Finite Group Databasebeta

← #17052102

Subgroups — #17052102

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic
2#14531667C211yesmaximal, characteristic
3#14529524C211yesmaximal, characteristic, complemented
4#14529509C211yesmaximal, characteristic, complemented
5#14529373C211yesmaximal, characteristic, complemented
6#14529369C211yesmaximal, characteristic, complemented
7#14529362C211yesmaximal, characteristic, complemented
8#14529352C211yesmaximal, characteristic, complemented
9#14528876C211yesmaximal, characteristic
10#14528874C211yesmaximal, characteristic
11#14528815C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13C4×#10587478C211yesmaximal, characteristic, complemented
14#14528568C211yesmaximal, characteristic
15C17×#2344C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587612C2211yescharacteristic
18#11762849C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×C4×#3622C2211yescharacteristic
28#11762707C2211yescharacteristic
29#11762707C2211yescharacteristic
30#11762697—21no—
31#11762697—21no—
32#11762673—21no—
33#11762669C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36#11762654—21no—
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×#117—21no—
50C22×C4⋊C4×C17C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#387—21no—
54C17×#387—21no—
55C17×#381—21no—
56C17×#380C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C22⋊C4×C17C2211yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C17×#117—44no—
63C2×C4⋊C4×C17—21no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2×C414yes—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17D411yescharacteristic
74C2×C22⋊C4×C17D411yescharacteristic
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17—21no—
81#10587612—22no—
82#10587612C2×C414yes—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223C2×C414yes—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95D411yescharacteristic
105C17×#95D411yescharacteristic
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111C22⋊#3622—22no—
112#10587479—22no—
113#10587479C2×C414yes—
114#10587479C2×C414yes—
115#10587478—22no—
116#10587478C2×C414yes—
117#10587478C2×C414yes—
118#10587474—22no—
119C4×#3622—22no—
120C24×C17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—24no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17#4112yes—
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C4⋊C4×C17—24no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—42no—
152C4⋊C4×C17—42no—
153C4⋊C4×C17C22×C412yes—
154C4⋊C4×C17C22×C412yes—
155C22⋊C4×C17—22no—
156C22⋊C4×C17—41no—
157C22⋊C4×C17—42no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—42no—
163C42×C17—42no—
164C22⋊Dic17—24no—
165C22⋊Dic17—81no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17C22×C412yes—
173C4⋊Dic17C22×C412yes—
174#63225—81no—
175#63225—81no—
176C4×Dic17—24no—
177C4×Dic17—42no—
178C4×Dic17—42no—
179#63223—28no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622C22×C412yes—
189C2×#3622C22×C412yes—
190C2×#3622#4112yes—
191C2×#3622#4112yes—
192#22624—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17C4×D412yes—
200C23×C17C22×D411yescharacteristic
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—48no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17C4×D412yes—
232C2×C4×C17AGL(1,ℤ/8)12yes—
233C2×C4×C17C23×C411yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C172−1+411yescharacteristic
238C2×C4×C172−1+411yescharacteristic
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251#3622—22no—
252#3622—41no—
253#3622—41no—
254#3622#11714yescomplemented
255C2×#428—171no—
256#2346—171no—
257#2344D1711yescharacteristic
258#1944—171no—
259#1941—171no—
260#1921—171no—
261#1920—171no—
262#1919—171no—
263#1918—171no—
264#1825—171no—
265#1824—171no—
266#1809—171no—
267#1776—171no—
268C4×#107—171no—
269#1753—171no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17C2×C4×D411yescharacteristic
278C22×C17#54711yescharacteristic
279C22×C17#54711yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57811yescharacteristic
284C4×C17—22no—
285C4×C17—22no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—41no—
289C4×C17—42no—
290C4×C17—42no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17C2×#11711yescharacteristic
294C4×C17#51911yescharacteristic
295C4×C17#55211yescharacteristic
296C4×C17#55211yescharacteristic
297Dic17—42no—
298Dic17—42no—
299Dic17—81no—
300Dic17—81no—
301C22×M16—171no—
302C2×#117—21no—
303C2×#117—171no—
304C22×C4⋊C4C2×D1711yescharacteristic
305#428—178no—
306C2×#107—171no—
307C2×#107—171no—
308C2×#107—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×#106—171no—
312C2×C22⋊C8—171no—
313C2×#97—171no—
314C2×C4×C8—171no—
315#389—171no—
316#389—171no—
317#389C2×D1711yescharacteristic
318#389C2×D1711yescharacteristic
319#388—341no—
320#388—341no—
321#387—21no—
322#387—21no—
323#382—341no—
324#381—21no—
325#380—171no—
326#380C2×D1711yescharacteristic
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C22⋊C4C2×D1711yescharacteristic
332#376—341no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#336—171no—
344#336—171no—
345C2×C17—42no—
346C2×C17#234411yescharacteristic
347C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
348C2×C17#294711yescharacteristic
349C2×C17#334311yescharacteristic
350C2×C17#334811yescharacteristic
351C2×C17#335211yescharacteristic
352C2×C17#335411yescharacteristic
353C23×C4—21no—
354C23×C4C22×D1711yescharacteristic
355C2×M16—174no—
356C2×M16—342no—
357C2×M16—342no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362#117—44no—
363#117—174no—
364#117—342no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C4×D1714yes—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4C22⋊D1711yescharacteristic
384C2×C22⋊C4C22⋊D1711yescharacteristic
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—341no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42C22×D1711yescharacteristic
391C2×C42C22×D1711yescharacteristic
392#107—174no—
393#107—174no—
394#107—342no—
395#106—174no—
396#106—174no—
397#106—342no—
398C22⋊C8—342no—
399#97—342no—
400C4×C8—342no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95C22⋊D1711yescharacteristic
412#95C22⋊D1711yescharacteristic
413C17#2262411yescharacteristic, complemented, Sylow
414C24C2×C22⋊D1711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4C2×C4×D1712yes—
431C22×C4#6325212yes—
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C2×C22⋊D1711yescharacteristic
436C22×C4C2×C22⋊D1711yescharacteristic
437C22×C4C23×D1711yescharacteristic
438M16—348no—
439C2×C8—172no—
440C2×C8—172no—
441C2×C8—172no—
442C2×C8—172no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—172no—
461C4⋊C4—172no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C4⋊C4C2×C4×D1712yes—
471C4⋊C4C2×C4×D1712yes—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—344no—
476C22⋊C4—1361no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—42no—
483C42—344no—
484C42—682no—
485C42—682no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D1712yes—
493C23C22×C22⋊D1711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—681no—
527C2×C4—681no—
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×C4C4×C22⋊D1712yes—
537C2×C4#1058761812yes—
538C2×C4C22×C4×D1711yescharacteristic
539C2×C4C2×#6325211yescharacteristic
540C2×C4#41×D1711yescharacteristic
541C2×C4#41×D1711yescharacteristic
542C2×C4#1058769711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—174no—
545C8—342no—
546C8—681no—
547C8—681no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D1711yescharacteristic
556C22#1176345811yescharacteristic
557C22#1176345811yescharacteristic
558C22C2×#1058761811yescharacteristic
559C22#1176360711yescharacteristic
560C22#1176361711yescharacteristic
561C22#1176362411yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—682no—
572C4—682no—
573C4—1361no—
574C4—1361no—
575C4#117×D1711yescharacteristic
576C4#1176328611yescharacteristic
577C4#1176349211yescharacteristic
578C4#1176349211yescharacteristic
579C2—42no—
580C2#1453165411yescharacteristic
581C2C4×#1058761811yescharacteristic
582C2#1453762711yescharacteristic
583C2#1454121811yescharacteristic
584C2#1454129311yescharacteristic
585C2#1454135311yescharacteristic
586C2#1454137911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2344C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2344D17
hypocenterC17#22624
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #14529524C2semidirect
4. #14529509C2semidirect
5. #14529373C2semidirect
6. #14529369C2semidirect
7. #14529362C2semidirect
8. #14529352C2semidirect
12. #14528682C2semidirect
13. C4×#10587478C2semidirect
18. #11762849C4semidirect
254. #3622#117semidirect
413. C17#22624semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
#145316672
#145295242
#145295092
#145293732
#145293692
#145293622
#145293522
#145288762
#145288742
#145288152
#145286822
C4×#105874782
#145285682
C17×#23442
#2262417