Finite Group Databasebeta

← #17042004

Subgroups — #17042004

Order 4352.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic
2C4×#10587556C211yesmaximal, characteristic, complemented
3#14529741C211yesmaximal, characteristic, complemented
4#14529552C211yesmaximal, characteristic, complemented
5#14529369C211yesmaximal, characteristic, complemented
6#14529355C211yesmaximal, characteristic, complemented
7#14529045C211yesmaximal, characteristic
8#14529031C211yesmaximal, characteristic
9#14528794C211yesmaximal, characteristic
10#14528682C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12#14528637C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14#14528609C211yesmaximal, characteristic
15C42×D4×C17C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17#11763109C418yescomplemented
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762693—21no—
33#11762693—21no—
34#11762675C2211yescharacteristic
35C4×#63225—21no—
36C4×#63225—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
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—
64C4×D4×C17—22no—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C4×D4×C17C2×C414yes—
69C4×D4×C17Q814yes—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17—44no—
80C2×C42×C17D411yescharacteristic
81C2×C42×C17D411yescharacteristic
82C2×C42×C17D411yescharacteristic
83C2×C42×C17D411yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587556—22no—
102#10587556C2×C414yescomplemented
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106#10587503—22no—
107#10587503—22no—
108#10587503C2×C414yes—
109#10587503C2×C414yes—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587479—22no—
117#10587479C2×C414yescomplemented
118#10587478—22no—
119#10587478C2×C414yescomplemented
120C4⋊#3622—22no—
121C4⋊#3622C2×C414yescomplemented
122#10587474—22no—
123#10587474—22no—
124C24×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17C22×C412yes—
129C2×D4×C17C2×Q812yes—
130C2×D4×C17#4112yes—
131C22×C4×C17—24no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—24no—
136C22×C4×C17—24no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17—42no—
142C22×C4×C17—42no—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17C22×C412yes—
158C4⋊C4×C17C2×Q812yes—
159C4⋊C4×C17#4112yes—
160C22⋊C4×C17—24no—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—24no—
163C42×C17—22no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—41no—
167C42×C17—41no—
168C42×C17—41no—
169C42×C17—42no—
170C42×C17—42no—
171C42×C17—42no—
172C42×C17C22×C412yes—
173C42×C17C2×Q812yes—
174C42×C17#4112yes—
175C42×C17#4112yes—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17C22×C412yes—
183C4⋊Dic17C22×C412yes—
184#63225—24no—
185#63225—24no—
186#63225—42no—
187#63225—42no—
188C4×Dic17—42no—
189C4×Dic17—42no—
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×#3622—21no—
199C2×#3622—21no—
200C2×#3622C22×C412yes—
201C2×#3622C22×C412yes—
202#21617—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17C22×D411yescharacteristic
211D4×C17—22no—
212D4×C17—22no—
213D4×C17—22no—
214D4×C17C4×Q814yes—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
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—22no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—48no—
232C2×C4×C17—48no—
233C2×C4×C17—48no—
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×C17C4×D412yes—
241C2×C4×C17C4×D412yes—
242C2×C4×C17C22⋊Q812yes—
243C2×C4×C17C22⋊Q812yes—
244C2×C4×C17AGL(1,ℤ/8)12yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C22×Q811yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263#3622—22no—
264#3622—22no—
265#3622—41no—
266#3622—41no—
267C2×#478—171no—
268C4×#125—171no—
269C42×D4D1711yescharacteristic
270#1989—171no—
271#1949—171no—
272#1920—171no—
273#1918—171no—
274#1864—171no—
275#1862—171no—
276#1804—171no—
277#1776—171no—
278#1776—171no—
279#1767—171no—
280#1762—171no—
281#1762—171no—
282C22×C17—24no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17C2×C4×D411yescharacteristic
290C22×C17C2×C22⋊Q811yescharacteristic
291C22×C17#52511yescharacteristic
292C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C17#57511yescharacteristic
294C22×C17#57511yescharacteristic
295C22×C17#57511yescharacteristic
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—42no—
304C4×C17—42no—
305C4×C17C2×C4×Q811yescharacteristic
306C4×C17C4×#4111yescharacteristic
307C4×C17#52911yescharacteristic
308C4×C17#53311yescharacteristic
309Dic17—42no—
310Dic17—42no—
311Dic17—81no—
312Dic17—81no—
313C2×#125—171no—
314C2×C4×D4C2×D1711yescharacteristic
315C2×C4×D4C2×D1711yescharacteristic
316C2×C4×D4C2×D1711yescharacteristic
317C22×C42—21no—
318#478—178no—
319C2×#107—171no—
320C2×#107—171no—
321C2×#106—171no—
322C2×#106—171no—
323C2×C4⋊C8—171no—
324C2×C4⋊C8—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102—171no—
328C2×#102—171no—
329C2×#97—171no—
330C2×#97—171no—
331#389—171no—
332#389—171no—
333#387—341no—
334#387—341no—
335#382—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—341no—
339C4×C4⋊C4—341no—
340C4×C4⋊C4C2×D1711yescharacteristic
341C4×C4⋊C4C2×D1711yescharacteristic
342C4×C4⋊C4C2×D1711yescharacteristic
343C4×C22⋊C4—21no—
344C4×C22⋊C4—21no—
345C4×C22⋊C4—21no—
346C43C2×D1711yescharacteristic
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#336—171no—
356#336—171no—
357C2×C17—44no—
358C2×C17C4×C22⋊Q811yescharacteristic
359C2×C17#294311yescharacteristic
360C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
361C2×C17#307711yescharacteristic
362C2×C17#308011yescharacteristic
363C2×C17#311111yescharacteristic
364C2×C17#311411yescharacteristic
365C22×D4C22×D1711yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C22×C8—171no—
370C22×C8—171no—
371C22×C8—171no—
372C22×C8—171no—
373#125—174no—
374#125—342no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4Dic3414yes—
380C4×D4C4×D1714yes—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—44no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42C22⋊D1711yescharacteristic
402C2×C42C22⋊D1711yescharacteristic
403C2×C42C22⋊D1711yescharacteristic
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—174no—
409#107—342no—
410#106—174no—
411#106—342no—
412C4⋊C8—174no—
413C4⋊C8—342no—
414#102—174no—
415#102—174no—
416#102—342no—
417#102—342no—
418#97—342no—
419#97—342no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429C17#2161711yescharacteristic, complemented, Sylow
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic3412yes—
435C2×D4C2×C4×D1712yes—
436C2×D4#6325212yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—172no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—172no—
476C4⋊C4—172no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—344no—
480C4⋊C4—344no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C4⋊C4C2×Dic3412yes—
488C4⋊C4C2×C4×D1712yes—
489C4⋊C4#6325212yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—682no—
503C42—682no—
504C42C2×Dic3412yes—
505C42C2×C4×D1712yes—
506C42#6325212yes—
507C42#6325212yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C22×C22⋊D1711yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic3414yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—342no—
546C2×C4—342no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4#1058761012yes—
558C2×C4#1058761012yes—
559C2×C4C4×C22⋊D1712yes—
560C2×C4C4×C22⋊D1712yes—
561C2×C4#1058761812yes—
562C2×C4C22×Dic3411yescharacteristic
563C2×C4C22×C4×D1711yescharacteristic
564C2×C4C2×#6325211yescharacteristic
565C2×C4C2×#6325211yescharacteristic
566C2×C4C2×#6325211yescharacteristic
567C2×C4C2×#6325211yescharacteristic
568C8—342no—
569C8—342no—
570C8—681no—
571C8—681no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D1711yescharacteristic
580C22C2×#1058761011yescharacteristic
581C22#1176332011yescharacteristic
582C22C2×#1058761811yescharacteristic
583C22#1176360711yescharacteristic
584C22#1176360711yescharacteristic
585C22#1176360711yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—682no—
596C4—682no—
597C4—1361no—
598C4—1361no—
599C4C2×C4×Dic3411yescharacteristic
600C4C4×#6325211yescharacteristic
601C4#1176334311yescharacteristic
602C4#1176336611yescharacteristic
603C2—44no—
604C2C4×#1058761011yescharacteristic
605C2#1453758211yescharacteristic
606C2C4×#1058761811yescharacteristic
607C2#1453858111yescharacteristic
608C2#1453860611yescharacteristic
609C2#1453889311yescharacteristic
610C2#1453892211yescharacteristic
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#21617
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. C4×#10587556C2semidirect
3. #14529741C2semidirect
4. #14529552C2semidirect
5. #14529369C2semidirect
6. #14529355C2semidirect
10. #14528682C2semidirect
11. #14528682C2semidirect
12. #14528637C2semidirect
17. #11763109C4semidirect
102. #10587556C2×C4semidirect
117. #10587479C2×C4semidirect
119. #10587478C2×C4semidirect
121. C4⋊#3622C2×C4semidirect
429. C17#21617semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
C4×#105875562
#145297412
#145295522
#145293692
#145293552
#145290452
#145290312
#145287942
#145286822
#145286372
#145286092
C42×D4×C172
#2161717