Finite Group Databasebeta

← #17068233

Subgroups — #17068233

Order 4352.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic, complemented
2C2×C4×#63230C211yesmaximal, characteristic, complemented
3#14533697C211yesmaximal, characteristic
4#14529977C212yesmaximal, complemented
5#14529785C212yesmaximal, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529101C212yesmaximal, complemented
9#14528977C212yesmaximal, complemented
10#14528789C211yesmaximal, characteristic, complemented
11C2×C17×#554C211yesmaximal, characteristic, complemented
12C22×#63230C2211yescharacteristic
13C2×#10587616C2212yes—
14C2×#10587624—21no—
15C2×C4×D68C2211yescharacteristic
16#11763096—24no—
17C4×#63230—24no—
18C2×#10587479C2212yes—
19C2×#10587479C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587506C2212yes—
23C2×#10587480C2212yes—
24C2×#10587480C2211yescharacteristic
25C2×C22⋊#3622C2212yes—
26C2×C22⋊#3622C2212yes—
27C2×C4×#3622C2211yescharacteristic
28#11762725—22no—
29#11762679C2211yescharacteristic
30#11762669C2212yes—
31#11762666—21no—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35C17×#554—24no—
36C17×#554—24no—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×#125—21no—
39C2×C17×C22⋊Q8—22no—
40C2×C17×C22⋊Q8C2212yes—
41C2×C4×Q8×C17C2211yescharacteristic
42C2×C4×D4×C17—21no—
43C2×C4×D4×C17C2211yescharacteristic
44C22×C4⋊C4×C17C2212yes—
45C22×Q8×C17—21no—
46C22×Q8×C17C2311yescharacteristic
47C22×D4×C17D411yescharacteristic
48C23×C4×C17—22no—
49C23×C4×C17C2312yes—
50C22×C22⋊D17—22no—
51C22×D68C2311yescharacteristic
52C22×C4×D17—21no—
53C17×C4⋊Q8—22no—
54C17×C4⋊Q8D414yescomplemented
55C17×#125—28no—
56C17×C22⋊Q8—216no—
57C17×C22⋊Q8—24no—
58C17×C22⋊Q8—24no—
59C4×Q8×C17—22no—
60C4×Q8×C17D414yescomplemented
61C4×D4×C17—28no—
62C4×D4×C17—22no—
63C4×D4×C17—22no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—24no—
69C2×C4⋊C4×C17—24no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2312yes—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17D412yes—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—22no—
78C2×#63230—22no—
79C2×#63230—22no—
80C2×#63230—22no—
81#10587624—44no—
82#10587616—24no—
83#10587616—42no—
84C2×#63227—21no—
85C2×#63227—22no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2312yes—
90C22×#3622C2311yescharacteristic
91C22×#3622C2312yes—
92C22×#3622C2311yescharacteristic
93C4×D68—22no—
94C4×D68—22no—
95#10587506—22no—
96#10587506—42no—
97#10587502—22no—
98C22⋊#3622—24no—
99C22⋊#3622—42no—
100#10587480—24no—
101#10587480—22no—
102#10587479—24no—
103#10587479—22no—
104C4⋊#3622—22no—
105C4×#3622—22no—
106C24×C17C2×D412yes—
107C23×D17—21no—
108C2×Q8×C17—24no—
109C2×Q8×C17—21no—
110C2×Q8×C17—21no—
111C2×Q8×C17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—21no—
114C2×Q8×C17—81no—
115C2×Q8×C17—81no—
116C2×Q8×C17C2×D412yes—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17SD1614yescomplemented
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—28no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—82no—
131C22×C4×C17—82no—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2411yescharacteristic
136C2×C22⋊D17—44no—
137C2×C22⋊D17—44no—
138C2×C22⋊D17—82no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68#4112yes—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C4⋊C4×C17—28no—
151C4⋊C4×C17—28no—
152C4⋊C4×C17—24no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—24no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17C2×D414yes—
158C4⋊C4×C17C2×D412yes—
159C22⋊C4×C17—28no—
160C22⋊C4×C17—24no—
161C42×C17—24no—
162C42×C17C2×D412yes—
163C42×C17C2×D412yes—
164C22⋊Dic17—44no—
165#63230—28no—
166#63227—44no—
167#63227—42no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—22no—
171C4⋊Dic17—22no—
172C4⋊Dic17—42no—
173C4⋊Dic17—81no—
174C2×#3622—21no—
175C2×#3622—21no—
176C2×#3622—21no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—21no—
180C2×#3622—22no—
181C2×#3622—21no—
182C2×#3622#4112yes—
183#24118—171nomaximal, Sylow
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17C2×SD1614yes—
191C23×C17C22×D411yescharacteristic
192C22×D17—22no—
193C22×D17—41no—
194C22×D17—41no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—41no—
199Q8×C17—22no—
200Q8×C17—22no—
201Q8×C17—41no—
202Q8×C17—41no—
203Q8×C17—84no—
204D4×C17—28no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—24no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—41no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—88no—
224C2×C4×C17#12012yes—
225C2×C4×C17#12012yes—
226C2×C4×C17C2×SD1612yes—
227C2×C4×C17#13712yes—
228C2×C4×C17C22×D411yescharacteristic
229C2×C4×C17C22×D411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17H(5,𝔽2)11yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C22⋊D17—88no—
235C2×Dic17—41no—
236C2×Dic17—42no—
237C2×Dic17—41no—
238C2×Dic17—42no—
239C2×Dic17—41no—
240C2×Dic17—42no—
241D68—22no—
242D68—22no—
243D68—41no—
244D68—41no—
245C4×D17—84no—
246#3622—22no—
247#3622—41no—
248#3622—42no—
249C2×#554D1711yescharacteristic, complemented
250C2×#476—171no—
251C2×C4×SD16—171no—
252#2537—171no—
253#2035—172no—
254#2000—172no—
255#1990—171no—
256#1954—171no—
257#1874—172no—
258#1853—172no—
259#1803—171no—
260C22×C17—24no—
261C22×C17—24no—
262C22×C17—24no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—42no—
266C22×C17—42no—
267C22×C17C2×#12011yescharacteristic
268C22×C17#54611yescharacteristic
269C22×C17#54711yescharacteristic
270C22×C17C22×SD1611yescharacteristic
271C22×C17C2×#13711yescharacteristic
272C22×C17#57611yescharacteristic
273C22×C17#57711yescharacteristic
274C2×D17—42no—
275C2×D17—42no—
276C2×D17—42no—
277C2×D17—81no—
278C2×D17—81no—
279C2×D17—81no—
280C2×D17—81no—
281C4×C17—22no—
282C4×C17—42no—
283C4×C17—42no—
284C4×C17—42no—
285C4×C17—44no—
286C4×C17#44814yes—
287C4×C17#52511yescharacteristic
288C4×C17#53711yescharacteristic
289C4×C17#54811yescharacteristic
290C4×C17#55011yescharacteristic
291Dic17—81no—
292Dic17—82no—
293C22×SD16—171no—
294#554—24no—
295#554—24no—
296C2×C4⋊Q8C2×D1711yescharacteristic
297C2×#125—21no—
298C2×C22⋊Q8—22no—
299C2×C22⋊Q8C2×D1712yes—
300C2×#121—172no—
301C2×#121—341no—
302C2×C4×Q8C2×D1711yescharacteristic
303C2×C4×D4—21no—
304C2×C4×D4—171no—
305C2×C4×D4C2×D1711yescharacteristic
306C22×C4⋊C4C2×D1712yes—
307#476—344no—
308C4×SD16—344no—
309C2×#106—171no—
310C2×#106—172no—
311C2×C4⋊C8—171no—
312C2×#103—171no—
313C2×#103—172no—
314C2×#102—171no—
315C2×#102—172no—
316C2×C22⋊C8—172no—
317C2×C22⋊C8—172no—
318C2×C4×C8—171no—
319#394—342no—
320#384—171no—
321#380—172no—
322#380—341no—
323#340—172no—
324#340—172no—
325#340—172no—
326C2×C17—44no—
327C2×C17#240711yescharacteristic
328C2×C17C2×#44811yescharacteristic
329C2×C17#301211yescharacteristic
330C2×C17#329511yescharacteristic
331C2×C17#333111yescharacteristic
332C2×C17#333511yescharacteristic
333C2×C17#335711yescharacteristic
334D17—82no—
335C22×Q8—21no—
336C22×Q8C22×D1711yescharacteristic
337C22×D4—171no—
338C22×D4—342no—
339C22×D4C22⋊D1711yescharacteristic
340C23×C4—22no—
341C23×C4—341no—
342C23×C4C22×D1712yes—
343C2×SD16—342no—
344C2×SD16—342no—
345C2×SD16—342no—
346C22×C8—171no—
347C22×C8—171no—
348C22×C8—172no—
349C4⋊Q8—22no—
350C4⋊Q8C22⋊D1714yescomplemented
351#125—28no—
352C22⋊Q8—216no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355#121—344no—
356#121—682no—
357#121—684no—
358C4×Q8—22no—
359C4×Q8C22⋊D1714yescomplemented
360C4×D4—28no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—342no—
364C4×D4—342no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—172no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C22⋊D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1712yes—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—342no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—342no—
382C2×C22⋊C4C22⋊D1712yes—
383C2×C42—21no—
384C2×C42C22×D1711yescharacteristic
385#106—342no—
386#106—344no—
387C4⋊C8—342no—
388#103—342no—
389#103—682no—
390#102—342no—
391#102—344no—
392C22⋊C8—344no—
393C22⋊C8—682no—
394C4×C8—342no—
395#95—342no—
396C17#2411811yescharacteristic, complemented, Sylow
397C24—341no—
398C24C2×C22⋊D1712yes—
399C2×Q8—21no—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×C22⋊D1712yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—172no—
411C2×D4—341no—
412C2×D4—341no—
413C2×D4—341no—
414C2×D4—341no—
415C2×D4—341no—
416C2×D4—684no—
417C2×D4—684no—
418C2×D4—1362no—
419C2×D4#6323014yescomplemented
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—341no—
433C22×C4—342no—
434C22×C4—682no—
435C22×C4—682no—
436C22×C4—682no—
437C22×C4C2×C22⋊D1712yes—
438C22×C4C2×C22⋊D1711yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C23×D1711yescharacteristic
441SD16—348no—
442C2×C8—172no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—342no—
446C2×C8—341no—
447C2×C8—342no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—342no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—342no—
459C4⋊C4—342no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—682no—
463C4⋊C4—1361no—
464C4⋊C4C2×C22⋊D1712yes—
465C4⋊C4C2×C22⋊D1714yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—684no—
469C22⋊C4—684no—
470C22⋊C4—682no—
471C42—24no—
472C42C2×C22⋊D1712yes—
473C42C2×C22⋊D1712yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—342no—
481C23—681no—
482C23—681no—
483C23—681no—
484C23—681no—
485C23—681no—
486C23—681no—
487C23C2×#6323014yes—
488C23C22×C22⋊D1711yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494D4—28no—
495D4—342no—
496D4—342no—
497D4—681no—
498D4—681no—
499D4—1368no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—681no—
520C2×C4—681no—
521C2×C4—682no—
522C2×C4—681no—
523C2×C4—682no—
524C2×C4—682no—
525C2×C4—1364no—
526C2×C4C2×#6323012yes—
527C2×C4#1058762812yes—
528C2×C4#1058763912yes—
529C2×C4#1058763912yes—
530C2×C4#41×D1711yescharacteristic
531C2×C4#41×D1711yescharacteristic
532C2×C4#1058769611yescharacteristic
533C2×C4#1058769711yescharacteristic
534C2×C4C22×C22⋊D1711yescharacteristic
535C2×C4C22×C22⋊D1711yescharacteristic
536C8—342no—
537C8—682no—
538C8—681no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—682no—
547C22—682no—
548C22—682no—
549C22—1361no—
550C22—1361no—
551C22—1361no—
552C22—1361no—
553C22C2×#1058763911yescharacteristic
554C22#1176344711yescharacteristic
555C22#1176345811yescharacteristic
556C22C22×#6323011yescharacteristic
557C22C2×#1058762811yescharacteristic
558C22#1176361211yescharacteristic
559C22#1176362011yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—1362no—
566C4—1361no—
567C4#1176294814yes—
568C4#1176331611yescharacteristic
569C4#1176338711yescharacteristic
570C4#1176346611yescharacteristic
571C4#1176347811yescharacteristic
572C2—44no—
573C2—1362no—
574C2#1453212311yescharacteristic
575C2C2×#1176294811yescharacteristic
576C2#1453808811yescharacteristic
577C2#1454060211yescharacteristic
578C2#1454109411yescharacteristic
579C2#1454113511yescharacteristic
580C2#1454141011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#554C2
socleC23×C17C22×D4
prime radical#14533697C2
hypercenterC2×#554D17
hypocenterC17#24118
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#11763096C2semidirect
2. C2×C4×#63230C2semidirect
4. #14529977C2semidirect
5. #14529785C2semidirect
6. #14529746C2semidirect
7. #14529579C2semidirect
8. #14529101C2semidirect
9. #14528977C2semidirect
10. #14528789C2semidirect
11. C2×C17×#554C2semidirect
54. C17×C4⋊Q8D4semidirect
60. C4×Q8×C17D4semidirect
119. C2×D4×C17SD16semidirect
249. C2×#554D17semidirect
350. C4⋊Q8C22⋊D17semidirect
359. C4×Q8C22⋊D17semidirect
396. C17#24118semidirect
419. C2×D4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×C4×#632302
#145336972
#145299772
#145297852
#145297462
#145295792
#145291012
#145289772
#145287892
C2×C17×#5542
#2411817