Finite Group Databasebeta

← #16956311

Subgroups — #16956311

Order 4352.

2222 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762673C212yesmaximal, complemented
3C2×#11762668C211yesmaximal, characteristic
4C2×#11762656C211yesmaximal, characteristic
5#14527043C211yesmaximal, characteristic, complemented
6#14527041C211yesmaximal, characteristic, complemented
7#14527040C211yesmaximal, characteristic, complemented
8#14527012C212yesmaximal, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D17C2212yes—
13C23×C4×D17C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225C2211yescharacteristic
16C22×#63225C2211yescharacteristic
17C22×#63225C2212yes—
18C2×C4⋊C4×D17—28no—
19C2×C4⋊C4×D17—24no—
20C22×#63227C2211yescharacteristic
21C22×#63227C2211yescharacteristic
22C22×#63227C2212yes—
23C22×C4×Dic17C2212yes—
24#11762673—28no—
25#11762668—24no—
26#11762656C418yescomplemented
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2211yescharacteristic
40C24×D17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
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×D17Q814yes—
63C2×C4⋊C4×C17—24no—
64C2×C4⋊C4×C17—22no—
65C2×#63227—22no—
66C2×#63227—22no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—24no—
71C2×#63227C2×C418yes—
72C2×C4⋊Dic17—22no—
73C2×C4⋊Dic17—24no—
74C2×#63225—24no—
75C2×#63225—24no—
76C2×#63225—22no—
77C2×#63225—22no—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225D418yes—
81C2×C4×Dic17—24no—
82C2×C4×Dic17—24no—
83C2×C4×Dic17C2×C418yescomplemented
84C4⋊C4×D17—232no—
85C17×#95—22no—
86C17×#95—24no—
87C17×#95—42no—
88C17×#95C2×C414yescomplemented
89#10587502—22no—
90#10587502—24no—
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—21no—
101C23×D17—21no—
102C23×D17—21no—
103C23×D17—21no—
104C23×D17C22×C412yes—
105C23×D17C2×D412yes—
106C23×D17C2×Q812yes—
107C23×D17#4114yes—
108C23×D17#4114yes—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
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—21no—
125C22×C4×C17—21no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17C22×C414yes—
134C22×C4×C17C22×C412yes—
135C22×C4×C17C2×D414yes—
136C22×C4×C17C2×D412yes—
137C22×C4×C17C2×Q812yes—
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—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17C22×C414yes—
161C22×Dic17C22×C412yes—
162C22×Dic17C2×D414yes—
163C22×Dic17C2×D412yes—
164C22×Dic17C2×D414yes—
165C22×Dic17C2×Q812yes—
166C22×Dic17#4114yes—
167C2×C4×D17—216no—
168C2×C4×D17—28no—
169C2×C4×D17—216no—
170C2×C4×D17—28no—
171C2×C4×D17—216no—
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C2×C4×D17—48no—
175C2×C4×D17—48no—
176C2×C4×D17—48no—
177C2×C4×D17—48no—
178C2×C4×D17C4⋊C4116yes—
179C4⋊C4×C17—28no—
180#63227—216no—
181#63227—44no—
182#63227—44no—
183C4⋊Dic17—28no—
184#63225—216no—
185#63225—44no—
186#63225—44no—
187C4×Dic17—216no—
188#13881—171nomaximal, Sylow
189C23×C17C23×C411yescharacteristic
190C23×C17C22×D411yescharacteristic
191C23×C17C22×D412yes—
192C23×C17C22×Q811yescharacteristic
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—24no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—24no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—22no—
208C22×D17—22no—
209C22×D17—22no—
210C22×D17—24no—
211C22×D17—24no—
212C22×D17—24no—
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—24no—
216C22×D17—24no—
217C22×D17—24no—
218C22×D17C2×C4⋊C414yes—
219C22×D17#11718yes—
220C22×D17#12414yes—
221C22×D17#12414yes—
222C22×D17#12518yes—
223C2×C4×C17—24no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—22no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17C2×C4⋊C414yes—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
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—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×Dic17C2×C4⋊C414yes—
277C2×Dic17C4×D418yes—
278C2×Dic17#12014yes—
279C2×Dic17C22⋊Q818yes—
280C2×Dic17#12314yes—
281C4×D17—232no—
282C4×D17—432no—
283C23×C4⋊C4—171no—
284C2×#382—172no—
285C2×#380—171no—
286C2×C4×C22⋊C4—171no—
287#1442—171no—
288#1442—171no—
289#1442—171no—
290#1442D1711yescharacteristic, complemented
291C4⋊#95—172no—
292C22⋊#95—172no—
293C22⋊#95—172no—
294C22×C17C22×C4⋊C411yescharacteristic
295C22×C17C2×#11712yes—
296C22×C17C2×C4×D412yes—
297C22×C17#52011yescharacteristic
298C22×C17#52011yescharacteristic
299C22×C17C2×#12011yescharacteristic
300C22×C17C2×C22⋊Q812yes—
301C22×C17C2×#12311yescharacteristic
302C22×C17C2×#12411yescharacteristic
303C22×C17C2×#12411yescharacteristic
304C22×C17C2×#12512yes—
305C22×C17#53511yescharacteristic
306C22×C17#53611yescharacteristic
307C22×C17#54411yescharacteristic
308C22×C17#54411yescharacteristic
309C22×C17#54612yes—
310C22×C17#54612yes—
311C22×C17#54812yes—
312C22×C17#54812yes—
313C22×C17#55212yes—
314C22×C17#55212yes—
315C22×C17#55312yes—
316C22×C17#55312yes—
317C2×D17—24no—
318C2×D17—28no—
319C2×D17—28no—
320C2×D17—24no—
321C2×D17—24no—
322C2×D17—28no—
323C2×D17—28no—
324C2×D17—24no—
325C2×D17—24no—
326C2×D17—24no—
327C2×D17#38218yescomplemented
328C4×C17—24no—
329C4×C17—44no—
330C4×C17—44no—
331C4×C17—81no—
332C4×C17—81no—
333Dic17—44no—
334Dic17—44no—
335Dic17—81no—
336Dic17—81no—
337Dic17#38518yes—
338C24×C4—171no—
339C24×C4—172no—
340C22×C4⋊C4—172no—
341C22×C4⋊C4—171no—
342C22×C4⋊C4—171no—
343C22×C4⋊C4—171no—
344C22×C4⋊C4—348no—
345C22×C4⋊C4—344no—
346C22×C4⋊C4C2×D1711yescharacteristic
347C22×C22⋊C4—172no—
348C22×C22⋊C4—171no—
349C22×C22⋊C4—171no—
350C22×C42—172no—
351#382—348no—
352#380—344no—
353C4×C22⋊C4—178no—
354C2×#95—171no—
355C2×#95—172no—
356C2×#95—172no—
357C2×#95—172no—
358C2×#95—171no—
359C2×#95—172no—
360C2×#95—172no—
361C2×#95—172no—
362C2×#95C2×D1712yes—
363C2×#95C2×D1711yescharacteristic
364C2×#95C2×D1712yes—
365C2×#95C2×D1711yescharacteristic
366C2×C17C2×#38211yescharacteristic
367C2×C17C2×#38511yescharacteristic
368C2×C17#231711yescharacteristic
369C2×C17#235212yes—
370C2×C17#235212yes—
371C2×C17#243511yescharacteristic
372C2×C17#243611yescharacteristic
373C2×C17#244212yes—
374C2×C17#244212yes—
375C2×C17#250311yescharacteristic
376C2×C17#250411yescharacteristic
377D17—28no—
378C25—171no—
379C23×C4—171no—
380C23×C4—174no—
381C23×C4—174no—
382C23×C4—174no—
383C23×C4—172no—
384C23×C4—172no—
385C23×C4—171no—
386C23×C4—171no—
387C23×C4—344no—
388C23×C4—344no—
389C23×C4—344no—
390C23×C4—344no—
391C23×C4—344no—
392C23×C4—344no—
393C23×C4—344no—
394C23×C4—682no—
395C23×C4—682no—
396C23×C4C22×D1711yescharacteristic
397C23×C4C22×D1711yescharacteristic
398C23×C4C22×D1712yes—
399C23×C4C22×D1712yes—
400C23×C4C22×D1711yescharacteristic
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—178no—
404C2×C4⋊C4—3432no—
405C2×C4⋊C4—344no—
406C2×C4⋊C4—344no—
407C2×C4⋊C4—344no—
408C2×C4⋊C4—342no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—344no—
412C2×C4⋊C4—344no—
413C2×C22⋊C4—178no—
414C2×C22⋊C4—344no—
415C2×C22⋊C4—344no—
416C2×C22⋊C4—342no—
417C2×C22⋊C4—342no—
418C2×C22⋊C4—344no—
419C2×C22⋊C4—344no—
420C2×C42—178no—
421C2×C42—344no—
422C2×C42—344no—
423#95—22no—
424#95—24no—
425#95—42no—
426#95—174no—
427#95—344no—
428#95—344no—
429#95—342no—
430#95—344no—
431#95—682no—
432#95—682no—
433#95—682no—
434#95C4×D1714yescomplemented
435C17#1388111yescharacteristic, complemented, Sylow
436C24—172no—
437C24—172no—
438C24—172no—
439C24—174no—
440C24—174no—
441C24—342no—
442C24—341no—
443C24—341no—
444C24—342no—
445C24—341no—
446C24—341no—
447C24C23×D1711yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—172no—
473C22×C4—174no—
474C22×C4—172no—
475C22×C4—172no—
476C22×C4—174no—
477C22×C4—1716no—
478C22×C4—174no—
479C22×C4—174no—
480C22×C4—348no—
481C22×C4—3416no—
482C22×C4—3416no—
483C22×C4—348no—
484C22×C4—3416no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—688no—
508C22×C4—688no—
509C22×C4—688no—
510C22×C4—688no—
511C22×C4—688no—
512C22×C4—688no—
513C22×C4C2×C4×D1714yes—
514C22×C4C2×C4×D1712yes—
515C22×C4D4×D1714yes—
516C22×C4D4×D1712yes—
517C22×C4Q8×D1712yes—
518C4⋊C4—28no—
519C4⋊C4—3416no—
520C4⋊C4—348no—
521C4⋊C4—684no—
522C4⋊C4—684no—
523C22⋊C4—3416no—
524C22⋊C4—684no—
525C22⋊C4—684no—
526C42—3416no—
527C23—174no—
528C23—178no—
529C23—174no—
530C23—174no—
531C23—178no—
532C23—344no—
533C23—344no—
534C23—344no—
535C23—344no—
536C23—344no—
537C23—344no—
538C23—344no—
539C23—342no—
540C23—342no—
541C23—342no—
542C23—344no—
543C23—342no—
544C23—342no—
545C23—342no—
546C23—342no—
547C23—342no—
548C23—344no—
549C23—344no—
550C23C22×C4×D1711yescharacteristic
551C23C2×D4×D1712yes—
552C23C2×D4×D1711yescharacteristic
553C23#1058769111yescharacteristic
554C23C2×Q8×D1711yescharacteristic
555C23#1058769411yescharacteristic
556C23#41×D1712yes—
557C23#41×D1712yes—
558C23#41×D1712yes—
559C23#1058769611yescharacteristic
560C23#1058769711yescharacteristic
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—174no—
590C2×C4—178no—
591C2×C4—174no—
592C2×C4—178no—
593C2×C4—174no—
594C2×C4—3432no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—344no—
600C2×C4—344no—
601C2×C4—6832no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4—682no—
608C2×C4—682no—
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—681no—
619C2×C4—681no—
620C2×C4C4⋊C4×D1714yes—
621C22—178no—
622C22—344no—
623C22—344no—
624C22—344no—
625C22—344no—
626C22—348no—
627C22—348no—
628C22—344no—
629C22—348no—
630C22—348no—
631C22—344no—
632C22C2×C4⋊C4×D1711yescharacteristic
633C22#117×D1712yes—
634C22C4×D4×D1712yes—
635C22#1176328811yescharacteristic
636C22#1176329111yescharacteristic
637C22#120×D1711yescharacteristic
638C22C22⋊Q8×D1712yes—
639C22#123×D1711yescharacteristic
640C22#124×D1711yescharacteristic
641C22#124×D1711yescharacteristic
642C22#125×D1712yes—
643C22#1176337511yescharacteristic
644C22#1176338211yescharacteristic
645C22#1176343611yescharacteristic
646C22#1176343911yescharacteristic
647C22#1176344812yes—
648C22#1176345112yes—
649C22#1176346712yes—
650C22#1176346912yes—
651C22#1176349112yes—
652C22#1176349512yes—
653C22#1176350412yes—
654C22#1176351012yes—
655C4—24no—
656C4—44no—
657C4—44no—
658C4—81no—
659C4—81no—
660C4—178no—
661C4—684no—
662C4—684no—
663C4—1361no—
664C4—1361no—
665C2—348no—
666C2D17×#38211yescharacteristic
667C2D17×#38511yescharacteristic
668C2#1453140611yescharacteristic
669C2#1453170812yes—
670C2#1453171212yes—
671C2#1453243011yescharacteristic
672C2#1453243711yescharacteristic
673C2#1453250112yes—
674C2#1453250312yes—
675C2#1453326411yescharacteristic
676C2#1453327111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1442C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1442D17
hypocenterC17#13881
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#11762673C2semidirect
5. #14527043C2semidirect
6. #14527041C2semidirect
7. #14527040C2semidirect
8. #14527012C2semidirect
11. C17×#1442C2semidirect
26. #11762656C4semidirect
83. C2×C4×Dic17C2×C4semidirect
88. C17×#95C2×C4semidirect
96. #10587502C2×C4semidirect
290. #1442D17semidirect
327. C2×D17#382semidirect
434. #95C4×D17semidirect
435. C17#13881semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D172
C2×#117626732
C2×#117626682
C2×#117626562
#145270432
#145270412
#145270402
#145270122
#145270052
C17×#14422
#1388117