Finite Group Databasebeta

← #17046582

Subgroups — #17046582

Order 4352.

740 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×#11763089C211yesmaximal, characteristic
2#14531530C211yesmaximal, characteristic, complemented
3#14531427C211yesmaximal, characteristic
4#14529731C211yesmaximal, characteristic, complemented
5#14529550C211yesmaximal, characteristic, complemented
6#14529521C211yesmaximal, characteristic, complemented
7#14529516C211yesmaximal, characteristic, complemented
8#14529478C211yesmaximal, characteristic
9#14529464C211yesmaximal, characteristic
10#14529414C211yesmaximal, characteristic
11#14529407C211yesmaximal, characteristic
12#14529379C211yesmaximal, characteristic, complemented
13#14529358C211yesmaximal, characteristic, complemented
14#14528643C211yesmaximal, characteristic, complemented
15C17×#1804C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17#11763089C418yescomplemented
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26C2×C8×Dic17C2211yescharacteristic
27#11762713—21no—
28#11762713—21no—
29#11762712C2211yescharacteristic
30#11762712C2211yescharacteristic
31#11762708C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762702—21no—
36#11762702—21no—
37#11762694—21no—
38#11762694—21no—
39#11762672—21no—
40#11762669—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×C4⋊Dic17C2211yescharacteristic
44#11762655C2211yescharacteristic
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×Dic34—21no—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C4×D4×C17—22no—
66C4×D4×C17C2×C414yes—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92#10587555—22no—
93#10587555C2×C414yes—
94C17×C4⋊C8—22no—
95C17×#102—41no—
96C17×#102—41no—
97C17×#95—21no—
98C17×#95—21no—
99#10587503C2×C414yes—
100#10587503C2×C414yes—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502D411yescharacteristic
109#10587502D411yescharacteristic
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587486—41no—
113C8×Dic17—41no—
114#10587479—41no—
115#10587478—41no—
116#10587478C2×C414yes—
117#10587478C2×C414yes—
118C4⋊#3622C2×C414yes—
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C22×C412yes—
126C22×C4×C17—21no—
127C22×C4×C17—24no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—81no—
131C22×C4×C17—81no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C2×Dic34—24no—
144C2×Dic34—81no—
145C2×Dic34—81no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17C22×C412yes—
158C22⋊C4×C17—24no—
159C42×C17C22×C412yes—
160C42×C17#4112yes—
161C22⋊Dic17—81no—
162C22⋊Dic17—81no—
163C22⋊Dic17—81no—
164C22⋊Dic17—81no—
165C4⋊Dic17—24no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—81no—
173C4⋊Dic17—81no—
174C4⋊Dic17C22×C412yes—
175C4⋊Dic17C22×C412yes—
176#63225—81no—
177#63225—81no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—81no—
183C4×Dic17—81no—
184C4×Dic17—81no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622C22×C412yes—
190C2×#3622C22×C412yes—
191#22127—171nomaximal, Sylow
192C23×C17—22no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—41no—
203D4×C17—41no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—41no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—84no—
218C2×C4×C17C4×D412yes—
219C2×C4×C17C4×D412yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17H(5,𝔽2)11yescharacteristic
223C2×C4×C172−1+411yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
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—
250Dic34—84no—
251C8×C17—41no—
252C8×C17—41no—
253#3622—22no—
254#3622—22no—
255C2×#474—171no—
256#2331—171no—
257#2319—171no—
258#1987—171no—
259#1949—171no—
260#1943—171no—
261#1942—171no—
262#1936—171no—
263#1934—171no—
264#1926—171no—
265#1925—171no—
266#1921—171no—
267#1918—171no—
268#1804D1711yescharacteristic
269#1768—171no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17C2×C4×D411yescharacteristic
278C22×C17#53611yescharacteristic
279C22×C17#53711yescharacteristic
280C22×C17#57611yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57811yescharacteristic
284C4×C17—22no—
285C4×C17—22no—
286C4×C17—42no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17#51911yescharacteristic
290C4×C17#52011yescharacteristic
291C4×C17#54111yescharacteristic
292C4×C17#54211yescharacteristic
293Dic17—42no—
294Dic17—42no—
295Dic17—41no—
296Dic17—41no—
297Dic17—41no—
298Dic17—41no—
299Dic17—81no—
300Dic17—81no—
301C2×C4⋊Q8—171no—
302C2×C4×D4C2×D1711yescharacteristic
303#474—178no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#107—171no—
307C2×#106—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8C2×D1711yescharacteristic
310C2×#102—171no—
311C2×#102—171no—
312C2×#102C2×D1711yescharacteristic
313C2×#102C2×D1711yescharacteristic
314C2×#97—171no—
315C2×C4×C8—171no—
316#391—341no—
317#391—341no—
318#390—171no—
319#390—171no—
320#389—171no—
321#389—171no—
322#389—171no—
323#389—171no—
324#388—341no—
325#388—341no—
326#387—341no—
327#387—341no—
328#382C2×D1711yescharacteristic
329#381—341no—
330#380—341no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334#376—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340C2×D1711yescharacteristic
342#340C2×D1711yescharacteristic
343#336—171no—
344#336—171no—
345C2×C17—82no—
346C2×C17#232311yescharacteristic
347C2×C17#294711yescharacteristic
348C2×C17#294811yescharacteristic
349C2×C17#324111yescharacteristic
350C2×C17#324311yescharacteristic
351C2×C17#325711yescharacteristic
352C2×C17#326211yescharacteristic
353C22×Q8—341no—
354C22×D4C22×D1711yescharacteristic
355C23×C4—21no—
356C22×C8—171no—
357C22×C8—171no—
358C22×C8C22×D1711yescharacteristic
359C22×C8C22×D1711yescharacteristic
360C4⋊Q8—174no—
361C4⋊Q8—342no—
362C4×D4—22no—
363C4×D4C4×D1714yes—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C22⋊C4—21no—
377C2×C22⋊C4—341no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C42—171no—
382C2×C42—171no—
383C2×C42—341no—
384C2×C42—341no—
385C2×C42—341no—
386C2×C42C22×D1711yescharacteristic
387#107—174no—
388#107—174no—
389#107—681no—
390#106—681no—
391C4⋊C8—22no—
392C4⋊C8—174no—
393#102—41no—
394#102—41no—
395#102—174no—
396#102—174no—
397#97—681no—
398C4×C8—681no—
399#95—21no—
400#95—21no—
401#95—171no—
402#95—171no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412C17#2212711yescharacteristic, complemented, Sylow
413C24—21no—
414C2×Q8—344no—
415C2×Q8—1361no—
416C2×Q8—1361no—
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—81no—
428C22×C4—81no—
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—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—1361no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C4⋊C4C2×C4×D1712yes—
471C22⋊C4—24no—
472C22⋊C4—1361no—
473C22⋊C4—1361no—
474C22⋊C4—1361no—
475C22⋊C4—1361no—
476C42—341no—
477C42—341no—
478C42—341no—
479C42—341no—
480C42—1361no—
481C42—1361no—
482C42—1361no—
483C42C2×C4×D1712yes—
484C42#6325412yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D1711yescharacteristic
493Q8—1364no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—342no—
513C2×C4—342no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
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×C4#1058756712yes—
537C2×C4#1058756712yes—
538C2×C4C22×C4×D1711yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4#1058769111yescharacteristic
541C2×C4#1058769411yescharacteristic
542C2×C4#1058769711yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—41no—
545C8—41no—
546C8—342no—
547C8—342no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×#1058756711yescharacteristic
556C22#1176338211yescharacteristic
557C22#1176338811yescharacteristic
558C22#1176361311yescharacteristic
559C22#1176361311yescharacteristic
560C22#1176361611yescharacteristic
561C22#1176362311yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—681no—
568C4—681no—
569C4—682no—
570C4—682no—
571C4—681no—
572C4—681no—
573C4—1361no—
574C4—1361no—
575C4#1176328611yescharacteristic
576C4#1176329011yescharacteristic
577C4#1176341911yescharacteristic
578C4#1176342611yescharacteristic
579C2—82no—
580C2#1453146211yescharacteristic
581C2#1453763011yescharacteristic
582C2#1453763211yescharacteristic
583C2#1454004211yescharacteristic
584C2#1454006611yescharacteristic
585C2#1454020011yescharacteristic
586C2#1454024211yescharacteristic
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#22127
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. #14531530C2semidirect
4. #14529731C2semidirect
5. #14529550C2semidirect
6. #14529521C2semidirect
7. #14529516C2semidirect
12. #14529379C2semidirect
13. #14529358C2semidirect
14. #14528643C2semidirect
17. #11763089C4semidirect
412. C17#22127semidirect

Maximal subgroups

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

subgroupindex
C2×#117630892
#145315302
#145314272
#145297312
#145295502
#145295212
#145295162
#145294782
#145294642
#145294142
#145294072
#145293792
#145293582
#145286432
C17×#18042
#2212717