Finite Group Databasebeta

← #16965615

Subgroups — #16965615

Order 4352.

2734 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587582C211yesmaximal, characteristic
2C22×C42×D17C211yesmaximal, characteristic
3C2×#11762713C212yesmaximal, complemented
4C2×#11762668C211yesmaximal, characteristic
5#14527019C211yesmaximal, characteristic, complemented
6#14527017C211yesmaximal, characteristic, complemented
7#14527016C211yesmaximal, characteristic, complemented
8#14527012C212yesmaximal, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1436C211yesmaximal, characteristic, complemented
12C23×Dic34C2211yescharacteristic
13C23×C4×D17C2212yes—
14C23×C4×D17C2211yescharacteristic
15C2×#10587582—24no—
16C2×#10587582—24no—
17C2×#10587582C418yescomplemented
18C22×#63225C2212yes—
19C22×#63225C2211yescharacteristic
20C22×#63225C2211yescharacteristic
21C22×#63227C2212yes—
22C22×#63227C2211yescharacteristic
23C22×#63227C2211yescharacteristic
24C22×C4×Dic17C2211yescharacteristic
25C22×C4×Dic17C2212yes—
26C2×C42×D17—28no—
27C2×C42×D17—24no—
28#11762713—28no—
29#11762668—24no—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C2×#10587502C2212yes—
36C2×#10587502C2211yescharacteristic
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C42×C17C2211yescharacteristic
39C2×C17×#95C2212yes—
40C2×C17×#95C2212yes—
41C2×C17×#95C2211yescharacteristic
42C24×D17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×C4×C17C2311yescharacteristic
47C23×C4×C17C2312yes—
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C23×Dic17C2312yes—
52C23×Dic17C2311yescharacteristic
53C22×C4×D17—24no—
54C22×C4×D17—24no—
55C22×C4×D17—24no—
56C22×C4×D17—24no—
57C22×C4×D17—24no—
58C22×C4×D17—24no—
59C22×C4×D17—24no—
60C22×C4×D17—24no—
61C22×C4×D17C2×C414yes—
62C22×C4×D17D414yes—
63C22×C4×D17D414yes—
64C22×C4×D17Q818yescomplemented
65C22×Dic34—24no—
66C22×Dic34—22no—
67C22×Dic34—22no—
68C22×Dic34—24no—
69C22×Dic34C2×C414yescomplemented
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17C2×C414yescomplemented
73C2×C42×C17—22no—
74C2×C42×C17—24no—
75C2×#63227—22no—
76C2×#63227—22no—
77C2×#63227—22no—
78C2×#63227—22no—
79C2×#63227—24no—
80C2×#63227—24no—
81C2×#63227—24no—
82C2×#63227C2×C414yes—
83C2×#63227C2×C414yes—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225—24no—
89C2×#63225—24no—
90C2×#63225—24no—
91C2×#63225C2×C414yescomplemented
92C2×#63225C2×C414yescomplemented
93C2×C4×Dic17—22no—
94C2×C4×Dic17—24no—
95C2×C4×Dic17—24no—
96C2×C4×Dic17—24no—
97C2×C4×Dic17D418yes—
98#10587582—232no—
99C42×D17—232no—
100C17×#95—24no—
101C17×#95—22no—
102C17×#95—24no—
103#10587502—24no—
104#10587502—22no—
105#10587502—24no—
106#10587502—24no—
107#10587502—24no—
108#10587502—24no—
109#10587502—24no—
110C24×C17C2411yescharacteristic
111C23×D17—22no—
112C23×D17—22no—
113C23×D17—22no—
114C23×D17—22no—
115C23×D17C22×C412yes—
116C23×D17C2×D412yes—
117C23×D17C2×D412yes—
118C23×D17C2×Q814yes—
119C23×D17#4114yes—
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—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
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×C17C22×C412yes—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D414yes—
147C22×C4×C17C2×Q814yes—
148C22×C4×C17#4114yes—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
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×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17C22×C412yes—
169C22×Dic17C22×C412yes—
170C22×Dic17C22×C412yes—
171C22×Dic17C2×D414yes—
172C22×Dic17C2×D412yes—
173C22×Dic17C2×D412yes—
174C22×Dic17C2×D414yes—
175C22×Dic17C2×Q814yes—
176C22×Dic17#4114yes—
177C22×Dic17#4114yes—
178C2×C4×D17—216no—
179C2×C4×D17—216no—
180C2×C4×D17—216no—
181C2×C4×D17—216no—
182C2×C4×D17—216no—
183C2×C4×D17—216no—
184C2×C4×D17—216no—
185C2×C4×D17—28no—
186C2×C4×D17—216no—
187C2×C4×D17—216no—
188C2×C4×D17—28no—
189C2×C4×D17C22⋊C4116yes—
190C2×Dic34—28no—
191C2×Dic34—216no—
192C2×Dic34—44no—
193C2×Dic34—48no—
194C2×Dic34—44no—
195C4⋊C4×C17—28no—
196C42×C17—28no—
197#63227—28no—
198#63227—28no—
199#63227—48no—
200#63225—28no—
201#63225—28no—
202#63225—48no—
203C4×Dic17—28no—
204C4×Dic17—216no—
205#14702—171nomaximal, Sylow
206C23×C17C23×C411yescharacteristic
207C23×C17C22×D411yescharacteristic
208C23×C17C22×D411yescharacteristic
209C23×C17C22×D412yes—
210C23×C17C22×Q812yes—
211C23×C17C2×#4112yes—
212C23×C17C2×#4112yes—
213C23×C17C2×#4112yes—
214C23×C17C2×#4112yes—
215C22×D17—24no—
216C22×D17—22no—
217C22×D17—22no—
218C22×D17—22no—
219C22×D17—24no—
220C22×D17—24no—
221C22×D17—24no—
222C22×D17—22no—
223C22×D17—24no—
224C22×D17—24no—
225C22×D17—24no—
226C22×D17—24no—
227C22×D17—24no—
228C22×D17—24no—
229C22×D17—24no—
230C22×D17—24no—
231C22×D17C2×C22⋊C414yes—
232C22×D17C4×Q818yes—
233C22×D17C22⋊Q818yes—
234C22×D17#12414yes—
235C22×D17C4⋊Q814yes—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—24no—
242C2×C4×C17—22no—
243C2×C4×C17—24no—
244C2×C4×C17—24no—
245C2×C4×C17—24no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—24no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—41no—
252C2×C4×C17—42no—
253C2×C4×C17—41no—
254C2×C4×C17—42no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—42no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—42no—
261C2×C4×C17—42no—
262C2×C4×C17—42no—
263C2×C4×C17C2×C22⋊C414yes—
264C2×C4×C17C22⋊Q818yescomplemented
265C2×Dic17—24no—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—24no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—41no—
277C2×Dic17—42no—
278C2×Dic17—41no—
279C2×Dic17—42no—
280C2×Dic17—42no—
281C2×Dic17—42no—
282C2×Dic17—42no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—42no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17C2×C22⋊C414yes—
289C2×Dic17C4×D418yes—
290C2×Dic17#12014yes—
291C2×Dic17#12118yes—
292C2×Dic17C22⋊Q818yescomplemented
293C2×Dic17#12314yes—
294C4×D17—264no—
295C4×D17—232no—
296Dic34—416no—
297C22×C22⋊Q8—171no—
298C23×C42—171no—
299C2×#391—172no—
300C2×#380—171no—
301#1436—171no—
302#1436—171no—
303#1436—171no—
304#1436D1711yescharacteristic, complemented
305C4⋊#95—172no—
306C22⋊#95—172no—
307C22⋊#95—172no—
308C22×C17C22×C22⋊C411yescharacteristic
309C22×C17C2×C4×D412yes—
310C22×C17C2×C4×Q812yes—
311C22×C17C4×#4111yescharacteristic
312C22×C17C4×#4111yescharacteristic
313C22×C17C2×#12011yescharacteristic
314C22×C17C2×#12112yes—
315C22×C17C2×C22⋊Q812yes—
316C22×C17C2×C22⋊Q812yes—
317C22×C17C2×C22⋊Q812yes—
318C22×C17C2×#12311yescharacteristic
319C22×C17#52512yes—
320C22×C17#52512yes—
321C22×C17#52512yes—
322C22×C17C2×#12411yescharacteristic
323C22×C17#52912yes—
324C22×C17#52912yes—
325C22×C17C2×C4⋊Q811yescharacteristic
326C22×C17#53212yes—
327C22×C17#53312yes—
328C22×C17#53312yes—
329C2×D17—28no—
330C2×D17—28no—
331C2×D17—28no—
332C2×D17—28no—
333C2×D17—24no—
334C2×D17—24no—
335C2×D17—28no—
336C2×D17—28no—
337C2×D17#39118yescomplemented
338C4×C17—28no—
339C4×C17—24no—
340C4×C17—42no—
341C4×C17—42no—
342C4×C17—82no—
343Dic17—28no—
344Dic17—42no—
345Dic17—42no—
346Dic17—82no—
347Dic17#38618yes—
348C23×Q8—171no—
349C24×C4—171no—
350C24×C4—172no—
351C2×C22⋊Q8—178no—
352C2×C22⋊Q8—344no—
353C2×C22⋊Q8—344no—
354C22×C4⋊C4—171no—
355C22×C4⋊C4—171no—
356C22×C4⋊C4—172no—
357C22×C4⋊C4C2×D1711yescharacteristic
358C22×C22⋊C4—171no—
359C22×C22⋊C4—171no—
360C22×C22⋊C4—172no—
361C22×C42—172no—
362C22×C42—171no—
363C22×C42—348no—
364C22×C42—344no—
365C22×C42C2×D1711yescharacteristic
366#391—348no—
367#380—344no—
368C2×#95—172no—
369C2×#95—172no—
370C2×#95—172no—
371C2×#95—172no—
372C2×#95—172no—
373C2×#95—172no—
374C2×#95—171no—
375C2×#95C2×D1712yes—
376C2×#95C2×D1712yes—
377C2×#95C2×D1711yescharacteristic
378C2×C17C2×#38611yescharacteristic
379C2×C17C2×#39111yescharacteristic
380C2×C17#229711yescharacteristic
381C2×C17C4×C22⋊Q812yes—
382C2×C17C4×C22⋊Q812yes—
383C2×C17#284812yes—
384C2×C17#285012yes—
385C2×C17#285512yes—
386C2×C17#285712yes—
387D17—28no—
388C25—171no—
389C22×Q8—174no—
390C22×Q8—342no—
391C22×Q8—342no—
392C22×Q8—344no—
393C22×Q8—344no—
394C23×C4—171no—
395C23×C4—172no—
396C23×C4—174no—
397C23×C4—174no—
398C23×C4—174no—
399C23×C4—171no—
400C23×C4—171no—
401C23×C4—178no—
402C23×C4—172no—
403C23×C4—344no—
404C23×C4—344no—
405C23×C4—344no—
406C23×C4—344no—
407C23×C4—344no—
408C23×C4—344no—
409C23×C4—344no—
410C23×C4—344no—
411C23×C4C22×D1712yes—
412C23×C4C22×D1711yescharacteristic
413C23×C4C22×D1711yescharacteristic
414C23×C4C22×D1711yescharacteristic
415C23×C4C22×D1712yes—
416C22⋊Q8—3432no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—174no—
420C2×C4⋊C4—174no—
421C2×C4⋊C4—342no—
422C2×C4⋊C4—342no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—342no—
425C2×C4⋊C4—344no—
426C2×C4⋊C4—344no—
427C2×C4⋊C4—344no—
428C2×C4⋊C4C4×D1714yescomplemented
429C2×C22⋊C4—174no—
430C2×C22⋊C4—174no—
431C2×C22⋊C4—342no—
432C2×C22⋊C4—342no—
433C2×C22⋊C4—342no—
434C2×C22⋊C4—342no—
435C2×C22⋊C4—344no—
436C2×C22⋊C4—344no—
437C2×C22⋊C4—344no—
438C2×C42—22no—
439C2×C42—24no—
440C2×C42—178no—
441C2×C42—3432no—
442C2×C42—344no—
443C2×C42—344no—
444C2×C42—344no—
445C2×C42—342no—
446#95—24no—
447#95—22no—
448#95—24no—
449#95—344no—
450#95—344no—
451#95—344no—
452#95—344no—
453#95—344no—
454#95—342no—
455#95—344no—
456C17#1470211yescharacteristic, complemented, Sylow
457C24—172no—
458C24—172no—
459C24—172no—
460C24—174no—
461C24—174no—
462C24—342no—
463C24—342no—
464C24—342no—
465C24—342no—
466C24C23×D1711yescharacteristic
467C2×Q8—348no—
468C2×Q8—3416no—
469C2×Q8—684no—
470C2×Q8—688no—
471C2×Q8—684no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
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—172no—
494C22×C4—174no—
495C22×C4—172no—
496C22×C4—172no—
497C22×C4—174no—
498C22×C4—174no—
499C22×C4—174no—
500C22×C4—174no—
501C22×C4—1716no—
502C22×C4—172no—
503C22×C4—172no—
504C22×C4—348no—
505C22×C4—3416no—
506C22×C4—3416no—
507C22×C4—3416no—
508C22×C4—348no—
509C22×C4—3416no—
510C22×C4—3416no—
511C22×C4—3416no—
512C22×C4—3416no—
513C22×C4—3416no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—3416no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—341no—
522C22×C4—341no—
523C22×C4—341no—
524C22×C4—341no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4C2×C4×D1712yes—
535C22×C4C2×C4×D1712yes—
536C22×C4C2×C4×D1712yes—
537C22×C4#6325214yes—
538C22×C4D4×D1712yes—
539C22×C4D4×D1712yes—
540C22×C4D4×D1714yes—
541C22×C4Q8×D1714yes—
542C4⋊C4—28no—
543C4⋊C4—348no—
544C4⋊C4—348no—
545C4⋊C4—688no—
546C22⋊C4—348no—
547C22⋊C4—348no—
548C22⋊C4—688no—
549C42—28no—
550C42—3416no—
551C42—348no—
552C23—174no—
553C23—178no—
554C23—178no—
555C23—174no—
556C23—174no—
557C23—344no—
558C23—342no—
559C23—342no—
560C23—342no—
561C23—344no—
562C23—344no—
563C23—344no—
564C23—342no—
565C23—344no—
566C23—344no—
567C23—344no—
568C23—344no—
569C23—344no—
570C23—344no—
571C23—344no—
572C23—344no—
573C23C22×C4×D1711yescharacteristic
574C23C2×#6325212yes—
575C23C2×D4×D1711yescharacteristic
576C23C2×D4×D1711yescharacteristic
577C23C2×D4×D1712yes—
578C23C2×Q8×D1712yes—
579C23#41×D1712yes—
580C23#41×D1712yes—
581C23#41×D1712yes—
582Q8—6816no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—174no—
611C2×C4—178no—
612C2×C4—174no—
613C2×C4—178no—
614C2×C4—174no—
615C2×C4—178no—
616C2×C4—3432no—
617C2×C4—3464no—
618C2×C4—344no—
619C2×C4—344no—
620C2×C4—344no—
621C2×C4—344no—
622C2×C4—344no—
623C2×C4—344no—
624C2×C4—342no—
625C2×C4—342no—
626C2×C4—344no—
627C2×C4—342no—
628C2×C4—342no—
629C2×C4—681no—
630C2×C4—682no—
631C2×C4—681no—
632C2×C4—682no—
633C2×C4—682no—
634C2×C4—682no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4—682no—
640C2×C4—682no—
641C2×C4C22⋊C4×D1714yes—
642C2×C4#1058758218yescomplemented
643C22—178no—
644C22—348no—
645C22—348no—
646C22—344no—
647C22—348no—
648C22—348no—
649C22—344no—
650C22—348no—
651C22—348no—
652C22C2×C22⋊C4×D1711yescharacteristic
653C22C4×D4×D1712yes—
654C22C4×Q8×D1712yes—
655C22#1176327711yescharacteristic
656C22C4×#6325211yescharacteristic
657C22#120×D1711yescharacteristic
658C22#121×D1712yes—
659C22C22⋊Q8×D1712yes—
660C22C22⋊Q8×D1712yes—
661C22C2×#1058758212yes—
662C22#123×D1711yescharacteristic
663C22#1176331712yes—
664C22#1176331812yes—
665C22#1176331812yes—
666C22#124×D1711yescharacteristic
667C22#1176333712yes—
668C22#1176334212yes—
669C22C4⋊Q8×D1711yescharacteristic
670C22#1176335612yes—
671C22#1176336312yes—
672C22#1176336512yes—
673C4—28no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—178no—
679C4—348no—
680C4—682no—
681C4—682no—
682C4—1362no—
683C2—348no—
684C2D17×#38611yescharacteristic
685C2D17×#39111yescharacteristic
686C2#1453127011yescharacteristic
687C2C4×#1058758212yes—
688C2#1453131812yes—
689C2#1453707012yes—
690C2#1453708212yes—
691C2#1453711912yes—
692C2#1453713512yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1436C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1436D17
hypocenterC17#14702
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. C2×#11762713C2semidirect
5. #14527019C2semidirect
6. #14527017C2semidirect
7. #14527016C2semidirect
8. #14527012C2semidirect
11. C17×#1436C2semidirect
17. C2×#10587582C4semidirect
64. C22×C4×D17Q8semidirect
69. C22×Dic34C2×C4semidirect
72. C2×C4⋊C4×C17C2×C4semidirect
91. C2×#63225C2×C4semidirect
92. C2×#63225C2×C4semidirect
264. C2×C4×C17C22⋊Q8semidirect
292. C2×Dic17C22⋊Q8semidirect
304. #1436D17semidirect
337. C2×D17#391semidirect
428. C2×C4⋊C4C4×D17semidirect
456. C17#14702semidirect
642. C2×C4#10587582semidirect

Maximal subgroups

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

subgroupindex
C22×#105875822
C22×C42×D172
C2×#117627132
C2×#117626682
#145270192
#145270172
#145270162
#145270122
#145270052
C17×#14362
#1470217