Finite Group Databasebeta

← #17039211

Subgroups — #17039211

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762864C211yesmaximal, characteristic
2#14531188C211yesmaximal, characteristic
3C4×C4⋊C4×D17C211yesmaximal, characteristic
4#14529546C211yesmaximal, characteristic, complemented
5#14529544C211yesmaximal, characteristic, complemented
6#14529541C211yesmaximal, characteristic, complemented
7#14529539C211yesmaximal, characteristic, complemented
8#14528978C211yesmaximal, characteristic
9#14528975C211yesmaximal, characteristic
10#14528973C211yesmaximal, characteristic
11#14528970C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13#14528682C211yesmaximal, characteristic, complemented
14#14528681C211yesmaximal, characteristic, complemented
15C17×#1776C211yesmaximal, characteristic, complemented
16C2×#10587558—21no—
17C2×C4⋊C4×D17C2211yescharacteristic
18C2×C4⋊C4×D17C2211yescharacteristic
19C2×C42×D17—21no—
20C2×C42×D17C2211yescharacteristic
21#11762864C418yescomplemented
22C2×#10587489C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587488C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×#10587485C2211yescharacteristic
30C2×#10587491C2211yescharacteristic
31C2×#10587491C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33#11762678C2211yescharacteristic
34#11762678C2211yescharacteristic
35#11762673—21no—
36#11762668—21no—
37#11762666C2211yescharacteristic
38#11762666C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×#63225—21no—
42C4×#63225—21no—
43C4×C4⋊Dic17C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762660—21no—
46#95×D17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×C4×D17—21no—
61C22×C4×D17—21no—
62C22×C4×D17D411yescharacteristic
63C22×C4×D17D411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×#63227—21no—
75C2×#63227—21no—
76C2×C4⋊Dic17C2311yescharacteristic
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
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×Dic17—21no—
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C4⋊C4×D17—22no—
96C4⋊C4×D17—22no—
97C4⋊C4×D17—22no—
98C4⋊C4×D17—22no—
99#10587558—44no—
100C42×D17—44no—
101C42×D17C2×C414yes—
102C42×D17Q814yes—
103C17×#107—22no—
104C17×#106—22no—
105C17×#97C2×C414yescomplemented
106C17×#95—21no—
107C17×#95—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587491C2×C414yes—
115#10587491C2×C414yes—
116#10587489—22no—
117#10587488—22no—
118#10587485C2×C414yescomplemented
119#10587485C2×C414yescomplemented
120#10587479—22no—
121#10587479—22no—
122#10587478—22no—
123#10587478—22no—
124#10587474C2×C414yescomplemented
125C23×D17C2×D411yescharacteristic
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C2×C4×D17—22no—
141C2×C4×D17—22no—
142C2×C4×D17—21no—
143C2×C4×D17—21no—
144C2×C4×D17—21no—
145C2×C4×D17—21no—
146C2×C4×D17—22no—
147C2×C4×D17—22no—
148C2×C4×D17—24no—
149C2×C4×D17—24no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C4×D17—42no—
154C2×C4×D17C4⋊C414yes—
155C2×C4×D17C4⋊C414yes—
156C2×C4×D17C22×C412yes—
157C2×C4×D17C2×Q812yes—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C22×C412yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17#4112yes—
171C4⋊C4×C17#4112yes—
172C42×C17—42no—
173C42×C17—42no—
174C42×C17C22×C412yes—
175C42×C17C2×Q812yes—
176#63227—42no—
177#63227—42no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17#4112yes—
185C4⋊Dic17#4112yes—
186#63225—24no—
187#63225—24no—
188#63225—42no—
189#63225—42no—
190#63225—42no—
191#63225—42no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C4×Dic17—42no—
195C4×Dic17—42no—
196C4×Dic17C4⋊C414yes—
197C4×Dic17C4⋊C414yes—
198C4×Dic17C22×C412yes—
199C4×Dic17C2×Q812yes—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622C22×C412yes—
205C2×#3622C22×C412yes—
206#21365—171nomaximal, Sylow
207C23×C17C22×D411yescharacteristic
208C22×D17—21no—
209C22×D17—21no—
210C22×D17—21no—
211C22×D17—21no—
212C22×D17C2×C4⋊C412yes—
213C22×D17AGL(1,ℤ/8)12yes—
214C22×D17#13712yes—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17C2×C4⋊C412yes—
232C2×C4×C17C2×C4⋊C412yes—
233C2×C4×C17C23×C411yescharacteristic
234C2×C4×C17C22×Q811yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17C2×C4⋊C412yes—
256C2×Dic17C2×C4⋊C412yes—
257C2×Dic17C2×C4⋊C412yes—
258C2×Dic17AGL(1,ℤ/8)12yes—
259C2×Dic17#13712yes—
260C4×D17—28no—
261C4×D17—22no—
262C4×D17—22no—
263C4×D17—22no—
264C4×D17—22no—
265C4×D17—28no—
266C4×D17—48no—
267C4×D17—48no—
268C8×C17—22no—
269C8×C17—22no—
270#3622—22no—
271#3622—22no—
272C2×#432—171no—
273#2285—171no—
274C2×C4×C4⋊C4—171no—
275#1948—171no—
276#1948—171no—
277#1947—171no—
278#1947—171no—
279#1853—171no—
280#1853—171no—
281#1852—171no—
282#1852—171no—
283#1776—171no—
284#1776—171no—
285#1776—171no—
286#1776D1711yescharacteristic, complemented
287C22×C17C22×C4⋊C411yescharacteristic
288C22×C17#52511yescharacteristic
289C22×C17#52511yescharacteristic
290C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C17C2×#13711yescharacteristic
293C22×C17C2×#13711yescharacteristic
294C2×D17—22no—
295C2×D17—22no—
296C2×D17—22no—
297C2×D17—22no—
298C2×D17—22no—
299C2×D17—22no—
300C2×D17#42814yes—
301C4×C17—22no—
302C4×C17—22no—
303C4×C17—42no—
304C4×C17—42no—
305C4×C17—81no—
306C4×C17—81no—
307C4×C17C4×#4111yescharacteristic
308C4×C17C4×#4111yescharacteristic
309C4×C17#53311yescharacteristic
310C4×C17#53311yescharacteristic
311Dic17—22no—
312Dic17—22no—
313Dic17—42no—
314Dic17—42no—
315Dic17—81no—
316Dic17—81no—
317Dic17#42814yes—
318C2×#117—341no—
319C22×C4⋊C4—171no—
320C22×C4⋊C4—171no—
321C22×C42—171no—
322C22×C42—341no—
323#432—178no—
324C2×#107—171no—
325C2×#107—171no—
326C2×#107—171no—
327C2×#107C2×D1711yescharacteristic
328C2×#106—171no—
329C2×#106—171no—
330C2×#106—171no—
331C2×#106C2×D1711yescharacteristic
332C2×C4⋊C8—171no—
333C2×C4⋊C8—171no—
334C2×C22⋊C8—171no—
335C2×C22⋊C8—171no—
336C2×#97—171no—
337C2×#97C2×D1711yescharacteristic
338#384—171no—
339#384—171no—
340#382—341no—
341#380—171no—
342#380—171no—
343#380—341no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—171no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—341no—
349C4×C4⋊C4—341no—
350C4×C4⋊C4—341no—
351C4×C4⋊C4C2×D1711yescharacteristic
352C4×C4⋊C4C2×D1711yescharacteristic
353C2×#95—341no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340C2×D1711yescharacteristic
361#340C2×D1711yescharacteristic
362C2×C17#229611yescharacteristic
363C2×C17C2×#42811yescharacteristic
364C2×C17C2×#42811yescharacteristic
365C2×C17#310911yescharacteristic
366C2×C17#311011yescharacteristic
367C2×C17#311211yescharacteristic
368C2×C17#311311yescharacteristic
369D17—24no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4—341no—
374C23×C4—341no—
375C22×C8—171no—
376C22×C8—171no—
377C22×C8C22×D1711yescharacteristic
378C22×C8C22×D1711yescharacteristic
379#117—684no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—342no—
385C2×C4⋊C4—342no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C4⋊C4C22×D1711yescharacteristic
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—171no—
403C2×C42—171no—
404C2×C42—171no—
405C2×C42—174no—
406C2×C42—174no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42—684no—
412C2×C42C22×D1711yescharacteristic
413#107—22no—
414#107—342no—
415#107—342no—
416#107—342no—
417#106—22no—
418#106—342no—
419#106—342no—
420#106—342no—
421C4⋊C8—174no—
422C4⋊C8—174no—
423C22⋊C8—174no—
424C22⋊C8—174no—
425#97—174no—
426#97C4×D1714yescomplemented
427#95—21no—
428#95—21no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435C17#2136511yescharacteristic, complemented, Sylow
436C24—171no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—172no—
445C22×C4—172no—
446C22×C4—174no—
447C22×C4—174no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—344no—
453C22×C4—344no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4D4×D1711yescharacteristic
467C22×C4D4×D1711yescharacteristic
468C22×C4C23×D1711yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—172no—
474C2×C8—172no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8C2×C4×D1712yes—
480C2×C8C2×C4×D1712yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—172no—
488C4⋊C4—172no—
489C4⋊C4—344no—
490C4⋊C4—344no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4—682no—
500C4⋊C4—682no—
501C4⋊C4#6325212yes—
502C4⋊C4#6325212yes—
503C22⋊C4—682no—
504C22⋊C4—682no—
505C42—42no—
506C42—42no—
507C42—172no—
508C42—172no—
509C42—174no—
510C42—174no—
511C42—682no—
512C42—682no—
513C42—682no—
514C42—682no—
515C42C2×C4×D1712yes—
516C42Q8×D1712yes—
517C23—172no—
518C23—172no—
519C23—172no—
520C23—341no—
521C23—341no—
522C23—341no—
523C23—341no—
524C23C2×D4×D1711yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—172no—
542C2×C4—172no—
543C2×C4—172no—
544C2×C4—172no—
545C2×C4—172no—
546C2×C4—348no—
547C2×C4—348no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—342no—
558C2×C4—688no—
559C2×C4—688no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4C4⋊C4×D1712yes—
571C2×C4C4⋊C4×D1712yes—
572C2×C4C22×C4×D1711yescharacteristic
573C2×C4C2×#6325211yescharacteristic
574C2×C4C2×#6325211yescharacteristic
575C2×C4C2×Q8×D1711yescharacteristic
576C2×C4#41×D1711yescharacteristic
577C2×C4#41×D1711yescharacteristic
578C8—22no—
579C8—22no—
580C8—342no—
581C8—342no—
582C22—174no—
583C22—342no—
584C22—342no—
585C22—342no—
586C22—342no—
587C22—342no—
588C22—342no—
589C22C2×C4⋊C4×D1711yescharacteristic
590C22#1176331811yescharacteristic
591C22#1176331811yescharacteristic
592C22AGL(1,ℤ/8)×D1711yescharacteristic
593C22AGL(1,ℤ/8)×D1711yescharacteristic
594C22#137×D1711yescharacteristic
595C22#137×D1711yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—174no—
603C4—342no—
604C4—342no—
605C4—682no—
606C4—682no—
607C4—1361no—
608C4—1361no—
609C4#1176327711yescharacteristic
610C4C4×#6325211yescharacteristic
611C4#1176336311yescharacteristic
612C4#1176336311yescharacteristic
613C2—344no—
614C2#1453126411yescharacteristic
615C2D17×#42811yescharacteristic
616C2D17×#42811yescharacteristic
617C2#1453887311yescharacteristic
618C2#1453888111yescharacteristic
619C2#1453890411yescharacteristic
620C2#1453891211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1776C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1776D17
hypocenterC17#21365
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #14529546C2semidirect
5. #14529544C2semidirect
6. #14529541C2semidirect
7. #14529539C2semidirect
12. #14528682C2semidirect
13. #14528682C2semidirect
14. #14528681C2semidirect
15. C17×#1776C2semidirect
21. #11762864C4semidirect
105. C17×#97C2×C4semidirect
118. #10587485C2×C4semidirect
119. #10587485C2×C4semidirect
124. #10587474C2×C4semidirect
286. #1776D17semidirect
426. #97C4×D17semidirect
435. C17#21365semidirect

Maximal subgroups

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

subgroupindex
C2×#117628642
#145311882
C4×C4⋊C4×D172
#145295462
#145295442
#145295412
#145295392
#145289782
#145289752
#145289732
#145289702
#145286822
#145286812
C17×#17762
#2136517