Finite Group Databasebeta

← #17034155

Subgroups — #17034155

Order 4352.

1470 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D17C211yesmaximal, characteristic
2C2×#11762772C211yesmaximal, characteristic
3#14531163C211yesmaximal, characteristic
4#14528882C218yesmaximal, complemented
5#14528495C211yesmaximal, characteristic, complemented
6#14528494C211yesmaximal, characteristic, complemented
7#14528493C211yesmaximal, characteristic, complemented
8C17×#1733C211yesmaximal, characteristic, complemented
9C23×C4×D17C2211yescharacteristic
10C22×#63223C2211yescharacteristic
11C2×#10587612C2211yescharacteristic
12C2×#10587612C2211yescharacteristic
13C2×#10587578C412yescomplemented
14C2×#10587558—21no—
15C2×#10587563—21no—
16#117×D17—22no—
17#117×D17—22no—
18#117×D17C414yescomplemented
19#117×D17C2214yes—
20C2×C4⋊C4×D17C412yescomplemented
21C2×C22⋊C4×D17—21no—
22C2×C42×D17—21no—
23#11762772—21no—
24#11762772—21no—
25#11762772C2214yes—
26C2×#10587491—22no—
27#11762670—21no—
28#11762666—22no—
29#11762664—21no—
30C4×#63227C2214yes—
31#11762570C2214yescomplemented
32#11762569C2214yescomplemented
33#11762568C2214yescomplemented
34C22×M16×C17C2211yescharacteristic
35C2×C17×#117C2211yescharacteristic
36C2×C17×#117C2211yescharacteristic
37C17×#339C2214yescomplemented
38C24×D17D411yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×Dic17D411yescharacteristic
41C22×C4×D17—22no—
42C22×C4×D17—22no—
43C22×C4×D17—22no—
44C22×C4×D17—22no—
45C22×C4×D17—22no—
46C22×C4×D17C2×C411yescharacteristic
47C22×C4×D17C2×C412yes—
48C22×C4×D17C2×C411yescharacteristic
49C22×C4×D17D412yes—
50C22×C4×D17D411yescharacteristic
51C22×C4×D17D412yes—
52C22×C4×D17D411yescharacteristic
53C22×C4×D17C2312yes—
54C2×M16×C17—21no—
55C2×M16×C17—22no—
56C2×M16×C17—22no—
57C2×M16×C17C2312yes—
58C22×C8×C17—21no—
59C17×#117—21no—
60C17×#117—21no—
61C17×#117—22no—
62C17×#117—22no—
63C17×#117C2×C412yes—
64C17×#117C2312yes—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2×C412yes—
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C42×C17—21no—
70C2×C42×C17C2312yes—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×#63227—22no—
74C2×#63227—24no—
75#10587612—21no—
76#10587612—21no—
77#10587612—22no—
78#10587612—22no—
79#10587612C2×C412yes—
80#10587612C2312yes—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2×C412yes—
83C2×#63225—22no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17C2312yes—
87C2×#63223—21no—
88C2×#63223—22no—
89C2×#63223—22no—
90C2×#63223C2312yes—
91C22×#3622—21no—
92#10587578—24no—
93#10587578C2×C418yes—
94C4⋊C4×D17—22no—
95C4⋊C4×D17—22no—
96C4⋊C4×D17C2×C414yes—
97C4⋊C4×D17C2×C414yes—
98C22⋊C4×D17—24no—
99C22⋊C4×D17—24no—
100#10587563—24no—
101#10587563—24no—
102#10587558—24no—
103#10587558—24no—
104C42×D17—24no—
105C42×D17—24no—
106#10587502—24no—
107#10587491—24no—
108#10587491—24no—
109C24×C17C2×D411yescharacteristic
110C23×D17—22no—
111C23×D17—22no—
112C23×D17—44no—
113C23×D17C22⋊C411yescharacteristic
114C23×D17C22⋊C412yes—
115C23×D17C22⋊C411yescharacteristic
116C23×D17C2×D412yes—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17C22×C411yescharacteristic
120C22×C4×C17C22×C411yescharacteristic
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17#4112yes—
125C22×C4×C17#4112yes—
126C22×C4×C17#4112yes—
127C22×C4×C17#4112yes—
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17C22⋊C412yes—
134C22×Dic17C22⋊C411yescharacteristic
135C22×Dic17C22⋊C411yescharacteristic
136C22×Dic17C2×D412yes—
137C2×C4×D17—22no—
138C2×C4×D17—24no—
139C2×C4×D17—22no—
140C2×C4×D17—22no—
141C2×C4×D17—24no—
142C2×C4×D17—22no—
143C2×C4×D17—24no—
144C2×C4×D17—22no—
145C2×C4×D17—22no—
146C2×C4×D17—24no—
147C2×C4×D17—24no—
148C2×C4×D17—22no—
149C2×C4×D17—22no—
150C2×C4×D17—24no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—22no—
154C2×C4×D17—22no—
155C2×C4×D17—24no—
156C2×C4×D17—24no—
157C2×C4×D17—48no—
158C2×C4×D17C22⋊C412yes—
159C2×C4×D17C22⋊C412yes—
160C2×C4×D17C22⋊C412yes—
161C2×C4×D17C22⋊C412yes—
162C2×C4×D17C22×C412yes—
163C2×C4×D17C22×C412yes—
164C2×C4×D17C2×D412yes—
165C2×C4×D17C2×D412yes—
166M16×C17—22no—
167M16×C17—22no—
168M16×C17—42no—
169C2×C8×C17—22no—
170C2×C8×C17—42no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17C22×C414yes—
174C22⋊C4×C17—22no—
175C22⋊C4×C17—22no—
176C22⋊C4×C17—42no—
177C42×C17—22no—
178C42×C17—22no—
179C42×C17—22no—
180C42×C17—22no—
181C22⋊Dic17—22no—
182C22⋊Dic17—22no—
183C22⋊Dic17—42no—
184#63227—28no—
185#63227—48no—
186C4⋊Dic17—22no—
187C4⋊Dic17—42no—
188C4⋊Dic17C22×C414yes—
189#63225—24no—
190#63225—24no—
191C4×Dic17—22no—
192C4×Dic17—22no—
193C4×Dic17—28no—
194C4×Dic17—22no—
195C4×Dic17—22no—
196#63223—22no—
197#63223—22no—
198#63223—42no—
199C2×#3622—22no—
200C2×#3622—42no—
201#20705—171nomaximal, Sylow
202C23×C17—42no—
203C23×C17C2×C22⋊C411yescharacteristic
204C23×C17C2×C22⋊C411yescharacteristic
205C23×C17#12312yes—
206C23×C17#12312yes—
207C23×C17C22×D411yescharacteristic
208C22×D17—22no—
209C22×D17—24no—
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—24no—
213C22×D17—22no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17—48no—
220C22×D17—48no—
221C22×D17C2×C22⋊C412yes—
222C22×D17C2×C22⋊C412yes—
223C2×C4×C17—24no—
224C2×C4×C17—44no—
225C2×C4×C17—44no—
226C2×C4×C17—42no—
227C2×C4×C17C2×C22⋊C411yescharacteristic
228C2×C4×C17C2×C22⋊C411yescharacteristic
229C2×C4×C17#11712yes—
230C2×C4×C17#11712yes—
231C2×C4×C17#11712yes—
232C2×C4×C17#11712yes—
233C2×C4×C17#12312yes—
234C2×C4×C17#12312yes—
235C2×C4×C17#12312yes—
236C2×C4×C17#12312yes—
237C2×C4×C17#12312yes—
238C2×C4×C17#12312yes—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—24no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—44no—
252C2×Dic17—44no—
253C2×Dic17—42no—
254C2×Dic17C2×C22⋊C412yes—
255C2×Dic17C2×C22⋊C412yes—
256C4×D17—22no—
257C4×D17—22no—
258C4×D17—22no—
259C4×D17—22no—
260C4×D17—22no—
261C4×D17—22no—
262C4×D17—22no—
263C4×D17—22no—
264C4×D17—24no—
265C4×D17—24no—
266C4×D17—24no—
267C4×D17—24no—
268C4×D17—48no—
269C4×D17—48no—
270C8×C17—42no—
271#3622—42no—
272C22×#117—171no—
273C2×#407—171no—
274#2280—171no—
275#1826—178no—
276#1733—171no—
277#1733—171no—
278#1733—171no—
279#1733D1711yescharacteristic, complemented
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17#38112yes—
286C22×C17#38112yes—
287C22×C17C22×C22⋊C411yescharacteristic
288C22×C17C2×#12311yescharacteristic
289C22×C17C2×#12311yescharacteristic
290C2×D17—28no—
291C2×D17—22no—
292C2×D17—22no—
293C2×D17—22no—
294C2×D17—22no—
295C2×D17—22no—
296C2×D17—44no—
297C2×D17—44no—
298C2×D17—48no—
299C2×D17—44no—
300C2×D17#42114yes—
301C4×C17—44no—
302C4×C17—82no—
303C4×C17#38112yes—
304C4×C17#38112yes—
305C4×C17C2×#11711yescharacteristic
306C4×C17C2×#11711yescharacteristic
307C4×C17C2×#12311yescharacteristic
308C4×C17C2×#12311yescharacteristic
309Dic17—22no—
310Dic17—22no—
311Dic17—44no—
312Dic17—82no—
313C24×C4—171no—
314C22×M16—171no—
315C22×M16C2×D1711yescharacteristic
316C2×#117—171no—
317C2×#117—171no—
318C2×#117—174no—
319C2×#117—174no—
320C2×#117—172no—
321C2×#117—341no—
322C2×#117—341no—
323C2×#117—342no—
324C2×#117—342no—
325C2×#117C2×D1711yescharacteristic
326C2×#117C2×D1711yescharacteristic
327C22×C4⋊C4—172no—
328C22×C22⋊C4—341no—
329C22×C42—341no—
330#407—174no—
331#407—341no—
332#407—341no—
333C2×C22⋊C8—342no—
334#381—341no—
335#380—342no—
336#379—341no—
337C4×C22⋊C4—174no—
338#339—174no—
339#339—174no—
340#339—174no—
341#339C2×D1714yescomplemented
342C2×C17—21no—
343C2×C17—21no—
344C2×C17—22no—
345C2×C17—22no—
346C2×C17C2×#38111yescharacteristic
347C2×C17C2×#42112yes—
348D17—24no—
349D17—42no—
350C25—171no—
351C23×C4—171no—
352C23×C4—171no—
353C23×C4—171no—
354C23×C4—172no—
355C23×C4—172no—
356C23×C4—172no—
357C23×C4—171no—
358C23×C4—171no—
359C23×C4—172no—
360C23×C4—342no—
361C23×C4—342no—
362C23×C4—342no—
363C23×C4—342no—
364C23×C4—342no—
365C23×C4C22×D1711yescharacteristic
366C2×M16—21no—
367C2×M16—22no—
368C2×M16—22no—
369C2×M16—172no—
370C2×M16—341no—
371C2×M16—342no—
372C2×M16—342no—
373C2×M16C22×D1712yes—
374C22×C8—21no—
375C22×C8—341no—
376#117—21no—
377#117—21no—
378#117—22no—
379#117—22no—
380#117—172no—
381#117—172no—
382#117—178no—
383#117—344no—
384#117—344no—
385#117—344no—
386#117—344no—
387#117—344no—
388#117—341no—
389#117—341no—
390#117—342no—
391#117—342no—
392#117C4×D1712yes—
393#117C22×D1712yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—172no—
396C2×C4⋊C4—174no—
397C2×C4⋊C4—174no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—342no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4C4×D1712yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—344no—
406C2×C22⋊C4—344no—
407C2×C22⋊C4—342no—
408C2×C22⋊C4—341no—
409C2×C22⋊C4—344no—
410C2×C22⋊C4—341no—
411C2×C42—21no—
412C2×C42—172no—
413C2×C42—344no—
414C2×C42—344no—
415C2×C42—342no—
416C2×C42—341no—
417C2×C42C22×D1712yes—
418C22⋊C8—344no—
419C22⋊C8—344no—
420#95—344no—
421C17#2070511yescharacteristic, complemented, Sylow
422C24—171no—
423C24—172no—
424C24—172no—
425C24—171no—
426C24—342no—
427C24—342no—
428C24—684no—
429C24D4×D1711yescharacteristic
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—172no—
433C22×C4—171no—
434C22×C4—171no—
435C22×C4—172no—
436C22×C4—172no—
437C22×C4—172no—
438C22×C4—172no—
439C22×C4—172no—
440C22×C4—172no—
441C22×C4—172no—
442C22×C4—172no—
443C22×C4—172no—
444C22×C4—344no—
445C22×C4—344no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—344no—
455C22×C4—344no—
456C22×C4—344no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—344no—
462C22×C4—344no—
463C22×C4—344no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—688no—
469C22×C4C2×C4×D1711yescharacteristic
470C22×C4C2×C4×D1711yescharacteristic
471C22×C4#6325212yes—
472C22×C4#6325212yes—
473C22×C4D4×D1711yescharacteristic
474C22×C4D4×D1711yescharacteristic
475C22×C4D4×D1711yescharacteristic
476C22×C4#6325412yes—
477C22×C4#6325612yes—
478C22×C4C23×D1711yescharacteristic
479M16—22no—
480M16—22no—
481M16—42no—
482M16—342no—
483M16—342no—
484M16—682no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—342no—
488C2×C8—682no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—174no—
492C4⋊C4—344no—
493C4⋊C4—344no—
494C4⋊C4—342no—
495C4⋊C4—682no—
496C4⋊C4C2×C4×D1714yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—348no—
501C22⋊C4—342no—
502C22⋊C4—342no—
503C22⋊C4—688no—
504C22⋊C4—682no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—348no—
510C42—342no—
511C42—342no—
512C42—342no—
513C42—342no—
514C23—42no—
515C23—172no—
516C23—172no—
517C23—344no—
518C23—344no—
519C23—344no—
520C23—342no—
521C23—342no—
522C23—342no—
523C23—344no—
524C23—342no—
525C23—682no—
526C23—682no—
527C23—682no—
528C23—688no—
529C23—688no—
530C23C22⋊C4×D1711yescharacteristic
531C23C22⋊C4×D1711yescharacteristic
532C23#1058756912yes—
533C23#1058758012yes—
534C23C2×D4×D1711yescharacteristic
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—172no—
540C2×C4—172no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—342no—
544C2×C4—342no—
545C2×C4—344no—
546C2×C4—342no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—342no—
552C2×C4—342no—
553C2×C4—342no—
554C2×C4—344no—
555C2×C4—344no—
556C2×C4—344no—
557C2×C4—344no—
558C2×C4—344no—
559C2×C4—688no—
560C2×C4—688no—
561C2×C4—684no—
562C2×C4—684no—
563C2×C4—682no—
564C2×C4#1058755812yes—
565C2×C4#1058755812yes—
566C2×C4C22⋊C4×D1711yescharacteristic
567C2×C4C22⋊C4×D1711yescharacteristic
568C2×C4#1058756912yes—
569C2×C4#1058756912yes—
570C2×C4#1058756912yes—
571C2×C4#1058757812yes—
572C2×C4#1058757812yes—
573C2×C4#1058758012yes—
574C2×C4#1058758012yes—
575C2×C4#1058758012yes—
576C2×C4C22×C4×D1711yescharacteristic
577C2×C4C2×#6325211yescharacteristic
578C2×C4C2×#6325211yescharacteristic
579C2×C4C2×D4×D1711yescharacteristic
580C2×C4C2×#6325411yescharacteristic
581C2×C4C2×#6325611yescharacteristic
582C8—42no—
583C8—682no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—174no—
590C22—348no—
591C22—342no—
592C22—342no—
593C22—342no—
594C22—342no—
595C22—342no—
596C22—684no—
597C22—684no—
598C22—688no—
599C22—684no—
600C22#1176267112yes—
601C22#1176267112yes—
602C22C2×C22⋊C4×D1711yescharacteristic
603C22C2×#1058756911yescharacteristic
604C22C2×#1058758011yescharacteristic
605C4—44no—
606C4—82no—
607C4—342no—
608C4—342no—
609C4—684no—
610C4—1362no—
611C4#1176267112yes—
612C4#1176267112yes—
613C4C2×#1058755811yescharacteristic
614C4C2×#1058757811yescharacteristic
615C4C2×#1058756911yescharacteristic
616C4C2×#1058758011yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—344no—
622C2—682no—
623C2C2×#1176267111yescharacteristic
624C2D17×#42112yes—
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1733C2
socleC22×C17C22×C22⋊C4
prime radicalC24×D17D4
cosocleC22×C4×C17C24
hypercenter#1733D17
hypocenterC17#20705
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #14528882C2semidirect
5. #14528495C2semidirect
6. #14528494C2semidirect
7. #14528493C2semidirect
8. C17×#1733C2semidirect
13. C2×#10587578C4semidirect
18. #117×D17C4semidirect
20. C2×C4⋊C4×D17C4semidirect
31. #11762570C22semidirect
32. #11762569C22semidirect
33. #11762568C22semidirect
37. C17×#339C22semidirect
279. #1733D17semidirect
341. #339C2×D17semidirect
421. C17#20705semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D172
C2×#117627722
#145311632
#145288822
#145284952
#145284942
#145284932
C17×#17332
#2070517