Finite Group Databasebeta

← #16960918

Subgroups — #16960918

Order 4352.

2086 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587582C211yesmaximal, characteristic
2C2×#11762666C211yesmaximal, characteristic
3C2×#95×D17C211yesmaximal, characteristic
4#14527054C212yesmaximal, complemented
5#14527050C212yesmaximal, complemented
6#14527038C211yesmaximal, characteristic, complemented
7#14527037C211yesmaximal, characteristic, complemented
8#14527034C211yesmaximal, characteristic, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×Dic34C2211yescharacteristic
13C23×C4×D17C2212yes—
14C23×C4×D17C2211yescharacteristic
15C2×#10587582—24no—
16C2×#10587582—24no—
17C2×#10587582C418yescomplemented
18C22×C4⋊Dic17C2211yescharacteristic
19C22×#63225C2211yescharacteristic
20C22×#63225C2211yescharacteristic
21C22×#63227C2212yes—
22C22×#63227C2211yescharacteristic
23C22×#63227C2211yescharacteristic
24#11762666—24no—
25C2×#10587502C2211yescharacteristic
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34#95×D17—24no—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C2×C17×#95C2212yes—
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—42no—
54C22×C4×D17—42no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17—42no—
58C22×C4×D17—42no—
59C22×C4×D17C2×C414yes—
60C22×C4×D17D414yes—
61C22×C4×D17D414yes—
62C22×C4×D17Q818yescomplemented
63C22×Dic34—22no—
64C22×Dic34—22no—
65C22×Dic34—42no—
66C22×Dic34—42no—
67C22×Dic34C2×C414yescomplemented
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—42no—
72C2×C4⋊C4×C17C2×C414yescomplemented
73C2×#63227—22no—
74C2×#63227—22no—
75C2×#63227—22no—
76C2×#63227—22no—
77C2×#63227—24no—
78C2×#63227—24no—
79C2×#63227—24no—
80C2×#63227C2×C414yes—
81C2×#63227C2×C414yes—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17—42no—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225C2×C414yescomplemented
89C2×#63225C2×C414yescomplemented
90#10587582—232no—
91C17×#95—22no—
92C17×#95—42no—
93C17×#95—42no—
94#10587502—22no—
95#10587502—24no—
96#10587502—24no—
97#10587502—42no—
98#10587502—42no—
99#10587502—42no—
100#10587502—42no—
101#10587502—42no—
102#10587502—42no—
103C24×C17C2411yescharacteristic
104C23×D17—22no—
105C23×D17—21no—
106C23×D17—22no—
107C23×D17—21no—
108C23×D17—21no—
109C23×D17—21no—
110C23×D17C22×C412yes—
111C23×D17C2×D412yes—
112C23×D17C2×D412yes—
113C23×D17C2×Q814yes—
114C23×D17#4114yes—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
121C22×C4×C17—21no—
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—21no—
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×C17C22×C412yes—
141C22×C4×C17C22×C412yes—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×Q814yes—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—22no—
165C22×Dic17—22no—
166C22×Dic17—22no—
167C22×Dic17—22no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17C22×C412yes—
172C22×Dic17C22×C412yes—
173C22×Dic17C22×C412yes—
174C22×Dic17C2×D412yes—
175C22×Dic17C2×D412yes—
176C22×Dic17C2×Q814yes—
177C2×C4×D17—216no—
178C2×C4×D17—48no—
179C2×C4×D17—44no—
180C2×C4×D17—48no—
181C2×C4×D17—44no—
182C2×C4×D17—48no—
183C2×C4×D17—48no—
184C2×C4×D17—48no—
185C2×C4×D17—48no—
186C2×C4×D17—48no—
187C2×C4×D17—48no—
188C2×C4×D17C22⋊C4116yes—
189C2×Dic34—28no—
190C2×Dic34—48no—
191C2×Dic34—48no—
192C2×Dic34—44no—
193C2×Dic34—44no—
194C4⋊C4×C17—28no—
195C4⋊C4×C17—44no—
196#63227—28no—
197#63227—28no—
198#63227—48no—
199C4⋊Dic17—44no—
200#63225—28no—
201#63225—28no—
202#14281—171nomaximal, Sylow
203C23×C17C23×C411yescharacteristic
204C23×C17C22×D411yescharacteristic
205C23×C17C22×D411yescharacteristic
206C23×C17C22×Q812yes—
207C23×C17C2×#4112yes—
208C23×C17H(5,𝔽2)12yes—
209C23×C17H(5,𝔽2)11yescharacteristic
210C23×C17H(5,𝔽2)12yes—
211C23×C17H(5,𝔽2)11yescharacteristic
212C23×C172−1+411yescharacteristic
213C23×C172−1+411yescharacteristic
214C22×D17—24no—
215C22×D17—22no—
216C22×D17—22no—
217C22×D17—24no—
218C22×D17—22no—
219C22×D17—22no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—24no—
225C22×D17—24no—
226C22×D17—24no—
227C22×D17—24no—
228C22×D17—24no—
229C22×D17—24no—
230C22×D17—24no—
231C22×D17—24no—
232C22×D17C2×C22⋊C414yes—
233C22×D17C4×Q818yes—
234C22×D17C22⋊Q818yes—
235C22×D17#12414yes—
236C22×D17C4⋊Q814yes—
237C2×C4×C17—24no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—24no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
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—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—42no—
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×C17—42no—
267C2×C4×C17C2×C22⋊C414yes—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—24no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—24no—
278C2×Dic17—24no—
279C2×Dic17—24no—
280C2×Dic17—42no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—42no—
284C2×Dic17—42no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—42no—
290C2×Dic17—42no—
291C2×Dic17—42no—
292C2×Dic17—42no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17—42no—
298C2×Dic17C2×C22⋊C414yes—
299C4×D17—416no—
300C4×D17—432no—
301Dic34—416no—
302C22×C22⋊Q8—171no—
303C2×#380—171no—
304C22×#95—171no—
305#1446—172no—
306#1445—172no—
307#1441—171no—
308#1441—171no—
309#1441—171no—
310#1441D1711yescharacteristic, complemented
311C22⋊#95—172no—
312C22⋊#95—172no—
313C22×C17C22×C22⋊C411yescharacteristic
314C22×C17C2×C4×Q812yes—
315C22×C17#51812yes—
316C22×C17#52011yescharacteristic
317C22×C17#52011yescharacteristic
318C22×C17C2×C22⋊Q812yes—
319C22×C17C2×#12411yescharacteristic
320C22×C17C2×C4⋊Q811yescharacteristic
321C22×C17#53412yes—
322C22×C17#53411yescharacteristic
323C22×C17#53411yescharacteristic
324C22×C17#53511yescharacteristic
325C22×C17#53611yescharacteristic
326C22×C17#53711yescharacteristic
327C22×C17#53711yescharacteristic
328C22×C17#53812yes—
329C22×C17#53812yes—
330C22×C17#53912yes—
331C22×C17#53912yes—
332C22×C17#54312yes—
333C22×C17#54312yes—
334C22×C17#54412yes—
335C22×C17#54412yes—
336C2×D17—24no—
337C2×D17—24no—
338C2×D17—28no—
339C2×D17—28no—
340C2×D17—28no—
341C2×D17—28no—
342C2×D17—24no—
343C2×D17—24no—
344C2×D17—24no—
345C2×D17—24no—
346C2×D17#39118yes—
347C4×C17—24no—
348C4×C17—42no—
349C4×C17—42no—
350C4×C17—44no—
351C4×C17—82no—
352Dic17—24no—
353Dic17—42no—
354Dic17—42no—
355Dic17—44no—
356Dic17—82no—
357C23×Q8—171no—
358C24×C4—171no—
359C24×C4—172no—
360C2×C22⋊Q8—178no—
361C2×C22⋊Q8—344no—
362C2×C22⋊Q8—344no—
363C22×C4⋊C4—171no—
364C22×C4⋊C4—171no—
365C22×C4⋊C4—171no—
366C22×C4⋊C4C2×D1711yescharacteristic
367C22×C4⋊C4C2×D1711yescharacteristic
368C22×C22⋊C4—171no—
369C22×C22⋊C4—171no—
370C22×C22⋊C4—172no—
371#380—344no—
372C2×#95—171no—
373C2×#95—172no—
374C2×#95—172no—
375C2×#95—172no—
376C2×#95—172no—
377C2×#95—172no—
378C2×#95—172no—
379C2×#95—172no—
380C2×#95—172no—
381C2×#95—344no—
382C2×#95C2×D1711yescharacteristic
383C2×#95C2×D1712yes—
384C2×#95C2×D1712yes—
385C2×C17C2×#39111yescharacteristic
386C2×C17#230811yescharacteristic
387C2×C17#231111yescharacteristic
388C2×C17#232912yes—
389C2×C17#232912yes—
390C2×C17#262612yes—
391C2×C17#262712yes—
392C2×C17#263911yescharacteristic
393C2×C17#263911yescharacteristic
394C2×C17#265911yescharacteristic
395C2×C17#265911yescharacteristic
396D17—28no—
397C25—171no—
398C22×Q8—174no—
399C22×Q8—342no—
400C22×Q8—342no—
401C22×Q8—682no—
402C22×Q8—682no—
403C23×C4—171no—
404C23×C4—174no—
405C23×C4—174no—
406C23×C4—174no—
407C23×C4—171no—
408C23×C4—171no—
409C23×C4—172no—
410C23×C4—178no—
411C23×C4—172no—
412C23×C4—344no—
413C23×C4—344no—
414C23×C4—682no—
415C23×C4—682no—
416C23×C4—682no—
417C23×C4—682no—
418C23×C4—682no—
419C23×C4—682no—
420C23×C4C22×D1712yes—
421C23×C4C22×D1711yescharacteristic
422C23×C4C22×D1711yescharacteristic
423C23×C4C22×D1712yes—
424C23×C4C22×D1711yescharacteristic
425C22⋊Q8—3432no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—42no—
430C2×C4⋊C4—174no—
431C2×C4⋊C4—174no—
432C2×C4⋊C4—342no—
433C2×C4⋊C4—342no—
434C2×C4⋊C4—342no—
435C2×C4⋊C4—342no—
436C2×C4⋊C4—342no—
437C2×C4⋊C4—682no—
438C2×C4⋊C4C4×D1714yescomplemented
439C2×C22⋊C4—174no—
440C2×C22⋊C4—174no—
441C2×C22⋊C4—342no—
442C2×C22⋊C4—342no—
443C2×C22⋊C4—342no—
444C2×C22⋊C4—342no—
445C2×C22⋊C4—344no—
446C2×C22⋊C4—344no—
447C2×C22⋊C4—344no—
448#95—22no—
449#95—42no—
450#95—42no—
451#95—344no—
452#95—344no—
453#95—342no—
454#95—682no—
455#95—682no—
456#95—682no—
457#95—682no—
458#95—682no—
459#95—682no—
460C17#1428111yescharacteristic, complemented, Sylow
461C24—172no—
462C24—172no—
463C24—172no—
464C24—174no—
465C24—174no—
466C24—342no—
467C24—341no—
468C24—341no—
469C24—342no—
470C24—341no—
471C24—341no—
472C24C23×D1711yescharacteristic
473C2×Q8—348no—
474C2×Q8—688no—
475C2×Q8—688no—
476C2×Q8—684no—
477C2×Q8—684no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—172no—
504C22×C4—172no—
505C22×C4—172no—
506C22×C4—1716no—
507C22×C4—172no—
508C22×C4—172no—
509C22×C4—174no—
510C22×C4—3416no—
511C22×C4—342no—
512C22×C4—342no—
513C22×C4—342no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—341no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—341no—
523C22×C4—342no—
524C22×C4—341no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4—342no—
528C22×C4—341no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—688no—
537C22×C4—688no—
538C22×C4—684no—
539C22×C4—684no—
540C22×C4—688no—
541C22×C4—688no—
542C22×C4—688no—
543C22×C4—688no—
544C22×C4—688no—
545C22×C4—688no—
546C22×C4C2×C4×D1712yes—
547C22×C4C2×C4×D1712yes—
548C22×C4C2×C4×D1712yes—
549C22×C4D4×D1712yes—
550C22×C4D4×D1712yes—
551C22×C4Q8×D1714yes—
552C4⋊C4—28no—
553C4⋊C4—44no—
554C4⋊C4—348no—
555C4⋊C4—348no—
556C4⋊C4—684no—
557C22⋊C4—348no—
558C22⋊C4—348no—
559C22⋊C4—688no—
560C23—174no—
561C23—178no—
562C23—178no—
563C23—174no—
564C23—174no—
565C23—344no—
566C23—342no—
567C23—342no—
568C23—344no—
569C23—342no—
570C23—342no—
571C23—342no—
572C23—342no—
573C23—342no—
574C23—342no—
575C23—344no—
576C23—344no—
577C23—344no—
578C23—344no—
579C23—344no—
580C23—344no—
581C23—344no—
582C23—344no—
583C23C22×C4×D1711yescharacteristic
584C23C2×D4×D1711yescharacteristic
585C23C2×D4×D1711yescharacteristic
586C23#1058769112yes—
587C23#1058769111yescharacteristic
588C23C2×Q8×D1712yes—
589C23#1058769411yescharacteristic
590C23#41×D1712yes—
591C23#1058769612yes—
592C23#1058769611yescharacteristic
593C23#1058769711yescharacteristic
594Q8—6816no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—174no—
626C2×C4—344no—
627C2×C4—344no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—342no—
631C2×C4—342no—
632C2×C4—344no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—344no—
636C2×C4—344no—
637C2×C4—344no—
638C2×C4—6816no—
639C2×C4—6832no—
640C2×C4—682no—
641C2×C4—682no—
642C2×C4—681no—
643C2×C4—681no—
644C2×C4—682no—
645C2×C4—681no—
646C2×C4—681no—
647C2×C4—681no—
648C2×C4—681no—
649C2×C4—682no—
650C2×C4—682no—
651C2×C4—682no—
652C2×C4—682no—
653C2×C4—682no—
654C2×C4—682no—
655C2×C4—682no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4C22⋊C4×D1714yes—
659C22—178no—
660C22—344no—
661C22—344no—
662C22—344no—
663C22—344no—
664C22—348no—
665C22—348no—
666C22—348no—
667C22—348no—
668C22—344no—
669C22—344no—
670C22C2×C22⋊C4×D1711yescharacteristic
671C22C4×Q8×D1712yes—
672C22#1176328112yes—
673C22#1176328911yescharacteristic
674C22#1176329011yescharacteristic
675C22C22⋊Q8×D1712yes—
676C22#124×D1711yescharacteristic
677C22C4⋊Q8×D1711yescharacteristic
678C22#1176336712yes—
679C22#1176336911yescharacteristic
680C22#1176337011yescharacteristic
681C22#1176337511yescharacteristic
682C22#1176338211yescharacteristic
683C22#1176338811yescharacteristic
684C22#1176338911yescharacteristic
685C22#1176339112yes—
686C22#1176339812yes—
687C22#1176340212yes—
688C22#1176340712yes—
689C22#1176343412yes—
690C22#1176343412yes—
691C22#1176343612yes—
692C22#1176343912yes—
693C4—24no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—344no—
699C4—682no—
700C4—682no—
701C4—684no—
702C4—1362no—
703C2—348no—
704C2D17×#39111yescharacteristic
705C2#1453135811yescharacteristic
706C2#1453137311yescharacteristic
707C2#1453151112yes—
708C2#1453151812yes—
709C2#1453459412yes—
710C2#1453460212yes—
711C2#1453471511yescharacteristic
712C2#1453471611yescharacteristic
713C2#1453489711yescharacteristic
714C2#1453489911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#14281
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. #14527054C2semidirect
5. #14527050C2semidirect
6. #14527038C2semidirect
7. #14527037C2semidirect
8. #14527034C2semidirect
11. C17×#1441C2semidirect
17. C2×#10587582C4semidirect
62. C22×C4×D17Q8semidirect
67. C22×Dic34C2×C4semidirect
72. C2×C4⋊C4×C17C2×C4semidirect
88. C2×#63225C2×C4semidirect
89. C2×#63225C2×C4semidirect
310. #1441D17semidirect
438. C2×C4⋊C4C4×D17semidirect
460. C17#14281semidirect

Maximal subgroups

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

subgroupindex
C22×#105875822
C2×#117626662
C2×#95×D172
#145270542
#145270502
#145270382
#145270372
#145270342
#145270052
C17×#14412
#1428117