Finite Group Databasebeta

← #16961334

Subgroups — #16961334

Order 4352.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762745C211yesmaximal, characteristic
2C2×#11762706C212yesmaximal, complemented
3C2×#11762705C211yesmaximal, characteristic, complemented
4C2×#11762702C211yesmaximal, characteristic
5C2×#11762697C212yesmaximal
6#14527051C211yesmaximal, characteristic, complemented
7#14527050C212yesmaximal, complemented
8#14527047C211yesmaximal, characteristic, complemented
9#14527027C212yesmaximal
10#95×Dic17C211yesmaximal, characteristic, complemented
11C17×#1439C211yesmaximal, characteristic
12C22×C4⋊Dic17C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14C22×#63225C2212yes—
15C22×#63225C2212yes—
16C22×#63225C2212yes—
17C22×C22⋊Dic17C2211yescharacteristic
18C22×C22⋊Dic17C2212yes—
19C22×C22⋊Dic17C2211yescharacteristic
20C22×C4×Dic17C2212yes—
21C22×C4×Dic17C2211yescharacteristic
22#11762745C418yescomplemented
23#11762706—28no—
24#11762705—24no—
25#11762702—24no—
26#11762697—28no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2212yes—
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2211yescharacteristic
33C2×#10587502C2212yes—
34C2×#10587502C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C22⋊C4×C17C2212yes—
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×C4×C17C2311yescharacteristic
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C23×Dic17C2312yes—
50C23×Dic17C2311yescharacteristic
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—42no—
53C2×C22⋊C4×C17—22no—
54C2×C22⋊C4×C17—22no—
55C2×C22⋊C4×C17—24no—
56C2×C22⋊C4×C17—42no—
57C2×C22⋊C4×C17—42no—
58C2×C22⋊C4×C17C2×C414yes—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—24no—
62C2×C22⋊Dic17—22no—
63C2×C22⋊Dic17—22no—
64C2×C22⋊Dic17—42no—
65C2×C22⋊Dic17C2×C418yes—
66C2×C4⋊Dic17—22no—
67C2×C4⋊Dic17—22no—
68C2×C4⋊Dic17—22no—
69C2×C4⋊Dic17—24no—
70C2×C4⋊Dic17C2×C414yes—
71C2×#63225—24no—
72C2×#63225—24no—
73C2×#63225—24no—
74C2×#63225—24no—
75C2×#63225—24no—
76C2×#63225—24no—
77C2×#63225—42no—
78C2×#63225—42no—
79C2×#63225C2×C418yes—
80C2×C4×Dic17—42no—
81C2×C4×Dic17—42no—
82C2×C4×Dic17—42no—
83C2×C4×Dic17D418yes—
84C2×C4×Dic17D414yes—
85C17×#95—24no—
86C17×#95—22no—
87#10587502—24no—
88#10587502—22no—
89#10587502—24no—
90#10587502—24no—
91#10587502—22no—
92#10587502—24no—
93#10587502—24no—
94#10587502C2×C414yes—
95C24×C17—22no—
96C24×C17—21no—
97C24×C17—22no—
98C24×C17—21no—
99C24×C17—22no—
100C24×C17—21no—
101C24×C17—21no—
102C24×C17—21no—
103C24×C17—21no—
104C24×C17—22no—
105C24×C17C22×C412yes—
106C24×C17C2411yescharacteristic
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—21no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
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×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D414yes—
129C22×C4×C17C2×D412yes—
130C22×C4×C17#4112yes—
131C22×Dic17—22no—
132C22×Dic17—21no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—21no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—22no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17C22×C414yes—
159C22×Dic17C22×C414yes—
160C22×Dic17C2×D412yes—
161C22×Dic17C2×D414yes—
162C22×Dic17C2×D414yes—
163C22×Dic17C2×D412yes—
164C22×Dic17#4114yes—
165C4⋊C4×C17—44no—
166C22⋊C4×C17—28no—
167C22⋊C4×C17—48no—
168C22⋊Dic17—216no—
169C22⋊Dic17—44no—
170C22⋊Dic17—44no—
171C4⋊Dic17—28no—
172C4⋊Dic17—44no—
173#63225—216no—
174#63225—48no—
175#63225—48no—
176C4×Dic17—48no—
177C4×Dic17—44no—
178#14322—171nomaximal, Sylow
179C23×C17—24no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17—24no—
183C23×C17—22no—
184C23×C17—41no—
185C23×C17—42no—
186C23×C17—41no—
187C23×C17—41no—
188C23×C17—42no—
189C23×C17—41no—
190C23×C17—42no—
191C23×C17—42no—
192C23×C17—41no—
193C23×C17—42no—
194C23×C17—41no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17—42no—
198C23×C17—42no—
199C23×C17—41no—
200C23×C17—42no—
201C23×C17—41no—
202C23×C17C23×C411yescharacteristic
203C23×C17C22×D412yes—
204C23×C17C22×D411yescharacteristic
205C23×C17C2×#4111yescharacteristic
206C23×C17C2×#4112yes—
207C23×C17H(5,𝔽2)12yes—
208C23×C17H(5,𝔽2)11yescharacteristic
209C23×C172−1+411yescharacteristic
210C23×C172−1+411yescharacteristic
211C23×C172−1+411yescharacteristic
212C23×C172−1+412yes—
213C2×C4×C17—24no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
217C2×C4×C17—22no—
218C2×C4×C17—24no—
219C2×C4×C17—24no—
220C2×C4×C17—24no—
221C2×C4×C17—24no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17—41no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—42no—
229C2×C4×C17—42no—
230C2×C4×C17—42no—
231C2×C4×C17—42no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17C4×D414yes—
237C2×Dic17—24no—
238C2×Dic17—24no—
239C2×Dic17—24no—
240C2×Dic17—22no—
241C2×Dic17—24no—
242C2×Dic17—24no—
243C2×Dic17—24no—
244C2×Dic17—22no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—42no—
263C2×Dic17—41no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—41no—
267C2×Dic17—42no—
268C2×Dic17—41no—
269C2×Dic17C4×D418yes—
270C2×Dic17#12714yes—
271C2×#399—171no—
272C2×#389—172no—
273C2×#389—171no—
274C2×#388—172no—
275C2×#388—171no—
276#1445—172no—
277#1445—171no—
278#1444—171no—
279#1439—172no—
280#1439D1711yescharacteristic
281C4×#95—171no—
282C22×C17—42no—
283C22×C17—44no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—44no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17—82no—
290C22×C17—81no—
291C22×C17—82no—
292C22×C17—81no—
293C22×C17C2×C4×D412yes—
294C22×C17C2×C4×D411yescharacteristic
295C22×C17#51911yescharacteristic
296C22×C17#52012yes—
297C22×C17#52011yescharacteristic
298C22×C17C2×#12711yescharacteristic
299C22×C17#53212yes—
300C22×C17#53211yescharacteristic
301C22×C17#53412yes—
302C22×C17#53411yescharacteristic
303C22×C17#53611yescharacteristic
304C22×C17#53612yes—
305C22×C17#53612yes—
306C22×C17#53611yescharacteristic
307C22×C17#53712yes—
308C22×C17#53711yescharacteristic
309C22×C17#53912yes—
310C22×C17#53912yes—
311C22×C17#53912yes—
312C22×C17#54112yes—
313C22×C17#54111yescharacteristic
314C22×C17#54212yes—
315C22×C17#54211yescharacteristic
316C4×C17—24no—
317C4×C17—42no—
318C4×C17—44no—
319C4×C17—82no—
320Dic17—28no—
321Dic17—24no—
322Dic17—44no—
323Dic17—42no—
324Dic17—81no—
325Dic17—81no—
326C22×C4⋊C4—172no—
327C22×C4⋊C4—171no—
328C22×C4⋊C4—172no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4—172no—
331C22×C4⋊C4C2×D1711yescharacteristic
332C22×C22⋊C4—172no—
333C22×C22⋊C4—171no—
334C22×C22⋊C4—171no—
335C22×C22⋊C4C2×D1711yescharacteristic
336C22×C22⋊C4C2×D1712yes—
337C22×C42—172no—
338C22×C42—171no—
339#399—178no—
340#389—348no—
341#389—344no—
342#388—348no—
343#388—344no—
344C2×#95—171no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—171no—
350C2×#95—171no—
351C2×#95—172no—
352C2×#95C2×D1711yescharacteristic
353C2×#95C2×D1712yes—
354C2×C17—82no—
355C2×C17C4×#12711yescharacteristic
356C2×C17#232212yes—
357C2×C17#232211yescharacteristic
358C2×C17#232311yescharacteristic
359C2×C17#232312yes—
360C2×C17#265611yescharacteristic
361C2×C17#265811yescharacteristic
362C2×C17#266312yes—
363C2×C17#266311yescharacteristic
364C2×C17#266412yes—
365C2×C17#266411yescharacteristic
366C25C22×D1711yescharacteristic
367C23×C4—171no—
368C23×C4—172no—
369C23×C4—172no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4C22×D1712yes—
374C23×C4C22×D1711yescharacteristic
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—178no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—344no—
382C2×C4⋊C4—344no—
383C2×C4⋊C4—342no—
384C2×C4⋊C4—342no—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4—344no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—344no—
390C2×C4⋊C4—344no—
391C2×C4⋊C4—682no—
392C2×C4⋊C4—682no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—178no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—342no—
403C2×C22⋊C4—342no—
404C2×C22⋊C4—682no—
405C2×C22⋊C4C4×D1714yes—
406C2×C42—178no—
407C2×C42—174no—
408C2×C42—682no—
409C2×C42—682no—
410C2×C42—682no—
411#95—24no—
412#95—22no—
413#95—174no—
414#95—344no—
415#95—342no—
416#95—344no—
417#95—344no—
418#95—344no—
419#95—342no—
420#95—344no—
421C17#1432211yescharacteristic, complemented, Sylow
422C24—22no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24—22no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24C2×C4×D1712yes—
433C24C23×D1711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—174no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—172no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—172no—
460C22×C4—342no—
461C22×C4—341no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—341no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—341no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—341no—
477C22×C4—342no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—342no—
486C22×C4—341no—
487C22×C4C2×C4×D1712yes—
488C22×C4C2×C4×D1712yes—
489C22×C4D4×D1714yes—
490C22×C4D4×D1712yes—
491C22×C4#6325412yes—
492C4⋊C4—44no—
493C4⋊C4—3416no—
494C4⋊C4—348no—
495C4⋊C4—688no—
496C4⋊C4—684no—
497C4⋊C4—688no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—3416no—
501C22⋊C4—684no—
502C22⋊C4—684no—
503C42—684no—
504C42—688no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23C22×C4×D1711yescharacteristic
529C23C2×D4×D1712yes—
530C23C2×D4×D1711yescharacteristic
531C23C2×#6325411yescharacteristic
532C23#1058769112yes—
533C23#1058769111yescharacteristic
534C23#1058769411yescharacteristic
535C23#41×D1712yes—
536C23#1058769711yescharacteristic
537C23#1058769711yescharacteristic
538C23#1058769712yes—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—178no—
563C2×C4—174no—
564C2×C4—344no—
565C2×C4—344no—
566C2×C4—344no—
567C2×C4—344no—
568C2×C4—342no—
569C2×C4—344no—
570C2×C4—344no—
571C2×C4—342no—
572C2×C4—344no—
573C2×C4—344no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—681no—
596C2×C4#1058756714yes—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—44no—
602C22—82no—
603C22—81no—
604C22—82no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C4×D4×D1712yes—
609C22C2×#1058756711yescharacteristic
610C22#1176328611yescharacteristic
611C22#1176328812yes—
612C22#1176329011yescharacteristic
613C22#127×D1711yescharacteristic
614C22#1176335412yes—
615C22#1176335811yescharacteristic
616C22#1176336912yes—
617C22#1176336911yescharacteristic
618C22#1176338112yes—
619C22#1176338111yescharacteristic
620C22#1176338211yescharacteristic
621C22#1176338212yes—
622C22#1176338812yes—
623C22#1176338811yescharacteristic
624C22#1176340212yes—
625C22#1176340412yes—
626C22#1176340512yes—
627C22#1176341612yes—
628C22#1176341911yescharacteristic
629C22#1176342312yes—
630C22#1176342611yescharacteristic
631C4—24no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—348no—
636C4—344no—
637C4—684no—
638C4—682no—
639C4—1361no—
640C4—1361no—
641C2—82no—
642C2#1453134811yescharacteristic
643C2#1453145112yes—
644C2#1453145811yescharacteristic
645C2#1453146211yescharacteristic
646C2#1453146312yes—
647C2#1453488511yescharacteristic
648C2#1453489311yescharacteristic
649C2#1453493112yes—
650C2#1453494011yescharacteristic
651C2#1453494412yes—
652C2#1453495211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1439C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1439D17
hypocenterC17#14322
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#11762706C2semidirect
3. C2×#11762705C2semidirect
6. #14527051C2semidirect
7. #14527050C2semidirect
8. #14527047C2semidirect
10. #95×Dic17C2semidirect
22. #11762745C4semidirect
421. C17#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#117627452
C2×#117627062
C2×#117627052
C2×#117627022
C2×#117626972
#145270512
#145270502
#145270472
#145270272
#95×Dic172
C17×#14392
#1432217