Finite Group Databasebeta

← #17046403

Subgroups — #17046403

Order 4352.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic
2#14531339C211yesmaximal, characteristic, complemented
3#14531249C211yesmaximal, characteristic
4#14529739C211yesmaximal, characteristic, complemented
5#14529550C211yesmaximal, characteristic, complemented
6#14529516C211yesmaximal, characteristic, complemented
7#14529511C211yesmaximal, characteristic, complemented
8#14529464C211yesmaximal, characteristic
9#14529464C211yesmaximal, characteristic
10#14529414C211yesmaximal, characteristic
11#14529407C211yesmaximal, characteristic
12#14529379C211yesmaximal, characteristic, complemented
13#14529372C211yesmaximal, characteristic, complemented
14C4⋊C8×Dic17C211yesmaximal, characteristic, complemented
15C17×#1804C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587612—21no—
18#11763109C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27C2×C8×Dic17C2211yescharacteristic
28#11762712C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762702—21no—
35#11762702—21no—
36#11762691—21no—
37#11762691—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41#11762660—21no—
42#11762660—21no—
43C4×C22⋊Dic17—21no—
44C42×Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#382C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C4×D4×C17—22no—
65C4×D4×C17C2×C414yes—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C22⋊C4×C17—21no—
70C2×C42×C17C2311yescharacteristic
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75#10587612—44no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587556—22no—
98#10587556C2×C414yes—
99C17×C4⋊C8—22no—
100C17×#102—41no—
101C17×#102—41no—
102C17×#95—21no—
103C17×#95—21no—
104#10587503C2×C414yes—
105#10587503C2×C414yes—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113C8×Dic17—41no—
114C8×Dic17—41no—
115#10587479—41no—
116#10587479C2×C414yes—
117#10587478—41no—
118#10587478C2×C414yes—
119C4⋊#3622C2×C414yes—
120C24×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17C22×C412yes—
127C22×C4×C17—21no—
128C22×C4×C17—24no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17C22×C412yes—
156C22⋊C4×C17—24no—
157C42×C17C22×C412yes—
158C42×C17#4112yes—
159C22⋊Dic17—42no—
160C22⋊Dic17—42no—
161C22⋊Dic17—42no—
162C22⋊Dic17—42no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—42no—
170C4⋊Dic17—42no—
171C4⋊Dic17C22×C412yes—
172C4⋊Dic17C22×C412yes—
173#63225—24no—
174#63225—24no—
175#63225—42no—
176#63225—42no—
177C4×Dic17—21no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—22no—
182C4×Dic17—22no—
183C4×Dic17—22no—
184C4×Dic17—41no—
185C4×Dic17—42no—
186C4×Dic17—42no—
187C4×Dic17—42no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622C22×C412yes—
193C2×#3622C22×C412yes—
194#22114—171nomaximal, Sylow
195C23×C17—22no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17—41no—
201C23×C17—41no—
202C23×C17C22×D411yescharacteristic
203D4×C17—22no—
204D4×C17—22no—
205D4×C17—41no—
206D4×C17—41no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—48no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17C4×D412yes—
222C2×C4×C17C4×D412yes—
223C2×C4×C17C23×C411yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C2×C4×C172−1+411yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—41no—
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#13512yes—
251C2×Dic17#13512yes—
252C2×Dic17AGL(1,ℤ/8)12yes—
253C8×C17—41no—
254C8×C17—41no—
255#3622—22no—
256#3622—22no—
257C2×#478—171no—
258C4×#125—171no—
259#2293—171no—
260#1989—171no—
261#1949—171no—
262#1942—171no—
263#1941—171no—
264#1934—171no—
265#1934—171no—
266#1926—171no—
267#1925—171no—
268#1921—171no—
269#1920—171no—
270#1804D1711yescharacteristic
271C4×C4⋊C8—171no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—42no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17—81no—
279C22×C17C2×C4×D411yescharacteristic
280C22×C17#53611yescharacteristic
281C22×C17#53711yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×#13511yescharacteristic
284C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C17#57511yescharacteristic
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—42no—
289C4×C17—81no—
290C4×C17—81no—
291C4×C17#51911yescharacteristic
292C4×C17#52011yescharacteristic
293C4×C17#54111yescharacteristic
294C4×C17#54211yescharacteristic
295Dic17—22no—
296Dic17—22no—
297Dic17—22no—
298Dic17—42no—
299Dic17—42no—
300Dic17—41no—
301Dic17—81no—
302Dic17—81no—
303C2×#125—171no—
304C2×C4×D4C2×D1711yescharacteristic
305C2×#117—341no—
306#478—178no—
307C2×#107—171no—
308C2×#107—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×C4⋊C8—171no—
312C2×C4⋊C8C2×D1711yescharacteristic
313C2×#102—171no—
314C2×#102—171no—
315C2×#102C2×D1711yescharacteristic
316C2×#102C2×D1711yescharacteristic
317C2×C4×C8—171no—
318C2×C4×C8—171no—
319#390—171no—
320#390—171no—
321#389—171no—
322#389—171no—
323#389—171no—
324#389—171no—
325#388—341no—
326#388—341no—
327#387—341no—
328#387—341no—
329#382C2×D1711yescharacteristic
330C4×C4⋊C4—171no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—341no—
334C4×C4⋊C4—341no—
335C4×C22⋊C4—341no—
336C43—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340C2×D1711yescharacteristic
344#340C2×D1711yescharacteristic
345#336—171no—
346#336—171no—
347C2×C17—82no—
348C2×C17#232311yescharacteristic
349C2×C17C4×#13511yescharacteristic
350C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
351C2×C17#322811yescharacteristic
352C2×C17#323211yescharacteristic
353C2×C17#324711yescharacteristic
354C2×C17#325011yescharacteristic
355C22×D4C22×D1711yescharacteristic
356C23×C4—21no—
357C22×C8—171no—
358C22×C8—171no—
359C22×C8C22×D1711yescharacteristic
360C22×C8C22×D1711yescharacteristic
361#125—174no—
362#125—342no—
363C4×D4—22no—
364C4×D4C4×D1714yes—
365#117—684no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—341no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C42—171no—
385C2×C42—171no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—171no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42C22×D1711yescharacteristic
394#107—174no—
395#107—681no—
396#106—174no—
397#106—681no—
398C4⋊C8—22no—
399C4⋊C8—174no—
400#102—41no—
401#102—41no—
402#102—174no—
403#102—174no—
404C4×C8—681no—
405C4×C8—681no—
406#95—21no—
407#95—21no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415C17#2211411yescharacteristic, complemented, Sylow
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D1712yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4D4×D1711yescharacteristic
438C22×C4D4×D1711yescharacteristic
439C22×C4C23×D1711yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—172no—
447C2×C8—172no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—172no—
458C4⋊C4—172no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—344no—
462C4⋊C4—344no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—682no—
468C4⋊C4—682no—
469C4⋊C4—682no—
470C4⋊C4—682no—
471C4⋊C4C2×C4×D1712yes—
472C22⋊C4—24no—
473C22⋊C4—682no—
474C22⋊C4—682no—
475C22⋊C4—682no—
476C22⋊C4—682no—
477C42—342no—
478C42—342no—
479C42—342no—
480C42—341no—
481C42—341no—
482C42—341no—
483C42—341no—
484C42—681no—
485C42—682no—
486C42—682no—
487C42—682no—
488C42C2×C4×D1712yes—
489C42#6325412yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D1711yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—172no—
517C2×C4—172no—
518C2×C4—172no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—342no—
526C2×C4—342no—
527C2×C4—341no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4#1058756712yes—
541C2×C4#1058756712yes—
542C2×C4C22×C4×D1711yescharacteristic
543C2×C4C2×#6325411yescharacteristic
544C2×C4#1058769111yescharacteristic
545C2×C4#1058769411yescharacteristic
546C2×C4#1058769711yescharacteristic
547C2×C4#1058769711yescharacteristic
548C8—41no—
549C8—41no—
550C8—342no—
551C8—342no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#1058756711yescharacteristic
560C22#1176338211yescharacteristic
561C22#1176338811yescharacteristic
562C22#135×D1711yescharacteristic
563C22#135×D1711yescharacteristic
564C22AGL(1,ℤ/8)×D1711yescharacteristic
565C22#1176360511yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—342no—
572C4—342no—
573C4—342no—
574C4—682no—
575C4—682no—
576C4—681no—
577C4—1361no—
578C4—1361no—
579C4#1176328611yescharacteristic
580C4#1176329011yescharacteristic
581C4#1176341911yescharacteristic
582C4#1176342611yescharacteristic
583C2—82no—
584C2#1453146211yescharacteristic
585C2#1453755211yescharacteristic
586C2#1453758811yescharacteristic
587C2#1453991911yescharacteristic
588C2#1453995011yescharacteristic
589C2#1454010711yescharacteristic
590C2#1454013311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1804C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1804D17
hypocenterC17#22114
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #14531339C2semidirect
4. #14529739C2semidirect
5. #14529550C2semidirect
6. #14529516C2semidirect
7. #14529511C2semidirect
12. #14529379C2semidirect
13. #14529372C2semidirect
14. C4⋊C8×Dic17C2semidirect
18. #11763109C4semidirect
415. C17#22114semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
#145313392
#145312492
#145297392
#145295502
#145295162
#145295112
#145294642
#145294142
#145294072
#145293792
#145293722
C4⋊C8×Dic172
C17×#18042
#2211417