Finite Group Databasebeta

← #16956143

Subgroups — #16956143

Order 4352.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762735C211yesmaximal, characteristic
2C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
3C2×#11762656C212yesmaximal
4C2×#11762656C211yesmaximal, characteristic
5#14527044C212yesmaximal, complemented
6#14527040C211yesmaximal, characteristic, complemented
7#14527032C212yesmaximal
8#14527030C211yesmaximal, characteristic
9#95×Dic17C212yesmaximal, complemented
10#95×Dic17C211yesmaximal, characteristic, complemented
11C17×#1445C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63227C2212yes—
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×#63227C2212yes—
18C22×C4×Dic17C2211yescharacteristic
19C22×C4×Dic17C2212yes—
20C22×C4×Dic17C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22#11762735C418yescomplemented
23C4⋊C4×Dic17—24no—
24#11762656—24no—
25#11762656—28no—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C22×C4⋊C4×C17C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C24×D17C2311yescharacteristic
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17D414yes—
55C2×C4⋊C4×C17—22no—
56C2×C4⋊C4×C17—22no—
57C2×C4⋊C4×C17C2×C414yescomplemented
58C2×#63227—24no—
59C2×#63227—24no—
60C2×#63227—22no—
61C2×#63227—24no—
62C2×#63227—22no—
63C2×#63227—22no—
64C2×#63227—22no—
65C2×#63227—24no—
66C2×#63227—22no—
67C2×#63227—24no—
68C2×#63227C2×C418yes—
69C2×#63227C2×C414yes—
70C2×C4⋊Dic17—22no—
71C2×C4⋊Dic17—24no—
72C2×C4×Dic17—24no—
73C2×C4×Dic17—24no—
74C2×C4×Dic17—22no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—24no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—24no—
79C2×C4×Dic17—24no—
80C2×C4×Dic17—24no—
81C2×C4×Dic17D414yes—
82C17×#95—22no—
83C17×#95—24no—
84C17×#95—24no—
85C17×#95—22no—
86#10587502—24no—
87#10587502—24no—
88#10587502—24no—
89#10587502—22no—
90#10587502—24no—
91#10587502—24no—
92#10587502C2×C418yescomplemented
93#10587502C2×C414yescomplemented
94C24×C17C2411yescharacteristic
95C23×D17—21no—
96C23×D17—21no—
97C23×D17—22no—
98C23×D17—22no—
99C23×D17—21no—
100C23×D17—21no—
101C23×D17—22no—
102C23×D17—22no—
103C23×D17C22×C412yes—
104C23×D17C2×D412yes—
105C23×D17#4112yes—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—21no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17C22×C414yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17#4112yes—
133C22×C4×C17#4112yes—
134C22×C4×C17#4114yes—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—21no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17C22×C414yes—
156C22×Dic17C22×C412yes—
157C22×Dic17C2×D412yes—
158C22×Dic17C2×D412yes—
159C22×Dic17C2×D412yes—
160C22×Dic17#4114yes—
161C22×Dic17#4112yes—
162C22×Dic17#4114yes—
163C22×Dic17#4114yes—
164C2×C4×D17—28no—
165C2×C4×D17—216no—
166C2×C4×D17—44no—
167C2×C4×D17—44no—
168C2×C4×D17—48no—
169C4⋊C4×C17—28no—
170#63227—28no—
171#63227—216no—
172#63227—216no—
173#63227—28no—
174C4⋊Dic17—28no—
175C4×Dic17—28no—
176C4×Dic17—48no—
177C4×Dic17—48no—
178C4×Dic17—44no—
179C4×#397—171nomaximal, Sylow
180C23×C17C23×C411yescharacteristic
181C23×C17C22×D411yescharacteristic
182C23×C17C22×D411yescharacteristic
183C23×C17C2×#4111yescharacteristic
184C23×C17C2×#4111yescharacteristic
185C23×C17C2×#4112yes—
186C23×C17C2×#4111yescharacteristic
187C23×C17C2×#4111yescharacteristic
188C23×C17C2×#4112yes—
189C23×C17C2×#4112yes—
190C23×C17C2×#4112yes—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—22no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—22no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—41no—
213C22×D17C4×D414yes—
214C2×C4×C17—24no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—22no—
222C2×C4×C17—24no—
223C2×C4×C17—22no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—42no—
229C2×C4×C17—41no—
230C2×C4×C17—42no—
231C2×C4×C17—41no—
232C2×C4×C17—42no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17C4×D414yes—
243C2×Dic17—24no—
244C2×Dic17—22no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—22no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—41no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—41no—
267C2×Dic17#12114yes—
268C2×Dic17#12318yes—
269C2×Dic17#12314yes—
270C2×Dic17#12414yes—
271C2×Dic17#12618yes—
272C4×D17—416no—
273C2×#397—171no—
274C2×C4×C4⋊C4—171no—
275C2×C4×C22⋊C4—171no—
276C2×C4×C22⋊C4—172no—
277#1445D1711yescharacteristic, complemented
278#1443—172no—
279#1442—171no—
280#1440—172no—
281#1439—171no—
282C4×#95—171no—
283C4×#95—172no—
284C22×C17C2×C4×D411yescharacteristic
285C22×C17C2×C4×D411yescharacteristic
286C22×C17C4×#4111yescharacteristic
287C22×C17C4×#4112yes—
288C22×C17C4×#4112yes—
289C22×C17C2×#12111yescharacteristic
290C22×C17C2×#12311yescharacteristic
291C22×C17C2×#12312yes—
292C22×C17#52511yescharacteristic
293C22×C17#52511yescharacteristic
294C22×C17#52512yes—
295C22×C17C2×#12411yescharacteristic
296C22×C17C2×#12612yes—
297C22×C17#52912yes—
298C22×C17#52912yes—
299C22×C17#52911yescharacteristic
300C22×C17#52912yes—
301C22×C17#52912yes—
302C22×C17#52912yes—
303C22×C17#52912yes—
304C22×C17#52912yes—
305C22×C17#53211yescharacteristic
306C22×C17#53211yescharacteristic
307C2×D17—24no—
308C2×D17—28no—
309C2×D17—44no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—44no—
314C2×D17—42no—
315C2×D17—44no—
316C2×D17—42no—
317C2×D17—42no—
318C4×C17—24no—
319C4×C17—42no—
320C4×C17—44no—
321C4×C17—42no—
322C4×C17—44no—
323Dic17—44no—
324Dic17—42no—
325Dic17—44no—
326Dic17—42no—
327Dic17#39718yescomplemented
328C24×C4—171no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4—171no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—171no—
335C22×C42—171no—
336C22×C42—171no—
337C22×C42—172no—
338C22×C42—172no—
339#397—178no—
340C4×C4⋊C4—344no—
341C4×C22⋊C4—344no—
342C4×C22⋊C4—348no—
343C2×#95—171no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—171no—
351C2×#95C2×D1711yescharacteristic
352C2×#95C2×D1712yes—
353C2×#95C2×D1711yescharacteristic
354C2×#95C2×D1712yes—
355C2×C17C4×#12111yescharacteristic
356C2×C17C4×#12311yescharacteristic
357C2×C17C4×#12312yes—
358C2×C17C4×#12411yescharacteristic
359C2×C17C4×#12612yes—
360C2×C17C2×#39711yescharacteristic
361C2×C17#239211yescharacteristic
362C2×C17#239511yescharacteristic
363C2×C17#239512yes—
364C2×C17#239811yescharacteristic
365C2×C17#240112yes—
366D17—44no—
367C25—171no—
368C23×C4—171no—
369C23×C4—172no—
370C23×C4—171no—
371C23×C4—172no—
372C23×C4—171no—
373C23×C4—174no—
374C23×C4—342no—
375C23×C4—344no—
376C23×C4—344no—
377C23×C4—342no—
378C23×C4C22×D1712yes—
379C23×C4C22×D1711yescharacteristic
380C23×C4C22×D1711yescharacteristic
381C23×C4C22×D1712yes—
382C23×C4C22×D1711yescharacteristic
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4C4×D1714yescomplemented
388C2×C22⋊C4—178no—
389C2×C22⋊C4—174no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—342no—
396C2×C22⋊C4—342no—
397C2×C22⋊C4—342no—
398C2×C22⋊C4—342no—
399C2×C22⋊C4—344no—
400C2×C42—174no—
401C2×C42—344no—
402C2×C42—342no—
403C2×C42—344no—
404C2×C42—344no—
405C2×C42—344no—
406C2×C42—344no—
407C2×C42—344no—
408C2×C42—344no—
409C2×C42—344no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—22no—
414#95—178no—
415#95—174no—
416#95—344no—
417#95—342no—
418#95—344no—
419#95—344no—
420#95—344no—
421#95—344no—
422C17C4×#39711yescharacteristic, complemented, Sylow
423C24—172no—
424C24—172no—
425C24—172no—
426C24—341no—
427C24—341no—
428C24—341no—
429C24—342no—
430C24—342no—
431C24—341no—
432C24—342no—
433C24—342no—
434C24C23×D1711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—172no—
458C22×C4—172no—
459C22×C4—172no—
460C22×C4—174no—
461C22×C4—174no—
462C22×C4—174no—
463C22×C4—172no—
464C22×C4—172no—
465C22×C4—174no—
466C22×C4—348no—
467C22×C4—3416no—
468C22×C4—342no—
469C22×C4—341no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—341no—
487C22×C4—342no—
488C22×C4—684no—
489C22×C4—684no—
490C22×C4—688no—
491C22×C4C2×C4×D1714yes—
492C22×C4C2×C4×D1712yes—
493C22×C4D4×D1712yes—
494C22×C4D4×D1712yes—
495C22×C4#6325412yes—
496C22×C4#6325412yes—
497C22×C4#6325614yes—
498C4⋊C4—28no—
499C4⋊C4—348no—
500C22⋊C4—3416no—
501C22⋊C4—348no—
502C22⋊C4—3416no—
503C22⋊C4—348no—
504C42—348no—
505C42—688no—
506C42—688no—
507C42—684no—
508C23—174no—
509C23—342no—
510C23—342no—
511C23—344no—
512C23—344no—
513C23—342no—
514C23—344no—
515C23—342no—
516C23—344no—
517C23—342no—
518C23—344no—
519C23—344no—
520C23—342no—
521C23—681no—
522C23—681no—
523C23—681no—
524C23—682no—
525C23—682no—
526C23—682no—
527C23—682no—
528C23—682no—
529C23—682no—
530C23—681no—
531C23C22×C4×D1711yescharacteristic
532C23C2×D4×D1711yescharacteristic
533C23C2×D4×D1711yescharacteristic
534C23C2×#6325411yescharacteristic
535C23C2×#6325411yescharacteristic
536C23C2×#6325612yes—
537C23#41×D1711yescharacteristic
538C23#41×D1711yescharacteristic
539C23#41×D1712yes—
540C23#41×D1712yes—
541C23#41×D1712yes—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—174no—
571C2×C4—174no—
572C2×C4—178no—
573C2×C4—174no—
574C2×C4—178no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—342no—
581C2×C4—344no—
582C2×C4—342no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—342no—
586C2×C4—6816no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—681no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—681no—
600C2×C4#1058756714yes—
601C22—344no—
602C22—348no—
603C22—684no—
604C22—682no—
605C22—682no—
606C22—682no—
607C22—684no—
608C22—682no—
609C22—682no—
610C22—684no—
611C22—682no—
612C22C4×D4×D1711yescharacteristic
613C22C2×#1058756711yescharacteristic
614C22C4×#6325411yescharacteristic
615C22#1176327712yes—
616C22C4×#6325612yes—
617C22#121×D1711yescharacteristic
618C22#123×D1711yescharacteristic
619C22#123×D1712yes—
620C22#1176331711yescharacteristic
621C22#1176331911yescharacteristic
622C22#1176332212yes—
623C22#124×D1711yescharacteristic
624C22#126×D1712yes—
625C22#1176333612yes—
626C22#1176333612yes—
627C22#1176333811yescharacteristic
628C22#1176334212yes—
629C22#1176334412yes—
630C22#1176334412yes—
631C22#1176334512yes—
632C22#1176334512yes—
633C22#1176335711yescharacteristic
634C22#1176336011yescharacteristic
635C4—24no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—178no—
641C4—684no—
642C4—682no—
643C4—684no—
644C4—682no—
645C2—684no—
646C2#1453130011yescharacteristic
647C2#1453130111yescharacteristic
648C2#1453130512yes—
649C2#1453132711yescharacteristic
650C2#1453133412yes—
651C2D17×#39711yescharacteristic
652C2#1453202511yescharacteristic
653C2#1453204811yescharacteristic
654C2#1453204912yes—
655C2#1453206911yescharacteristic
656C2#1453208312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1445C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1445D17
hypocenterC17C4×#397
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic17C2semidirect
5. #14527044C2semidirect
6. #14527040C2semidirect
9. #95×Dic17C2semidirect
10. #95×Dic17C2semidirect
11. C17×#1445C2semidirect
22. #11762735C4semidirect
57. C2×C4⋊C4×C17C2×C4semidirect
92. #10587502C2×C4semidirect
93. #10587502C2×C4semidirect
277. #1445D17semidirect
327. Dic17#397semidirect
387. C2×C4⋊C4C4×D17semidirect
422. C17C4×#397semidirect

Maximal subgroups

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

subgroupindex
C2×#117627352
C2×C4⋊C4×Dic172
C2×#117626562
#145270442
#145270402
#145270322
#145270302
#95×Dic172
C17×#14452
C4×#39717