Finite Group Databasebeta

← #16957027

Subgroups — #16957027

Order 4352.

2078 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×#10587558C211yesmaximal, characteristic
2C2×#11762671C211yesmaximal, characteristic
3C2×#95×D17C211yesmaximal, characteristic
4#14527050C212yesmaximal, complemented
5#14527050C212yesmaximal, complemented
6#14527011C211yesmaximal, characteristic, complemented
7#14527008C211yesmaximal, characteristic, complemented
8#14527008C211yesmaximal, characteristic, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2212yes—
13C23×C4×D17C2211yescharacteristic
14C2×#10587558—24no—
15C2×#10587558—24no—
16C2×#10587558C418yescomplemented
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×#63227C2212yes—
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×C4×Dic17C2211yescharacteristic
23#11762671—24no—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34#95×D17—24no—
35C22×C42×C17C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17—24no—
55C22×C4×D17—24no—
56C22×C4×D17—24no—
57C22×C4×D17—24no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17C2×C414yes—
61C22×C4×D17D414yes—
62C22×C4×D17D414yes—
63C2×C42×C17—22no—
64C2×C42×C17—22no—
65C2×C42×C17C2×C414yescomplemented
66C2×#63227—22no—
67C2×#63227—22no—
68C2×#63227—22no—
69C2×#63227—22no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—24no—
73C2×#63227C2×C414yes—
74C2×#63227C2×C414yes—
75C2×#63225—22no—
76C2×#63225—22no—
77C2×#63225—22no—
78C2×#63225—22no—
79C2×#63225C2×C414yescomplemented
80C2×#63225C2×C414yescomplemented
81C2×C4×Dic17—22no—
82C2×C4×Dic17—22no—
83C2×C4×Dic17C2×C414yescomplemented
84#10587558—232no—
85C17×#95—22no—
86C17×#95—22no—
87C17×#95—24no—
88C17×#95—42no—
89#10587502—22no—
90#10587502—22no—
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×C17C2411yescharacteristic
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—21no—
103C23×D17—21no—
104C23×D17—21no—
105C23×D17—21no—
106C23×D17C22×C412yes—
107C23×D17C2×D412yes—
108C23×D17C2×D412yes—
109C23×D17#4114yes—
110C23×D17#4114yes—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
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—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
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×C17C22×C412yes—
137C22×C4×C17C22×C412yes—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C2×D412yes—
140C22×C4×C17C2×D412yes—
141C22×C4×C17#4114yes—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17C22×C412yes—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C2×D412yes—
171C22×Dic17C2×D412yes—
172C22×Dic17#4114yes—
173C2×C4×D17—28no—
174C2×C4×D17—28no—
175C2×C4×D17—216no—
176C2×C4×D17—216no—
177C2×C4×D17—216no—
178C2×C4×D17—48no—
179C2×C4×D17—48no—
180C2×C4×D17—48no—
181C2×C4×D17—48no—
182C2×C4×D17—48no—
183C2×C4×D17—48no—
184C2×C4×D17C22⋊C4116yes—
185C42×C17—28no—
186#63227—28no—
187#63227—28no—
188#63227—48no—
189#63225—28no—
190#63225—28no—
191C4×Dic17—28no—
192#13958—171nomaximal, Sylow
193C23×C17C23×C411yescharacteristic
194C23×C17C22×D411yescharacteristic
195C23×C17C22×D411yescharacteristic
196C23×C17C2×#4112yes—
197C23×C17C2×#4112yes—
198C23×C17C2×#4112yes—
199C23×C17C2×#4112yes—
200C23×C17H(5,𝔽2)11yescharacteristic
201C23×C17H(5,𝔽2)11yescharacteristic
202C23×C172−1+411yescharacteristic
203C23×C172−1+411yescharacteristic
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—22no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17—24no—
218C22×D17—24no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17C2×C22⋊C414yes—
223C22×D17#11714yes—
224C22×D17#11714yes—
225C22×D17#12318yes—
226C22×D17#12318yes—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
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—22no—
259C2×Dic17—22no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
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—232no—
290C4×D17—432no—
291C22×#117—171no—
292C2×#381—171no—
293C22×#95—171no—
294#1445—172no—
295#1445—172no—
296#1434—171no—
297#1434—171no—
298#1434—171no—
299#1434D1711yescharacteristic, complemented
300C22⋊#95—172no—
301C22⋊#95—172no—
302C22×C17C22×C22⋊C411yescharacteristic
303C22×C17C2×#11711yescharacteristic
304C22×C17C2×#11711yescharacteristic
305C22×C17C4×#4111yescharacteristic
306C22×C17C4×#4111yescharacteristic
307C22×C17#52011yescharacteristic
308C22×C17#52011yescharacteristic
309C22×C17C2×#12312yes—
310C22×C17C2×#12312yes—
311C22×C17#52512yes—
312C22×C17#53212yes—
313C22×C17#53511yescharacteristic
314C22×C17#53611yescharacteristic
315C22×C17#53711yescharacteristic
316C22×C17#53711yescharacteristic
317C22×C17#55112yes—
318C22×C17#55112yes—
319C22×C17#55112yes—
320C22×C17#55112yes—
321C22×C17#55212yes—
322C22×C17#55212yes—
323C22×C17#55212yes—
324C22×C17#55212yes—
325C2×D17—24no—
326C2×D17—24no—
327C2×D17—24no—
328C2×D17—24no—
329C2×D17—24no—
330C2×D17—28no—
331C2×D17—28no—
332C2×D17—28no—
333C2×D17—28no—
334C2×D17—24no—
335C2×D17#38118yes—
336C4×C17—24no—
337C4×C17—42no—
338C4×C17—44no—
339C4×C17—42no—
340C4×C17—82no—
341Dic17—24no—
342Dic17—42no—
343Dic17—44no—
344Dic17—42no—
345Dic17—82no—
346C24×C4—171no—
347C24×C4—172no—
348C2×#117—178no—
349C2×#117—344no—
350C2×#117—344no—
351C22×C4⋊C4—171no—
352C22×C4⋊C4—171no—
353C22×C22⋊C4—171no—
354C22×C22⋊C4—171no—
355C22×C22⋊C4—172no—
356C22×C42—171no—
357C22×C42C2×D1711yescharacteristic
358#381—344no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—171no—
363C2×#95—171no—
364C2×#95—172no—
365C2×#95—172no—
366C2×#95—172no—
367C2×#95—172no—
368C2×#95—172no—
369C2×#95—344no—
370C2×#95C2×D1712yes—
371C2×#95C2×D1711yescharacteristic
372C2×#95C2×D1711yescharacteristic
373C2×#95C2×D1712yes—
374C2×C17C2×#38111yescharacteristic
375C2×C17#229711yescharacteristic
376C2×C17#231111yescharacteristic
377C2×C17#234311yescharacteristic
378C2×C17#234311yescharacteristic
379C2×C17#234311yescharacteristic
380C2×C17#234311yescharacteristic
381C2×C17#248212yes—
382C2×C17#248412yes—
383C2×C17#249812yes—
384C2×C17#249812yes—
385D17—28no—
386C25—171no—
387C23×C4—171no—
388C23×C4—174no—
389C23×C4—174no—
390C23×C4—174no—
391C23×C4—171no—
392C23×C4—171no—
393C23×C4—172no—
394C23×C4—172no—
395C23×C4—344no—
396C23×C4—344no—
397C23×C4—344no—
398C23×C4—344no—
399C23×C4—344no—
400C23×C4—344no—
401C23×C4—344no—
402C23×C4—682no—
403C23×C4—682no—
404C23×C4C22×D1712yes—
405C23×C4C22×D1711yescharacteristic
406C23×C4C22×D1711yescharacteristic
407C23×C4C22×D1712yes—
408C23×C4C22×D1711yescharacteristic
409#117—3432no—
410C2×C4⋊C4—174no—
411C2×C4⋊C4—174no—
412C2×C4⋊C4—342no—
413C2×C4⋊C4—342no—
414C2×C4⋊C4—342no—
415C2×C4⋊C4—342no—
416C2×C22⋊C4—174no—
417C2×C22⋊C4—174no—
418C2×C22⋊C4—342no—
419C2×C22⋊C4—342no—
420C2×C22⋊C4—342no—
421C2×C22⋊C4—342no—
422C2×C22⋊C4—344no—
423C2×C22⋊C4—344no—
424C2×C22⋊C4—344no—
425C2×C42—22no—
426C2×C42—22no—
427C2×C42—174no—
428C2×C42—342no—
429C2×C42—342no—
430C2×C42C4×D1714yescomplemented
431#95—24no—
432#95—22no—
433#95—22no—
434#95—42no—
435#95—344no—
436#95—344no—
437#95—344no—
438#95—344no—
439#95—344no—
440#95—342no—
441#95—342no—
442#95—682no—
443#95—682no—
444#95—682no—
445C17#1395811yescharacteristic, complemented, Sylow
446C24—172no—
447C24—172no—
448C24—172no—
449C24—174no—
450C24—174no—
451C24—342no—
452C24—341no—
453C24—341no—
454C24—342no—
455C24—341no—
456C24—341no—
457C24C23×D1711yescharacteristic
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—172no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—1716no—
487C22×C4—172no—
488C22×C4—172no—
489C22×C4—174no—
490C22×C4—348no—
491C22×C4—348no—
492C22×C4—3416no—
493C22×C4—3416no—
494C22×C4—3416no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—342no—
499C22×C4—342no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—341no—
511C22×C4—342no—
512C22×C4—341no—
513C22×C4—342no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—688no—
521C22×C4—688no—
522C22×C4—688no—
523C22×C4—688no—
524C22×C4—688no—
525C22×C4—688no—
526C22×C4C2×C4×D1712yes—
527C22×C4C2×C4×D1712yes—
528C22×C4C2×C4×D1712yes—
529C22×C4#6325214yes—
530C22×C4D4×D1712yes—
531C22×C4D4×D1712yes—
532C4⋊C4—348no—
533C4⋊C4—348no—
534C22⋊C4—348no—
535C22⋊C4—348no—
536C22⋊C4—688no—
537C42—28no—
538C42—348no—
539C23—174no—
540C23—174no—
541C23—174no—
542C23—178no—
543C23—178no—
544C23—344no—
545C23—344no—
546C23—344no—
547C23—344no—
548C23—344no—
549C23—344no—
550C23—344no—
551C23—344no—
552C23—342no—
553C23—342no—
554C23—342no—
555C23—342no—
556C23—344no—
557C23—344no—
558C23—342no—
559C23—342no—
560C23—342no—
561C23—342no—
562C23C22×C4×D1711yescharacteristic
563C23C2×#6325212yes—
564C23C2×D4×D1711yescharacteristic
565C23C2×D4×D1711yescharacteristic
566C23#1058769111yescharacteristic
567C23#1058769411yescharacteristic
568C23#41×D1712yes—
569C23#41×D1712yes—
570C23#41×D1712yes—
571C23#1058769611yescharacteristic
572C23#1058769711yescharacteristic
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—174no—
604C2×C4—3432no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—344no—
608C2×C4—344no—
609C2×C4—344no—
610C2×C4—344no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—344no—
618C2×C4—6832no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
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—178no—
638C22—344no—
639C22—344no—
640C22—344no—
641C22—344no—
642C22—344no—
643C22—348no—
644C22—348no—
645C22—348no—
646C22—344no—
647C22—348no—
648C22C2×C22⋊C4×D1711yescharacteristic
649C22#117×D1711yescharacteristic
650C22#117×D1711yescharacteristic
651C22#1176327711yescharacteristic
652C22C4×#6325211yescharacteristic
653C22#1176328911yescharacteristic
654C22#1176329011yescharacteristic
655C22#123×D1712yes—
656C22#123×D1712yes—
657C22#1176331812yes—
658C22#1176335612yes—
659C22#1176337511yescharacteristic
660C22#1176338211yescharacteristic
661C22#1176338811yescharacteristic
662C22#1176338911yescharacteristic
663C22#1176348312yes—
664C22#1176348412yes—
665C22#1176348812yes—
666C22#1176348912yes—
667C22#1176349112yes—
668C22#1176349512yes—
669C22#1176349812yes—
670C22#1176350012yes—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—82no—
676C4—344no—
677C4—682no—
678C4—684no—
679C4—682no—
680C4—1362no—
681C2—348no—
682C2D17×#38111yescharacteristic
683C2#1453127011yescharacteristic
684C2#1453137311yescharacteristic
685C2#1453163711yescharacteristic
686C2#1453164111yescharacteristic
687C2#1453164511yescharacteristic
688C2#1453164711yescharacteristic
689C2#1453305612yes—
690C2#1453306912yes—
691C2#1453322512yes—
692C2#1453322712yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13958
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #14527050C2semidirect
5. #14527050C2semidirect
6. #14527011C2semidirect
7. #14527008C2semidirect
8. #14527008C2semidirect
11. C17×#1434C2semidirect
16. C2×#10587558C4semidirect
65. C2×C42×C17C2×C4semidirect
79. C2×#63225C2×C4semidirect
80. C2×#63225C2×C4semidirect
83. C2×C4×Dic17C2×C4semidirect
299. #1434D17semidirect
430. C2×C42C4×D17semidirect
445. C17#13958semidirect

Maximal subgroups

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

subgroupindex
C22×#105875582
C2×#117626712
C2×#95×D172
#145270502
#145270112
#145270082
#145270052
C17×#14342
#1395817