Finite Group Databasebeta

← #16960864

Subgroups — #16960864

Order 4352.

1926 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762683C212yesmaximal
2C2×#11762668C211yesmaximal, characteristic
3C2×#11762666C211yesmaximal, characteristic
4C2×#95×D17C211yesmaximal, characteristic
5#14527049C211yesmaximal, characteristic, complemented
6#14527048C211yesmaximal, characteristic, complemented
7#14527043C211yesmaximal, characteristic, complemented
8#14527038C212yesmaximal, complemented
9#14527035C212yesmaximal, complemented
10#14527005C212yesmaximal
11C17×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C22×C4⋊Dic17C2212yes—
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2212yes—
18C22×#63225C2212yes—
19C22×#63227C2212yes—
20C22×#63227C2212yes—
21#11762683—28no—
22#11762668C418yescomplemented
23#11762666—24no—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33#95×D17—24no—
34C22×C4⋊C4×C17C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2212yes—
38C2×C17×#95C2211yescharacteristic
39C24×D17C2311yescharacteristic
40C23×C4×C17C2312yes—
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—42no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17—42no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17C2×C414yes—
61C22×C4×D17D414yes—
62C22×C4×D17D414yes—
63C22×C4×D17Q814yes—
64C22×C4×D17Q814yes—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—42no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227C2×C418yes—
73C2×C4⋊Dic17—22no—
74C2×C4⋊Dic17—24no—
75C2×C4⋊Dic17—42no—
76C2×C4⋊Dic17—42no—
77C2×C4⋊Dic17—42no—
78C2×#63225—24no—
79C2×#63225—42no—
80C2×#63225—42no—
81C2×#63225—42no—
82C2×#63225—42no—
83C2×#63225C2×C418yescomplemented
84C17×#95—22no—
85C17×#95—24no—
86C17×#95—42no—
87C17×#95C2×C414yescomplemented
88#10587502—22no—
89#10587502—22no—
90#10587502—22no—
91#10587502—24no—
92#10587502—24no—
93#10587502—42no—
94#10587502—42no—
95#10587502—42no—
96#10587502C2×C414yescomplemented
97C24×C17C2411yescharacteristic
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17C22×C412yes—
103C23×D17C2×D412yes—
104C23×D17C2×D412yes—
105C23×D17C2×Q812yes—
106C23×D17C2×Q812yes—
107C23×D17#4112yes—
108C23×D17#4112yes—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—21no—
112C22×C4×C17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
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×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17C22×C414yes—
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D412yes—
135C22×C4×C17C2×D412yes—
136C22×C4×C17C2×Q812yes—
137C22×C4×C17C2×Q812yes—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—22no—
143C22×Dic17—21no—
144C22×Dic17—21no—
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×Dic17—22no—
161C22×Dic17C22×C414yes—
162C22×Dic17C22×C412yes—
163C22×Dic17C2×D412yes—
164C22×Dic17C2×D412yes—
165C22×Dic17C2×Q812yes—
166C22×Dic17C2×Q812yes—
167C2×C4×D17—28no—
168C2×C4×D17—28no—
169C2×C4×D17—48no—
170C2×C4×D17—44no—
171C2×C4×D17—44no—
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17—44no—
176C2×C4×D17—44no—
177C2×C4×D17—44no—
178C2×C4×D17—44no—
179C2×C4×D17—48no—
180C2×C4×D17—48no—
181C2×C4×D17C4⋊C4116yescomplemented
182C4⋊C4×C17—44no—
183#63227—216no—
184#63227—48no—
185C4⋊Dic17—48no—
186C4⋊Dic17—44no—
187#63225—48no—
188#63225—48no—
189#14275—171nomaximal, Sylow
190C23×C17C23×C411yescharacteristic
191C23×C17C22×D411yescharacteristic
192C23×C17C22×D411yescharacteristic
193C23×C17C22×Q811yescharacteristic
194C23×C17C22×Q811yescharacteristic
195C23×C17C2×#4111yescharacteristic
196C23×C17C2×#4111yescharacteristic
197C23×C17H(5,𝔽2)12yes—
198C23×C17H(5,𝔽2)12yes—
199C23×C17H(5,𝔽2)12yes—
200C23×C172−1+412yes—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—22no—
204C22×D17—22no—
205C22×D17—24no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17C2×C4⋊C414yes—
218C22×D17C4×D414yes—
219C22×D17C4×Q814yes—
220C22×D17#12114yes—
221C22×D17C22⋊Q814yes—
222C22×D17#12514yes—
223C22×D17C4⋊Q814yes—
224C2×C4×C17—24no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—42no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17—42no—
252C2×C4×C17C2×C4⋊C414yes—
253C2×Dic17—24no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—24no—
263C2×Dic17—24no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—42no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17C2×C4⋊C414yes—
282C4×D17—416no—
283C4×D17—416no—
284C4×D17—416no—
285C2×#385—172no—
286C2×#380—171no—
287C2×#380—171no—
288C22×#95—171no—
289#1444—171no—
290#1444—171no—
291#1442—171no—
292#1442D1711yescharacteristic, complemented
293#1441—172no—
294#1441—172no—
295C22⋊#95—172no—
296C22×C17C22×C4⋊C411yescharacteristic
297C22×C17C2×C4×D411yescharacteristic
298C22×C17C2×C4×Q811yescharacteristic
299C22×C17#51812yes—
300C22×C17#52012yes—
301C22×C17C2×#12111yescharacteristic
302C22×C17C2×C22⋊Q811yescharacteristic
303C22×C17C2×#12511yescharacteristic
304C22×C17C2×C4⋊Q811yescharacteristic
305C22×C17#53411yescharacteristic
306C22×C17#53411yescharacteristic
307C22×C17#53412yes—
308C22×C17#53511yescharacteristic
309C22×C17#53611yescharacteristic
310C22×C17#53712yes—
311C22×C17#53812yes—
312C22×C17#53812yes—
313C22×C17#53912yes—
314C22×C17#53912yes—
315C22×C17#54312yes—
316C22×C17#54312yes—
317C22×C17#54412yes—
318C22×C17#54412yes—
319C2×D17—24no—
320C2×D17—24no—
321C2×D17—28no—
322C2×D17—28no—
323C2×D17—24no—
324C2×D17—24no—
325C2×D17—24no—
326C2×D17—24no—
327C2×D17—28no—
328C2×D17—28no—
329C2×D17#38918yes—
330C4×C17—24no—
331C4×C17—44no—
332C4×C17—42no—
333C4×C17—42no—
334C4×C17—82no—
335Dic17—24no—
336Dic17—44no—
337Dic17—42no—
338Dic17—42no—
339Dic17—82no—
340C24×C4—171no—
341C24×C4—171no—
342C24×C4—171no—
343C22×C4⋊C4—172no—
344C22×C4⋊C4—172no—
345C22×C4⋊C4—172no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4C2×D1711yescharacteristic
348C22×C22⋊C4—172no—
349C22×C22⋊C4—172no—
350#385—348no—
351#380—178no—
352#380—344no—
353C2×#95—171no—
354C2×#95—171no—
355C2×#95—171no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—171no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—344no—
363C2×#95C2×D1712yes—
364C2×#95C2×D1711yescharacteristic
365C2×#95C2×D1712yes—
366C2×#95C2×D1711yescharacteristic
367C2×C17C2×#38911yescharacteristic
368C2×C17#231511yescharacteristic
369C2×C17#231711yescharacteristic
370C2×C17#232512yes—
371C2×C17#232912yes—
372C2×C17#261711yescharacteristic
373C2×C17#261811yescharacteristic
374C2×C17#262611yescharacteristic
375C2×C17#262711yescharacteristic
376C2×C17#264612yes—
377C2×C17#265912yes—
378D17—28no—
379C25—171no—
380C23×C4—171no—
381C23×C4—174no—
382C23×C4—174no—
383C23×C4—174no—
384C23×C4—172no—
385C23×C4—171no—
386C23×C4—171no—
387C23×C4—174no—
388C23×C4—174no—
389C23×C4—172no—
390C23×C4—342no—
391C23×C4—342no—
392C23×C4—342no—
393C23×C4—342no—
394C23×C4—682no—
395C23×C4—682no—
396C23×C4—682no—
397C23×C4—682no—
398C23×C4—682no—
399C23×C4—682no—
400C23×C4C22×D1712yes—
401C23×C4C22×D1712yes—
402C23×C4C22×D1711yescharacteristic
403C23×C4C22×D1711yescharacteristic
404C23×C4C22×D1711yescharacteristic
405C2×C4⋊C4—22no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—178no—
408C2×C4⋊C4—344no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—344no—
411C2×C4⋊C4—682no—
412C2×C4⋊C4—682no—
413C2×C4⋊C4—682no—
414C2×C4⋊C4—682no—
415C2×C4⋊C4—682no—
416C2×C4⋊C4—682no—
417C2×C4⋊C4—682no—
418C2×C22⋊C4—178no—
419C2×C22⋊C4—344no—
420C2×C22⋊C4—344no—
421C2×C22⋊C4—344no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—344no—
424#95—22no—
425#95—24no—
426#95—42no—
427#95—174no—
428#95—342no—
429#95—342no—
430#95—344no—
431#95—344no—
432#95—342no—
433#95—682no—
434#95—682no—
435#95—682no—
436#95C4×D1714yescomplemented
437C17#1427511yescharacteristic, complemented, Sylow
438C24—172no—
439C24—172no—
440C24—172no—
441C24—172no—
442C24—172no—
443C24—172no—
444C24—172no—
445C24—342no—
446C24—342no—
447C24—342no—
448C24—342no—
449C24C23×D1711yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—172no—
474C22×C4—172no—
475C22×C4—172no—
476C22×C4—1716no—
477C22×C4—174no—
478C22×C4—172no—
479C22×C4—172no—
480C22×C4—348no—
481C22×C4—348no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—688no—
506C22×C4—684no—
507C22×C4—684no—
508C22×C4—688no—
509C22×C4—688no—
510C22×C4—688no—
511C22×C4—684no—
512C22×C4—684no—
513C22×C4—684no—
514C22×C4—684no—
515C22×C4—688no—
516C22×C4—688no—
517C22×C4C2×C4×D1714yes—
518C22×C4C2×C4×D1712yes—
519C22×C4D4×D1712yes—
520C22×C4D4×D1712yes—
521C22×C4Q8×D1712yes—
522C22×C4Q8×D1712yes—
523C4⋊C4—44no—
524C4⋊C4—688no—
525C4⋊C4—688no—
526C4⋊C4—684no—
527C4⋊C4—688no—
528C22⋊C4—3416no—
529C22⋊C4—688no—
530C23—174no—
531C23—174no—
532C23—174no—
533C23—174no—
534C23—174no—
535C23—174no—
536C23—174no—
537C23—344no—
538C23—344no—
539C23—342no—
540C23—342no—
541C23—344no—
542C23—342no—
543C23—342no—
544C23—344no—
545C23—344no—
546C23—344no—
547C23—344no—
548C23—344no—
549C23—344no—
550C23—344no—
551C23—344no—
552C23—344no—
553C23C22×C4×D1711yescharacteristic
554C23C2×D4×D1711yescharacteristic
555C23C2×D4×D1711yescharacteristic
556C23#1058769112yes—
557C23C2×Q8×D1711yescharacteristic
558C23C2×Q8×D1711yescharacteristic
559C23#41×D1711yescharacteristic
560C23#41×D1711yescharacteristic
561C23#1058769612yes—
562C23#1058769612yes—
563C23#1058769712yes—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—174no—
593C2×C4—344no—
594C2×C4—342no—
595C2×C4—342no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—344no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—344no—
603C2×C4—344no—
604C2×C4—6816no—
605C2×C4—6816no—
606C2×C4—6816no—
607C2×C4—681no—
608C2×C4—682no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—682no—
613C2×C4—682no—
614C2×C4—682no—
615C2×C4—682no—
616C2×C4—682no—
617C2×C4—682no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4C4⋊C4×D1714yes—
625C22—178no—
626C22—344no—
627C22—344no—
628C22—348no—
629C22—344no—
630C22—344no—
631C22—348no—
632C22—348no—
633C22—344no—
634C22—344no—
635C22—348no—
636C22C2×C4⋊C4×D1711yescharacteristic
637C22C4×D4×D1711yescharacteristic
638C22C4×Q8×D1711yescharacteristic
639C22#1176328112yes—
640C22#1176328912yes—
641C22#121×D1711yescharacteristic
642C22C22⋊Q8×D1711yescharacteristic
643C22#125×D1711yescharacteristic
644C22C4⋊Q8×D1711yescharacteristic
645C22#1176336911yescharacteristic
646C22#1176337011yescharacteristic
647C22#1176337012yes—
648C22#1176337711yescharacteristic
649C22#1176337911yescharacteristic
650C22#1176338812yes—
651C22#1176339312yes—
652C22#1176339812yes—
653C22#1176340012yes—
654C22#1176340212yes—
655C22#1176343412yes—
656C22#1176343412yes—
657C22#1176343912yes—
658C22#1176343912yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—344no—
665C4—682no—
666C4—684no—
667C4—682no—
668C4—1362no—
669C2—348no—
670C2D17×#38911yescharacteristic
671C2#1453139511yescharacteristic
672C2#1453140911yescharacteristic
673C2#1453148812yes—
674C2#1453151112yes—
675C2#1453451911yescharacteristic
676C2#1453452511yescharacteristic
677C2#1453458811yescharacteristic
678C2#1453459811yescharacteristic
679C2#1453475512yes—
680C2#1453489912yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1442C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1442D17
hypocenterC17#14275
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
5. #14527049C2semidirect
6. #14527048C2semidirect
7. #14527043C2semidirect
8. #14527038C2semidirect
9. #14527035C2semidirect
11. C17×#1442C2semidirect
22. #11762668C4semidirect
83. C2×#63225C2×C4semidirect
87. C17×#95C2×C4semidirect
96. #10587502C2×C4semidirect
181. C2×C4×D17C4⋊C4semidirect
292. #1442D17semidirect
436. #95C4×D17semidirect
437. C17#14275semidirect

Maximal subgroups

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

subgroupindex
C2×#117626832
C2×#117626682
C2×#117626662
C2×#95×D172
#145270492
#145270482
#145270432
#145270382
#145270352
#145270052
C17×#14422
#1427517