Finite Group Databasebeta

← #17037340

Subgroups — #17037340

Order 4352.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2#14531261C211yesmaximal, characteristic, complemented
3#14529425C211yesmaximal, characteristic, complemented
4#14529418C211yesmaximal, characteristic, complemented
5#14529376C211yesmaximal, characteristic, complemented
6#14529355C211yesmaximal, characteristic, complemented
7#14529118C211yesmaximal, characteristic
8#14529118C211yesmaximal, characteristic
9#14529045C211yesmaximal, characteristic
10#14529031C211yesmaximal, characteristic
11#14528682C211yesmaximal, characteristic, complemented
12#14528617C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14#14528533C211yesmaximal, characteristic, complemented
15C42×D4×C17C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11762909C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30#11762715C2211yescharacteristic
31#11762715C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762693—21no—
35#11762693—21no—
36C4×#63225—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38#11762655—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C2×C4×D4×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C22×C42×C17—21no—
53C4×C4⋊C4×C17C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C22⋊C4×C17—21no—
57C4×C22⋊C4×C17—21no—
58C4×C22⋊C4×C17—21no—
59C43×C17C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17—21no—
63C23×C4×C17—21no—
64C22×Dic34C2311yescharacteristic
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C4×D4×C17C2×C414yes—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—44no—
81C2×C42×C17D411yescharacteristic
82C2×C42×C17D411yescharacteristic
83C2×C42×C17D411yescharacteristic
84C2×C42×C17D411yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×#63229—22no—
89C2×#63229—22no—
90C2×#63229C2×C414yes—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C4×Dic34—22no—
104C4×Dic34C2×C414yescomplemented
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108#10587503—22no—
109#10587503—22no—
110#10587503C2×C414yes—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587481—22no—
119#10587481—22no—
120#10587481C2×C414yescomplemented
121#10587479—22no—
122#10587478—22no—
123#10587478C2×C414yescomplemented
124#10587474—22no—
125#10587474C2×C414yescomplemented
126C24×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17C22×C412yes—
131C2×D4×C17#4112yes—
132C2×D4×C17#4112yes—
133C22×C4×C17—24no—
134C22×C4×C17—24no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—24no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17—42no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17—42no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34C22×C412yes—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17C22×C412yes—
166C4⋊C4×C17#4112yes—
167C4⋊C4×C17#4112yes—
168C22⋊C4×C17—24no—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—24no—
171C42×C17—22no—
172C42×C17—22no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—41no—
178C42×C17—41no—
179C42×C17—42no—
180C42×C17—42no—
181C42×C17—42no—
182C42×C17C22×C412yes—
183C42×C17#4112yes—
184#63229—28no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17C22×C412yes—
191#63225—24no—
192#63225—42no—
193#63225—42no—
194C4×Dic17—24no—
195C4×Dic17—42no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622C22×C412yes—
207C2×#3622C22×C412yes—
208#21121—171nomaximal, Sylow
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—22no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17C22×D411yescharacteristic
217D4×C17—22no—
218D4×C17—22no—
219D4×C17—22no—
220D4×C17#11714yes—
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—21no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—48no—
239C2×C4×C17—48no—
240C2×C4×C17—48no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17C4×D412yes—
248C2×C4×C17C4×D412yes—
249C2×C4×C17AGL(1,ℤ/8)12yes—
250C2×C4×C17#13712yes—
251C2×C4×C17C23×C411yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269Dic34—22no—
270Dic34—22no—
271Dic34—41no—
272Dic34—41no—
273#3622—22no—
274#3622—22no—
275#3622—41no—
276#3622—41no—
277C2×#440—171no—
278#2295—171no—
279C42×D4D1711yescharacteristic
280#1928—171no—
281#1927—171no—
282#1921—171no—
283#1918—171no—
284#1877—171no—
285#1877—171no—
286#1864—171no—
287#1862—171no—
288#1776—171no—
289#1763—171no—
290#1762—171no—
291#1745—171no—
292C22×C17—24no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17—42no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17C2×C4×D411yescharacteristic
300C22×C17#52511yescharacteristic
301C22×C17#52511yescharacteristic
302C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C17C2×#13711yescharacteristic
304C22×C17#57511yescharacteristic
305C22×C17#57511yescharacteristic
306C4×C17—22no—
307C4×C17—22no—
308C4×C17—22no—
309C4×C17—22no—
310C4×C17—42no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—41no—
314C4×C17—42no—
315C4×C17C2×#11711yescharacteristic
316C4×C17C4×#4111yescharacteristic
317C4×C17#52911yescharacteristic
318C4×C17#52911yescharacteristic
319Dic17—42no—
320Dic17—42no—
321Dic17—81no—
322Dic17—81no—
323C22×SD16—171no—
324C2×C4×Q8—171no—
325C2×C4×D4C2×D1711yescharacteristic
326C2×C4×D4C2×D1711yescharacteristic
327C2×C4×D4C2×D1711yescharacteristic
328C22×C42—21no—
329#440—178no—
330C2×#107—171no—
331C2×#107—171no—
332C2×#106—171no—
333C2×#103—171no—
334C2×#103—171no—
335C2×#103—171no—
336C2×#102—171no—
337C2×#102—171no—
338C2×#102—171no—
339C2×#97—171no—
340C2×#97—171no—
341#391—171no—
342#391—171no—
343#389—171no—
344#389—171no—
345#387—341no—
346#387—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—341no—
349C4×C4⋊C4C2×D1711yescharacteristic
350C4×C4⋊C4C2×D1711yescharacteristic
351C4×C4⋊C4C2×D1711yescharacteristic
352C4×C22⋊C4—21no—
353C4×C22⋊C4—21no—
354C4×C22⋊C4—21no—
355#376—341no—
356C43C2×D1711yescharacteristic
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#336—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367C2×C17—44no—
368C2×C17#229011yescharacteristic
369C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
370C2×C17C4×#13711yescharacteristic
371C2×C17#311011yescharacteristic
372C2×C17#311111yescharacteristic
373C2×C17#311411yescharacteristic
374C2×C17#311611yescharacteristic
375C22×Q8—171no—
376C22×D4C22×D1711yescharacteristic
377C23×C4—21no—
378C23×C4—21no—
379C23×C4—21no—
380C2×SD16—174no—
381C2×SD16—342no—
382C2×SD16—342no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386C22×C8—171no—
387C4×Q8—174no—
388C4×Q8—342no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4C4×D1714yes—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—44no—
411C2×C42—341no—
412C2×C42—341no—
413C2×C42C22⋊D1711yescharacteristic
414C2×C42C22⋊D1711yescharacteristic
415C2×C42C22⋊D1711yescharacteristic
416C2×C42C22⋊D1711yescharacteristic
417C2×C42C22×D1711yescharacteristic
418C2×C42C22×D1711yescharacteristic
419C2×C42C22×D1711yescharacteristic
420#107—174no—
421#107—342no—
422#106—342no—
423#103—174no—
424#103—342no—
425#103—342no—
426#102—174no—
427#102—342no—
428#102—342no—
429#97—174no—
430#97—342no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441C17#2112111yescharacteristic, complemented, Sylow
442C24—21no—
443C2×Q8—172no—
444C2×Q8—341no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D1712yes—
453C2×D4#6325212yes—
454C2×D4#6325212yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C23×D1711yescharacteristic
478SD16—348no—
479C2×C8—172no—
480C2×C8—172no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—172no—
495C4⋊C4—341no—
496C4⋊C4—344no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—682no—
502C4⋊C4—682no—
503C4⋊C4C2×C4×D1712yes—
504C4⋊C4#6325212yes—
505C4⋊C4#6325212yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—344no—
521C42—682no—
522C42C2×C4×D1712yes—
523C42#6325212yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C22⋊D1711yescharacteristic
532Q8—342no—
533Q8—342no—
534Q8—681no—
535Q8—681no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1058755814yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—342no—
567C2×C4—342no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4C4×C22⋊D1712yes—
579C2×C4C4×C22⋊D1712yes—
580C2×C4#1058761812yes—
581C2×C4#1058763712yes—
582C2×C4C22×C4×D1711yescharacteristic
583C2×C4C2×#6325211yescharacteristic
584C2×C4C2×#6325211yescharacteristic
585C2×C4C2×#6325211yescharacteristic
586C2×C4C2×#6325211yescharacteristic
587C2×C4C2×#6325211yescharacteristic
588C8—342no—
589C8—342no—
590C8—681no—
591C8—681no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×C4×C22⋊D1711yescharacteristic
600C22#1176332011yescharacteristic
601C22#1176332011yescharacteristic
602C22C2×#1058761811yescharacteristic
603C22C2×#1058763711yescharacteristic
604C22#1176360711yescharacteristic
605C22#1176360711yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—682no—
616C4—682no—
617C4—1361no—
618C4—1361no—
619C4C2×#1058755811yescharacteristic
620C4C4×#6325211yescharacteristic
621C4#1176334311yescharacteristic
622C4#1176334311yescharacteristic
623C2—44no—
624C2#1453122711yescharacteristic
625C2C4×#1058761811yescharacteristic
626C2C4×#1058763711yescharacteristic
627C2#1453888311yescharacteristic
628C2#1453889311yescharacteristic
629C2#1453892211yescharacteristic
630C2#1453894611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC42×D4×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC42×D4D17
hypocenterC17#21121
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #14531261C2semidirect
3. #14529425C2semidirect
4. #14529418C2semidirect
5. #14529376C2semidirect
6. #14529355C2semidirect
11. #14528682C2semidirect
12. #14528617C2semidirect
14. #14528533C2semidirect
18. #11762909C4semidirect
104. C4×Dic34C2×C4semidirect
120. #10587481C2×C4semidirect
123. #10587478C2×C4semidirect
125. #10587474C2×C4semidirect
441. C17#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
#145312612
#145294252
#145294182
#145293762
#145293552
#145291182
#145290452
#145290312
#145286822
#145286172
#145286092
#145285332
C42×D4×C172
#2112117