Finite Group Databasebeta

← #16960702

Subgroups — #16960702

Order 4352.

2142 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587583C211yesmaximal, characteristic
2C22×#10587558C211yesmaximal, characteristic
3C2×#11762715C211yesmaximal, characteristic, complemented
4C2×#11762714C211yesmaximal, characteristic, complemented
5C2×#11762666C211yesmaximal, characteristic
6#14527036C214yesmaximal, complemented
7#14527032C214yesmaximal
8#14527013C211yesmaximal, characteristic, complemented
9C17×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Dic34C2211yescharacteristic
11C23×C4×D17C2211yescharacteristic
12C2×#10587583—24no—
13C2×#10587583—24no—
14C2×#10587583C418yescomplemented
15C2×#10587558—24no—
16C2×#10587558—24no—
17C2×#10587558—24no—
18C22×C4⋊Dic17C2211yescharacteristic
19C22×C4⋊Dic17C2212yes—
20C22×#63225C2212yes—
21C22×#63227C2212yes—
22C22×#63227C2212yes—
23C22×#63227C2212yes—
24C22×C4×Dic17C2211yescharacteristic
25#11762715—24no—
26#11762714—24no—
27#11762666—24no—
28C2×#10587502C2214yes—
29C2×#10587502C2214yes—
30C2×#10587502C2212yes—
31C2×#10587502C2214yes—
32C22×C4⋊C4×C17C2211yescharacteristic
33C22×C4⋊C4×C17C2211yescharacteristic
34C22×C42×C17C2211yescharacteristic
35C2×C17×#95C2214yes—
36C24×D17C2311yescharacteristic
37C23×C4×C17C2312yes—
38C23×C4×C17C2312yes—
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C23×Dic17C2312yes—
44C23×Dic17C2311yescharacteristic
45C22×C4×D17—22no—
46C22×C4×D17—22no—
47C22×C4×D17—22no—
48C22×C4×D17—22no—
49C22×C4×D17C2×C414yes—
50C22×C4×D17D414yes—
51C22×C4×D17D414yes—
52C22×Dic34—22no—
53C22×Dic34—22no—
54C22×Dic34—22no—
55C22×Dic34—22no—
56C22×Dic34—22no—
57C22×Dic34—22no—
58C22×Dic34C2×C414yes—
59C2×C4⋊C4×C17—22no—
60C2×C4⋊C4×C17—22no—
61C2×C4⋊C4×C17—22no—
62C2×C4⋊C4×C17—22no—
63C2×C4⋊C4×C17—22no—
64C2×C4⋊C4×C17C2×C414yes—
65C2×C42×C17—22no—
66C2×C42×C17—22no—
67C2×C42×C17D414yes—
68C2×#63227—24no—
69C2×#63227—24no—
70C2×#63227—24no—
71C2×#63227—24no—
72C2×#63227—24no—
73C2×#63227—24no—
74C2×#63227—44no—
75C2×#63227C2×C418yes—
76C2×C4⋊Dic17—22no—
77C2×C4⋊Dic17—22no—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17—24no—
80C2×C4⋊Dic17—24no—
81C2×C4⋊Dic17C2×C418yes—
82C2×#63225—24no—
83C2×#63225—24no—
84C2×#63225—24no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17D414yes—
88#10587583—232no—
89#10587558—232no—
90C17×#95—44no—
91#10587502—28no—
92#10587502—28no—
93#10587502—24no—
94#10587502—44no—
95C24×C17C2411yescharacteristic
96C23×D17—21no—
97C23×D17—21no—
98C23×D17—21no—
99C23×D17—21no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—22no—
103C23×D17—22no—
104C23×D17C22×C412yes—
105C23×D17C2×D412yes—
106C23×D17C2×D412yes—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—24no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—24no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—24no—
121C22×C4×C17—24no—
122C22×C4×C17C22×C414yes—
123C22×C4×C17C22×C412yes—
124C22×C4×C17C2×D414yes—
125C22×C4×C17C2×D412yes—
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2×D412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17#4112yes—
130C22×C4×C17#4112yes—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
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—24no—
151C22×Dic17—24no—
152C22×Dic17C22×C414yes—
153C22×Dic17C22×C412yes—
154C22×Dic17C2×D414yes—
155C22×Dic17C2×D412yes—
156C22×Dic17C2×D412yes—
157C2×C4×D17—28no—
158C2×C4×D17—28no—
159C2×C4×D17—28no—
160C2×C4×D17—28no—
161C2×C4×D17—28no—
162C2×C4×D17—28no—
163C2×C4×D17C22⋊C4116yes—
164C2×Dic34—28no—
165C2×Dic34—28no—
166C2×Dic34—28no—
167C2×Dic34—44no—
168C2×Dic34—44no—
169C2×Dic34—44no—
170C2×Dic34—44no—
171C4⋊C4×C17—28no—
172C4⋊C4×C17—44no—
173C42×C17—28no—
174#63227—216no—
175#63227—216no—
176#63227—48no—
177C4⋊Dic17—216no—
178C4⋊Dic17—44no—
179#63225—216no—
180C4×Dic17—28no—
181#14258—171nomaximal, Sylow
182C23×C17C23×C411yescharacteristic
183C23×C17C22×D411yescharacteristic
184C23×C17C22×D411yescharacteristic
185C23×C17C22×D411yescharacteristic
186C23×C17C22×D411yescharacteristic
187C23×C17C2×#4111yescharacteristic
188C23×C17C2×#4111yescharacteristic
189C23×C17H(5,𝔽2)12yes—
190C23×C172−1+412yes—
191C23×C172−1+412yes—
192C23×C172−1+412yes—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—22no—
203C22×D17—22no—
204C22×D17—24no—
205C22×D17—22no—
206C22×D17—24no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17C2×C22⋊C414yes—
216C2×C4×C17—24no—
217C2×C4×C17—28no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—44no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—44no—
232C2×C4×C17—44no—
233C2×C4×C17C2×C22⋊C414yes—
234C2×C4×C17C4×D414yes—
235C2×C4×C17C4×D414yes—
236C2×C4×C17#12014yes—
237C2×C4×C17#12114yes—
238C2×C4×C17#12114yes—
239C2×C4×C17#12314yes—
240C2×Dic17—22no—
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—44no—
264C2×Dic17—44no—
265C2×Dic17C2×C22⋊C414yes—
266C4×D17—232no—
267Dic34—416no—
268C22×C22⋊Q8—171no—
269C22×#117—171no—
270C2×#391—171no—
271C2×#391—171no—
272C2×#380—171no—
273#1441—174no—
274#1440—174no—
275C4⋊#95—171no—
276C4⋊#95D1711yescharacteristic, complemented
277C22×C17C22×C22⋊C411yescharacteristic
278C22×C17C2×C4×D411yescharacteristic
279C22×C17C2×C4×D411yescharacteristic
280C22×C17#51912yes—
281C22×C17#52012yes—
282C22×C17C2×#12011yescharacteristic
283C22×C17C2×#12111yescharacteristic
284C22×C17C2×#12111yescharacteristic
285C22×C17C2×#12311yescharacteristic
286C22×C17#53511yescharacteristic
287C22×C17#53611yescharacteristic
288C22×C17#53611yescharacteristic
289C22×C17#53612yes—
290C22×C17#53612yes—
291C22×C17#53611yescharacteristic
292C22×C17#53612yes—
293C22×C17#53712yes—
294C22×C17#53712yes—
295C22×C17#53712yes—
296C22×C17#54112yes—
297C22×C17#54112yes—
298C22×C17#54212yes—
299C22×C17#54212yes—
300C2×D17—24no—
301C2×D17—24no—
302C2×D17—24no—
303C2×D17—44no—
304C2×D17—44no—
305C2×D17—44no—
306C2×D17—42no—
307C2×D17—42no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—44no—
311C4×C17—44no—
312C4×C17—44no—
313C4×C17—82no—
314C4×C17#38618yes—
315Dic17—24no—
316Dic17—44no—
317Dic17—44no—
318Dic17—82no—
319C23×Q8—171no—
320C24×C4—171no—
321C2×C22⋊Q8—178no—
322C2×C22⋊Q8—344no—
323C2×C22⋊Q8—344no—
324C2×#117—344no—
325C2×#117—344no—
326C2×#117—344no—
327C22×C4⋊C4—172no—
328C22×C4⋊C4—172no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C4⋊C4C2×D1711yescharacteristic
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C42—171no—
336C22×C42C2×D1711yescharacteristic
337#391—344no—
338#391—344no—
339#380—344no—
340C2×#95—174no—
341C2×#95—174no—
342C2×#95—172no—
343C2×#95—174no—
344C2×#95C2×D1714yes—
345C2×C17C2×#38611yescharacteristic
346C2×C17#231011yescharacteristic
347C2×C17#231111yescharacteristic
348C2×C17#232312yes—
349C2×C17#232312yes—
350C2×C17#261612yes—
351C2×C17#262011yescharacteristic
352C2×C17#262011yescharacteristic
353C2×C17#262111yescharacteristic
354C2×C17#262111yescharacteristic
355C2×C17#262512yes—
356D17—44no—
357C25—171no—
358C22×Q8—174no—
359C22×Q8—342no—
360C22×Q8—342no—
361C22×Q8—342no—
362C22×Q8—342no—
363C22×Q8—342no—
364C22×Q8—342no—
365C23×C4—171no—
366C23×C4—174no—
367C23×C4—174no—
368C23×C4—174no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4—172no—
372C23×C4—342no—
373C23×C4—342no—
374C23×C4—342no—
375C23×C4—342no—
376C23×C4C22×D1712yes—
377C23×C4C22×D1712yes—
378C23×C4C22×D1712yes—
379C23×C4C22×D1711yescharacteristic
380C22⋊Q8—3432no—
381#117—3432no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—178no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—344no—
392C2×C4⋊C4—344no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4—342no—
395C2×C4⋊C4—342no—
396C2×C4⋊C4C4×D1714yes—
397C2×C22⋊C4—178no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—344no—
403C2×C22⋊C4—344no—
404C2×C22⋊C4—684no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—174no—
408C2×C42—342no—
409C2×C42—342no—
410C2×C42D6814yes—
411#95—44no—
412#95—344no—
413#95—348no—
414#95—348no—
415#95—684no—
416C17#1425811yescharacteristic, complemented, Sylow
417C24—172no—
418C24—172no—
419C24—172no—
420C24—341no—
421C24—341no—
422C24—341no—
423C24—342no—
424C24—341no—
425C24—342no—
426C24—342no—
427C24—342no—
428C24C23×D1711yescharacteristic
429C2×Q8—348no—
430C2×Q8—348no—
431C2×Q8—348no—
432C2×Q8—684no—
433C2×Q8—684no—
434C2×Q8—684no—
435C2×Q8—684no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—172no—
452C22×C4—172no—
453C22×C4—172no—
454C22×C4—1716no—
455C22×C4—174no—
456C22×C4—174no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—348no—
462C22×C4—348no—
463C22×C4—348no—
464C22×C4—348no—
465C22×C4—348no—
466C22×C4—348no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—344no—
482C22×C4—342no—
483C22×C4—344no—
484C22×C4C2×C4×D1714yes—
485C22×C4C2×C4×D1712yes—
486C22×C4C2×D6814yes—
487C22×C4C2×D6812yes—
488C22×C4#6325212yes—
489C22×C4D4×D1712yes—
490C22×C4D4×D1712yes—
491C22×C4D4×D1712yes—
492C22×C4#6325612yes—
493C4⋊C4—28no—
494C4⋊C4—44no—
495C4⋊C4—3416no—
496C4⋊C4—3416no—
497C4⋊C4—684no—
498C22⋊C4—3416no—
499C22⋊C4—3416no—
500C22⋊C4—688no—
501C42—28no—
502C42—348no—
503C23—174no—
504C23—342no—
505C23—342no—
506C23—344no—
507C23—344no—
508C23—342no—
509C23—342no—
510C23—342no—
511C23—342no—
512C23—342no—
513C23—342no—
514C23—342no—
515C23—344no—
516C23—342no—
517C23—344no—
518C23—682no—
519C23—682no—
520C23—682no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23—682no—
525C23—682no—
526C23C22×C4×D1711yescharacteristic
527C23C22×D6811yescharacteristic
528C23C2×#6325211yescharacteristic
529C23C2×D4×D1711yescharacteristic
530C23C2×D4×D1711yescharacteristic
531C23C2×D4×D1711yescharacteristic
532C23#1058769112yes—
533C23C2×#6325611yescharacteristic
534C23#1058769412yes—
535C23#1058769712yes—
536C23#1058769712yes—
537Q8—6816no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—174no—
556C2×C4—342no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—342no—
561C2×C4—342no—
562C2×C4—3432no—
563C2×C4—344no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4—682no—
579C2×C4—682no—
580C2×C4—684no—
581C2×C4—684no—
582C2×C4C4×D6814yes—
583C2×C4C22⋊C4×D1714yes—
584C2×C4#1058756814yes—
585C2×C4#1058757014yes—
586C2×C4#1058757914yes—
587C2×C4#1058758014yes—
588C2×C4#1058758114yes—
589C22—344no—
590C22—344no—
591C22—344no—
592C22—684no—
593C22—684no—
594C22—684no—
595C22—682no—
596C22—682no—
597C22—682no—
598C22—682no—
599C22—684no—
600C22C2×C22⋊C4×D1711yescharacteristic
601C22C2×#1058757911yescharacteristic
602C22C2×C4×D6811yescharacteristic
603C22#1176328512yes—
604C22#1176328812yes—
605C22C2×#1058756811yescharacteristic
606C22C2×#1058757011yescharacteristic
607C22C2×#1058758111yescharacteristic
608C22C2×#1058758011yescharacteristic
609C22#1176337211yescharacteristic
610C22#1176337911yescharacteristic
611C22#1176337911yescharacteristic
612C22#1176338212yes—
613C22#1176338212yes—
614C22#1176338411yescharacteristic
615C22#1176338512yes—
616C22#1176338812yes—
617C22#1176338812yes—
618C22#1176339012yes—
619C22#1176341712yes—
620C22#1176342212yes—
621C22#1176342412yes—
622C22#1176343312yes—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—344no—
627C4—684no—
628C4—684no—
629C4—1362no—
630C4#1176268718yes—
631C2—684no—
632C2C2×#1176268711yescharacteristic
633C2#1453136711yescharacteristic
634C2#1453137411yescharacteristic
635C2#1453146412yes—
636C2#1453146712yes—
637C2#1453451512yes—
638C2#1453454411yescharacteristic
639C2#1453454711yescharacteristic
640C2#1453455311yescharacteristic
641C2#1453455511yescharacteristic
642C2#1453458612yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C4⋊#95C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenterC4⋊#95D17
hypocenterC17#14258
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#11762715C2semidirect
4. C2×#11762714C2semidirect
6. #14527036C2semidirect
8. #14527013C2semidirect
9. C17×C4⋊#95C2semidirect
14. C2×#10587583C4semidirect
276. C4⋊#95D17semidirect
416. C17#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#105875832
C22×#105875582
C2×#117627152
C2×#117627142
C2×#117626662
#145270362
#145270322
#145270132
C17×C4⋊#952
#1425817