Finite Group Databasebeta

← #17000249

Subgroups — #17000249

Order 4352.

1206 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763288C211yesmaximal, characteristic
2#14537823C212yesmaximal, complemented
3#14527122C211yesmaximal, characteristic, complemented
4#14527119C212yesmaximal, complemented
5#14527113C212yesmaximal, complemented
6#14527112C212yesmaximal, complemented
7#14527106C212yesmaximal, complemented
8#14527095C212yesmaximal, complemented
9C17×#1465C211yesmaximal, characteristic, complemented
10C22×#63254C411yescharacteristic
11#11763288—24no—
12#11763288—24no—
13C2×C4×Dic34C411yescharacteristic
14C2×#10587573C2211yescharacteristic
15C2×D4×Dic17C411yescharacteristic
16C2×C4×C22⋊D17C411yescharacteristic
17C2×C4×C22⋊D17C2211yescharacteristic
18C2×#10587567C411yescharacteristic
19C2×#10587567C2211yescharacteristic
20C2×#10587558C411yescharacteristic
21C2×#10587563C411yescharacteristic
22C2×#10587563C2211yescharacteristic
23C22×#63225C411yescharacteristic
24C22×#63225C2211yescharacteristic
25C2×C4⋊C4×D17C2211yescharacteristic
26#11762885—24no—
27#11762884—24no—
28#11762859C2214yes—
29#11762773C2212yes—
30#11762771C2214yes—
31#11762535C2212yes—
32#11762534C2214yes—
33#11762534C2214yes—
34#11762533C2212yes—
35C2×C4×D4×C17C2211yescharacteristic
36C17×#407C2212yes—
37C17×#324C2212yes—
38C17×#323C2212yes—
39C22×D4×C17C2×C411yescharacteristic
40C23×C4×C17C2×C411yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C22×C22⋊D17C2×C411yescharacteristic
43C22×C22⋊D17C2×C411yescharacteristic
44C23×Dic17C2×C411yescharacteristic
45C23×Dic17C2×C411yescharacteristic
46C23×Dic17C2×C411yescharacteristic
47C23×Dic17C2×C411yescharacteristic
48C2×#63254—24no—
49C2×#63254—24no—
50C2×#63254—24no—
51C22×C4×D17C2×C411yescharacteristic
52C22×C4×D17C2×C411yescharacteristic
53C22×C4×D17C2311yescharacteristic
54C22×Dic34C2×C411yescharacteristic
55C2×M16×C17—22no—
56C4×D4×C17—22no—
57C4×D4×C17—22no—
58C2×C4⋊C4×C17C2311yescharacteristic
59C2×C22⋊C4×C17C2×C411yescharacteristic
60C2×C22⋊C4×C17C2311yescharacteristic
61C2×C42×C17C2×C411yescharacteristic
62C2×C22⋊Dic17C2×C411yescharacteristic
63C2×C22⋊Dic17C2×C411yescharacteristic
64D4×Dic17—24no—
65C4×C22⋊D17—22no—
66C4×C22⋊D17—22no—
67C4×C22⋊D17D414yes—
68C4×C22⋊D17D414yes—
69C2×#63227C2×C411yescharacteristic
70C2×#63227C2×C411yescharacteristic
71C2×C4⋊Dic17C2×C411yescharacteristic
72C2×#63225—22no—
73C2×#63225—22no—
74C2×#63225—22no—
75C2×#63225—22no—
76C2×#63225C2×C411yescharacteristic
77C2×#63225C2×C411yescharacteristic
78C2×#63225C2×C411yescharacteristic
79C2×#63225C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×#63225C2311yescharacteristic
83C2×#63225C2311yescharacteristic
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×#63223—22no—
90C2×#63218—24no—
91C4⋊C4×D17—24no—
92#10587573—24no—
93#10587567—24no—
94#10587567—24no—
95#10587563—24no—
96#10587563—24no—
97#10587558—22no—
98#10587558—22no—
99C4×Dic34—22no—
100C4×Dic34—22no—
101C17×C22⋊C8—22no—
102C17×C22⋊C8C2314yes—
103C22⋊#3622—22no—
104C22⋊#3622C2314yes—
105#10587491—24no—
106#10587485—24no—
107C8×Dic17—24no—
108C24×C17C22×C411yescharacteristic
109C24×C17C22×C411yescharacteristic
110C23×D17C22×C411yescharacteristic
111C2×D4×C17—22no—
112C2×D4×C17—22no—
113C2×D4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—81no—
117C22×C4×C17—81no—
118C22×C4×C17C22×C411yescharacteristic
119C22×C4×C17C22×C411yescharacteristic
120C22×C4×C17C22×C411yescharacteristic
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C2×D412yes—
123C22×C4×C17C2×D412yes—
124C22×C4×C17C2411yescharacteristic
125C2×C22⋊D17—22no—
126C2×C22⋊D17—22no—
127C2×C22⋊D17—22no—
128C2×C22⋊D17—22no—
129C2×C22⋊D17—22no—
130C2×C22⋊D17C22⋊C414yes—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—81no—
139C22×Dic17—81no—
140C22×Dic17M1614yescomplemented
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148#63254—232no—
149C2×C4×D17—22no—
150C2×C4×D17—22no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—81no—
154C2×C4×D17—81no—
155C2×C4×D17C2×D414yes—
156C2×Dic34—22no—
157C2×Dic34—22no—
158C2×Dic34M1614yescomplemented
159M16×C17—82no—
160C2×C8×C17—24no—
161C4⋊C4×C17—22no—
162C22⋊C4×C17—22no—
163C22⋊C4×C17—22no—
164C42×C17—21no—
165C42×C17—21no—
166C22⋊Dic17—22no—
167C22⋊Dic17C22⋊C414yes—
168#63227—22no—
169#63227C22⋊C412yes—
170#63227C22⋊C412yes—
171C4⋊Dic17—22no—
172#63225—21no—
173#63225—21no—
174#63225—21no—
175#63225—21no—
176#63225—22no—
177#63225—22no—
178#63225—22no—
179#63225—22no—
180#63225—22no—
181#63225C22⋊C412yes—
182#63225C22⋊C412yes—
183C4×Dic17—22no—
184C4×Dic17—22no—
185C4×Dic17—22no—
186C4×Dic17—22no—
187C4×Dic17C2×D414yes—
188#63223—82no—
189C2×#3622—24no—
190#63218—84no—
191#17370—171nomaximal, Sylow
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C2×C22⋊C412yes—
201C23×C17C2×M1612yes—
202C23×C17C23×C411yescharacteristic
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—41no—
206C22×D17—41no—
207C22×D17C2×C22⋊C412yes—
208D4×C17—28no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—42no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—42no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—81no—
220C2×C4×C17—81no—
221C2×C4×C17C2×C22⋊C411yescharacteristic
222C2×C4×C17C2×C22⋊C411yescharacteristic
223C2×C4×C17C2×M1612yes—
224C2×C4×C17C22×D412yes—
225C22⋊D17—28no—
226C22⋊D17—28no—
227C2×Dic17—28no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—28no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—82no—
240C2×Dic17—82no—
241C2×Dic17C2×C22⋊C412yes—
242C2×Dic17C2×C22⋊C412yes—
243C2×Dic17C2×C22⋊C412yes—
244C2×Dic17C2×M1612yes—
245C2×Dic17C2×M1612yes—
246C2×Dic17C2×M1612yes—
247C2×Dic17C2×M1612yes—
248C4×D17—28no—
249C4×D17—44no—
250C4×D17—82no—
251Dic34—28no—
252C8×C17—44no—
253#3622—44no—
254C2×#520—171no—
255#2975—172no—
256#1465—172no—
257#1465—171no—
258#1465D1711yescharacteristic, complemented
259#1463—172no—
260#1463—172no—
261#1461—172no—
262#1458—172no—
263C22×C17—21no—
264C22×C17—21no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—22no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17C22×C22⋊C411yescharacteristic
276C22×C17C22×M1611yescharacteristic
277C22×C17C22×M1611yescharacteristic
278C2×D17—22no—
279C2×D17—22no—
280C2×D17—22no—
281C2×D17—42no—
282C2×D17—42no—
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—21no—
286C4×C17—21no—
287C4×C17—41no—
288C4×C17—41no—
289Dic17—22no—
290Dic17—22no—
291Dic17—22no—
292Dic17—22no—
293Dic17—41no—
294Dic17—41no—
295Dic17#40714yes—
296C22×#41—171no—
297#520—344no—
298#520—344no—
299C2×C4×Q8—171no—
300C2×C4×Q8—171no—
301C2×C4×D4—171no—
302C2×C4×D4—171no—
303C2×C4×D4—171no—
304C2×C4×D4—171no—
305C2×C4×D4—171no—
306C2×C4×D4C2×D1711yescharacteristic
307C2×#117—171no—
308C2×#117—171no—
309C2×#117—171no—
310C22×C4⋊C4—171no—
311C22×C4⋊C4—171no—
312C22×C4⋊C4—171no—
313#436—344no—
314#436—344no—
315#431—174no—
316#407—172no—
317#407—174no—
318#407C2×D1712yes—
319#324—172no—
320#324—174no—
321#324—174no—
322#324C2×D1712yes—
323#323—172no—
324#323C2×D1712yes—
325C2×C17—21no—
326C2×C17—21no—
327C2×C17—42no—
328C2×C17—42no—
329C2×C17C2×#40711yescharacteristic
330C2×C17#288311yescharacteristic
331C2×C17#288911yescharacteristic
332D17—42no—
333C2×#41—344no—
334C2×#41—344no—
335C2×#41—344no—
336C22×Q8—171no—
337C22×D4—171no—
338C22×D4—171no—
339C22×D4C4×D1711yescharacteristic
340C23×C4—171no—
341C23×C4—171no—
342C23×C4—171no—
343C23×C4—171no—
344C23×C4—171no—
345C23×C4—171no—
346C23×C4—171no—
347C23×C4C4×D1711yescharacteristic
348C23×C4C22×D1711yescharacteristic
349C2×M16—22no—
350C2×M16—344no—
351C2×M16—342no—
352C4×Q8—342no—
353C4×Q8—342no—
354C4×Q8—344no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—174no—
358C4×D4—174no—
359C4×D4—344no—
360C4×D4—344no—
361C4×D4—342no—
362C4×D4—342no—
363C4×D4—344no—
364#117—342no—
365#117—342no—
366#117—344no—
367#117—344no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—344no—
378C2×C4⋊C4—342no—
379C2×C4⋊C4—342no—
380C2×C4⋊C4—342no—
381C2×C4⋊C4—342no—
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C22⋊C4—171no—
384C2×C22⋊C4—171no—
385C2×C22⋊C4—171no—
386C2×C22⋊C4—171no—
387C2×C22⋊C4C4×D1711yescharacteristic
388C2×C22⋊C4C22×D1711yescharacteristic
389C2×C42—171no—
390C2×C42—171no—
391C2×C42—171no—
392C2×C42—171no—
393C2×C42—171no—
394C2×C42C4×D1711yescharacteristic
395C4⋊C8—344no—
396C22⋊C8—22no—
397C22⋊C8—174no—
398C22⋊C8—344no—
399C22⋊C8—342no—
400C22⋊C8C22×D1714yes—
401C4×C8—344no—
402C17#1737011yescharacteristic, complemented, Sylow
403C24—171no—
404C24C2×C4×D1711yescharacteristic
405C24C2×C4×D1711yescharacteristic
406#41—3432no—
407C2×Q8—174no—
408C2×Q8—342no—
409C2×Q8—342no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—174no—
414C2×D4—342no—
415C2×D4—342no—
416C2×D4—342no—
417C2×D4—342no—
418C2×D4—342no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—174no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—174no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—1361no—
444C22×C4—1361no—
445C22×C4—1361no—
446C22×C4—1361no—
447C22×C4C2×C4×D1711yescharacteristic
448C22×C4C2×C4×D1711yescharacteristic
449C22×C4C2×C4×D1711yescharacteristic
450C22×C4C2×C4×D1711yescharacteristic
451C22×C4D4×D1712yes—
452C22×C4D4×D1712yes—
453C22×C4C23×D1711yescharacteristic
454M16—82no—
455M16—1364no—
456M16—1362no—
457C2×C8—24no—
458C2×C8—344no—
459C4⋊C4—22no—
460C4⋊C4—172no—
461C4⋊C4—172no—
462C4⋊C4—342no—
463C4⋊C4—342no—
464C4⋊C4—342no—
465C4⋊C4—342no—
466C4⋊C4—342no—
467C4⋊C4—342no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—174no—
475C22⋊C4—172no—
476C22⋊C4—172no—
477C22⋊C4—342no—
478C22⋊C4—342no—
479C42—21no—
480C42—21no—
481C42—174no—
482C42—342no—
483C42—342no—
484C42—342no—
485C42—342no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—172no—
495C23—341no—
496C23—341no—
497C23—681no—
498C23—681no—
499C23C22⋊C4×D1712yes—
500C23M16×D1712yes—
501C23C22×C4×D1711yescharacteristic
502Q8—348no—
503D4—28no—
504D4—348no—
505D4—348no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—172no—
519C2×C4—172no—
520C2×C4—172no—
521C2×C4—172no—
522C2×C4—172no—
523C2×C4—172no—
524C2×C4—172no—
525C2×C4—348no—
526C2×C4—348no—
527C2×C4—348no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—684no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—1362no—
540C2×C4—1362no—
541C2×C4—1362no—
542C2×C4C22⋊C4×D1711yescharacteristic
543C2×C4C22⋊C4×D1711yescharacteristic
544C2×C4M16×D1712yes—
545C2×C4C2×D4×D1712yes—
546C8—44no—
547C8—684no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—342no—
561C22—342no—
562C22—342no—
563C22—682no—
564C22—682no—
565C22C2×C22⋊C4×D1711yescharacteristic
566C22C2×M16×D1711yescharacteristic
567C22C2×M16×D1711yescharacteristic
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—174no—
575C4—342no—
576C4—342no—
577C4—342no—
578C4—342no—
579C4—681no—
580C4—681no—
581C2—21no—
582C2—21no—
583C2—42no—
584C2—42no—
585C2—682no—
586C2D17×#40711yescharacteristic
587C2#1453724211yescharacteristic
588C2#1453726511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1465C2
socleC22×C17C22×C22⋊C4
prime radicalC22×#63254C4
cosocleC22×C4×C17C24
hypercenter#1465D17
hypocenterC17#17370
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #14537823C2semidirect
3. #14527122C2semidirect
4. #14527119C2semidirect
5. #14527113C2semidirect
6. #14527112C2semidirect
7. #14527106C2semidirect
8. #14527095C2semidirect
9. C17×#1465C2semidirect
140. C22×Dic17M16semidirect
158. C2×Dic34M16semidirect
258. #1465D17semidirect
402. C17#17370semidirect

Maximal subgroups

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

subgroupindex
C2×#117632882
#145378232
#145271222
#145271192
#145271132
#145271122
#145271062
#145270952
C17×#14652
#1737017