Finite Group Databasebeta

← #16956905

Subgroups — #16956905

Order 4352.

1846 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587612C211yesmaximal, characteristic
2C2×#11762694C212yesmaximal, complemented
3#14527053C212yesmaximal, complemented
4#14527027C212yesmaximal
5#14527021C212yesmaximal
6#14527012C211yesmaximal, characteristic, complemented
7#14527010C211yesmaximal, characteristic, complemented
8#14527009C211yesmaximal, characteristic, complemented
9#14527007C211yesmaximal, characteristic
10#95×Dic17C211yesmaximal, characteristic, complemented
11C17×C22⋊#95C211yesmaximal, characteristic
12C2×#10587612—28no—
13C2×#10587612C418yescomplemented
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225C2212yes—
17C22×C22⋊Dic17C2212yes—
18C22×C22⋊Dic17C2211yescharacteristic
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C4×Dic17C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22C22×C4×Dic17C2211yescharacteristic
23#11762694—28no—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C24×C4×C17C2211yescharacteristic
34C22×C22⋊C4×C17C2212yes—
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2212yes—
38C25×C17C2311yescharacteristic
39C23×C4×C17—24no—
40C23×C4×C17—24no—
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17C2×C414yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2312yes—
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2312yes—
52C23×Dic17C2311yescharacteristic
53C23×Dic17C2311yescharacteristic
54C2×C22⋊C4×C17—24no—
55C2×C22⋊C4×C17—24no—
56C2×C22⋊C4×C17—42no—
57C2×C22⋊Dic17—24no—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17C2×C414yes—
63C2×C22⋊Dic17C2×C414yes—
64#10587612—232no—
65C2×C4⋊Dic17—24no—
66C2×C4⋊Dic17C2×C414yes—
67C2×C4⋊Dic17C2×C414yes—
68C2×C4⋊Dic17D418yes—
69C2×#63225—24no—
70C2×#63225—24no—
71C2×#63225—42no—
72C2×C4×Dic17—24no—
73C2×C4×Dic17—24no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—42no—
77C2×C4×Dic17C2×C414yes—
78C2×C4×Dic17C2×C414yes—
79C17×#95—24no—
80C17×#95—22no—
81C17×#95—22no—
82#10587502—24no—
83#10587502—22no—
84#10587502—22no—
85#10587502—24no—
86#10587502—24no—
87#10587502—24no—
88#10587502—24no—
89#10587502—42no—
90#10587502—42no—
91C24×C17—22no—
92C24×C17—21no—
93C24×C17—21no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—21no—
97C24×C17—21no—
98C24×C17—21no—
99C24×C17—21no—
100C24×C17C22×C412yes—
101C24×C17#4114yes—
102C24×C17C2411yescharacteristic
103C22×C4×C17—216no—
104C22×C4×C17—216no—
105C22×C4×C17—28no—
106C22×C4×C17—22no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—28no—
112C22×C4×C17—28no—
113C22×C4×C17—22no—
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×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D414yes—
129C22×C4×C17#4114yes—
130C22×C4×C17#4112yes—
131C22×C4×C17#4112yes—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—21no—
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—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—21no—
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×Dic17C22×C412yes—
160C22×Dic17C22×C412yes—
161C22×Dic17C22×C412yes—
162C22×Dic17C22×C412yes—
163C22×Dic17C2×D414yes—
164C22×Dic17C2×D414yes—
165C22×Dic17#4114yes—
166C22⋊C4×C17—48no—
167C22⋊Dic17—28no—
168C22⋊Dic17—28no—
169C22⋊Dic17—48no—
170C4⋊Dic17—28no—
171C4⋊Dic17C22⋊C4116yescomplemented
172#63225—48no—
173C4×Dic17—28no—
174C4×Dic17—28no—
175C4×Dic17—48no—
176#13944—171nomaximal, Sylow
177C23×C17—22no—
178C23×C17—22no—
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—24no—
187C23×C17—24no—
188C23×C17—24no—
189C23×C17—42no—
190C23×C17—42no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17#11714yes—
200C23×C17C23×C411yescharacteristic
201C23×C17C22×D412yes—
202C23×C17C2×#4112yes—
203C23×C17C2×#4111yescharacteristic
204C23×C17C2×#4111yescharacteristic
205C23×C17C2×#4112yes—
206C23×C17C2×#4112yes—
207C23×C17H(5,𝔽2)11yescharacteristic
208C23×C17H(5,𝔽2)11yescharacteristic
209C23×C172−1+411yescharacteristic
210C23×C172−1+411yescharacteristic
211C2×C4×C17—232no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—24no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
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×C17C2×C22⋊C414yes—
235C2×C4×C17#11714yes—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—22no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
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—
270C22×#117—171no—
271C2×#387—172no—
272#1445—172no—
273#1439—172no—
274#1437—172no—
275C4⋊#95—171no—
276#1434—171no—
277#1434—171no—
278C22⋊#95—171no—
279C22⋊#95D1711yescharacteristic
280C4×#95—171no—
281C22×C17—24no—
282C22×C17—24no—
283C22×C17—24no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—44no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—42no—
292C22×C17C22×C22⋊C411yescharacteristic
293C22×C17C2×#11711yescharacteristic
294C22×C17C2×#11711yescharacteristic
295C22×C17C4×#4111yescharacteristic
296C22×C17C4×#4111yescharacteristic
297C22×C17#52011yescharacteristic
298C22×C17#52011yescharacteristic
299C22×C17#52512yes—
300C22×C17#52512yes—
301C22×C17#52912yes—
302C22×C17#52912yes—
303C22×C17#54011yescharacteristic
304C22×C17#54111yescharacteristic
305C22×C17#54211yescharacteristic
306C22×C17#54211yescharacteristic
307C22×C17#54612yes—
308C22×C17#54712yes—
309C22×C17#54812yes—
310C22×C17#55012yes—
311C22×C17#55112yes—
312C22×C17#55112yes—
313C22×C17#55212yes—
314C22×C17#55212yes—
315C4×C17—24no—
316C4×C17—24no—
317C4×C17—44no—
318C4×C17—82no—
319Dic17—24no—
320Dic17—24no—
321Dic17—42no—
322Dic17—42no—
323Dic17—44no—
324Dic17—82no—
325C24×C4C2×D1711yescharacteristic
326C2×#117—178no—
327C2×#117—348no—
328C22×C4⋊C4—171no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4—172no—
331C22×C22⋊C4—171no—
332C22×C22⋊C4—171no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4C2×D1712yes—
335C22×C42—171no—
336C22×C42—171no—
337C22×C42—172no—
338#387—348no—
339C2×#95—171no—
340C2×#95—172no—
341C2×#95—171no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95C2×D1711yescharacteristic
349C2×#95C2×D1711yescharacteristic
350C2×#95C2×D1712yes—
351C2×C17—44no—
352C2×C17#228011yescharacteristic
353C2×C17#229811yescharacteristic
354C2×C17#231411yescharacteristic
355C2×C17#233911yescharacteristic
356C2×C17#234111yescharacteristic
357C2×C17#234311yescharacteristic
358C2×C17#234311yescharacteristic
359C2×C17#247112yes—
360C2×C17#247512yes—
361C2×C17#247712yes—
362C2×C17#247912yes—
363C25C22×D1711yescharacteristic
364C23×C4—24no—
365C23×C4—24no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—171no—
369C23×C4—171no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—172no—
373C23×C4—172no—
374C23×C4C4×D1714yes—
375C23×C4C22×D1712yes—
376C23×C4C22×D1712yes—
377C23×C4C22×D1711yescharacteristic
378C23×C4C22×D1711yescharacteristic
379#117—3432no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—178no—
382C2×C4⋊C4—174no—
383C2×C4⋊C4—344no—
384C2×C4⋊C4—344no—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—682no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—174no—
391C2×C22⋊C4—174no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—682no—
397C2×C42—174no—
398C2×C42—174no—
399C2×C42—344no—
400C2×C42—344no—
401C2×C42—344no—
402C2×C42—344no—
403C2×C42—682no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—344no—
408#95—344no—
409#95—344no—
410#95—344no—
411#95—344no—
412#95—342no—
413#95—342no—
414#95—682no—
415#95—682no—
416C17#1394411yescharacteristic, complemented, Sylow
417C24—22no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—21no—
422C24—21no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24C2×C4×D1712yes—
427C24#6325414yes—
428C24C23×D1711yescharacteristic
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
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—172no—
453C22×C4—172no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—172no—
457C22×C4—172no—
458C22×C4—174no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—342no—
474C22×C4—341no—
475C22×C4—341no—
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×C4C2×C4×D1712yes—
487C22×C4C2×C4×D1712yes—
488C22×C4#6325214yes—
489C22×C4D4×D1714yes—
490C22×C4#6325412yes—
491C22×C4#6325412yes—
492C4⋊C4—1716no—
493C4⋊C4—348no—
494C4⋊C4—688no—
495C22⋊C4—48no—
496C22⋊C4—348no—
497C22⋊C4—348no—
498C22⋊C4—688no—
499C42—348no—
500C42—348no—
501C42—688no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23#1058756314yes—
525C23C22×C4×D1711yescharacteristic
526C23C2×#6325212yes—
527C23C2×D4×D1712yes—
528C23C2×#6325412yes—
529C23C2×#6325411yescharacteristic
530C23C2×#6325411yescharacteristic
531C23#1058769111yescharacteristic
532C23#1058769411yescharacteristic
533C23#41×D1712yes—
534C23#1058769611yescharacteristic
535C23#1058769711yescharacteristic
536C2×C4—232no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—174no—
560C2×C4—174no—
561C2×C4—344no—
562C2×C4—344no—
563C2×C4—342no—
564C2×C4—342no—
565C2×C4—342no—
566C2×C4—342no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4#1058756314yes—
594C2×C4C22⋊C4×D1714yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22C2×C22⋊C4×D1711yescharacteristic
607C22C2×#1058756311yescharacteristic
608C22C2×#1058756311yescharacteristic
609C22#1176327711yescharacteristic
610C22C4×#6325211yescharacteristic
611C22#1176328911yescharacteristic
612C22#1176329011yescharacteristic
613C22#1176331512yes—
614C22#1176331512yes—
615C22#1176333912yes—
616C22#1176334112yes—
617C22#1176341211yescharacteristic
618C22#1176341911yescharacteristic
619C22#1176342611yescharacteristic
620C22#1176343211yescharacteristic
621C22#1176345512yes—
622C22#1176346412yes—
623C22#1176346912yes—
624C22#1176347712yes—
625C22#1176348212yes—
626C22#1176348512yes—
627C22#1176349412yes—
628C22#1176349612yes—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—82no—
633C4—344no—
634C4—344no—
635C4—682no—
636C4—682no—
637C4—684no—
638C4—1362no—
639C2—44no—
640C2#1453116211yescharacteristic
641C2#1453128011yescharacteristic
642C2#1453139111yescharacteristic
643C2#1453161511yescharacteristic
644C2#1453162111yescharacteristic
645C2#1453164011yescharacteristic
646C2#1453164611yescharacteristic
647C2#1453289212yes—
648C2#1453295212yes—
649C2#1453298412yes—
650C2#1453301412yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C22⋊#95C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC22⋊#95D17
hypocenterC17#13944
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11762694C2semidirect
3. #14527053C2semidirect
6. #14527012C2semidirect
7. #14527010C2semidirect
8. #14527009C2semidirect
10. #95×Dic17C2semidirect
13. C2×#10587612C4semidirect
171. C4⋊Dic17C22⋊C4semidirect
416. C17#13944semidirect

Maximal subgroups

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

subgroupindex
C22×#105876122
C2×#117626942
#145270532
#145270272
#145270212
#145270122
#145270102
#145270092
#145270072
#95×Dic172
C17×C22⋊#952
#1394417