Finite Group Databasebeta

← #16965359

Subgroups — #16965359

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587582C211yesmaximal, characteristic
2C2×#11762671C211yesmaximal, characteristic
3C2×#11762666C211yesmaximal, characteristic
4#14527054C212yesmaximal, complemented
5#14527050C212yesmaximal, complemented
6#14527038C211yesmaximal, characteristic, complemented
7#14527037C211yesmaximal, characteristic, complemented
8#14527034C211yesmaximal, characteristic, complemented
9#14527032C212yesmaximal
10#14527032C212yesmaximal
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×Dic34C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C2×#10587582—24no—
15C2×#10587582—24no—
16C2×#10587582C418yescomplemented
17C22×C4⋊Dic17C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×#63225C2211yescharacteristic
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×#63227C2212yes—
23C22×#63227C2212yes—
24#11762671—24no—
25#11762666—24no—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—42no—
52C22×C4×D17—42no—
53C22×C4×D17C2×C414yes—
54C22×C4×D17D414yes—
55C22×C4×D17D414yes—
56C22×Dic34—22no—
57C22×Dic34—22no—
58C22×Dic34—42no—
59C22×Dic34—42no—
60C22×Dic34C2×C414yes—
61C2×C4⋊C4×C17—22no—
62C2×C4⋊C4×C17—22no—
63C2×C4⋊C4×C17—22no—
64C2×C4⋊C4×C17—42no—
65C2×C4⋊C4×C17C2×C414yes—
66C2×#63227—22no—
67C2×#63227—22no—
68C2×#63227—22no—
69C2×#63227—22no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—42no—
73C2×#63227—42no—
74C2×#63227—42no—
75C2×#63227—42no—
76C2×#63227C2×C414yes—
77C2×#63227C2×C414yes—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17—42no—
80C2×#63225—22no—
81C2×#63225—22no—
82C2×#63225—22no—
83C2×#63225—22no—
84C2×#63225C2×C414yes—
85C2×#63225C2×C414yes—
86#10587582—232no—
87C17×#95—22no—
88C17×#95—42no—
89C17×#95—42no—
90#10587502—22no—
91#10587502—24no—
92#10587502—24no—
93#10587502—42no—
94#10587502—42no—
95#10587502—42no—
96#10587502—42no—
97#10587502—42no—
98#10587502—42no—
99C24×C17C2411yescharacteristic
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—21no—
103C23×D17—21no—
104C23×D17—22no—
105C23×D17—21no—
106C23×D17—22no—
107C23×D17—21no—
108C23×D17C22×C412yes—
109C23×D17C2×D412yes—
110C23×D17C2×D412yes—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—21no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
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×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C22×C412yes—
140C22×C4×C17C2×D412yes—
141C22×C4×C17C2×D412yes—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C22×C412yes—
171C22×Dic17C2×D412yes—
172C22×Dic17C2×D412yes—
173C2×C4×D17—48no—
174C2×C4×D17—44no—
175C2×C4×D17—48no—
176C2×C4×D17—44no—
177C2×C4×D17C22⋊C4116yes—
178C2×Dic34—28no—
179C2×Dic34—48no—
180C2×Dic34—48no—
181C2×Dic34—44no—
182C2×Dic34—44no—
183C4⋊C4×C17—28no—
184C4⋊C4×C17—44no—
185#63227—28no—
186#63227—28no—
187#63227—48no—
188#63227—48no—
189C4⋊Dic17—44no—
190#63225—28no—
191#63225—28no—
192#14660—171nomaximal, Sylow
193C23×C17C23×C411yescharacteristic
194C23×C17C22×D411yescharacteristic
195C23×C17C22×D411yescharacteristic
196C23×C17H(5,𝔽2)12yes—
197C23×C17H(5,𝔽2)11yescharacteristic
198C23×C17H(5,𝔽2)12yes—
199C23×C17H(5,𝔽2)11yescharacteristic
200C23×C172−1+412yes—
201C23×C172−1+411yescharacteristic
202C23×C172−1+412yes—
203C23×C172−1+411yescharacteristic
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—22no—
209C22×D17—22no—
210C22×D17—24no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—24no—
214C22×D17—22no—
215C22×D17—22no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—42no—
223C22×D17—42no—
224C22×D17—42no—
225C22×D17—42no—
226C22×D17C2×C22⋊C414yes—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—42no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
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×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17C2×C22⋊C414yes—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—24no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—42no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17—42no—
277C2×Dic17—42no—
278C2×Dic17—42no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17C2×C22⋊C414yes—
289C4×D17—416no—
290Dic34—416no—
291C22×C22⋊Q8—171no—
292C2×#381—171no—
293C2×#380—171no—
294#1446—172no—
295#1445—172no—
296#1441—171no—
297#1441—171no—
298#1441—171no—
299#1441D1711yescharacteristic, complemented
300#1440—172no—
301#1440—172no—
302C22×C17C22×C22⋊C411yescharacteristic
303C22×C17#51812yes—
304C22×C17#51912yes—
305C22×C17#52011yescharacteristic
306C22×C17#52011yescharacteristic
307C22×C17#53412yes—
308C22×C17#53411yescharacteristic
309C22×C17#53411yescharacteristic
310C22×C17#53511yescharacteristic
311C22×C17#53611yescharacteristic
312C22×C17#53611yescharacteristic
313C22×C17#53612yes—
314C22×C17#53611yescharacteristic
315C22×C17#53711yescharacteristic
316C22×C17#53711yescharacteristic
317C22×C17#56212yes—
318C22×C17#56212yes—
319C22×C17#56212yes—
320C22×C17#56212yes—
321C22×C17#56312yes—
322C22×C17#56312yes—
323C22×C17#56312yes—
324C22×C17#56312yes—
325C2×D17—24no—
326C2×D17—24no—
327C2×D17—24no—
328C2×D17—42no—
329C2×D17—42no—
330C2×D17—42no—
331C2×D17—42no—
332C2×D17—44no—
333C2×D17—44no—
334C2×D17—44no—
335C2×D17—44no—
336C4×C17—24no—
337C4×C17—42no—
338C4×C17—42no—
339C4×C17—82no—
340C4×C17—82no—
341Dic17—24no—
342Dic17—42no—
343Dic17—42no—
344Dic17—82no—
345Dic17—82no—
346C23×Q8—171no—
347C24×C4—171no—
348C2×C22⋊Q8—178no—
349C2×C22⋊Q8—344no—
350C2×C22⋊Q8—344no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4—171no—
353C22×C4⋊C4—171no—
354C22×C4⋊C4C2×D1711yescharacteristic
355C22×C4⋊C4C2×D1711yescharacteristic
356C22×C22⋊C4—171no—
357C22×C22⋊C4—171no—
358C22×C22⋊C4—172no—
359C22×C22⋊C4—172no—
360#381—344no—
361#380—344no—
362C2×#95—171no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95C2×D1711yescharacteristic
372C2×#95C2×D1712yes—
373C2×#95C2×D1712yes—
374C2×C17#230811yescharacteristic
375C2×C17#231011yescharacteristic
376C2×C17#231111yescharacteristic
377C2×C17#237812yes—
378C2×C17#237812yes—
379C2×C17#281412yes—
380C2×C17#281412yes—
381C2×C17#281611yescharacteristic
382C2×C17#281611yescharacteristic
383C2×C17#281711yescharacteristic
384C2×C17#281711yescharacteristic
385D17—44no—
386C25—171no—
387C22×Q8—174no—
388C22×Q8—342no—
389C22×Q8—342no—
390C22×Q8—682no—
391C22×Q8—682no—
392C23×C4—171no—
393C23×C4—174no—
394C23×C4—174no—
395C23×C4—174no—
396C23×C4—171no—
397C23×C4—171no—
398C23×C4—172no—
399C23×C4—172no—
400C23×C4—682no—
401C23×C4—682no—
402C23×C4C22×D1712yes—
403C23×C4C22×D1712yes—
404C23×C4C22×D1711yescharacteristic
405C23×C4C22×D1711yescharacteristic
406C23×C4C22×D1711yescharacteristic
407C22⋊Q8—3432no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—42no—
412C2×C4⋊C4—174no—
413C2×C4⋊C4—174no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C4⋊C4—342no—
417C2×C4⋊C4—342no—
418C2×C4⋊C4—342no—
419C2×C4⋊C4—682no—
420C2×C4⋊C4C4×D1714yes—
421C2×C22⋊C4—174no—
422C2×C22⋊C4—174no—
423C2×C22⋊C4—342no—
424C2×C22⋊C4—342no—
425C2×C22⋊C4—342no—
426C2×C22⋊C4—342no—
427C2×C22⋊C4—344no—
428C2×C22⋊C4—344no—
429C2×C22⋊C4—682no—
430C2×C22⋊C4—682no—
431C2×C22⋊C4—682no—
432C2×C22⋊C4—682no—
433#95—22no—
434#95—42no—
435#95—42no—
436#95—344no—
437#95—344no—
438#95—342no—
439#95—682no—
440#95—682no—
441#95—682no—
442#95—682no—
443#95—682no—
444#95—682no—
445C17#1466011yescharacteristic, complemented, Sylow
446C24—172no—
447C24—172no—
448C24—172no—
449C24—341no—
450C24—342no—
451C24—342no—
452C24—341no—
453C24—341no—
454C24—342no—
455C24—342no—
456C24—341no—
457C24C23×D1711yescharacteristic
458C2×Q8—348no—
459C2×Q8—688no—
460C2×Q8—688no—
461C2×Q8—684no—
462C2×Q8—684no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—172no—
490C22×C4—172no—
491C22×C4—172no—
492C22×C4—1716no—
493C22×C4—172no—
494C22×C4—172no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—688no—
522C22×C4—688no—
523C22×C4—684no—
524C22×C4—684no—
525C22×C4C2×C4×D1712yes—
526C22×C4C2×C4×D1712yes—
527C22×C4C2×C4×D1712yes—
528C22×C4D4×D1712yes—
529C22×C4D4×D1712yes—
530C4⋊C4—28no—
531C4⋊C4—44no—
532C4⋊C4—348no—
533C4⋊C4—348no—
534C4⋊C4—684no—
535C22⋊C4—348no—
536C22⋊C4—348no—
537C22⋊C4—688no—
538C22⋊C4—688no—
539C23—174no—
540C23—344no—
541C23—344no—
542C23—342no—
543C23—342no—
544C23—344no—
545C23—342no—
546C23—342no—
547C23—342no—
548C23—342no—
549C23—344no—
550C23—342no—
551C23—342no—
552C23—342no—
553C23—342no—
554C23—682no—
555C23—682no—
556C23—682no—
557C23—682no—
558C23—682no—
559C23—682no—
560C23—682no—
561C23—682no—
562C23C22×C4×D1711yescharacteristic
563C23C2×D4×D1711yescharacteristic
564C23C2×D4×D1711yescharacteristic
565C23#1058769112yes—
566C23#1058769111yescharacteristic
567C23#1058769412yes—
568C23#1058769411yescharacteristic
569C23#1058769612yes—
570C23#1058769611yescharacteristic
571C23#1058769712yes—
572C23#1058769711yescharacteristic
573Q8—6816no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—174no—
605C2×C4—344no—
606C2×C4—344no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—342no—
610C2×C4—342no—
611C2×C4—344no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—6816no—
615C2×C4—682no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4—682no—
627C2×C4—682no—
628C2×C4—682no—
629C2×C4—682no—
630C2×C4—682no—
631C2×C4—682no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4C22⋊C4×D1714yes—
637C22—344no—
638C22—344no—
639C22—344no—
640C22—684no—
641C22—684no—
642C22—684no—
643C22—684no—
644C22—682no—
645C22—682no—
646C22—682no—
647C22—682no—
648C22C2×C22⋊C4×D1711yescharacteristic
649C22#1176328112yes—
650C22#1176328512yes—
651C22#1176328911yescharacteristic
652C22#1176329011yescharacteristic
653C22#1176336712yes—
654C22#1176336911yescharacteristic
655C22#1176337011yescharacteristic
656C22#1176337511yescharacteristic
657C22#1176337911yescharacteristic
658C22#1176338212yes—
659C22#1176338211yescharacteristic
660C22#1176338411yescharacteristic
661C22#1176338811yescharacteristic
662C22#1176338911yescharacteristic
663C22#1176355412yes—
664C22#1176355512yes—
665C22#1176355612yes—
666C22#1176355712yes—
667C22#1176355812yes—
668C22#1176356012yes—
669C22#1176356112yes—
670C22#1176356312yes—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—344no—
677C4—682no—
678C4—682no—
679C4—1362no—
680C4—1362no—
681C2—684no—
682C2#1453135811yescharacteristic
683C2#1453136711yescharacteristic
684C2#1453137311yescharacteristic
685C2#1453194612yes—
686C2#1453194912yes—
687C2#1453680212yes—
688C2#1453680412yes—
689C2#1453681511yescharacteristic
690C2#1453681711yescharacteristic
691C2#1453681811yescharacteristic
692C2#1453682011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#14660
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. #14527054C2semidirect
5. #14527050C2semidirect
6. #14527038C2semidirect
7. #14527037C2semidirect
8. #14527034C2semidirect
11. C17×#1441C2semidirect
16. C2×#10587582C4semidirect
299. #1441D17semidirect
445. C17#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#105875822
C2×#117626712
C2×#117626662
#145270542
#145270502
#145270382
#145270372
#145270342
#145270322
C17×#14412
#1466017