Finite Group Databasebeta

← #16960701

Subgroups — #16960701

Order 4352.

2142 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587582C211yesmaximal, characteristic
2C22×#10587578C211yesmaximal, characteristic
3C2×#11762713C212yesmaximal, complemented
4C2×#11762668C211yesmaximal, characteristic
5#14527038C211yesmaximal, characteristic, complemented
6#14527037C211yesmaximal, characteristic, complemented
7#14527034C211yesmaximal, characteristic, complemented
8#14527032C212yesmaximal
9#14527032C212yesmaximal
10#14527012C212yesmaximal, complemented
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×Dic34C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C2×#10587582—24no—
15C2×#10587582—24no—
16C2×#10587582C418yescomplemented
17C2×#10587578—28no—
18C2×#10587578—24no—
19C22×C4⋊Dic17C2211yescharacteristic
20C22×#63225C2212yes—
21C22×#63225C2211yescharacteristic
22C22×#63225C2211yescharacteristic
23C22×#63227C2212yes—
24C22×#63227C2211yescharacteristic
25C22×#63227C2211yescharacteristic
26C22×#63227C2212yes—
27C22×C4×Dic17C2212yes—
28#11762713—28no—
29#11762668—24no—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2211yescharacteristic
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C2×#10587502C2212yes—
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C4⋊C4×C17C2211yescharacteristic
39C2×C17×#95C2212yes—
40C2×C17×#95C2211yescharacteristic
41C2×C17×#95C2212yes—
42C24×D17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2312yes—
47C23×C4×C17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2312yes—
52C23×Dic17C2311yescharacteristic
53C22×C4×D17—24no—
54C22×C4×D17—24no—
55C22×C4×D17C2×C414yes—
56C22×C4×D17D414yes—
57C22×C4×D17D414yes—
58C22×Dic34—24no—
59C22×Dic34—22no—
60C22×Dic34—22no—
61C22×Dic34—24no—
62C22×Dic34C2×C414yes—
63C2×C4⋊C4×C17—24no—
64C2×C4⋊C4×C17—22no—
65C2×C4⋊C4×C17—22no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17C2×C414yes—
68C2×#63227—24no—
69C2×#63227—22no—
70C2×#63227—22no—
71C2×#63227—22no—
72C2×#63227—22no—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—24no—
76C2×#63227—42no—
77C2×#63227—42no—
78C2×#63227C2×C414yes—
79C2×#63227C2×C414yes—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17—24no—
82C2×#63225—22no—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—24no—
87C2×#63225—24no—
88C2×#63225—24no—
89C2×#63225C2×C414yes—
90C2×#63225C2×C414yes—
91C2×C4×Dic17—24no—
92C2×C4×Dic17—24no—
93C2×C4×Dic17D418yes—
94#10587582—232no—
95#10587578—232no—
96C17×#95—24no—
97C17×#95—22no—
98C17×#95—42no—
99#10587502—24no—
100#10587502—22no—
101#10587502—24no—
102#10587502—24no—
103#10587502—42no—
104#10587502—42no—
105#10587502—42no—
106C24×C17C2411yescharacteristic
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—21no—
110C23×D17—21no—
111C23×D17—22no—
112C23×D17—21no—
113C23×D17—22no—
114C23×D17—21no—
115C23×D17C22×C412yes—
116C23×D17C2×D412yes—
117C23×D17C2×D412yes—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—22no—
128C22×C4×C17—21no—
129C22×C4×C17—22no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—22no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C22×C412yes—
145C22×C4×C17C22×C412yes—
146C22×C4×C17C2×D414yes—
147C22×C4×C17C2×D412yes—
148C22×C4×C17C2×D412yes—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—22no—
160C22×Dic17—21no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17—22no—
164C22×Dic17—21no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17C22×C412yes—
173C22×Dic17C22×C412yes—
174C22×Dic17C22×C412yes—
175C22×Dic17C2×D414yes—
176C22×Dic17C2×D412yes—
177C22×Dic17C2×D412yes—
178C22×Dic17C2×D414yes—
179C22×Dic17#4114yes—
180C2×C4×D17—216no—
181C2×C4×D17—28no—
182C2×C4×D17—216no—
183C2×C4×D17—28no—
184C2×C4×D17C22⋊C4116yes—
185C2×Dic34—28no—
186C2×Dic34—216no—
187C2×Dic34—44no—
188C2×Dic34—48no—
189C2×Dic34—44no—
190C4⋊C4×C17—28no—
191C4⋊C4×C17—28no—
192#63227—216no—
193#63227—28no—
194#63227—28no—
195#63227—48no—
196C4⋊Dic17—28no—
197#63225—28no—
198#63225—28no—
199#63225—48no—
200C4×Dic17—216no—
201#14258—171nomaximal, Sylow
202C23×C17C23×C411yescharacteristic
203C23×C17C22×D411yescharacteristic
204C23×C17C22×D411yescharacteristic
205C23×C17C22×D412yes—
206C23×C17C2×#4112yes—
207C23×C17H(5,𝔽2)11yescharacteristic
208C23×C17H(5,𝔽2)11yescharacteristic
209C23×C172−1+412yes—
210C23×C172−1+411yescharacteristic
211C23×C172−1+412yes—
212C23×C172−1+411yescharacteristic
213C22×D17—24no—
214C22×D17—24no—
215C22×D17—22no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—24no—
219C22×D17—22no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—24no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—42no—
228C22×D17—42no—
229C22×D17—42no—
230C22×D17—42no—
231C22×D17—42no—
232C22×D17—42no—
233C22×D17—42no—
234C22×D17—42no—
235C22×D17C2×C22⋊C414yes—
236C2×C4×C17—24no—
237C2×C4×C17—22no—
238C2×C4×C17—24no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—24no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—24no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—41no—
249C2×C4×C17—42no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17—42no—
254C2×C4×C17—41no—
255C2×C4×C17—42no—
256C2×C4×C17—41no—
257C2×C4×C17—42no—
258C2×C4×C17—41no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17—42no—
264C2×C4×C17—42no—
265C2×C4×C17—42no—
266C2×C4×C17C2×C22⋊C414yes—
267C2×Dic17—22no—
268C2×Dic17—22no—
269C2×Dic17—24no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—41no—
276C2×Dic17—42no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—42no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17C2×C22⋊C414yes—
294C2×Dic17C4×D418yes—
295C2×Dic17#12014yes—
296C2×Dic17#12118yes—
297C2×Dic17#12314yes—
298C4×D17—232no—
299Dic34—416no—
300C22×C22⋊Q8—171no—
301C22×#117—171no—
302C2×#391—172no—
303C2×#380—171no—
304#1441—171no—
305#1441—171no—
306#1441—171no—
307#1441D1711yescharacteristic, complemented
308#1440—172no—
309#1440—172no—
310C4⋊#95—172no—
311C22×C17C22×C22⋊C411yescharacteristic
312C22×C17C2×C4×D412yes—
313C22×C17#51912yes—
314C22×C17#52011yescharacteristic
315C22×C17#52011yescharacteristic
316C22×C17C2×#12011yescharacteristic
317C22×C17C2×#12112yes—
318C22×C17C2×#12311yescharacteristic
319C22×C17#53511yescharacteristic
320C22×C17#53611yescharacteristic
321C22×C17#53612yes—
322C22×C17#53612yes—
323C22×C17#53611yescharacteristic
324C22×C17#53611yescharacteristic
325C22×C17#53612yes—
326C22×C17#53712yes—
327C22×C17#53711yescharacteristic
328C22×C17#53711yescharacteristic
329C22×C17#53712yes—
330C22×C17#54112yes—
331C22×C17#54112yes—
332C22×C17#54212yes—
333C22×C17#54212yes—
334C2×D17—24no—
335C2×D17—24no—
336C2×D17—24no—
337C2×D17—44no—
338C2×D17—44no—
339C2×D17—42no—
340C2×D17—42no—
341C2×D17—44no—
342C2×D17—44no—
343C2×D17—42no—
344C2×D17—42no—
345C4×C17—24no—
346C4×C17—42no—
347C4×C17—44no—
348C4×C17—42no—
349C4×C17—82no—
350Dic17—42no—
351Dic17—44no—
352Dic17—42no—
353Dic17—82no—
354Dic17#38618yes—
355C23×Q8—171no—
356C24×C4—171no—
357C2×C22⋊Q8—178no—
358C2×C22⋊Q8—344no—
359C2×C22⋊Q8—344no—
360C2×#117—348no—
361C2×#117—344no—
362C22×C4⋊C4—171no—
363C22×C4⋊C4—171no—
364C22×C4⋊C4—172no—
365C22×C4⋊C4—171no—
366C22×C4⋊C4C2×D1711yescharacteristic
367C22×C4⋊C4C2×D1711yescharacteristic
368C22×C22⋊C4—171no—
369C22×C22⋊C4—171no—
370C22×C22⋊C4—172no—
371C22×C22⋊C4—172no—
372C22×C42—172no—
373#391—348no—
374#380—344no—
375C2×#95—172no—
376C2×#95—172no—
377C2×#95—172no—
378C2×#95—171no—
379C2×#95—172no—
380C2×#95—172no—
381C2×#95—172no—
382C2×#95C2×D1712yes—
383C2×#95C2×D1711yescharacteristic
384C2×#95C2×D1712yes—
385C2×C17C2×#38611yescharacteristic
386C2×C17#231011yescharacteristic
387C2×C17#231111yescharacteristic
388C2×C17#232312yes—
389C2×C17#232312yes—
390C2×C17#261611yescharacteristic
391C2×C17#261611yescharacteristic
392C2×C17#262012yes—
393C2×C17#262112yes—
394C2×C17#262511yescharacteristic
395C2×C17#262511yescharacteristic
396D17—44no—
397C25—171no—
398C22×Q8—174no—
399C22×Q8—342no—
400C22×Q8—342no—
401C22×Q8—344no—
402C22×Q8—344no—
403C23×C4—171no—
404C23×C4—174no—
405C23×C4—174no—
406C23×C4—174no—
407C23×C4—171no—
408C23×C4—171no—
409C23×C4—172no—
410C23×C4—172no—
411C23×C4—344no—
412C23×C4—344no—
413C23×C4C22×D1712yes—
414C23×C4C22×D1711yescharacteristic
415C23×C4C22×D1711yescharacteristic
416C23×C4C22×D1712yes—
417C23×C4C22×D1711yescharacteristic
418C22⋊Q8—3432no—
419#117—3432no—
420C2×C4⋊C4—24no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—174no—
425C2×C4⋊C4—174no—
426C2×C4⋊C4—344no—
427C2×C4⋊C4—342no—
428C2×C4⋊C4—342no—
429C2×C4⋊C4—342no—
430C2×C4⋊C4—342no—
431C2×C4⋊C4—342no—
432C2×C4⋊C4—344no—
433C2×C4⋊C4—344no—
434C2×C4⋊C4—344no—
435C2×C4⋊C4C4×D1714yes—
436C2×C22⋊C4—174no—
437C2×C22⋊C4—174no—
438C2×C22⋊C4—344no—
439C2×C22⋊C4—344no—
440C2×C22⋊C4—342no—
441C2×C22⋊C4—342no—
442C2×C22⋊C4—342no—
443C2×C22⋊C4—342no—
444C2×C22⋊C4—344no—
445C2×C22⋊C4—344no—
446C2×C22⋊C4—682no—
447C2×C22⋊C4—682no—
448C2×C42—178no—
449C2×C42—344no—
450C2×C42—344no—
451#95—22no—
452#95—24no—
453#95—42no—
454#95—344no—
455#95—344no—
456#95—342no—
457#95—344no—
458#95—682no—
459#95—682no—
460#95—682no—
461C17#1425811yescharacteristic, complemented, Sylow
462C24—172no—
463C24—172no—
464C24—172no—
465C24—342no—
466C24—341no—
467C24—342no—
468C24—341no—
469C24—342no—
470C24—341no—
471C24—342no—
472C24—341no—
473C24C23×D1711yescharacteristic
474C2×Q8—348no—
475C2×Q8—3416no—
476C2×Q8—684no—
477C2×Q8—688no—
478C2×Q8—684no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—172no—
505C22×C4—174no—
506C22×C4—172no—
507C22×C4—172no—
508C22×C4—174no—
509C22×C4—1716no—
510C22×C4—172no—
511C22×C4—172no—
512C22×C4—174no—
513C22×C4—348no—
514C22×C4—3416no—
515C22×C4—3416no—
516C22×C4—348no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—341no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4—341no—
525C22×C4—341no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—341no—
531C22×C4—342no—
532C22×C4—341no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4C2×C4×D1712yes—
541C22×C4C2×C4×D1712yes—
542C22×C4C2×C4×D1712yes—
543C22×C4D4×D1714yes—
544C22×C4D4×D1712yes—
545C22×C4D4×D1712yes—
546C4⋊C4—28no—
547C4⋊C4—28no—
548C4⋊C4—348no—
549C4⋊C4—348no—
550C4⋊C4—348no—
551C4⋊C4—688no—
552C22⋊C4—348no—
553C22⋊C4—348no—
554C22⋊C4—3416no—
555C22⋊C4—688no—
556C42—3416no—
557C23—174no—
558C23—344no—
559C23—344no—
560C23—342no—
561C23—342no—
562C23—342no—
563C23—344no—
564C23—342no—
565C23—342no—
566C23—342no—
567C23—344no—
568C23—342no—
569C23—342no—
570C23—342no—
571C23—342no—
572C23—682no—
573C23—682no—
574C23—682no—
575C23—682no—
576C23—682no—
577C23—682no—
578C23—682no—
579C23—682no—
580C23C22×C4×D1711yescharacteristic
581C23C2×D4×D1711yescharacteristic
582C23C2×D4×D1711yescharacteristic
583C23C2×D4×D1712yes—
584C23#1058769111yescharacteristic
585C23#1058769412yes—
586C23#1058769411yescharacteristic
587C23#41×D1712yes—
588C23#1058769611yescharacteristic
589C23#1058769712yes—
590C23#1058769711yescharacteristic
591Q8—6816no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—174no—
623C2×C4—178no—
624C2×C4—174no—
625C2×C4—178no—
626C2×C4—174no—
627C2×C4—3432no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—344no—
631C2×C4—342no—
632C2×C4—342no—
633C2×C4—344no—
634C2×C4—344no—
635C2×C4—344no—
636C2×C4—681no—
637C2×C4—682no—
638C2×C4—681no—
639C2×C4—681no—
640C2×C4—681no—
641C2×C4—682no—
642C2×C4—682no—
643C2×C4—681no—
644C2×C4—681no—
645C2×C4—682no—
646C2×C4—682no—
647C2×C4—682no—
648C2×C4—682no—
649C2×C4—682no—
650C2×C4—682no—
651C2×C4—682no—
652C2×C4—682no—
653C2×C4—682no—
654C2×C4C22⋊C4×D1714yes—
655C22—344no—
656C22—344no—
657C22—344no—
658C22—682no—
659C22—682no—
660C22—684no—
661C22—684no—
662C22—684no—
663C22—684no—
664C22—682no—
665C22—682no—
666C22C2×C22⋊C4×D1711yescharacteristic
667C22C4×D4×D1712yes—
668C22#1176328512yes—
669C22#1176328911yescharacteristic
670C22#1176329011yescharacteristic
671C22#120×D1711yescharacteristic
672C22#121×D1712yes—
673C22#123×D1711yescharacteristic
674C22#1176337511yescharacteristic
675C22#1176337911yescharacteristic
676C22#1176338112yes—
677C22#1176338212yes—
678C22#1176338211yescharacteristic
679C22#1176338411yescharacteristic
680C22#1176338412yes—
681C22#1176338612yes—
682C22#1176338811yescharacteristic
683C22#1176338911yescharacteristic
684C22#1176338912yes—
685C22#1176341812yes—
686C22#1176341812yes—
687C22#1176342512yes—
688C22#1176342912yes—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—82no—
694C4—178no—
695C4—682no—
696C4—684no—
697C4—682no—
698C4—1362no—
699C2—684no—
700C2D17×#38611yescharacteristic
701C2#1453136711yescharacteristic
702C2#1453137311yescharacteristic
703C2#1453146112yes—
704C2#1453146512yes—
705C2#1453451211yescharacteristic
706C2#1453451611yescharacteristic
707C2#1453454312yes—
708C2#1453455212yes—
709C2#1453458511yescharacteristic
710C2#1453458711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#14258
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#11762713C2semidirect
5. #14527038C2semidirect
6. #14527037C2semidirect
7. #14527034C2semidirect
10. #14527012C2semidirect
11. C17×#1441C2semidirect
16. C2×#10587582C4semidirect
307. #1441D17semidirect
461. C17#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#105875822
C22×#105875782
C2×#117627132
C2×#117626682
#145270382
#145270372
#145270342
#145270322
#145270122
C17×#14412
#1425817