidirect
7. #14529708C2semidirect
8. #14529545C2semidirect
9. #14529208C2semidirect
10. #14528781C2semidirect
11. C17×#2857C2semidirect
105. C4⋊#3622D4semidirect
125. C22×C4×C17SD16semidirect
257. #2857D17semidirect
271. C22×C17#461semidirect
292. C4×C17#465semidirect
403. C17#24751semidirect
438. C22×C4#63230semidirect
564. C22#11763011semidirect
581. C4#11763040semidirect

Maximal subgroups

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

subgroupindex
C2×#117630402
C2×#117630112
C22×C4⋊#36222
C4×#105876162
#145301652
#145300992
#145297082
#145295452
#145292082
#145287812
C17×#28572
#2475117