Finite Group Databasebeta

← #16956393

Subgroups — #16956393

Order 4352.

1602 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×#11762675C211yesmaximal, characteristic, complemented
2C2×C4×#63227C211yesmaximal, characteristic
3C2×#11762656C211yesmaximal, characteristic
4#14527032C212yesmaximal
5#14527030C212yesmaximal
6#14527018C212yesmaximal, complemented
7#14527010C212yesmaximal, complemented
8#14527010C211yesmaximal, characteristic, complemented
9#14527009C211yesmaximal, characteristic, complemented
10#14527005C211yesmaximal, characteristic
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C22×C4⋊Dic17C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×#63227C2212yes—
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20#11762675—24no—
21C4×#63227C418yescomplemented
22#11762656—24no—
23C2×#10587502C2212yes—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32C2×#10587502C2211yescharacteristic
33C22×C42×C17C2211yescharacteristic
34C2×C17×#95C2211yescharacteristic
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2212yes—
37C2×C17×#95C2212yes—
38C24×D17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2311yescharacteristic
44C23×Dic17C2312yes—
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C22×C4×D17—22no—
50C22×C4×D17—22no—
51C22×C4×D17—24no—
52C22×C4×D17—42no—
53C22×C4×D17C2×C414yes—
54C2×C42×C17—22no—
55C2×C42×C17—22no—
56C2×C42×C17C2×C414yes—
57C2×#63227—24no—
58C2×#63227—24no—
59C2×#63227—24no—
60C2×#63227—24no—
61C2×#63227—22no—
62C2×#63227—22no—
63C2×#63227—24no—
64C2×#63227—42no—
65C2×#63227—42no—
66C2×#63227C2×C414yes—
67C2×#63227C2×C414yes—
68C2×C4⋊Dic17—24no—
69C2×C4⋊Dic17—24no—
70C2×C4⋊Dic17—42no—
71C2×C4×Dic17—24no—
72C2×C4×Dic17—24no—
73C2×C4×Dic17—42no—
74C2×C4×Dic17—42no—
75C2×C4×Dic17C2×C414yes—
76C17×#95—22no—
77C17×#95—24no—
78C17×#95—24no—
79C17×#95—22no—
80#10587502—22no—
81#10587502—24no—
82#10587502—24no—
83#10587502—24no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502C2×C414yes—
88#10587502C2×C414yes—
89#10587502D414yes—
90C24×C17C2411yescharacteristic
91C23×D17—21no—
92C23×D17—21no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17—22no—
98C23×D17—21no—
99C23×D17—21no—
100C23×D17C22×C412yes—
101C23×D17#4114yes—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—21no—
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×C17C22×C412yes—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C22×C412yes—
128C22×C4×C17C2×D412yes—
129C22×C4×C17#4114yes—
130C22×C4×C17#4112yes—
131C22×C4×C17#4112yes—
132C22×C4×C17#4112yes—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C412yes—
160C22×Dic17C2×D414yes—
161C22×Dic17#4114yes—
162C2×C4×D17—28no—
163C2×C4×D17—28no—
164C2×C4×D17—44no—
165C2×C4×D17—48no—
166C2×C4×D17—48no—
167C42×C17—28no—
168#63227—216no—
169#63227—28no—
170#63227—28no—
171#63227—48no—
172C4⋊Dic17—48no—
173C4×Dic17—44no—
174C4×Dic17—48no—
175#13888—171nomaximal, Sylow
176C23×C17C23×C411yescharacteristic
177C23×C17C22×D411yescharacteristic
178C23×C17C2×#4112yes—
179C23×C17C2×#4111yescharacteristic
180C23×C17C2×#4111yescharacteristic
181C23×C17C2×#4111yescharacteristic
182C23×C17C2×#4112yes—
183C23×C17C2×#4112yes—
184C23×C17H(5,𝔽2)12yes—
185C23×C17H(5,𝔽2)11yescharacteristic
186C23×C172−1+411yescharacteristic
187C22×D17—22no—
188C22×D17—22no—
189C22×D17—22no—
190C22×D17—22no—
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—42no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—41no—
205C22×D17—42no—
206C22×D17—41no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17#11714yes—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—24no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
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×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17#11714yes—
240C2×C4×C17C4×D414yes—
241C2×Dic17—22no—
242C2×Dic17—22no—
243C2×Dic17—24no—
244C2×Dic17—24no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—42no—
266C2×Dic17—42no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17#11714yes—
270C4×D17—416no—
271C2×#382—171no—
272C2×C4×C22⋊C4—171no—
273C2×C4×C22⋊C4—171no—
274#1440—172no—
275#1439—172no—
276#1436—172no—
277#1434—171no—
278#1434—172no—
279#1434—171no—
280#1434D1711yescharacteristic, complemented
281C22⋊#95—171no—
282C22×C17C2×#11711yescharacteristic
283C22×C17C2×#11711yescharacteristic
284C22×C17C2×#11711yescharacteristic
285C22×C17C2×C4×D411yescharacteristic
286C22×C17C4×#4111yescharacteristic
287C22×C17#51811yescharacteristic
288C22×C17#52011yescharacteristic
289C22×C17#52512yes—
290C22×C17#52912yes—
291C22×C17#52912yes—
292C22×C17#52912yes—
293C22×C17#53812yes—
294C22×C17#53912yes—
295C22×C17#54611yescharacteristic
296C22×C17#54612yes—
297C22×C17#54811yescharacteristic
298C22×C17#55112yes—
299C22×C17#55212yes—
300C22×C17#55211yescharacteristic
301C22×C17#55312yes—
302C22×C17#55311yescharacteristic
303C22×C17#55512yes—
304C22×C17#55912yes—
305C2×D17—24no—
306C2×D17—24no—
307C2×D17—24no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—44no—
313C2×D17—44no—
314C2×D17—44no—
315C2×D17—44no—
316C4×C17—24no—
317C4×C17—24no—
318C4×C17—44no—
319C4×C17—42no—
320C4×C17—82no—
321Dic17—24no—
322Dic17—42no—
323Dic17—42no—
324Dic17—44no—
325Dic17—82no—
326C24×C4—171no—
327C22×C4⋊C4—172no—
328C22×C22⋊C4—171no—
329C22×C22⋊C4—171no—
330C22×C22⋊C4—172no—
331C22×C22⋊C4—172no—
332C22×C42—171no—
333C22×C42—172no—
334C22×C42C2×D1711yescharacteristic
335#382—344no—
336C4×C22⋊C4—178no—
337C4×C22⋊C4—344no—
338C2×#95—171no—
339C2×#95—171no—
340C2×#95—171no—
341C2×#95—172no—
342C2×#95—171no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95C2×D1711yescharacteristic
349C2×#95C2×D1712yes—
350C2×#95C2×D1711yescharacteristic
351C2×#95C2×D1712yes—
352C2×C17#228311yescharacteristic
353C2×C17#229011yescharacteristic
354C2×C17#231811yescharacteristic
355C2×C17#234211yescharacteristic
356C2×C17#234611yescharacteristic
357C2×C17#235211yescharacteristic
358C2×C17#235711yescharacteristic
359C2×C17#245912yes—
360C2×C17#246012yes—
361C2×C17#251712yes—
362C2×C17#251812yes—
363D17—44no—
364C25—171no—
365C23×C4—171no—
366C23×C4—172no—
367C23×C4—171no—
368C23×C4—171no—
369C23×C4—174no—
370C23×C4—172no—
371C23×C4—344no—
372C23×C4—342no—
373C23×C4—342no—
374C23×C4—682no—
375C23×C4C22×D1712yes—
376C23×C4C22×D1711yescharacteristic
377C23×C4C22×D1712yes—
378C23×C4C22×D1711yescharacteristic
379C23×C4C22×D1711yescharacteristic
380C2×C4⋊C4—344no—
381C2×C4⋊C4—344no—
382C2×C4⋊C4—682no—
383C2×C22⋊C4—174no—
384C2×C22⋊C4—174no—
385C2×C22⋊C4—344no—
386C2×C22⋊C4—344no—
387C2×C22⋊C4—344no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—342no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—682no—
393C2×C22⋊C4—682no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—174no—
397C2×C42—344no—
398C2×C42—344no—
399C2×C42—682no—
400C2×C42—682no—
401C2×C42C4×D1714yes—
402#95—24no—
403#95—22no—
404#95—24no—
405#95—22no—
406#95—174no—
407#95—174no—
408#95—174no—
409#95—344no—
410#95—344no—
411#95—344no—
412#95—342no—
413#95—682no—
414#95—682no—
415#95—682no—
416C17#1388811yescharacteristic, complemented, Sylow
417C24—172no—
418C24—174no—
419C24—342no—
420C24—341no—
421C24—341no—
422C24—341no—
423C24—341no—
424C24—342no—
425C24—341no—
426C24—342no—
427C24—341no—
428C24C23×D1711yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
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—172no—
453C22×C4—174no—
454C22×C4—172no—
455C22×C4—172no—
456C22×C4—174no—
457C22×C4—348no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—341no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—341no—
475C22×C4—348no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—684no—
484C22×C4—688no—
485C22×C4—688no—
486C22×C4C2×C4×D1712yes—
487C22×C4C2×C4×D1712yes—
488C22×C4C2×C4×D1712yes—
489C22×C4#6325214yes—
490C22×C4D4×D1712yes—
491C22×C4#6325412yes—
492C22×C4#6325412yes—
493C22×C4#6325612yes—
494C4⋊C4—688no—
495C22⋊C4—348no—
496C22⋊C4—3416no—
497C22⋊C4—348no—
498C22⋊C4—688no—
499C42—28no—
500C42—684no—
501C42—688no—
502C23—174no—
503C23—342no—
504C23—342no—
505C23—342no—
506C23—342no—
507C23—342no—
508C23—342no—
509C23—344no—
510C23—344no—
511C23—344no—
512C23—344no—
513C23—344no—
514C23—344no—
515C23—681no—
516C23—682no—
517C23—682no—
518C23—681no—
519C23—682no—
520C23—681no—
521C23—682no—
522C23—682no—
523C23—681no—
524C23—682no—
525C23C22×C4×D1711yescharacteristic
526C23C2×#6325212yes—
527C23C2×D4×D1711yescharacteristic
528C23C2×#6325411yescharacteristic
529C23C2×#6325411yescharacteristic
530C23C2×#6325611yescharacteristic
531C23#41×D1712yes—
532C23#41×D1712yes—
533C23#1058769612yes—
534C23#1058769611yescharacteristic
535C23#1058769711yescharacteristic
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
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—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—174no—
566C2×C4—344no—
567C2×C4—342no—
568C2×C4—342no—
569C2×C4—344no—
570C2×C4—342no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—344no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—6816no—
580C2×C4—682no—
581C2×C4—682no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—682no—
586C2×C4—682no—
587C2×C4—682no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4#1058756714yes—
596C2×C4#1058757814yes—
597C22—344no—
598C22—344no—
599C22—344no—
600C22—682no—
601C22—684no—
602C22—684no—
603C22—682no—
604C22—684no—
605C22—682no—
606C22—684no—
607C22—682no—
608C22#117×D1711yescharacteristic
609C22#117×D1711yescharacteristic
610C22C2×#1058757811yescharacteristic
611C22C2×#1058756711yescharacteristic
612C22C4×#6325211yescharacteristic
613C22#1176328311yescharacteristic
614C22#1176328911yescharacteristic
615C22#1176331812yes—
616C22#1176333812yes—
617C22#1176334112yes—
618C22#1176334412yes—
619C22#1176339812yes—
620C22#1176340012yes—
621C22#1176345011yescharacteristic
622C22#1176345112yes—
623C22#1176346511yescharacteristic
624C22#1176348212yes—
625C22#1176349212yes—
626C22#1176350111yescharacteristic
627C22#1176350512yes—
628C22#1176351111yescharacteristic
629C22#1176352212yes—
630C22#1176354412yes—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—82no—
636C4—344no—
637C4—682no—
638C4—684no—
639C4—682no—
640C4—1362no—
641C2—684no—
642C2#1453117711yescharacteristic
643C2#1453123011yescharacteristic
644C2#1453141511yescharacteristic
645C2#1453162611yescharacteristic
646C2#1453167011yescharacteristic
647C2#1453171511yescharacteristic
648C2#1453175511yescharacteristic
649C2#1453272512yes—
650C2#1453274012yes—
651C2#1453343112yes—
652C2#1453344612yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13888
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#11762675C2semidirect
6. #14527018C2semidirect
7. #14527010C2semidirect
8. #14527010C2semidirect
9. #14527009C2semidirect
11. C17×#1434C2semidirect
21. C4×#63227C4semidirect
280. #1434D17semidirect
416. C17#13888semidirect

Maximal subgroups

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

subgroupindex
C2×#117626752
C2×C4×#632272
C2×#117626562
#145270322
#145270302
#145270182
#145270102
#145270092
#145270052
C17×#14342
#1388817