Finite Group Databasebeta

← #17042437

Subgroups — #17042437

Order 4352.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic
2#14531532C211yesmaximal, characteristic, complemented
3#14529744C211yesmaximal, characteristic, complemented
4#14529596C211yesmaximal, characteristic
5#14529484C211yesmaximal, characteristic
6#14529477C211yesmaximal, characteristic, complemented
7#14529470C211yesmaximal, characteristic
8#14529463C211yesmaximal, characteristic, complemented
9#14528919C211yesmaximal, characteristic, complemented
10#14528614C211yesmaximal, characteristic, complemented
11#14528609C211yesmaximal, characteristic
12C4×#10587506C211yesmaximal, characteristic, complemented
13C4×#10587503C211yesmaximal, characteristic
14#14528471C211yesmaximal, characteristic, complemented
15C17×#2326C211yesmaximal, characteristic
16C2×#10587555C2211yescharacteristic
17#11763168C418yescomplemented
18C2×#10587506C2211yescharacteristic
19C2×#10587506C2211yescharacteristic
20C2×#10587506C2211yescharacteristic
21C2×#10587506C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32#11762713—21no—
33#11762713—21no—
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762692—21no—
37#11762692—21no—
38#11762675C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49C2×C17×#124C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#391C2211yescharacteristic
53C17×#390C2211yescharacteristic
54C17×#388—21no—
55C17×#388—21no—
56C17×#385—21no—
57C17×#385—21no—
58C17×#384C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C22×Dic34—21no—
64C17×#124—22no—
65C17×#124C2×C414yes—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587555—22no—
98#10587555C2×C414yescomplemented
99C17×#95—21no—
100C17×#95—21no—
101C17×#95D411yescharacteristic
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104C17×#95D411yescharacteristic
105#10587506—22no—
106#10587506—22no—
107#10587506C2×C414yescomplemented
108#10587506C2×C414yescomplemented
109#10587503—22no—
110#10587503—22no—
111#10587503C2×C414yes—
112#10587503C2×C414yes—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587474—22no—
120#10587474—22no—
121#10587474C2×C414yescomplemented
122C4×#3622—22no—
123C4×#3622—22no—
124C4×#3622—22no—
125C24×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17C22×C412yes—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C22×C412yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—42no—
142C22×C4×C17—81no—
143C22×C4×C17—81no—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C2×Dic34—24no—
156C2×Dic34—81no—
157C2×Dic34—81no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17—81no—
164C4⋊C4×C17—81no—
165C22⋊C4×C17—24no—
166C22⋊C4×C17—24no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—81no—
169C22⋊C4×C17—81no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—42no—
173C42×C17C22×C412yes—
174C42×C17#4112yes—
175C42×C17#4112yes—
176C42×C17#4112yes—
177C4⋊Dic17—24no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17C22×C412yes—
186#63225—81no—
187#63225—81no—
188C4×Dic17—81no—
189C4×Dic17—81no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199C2×#3622C22×C412yes—
200C2×#3622#4112yes—
201C2×#3622#4112yes—
202#21654—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17C22×D411yescharacteristic
211Q8×C17—22no—
212Q8×C17—22no—
213Q8×C17—41no—
214Q8×C17—41no—
215D4×C17—22no—
216D4×C17—22no—
217D4×C17—41no—
218D4×C17—41no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—84no—
243C2×C4×C17C4×D412yes—
244C2×C4×C17C4×D412yes—
245C2×C4×C17#12412yes—
246C2×C4×C17#12412yes—
247C2×C4×C17C23×C411yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265Dic34—84no—
266#3622—22no—
267#3622—22no—
268#3622—22no—
269#3622—22no—
270C2×#489—171no—
271#2331—171no—
272#2326D1711yescharacteristic
273#1990—171no—
274#1958—171no—
275#1936—171no—
276#1935—171no—
277#1934—171no—
278#1933—171no—
279#1836—171no—
280#1763—171no—
281#1762—171no—
282C4×#103—171no—
283C4×#102—171no—
284#1726—171no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17C2×C4×D411yescharacteristic
293C22×C17C2×#12411yescharacteristic
294C22×C17#53211yescharacteristic
295C22×C17#57611yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57711yescharacteristic
298C22×C17#57811yescharacteristic
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—22no—
302C4×C17—22no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—42no—
306C4×C17—81no—
307C4×C17—81no—
308C4×C17C4×#4111yescharacteristic
309C4×C17C4×#4111yescharacteristic
310C4×C17#52911yescharacteristic
311C4×C17#52911yescharacteristic
312Dic17—42no—
313Dic17—42no—
314Dic17—81no—
315Dic17—81no—
316C2×C4⋊Q8—171no—
317C2×#124C2×D1711yescharacteristic
318C2×C4×Q8C2×D1711yescharacteristic
319C2×C4×D4C2×D1711yescharacteristic
320#489—178no—
321C2×#103—171no—
322C2×#103—171no—
323C2×#103—171no—
324C2×#103—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102—171no—
328C2×#102—171no—
329C2×#97—171no—
330C2×#97—171no—
331C2×#97—171no—
332C2×C4×C8—171no—
333C2×C4×C8—171no—
334C2×C4×C8—171no—
335#391—341no—
336#391—341no—
337#391C2×D1711yescharacteristic
338#390C2×D1711yescharacteristic
339#389—171no—
340#389—171no—
341#388—21no—
342#388—21no—
343#387—341no—
344#387—341no—
345#385—21no—
346#385—21no—
347#384C2×D1711yescharacteristic
348#382—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351#376C2×D1711yescharacteristic
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360C2×C17—82no—
361C2×C17C4×#12411yescharacteristic
362C2×C17#294611yescharacteristic
363C2×C17#294711yescharacteristic
364C2×C17#313611yescharacteristic
365C2×C17#313711yescharacteristic
366C2×C17#317311yescharacteristic
367C2×C17#317511yescharacteristic
368C22×Q8—341no—
369C22×Q8C22×D1711yescharacteristic
370C22×D4C22×D1711yescharacteristic
371C23×C4—21no—
372C22×C8—171no—
373C22×C8—171no—
374C22×C8—171no—
375C22×C8—171no—
376C4⋊Q8—174no—
377C4⋊Q8—342no—
378#124—22no—
379#124C4×D1714yes—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×D4—22no—
383C4×D4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—21no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42C22×D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407#103—174no—
408#103—174no—
409#103—342no—
410#103—342no—
411#102—174no—
412#102—174no—
413#102—342no—
414#102—342no—
415#97—174no—
416#97—342no—
417#97—342no—
418C4×C8—342no—
419C4×C8—342no—
420C4×C8—342no—
421#95—21no—
422#95—21no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95C22⋊D1711yescharacteristic
430#95C22⋊D1711yescharacteristic
431#95C22⋊D1711yescharacteristic
432#95C22⋊D1711yescharacteristic
433C17#2165411yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—344no—
441C2×Q8—1361no—
442C2×Q8—1361no—
443C2×Q8C2×C4×D1712yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D1712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C23×D1711yescharacteristic
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—172no—
470C2×C8—172no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—172no—
487C4⋊C4—172no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—344no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—1361no—
496C4⋊C4—1361no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—1361no—
506C42—1361no—
507C42C2×C4×D1712yes—
508C42#6325212yes—
509C42#6325412yes—
510C42#6325412yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C22×C22⋊D1711yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—1364no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4C4×C22⋊D1712yes—
565C2×C4C4×C22⋊D1712yes—
566C2×C4#1058762212yes—
567C2×C4#1058762212yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4C2×#6325411yescharacteristic
571C2×C4C2×#6325411yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—342no—
575C8—342no—
576C8—342no—
577C8—342no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C4×C22⋊D1711yescharacteristic
586C22C2×#1058762211yescharacteristic
587C22#1176335511yescharacteristic
588C22#1176361211yescharacteristic
589C22#1176361711yescharacteristic
590C22#1176362011yescharacteristic
591C22#1176362411yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—682no—
602C4—682no—
603C4—1361no—
604C4—1361no—
605C4C4×#6325411yescharacteristic
606C4C4×#6325411yescharacteristic
607C4#1176334111yescharacteristic
608C4#1176334111yescharacteristic
609C2—82no—
610C2C4×#1058762211yescharacteristic
611C2#1453761211yescharacteristic
612C2#1453762311yescharacteristic
613C2#1453913811yescharacteristic
614C2#1453914911yescharacteristic
615C2#1453939511yescharacteristic
616C2#1453941711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2326C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2326D17
hypocenterC17#21654
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #14531532C2semidirect
3. #14529744C2semidirect
6. #14529477C2semidirect
8. #14529463C2semidirect
9. #14528919C2semidirect
10. #14528614C2semidirect
12. C4×#10587506C2semidirect
14. #14528471C2semidirect
17. #11763168C4semidirect
98. #10587555C2×C4semidirect
107. #10587506C2×C4semidirect
108. #10587506C2×C4semidirect
121. #10587474C2×C4semidirect
433. C17#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
#145315322
#145297442
#145295962
#145294842
#145294772
#145294702
#145294632
#145289192
#145286142
#145286092
C4×#105875062
C4×#105875032
#145284712
C17×#23262
#2165417