Finite Group Databasebeta

← #16956537

Subgroups — #16956537

Order 4352.

1818 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762676C211yesmaximal, characteristic, complemented
2C2×C4×#63227C211yesmaximal, characteristic
3C2×#95×D17C211yesmaximal, characteristic
4#14527045C212yesmaximal, complemented
5#14527045C212yesmaximal, complemented
6#14527011C211yesmaximal, characteristic, complemented
7#14527008C211yesmaximal, characteristic, complemented
8#14527005C211yesmaximal, characteristic
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2212yes—
13C23×C4×D17C2211yescharacteristic
14C22×#63227C2212yes—
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18#11762676—24no—
19C4×#63227C418yescomplemented
20C2×#10587502C2211yescharacteristic
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32#95×D17—24no—
33C22×C42×C17C2211yescharacteristic
34C2×C17×#95C2212yes—
35C2×C17×#95C2211yescharacteristic
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2212yes—
38C24×D17C2311yescharacteristic
39C23×C4×C17C2312yes—
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×Dic17C2312yes—
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2311yescharacteristic
48C23×Dic17C2312yes—
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—22no—
52C22×C4×D17—24no—
53C22×C4×D17—22no—
54C22×C4×D17—24no—
55C22×C4×D17—24no—
56C22×C4×D17—24no—
57C22×C4×D17—42no—
58C22×C4×D17—42no—
59C22×C4×D17—42no—
60C22×C4×D17C2×C414yes—
61C2×C42×C17—42no—
62C2×C42×C17C2×C414yescomplemented
63C2×#63227—22no—
64C2×#63227—24no—
65C2×#63227—22no—
66C2×#63227—24no—
67C2×#63227—24no—
68C2×#63227—24no—
69C2×#63227C2×C414yes—
70C2×#63227C2×C414yes—
71C2×C4×Dic17—22no—
72C2×C4×Dic17—22no—
73C2×C4×Dic17C2×C414yescomplemented
74C17×#95—22no—
75C17×#95—22no—
76C17×#95—24no—
77C17×#95—42no—
78#10587502—24no—
79#10587502—22no—
80#10587502—24no—
81#10587502—24no—
82#10587502—24no—
83#10587502—42no—
84#10587502—42no—
85#10587502—42no—
86#10587502—42no—
87#10587502C2×C414yescomplemented
88#10587502C2×C414yescomplemented
89#10587502D414yes—
90C24×C17C2411yescharacteristic
91C23×D17—21no—
92C23×D17—22no—
93C23×D17—21no—
94C23×D17—22no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17C22×C412yes—
98C23×D17#4114yes—
99C23×D17#4112yes—
100C23×D17#4114yes—
101C23×D17#4112yes—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—21no—
111C22×C4×C17—21no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
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×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—22no—
135C22×Dic17—21no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—22no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C412yes—
160C22×Dic17C22×C412yes—
161C22×Dic17C2×D414yes—
162C2×C4×D17—216no—
163C2×C4×D17—28no—
164C2×C4×D17—28no—
165C2×C4×D17—44no—
166C2×C4×D17—48no—
167C2×C4×D17—48no—
168C2×C4×D17—48no—
169C2×C4×D17—48no—
170C2×C4×D17—48no—
171C2×C4×D17—48no—
172C2×C4×D17—48no—
173C2×C4×D17—48no—
174C42×C17—44no—
175#63227—28no—
176#63227—28no—
177#63227—48no—
178C4×Dic17—28no—
179#13901—171nomaximal, Sylow
180C23×C17C23×C411yescharacteristic
181C23×C17C22×D411yescharacteristic
182C23×C17C2×#4112yes—
183C23×C17C2×#4111yescharacteristic
184C23×C17C2×#4112yes—
185C23×C17C2×#4112yes—
186C23×C17C2×#4111yescharacteristic
187C23×C17C2×#4111yescharacteristic
188C23×C17H(5,𝔽2)12yes—
189C23×C17H(5,𝔽2)11yescharacteristic
190C23×C172−1+411yescharacteristic
191C22×D17—22no—
192C22×D17—22no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—22no—
201C22×D17—22no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—24no—
205C22×D17—22no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17—22no—
209C22×D17#11714yes—
210C22×D17#11714yes—
211C22×D17#11714yes—
212C22×D17#12618yes—
213C22×D17#12618yes—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—24no—
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—22no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
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—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17C4×D414yes—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—42no—
268C2×Dic17—42no—
269C2×Dic17—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C4×D17—416no—
275C4×D17—432no—
276C2×#383—171no—
277C2×C4×C22⋊C4—171no—
278C22×#95—171no—
279#1443—172no—
280#1443—172no—
281#1434—171no—
282#1434—171no—
283#1434D1711yescharacteristic, complemented
284C22⋊#95—172no—
285C22⋊#95—172no—
286C22⋊#95—171no—
287C22×C17C2×#11711yescharacteristic
288C22×C17C2×#11711yescharacteristic
289C22×C17C2×#11711yescharacteristic
290C22×C17C2×C4×D411yescharacteristic
291C22×C17C4×#4111yescharacteristic
292C22×C17#51811yescharacteristic
293C22×C17#52011yescharacteristic
294C22×C17#52512yes—
295C22×C17C2×#12612yes—
296C22×C17C2×#12612yes—
297C22×C17#52912yes—
298C22×C17#53812yes—
299C22×C17#53912yes—
300C22×C17#54612yes—
301C22×C17#54611yescharacteristic
302C22×C17#54811yescharacteristic
303C22×C17#55112yes—
304C22×C17#55112yes—
305C22×C17#55112yes—
306C22×C17#55211yescharacteristic
307C22×C17#55212yes—
308C22×C17#55312yes—
309C22×C17#55311yescharacteristic
310C2×D17—24no—
311C2×D17—24no—
312C2×D17—24no—
313C2×D17—24no—
314C2×D17—24no—
315C2×D17—28no—
316C2×D17—28no—
317C2×D17—28no—
318C2×D17—28no—
319C2×D17—24no—
320C2×D17#38318yescomplemented
321C4×C17—24no—
322C4×C17—42no—
323C4×C17—42no—
324C4×C17—44no—
325C4×C17—82no—
326Dic17—42no—
327Dic17—42no—
328Dic17—44no—
329Dic17—42no—
330Dic17—82no—
331C24×C4—171no—
332C24×C4—172no—
333C22×C22⋊C4—171no—
334C22×C22⋊C4—171no—
335C22×C22⋊C4—172no—
336C22×C42—171no—
337C22×C42C2×D1711yescharacteristic
338#383—344no—
339C4×C22⋊C4—178no—
340C2×#95—171no—
341C2×#95—171no—
342C2×#95—171no—
343C2×#95—171no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—344no—
353C2×#95C2×D1712yes—
354C2×#95C2×D1711yescharacteristic
355C2×#95C2×D1711yescharacteristic
356C2×#95C2×D1712yes—
357C2×C17C2×#38311yescharacteristic
358C2×C17#229011yescharacteristic
359C2×C17#231811yescharacteristic
360C2×C17#234211yescharacteristic
361C2×C17#234311yescharacteristic
362C2×C17#235211yescharacteristic
363C2×C17#235311yescharacteristic
364C2×C17#248012yes—
365C2×C17#248112yes—
366C2×C17#249312yes—
367C2×C17#251012yes—
368D17—28no—
369C25—171no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4—172no—
374C23×C4—174no—
375C23×C4—172no—
376C23×C4—344no—
377C23×C4—344no—
378C23×C4—344no—
379C23×C4—344no—
380C23×C4—344no—
381C23×C4—344no—
382C23×C4—342no—
383C23×C4—342no—
384C23×C4—682no—
385C23×C4—682no—
386C23×C4—682no—
387C23×C4C22×D1712yes—
388C23×C4C22×D1711yescharacteristic
389C23×C4C22×D1711yescharacteristic
390C23×C4C22×D1712yes—
391C23×C4C22×D1711yescharacteristic
392C2×C22⋊C4—174no—
393C2×C22⋊C4—174no—
394C2×C22⋊C4—342no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—342no—
397C2×C22⋊C4—344no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C42—42no—
401C2×C42—174no—
402C2×C42—342no—
403C2×C42—342no—
404C2×C42C4×D1714yescomplemented
405#95—24no—
406#95—22no—
407#95—22no—
408#95—42no—
409#95—174no—
410#95—174no—
411#95—174no—
412#95—344no—
413#95—344no—
414#95—344no—
415#95—344no—
416#95—342no—
417#95—682no—
418#95—682no—
419#95—682no—
420#95—682no—
421C17#1390111yescharacteristic, complemented, Sylow
422C24—172no—
423C24—172no—
424C24—174no—
425C24—172no—
426C24—174no—
427C24—342no—
428C24—341no—
429C24—342no—
430C24—341no—
431C24—341no—
432C24—341no—
433C24C23×D1711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—172no—
458C22×C4—172no—
459C22×C4—172no—
460C22×C4—174no—
461C22×C4—348no—
462C22×C4—3416no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4—342no—
482C22×C4—341no—
483C22×C4—348no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—684no—
492C22×C4—688no—
493C22×C4—688no—
494C22×C4—688no—
495C22×C4—688no—
496C22×C4—688no—
497C22×C4—688no—
498C22×C4—688no—
499C22×C4—688no—
500C22×C4C2×C4×D1712yes—
501C22×C4C2×C4×D1712yes—
502C22×C4C2×C4×D1712yes—
503C22×C4#6325214yes—
504C22×C4D4×D1712yes—
505C22×C4#6325612yes—
506C22⋊C4—348no—
507C22⋊C4—348no—
508C22⋊C4—688no—
509C42—44no—
510C42—348no—
511C23—174no—
512C23—174no—
513C23—174no—
514C23—178no—
515C23—178no—
516C23—342no—
517C23—344no—
518C23—344no—
519C23—344no—
520C23—342no—
521C23—344no—
522C23—344no—
523C23—342no—
524C23—342no—
525C23—342no—
526C23—342no—
527C23—342no—
528C23—344no—
529C23—344no—
530C23—344no—
531C23—344no—
532C23—344no—
533C23—342no—
534C23C22×C4×D1711yescharacteristic
535C23C2×#6325212yes—
536C23C2×D4×D1711yescharacteristic
537C23#1058769112yes—
538C23C2×#6325611yescharacteristic
539C23#41×D1711yescharacteristic
540C23#41×D1712yes—
541C23#41×D1711yescharacteristic
542C23#41×D1712yes—
543C23#1058769611yescharacteristic
544C23#1058769711yescharacteristic
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—342no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—6816no—
586C2×C4—6832no—
587C2×C4—681no—
588C2×C4—682no—
589C2×C4—682no—
590C2×C4—682no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—682no—
594C2×C4—682no—
595C2×C4—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—681no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4#1058757914yes—
607C22—178no—
608C22—344no—
609C22—348no—
610C22—344no—
611C22—344no—
612C22—344no—
613C22—344no—
614C22—344no—
615C22—348no—
616C22—348no—
617C22—348no—
618C22#117×D1711yescharacteristic
619C22#117×D1711yescharacteristic
620C22#117×D1711yescharacteristic
621C22C2×#1058757911yescharacteristic
622C22C4×#6325211yescharacteristic
623C22#1176328211yescharacteristic
624C22#1176328911yescharacteristic
625C22#1176331812yes—
626C22#126×D1712yes—
627C22#126×D1712yes—
628C22#1176334612yes—
629C22#1176339512yes—
630C22#1176340212yes—
631C22#1176344812yes—
632C22#1176345211yescharacteristic
633C22#1176346711yescharacteristic
634C22#1176348312yes—
635C22#1176348612yes—
636C22#1176348812yes—
637C22#1176349711yescharacteristic
638C22#1176349812yes—
639C22#1176350912yes—
640C22#1176351211yescharacteristic
641C4—24no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—682no—
647C4—682no—
648C4—682no—
649C4—684no—
650C4—1362no—
651C2—348no—
652C2D17×#38311yescharacteristic
653C2#1453122511yescharacteristic
654C2#1453141311yescharacteristic
655C2#1453163011yescharacteristic
656C2#1453164511yescharacteristic
657C2#1453171011yescharacteristic
658C2#1453172711yescharacteristic
659C2#1453303612yes—
660C2#1453305112yes—
661C2#1453317612yes—
662C2#1453334112yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13901
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#11762676C2semidirect
4. #14527045C2semidirect
5. #14527045C2semidirect
6. #14527011C2semidirect
7. #14527008C2semidirect
11. C17×#1434C2semidirect
19. C4×#63227C4semidirect
62. C2×C42×C17C2×C4semidirect
73. C2×C4×Dic17C2×C4semidirect
87. #10587502C2×C4semidirect
88. #10587502C2×C4semidirect
283. #1434D17semidirect
320. C2×D17#383semidirect
404. C2×C42C4×D17semidirect
421. C17#13901semidirect

Maximal subgroups

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

subgroupindex
C2×#117626762
C2×C4×#632272
C2×#95×D172
#145270452
#145270112
#145270082
#145270052
C17×#14342
#1390117