Finite Group Databasebeta

← #16956857

Subgroups — #16956857

Order 4352.

1862 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587558C211yesmaximal, characteristic
2C2×#11762668C211yesmaximal, characteristic
3C2×#11762656C211yesmaximal, characteristic
4#14527032C212yesmaximal
5#14527032C212yesmaximal
6#14527012C212yesmaximal, complemented
7#14527012C212yesmaximal, complemented
8#14527011C211yesmaximal, characteristic, complemented
9#14527008C211yesmaximal, characteristic, complemented
10#14527008C211yesmaximal, characteristic, complemented
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587558—24no—
14C2×#10587558—24no—
15C2×#10587558C418yescomplemented
16C22×#63225C2212yes—
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C22×#63227C2212yes—
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×#63227C2212yes—
23C22×C4×Dic17C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25#11762668—24no—
26#11762656—24no—
27C2×#10587502C2211yescharacteristic
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C42×C17C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
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×D17C2×C414yes—
54C22×C4×D17D414yes—
55C22×C4×D17D414yes—
56C2×C42×C17—22no—
57C2×C42×C17—22no—
58C2×C42×C17C2×C414yes—
59C2×#63227—22no—
60C2×#63227—22no—
61C2×#63227—22no—
62C2×#63227—22no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227C2×C414yes—
70C2×#63227C2×C414yes—
71C2×#63225—22no—
72C2×#63225—22no—
73C2×#63225—22no—
74C2×#63225—22no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—24no—
78C2×#63225C2×C414yes—
79C2×#63225C2×C414yes—
80C2×C4×Dic17—22no—
81C2×C4×Dic17—22no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17—24no—
84C2×C4×Dic17—24no—
85C2×C4×Dic17C2×C414yes—
86#10587558—232no—
87C17×#95—22no—
88C17×#95—22no—
89C17×#95—24no—
90C17×#95—42no—
91#10587502—22no—
92#10587502—22no—
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—22no—
103C23×D17—22no—
104C23×D17—21no—
105C23×D17—21no—
106C23×D17—21no—
107C23×D17—21no—
108C23×D17C22×C412yes—
109C23×D17C2×D412yes—
110C23×D17C2×D412yes—
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—21no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
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×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—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—22no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17C22×C412yes—
166C22×Dic17C22×C412yes—
167C22×Dic17C22×C412yes—
168C22×Dic17C2×D412yes—
169C22×Dic17C2×D412yes—
170C22×Dic17#4114yes—
171C22×Dic17#4114yes—
172C22×Dic17#4114yes—
173C2×C4×D17—28no—
174C2×C4×D17—28no—
175C2×C4×D17—216no—
176C2×C4×D17—216no—
177C2×C4×D17C22⋊C4116yes—
178C42×C17—28no—
179#63227—216no—
180#63227—28no—
181#63227—28no—
182#63227—48no—
183#63225—28no—
184#63225—28no—
185#63225—48no—
186C4×Dic17—28no—
187C4×Dic17—48no—
188#13939—171nomaximal, Sylow
189C23×C17C23×C411yescharacteristic
190C23×C17C22×D411yescharacteristic
191C23×C17C22×D411yescharacteristic
192C23×C17C2×#4112yes—
193C23×C17C2×#4112yes—
194C23×C17C2×#4112yes—
195C23×C17C2×#4112yes—
196C23×C17H(5,𝔽2)11yescharacteristic
197C23×C17H(5,𝔽2)11yescharacteristic
198C23×C172−1+411yescharacteristic
199C23×C172−1+411yescharacteristic
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17—24no—
209C22×D17—24no—
210C22×D17—22no—
211C22×D17—22no—
212C22×D17—22no—
213C22×D17—22no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17C2×C22⋊C414yes—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—24no—
234C2×C4×C17—24no—
235C2×C4×C17—24no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—42no—
243C2×C4×C17—41no—
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×C17C2×C22⋊C414yes—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
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×Dic17C2×C22⋊C414yes—
281C2×Dic17#11714yes—
282C2×Dic17#11714yes—
283C2×Dic17#12318yes—
284C2×Dic17#12318yes—
285C4×D17—232no—
286C22×#117—171no—
287C2×#380—171no—
288C2×C4×C22⋊C4—171no—
289#1440—172no—
290#1440—172no—
291C4⋊#95—172no—
292C4⋊#95—172no—
293#1434—171no—
294#1434—171no—
295#1434—171no—
296#1434D1711yescharacteristic, complemented
297C22×C17C22×C22⋊C411yescharacteristic
298C22×C17C2×#11711yescharacteristic
299C22×C17C2×#11711yescharacteristic
300C22×C17C4×#4111yescharacteristic
301C22×C17C4×#4111yescharacteristic
302C22×C17#52011yescharacteristic
303C22×C17#52011yescharacteristic
304C22×C17C2×#12312yes—
305C22×C17C2×#12312yes—
306C22×C17#52512yes—
307C22×C17#53212yes—
308C22×C17#53511yescharacteristic
309C22×C17#53611yescharacteristic
310C22×C17#53711yescharacteristic
311C22×C17#53711yescharacteristic
312C22×C17#55212yes—
313C22×C17#55212yes—
314C22×C17#55312yes—
315C22×C17#55312yes—
316C22×C17#55512yes—
317C22×C17#55512yes—
318C22×C17#55612yes—
319C22×C17#55612yes—
320C2×D17—24no—
321C2×D17—24no—
322C2×D17—24no—
323C2×D17—42no—
324C2×D17—42no—
325C2×D17—42no—
326C2×D17—42no—
327C2×D17—44no—
328C2×D17—44no—
329C2×D17—44no—
330C2×D17—44no—
331C4×C17—24no—
332C4×C17—42no—
333C4×C17—44no—
334C4×C17—42no—
335C4×C17—82no—
336Dic17—42no—
337Dic17—44no—
338Dic17—42no—
339Dic17—82no—
340Dic17#38118yes—
341C24×C4—171no—
342C2×#117—178no—
343C2×#117—344no—
344C2×#117—344no—
345C22×C4⋊C4—171no—
346C22×C4⋊C4—171no—
347C22×C4⋊C4—172no—
348C22×C22⋊C4—171no—
349C22×C22⋊C4—171no—
350C22×C22⋊C4—172no—
351C22×C22⋊C4—172no—
352C22×C42—171no—
353C22×C42—172no—
354C22×C42C2×D1711yescharacteristic
355#380—344no—
356C4×C22⋊C4—344no—
357C2×#95—171no—
358C2×#95—171no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95—172no—
363C2×#95—172no—
364C2×#95—172no—
365C2×#95C2×D1711yescharacteristic
366C2×#95C2×D1711yescharacteristic
367C2×#95C2×D1712yes—
368C2×#95C2×D1712yes—
369C2×C17C2×#38111yescharacteristic
370C2×C17#229711yescharacteristic
371C2×C17#231111yescharacteristic
372C2×C17#234611yescharacteristic
373C2×C17#234611yescharacteristic
374C2×C17#234711yescharacteristic
375C2×C17#234711yescharacteristic
376C2×C17#248512yes—
377C2×C17#248512yes—
378C2×C17#249912yes—
379C2×C17#250012yes—
380D17—44no—
381C25—171no—
382C23×C4—171no—
383C23×C4—174no—
384C23×C4—174no—
385C23×C4—174no—
386C23×C4—171no—
387C23×C4—171no—
388C23×C4—172no—
389C23×C4—172no—
390C23×C4—344no—
391C23×C4—344no—
392C23×C4C22×D1712yes—
393C23×C4C22×D1711yescharacteristic
394C23×C4C22×D1711yescharacteristic
395C23×C4C22×D1712yes—
396C23×C4C22×D1711yescharacteristic
397#117—3432no—
398C2×C4⋊C4—174no—
399C2×C4⋊C4—174no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—342no—
402C2×C4⋊C4—342no—
403C2×C4⋊C4—342no—
404C2×C4⋊C4—344no—
405C2×C4⋊C4—344no—
406C2×C4⋊C4—344no—
407C2×C22⋊C4—174no—
408C2×C22⋊C4—174no—
409C2×C22⋊C4—344no—
410C2×C22⋊C4—342no—
411C2×C22⋊C4—342no—
412C2×C22⋊C4—342no—
413C2×C22⋊C4—342no—
414C2×C22⋊C4—344no—
415C2×C22⋊C4—344no—
416C2×C22⋊C4—344no—
417C2×C22⋊C4—682no—
418C2×C22⋊C4—682no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—174no—
422C2×C42—342no—
423C2×C42—342no—
424C2×C42—344no—
425C2×C42—344no—
426C2×C42—344no—
427C2×C42C4×D1714yes—
428#95—24no—
429#95—22no—
430#95—22no—
431#95—42no—
432#95—344no—
433#95—344no—
434#95—344no—
435#95—342no—
436#95—342no—
437#95—682no—
438#95—682no—
439#95—682no—
440C17#1393911yescharacteristic, complemented, Sylow
441C24—172no—
442C24—172no—
443C24—172no—
444C24—342no—
445C24—341no—
446C24—341no—
447C24—342no—
448C24—342no—
449C24—342no—
450C24—341no—
451C24—341no—
452C24C23×D1711yescharacteristic
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—172no—
479C22×C4—172no—
480C22×C4—172no—
481C22×C4—174no—
482C22×C4—174no—
483C22×C4—1716no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—174no—
487C22×C4—348no—
488C22×C4—348no—
489C22×C4—3416no—
490C22×C4—3416no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—342no—
500C22×C4—341no—
501C22×C4—342no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—341no—
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×C4C2×C4×D1712yes—
515C22×C4C2×C4×D1712yes—
516C22×C4C2×C4×D1712yes—
517C22×C4#6325214yes—
518C22×C4D4×D1712yes—
519C22×C4D4×D1712yes—
520C4⋊C4—348no—
521C4⋊C4—348no—
522C4⋊C4—688no—
523C22⋊C4—348no—
524C22⋊C4—348no—
525C22⋊C4—3416no—
526C22⋊C4—688no—
527C42—28no—
528C42—348no—
529C42—688no—
530C23—174no—
531C23—342no—
532C23—342no—
533C23—342no—
534C23—342no—
535C23—342no—
536C23—342no—
537C23—344no—
538C23—344no—
539C23—344no—
540C23—344no—
541C23—342no—
542C23—342no—
543C23—342no—
544C23—342no—
545C23—682no—
546C23—682no—
547C23—682no—
548C23—682no—
549C23—682no—
550C23—682no—
551C23—682no—
552C23—682no—
553C23C22×C4×D1711yescharacteristic
554C23C2×#6325212yes—
555C23C2×D4×D1711yescharacteristic
556C23C2×D4×D1711yescharacteristic
557C23#1058769111yescharacteristic
558C23#1058769411yescharacteristic
559C23#41×D1712yes—
560C23#41×D1712yes—
561C23#41×D1712yes—
562C23#1058769611yescharacteristic
563C23#1058769711yescharacteristic
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—174no—
595C2×C4—174no—
596C2×C4—174no—
597C2×C4—178no—
598C2×C4—178no—
599C2×C4—3432no—
600C2×C4—344no—
601C2×C4—344no—
602C2×C4—342no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—344no—
609C2×C4—682no—
610C2×C4—682no—
611C2×C4—682no—
612C2×C4—682no—
613C2×C4—682no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—682no—
619C2×C4—682no—
620C2×C4—682no—
621C2×C4—682no—
622C2×C4—682no—
623C2×C4—682no—
624C2×C4—682no—
625C2×C4—682no—
626C2×C4C22⋊C4×D1714yes—
627C22—344no—
628C22—344no—
629C22—344no—
630C22—682no—
631C22—682no—
632C22—682no—
633C22—682no—
634C22—684no—
635C22—684no—
636C22—684no—
637C22—684no—
638C22C2×C22⋊C4×D1711yescharacteristic
639C22#117×D1711yescharacteristic
640C22#117×D1711yescharacteristic
641C22#1176327711yescharacteristic
642C22C4×#6325211yescharacteristic
643C22#1176328911yescharacteristic
644C22#1176329011yescharacteristic
645C22#123×D1712yes—
646C22#123×D1712yes—
647C22#1176331812yes—
648C22#1176335612yes—
649C22#1176337511yescharacteristic
650C22#1176338211yescharacteristic
651C22#1176338811yescharacteristic
652C22#1176338911yescharacteristic
653C22#1176349112yes—
654C22#1176349212yes—
655C22#1176350812yes—
656C22#1176351012yes—
657C22#1176352112yes—
658C22#1176352212yes—
659C22#1176352612yes—
660C22#1176353112yes—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—178no—
667C4—682no—
668C4—682no—
669C4—684no—
670C4—1362no—
671C2—684no—
672C2D17×#38111yescharacteristic
673C2#1453127011yescharacteristic
674C2#1453137311yescharacteristic
675C2#1453166411yescharacteristic
676C2#1453167011yescharacteristic
677C2#1453167211yescharacteristic
678C2#1453167811yescharacteristic
679C2#1453307812yes—
680C2#1453308012yes—
681C2#1453323512yes—
682C2#1453324412yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13939
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
6. #14527012C2semidirect
7. #14527012C2semidirect
8. #14527011C2semidirect
9. #14527008C2semidirect
10. #14527008C2semidirect
11. C17×#1434C2semidirect
15. C2×#10587558C4semidirect
296. #1434D17semidirect
440. C17#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#105875582
C2×#117626682
C2×#117626562
#145270322
#145270122
#145270112
#145270082
C17×#14342
#1393917