Finite Group Databasebeta

← #16964607

Subgroups — #16964607

Order 4352.

1650 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762726C211yesmaximal, characteristic
2C2×#11762701C212yesmaximal, complemented
3#14527026C212yesmaximal, complemented
4#14527024C212yesmaximal
5#14527020C212yesmaximal, complemented
6#14527020C212yesmaximal, complemented
7#14527013C211yesmaximal, characteristic, complemented
8#14527010C212yesmaximal, complemented
9C17×#1441C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×#63227C2212yes—
13C22×#63227C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×C4×Dic17C2212yes—
19C22×C4×Dic17C2211yescharacteristic
20#11762726C418yescomplemented
21#11762701—28no—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2211yescharacteristic
28C22×C4⋊C4×C17C2211yescharacteristic
29C22×C4⋊C4×C17C2211yescharacteristic
30C2×C17×#95C2212yes—
31C2×C17×#95C2212yes—
32C2×C17×#95C2211yescharacteristic
33C24×D17C2312yes—
34C23×C4×C17C2312yes—
35C23×C4×C17C2311yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2312yes—
38C23×C4×C17C2311yescharacteristic
39C23×Dic17C2312yes—
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C22×D68—42no—
43C22×D68—42no—
44C22×D68—41no—
45C22×D68—41no—
46C22×D68C2×C414yes—
47C2×C4⋊C4×C17—41no—
48C2×C4⋊C4×C17—41no—
49C2×C4⋊C4×C17—42no—
50C2×C4⋊C4×C17—41no—
51C2×C4⋊C4×C17C2×C414yes—
52C2×#63227—24no—
53C2×#63227—24no—
54C2×#63227—24no—
55C2×#63227—24no—
56C2×#63227—24no—
57C2×#63227—24no—
58C2×#63227—24no—
59C2×#63227—24no—
60C2×#63227—42no—
61C2×#63227—42no—
62C2×#63227—42no—
63C2×#63227—42no—
64C2×#63227—42no—
65C2×#63227—42no—
66C2×#63227—42no—
67C2×#63227—42no—
68C2×#63227C2×C418yes—
69C2×#63227C2×C418yes—
70C2×C4⋊Dic17—41no—
71C2×C4⋊Dic17—41no—
72C2×C4⋊Dic17D414yescomplemented
73C2×C4×Dic17—24no—
74C2×C4×Dic17—22no—
75C2×C4×Dic17—24no—
76C2×C4×Dic17—42no—
77C2×C4×Dic17—41no—
78C2×C4×Dic17—41no—
79C17×#95—24no—
80C17×#95—42no—
81C17×#95—41no—
82#10587502—24no—
83#10587502—24no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502C2×C414yes—
88C24×C17C2411yescharacteristic
89C23×D17—22no—
90C23×D17—22no—
91C23×D17—22no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—22no—
96C23×D17—22no—
97C23×D17—22no—
98C23×D17—22no—
99C23×D17—22no—
100C23×D17—22no—
101C23×D17—22no—
102C23×D17—22no—
103C23×D17C22×C414yes—
104C22×C4×C17—21no—
105C22×C4×C17—21no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—21no—
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—22no—
128C22×C4×C17—22no—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×C4×C17C22×C412yes—
132C22×C4×C17C2×D412yes—
133C22×C4×C17#4114yes—
134C22×C4×C17#4112yes—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
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—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17C22×C414yes—
154C22×Dic17C2×D414yes—
155C22×Dic17#4114yes—
156C2×D68—44no—
157C2×D68—48no—
158C2×D68—48no—
159C2×D68—44no—
160C2×D68—44no—
161C4⋊C4×C17—44no—
162C4⋊C4×C17—44no—
163#63227—216no—
164#63227—48no—
165#63227—48no—
166#63227—48no—
167#63227—48no—
168#63227—48no—
169C4⋊Dic17—44no—
170C4×Dic17—48no—
171C4×Dic17—44no—
172#14589—171nomaximal, Sylow
173C23×C17C23×C411yescharacteristic
174C23×C17C22×D411yescharacteristic
175C23×C17C2×#4111yescharacteristic
176C23×C17C2×#4112yes—
177C23×C17C2×#4112yes—
178C23×C17H(5,𝔽2)11yescharacteristic
179C23×C17H(5,𝔽2)11yescharacteristic
180C23×C17H(5,𝔽2)12yes—
181C23×C17H(5,𝔽2)12yes—
182C23×C17H(5,𝔽2)11yescharacteristic
183C23×C172−1+411yescharacteristic
184C22×D17—24no—
185C22×D17—24no—
186C22×D17—24no—
187C22×D17—24no—
188C22×D17—24no—
189C22×D17—24no—
190C22×D17—24no—
191C22×D17—42no—
192C22×D17—42no—
193C22×D17—42no—
194C22×D17—42no—
195C22×D17—42no—
196C22×D17—42no—
197C22×D17—42no—
198C22×D17—42no—
199C22×D17—42no—
200C22×D17—42no—
201C22×D17—42no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—42no—
211C22×D17—42no—
212C22×D17—42no—
213C22×D17—42no—
214C22×D17—42no—
215C22×D17—42no—
216C22×D17—42no—
217C22×D17—42no—
218C22×D17—42no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—42no—
233C2×C4×C17—42no—
234C2×C4×C17—42no—
235C2×C4×C17—42no—
236C2×C4×C17—41no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17C4×D414yes—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—24no—
256C2×Dic17—24no—
257C2×Dic17—24no—
258C2×Dic17—24no—
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—42no—
270C2×Dic17—42no—
271D68—416no—
272C2×#394—171no—
273C2×#388—172no—
274#1441D1711yescharacteristic, complemented
275#1439—172no—
276#1438—172no—
277#1437—172no—
278#1437—172no—
279C4⋊#95—171no—
280#1434—172no—
281C22×C17C2×C4×D411yescharacteristic
282C22×C17C4×#4112yes—
283C22×C17#51812yes—
284C22×C17#51811yescharacteristic
285C22×C17#52011yescharacteristic
286C22×C17#53411yescharacteristic
287C22×C17#53711yescharacteristic
288C22×C17#53812yes—
289C22×C17#53811yescharacteristic
290C22×C17#53911yescharacteristic
291C22×C17#53912yes—
292C22×C17#54012yes—
293C22×C17#54212yes—
294C22×C17#54612yes—
295C22×C17#55012yes—
296C22×C17#55112yes—
297C22×C17#55312yes—
298C22×C17#55511yescharacteristic
299C22×C17#55512yes—
300C22×C17#55511yescharacteristic
301C22×C17#56012yes—
302C22×C17#56011yescharacteristic
303C22×C17H(3,𝔽4)11yescharacteristic
304C2×D17—44no—
305C2×D17—44no—
306C2×D17—44no—
307C2×D17—44no—
308C2×D17—44no—
309C2×D17—44no—
310C2×D17—44no—
311C2×D17—82no—
312C2×D17—82no—
313C2×D17—82no—
314C2×D17—82no—
315C2×D17—82no—
316C2×D17—82no—
317C2×D17—82no—
318C2×D17—82no—
319C4×C17—24no—
320C4×C17—42no—
321C4×C17—42no—
322C4×C17—44no—
323C4×C17—82no—
324Dic17—44no—
325Dic17—44no—
326Dic17—44no—
327C23×D4—171no—
328C22×C4⋊C4—171no—
329C22×C4⋊C4C2×D1711yescharacteristic
330C22×C4⋊C4C2×D1711yescharacteristic
331C22×C22⋊C4—172no—
332C22×C22⋊C4—172no—
333C22×C22⋊C4—172no—
334C22×C22⋊C4—172no—
335C22×C22⋊C4—172no—
336C22×C22⋊C4—172no—
337C22×C42—172no—
338C22×C42—171no—
339#394—178no—
340#388—348no—
341C2×#95—171no—
342C2×#95—172no—
343C2×#95—172no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95C2×D1712yes—
348C2×#95C2×D1712yes—
349C2×#95C2×D1711yescharacteristic
350C2×C17#232511yescharacteristic
351C2×C17#233512yes—
352C2×C17#236312yes—
353C2×C17#237211yescharacteristic
354C2×C17#237511yescharacteristic
355C2×C17#269911yescharacteristic
356C2×C17#270512yes—
357C2×C17#273311yescharacteristic
358C2×C17#275612yes—
359C2×C17#278511yescharacteristic
360C2×C17#279511yescharacteristic
361D17—84no—
362C25—172no—
363C22×D4—174no—
364C22×D4—682no—
365C22×D4—681no—
366C22×D4—681no—
367C22×D4—682no—
368C23×C4—172no—
369C23×C4—172no—
370C23×C4—172no—
371C23×C4C22×D1712yes—
372C23×C4C22×D1711yescharacteristic
373C23×C4C22×D1711yescharacteristic
374C23×C4C22×D1712yes—
375C23×C4C22×D1711yescharacteristic
376C2×C4⋊C4—41no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—174no—
381C2×C4⋊C4—681no—
382C2×C4⋊C4—681no—
383C2×C4⋊C4C4×D1714yes—
384C2×C22⋊C4—178no—
385C2×C22⋊C4—178no—
386C2×C22⋊C4—344no—
387C2×C22⋊C4—344no—
388C2×C22⋊C4—344no—
389C2×C22⋊C4—344no—
390C2×C22⋊C4—344no—
391C2×C22⋊C4—344no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—682no—
395C2×C22⋊C4—682no—
396C2×C22⋊C4—682no—
397C2×C22⋊C4—682no—
398C2×C22⋊C4—682no—
399C2×C22⋊C4—682no—
400C2×C22⋊C4—682no—
401C2×C22⋊C4—682no—
402C2×C42—344no—
403C2×C42—344no—
404C2×C42—342no—
405C2×C42—682no—
406C2×C42—681no—
407C2×C42—681no—
408#95—24no—
409#95—42no—
410#95—41no—
411#95—174no—
412#95—344no—
413#95—344no—
414#95—682no—
415#95—682no—
416#95—682no—
417C17#1458911yescharacteristic, complemented, Sylow
418C24—174no—
419C24—342no—
420C24—342no—
421C24—342no—
422C24—342no—
423C24—342no—
424C24—342no—
425C24—342no—
426C24—342no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24—342no—
432C24—342no—
433C24C23×D1711yescharacteristic
434C2×D4—688no—
435C2×D4—684no—
436C2×D4—688no—
437C2×D4—684no—
438C2×D4—684no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—174no—
465C22×C4—174no—
466C22×C4—174no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4C2×C4×D1712yes—
486C22×C4C2×C4×D1712yes—
487C22×C4C2×C4×D1712yes—
488C22×C4D4×D1712yes—
489C22×C4#6325614yes—
490C22×C4#6325612yes—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—684no—
494C22⋊C4—3416no—
495C22⋊C4—688no—
496C22⋊C4—688no—
497C22⋊C4—688no—
498C22⋊C4—688no—
499C22⋊C4—688no—
500C42—688no—
501C42—684no—
502C23—344no—
503C23—344no—
504C23—344no—
505C23—344no—
506C23—344no—
507C23—344no—
508C23—344no—
509C23—682no—
510C23—682no—
511C23—682no—
512C23—682no—
513C23—682no—
514C23—682no—
515C23—682no—
516C23—682no—
517C23—682no—
518C23—682no—
519C23—682no—
520C23—682no—
521C23—682no—
522C23—682no—
523C23—682no—
524C23—682no—
525C23—682no—
526C23—682no—
527C23—682no—
528C23—682no—
529C23—682no—
530C23—682no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23—682no—
537C23C22×C4×D1711yescharacteristic
538C23C2×D4×D1711yescharacteristic
539C23#1058769112yes—
540C23#1058769111yescharacteristic
541C23#1058769111yescharacteristic
542C23C2×#6325612yes—
543C23C2×#6325611yescharacteristic
544C23#41×D1712yes—
545C23#1058769612yes—
546C23#1058769611yescharacteristic
547C23#1058769711yescharacteristic
548D4—6816no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—344no—
580C2×C4—344no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—344no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—682no—
592C2×C4—682no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4#1058757914yes—
601C22—684no—
602C22—684no—
603C22—684no—
604C22—684no—
605C22—684no—
606C22—684no—
607C22—684no—
608C22—1362no—
609C22—1362no—
610C22—1362no—
611C22—1362no—
612C22—1362no—
613C22—1362no—
614C22—1362no—
615C22—1362no—
616C22C2×#1058757911yescharacteristic
617C22C4×#6325612yes—
618C22#1176328112yes—
619C22#1176328211yescharacteristic
620C22#1176328911yescharacteristic
621C22#1176336711yescharacteristic
622C22#1176338611yescharacteristic
623C22#1176339512yes—
624C22#1176339911yescharacteristic
625C22#1176340812yes—
626C22#1176340811yescharacteristic
627C22#1176341512yes—
628C22#1176342512yes—
629C22#1176345012yes—
630C22#1176347712yes—
631C22#1176348612yes—
632C22#1176351212yes—
633C22#1176352111yescharacteristic
634C22#1176352511yescharacteristic
635C22#1176352512yes—
636C22#1176354712yes—
637C22#1176354911yescharacteristic
638C22#1176355211yescharacteristic
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—684no—
645C4—684no—
646C4—684no—
647C2—1364no—
648C2#1453148411yescharacteristic
649C2#1453157312yes—
650C2#1453182112yes—
651C2#1453189511yescharacteristic
652C2#1453192211yescharacteristic
653C2#1453537011yescharacteristic
654C2#1453545612yes—
655C2#1453580711yescharacteristic
656C2#1453611212yes—
657C2#1453649211yescharacteristic
658C2#1453661811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1441C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1441D17
hypocenterC17#14589
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×#11762701C2semidirect
3. #14527026C2semidirect
5. #14527020C2semidirect
6. #14527020C2semidirect
7. #14527013C2semidirect
8. #14527010C2semidirect
9. C17×#1441C2semidirect
20. #11762726C4semidirect
72. C2×C4⋊Dic17D4semidirect
274. #1441D17semidirect
417. C17#14589semidirect

Maximal subgroups

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

subgroupindex
C2×#117627262
C2×#117627012
#145270262
#145270242
#145270202
#145270132
#145270102
C17×#14412
#1458917