Finite Group Databasebeta

← #16956855

Subgroups — #16956855

Order 4352.

1862 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587563C211yesmaximal, characteristic
2C2×#11762667C211yesmaximal, characteristic
3C2×C22⋊C4×Dic17C211yesmaximal, characteristic
4#14527033C211yesmaximal, characteristic
5#14527031C211yesmaximal, characteristic
6#14527031C211yesmaximal, characteristic
7#14527012C212yesmaximal, complemented
8#14527012C212yesmaximal, complemented
9#14527010C212yesmaximal, complemented
10#14527009C212yesmaximal, complemented
11C17×#1440C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C2×#10587563—24no—
14C2×#10587563—24no—
15C2×#10587563C418yescomplemented
16C22×#63225C2212yes—
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C4×Dic17C2212yes—
23C22×C4×Dic17C2211yescharacteristic
24C22×C4×Dic17C2211yescharacteristic
25#11762667—24no—
26C22⋊C4×Dic17—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C22⋊C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×Dic17—24no—
46C23×Dic17—24no—
47C23×Dic17C2×C414yes—
48C23×Dic17D414yes—
49C23×Dic17D414yes—
50C23×Dic17C2312yes—
51C23×Dic17C2311yescharacteristic
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2312yes—
54C23×Dic17C2311yescharacteristic
55C23×Dic17C2311yescharacteristic
56C2×C22⋊C4×C17—22no—
57C2×C22⋊C4×C17—22no—
58C2×C22⋊C4×C17—22no—
59C2×C22⋊C4×C17—24no—
60C2×C22⋊C4×C17—22no—
61C2×C22⋊C4×C17—42no—
62C2×C22⋊C4×C17C2×C414yes—
63C2×C22⋊Dic17—22no—
64C2×C22⋊Dic17—22no—
65C2×C22⋊Dic17—22no—
66C2×C22⋊Dic17—24no—
67C2×C22⋊Dic17—22no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17C2×C414yes—
70C2×#63225—22no—
71C2×#63225—22no—
72C2×#63225—22no—
73C2×#63225—22no—
74C2×#63225—24no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225C2×C414yes—
78C2×#63225C2×C414yes—
79C2×C4×Dic17—22no—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—22no—
83C2×C4×Dic17—24no—
84C2×C4×Dic17—24no—
85C2×C4×Dic17—24no—
86C2×C4×Dic17C2×C414yes—
87C2×C4×Dic17C2×C414yes—
88#10587563—232no—
89C17×#95—24no—
90C17×#95—42no—
91#10587502—24no—
92#10587502—24no—
93#10587502—24no—
94#10587502—24no—
95#10587502—24no—
96#10587502—42no—
97#10587502—42no—
98#10587502—42no—
99C24×C17—22no—
100C24×C17—22no—
101C24×C17—22no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—22no—
105C24×C17C22×C412yes—
106C24×C17C2×D412yes—
107C24×C17C2×D412yes—
108C24×C17C2411yescharacteristic
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
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—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17#4114yes—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—28no—
136C22×Dic17—28no—
137C22×Dic17—216no—
138C22×Dic17—216no—
139C22×Dic17—22no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17C22⋊C4116yes—
161C22×Dic17C22×C412yes—
162C22×Dic17C22×C412yes—
163C22×Dic17C22×C412yes—
164C22×Dic17C22×C412yes—
165C22×Dic17C2×D412yes—
166C22×Dic17C2×D412yes—
167C22×Dic17C2×D412yes—
168C22×Dic17C2×D412yes—
169C22×Dic17#4114yes—
170C22×Dic17#4114yes—
171C22×Dic17#4114yes—
172C22⋊C4×C17—28no—
173C22⋊C4×C17—28no—
174C22⋊C4×C17—44no—
175C22⋊Dic17—28no—
176C22⋊Dic17—28no—
177C22⋊Dic17—44no—
178#63225—28no—
179#63225—28no—
180#63225—48no—
181C4×Dic17—28no—
182C4×Dic17—28no—
183C4×Dic17—48no—
184#13939—171nomaximal, Sylow
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—24no—
190C23×C17—24no—
191C23×C17—22no—
192C23×C17—22no—
193C23×C17—24no—
194C23×C17—22no—
195C23×C17—22no—
196C23×C17—24no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—42no—
200C23×C17—42no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17C2×C22⋊C414yes—
206C23×C17C23×C411yescharacteristic
207C23×C17C22×D411yescharacteristic
208C23×C17C22×D411yescharacteristic
209C23×C17C2×#4112yes—
210C23×C17C2×#4112yes—
211C23×C17C2×#4112yes—
212C23×C17C2×#4112yes—
213C23×C17H(5,𝔽2)12yes—
214C23×C172−1+412yes—
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—24no—
222C2×C4×C17—42no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—42no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—232no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—42no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—42no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17C2×C22⋊C414yes—
269C2×Dic17C2×C22⋊C414yes—
270C2×Dic17#11714yes—
271C2×Dic17#11714yes—
272C2×Dic17#12318yes—
273C2×Dic17#12318yes—
274C22×#117—171no—
275C2×#380—171no—
276C2×C4×C22⋊C4—171no—
277#1440—171no—
278#1440—171no—
279#1440—171no—
280#1440D1711yescharacteristic
281C4⋊#95—172no—
282C4⋊#95—172no—
283#1434—172no—
284#1434—172no—
285C22×C17—24no—
286C22×C17—24no—
287C22×C17—24no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17C22×C22⋊C411yescharacteristic
295C22×C17C2×#11711yescharacteristic
296C22×C17C2×#11711yescharacteristic
297C22×C17C4×#4112yes—
298C22×C17#52012yes—
299C22×C17C2×#12312yes—
300C22×C17C2×#12312yes—
301C22×C17#52511yescharacteristic
302C22×C17#52511yescharacteristic
303C22×C17#53212yes—
304C22×C17#53511yescharacteristic
305C22×C17#53611yescharacteristic
306C22×C17#53712yes—
307C22×C17#55212yes—
308C22×C17#55212yes—
309C22×C17#55312yes—
310C22×C17#55312yes—
311C22×C17#55512yes—
312C22×C17#55512yes—
313C22×C17#55612yes—
314C22×C17#55612yes—
315C4×C17—42no—
316C4×C17—44no—
317C4×C17—42no—
318C4×C17—82no—
319Dic17—24no—
320Dic17—42no—
321Dic17—44no—
322Dic17—42no—
323Dic17—82no—
324Dic17#38118yes—
325C24×C4—171no—
326C2×#117—178no—
327C2×#117—344no—
328C2×#117—344no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4—171no—
331C22×C4⋊C4—172no—
332C22×C22⋊C4—171no—
333C22×C22⋊C4—171no—
334C22×C22⋊C4—171no—
335C22×C22⋊C4C2×D1711yescharacteristic
336C22×C22⋊C4C2×D1711yescharacteristic
337C22×C22⋊C4C2×D1711yescharacteristic
338C22×C42—171no—
339C22×C42—171no—
340C22×C42—172no—
341#380—344no—
342C4×C22⋊C4—344no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95C2×D1712yes—
352C2×#95C2×D1712yes—
353C2×C17—44no—
354C2×C17C2×#38111yescharacteristic
355C2×C17#229711yescharacteristic
356C2×C17#231111yescharacteristic
357C2×C17#234612yes—
358C2×C17#234712yes—
359C2×C17#248512yes—
360C2×C17#248512yes—
361C2×C17#249912yes—
362C2×C17#250012yes—
363C25C22×D1711yescharacteristic
364C23×C4—171no—
365C23×C4—174no—
366C23×C4—174no—
367C23×C4—174no—
368C23×C4—171no—
369C23×C4—171no—
370C23×C4—171no—
371C23×C4—172no—
372C23×C4—172no—
373C23×C4—344no—
374C23×C4—344no—
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1712yes—
379#117—3432no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—174no—
382C2×C4⋊C4—342no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4—342no—
385C2×C4⋊C4—342no—
386C2×C4⋊C4—344no—
387C2×C4⋊C4—344no—
388C2×C4⋊C4—344no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—174no—
396C2×C22⋊C4—344no—
397C2×C22⋊C4—342no—
398C2×C22⋊C4—342no—
399C2×C22⋊C4—342no—
400C2×C22⋊C4—342no—
401C2×C22⋊C4—682no—
402C2×C22⋊C4C4×D1714yes—
403C2×C42—174no—
404C2×C42—174no—
405C2×C42—342no—
406C2×C42—342no—
407C2×C42—342no—
408C2×C42—342no—
409C2×C42—344no—
410C2×C42—344no—
411C2×C42—344no—
412#95—24no—
413#95—42no—
414#95—344no—
415#95—344no—
416#95—344no—
417#95—344no—
418#95—344no—
419#95—682no—
420#95—682no—
421#95—682no—
422C17#1393911yescharacteristic, complemented, Sylow
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D1712yes—
430C24D4×D1712yes—
431C24D4×D1712yes—
432C24C23×D1711yescharacteristic
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
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—22no—
453C22×C4—22no—
454C22×C4—172no—
455C22×C4—172no—
456C22×C4—172no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—172no—
460C22×C4—172no—
461C22×C4—172no—
462C22×C4—1716no—
463C22×C4—172no—
464C22×C4—172no—
465C22×C4—174no—
466C22×C4—348no—
467C22×C4—348no—
468C22×C4—3416no—
469C22×C4—3416no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
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—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4C2×C4×D1712yes—
494C22×C4C2×C4×D1712yes—
495C22×C4#6325414yes—
496C4⋊C4—348no—
497C4⋊C4—348no—
498C4⋊C4—688no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—348no—
503C22⋊C4—348no—
504C22⋊C4—684no—
505C42—348no—
506C42—348no—
507C42—688no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C22⋊C4×D1714yes—
529C23C22×C4×D1711yescharacteristic
530C23C2×D4×D1711yescharacteristic
531C23C2×D4×D1711yescharacteristic
532C23C2×#6325412yes—
533C23#1058769112yes—
534C23#41×D1712yes—
535C23#41×D1712yes—
536C23#41×D1712yes—
537C23#1058769712yes—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—174no—
563C2×C4—174no—
564C2×C4—174no—
565C2×C4—178no—
566C2×C4—178no—
567C2×C4—174no—
568C2×C4—3432no—
569C2×C4—342no—
570C2×C4—342no—
571C2×C4—344no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—682no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22C2×C22⋊C4×D1711yescharacteristic
607C22#117×D1711yescharacteristic
608C22#117×D1711yescharacteristic
609C22C4×#6325412yes—
610C22#1176328812yes—
611C22#123×D1712yes—
612C22#123×D1712yes—
613C22#1176331511yescharacteristic
614C22#1176331711yescharacteristic
615C22#1176335412yes—
616C22#1176337411yescharacteristic
617C22#1176338111yescharacteristic
618C22#1176338612yes—
619C22#1176349512yes—
620C22#1176349612yes—
621C22#1176350212yes—
622C22#1176350412yes—
623C22#1176352112yes—
624C22#1176352312yes—
625C22#1176352612yes—
626C22#1176352812yes—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—82no—
631C4—178no—
632C4—344no—
633C4—682no—
634C4—682no—
635C4—684no—
636C4—1362no—
637C2—44no—
638C2D17×#38111yescharacteristic
639C2#1453126911yescharacteristic
640C2#1453137111yescharacteristic
641C2#1453166612yes—
642C2#1453167412yes—
643C2#1453307412yes—
644C2#1453307612yes—
645C2#1453323212yes—
646C2#1453324112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1440C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1440D17
hypocenterC17#13939
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
7. #14527012C2semidirect
8. #14527012C2semidirect
9. #14527010C2semidirect
10. #14527009C2semidirect
15. C2×#10587563C4semidirect
422. C17#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#105875632
C2×#117626672
C2×C22⋊C4×Dic172
#145270332
#145270312
#145270122
#145270102
#145270092
C17×#14402
#1393917