Finite Group Databasebeta

← #16957035

Subgroups — #16957035

Order 4352.

2230 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587567C211yesmaximal, characteristic
2C2×#11762676C211yesmaximal, characteristic, complemented
3C2×#11762655C211yesmaximal, characteristic, complemented
4#14527050C212yesmaximal, complemented
5#14527023C212yesmaximal
6#14527022C211yesmaximal, characteristic, complemented
7#14527021C211yesmaximal, characteristic, complemented
8#14527020C211yesmaximal, characteristic, complemented
9#14527006C212yesmaximal, complemented
10#14527005C212yesmaximal, complemented
11C17×#1437C211yesmaximal, characteristic, complemented
12C23×C22⋊D17C2211yescharacteristic
13C24×Dic17C2211yescharacteristic
14C23×C4×D17C2211yescharacteristic
15C2×#10587567—24no—
16C2×#10587567—24no—
17C2×#10587567C418yescomplemented
18C22×#63225C2211yescharacteristic
19C22×C22⋊Dic17C2212yes—
20C22×#63227C2212yes—
21C22×#63227C2212yes—
22C22×#63227C2211yescharacteristic
23C22×C4×Dic17C2211yescharacteristic
24C22×C4×Dic17C2211yescharacteristic
25#11762676—24no—
26#11762655—24no—
27C2×#10587502C2212yes—
28C2×#10587502C2212yes—
29C2×#10587502C2212yes—
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2212yes—
32C2×#10587502C2212yes—
33C2×#10587502C2212yes—
34C2×#10587502C2212yes—
35C22×C22⋊C4×C17C2212yes—
36C22×C22⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C2×C17×#95C2211yescharacteristic
39C2×C17×#95C2212yes—
40C25×C17C2311yescharacteristic
41C24×D17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C22×C22⋊D17—24no—
47C22×C22⋊D17—24no—
48C22×C22⋊D17—22no—
49C22×C22⋊D17—22no—
50C22×C22⋊D17C2×C414yes—
51C23×Dic17—22no—
52C23×Dic17—22no—
53C23×Dic17—42no—
54C23×Dic17—42no—
55C23×Dic17C2×C414yes—
56C23×Dic17C2312yes—
57C23×Dic17C2311yescharacteristic
58C23×Dic17C2312yes—
59C23×Dic17C2311yescharacteristic
60C23×Dic17C2311yescharacteristic
61C22×C4×D17—22no—
62C22×C4×D17—22no—
63C22×C4×D17—42no—
64C22×C4×D17—42no—
65C22×C4×D17C2×C414yes—
66C2×C22⋊C4×C17—22no—
67C2×C22⋊C4×C17—22no—
68C2×C22⋊C4×C17—24no—
69C2×C22⋊C4×C17—24no—
70C2×C22⋊C4×C17—42no—
71C2×C22⋊C4×C17C2×C414yes—
72C2×C42×C17—22no—
73C2×C42×C17—24no—
74C2×C22⋊Dic17—24no—
75C2×C22⋊Dic17—24no—
76C2×C22⋊Dic17—42no—
77C2×#63227—24no—
78C2×#63227—24no—
79C2×#63227—22no—
80C2×#63227—22no—
81C2×#63227—24no—
82C2×#63227—24no—
83C2×#63227—42no—
84C2×#63227—42no—
85C2×#63227C2×C414yes—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225C2×C414yes—
89C2×C4×Dic17—22no—
90C2×C4×Dic17—24no—
91C2×C4×Dic17C2×C414yes—
92C2×C4×Dic17D414yes—
93C2×C4×Dic17D414yes—
94#10587567—232no—
95C17×#95—22no—
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×C17—21no—
106C24×C17—21no—
107C24×C17—22no—
108C24×C17—22no—
109C24×C17—22no—
110C24×C17—22no—
111C24×C17—22no—
112C24×C17C22×C412yes—
113C24×C17#4114yes—
114C24×C17C2411yescharacteristic
115C23×D17—21no—
116C23×D17—21no—
117C23×D17—22no—
118C23×D17—22no—
119C23×D17—22no—
120C23×D17—22no—
121C23×D17—22no—
122C23×D17C22×C412yes—
123C23×D17#4114yes—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
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×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—22no—
143C22×C4×C17C22×C412yes—
144C22×C4×C17C22×C412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D412yes—
147C22×C4×C17#4114yes—
148C2×C22⋊D17—216no—
149C2×C22⋊D17—216no—
150C2×C22⋊D17—28no—
151C2×C22⋊D17—28no—
152C2×C22⋊D17—28no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—21no—
158C22×Dic17—21no—
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—28no—
168C22×Dic17—22no—
169C22×Dic17—22no—
170C22×Dic17—22no—
171C22×Dic17—22no—
172C22×Dic17—22no—
173C22×Dic17—22no—
174C22×Dic17—22no—
175C22×Dic17—48no—
176C22×Dic17—44no—
177C22×Dic17—48no—
178C22×Dic17—44no—
179C22×Dic17C22×C412yes—
180C22×Dic17C22×C412yes—
181C22×Dic17C22×C412yes—
182C22×Dic17C2×D412yes—
183C22×Dic17C2×D412yes—
184C22×Dic17C2×D412yes—
185C22×Dic17C2×D412yes—
186C22×Dic17#4114yes—
187C2×C4×D17—28no—
188C2×C4×D17—48no—
189C2×C4×D17—44no—
190C2×C4×D17—48no—
191C2×C4×D17—44no—
192C22⋊C4×C17—28no—
193C22⋊C4×C17—48no—
194C42×C17—28no—
195C22⋊Dic17—48no—
196#63227—28no—
197#63227—48no—
198#63227—48no—
199#63225—28no—
200C4×Dic17—28no—
201C4×Dic17C22⋊C4116yescomplemented
202#13959—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—24no—
206C23×C17—24no—
207C23×C17—24no—
208C23×C17—24no—
209C23×C17—24no—
210C23×C17—24no—
211C23×C17—24no—
212C23×C17—24no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—42no—
216C23×C17—41no—
217C23×C17—42no—
218C23×C17—41no—
219C23×C17—42no—
220C23×C17—42no—
221C23×C17—42no—
222C23×C17—42no—
223C23×C17#11714yes—
224C23×C17C23×C411yescharacteristic
225C23×C17C22×D411yescharacteristic
226C23×C17C22×D411yescharacteristic
227C23×C17C2×#4112yes—
228C23×C17C2×#4112yes—
229C23×C17C2×#4112yes—
230C23×C17C2×#4112yes—
231C23×C17H(5,𝔽2)12yes—
232C23×C17H(5,𝔽2)12yes—
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—24no—
236C22×D17—24no—
237C22×D17—24no—
238C22×D17—24no—
239C22×D17—24no—
240C22×D17—24no—
241C22×D17—24no—
242C22×D17—24no—
243C22×D17—41no—
244C22×D17—41no—
245C22×D17—42no—
246C22×D17—41no—
247C22×D17—42no—
248C22×D17—41no—
249C22×D17—42no—
250C22×D17—42no—
251C22×D17—42no—
252C22×D17—42no—
253C22×D17#11714yes—
254C2×C4×C17—24no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—24no—
258C2×C4×C17—22no—
259C2×C4×C17—24no—
260C2×C4×C17—24no—
261C2×C4×C17—24no—
262C2×C4×C17—24no—
263C2×C4×C17—42no—
264C2×C4×C17—41no—
265C2×C4×C17—42no—
266C2×C4×C17—41no—
267C2×C4×C17—42no—
268C2×C4×C17—42no—
269C2×C4×C17—42no—
270C2×C4×C17—42no—
271C2×C4×C17—42no—
272C2×C4×C17—42no—
273C2×C4×C17—42no—
274C2×C4×C17—42no—
275C2×C4×C17—42no—
276C2×C4×C17—42no—
277C2×C4×C17C2×C22⋊C414yes—
278C22⋊D17—232no—
279C2×Dic17—22no—
280C2×Dic17—22no—
281C2×Dic17—24no—
282C2×Dic17—24no—
283C2×Dic17—24no—
284C2×Dic17—22no—
285C2×Dic17—24no—
286C2×Dic17—22no—
287C2×Dic17—22no—
288C2×Dic17—24no—
289C2×Dic17—24no—
290C2×Dic17—24no—
291C2×Dic17—24no—
292C2×Dic17—416no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—41no—
296C2×Dic17—42no—
297C2×Dic17—41no—
298C2×Dic17—42no—
299C2×Dic17—42no—
300C2×Dic17—42no—
301C2×Dic17—42no—
302C2×Dic17—42no—
303C2×Dic17—42no—
304C2×Dic17—42no—
305C2×Dic17—42no—
306C2×Dic17—42no—
307C2×Dic17C2×C22⋊C414yes—
308C2×Dic17C2×C22⋊C414yes—
309C4×D17—416no—
310C22×C4×D4—171no—
311C2×#383—171no—
312C2×#376—171no—
313#1445—172no—
314#1438—172no—
315#1437—171no—
316#1437—171no—
317#1437—171no—
318#1437D1711yescharacteristic, complemented
319C22⋊#95—172no—
320C22⋊#95—172no—
321C22×C17—28no—
322C22×C17—24no—
323C22×C17—42no—
324C22×C17—44no—
325C22×C17—44no—
326C22×C17—44no—
327C22×C17—44no—
328C22×C17—44no—
329C22×C17—42no—
330C22×C17C22×C22⋊C411yescharacteristic
331C22×C17C2×#11711yescharacteristic
332C22×C17C2×#11711yescharacteristic
333C22×C17C4×#4112yes—
334C22×C17#51812yes—
335C22×C17#52511yescharacteristic
336C22×C17#52511yescharacteristic
337C22×C17#52512yes—
338C22×C17#52512yes—
339C22×C17#53212yes—
340C22×C17#53511yescharacteristic
341C22×C17#53512yes—
342C22×C17#53511yescharacteristic
343C22×C17#55112yes—
344C22×C17#55112yes—
345C22×C17#55112yes—
346C22×C17#55112yes—
347C22×C17#55312yes—
348C22×C17#55312yes—
349C22×C17#55312yes—
350C22×C17#55312yes—
351C2×D17—28no—
352C2×D17—24no—
353C2×D17—42no—
354C2×D17—44no—
355C2×D17—44no—
356C2×D17—44no—
357C2×D17—44no—
358C2×D17—44no—
359C2×D17—42no—
360C4×C17—24no—
361C4×C17—42no—
362C4×C17—44no—
363C4×C17—82no—
364Dic17—24no—
365Dic17—24no—
366Dic17—42no—
367Dic17—44no—
368Dic17—82no—
369C23×D4—171no—
370C24×C4—171no—
371C24×C4—171no—
372C2×C4×D4—178no—
373C2×C4×D4—344no—
374C2×C4×D4—344no—
375C22×C4⋊C4—171no—
376C22×C22⋊C4—171no—
377C22×C22⋊C4—172no—
378C22×C22⋊C4—172no—
379C22×C22⋊C4—172no—
380C22×C22⋊C4C2×D1712yes—
381C22×C22⋊C4C2×D1711yescharacteristic
382C22×C42—171no—
383C22×C42—171no—
384C22×C42C2×D1711yescharacteristic
385#383—344no—
386#376—344no—
387C2×#95—172no—
388C2×#95—172no—
389C2×#95—171no—
390C2×#95—172no—
391C2×#95—172no—
392C2×#95—172no—
393C2×#95—172no—
394C2×#95—172no—
395C2×#95C2×D1711yescharacteristic
396C2×#95C2×D1712yes—
397C2×C17—44no—
398C2×C17#228011yescharacteristic
399C2×C17#229711yescharacteristic
400C2×C17#230911yescharacteristic
401C2×C17#235312yes—
402C2×C17#235312yes—
403C2×C17#259812yes—
404C2×C17#259812yes—
405C2×C17#260312yes—
406C2×C17#260312yes—
407D17—44no—
408C25—171no—
409C25C22×D1711yescharacteristic
410C22×D4—174no—
411C22×D4—342no—
412C22×D4—342no—
413C22×D4—344no—
414C22×D4—344no—
415C23×C4—171no—
416C23×C4—171no—
417C23×C4—172no—
418C23×C4—171no—
419C23×C4—174no—
420C23×C4—174no—
421C23×C4—172no—
422C23×C4—342no—
423C23×C4—342no—
424C23×C4—342no—
425C23×C4—342no—
426C23×C4—682no—
427C23×C4—682no—
428C23×C4—682no—
429C23×C4—682no—
430C23×C4C22×D1712yes—
431C23×C4C22×D1711yescharacteristic
432C23×C4C22×D1712yes—
433C23×C4C22×D1711yescharacteristic
434C4×D4—3432no—
435C2×C4⋊C4—174no—
436C2×C4⋊C4—342no—
437C2×C4⋊C4—342no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—24no—
441C2×C22⋊C4—24no—
442C2×C22⋊C4—42no—
443C2×C22⋊C4—174no—
444C2×C22⋊C4—342no—
445C2×C22⋊C4—342no—
446C2×C22⋊C4—344no—
447C2×C22⋊C4—344no—
448C2×C22⋊C4—344no—
449C2×C22⋊C4—344no—
450C2×C22⋊C4—344no—
451C2×C22⋊C4—344no—
452C2×C22⋊C4—682no—
453C2×C22⋊C4—682no—
454C2×C22⋊C4—682no—
455C2×C22⋊C4C4×D1714yes—
456C2×C42—22no—
457C2×C42—24no—
458C2×C42—174no—
459C2×C42—174no—
460C2×C42—174no—
461C2×C42—344no—
462C2×C42—342no—
463#95—22no—
464#95—42no—
465#95—344no—
466#95—344no—
467#95—344no—
468#95—344no—
469#95—342no—
470#95—682no—
471#95—682no—
472#95—682no—
473C17#1395911yescharacteristic, complemented, Sylow
474C24—21no—
475C24—21no—
476C24—22no—
477C24—22no—
478C24—22no—
479C24—22no—
480C24—22no—
481C24—172no—
482C24—174no—
483C24—342no—
484C24—341no—
485C24—341no—
486C24—342no—
487C24—342no—
488C24—342no—
489C24—342no—
490C24C2×C4×D1712yes—
491C24#6325214yes—
492C24C23×D1711yescharacteristic
493C2×D4—348no—
494C2×D4—348no—
495C2×D4—3416no—
496C2×D4—3416no—
497C2×D4—348no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—172no—
518C22×C4—172no—
519C22×C4—172no—
520C22×C4—172no—
521C22×C4—172no—
522C22×C4—172no—
523C22×C4—172no—
524C22×C4—174no—
525C22×C4—348no—
526C22×C4—348no—
527C22×C4—342no—
528C22×C4—342no—
529C22×C4—342no—
530C22×C4—342no—
531C22×C4—342no—
532C22×C4—342no—
533C22×C4—341no—
534C22×C4—341no—
535C22×C4—342no—
536C22×C4—342no—
537C22×C4—342no—
538C22×C4—342no—
539C22×C4—342no—
540C22×C4—342no—
541C22×C4—342no—
542C22×C4—342no—
543C22×C4—342no—
544C22×C4—342no—
545C22×C4—342no—
546C22×C4—342no—
547C22×C4—342no—
548C22×C4—688no—
549C22×C4—684no—
550C22×C4—688no—
551C22×C4—684no—
552C22×C4—688no—
553C22×C4—684no—
554C22×C4—688no—
555C22×C4—684no—
556C22×C4C2×C4×D1712yes—
557C22×C4C2×C4×D1712yes—
558C22×C4D4×D1712yes—
559C22×C4D4×D1712yes—
560C22×C4#6325414yes—
561C4⋊C4—348no—
562C22⋊C4—28no—
563C22⋊C4—48no—
564C22⋊C4—348no—
565C22⋊C4—688no—
566C22⋊C4—688no—
567C22⋊C4—688no—
568C42—28no—
569C42—1716no—
570C42—348no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—42no—
588C23—41no—
589C23—42no—
590C23—41no—
591C23—174no—
592C23—344no—
593C23—344no—
594C23—344no—
595C23—344no—
596C23—342no—
597C23—342no—
598C23—344no—
599C23—344no—
600C23—344no—
601C23—344no—
602C23—682no—
603C23—682no—
604C23—682no—
605C23—682no—
606C23—681no—
607C23—681no—
608C23—682no—
609C23—681no—
610C23—682no—
611C23—681no—
612C23#1058755814yes—
613C23C22×C4×D1711yescharacteristic
614C23C2×#6325212yes—
615C23C2×D4×D1711yescharacteristic
616C23C2×D4×D1711yescharacteristic
617C23C2×#6325412yes—
618C23#1058769112yes—
619C23#41×D1712yes—
620C23#41×D1712yes—
621C23#1058769612yes—
622D4—3432no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—174no—
647C2×C4—174no—
648C2×C4—342no—
649C2×C4—342no—
650C2×C4—344no—
651C2×C4—344no—
652C2×C4—344no—
653C2×C4—342no—
654C2×C4—344no—
655C2×C4—342no—
656C2×C4—344no—
657C2×C4—344no—
658C2×C4—342no—
659C2×C4—344no—
660C2×C4—344no—
661C2×C4—6816no—
662C2×C4—6816no—
663C2×C4—682no—
664C2×C4—682no—
665C2×C4—682no—
666C2×C4—681no—
667C2×C4—682no—
668C2×C4—682no—
669C2×C4—681no—
670C2×C4—682no—
671C2×C4—682no—
672C2×C4—682no—
673C2×C4—682no—
674C2×C4—682no—
675C2×C4—682no—
676C2×C4—682no—
677C2×C4C22⋊C4×D1714yes—
678C22—24no—
679C22—28no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—344no—
688C22—348no—
689C22—684no—
690C22—684no—
691C22—682no—
692C22—684no—
693C22—682no—
694C22—684no—
695C22—684no—
696C22C2×C22⋊C4×D1711yescharacteristic
697C22#117×D1711yescharacteristic
698C22C2×#1058755811yescharacteristic
699C22C4×#6325412yes—
700C22#1176328112yes—
701C22#1176331511yescharacteristic
702C22#1176331711yescharacteristic
703C22#1176331812yes—
704C22#1176331812yes—
705C22#1176335412yes—
706C22#1176337211yescharacteristic
707C22#1176337512yes—
708C22#1176337711yescharacteristic
709C22#1176348212yes—
710C22#1176348312yes—
711C22#1176348512yes—
712C22#1176348912yes—
713C22#1176350512yes—
714C22#1176350612yes—
715C22#1176350812yes—
716C22#1176351012yes—
717C4—24no—
718C4—42no—
719C4—44no—
720C4—82no—
721C4—344no—
722C4—344no—
723C4—682no—
724C4—684no—
725C4—1362no—
726C2—44no—
727C2—684no—
728C2#1453116511yescharacteristic
729C2#1453126911yescharacteristic
730C2#1453136311yescharacteristic
731C2#1453171812yes—
732C2#1453171912yes—
733C2#1453434612yes—
734C2#1453434812yes—
735C2#1453440712yes—
736C2#1453440912yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1437C2
socle, cosocleC24×C17C24
prime radicalC23×C22⋊D17C22
hypercenter#1437D17
hypocenterC17#13959
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#11762676C2semidirect
3. C2×#11762655C2semidirect
4. #14527050C2semidirect
6. #14527022C2semidirect
7. #14527021C2semidirect
8. #14527020C2semidirect
9. #14527006C2semidirect
10. #14527005C2semidirect
11. C17×#1437C2semidirect
17. C2×#10587567C4semidirect
201. C4×Dic17C22⋊C4semidirect
318. #1437D17semidirect
473. C17#13959semidirect

Maximal subgroups

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

subgroupindex
C22×#105875672
C2×#117626762
C2×#117626552
#145270502
#145270232
#145270222
#145270212
#145270202
#145270062
#145270052
C17×#14372
#1395917