Finite Group Databasebeta

← #17033909

Subgroups — #17033909

Order 4352.

1590 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D17C211yesmaximal, characteristic
2C2×#11762852C211yesmaximal, characteristic, complemented
3C2×#11762849C211yesmaximal, characteristic, complemented
4#14531163C211yesmaximal, characteristic
5C2×#11762666C211yesmaximal, characteristic
6#14528816C212yesmaximal, complemented
7#14528816C212yesmaximal, complemented
8#14528812C212yesmaximal, complemented
9#14528808C212yesmaximal, complemented
10#14528504C211yesmaximal, characteristic, complemented
11C17×#1736C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C22×#63218C2212yes—
15C2×M16×D17—24no—
16C2×M16×D17—24no—
17C2×M16×D17C418yescomplemented
18C22×C8×D17C2212yes—
19C2×#10587612C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21C22×#63227—21no—
22#11762852—24no—
23#11762849—24no—
24C2×#10587478C2212yescomplemented
25C2×#10587478C2212yescomplemented
26C2×#10587479C2212yescomplemented
27C2×#10587479C2212yescomplemented
28#11762670—21no—
29#11762666—21no—
30#11762666—21no—
31#11762666C2211yescharacteristic
32#11762666C2212yes—
33#11762666C2211yescharacteristic
34#11762666C2212yes—
35#11762666C2211yescharacteristic
36#11762666C2211yescharacteristic
37#11762664—21no—
38C4×#63227—21no—
39C4×#63227—21no—
40C2×#10587502—21no—
41#11762572C2212yescomplemented
42#11762572C2212yescomplemented
43C22×M16×C17C2211yescharacteristic
44C2×C17×#117C2211yescharacteristic
45C22×C4⋊C4×C17C2211yescharacteristic
46C17×#340C2212yescomplemented
47C17×#340C2212yescomplemented
48C24×D17D411yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17C2311yescharacteristic
51C23×Dic17—21no—
52C23×Dic17D411yescharacteristic
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17—22no—
57C22×C4×D17C2×C414yes—
58C22×C4×D17D411yescharacteristic
59C22×C4×D17D411yescharacteristic
60C22×C4×D17Q814yes—
61C22×C4×D17C2311yescharacteristic
62C22×C4×D17C2311yescharacteristic
63C2×M16×C17—22no—
64C2×M16×C17—22no—
65C2×M16×C17C2×C414yes—
66C22×C8×C17C2312yes—
67C17×#117—22no—
68C17×#117—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2312yes—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C42×C17—21no—
77C2×C22⋊Dic17—21no—
78C2×#63227—22no—
79C2×#63227—22no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—44no—
85#10587612—22no—
86#10587612—22no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—22no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17C2312yes—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4×Dic17—21no—
94C2×#63223C2×C414yes—
95C2×#63223D414yes—
96C2×#63223D414yes—
97C22×#3622C2312yes—
98M16×D17—232no—
99C2×#63218—24no—
100C2×#63218—24no—
101C2×#63218C2×C418yescomplemented
102C2×C8×D17—24no—
103C2×C8×D17—24no—
104C2×C8×D17C2×C418yescomplemented
105#10587502—22no—
106#10587502—22no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587479—24no—
112#10587479—24no—
113#10587478—24no—
114#10587478—24no—
115C24×C17C2×D411yescharacteristic
116C23×D17—22no—
117C23×D17—22no—
118C23×D17—22no—
119C23×D17—22no—
120C23×D17—41no—
121C23×D17—41no—
122C23×D17C4⋊C414yes—
123C23×D17C2×D411yescharacteristic
124C23×D17C2×D411yescharacteristic
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—42no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×Q812yes—
136C22×C4×C17#4112yes—
137C22×C4×C17#4112yes—
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—42no—
145C22×Dic17—42no—
146C22×Dic17—42no—
147C22×Dic17C4⋊C414yes—
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C2×C4×D17—28no—
151C2×C4×D17—28no—
152C2×C4×D17—22no—
153C2×C4×D17—22no—
154C2×C4×D17—22no—
155C2×C4×D17—22no—
156C2×C4×D17—28no—
157C2×C4×D17—28no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17C4⋊C414yes—
167C2×C4×D17C4⋊C414yes—
168C2×C4×D17C22×C412yes—
169C2×C4×D17C22×C412yes—
170C2×C4×D17C2×Q812yes—
171C2×C4×D17C2×Q812yes—
172M16×C17—28no—
173C2×C8×C17—22no—
174C2×C8×C17—22no—
175C2×C8×C17C22×C414yes—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—22no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—42no—
183C22⋊C4×C17—42no—
184C42×C17—42no—
185C22⋊Dic17—42no—
186#63227—42no—
187#63227—42no—
188#63227—44no—
189#63227—42no—
190#63227—44no—
191#63227—42no—
192C4⋊Dic17—21no—
193C4⋊Dic17—22no—
194C4⋊Dic17—21no—
195C4⋊Dic17—22no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—42no—
199C4×Dic17—42no—
200#63223C22⋊C4116yescomplemented
201C2×#3622C22×C414yes—
202C2×#3622C2×D414yes—
203C2×#3622C2×D414yes—
204#63218—216no—
205C8×D17—216no—
206#20661—171nomaximal, Sylow
207C23×C17—21no—
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17C2×C4⋊C412yes—
212C23×C17#12112yes—
213C23×C17#12112yes—
214C23×C17C22×D411yescharacteristic
215C22×D17—21no—
216C22×D17—21no—
217C22×D17—21no—
218C22×D17—21no—
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—28no—
226C22×D17—28no—
227C22×D17—44no—
228C22×D17—44no—
229C22×D17—44no—
230C22×D17—44no—
231C22×D17—44no—
232C22×D17C2×C4⋊C412yes—
233C22×D17C2×C4⋊C412yes—
234C22×D17AGL(1,ℤ/8)12yes—
235C22×D17#13712yes—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—42no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—84no—
246C2×C4×C17C2×C22⋊C414yes—
247C2×C4×C17C2×C4⋊C412yes—
248C2×C4×C17#11714yes—
249C2×C4×C17#12112yes—
250C2×C4×C17#12112yes—
251C2×C4×C17C22⋊Q814yes—
252C2×C4×C17C22⋊Q814yes—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C22×Q811yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—28no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—42no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—84no—
274C2×Dic17C2×C4⋊C412yes—
275C2×Dic17C2×C4⋊C412yes—
276C4×D17—22no—
277C4×D17—22no—
278C4×D17—22no—
279C4×D17—22no—
280C4×D17—22no—
281C4×D17—22no—
282C4×D17—22no—
283C4×D17—22no—
284C4×D17—28no—
285C4×D17—28no—
286C8×C17—24no—
287#3622C2×C22⋊C418yescomplemented
288C23×M16—171no—
289C2×#428—171no—
290C2×#428—171no—
291#2280—171no—
292C2×#380—171no—
293#1809—172no—
294#1809—172no—
295#1808—172no—
296#1807—172no—
297#1736—171no—
298#1736D1711yescharacteristic, complemented
299C22×C17—22no—
300C22×C17—22no—
301C22×C17—22no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—22no—
305C22×C17#38014yes—
306C22×C17C22×C4⋊C411yescharacteristic
307C22×C17C2×#12111yescharacteristic
308C22×C17C2×#12111yescharacteristic
309C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C17C2×#13711yescharacteristic
311C22×C17#57511yescharacteristic
312C22×C17#57511yescharacteristic
313C2×D17—22no—
314C2×D17—22no—
315C2×D17—22no—
316C2×D17—22no—
317C2×D17—22no—
318C2×D17—22no—
319C2×D17—22no—
320C2×D17—22no—
321C2×D17—22no—
322C2×D17—416no—
323C2×D17—41no—
324C2×D17—41no—
325C2×D17—41no—
326C2×D17—41no—
327C2×D17#42814yescomplemented
328C4×C17—81no—
329C4×C17—82no—
330C4×C17—81no—
331C4×C17#38014yes—
332C4×C17C22×C22⋊C411yescharacteristic
333C4×C17C2×#11711yescharacteristic
334C4×C17C2×C22⋊Q811yescharacteristic
335C4×C17C2×C22⋊Q811yescharacteristic
336Dic17—22no—
337Dic17—22no—
338Dic17—81no—
339Dic17—82no—
340Dic17—81no—
341C24×C4—171no—
342C22×M16—171no—
343C22×M16—178no—
344C22×M16—172no—
345C22×M16—344no—
346C22×M16—344no—
347C22×M16C2×D1711yescharacteristic
348C23×C8—172no—
349C2×#117—171no—
350C2×#117C2×D1711yescharacteristic
351C22×C4⋊C4—171no—
352C22×C4⋊C4C2×D1711yescharacteristic
353C22×C22⋊C4—341no—
354#428—344no—
355#428—344no—
356C2×#107—172no—
357C2×#107—172no—
358C2×#106—172no—
359C2×#106—172no—
360#381—341no—
361#380—171no—
362#380—172no—
363#380—171no—
364#380—172no—
365#380—171no—
366#380—171no—
367#380—341no—
368#380—341no—
369#379—341no—
370C4×C22⋊C4—341no—
371C4×C22⋊C4—341no—
372C2×#95—341no—
373#340—172no—
374#340—172no—
375#340C2×D1712yescomplemented
376#340C2×D1712yescomplemented
377C2×C17—24no—
378C2×C17C2×#38011yescharacteristic
379C2×C17C2×#42811yescharacteristic
380C2×C17#291111yescharacteristic
381C2×C17#305511yescharacteristic
382C2×C17#305511yescharacteristic
383C2×C17#305611yescharacteristic
384C2×C17#305611yescharacteristic
385D17—24no—
386D17—42no—
387C25—171no—
388C23×C4—21no—
389C23×C4—171no—
390C23×C4—174no—
391C23×C4—171no—
392C23×C4—171no—
393C23×C4—174no—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4—342no—
397C23×C4—342no—
398C23×C4—342no—
399C23×C4—342no—
400C23×C4—341no—
401C23×C4C22×D1711yescharacteristic
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—174no—
405C2×M16—174no—
406C2×M16—174no—
407C2×M16—178no—
408C2×M16—344no—
409C2×M16—344no—
410C2×M16—3432no—
411C2×M16C4×D1714yes—
412C22×C8—172no—
413C22×C8—178no—
414C22×C8—344no—
415C22×C8—344no—
416C22×C8C22×D1712yes—
417#117—22no—
418#117—22no—
419#117—342no—
420#117—342no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—172no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—342no—
428C2×C4⋊C4—342no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22×D1712yes—
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4—342no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—684no—
442C2×C42—21no—
443C2×C42—341no—
444#107—344no—
445#107—344no—
446#106—344no—
447#106—344no—
448#95—342no—
449#95—342no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454C17#2066111yescharacteristic, complemented, Sylow
455C24—171no—
456C24—171no—
457C24—174no—
458C24—342no—
459C24—342no—
460C24—342no—
461C24—342no—
462C24—681no—
463C24—681no—
464C24D4×D1711yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—174no—
474C22×C4—172no—
475C22×C4—172no—
476C22×C4—172no—
477C22×C4—172no—
478C22×C4—174no—
479C22×C4—174no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—348no—
491C22×C4—348no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—348no—
497C22×C4—348no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—342no—
501C22×C4—682no—
502C22×C4—682no—
503C22×C4—682no—
504C22×C4C2×C4×D1712yes—
505C22×C4#6325212yes—
506C22×C4#6325212yes—
507C22×C4D4×D1712yes—
508C22×C4D4×D1712yes—
509C22×C4D4×D1711yescharacteristic
510C22×C4Q8×D1712yes—
511C22×C4C23×D1711yescharacteristic
512M16—28no—
513M16—1716no—
514M16—3416no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—174no—
518C2×C8—174no—
519C2×C8—174no—
520C2×C8—3416no—
521C2×C8C2×C4×D1714yes—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—342no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—342no—
534C4⋊C4—341no—
535C4⋊C4—682no—
536C22⋊C4—42no—
537C22⋊C4—684no—
538C22⋊C4—682no—
539C22⋊C4—682no—
540C22⋊C4—684no—
541C22⋊C4—682no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C42—42no—
545C42—682no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—172no—
551C23—172no—
552C23—172no—
553C23—172no—
554C23—348no—
555C23—341no—
556C23—341no—
557C23—341no—
558C23—341no—
559C23—341no—
560C23—341no—
561C23—341no—
562C23—341no—
563C23—341no—
564C23—341no—
565C23—348no—
566C23—684no—
567C23—684no—
568C23—684no—
569C23—684no—
570C23—684no—
571C23#1058757012yes—
572C23#1058757012yes—
573C23C4⋊C4×D1712yes—
574C23C2×D4×D1711yescharacteristic
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—172no—
586C2×C4—172no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—348no—
592C2×C4—342no—
593C2×C4—342no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—348no—
601C2×C4—348no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—682no—
611C2×C4—1364no—
612C2×C4#1058755814yes—
613C2×C4C22⋊C4×D1714yes—
614C2×C4#1058757012yes—
615C2×C4#1058757012yes—
616C2×C4C4⋊C4×D1712yes—
617C2×C4#1058758214yes—
618C2×C4#1058758214yes—
619C2×C4C22×C4×D1711yescharacteristic
620C2×C4C2×#6325211yescharacteristic
621C2×C4C2×#6325211yescharacteristic
622C2×C4C2×D4×D1711yescharacteristic
623C2×C4C2×D4×D1711yescharacteristic
624C2×C4C2×Q8×D1711yescharacteristic
625C8—24no—
626C8—178no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—174no—
634C22—342no—
635C22—342no—
636C22—342no—
637C22—342no—
638C22—342no—
639C22—342no—
640C22—342no—
641C22—342no—
642C22—342no—
643C22—6816no—
644C22—681no—
645C22—681no—
646C22—681no—
647C22—681no—
648C22#1176266814yes—
649C22C2×C4⋊C4×D1711yescharacteristic
650C22C2×#1058757011yescharacteristic
651C22C2×#1058757011yescharacteristic
652C22AGL(1,ℤ/8)×D1711yescharacteristic
653C22#137×D1711yescharacteristic
654C22#1176360511yescharacteristic
655C22#1176360811yescharacteristic
656C4—82no—
657C4—81no—
658C4—81no—
659C4—342no—
660C4—342no—
661C4—1361no—
662C4—1362no—
663C4—1361no—
664C4#1176266814yes—
665C4C2×C22⋊C4×D1711yescharacteristic
666C4C2×#1058755811yescharacteristic
667C4C2×#1058758211yescharacteristic
668C4C2×#1058758211yescharacteristic
669C2—24no—
670C2—344no—
671C2—682no—
672C2C2×#1176266811yescharacteristic
673C2D17×#42811yescharacteristic
674C2#1453737111yescharacteristic
675C2#1453840111yescharacteristic
676C2#1453840611yescharacteristic
677C2#1453841211yescharacteristic
678C2#1453841711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1736C2
socleC23×C17C22×D4
prime radicalC24×D17D4
cosocleC22×C4×C17C24
hypercenter#1736D17
hypocenterC17#20661
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#11762852C2semidirect
3. C2×#11762849C2semidirect
6. #14528816C2semidirect
7. #14528816C2semidirect
8. #14528812C2semidirect
9. #14528808C2semidirect
10. #14528504C2semidirect
11. C17×#1736C2semidirect
17. C2×M16×D17C4semidirect
24. C2×#10587478C22semidirect
25. C2×#10587478C22semidirect
26. C2×#10587479C22semidirect
27. C2×#10587479C22semidirect
41. #11762572C22semidirect
42. #11762572C22semidirect
46. C17×#340C22semidirect
47. C17×#340C22semidirect
101. C2×#63218C2×C4semidirect
104. C2×C8×D17C2×C4semidirect
200. #63223C22⋊C4semidirect
287. #3622C2×C22⋊C4semidirect
298. #1736D17semidirect
327. C2×D17#428semidirect
375. #340C2×D17semidirect
376. #340C2×D17semidirect
454. C17#20661semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D172
C2×#117628522
C2×#117628492
#145311632
C2×#117626662
#145288162
#145288122
#145288082
#145285042
C17×#17362
#2066117