Finite Group Databasebeta

← #16957059

Subgroups — #16957059

Order 4352.

2222 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587563C211yesmaximal, characteristic
2C2×#11762682C211yesmaximal, characteristic
3C2×#11762681C211yesmaximal, characteristic
4C2×#11762670C211yesmaximal, characteristic
5C22×#10587502C211yesmaximal, characteristic
6#14527035C212yesmaximal, complemented
7#14527035C212yesmaximal, complemented
8#14527012C212yesmaximal, complemented
9#14527009C212yesmaximal, complemented
10#14527006C211yesmaximal, characteristic
11C17×C22⋊#95C211yesmaximal, characteristic
12C24×Dic17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C2×#10587563—24no—
15C2×#10587563—24no—
16C2×#10587563C418yescomplemented
17C22×#63225C2212yes—
18C22×#63225C2212yes—
19C22×#63225C2212yes—
20C22×C22⋊Dic17C2211yescharacteristic
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C4×Dic17C2212yes—
23#11762682—24no—
24#11762681—24no—
25#11762670—24no—
26C2×#10587502—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C2×#10587502C2211yescharacteristic
36C2×#10587502C2212yes—
37C24×C4×C17C2211yescharacteristic
38C22×C22⋊C4×C17C2211yescharacteristic
39C22×C22⋊C4×C17C2211yescharacteristic
40C2×C17×#95C2212yes—
41C2×C17×#95C2212yes—
42C25×C17C2311yescharacteristic
43C23×C4×C17—22no—
44C23×C4×C17—24no—
45C23×C4×C17—22no—
46C23×C4×C17—22no—
47C23×C4×C17—42no—
48C23×C4×C17C2311yescharacteristic
49C23×C4×C17C2311yescharacteristic
50C23×C4×C17C2312yes—
51C23×C4×C17C2312yes—
52C23×Dic17—22no—
53C23×Dic17—22no—
54C23×Dic17—22no—
55C23×Dic17—22no—
56C23×Dic17—22no—
57C23×Dic17—24no—
58C23×Dic17—22no—
59C23×Dic17—22no—
60C23×Dic17—42no—
61C23×Dic17C2×C414yes—
62C23×Dic17D414yes—
63C23×Dic17Q814yes—
64C23×Dic17C2311yescharacteristic
65C23×Dic17C2311yescharacteristic
66C23×Dic17C2312yes—
67C23×Dic17C2312yes—
68C23×Dic17C2311yescharacteristic
69C23×Dic17C2311yescharacteristic
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17—22no—
72C2×C22⋊C4×C17—22no—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17—22no—
75C2×C22⋊C4×C17C2×C414yes—
76C2×C22⋊Dic17—22no—
77C2×C22⋊Dic17—22no—
78C2×C22⋊Dic17—22no—
79C2×C22⋊Dic17C2×C414yes—
80C2×C22⋊Dic17D414yes—
81C2×C22⋊Dic17D414yes—
82C2×#63225—24no—
83C2×#63225—24no—
84C2×#63225—24no—
85C2×#63225—24no—
86C2×#63225—24no—
87C2×#63225—24no—
88C2×#63225—42no—
89C2×#63225—42no—
90C2×#63225C2×C418yescomplemented
91C2×C4×Dic17—24no—
92C2×C4×Dic17—24no—
93C2×C4×Dic17C2×C418yescomplemented
94#10587563—232no—
95C17×#95—24no—
96C17×#95—42no—
97#10587502—24no—
98#10587502—24no—
99#10587502—22no—
100#10587502—22no—
101#10587502—22no—
102#10587502—22no—
103#10587502—24no—
104#10587502—24no—
105#10587502—24no—
106#10587502—42no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17C22×C412yes—
112C24×C17C2×D412yes—
113C24×C17C2×D412yes—
114C24×C17C2×D412yes—
115C24×C17C2×Q812yes—
116C24×C17#4112yes—
117C24×C17#4112yes—
118C24×C17C2411yescharacteristic
119C22×C4×C17—28no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
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—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—48no—
143C22×C4×C17—48no—
144C22×C4×C17—44no—
145C22×C4×C17—44no—
146C22×C4×C17C22×C414yes—
147C22×C4×C17#4114yes—
148C22×Dic17—28no—
149C22×Dic17—28no—
150C22×Dic17—28no—
151C22×Dic17—28no—
152C22×Dic17—28no—
153C22×Dic17—28no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—28no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—21no—
174C22×Dic17—21no—
175C22×Dic17—21no—
176C22×Dic17—21no—
177C22×Dic17—22no—
178C22×Dic17—22no—
179C22×Dic17—21no—
180C22×Dic17—21no—
181C22×Dic17—22no—
182C22×Dic17—22no—
183C22×Dic17—48no—
184C22×Dic17—48no—
185C22×Dic17—44no—
186C22×Dic17—44no—
187C22×Dic17C4⋊C4116yescomplemented
188C22×Dic17C22×C414yes—
189C22×Dic17C22×C412yes—
190C22×Dic17C22×C412yes—
191C22×Dic17C2×D414yes—
192C22×Dic17C2×D414yes—
193C22×Dic17C2×D412yes—
194C22×Dic17C2×D412yes—
195C22×Dic17C2×Q812yes—
196C22×Dic17C2×Q812yes—
197C22×Dic17#4114yes—
198C22⋊C4×C17—28no—
199C22⋊C4×C17—44no—
200C22⋊Dic17—44no—
201C22⋊Dic17C22⋊C4116yes—
202#63225—216no—
203#63225—48no—
204#63225—48no—
205C4×Dic17—216no—
206#13962—171nomaximal, Sylow
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—22no—
210C23×C17—22no—
211C23×C17—24no—
212C23×C17—22no—
213C23×C17—22no—
214C23×C17—24no—
215C23×C17—24no—
216C23×C17—24no—
217C23×C17—24no—
218C23×C17—24no—
219C23×C17—24no—
220C23×C17—24no—
221C23×C17—24no—
222C23×C17—24no—
223C23×C17C2×C22⋊C414yes—
224C23×C17C2×C4⋊C414yes—
225C23×C17#11714yes—
226C23×C17#12114yes—
227C23×C17#12114yes—
228C23×C17C22⋊Q814yes—
229C23×C17C22⋊Q814yes—
230C23×C17C23×C411yescharacteristic
231C23×C17C22×D411yescharacteristic
232C23×C17C22×D411yescharacteristic
233C23×C17C22×D411yescharacteristic
234C23×C17C22×Q811yescharacteristic
235C23×C17C2×#4111yescharacteristic
236C23×C17C2×#4111yescharacteristic
237C23×C17C2×#4112yes—
238C23×C17C2×#4112yes—
239C23×C17H(5,𝔽2)12yes—
240C23×C172−1+412yes—
241C2×C4×C17—24no—
242C2×C4×C17—24no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—24no—
246C2×C4×C17—24no—
247C2×C4×C17—416no—
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—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×Dic17—22no—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—24no—
271C2×Dic17—24no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—24no—
281C2×Dic17—24no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—232no—
285C2×Dic17—24no—
286C2×Dic17—24no—
287C2×Dic17—24no—
288C2×Dic17—416no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—41no—
294C2×Dic17—41no—
295C2×Dic17—41no—
296C2×Dic17—41no—
297C2×Dic17—41no—
298C2×Dic17—41no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17C2×C22⋊C418yes—
304C2×Dic17C2×C4⋊C414yes—
305C2×Dic17C2×C4⋊C414yes—
306C22×#117—171no—
307C2×#385—171no—
308C2×#385—171no—
309C2×#381—171no—
310C22×#95—171no—
311#1441—172no—
312#1441—172no—
313C4⋊#95—172no—
314#1434—172no—
315C22⋊#95—171no—
316C22⋊#95D1711yescharacteristic
317C22×C17—24no—
318C22×C17—28no—
319C22×C17—28no—
320C22×C17—24no—
321C22×C17—24no—
322C22×C17—24no—
323C22×C17—24no—
324C22×C17—28no—
325C22×C17—28no—
326C22×C17—24no—
327C22×C17#38018yes—
328C22×C17C22×C22⋊C411yescharacteristic
329C22×C17C22×C4⋊C411yescharacteristic
330C22×C17C2×#11711yescharacteristic
331C22×C17C4×#4112yes—
332C22×C17#52012yes—
333C22×C17C2×#12111yescharacteristic
334C22×C17C2×#12111yescharacteristic
335C22×C17C2×C22⋊Q811yescharacteristic
336C22×C17C2×C22⋊Q811yescharacteristic
337C22×C17#52511yescharacteristic
338C22×C17#52511yescharacteristic
339C22×C17#53312yes—
340C22×C17#53511yescharacteristic
341C22×C17#53611yescharacteristic
342C22×C17#54412yes—
343C22×C17#54612yes—
344C22×C17#54612yes—
345C22×C17#54712yes—
346C22×C17#54712yes—
347C22×C17#55112yes—
348C22×C17#55112yes—
349C22×C17#55212yes—
350C22×C17#55212yes—
351C4×C17—44no—
352C4×C17—44no—
353C4×C17—81no—
354C4×C17—81no—
355Dic17—28no—
356Dic17—24no—
357Dic17—24no—
358Dic17—44no—
359Dic17—81no—
360Dic17—81no—
361C24×C4—171no—
362C24×C4—171no—
363C24×C4C2×D1711yescharacteristic
364C2×#117—178no—
365C2×#117—344no—
366C2×#117—344no—
367C22×C4⋊C4—172no—
368C22×C4⋊C4—172no—
369C22×C4⋊C4—172no—
370C22×C22⋊C4—171no—
371C22×C22⋊C4—171no—
372C22×C22⋊C4C2×D1711yescharacteristic
373C22×C22⋊C4C2×D1711yescharacteristic
374C22×C42—172no—
375#385—344no—
376#385—344no—
377#381—344no—
378C2×#95—172no—
379C2×#95—171no—
380C2×#95—172no—
381C2×#95—172no—
382C2×#95—172no—
383C2×#95—171no—
384C2×#95—172no—
385C2×#95—171no—
386C2×#95—171no—
387C2×#95—172no—
388C2×#95—344no—
389C2×#95C2×D1712yes—
390C2×#95C2×D1712yes—
391C2×C17—28no—
392C2×C17C2×#38011yescharacteristic
393C2×C17#229611yescharacteristic
394C2×C17#231711yescharacteristic
395C2×C17#233912yes—
396C2×C17#234312yes—
397C2×C17#241111yescharacteristic
398C2×C17#241111yescharacteristic
399C2×C17#241311yescharacteristic
400C2×C17#241311yescharacteristic
401C2×C17#243812yes—
402C2×C17#243812yes—
403C25C22×D1711yescharacteristic
404C23×C4—22no—
405C23×C4—24no—
406C23×C4—22no—
407C23×C4—22no—
408C23×C4—42no—
409C23×C4—174no—
410C23×C4—174no—
411C23×C4—174no—
412C23×C4—172no—
413C23×C4—171no—
414C23×C4—171no—
415C23×C4—172no—
416C23×C4—171no—
417C23×C4—171no—
418C23×C4—344no—
419C23×C4—342no—
420C23×C4—342no—
421C23×C4—342no—
422C23×C4—342no—
423C23×C4—342no—
424C23×C4—342no—
425C23×C4—342no—
426C23×C4—682no—
427C23×C4C22×D1711yescharacteristic
428C23×C4C22×D1711yescharacteristic
429C23×C4C22×D1712yes—
430C23×C4C22×D1712yes—
431#117—3432no—
432C2×C4⋊C4—178no—
433C2×C4⋊C4—344no—
434C2×C4⋊C4—344no—
435C2×C4⋊C4—344no—
436C2×C4⋊C4—344no—
437C2×C4⋊C4—344no—
438C2×C4⋊C4—344no—
439C2×C4⋊C4—682no—
440C2×C4⋊C4—682no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4—22no—
445C2×C22⋊C4—22no—
446C2×C22⋊C4—174no—
447C2×C22⋊C4—174no—
448C2×C22⋊C4—174no—
449C2×C22⋊C4—342no—
450C2×C22⋊C4—342no—
451C2×C22⋊C4—342no—
452C2×C22⋊C4C4×D1714yes—
453C2×C42—178no—
454C2×C42—344no—
455C2×C42—344no—
456#95—24no—
457#95—42no—
458#95—344no—
459#95—344no—
460#95—344no—
461#95—342no—
462#95—342no—
463#95—342no—
464#95—342no—
465#95—344no—
466#95—344no—
467#95—682no—
468C17#1396211yescharacteristic, complemented, Sylow
469C24—22no—
470C24—22no—
471C24—22no—
472C24—22no—
473C24C2×C4×D1712yes—
474C24#6325212yes—
475C24#6325212yes—
476C24D4×D1712yes—
477C24D4×D1712yes—
478C24D4×D1712yes—
479C24Q8×D1712yes—
480C24C23×D1711yescharacteristic
481C22×C4—28no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—172no—
509C22×C4—174no—
510C22×C4—172no—
511C22×C4—172no—
512C22×C4—172no—
513C22×C4—174no—
514C22×C4—174no—
515C22×C4—172no—
516C22×C4—172no—
517C22×C4—1716no—
518C22×C4—174no—
519C22×C4—341no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4—341no—
525C22×C4—341no—
526C22×C4—341no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—348no—
532C22×C4—348no—
533C22×C4—348no—
534C22×C4—348no—
535C22×C4—348no—
536C22×C4—348no—
537C22×C4—348no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—341no—
545C22×C4—341no—
546C22×C4—341no—
547C22×C4—341no—
548C22×C4—341no—
549C22×C4—341no—
550C22×C4—342no—
551C22×C4—342no—
552C22×C4—342no—
553C22×C4—342no—
554C22×C4—688no—
555C22×C4—688no—
556C22×C4—684no—
557C22×C4—684no—
558C22×C4C2×C4×D1714yes—
559C22×C4#6325414yes—
560C4⋊C4—3416no—
561C4⋊C4—688no—
562C4⋊C4—688no—
563C22⋊C4—28no—
564C22⋊C4—44no—
565C22⋊C4—1716no—
566C22⋊C4—684no—
567C42—3416no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23—22no—
581C23—24no—
582C23—24no—
583C23—24no—
584C23#1058755814yes—
585C23C22⋊C4×D1714yes—
586C23#1058757014yes—
587C23#1058757014yes—
588C23C4⋊C4×D1714yes—
589C23#1058758214yes—
590C23#1058758214yes—
591C23C22×C4×D1711yescharacteristic
592C23C2×#6325211yescharacteristic
593C23C2×#6325211yescharacteristic
594C23C2×D4×D1711yescharacteristic
595C23C2×D4×D1711yescharacteristic
596C23C2×D4×D1711yescharacteristic
597C23C2×#6325412yes—
598C23#1058769112yes—
599C23C2×Q8×D1711yescharacteristic
600C23#41×D1712yes—
601C23#1058769712yes—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—178no—
628C2×C4—174no—
629C2×C4—174no—
630C2×C4—344no—
631C2×C4—344no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—342no—
636C2×C4—344no—
637C2×C4—344no—
638C2×C4—342no—
639C2×C4—342no—
640C2×C4—342no—
641C2×C4—342no—
642C2×C4—342no—
643C2×C4—342no—
644C2×C4—342no—
645C2×C4—342no—
646C2×C4—3432no—
647C2×C4—344no—
648C2×C4—344no—
649C2×C4—344no—
650C2×C4—344no—
651C2×C4—344no—
652C2×C4—6816no—
653C2×C4—682no—
654C2×C4—682no—
655C2×C4—682no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4—682no—
659C2×C4—682no—
660C2×C4—681no—
661C2×C4—681no—
662C2×C4—681no—
663C2×C4—681no—
664C2×C4—681no—
665C2×C4—681no—
666C2×C4—682no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—28no—
675C22—28no—
676C22—24no—
677C22#1176266818yes—
678C22C2×C22⋊C4×D1711yescharacteristic
679C22C2×C4⋊C4×D1711yescharacteristic
680C22C2×#1058755811yescharacteristic
681C22C4×#6325412yes—
682C22#1176328812yes—
683C22C2×#1058757011yescharacteristic
684C22C2×#1058757011yescharacteristic
685C22C2×#1058758211yescharacteristic
686C22C2×#1058758211yescharacteristic
687C22#1176331511yescharacteristic
688C22#1176331711yescharacteristic
689C22#1176336112yes—
690C22#1176337411yescharacteristic
691C22#1176338111yescharacteristic
692C22#1176343812yes—
693C22#1176344612yes—
694C22#1176344812yes—
695C22#1176345712yes—
696C22#1176345912yes—
697C22#1176348512yes—
698C22#1176348812yes—
699C22#1176349212yes—
700C22#1176349912yes—
701C4—44no—
702C4—44no—
703C4—81no—
704C4—81no—
705C4—348no—
706C4—344no—
707C4—344no—
708C4—684no—
709C4—1361no—
710C4—1361no—
711C2—28no—
712C2C2×#1176266811yescharacteristic
713C2#1453126311yescharacteristic
714C2#1453140311yescharacteristic
715C2#1453161012yes—
716C2#1453163812yes—
717C2#1453217111yescharacteristic
718C2#1453217211yescharacteristic
719C2#1453219311yescharacteristic
720C2#1453219411yescharacteristic
721C2#1453246412yes—
722C2#1453246612yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C22⋊#95C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC22⋊#95D17
hypocenterC17#13962
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #14527035C2semidirect
7. #14527035C2semidirect
8. #14527012C2semidirect
9. #14527009C2semidirect
16. C2×#10587563C4semidirect
90. C2×#63225C2×C4semidirect
93. C2×C4×Dic17C2×C4semidirect
187. C22×Dic17C4⋊C4semidirect
468. C17#13962semidirect

Maximal subgroups

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

subgroupindex
C22×#105875632
C2×#117626822
C2×#117626812
C2×#117626702
C22×#105875022
#145270352
#145270122
#145270092
#145270062
C17×C22⋊#952
#1396217