Finite Group Databasebeta

← #17068244

Subgroups — #17068244

Order 4352.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763086C211yesmaximal, characteristic, complemented
2C2×C4×#63230C211yesmaximal, characteristic, complemented
3#14533697C211yesmaximal, characteristic
4#14529984C212yesmaximal, complemented
5#14529785C212yesmaximal, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529106C212yesmaximal, complemented
9#14528983C212yesmaximal, complemented
10#14528789C211yesmaximal, characteristic, complemented
11C2×C17×#556C211yesmaximal, characteristic, complemented
12C22×#63230C2211yescharacteristic
13C2×#10587616C2212yes—
14C2×#10587624—21no—
15C2×C4×D68C2211yescharacteristic
16#11763086—24no—
17C4×#63230—24no—
18C2×#10587478C2212yes—
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×#556—24no—
36C17×#556—24no—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×#126—22no—
39C2×C17×#124—21no—
40C2×C17×C22⋊Q8C2212yes—
41C2×C4×Q8×C17—21no—
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C2×C17×#117C2212yes—
45C22×Q8×C17—21no—
46C22×Q8×C17C2311yescharacteristic
47C22×D4×C17D411yescharacteristic
48C23×C4×C17C2312yes—
49C22×C22⋊D17—22no—
50C22×D68C2311yescharacteristic
51C22×C4×D17—21no—
52C17×C4⋊Q8—22no—
53C17×C4⋊Q8D414yescomplemented
54C17×#126—216no—
55C17×#124—28no—
56C17×C22⋊Q8—24no—
57C17×C22⋊Q8—24no—
58C4×Q8×C17—28no—
59C4×Q8×C17—22no—
60C4×Q8×C17D414yescomplemented
61C4×D4×C17—22no—
62C4×D4×C17—22no—
63C17×#117—24no—
64C17×#117—24no—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17D411yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2312yes—
71C2×C22⋊C4×C17—22no—
72C2×C22⋊C4×C17—22no—
73C2×C22⋊C4×C17D412yes—
74C2×C42×C17—22no—
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—22no—
103#10587478—24no—
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—42no—
115C2×Q8×C17—42no—
116C2×Q8×C17C2×D412yes—
117C2×D4×C17—22no—
118C2×D4×C17—22no—
119C2×D4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17C2×D412yes—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C2×C22⋊D17—44no—
134C2×C22⋊D17—44no—
135C2×C22⋊D17—82no—
136C22×Dic17—21no—
137C22×Dic17—22no—
138C2×D68—21no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68#4112yes—
144C2×C4×D17—42no—
145C2×C4×D17—42no—
146C2×C4×D17—42no—
147C4⋊C4×C17—28no—
148C4⋊C4×C17—24no—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17C2×D414yes—
154C4⋊C4×C17C2×D412yes—
155C22⋊C4×C17—28no—
156C22⋊C4×C17—28no—
157C22⋊C4×C17—24no—
158C42×C17—28no—
159C42×C17—24no—
160C42×C17C2×D412yes—
161C42×C17C2×D412yes—
162C22⋊Dic17—44no—
163#63230—28no—
164#63227—44no—
165#63227—42no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—22no—
169C4⋊Dic17—22no—
170C4⋊Dic17—42no—
171C4⋊Dic17—81no—
172C2×#3622—21no—
173C2×#3622—21no—
174C2×#3622—21no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—21no—
178C2×#3622—22no—
179C2×#3622—21no—
180C2×#3622#4112yes—
181#24119—171nomaximal, Sylow
182C23×C17—22no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17C22×D411yescharacteristic
190C22×D17—22no—
191C22×D17—41no—
192C22×D17—41no—
193C22×D17—41no—
194C22×D17—41no—
195C22×D17—41no—
196C22×D17—41no—
197Q8×C17—22no—
198Q8×C17—22no—
199Q8×C17—41no—
200Q8×C17—41no—
201Q8×C17—48no—
202D4×C17—44no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—24no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17#12012yes—
222C2×C4×C17#12012yes—
223C2×C4×C17#13512yes—
224C2×C4×C17AGL(1,ℤ/8)12yes—
225C2×C4×C17C22×D411yescharacteristic
226C2×C4×C17C22×D411yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17H(5,𝔽2)11yescharacteristic
230C2×C4×C172−1+411yescharacteristic
231C22⋊D17—88no—
232C2×Dic17—41no—
233C2×Dic17—42no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238D68—22no—
239D68—22no—
240D68—41no—
241D68—41no—
242C4×D17—84no—
243#3622—22no—
244#3622—41no—
245#3622—42no—
246C2×#556D1711yescharacteristic, complemented
247C2×#474—171no—
248C2×C4×SD16—171no—
249#2537—171no—
250#2036—172no—
251#2000—172no—
252#1990—171no—
253#1954—171no—
254#1875—172no—
255#1854—172no—
256#1803—171no—
257C22×C17—42no—
258C22×C17—42no—
259C22×C17—42no—
260C22×C17—42no—
261C22×C17—42no—
262C22×C17—42no—
263C22×C17—42no—
264C22×C17C2×#12011yescharacteristic
265C22×C17#54611yescharacteristic
266C22×C17#54711yescharacteristic
267C22×C17C2×#13511yescharacteristic
268C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C17#57711yescharacteristic
270C22×C17#57811yescharacteristic
271C2×D17—42no—
272C2×D17—42no—
273C2×D17—42no—
274C2×D17—81no—
275C2×D17—81no—
276C2×D17—81no—
277C2×D17—81no—
278C4×C17—22no—
279C4×C17—42no—
280C4×C17—42no—
281C4×C17—44no—
282C4×C17—42no—
283C4×C17#44914yes—
284C4×C17#52511yescharacteristic
285C4×C17#53711yescharacteristic
286C4×C17#54811yescharacteristic
287C4×C17#55011yescharacteristic
288Dic17—81no—
289Dic17—82no—
290C22×SD16—171no—
291#556—24no—
292#556—24no—
293C2×C4⋊Q8C2×D1711yescharacteristic
294C2×#126—22no—
295C2×#124—21no—
296C2×C22⋊Q8C2×D1712yes—
297C2×#121—172no—
298C2×#121—341no—
299C2×C4×Q8—21no—
300C2×C4×Q8C2×D1711yescharacteristic
301C2×C4×D4—171no—
302C2×C4×D4C2×D1711yescharacteristic
303C2×#117C2×D1712yes—
304#474—344no—
305C4×SD16—344no—
306C2×#107—172no—
307C2×#106—171no—
308C2×C4⋊C8—171no—
309C2×#103—171no—
310C2×#103—172no—
311C2×#102—171no—
312C2×#102—172no—
313C2×C22⋊C8—172no—
314C2×C22⋊C8—172no—
315C2×C4×C8—171no—
316#394—342no—
317#384—171no—
318#380—172no—
319#380—341no—
320#340—172no—
321#340—172no—
322#340—172no—
323C2×C17—82no—
324C2×C17#240711yescharacteristic
325C2×C17C2×#44911yescharacteristic
326C2×C17#301311yescharacteristic
327C2×C17#330211yescharacteristic
328C2×C17#332411yescharacteristic
329C2×C17#334011yescharacteristic
330C2×C17#335311yescharacteristic
331D17—82no—
332C22×Q8—21no—
333C22×Q8C22×D1711yescharacteristic
334C22×D4—171no—
335C22×D4—342no—
336C22×D4C22⋊D1711yescharacteristic
337C23×C4—341no—
338C23×C4C22×D1712yes—
339C2×SD16—342no—
340C2×SD16—342no—
341C2×SD16—342no—
342C22×C8—171no—
343C22×C8—171no—
344C22×C8—172no—
345C4⋊Q8—22no—
346C4⋊Q8C22⋊D1714yescomplemented
347#126—216no—
348#124—28no—
349C22⋊Q8—24no—
350C22⋊Q8—24no—
351#121—344no—
352#121—682no—
353#121—684no—
354C4×Q8—28no—
355C4×Q8—22no—
356C4×Q8C22⋊D1714yescomplemented
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—342no—
360C4×D4—342no—
361#117—24no—
362#117—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—172no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4C22⋊D1711yescharacteristic
371C2×C4⋊C4C22×D1711yescharacteristic
372C2×C4⋊C4C22×D1712yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—342no—
376C2×C22⋊C4—341no—
377C2×C22⋊C4—342no—
378C2×C22⋊C4C22⋊D1712yes—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42C22×D1711yescharacteristic
382#107—344no—
383#106—342no—
384C4⋊C8—342no—
385#103—342no—
386#103—682no—
387#102—342no—
388#102—344no—
389C22⋊C8—344no—
390C22⋊C8—682no—
391C4×C8—342no—
392#95—342no—
393C17#2411911yescharacteristic, complemented, Sylow
394C24—341no—
395C24C2×C22⋊D1712yes—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×C22⋊D1712yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—172no—
409C2×D4—341no—
410C2×D4—341no—
411C2×D4—341no—
412C2×D4—341no—
413C2×D4—341no—
414C2×D4—684no—
415C2×D4—684no—
416C2×D4—1362no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—341no—
427C22×C4—342no—
428C22×C4—682no—
429C22×C4—682no—
430C22×C4—682no—
431C22×C4C2×C22⋊D1712yes—
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C23×D1711yescharacteristic
435SD16—348no—
436C2×C8—172no—
437C2×C8—341no—
438C2×C8—341no—
439C2×C8—342no—
440C2×C8—341no—
441C2×C8—342no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—342no—
445C4⋊C4—28no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—342no—
452C4⋊C4—342no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—682no—
456C4⋊C4—1361no—
457C4⋊C4C2×C22⋊D1712yes—
458C4⋊C4C2×C22⋊D1714yes—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—684no—
463C22⋊C4—684no—
464C22⋊C4—682no—
465C42—28no—
466C42—24no—
467C42C2×C22⋊D1712yes—
468C42C2×C22⋊D1712yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—342no—
477C23—681no—
478C23—681no—
479C23—681no—
480C23—681no—
481C23—681no—
482C23—681no—
483C23C22×C22⋊D1711yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—41no—
489D4—44no—
490D4—342no—
491D4—342no—
492D4—681no—
493D4—681no—
494D4—1368no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—681no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4—682no—
517C2×C4—682no—
518C2×C4—682no—
519C2×C4—1364no—
520C2×C4#1058763512yes—
521C2×C4#1058763612yes—
522C2×C4#1058763912yes—
523C2×C4#1058763912yes—
524C2×C4#41×D1711yescharacteristic
525C2×C4#41×D1711yescharacteristic
526C2×C4#1058769611yescharacteristic
527C2×C4#1058769711yescharacteristic
528C2×C4C22×C22⋊D1711yescharacteristic
529C2×C4C22×C22⋊D1711yescharacteristic
530C8—342no—
531C8—682no—
532C8—681no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—682no—
541C22—682no—
542C22—682no—
543C22—1361no—
544C22—1361no—
545C22—1361no—
546C22—1361no—
547C22C2×#1058763911yescharacteristic
548C22#1176344711yescharacteristic
549C22#1176345811yescharacteristic
550C22C2×#1058763611yescharacteristic
551C22C2×#1058763511yescharacteristic
552C22#1176361711yescharacteristic
553C22#1176362411yescharacteristic
554C4—22no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—1361no—
560C4—1362no—
561C4#1176295414yes—
562C4#1176331611yescharacteristic
563C4#1176338711yescharacteristic
564C4#1176346611yescharacteristic
565C4#1176347811yescharacteristic
566C2—82no—
567C2—1362no—
568C2#1453212311yescharacteristic
569C2C2×#1176295411yescharacteristic
570C2#1453809911yescharacteristic
571C2#1454069511yescharacteristic
572C2#1454100911yescharacteristic
573C2#1454119111yescharacteristic
574C2#1454136211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#556C2
socleC23×C17C22×D4
prime radical#14533697C2
hypercenterC2×#556D17
hypocenterC17#24119
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#11763086C2semidirect
2. C2×C4×#63230C2semidirect
4. #14529984C2semidirect
5. #14529785C2semidirect
6. #14529746C2semidirect
7. #14529579C2semidirect
8. #14529106C2semidirect
9. #14528983C2semidirect
10. #14528789C2semidirect
11. C2×C17×#556C2semidirect
53. C17×C4⋊Q8D4semidirect
60. C4×Q8×C17D4semidirect
246. C2×#556D17semidirect
346. C4⋊Q8C22⋊D17semidirect
356. C4×Q8C22⋊D17semidirect
393. C17#24119semidirect

Maximal subgroups

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

subgroupindex
C2×#117630862
C2×C4×#632302
#145336972
#145299842
#145297852
#145297462
#145295792
#145291062
#145289832
#145287892
C2×C17×#5562
#2411917