Finite Group Databasebeta

← #16956714

Subgroups — #16956714

Order 4352.

2166 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D17C211yesmaximal, characteristic
2C2×#11762666C211yesmaximal, characteristic
3C2×#11762656C211yesmaximal, characteristic
4#14527038C211yesmaximal, characteristic, complemented
5#14527037C211yesmaximal, characteristic, complemented
6#14527034C211yesmaximal, characteristic, complemented
7#14527018C212yesmaximal, complemented
8#14527015C212yesmaximal, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1441C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C23×C4×D17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×C4⋊Dic17C2211yescharacteristic
17C22×#63225C2211yescharacteristic
18C22×#63225C2211yescharacteristic
19C2×C4⋊C4×D17—24no—
20C2×C4⋊C4×D17—24no—
21C2×C4⋊C4×D17C418yescomplemented
22C22×#63227C2212yes—
23C22×#63227C2212yes—
24C22×C4×Dic17C2212yes—
25#11762666—24no—
26#11762656—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C2×C17×#95C2212yes—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C24×D17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×C4×C17C2311yescharacteristic
46C23×Dic17C2312yes—
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C23×Dic17C2311yescharacteristic
50C23×Dic17C2311yescharacteristic
51C22×C4×D17—22no—
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17—22no—
57C22×C4×D17—22no—
58C22×C4×D17—22no—
59C22×C4×D17—42no—
60C22×C4×D17—42no—
61C22×C4×D17—42no—
62C22×C4×D17—42no—
63C22×C4×D17C2×C414yes—
64C22×C4×D17C2×C414yes—
65C22×C4×D17C2×C414yes—
66C22×C4×D17D414yes—
67C22×C4×D17D414yes—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—42no—
72C2×C4⋊C4×C17C2×C414yes—
73C2×#63227—24no—
74C2×#63227—24no—
75C2×#63227—24no—
76C2×#63227—24no—
77C2×#63227—24no—
78C2×#63227Q818yes—
79C2×C4⋊Dic17—22no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—42no—
83C2×C4⋊Dic17C2×C414yes—
84C2×#63225—22no—
85C2×#63225—22no—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225C2×C414yes—
89C2×#63225C2×C414yes—
90C2×C4×Dic17—24no—
91C2×C4×Dic17—42no—
92C2×C4×Dic17—42no—
93C4⋊C4×D17—232no—
94C17×#95—22no—
95C17×#95—24no—
96C17×#95—42no—
97#10587502—22no—
98#10587502—24no—
99#10587502—24no—
100#10587502—24no—
101#10587502—24no—
102#10587502—42no—
103#10587502—42no—
104#10587502—42no—
105C24×C17C2411yescharacteristic
106C23×D17—21no—
107C23×D17—21no—
108C23×D17—21no—
109C23×D17—21no—
110C23×D17—22no—
111C23×D17—22no—
112C23×D17C22×C412yes—
113C23×D17C2×D412yes—
114C23×D17C2×D412yes—
115C23×D17C2×Q814yes—
116C23×D17#4114yes—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
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—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×C17C22×C412yes—
141C22×C4×C17C22×C412yes—
142C22×C4×C17C22×C412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×Q814yes—
146C22×C4×C17#4112yes—
147C22×C4×C17#4112yes—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—22no—
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×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—
177C22×Dic17#4112yes—
178C22×Dic17#4112yes—
179C2×C4×D17—28no—
180C2×C4×D17—28no—
181C2×C4×D17—28no—
182C2×C4×D17—28no—
183C2×C4×D17—28no—
184C2×C4×D17—28no—
185C2×C4×D17—28no—
186C2×C4×D17—28no—
187C2×C4×D17—48no—
188C2×C4×D17—48no—
189C2×C4×D17—44no—
190C2×C4×D17—44no—
191C2×C4×D17—48no—
192C2×C4×D17—48no—
193C2×C4×D17—44no—
194C2×C4×D17—44no—
195C2×C4×D17C22⋊C4116yes—
196C4⋊C4×C17—28no—
197C4⋊C4×C17—44no—
198#63227—216no—
199#63227—48no—
200C4⋊Dic17—28no—
201C4⋊Dic17—44no—
202#63225—28no—
203#63225—28no—
204C4×Dic17—48no—
205#13923—171nomaximal, Sylow
206C23×C17C23×C411yescharacteristic
207C23×C17C22×D411yescharacteristic
208C23×C17C22×D411yescharacteristic
209C23×C17C22×Q812yes—
210C23×C17C2×#4111yescharacteristic
211C23×C17C2×#4111yescharacteristic
212C23×C17C2×#4111yescharacteristic
213C23×C17C2×#4111yescharacteristic
214C23×C17C2×#4112yes—
215C23×C17H(5,𝔽2)12yes—
216C23×C17H(5,𝔽2)12yes—
217C22×D17—22no—
218C22×D17—22no—
219C22×D17—24no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—24no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—24no—
228C22×D17—24no—
229C22×D17—24no—
230C22×D17—24no—
231C22×D17—24no—
232C22×D17—24no—
233C22×D17—24no—
234C22×D17—24no—
235C22×D17C2×C22⋊C414yes—
236C22×D17C4×Q818yes—
237C22×D17C22⋊Q818yes—
238C22×D17#12414yes—
239C22×D17C4⋊Q814yes—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—24no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—24no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—24no—
252C2×C4×C17—24no—
253C2×C4×C17—24no—
254C2×C4×C17—24no—
255C2×C4×C17—42no—
256C2×C4×C17—42no—
257C2×C4×C17—41no—
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×C17—42no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17C2×C22⋊C414yes—
270C2×C4×C17#11714yes—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—24no—
274C2×Dic17—22no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—24no—
281C2×Dic17—22no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—24no—
285C2×Dic17—24no—
286C2×Dic17—42no—
287C2×Dic17—42no—
288C2×Dic17—41no—
289C2×Dic17—41no—
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×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17C2×C22⋊C414yes—
301C2×Dic17#11714yes—
302C4×D17—232no—
303C4×D17—416no—
304C4×D17—416no—
305C23×C4⋊C4—171no—
306C2×#380—171no—
307C2×C4×C22⋊C4—171no—
308#1441—171no—
309#1441—171no—
310#1441—171no—
311#1441D1711yescharacteristic, complemented
312#1436—172no—
313#1436—172no—
314C22⋊#95—172no—
315C22⋊#95—172no—
316C22×C17C22×C22⋊C411yescharacteristic
317C22×C17C2×#11711yescharacteristic
318C22×C17C2×#11711yescharacteristic
319C22×C17C2×C4×Q812yes—
320C22×C17#51812yes—
321C22×C17C2×C22⋊Q812yes—
322C22×C17#52511yescharacteristic
323C22×C17#52511yescharacteristic
324C22×C17#52511yescharacteristic
325C22×C17#52511yescharacteristic
326C22×C17C2×#12411yescharacteristic
327C22×C17C2×C4⋊Q811yescharacteristic
328C22×C17#53412yes—
329C22×C17#53411yescharacteristic
330C22×C17#53411yescharacteristic
331C22×C17#55312yes—
332C22×C17#55312yes—
333C22×C17#55312yes—
334C22×C17#55312yes—
335C22×C17#55412yes—
336C22×C17#55412yes—
337C22×C17#55412yes—
338C22×C17#55412yes—
339C2×D17—24no—
340C2×D17—28no—
341C2×D17—28no—
342C2×D17—24no—
343C2×D17—24no—
344C2×D17—24no—
345C2×D17—24no—
346C2×D17—28no—
347C2×D17—28no—
348C2×D17—24no—
349C2×D17#39118yes—
350C4×C17—24no—
351C4×C17—24no—
352C4×C17—42no—
353C4×C17—44no—
354C4×C17—82no—
355Dic17—24no—
356Dic17—24no—
357Dic17—42no—
358Dic17—44no—
359Dic17—82no—
360C24×C4—171no—
361C24×C4—171no—
362C24×C4—171no—
363C22×C4⋊C4—171no—
364C22×C4⋊C4—171no—
365C22×C4⋊C4—171no—
366C22×C4⋊C4—178no—
367C22×C4⋊C4—171no—
368C22×C4⋊C4—344no—
369C22×C4⋊C4—344no—
370C22×C4⋊C4C2×D1711yescharacteristic
371C22×C4⋊C4C2×D1711yescharacteristic
372C22×C22⋊C4—172no—
373C22×C22⋊C4—172no—
374C22×C42—172no—
375#380—344no—
376C4×C22⋊C4—344no—
377C2×#95—172no—
378C2×#95—172no—
379C2×#95—172no—
380C2×#95—172no—
381C2×#95—171no—
382C2×#95—172no—
383C2×#95—172no—
384C2×#95—172no—
385C2×#95C2×D1712yes—
386C2×#95C2×D1711yescharacteristic
387C2×#95C2×D1712yes—
388C2×C17#228011yescharacteristic
389C2×C17C2×#39111yescharacteristic
390C2×C17#230811yescharacteristic
391C2×C17#234512yes—
392C2×C17#234512yes—
393C2×C17#257912yes—
394C2×C17#257912yes—
395C2×C17#258311yescharacteristic
396C2×C17#258311yescharacteristic
397C2×C17#259111yescharacteristic
398C2×C17#259111yescharacteristic
399D17—28no—
400C25—171no—
401C23×C4—171no—
402C23×C4—171no—
403C23×C4—174no—
404C23×C4—174no—
405C23×C4—174no—
406C23×C4—172no—
407C23×C4—171no—
408C23×C4—174no—
409C23×C4—174no—
410C23×C4—172no—
411C23×C4—342no—
412C23×C4—342no—
413C23×C4—342no—
414C23×C4—342no—
415C23×C4—342no—
416C23×C4—342no—
417C23×C4—342no—
418C23×C4—342no—
419C23×C4—682no—
420C23×C4—682no—
421C23×C4—682no—
422C23×C4—682no—
423C23×C4C22×D1712yes—
424C23×C4C22×D1711yescharacteristic
425C23×C4C22×D1712yes—
426C23×C4C22×D1711yescharacteristic
427C23×C4C22×D1711yescharacteristic
428C2×C4⋊C4—22no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—42no—
432C2×C4⋊C4—174no—
433C2×C4⋊C4—174no—
434C2×C4⋊C4—174no—
435C2×C4⋊C4—3432no—
436C2×C4⋊C4—342no—
437C2×C4⋊C4—342no—
438C2×C4⋊C4—342no—
439C2×C4⋊C4—342no—
440C2×C4⋊C4—342no—
441C2×C4⋊C4—342no—
442C2×C4⋊C4—342no—
443C2×C4⋊C4—682no—
444C2×C4⋊C4C4×D1714yes—
445C2×C22⋊C4—178no—
446C2×C22⋊C4—344no—
447C2×C22⋊C4—344no—
448C2×C22⋊C4—344no—
449C2×C22⋊C4—344no—
450C2×C22⋊C4—344no—
451C2×C42—344no—
452C2×C42—682no—
453C2×C42—682no—
454#95—22no—
455#95—24no—
456#95—42no—
457#95—344no—
458#95—344no—
459#95—344no—
460#95—342no—
461#95—344no—
462#95—682no—
463#95—682no—
464#95—682no—
465C17#1392311yescharacteristic, complemented, Sylow
466C24—172no—
467C24—172no—
468C24—172no—
469C24—174no—
470C24—174no—
471C24—341no—
472C24—342no—
473C24—341no—
474C24—341no—
475C24—341no—
476C24—342no—
477C24C23×D1711yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—172no—
502C22×C4—172no—
503C22×C4—172no—
504C22×C4—172no—
505C22×C4—172no—
506C22×C4—172no—
507C22×C4—1716no—
508C22×C4—172no—
509C22×C4—174no—
510C22×C4—348no—
511C22×C4—348no—
512C22×C4—348no—
513C22×C4—348no—
514C22×C4—348no—
515C22×C4—348no—
516C22×C4—348no—
517C22×C4—341no—
518C22×C4—341no—
519C22×C4—341no—
520C22×C4—342no—
521C22×C4—341no—
522C22×C4—342no—
523C22×C4—348no—
524C22×C4—342no—
525C22×C4—341no—
526C22×C4—341no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—342no—
534C22×C4—342no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—688no—
542C22×C4—688no—
543C22×C4—684no—
544C22×C4—684no—
545C22×C4—688no—
546C22×C4—688no—
547C22×C4—684no—
548C22×C4—684no—
549C22×C4C2×C4×D1712yes—
550C22×C4C2×C4×D1712yes—
551C22×C4C2×C4×D1712yes—
552C22×C4D4×D1712yes—
553C22×C4D4×D1712yes—
554C22×C4#6325412yes—
555C22×C4Q8×D1714yes—
556C22×C4#6325612yes—
557C4⋊C4—28no—
558C4⋊C4—44no—
559C4⋊C4—348no—
560C4⋊C4—348no—
561C4⋊C4—348no—
562C4⋊C4—684no—
563C22⋊C4—3416no—
564C22⋊C4—688no—
565C42—688no—
566C23—174no—
567C23—178no—
568C23—178no—
569C23—174no—
570C23—174no—
571C23—344no—
572C23—344no—
573C23—344no—
574C23—344no—
575C23—344no—
576C23—344no—
577C23—342no—
578C23—344no—
579C23—342no—
580C23—344no—
581C23—344no—
582C23—342no—
583C23—342no—
584C23—342no—
585C23—342no—
586C23—344no—
587C23—342no—
588C23—342no—
589C23C22×C4×D1711yescharacteristic
590C23C2×D4×D1711yescharacteristic
591C23C2×D4×D1711yescharacteristic
592C23C2×#6325411yescharacteristic
593C23#1058769112yes—
594C23C2×Q8×D1712yes—
595C23C2×#6325611yescharacteristic
596C23#41×D1711yescharacteristic
597C23#41×D1712yes—
598C23#41×D1711yescharacteristic
599C23#1058769612yes—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—174no—
630C2×C4—174no—
631C2×C4—3432no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—344no—
635C2×C4—342no—
636C2×C4—342no—
637C2×C4—342no—
638C2×C4—342no—
639C2×C4—344no—
640C2×C4—342no—
641C2×C4—342no—
642C2×C4—342no—
643C2×C4—344no—
644C2×C4—344no—
645C2×C4—344no—
646C2×C4—344no—
647C2×C4—6816no—
648C2×C4—6816no—
649C2×C4—682no—
650C2×C4—682no—
651C2×C4—682no—
652C2×C4—682no—
653C2×C4—682no—
654C2×C4—681no—
655C2×C4—681no—
656C2×C4—682no—
657C2×C4—682no—
658C2×C4—682no—
659C2×C4—682no—
660C2×C4—682no—
661C2×C4—682no—
662C2×C4—682no—
663C2×C4C22⋊C4×D1714yes—
664C2×C4#1058757814yes—
665C22—178no—
666C22—344no—
667C22—344no—
668C22—344no—
669C22—348no—
670C22—348no—
671C22—344no—
672C22—344no—
673C22—344no—
674C22—348no—
675C22—348no—
676C22C2×C22⋊C4×D1711yescharacteristic
677C22#117×D1711yescharacteristic
678C22C2×#1058757811yescharacteristic
679C22C4×Q8×D1712yes—
680C22#1176328112yes—
681C22C22⋊Q8×D1712yes—
682C22#1176331911yescharacteristic
683C22#1176331911yescharacteristic
684C22#1176332211yescharacteristic
685C22#1176332211yescharacteristic
686C22#124×D1711yescharacteristic
687C22C4⋊Q8×D1711yescharacteristic
688C22#1176336712yes—
689C22#1176336911yescharacteristic
690C22#1176337011yescharacteristic
691C22#1176350412yes—
692C22#1176350612yes—
693C22#1176351012yes—
694C22#1176351212yes—
695C22#1176351312yes—
696C22#1176351512yes—
697C22#1176351712yes—
698C22#1176352012yes—
699C4—24no—
700C4—24no—
701C4—42no—
702C4—44no—
703C4—82no—
704C4—344no—
705C4—344no—
706C4—682no—
707C4—684no—
708C4—1362no—
709C2—348no—
710C2#1453116611yescharacteristic
711C2D17×#39111yescharacteristic
712C2#1453135811yescharacteristic
713C2#1453165812yes—
714C2#1453166012yes—
715C2#1453412612yes—
716C2#1453413212yes—
717C2#1453415811yescharacteristic
718C2#1453416011yescharacteristic
719C2#1453425811yescharacteristic
720C2#1453426011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1441D17
hypocenterC17#13923
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. #14527038C2semidirect
5. #14527037C2semidirect
6. #14527034C2semidirect
7. #14527018C2semidirect
8. #14527015C2semidirect
11. C17×#1441C2semidirect
21. C2×C4⋊C4×D17C4semidirect
311. #1441D17semidirect
465. C17#13923semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D172
C2×#117626662
C2×#117626562
#145270382
#145270372
#145270342
#145270182
#145270152
#145270052
C17×#14412
#1392317