Finite Group Databasebeta

← #16962364

Subgroups — #16962364

Order 4352.

1866 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762702C212yesmaximal, complemented
2C2×#11762669C212yesmaximal, complemented
3#14527041C212yesmaximal, complemented
4#14527029C212yesmaximal, complemented
5#14527025C211yesmaximal, characteristic
6#14527025C211yesmaximal, characteristic
7#14527021C212yesmaximal, complemented
8#14527018C212yesmaximal, complemented
9C2×C17×#386C211yesmaximal, characteristic
10C22×C4⋊Dic17C2212yes—
11C22×C4⋊Dic17C2212yes—
12C22×C22⋊Dic17C2212yes—
13C22×C22⋊Dic17C2212yes—
14C22×C22⋊Dic17C2211yescharacteristic
15C22×C22⋊Dic17C2212yes—
16C22×C22⋊Dic17C2211yescharacteristic
17C22×C4×Dic17C2212yes—
18#11762702—28no—
19#11762669—28no—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2211yescharacteristic
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2211yescharacteristic
27C2×#10587502C2212yes—
28C23×D4×C17C2211yescharacteristic
29C24×C4×C17C2211yescharacteristic
30C22×C22⋊C4×C17C2212yes—
31C22×C22⋊C4×C17C2211yescharacteristic
32C17×#386C418yescomplemented
33C2×C17×#95C2212yes—
34C25×C17C2311yescharacteristic
35C25×C17C2311yescharacteristic
36C22×D4×C17—24no—
37C22×D4×C17—41no—
38C22×D4×C17—41no—
39C22×D4×C17—42no—
40C22×D4×C17C2×C414yes—
41C23×C4×C17—24no—
42C23×C4×C17—24no—
43C23×C4×C17—41no—
44C23×C4×C17—41no—
45C23×C4×C17C2×C414yes—
46C23×C4×C17C2312yes—
47C23×C4×C17C2311yescharacteristic
48C23×C4×C17C2312yes—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C2×C22⋊C4×C17—24no—
54C2×C22⋊C4×C17—24no—
55C2×C22⋊C4×C17—42no—
56C2×C22⋊C4×C17C2×C418yes—
57C2×C22⋊C4×C17C2×C414yes—
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—24no—
62C2×C22⋊Dic17—24no—
63C2×C22⋊Dic17—24no—
64C2×C22⋊Dic17—41no—
65C2×C22⋊Dic17—42no—
66C2×C22⋊Dic17—42no—
67C2×C22⋊Dic17—42no—
68C2×C22⋊Dic17—42no—
69C2×C22⋊Dic17—42no—
70C2×C22⋊Dic17—41no—
71C2×C4⋊Dic17—24no—
72C2×C4⋊Dic17—24no—
73C2×C4⋊Dic17—24no—
74C2×C4⋊Dic17—24no—
75C2×C4⋊Dic17—42no—
76C2×C4⋊Dic17—42no—
77C2×C4×Dic17—24no—
78C2×C4×Dic17—42no—
79C2×C4×Dic17—42no—
80C17×#95C2×C418yes—
81#10587502—24no—
82#10587502—24no—
83#10587502—42no—
84#10587502—41no—
85#10587502—42no—
86#10587502—42no—
87#10587502—41no—
88#10587502D418yescomplemented
89C24×C17—22no—
90C24×C17—22no—
91C24×C17—21no—
92C24×C17—21no—
93C24×C17—22no—
94C24×C17—22no—
95C24×C17—22no—
96C24×C17—22no—
97C24×C17—22no—
98C24×C17—22no—
99C24×C17—22no—
100C24×C17—21no—
101C24×C17—21no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17C22×C412yes—
105C24×C17C22×C412yes—
106C24×C17#4114yes—
107C24×C17C2411yescharacteristic
108C2×D4×C17—28no—
109C2×D4×C17—44no—
110C2×D4×C17—44no—
111C2×D4×C17—48no—
112C2×D4×C17—48no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
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—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—216no—
128C22×C4×C17—44no—
129C22×C4×C17—44no—
130C22×C4×C17—48no—
131C22×C4×C17—44no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C22×C414yes—
134C22×C4×C17C22×C414yes—
135C22×C4×C17C2×D414yes—
136C22×C4×C17#4114yes—
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×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—22no—
160C22×Dic17—22no—
161C22×Dic17—22no—
162C22×Dic17—22no—
163C22×Dic17C2×D414yes—
164C22×Dic17C2×D414yes—
165C22⋊C4×C17—216no—
166C22⋊C4×C17—44no—
167C22⋊Dic17—44no—
168C22⋊Dic17—48no—
169C22⋊Dic17—48no—
170C22⋊Dic17—48no—
171C22⋊Dic17—44no—
172C4⋊Dic17—48no—
173C4⋊Dic17—48no—
174C4×Dic17—48no—
175#14410—171nomaximal, Sylow
176C23×C17—24no—
177C23×C17—22no—
178C23×C17—22no—
179C23×C17—24no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—24no—
185C23×C17—24no—
186C23×C17—22no—
187C23×C17—24no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17—24no—
191C23×C17—42no—
192C23×C17—42no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—41no—
196C23×C17—42no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—42no—
200C23×C17—41no—
201C23×C17—42no—
202C23×C17—42no—
203C23×C17—42no—
204C23×C17—42no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—41no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17—41no—
212C23×C17—42no—
213C23×C17—42no—
214C23×C17—41no—
215C23×C17—42no—
216C23×C17—41no—
217C23×C17#11714yes—
218C23×C17C23×C411yescharacteristic
219C23×C17C22×D412yes—
220C23×C17C2×#4112yes—
221C23×C17C2×#4112yes—
222C23×C17H(5,𝔽2)12yes—
223C23×C17H(5,𝔽2)11yescharacteristic
224C23×C17H(5,𝔽2)11yescharacteristic
225C23×C17H(5,𝔽2)12yes—
226C23×C172−1+412yes—
227D4×C17—416no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—24no—
233C2×C4×C17—24no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—416no—
241C2×C4×C17—42no—
242C2×C4×C17—42no—
243C2×C4×C17—42no—
244C2×C4×C17—42no—
245C2×C4×C17—42no—
246C2×C4×C17—42no—
247C2×C4×C17C4×D418yes—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
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—42no—
270C2×Dic17—42no—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274C2×Dic17—42no—
275C2×Dic17—42no—
276C2×#388—172no—
277C2×#386D1711yescharacteristic
278C2×#380—172no—
279#1442—172no—
280#1439—172no—
281#1438—171no—
282#1438—171no—
283#1437—172no—
284#1436—172no—
285C22×C17—24no—
286C22×C17—28no—
287C22×C17—44no—
288C22×C17—42no—
289C22×C17—42no—
290C22×C17—44no—
291C22×C17—44no—
292C22×C17—44no—
293C22×C17—44no—
294C22×C17—42no—
295C22×C17—44no—
296C22×C17—44no—
297C22×C17—42no—
298C22×C17—42no—
299C22×C17—82no—
300C22×C17—82no—
301C22×C17—82no—
302C22×C17—82no—
303C22×C17C2×#11711yescharacteristic
304C22×C17C2×C4×D412yes—
305C22×C17#51811yescharacteristic
306C22×C17#51811yescharacteristic
307C22×C17#52012yes—
308C22×C17#53512yes—
309C22×C17#53712yes—
310C22×C17#53812yes—
311C22×C17#53812yes—
312C22×C17#54012yes—
313C22×C17#54212yes—
314C22×C17D4211yescharacteristic
315C22×C17#54612yes—
316C22×C17#54612yes—
317C22×C17#54812yes—
318C22×C17#55111yescharacteristic
319C22×C17#55212yes—
320C22×C17#55312yes—
321C22×C17#56011yescharacteristic
322C22×C17#56212yes—
323C22×C17#56311yescharacteristic
324C4×C17—28no—
325C4×C17—42no—
326C4×C17—44no—
327Dic17—44no—
328Dic17—44no—
329Dic17—82no—
330Dic17—82no—
331C23×D4C2×D1711yescharacteristic
332C24×C4C2×D1711yescharacteristic
333C22×C4⋊C4—172no—
334C22×C4⋊C4—172no—
335C22×C22⋊C4—172no—
336C22×C22⋊C4—172no—
337C22×C22⋊C4—171no—
338C22×C22⋊C4—172no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4C2×D1712yes—
341C22×C22⋊C4C2×D1711yescharacteristic
342C22×C42—172no—
343#388—348no—
344#386Dic1718yescomplemented
345#380—348no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—171no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95—172no—
353C2×#95—171no—
354C2×#95C2×D1712yes—
355C2×C17—44no—
356C2×C17—82no—
357C2×C17#231211yescharacteristic
358C2×C17#232412yes—
359C2×C17#235212yes—
360C2×C17#235811yescharacteristic
361C2×C17#237611yescharacteristic
362C2×C17#267612yes—
363C2×C17#270212yes—
364C2×C17#270312yes—
365C2×C17#274112yes—
366C25C22×D1711yescharacteristic
367C25C22×D1711yescharacteristic
368C22×D4—24no—
369C22×D4—42no—
370C22×D4—41no—
371C22×D4—41no—
372C22×D4C2×Dic1714yes—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—41no—
376C23×C4—41no—
377C23×C4—172no—
378C23×C4—172no—
379C23×C4—172no—
380C23×C4—172no—
381C23×C4C2×Dic1714yes—
382C23×C4C22×D1712yes—
383C23×C4C22×D1711yescharacteristic
384C23×C4C22×D1712yes—
385C2×C4⋊C4—344no—
386C2×C4⋊C4—344no—
387C2×C4⋊C4—344no—
388C2×C4⋊C4—344no—
389C2×C4⋊C4—682no—
390C2×C4⋊C4—682no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—344no—
396C2×C22⋊C4—344no—
397C2×C22⋊C4—344no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—681no—
401C2×C22⋊C4—682no—
402C2×C22⋊C4—682no—
403C2×C22⋊C4—682no—
404C2×C22⋊C4—682no—
405C2×C22⋊C4—681no—
406C2×C22⋊C4—682no—
407C2×C22⋊C4C2×Dic1718yes—
408C2×C22⋊C4C2×Dic1714yes—
409C2×C42—344no—
410C2×C42—682no—
411C2×C42—682no—
412#95—178no—
413#95—344no—
414#95—344no—
415#95—682no—
416#95—682no—
417#95—681no—
418#95—681no—
419#95—682no—
420#95C2×Dic1718yes—
421C17#1441011yescharacteristic, complemented, Sylow
422C24—22no—
423C24—22no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—21no—
433C24—21no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24#6325214yes—
438C24C22×Dic1712yes—
439C24C22×Dic1712yes—
440C24C23×D1711yescharacteristic
441C2×D4—28no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—48no—
445C2×D4—48no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—216no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
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×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4D4×D1714yes—
494C22×C4#6325414yes—
495C22×C4C22×Dic1712yes—
496C22×C4C22×Dic1714yes—
497C22×C4C22×Dic1714yes—
498C4⋊C4—688no—
499C4⋊C4—688no—
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—684no—
503C22⋊C4—688no—
504C22⋊C4—688no—
505C22⋊C4—688no—
506C22⋊C4—684no—
507C42—688no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23#1058761214yes—
550C23C2×#6325212yes—
551C23C2×D4×D1712yes—
552C23C2×#6325412yes—
553C23#1058769111yescharacteristic
554C23#1058769111yescharacteristic
555C23#1058769112yes—
556C23#1058769612yes—
557C23#1058769712yes—
558C23C23×Dic1711yescharacteristic
559D4—416no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—416no—
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—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—682no—
596C2×C4—682no—
597C2×C4—682no—
598C2×C4—682no—
599C2×C4—682no—
600C2×C4—682no—
601C2×C4—682no—
602C2×C4—682no—
603C2×C4—682no—
604C2×C4—682no—
605C2×C4—682no—
606C2×C4—682no—
607C2×C4D4×Dic1718yes—
608C22—24no—
609C22—28no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#1058761211yescharacteristic
627C22C2×D4×Dic1712yes—
628C22#1176328411yescharacteristic
629C22#1176328411yescharacteristic
630C22#1176329212yes—
631C22#1176337512yes—
632C22#1176338812yes—
633C22#1176339712yes—
634C22#1176339712yes—
635C22#1176341212yes—
636C22#1176342612yes—
637C22#1176344111yescharacteristic
638C22#1176344612yes—
639C22#1176345512yes—
640C22#1176346812yes—
641C22#1176349011yescharacteristic
642C22#1176349912yes—
643C22#1176350512yes—
644C22#1176354911yescharacteristic
645C22#1176355612yes—
646C22#1176356211yescharacteristic
647C4—28no—
648C4—42no—
649C4—44no—
650C4—684no—
651C4—684no—
652C4—1362no—
653C4—1362no—
654C2—44no—
655C2—82no—
656C2#1453138211yescharacteristic
657C2#1453147912yes—
658C2#1453171312yes—
659C2#1453176811yescharacteristic
660C2#1453193211yescharacteristic
661C2#1453508912yes—
662C2#1453541612yes—
663C2#1453543112yes—
664C2#1453590312yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#386C2
socle, cosocleC24×C17C24
prime radicalC23×D4×C17C22
hypercenterC2×#386D17
hypocenterC17#14410
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C2×#11762702C2semidirect
2. C2×#11762669C2semidirect
3. #14527041C2semidirect
4. #14527029C2semidirect
7. #14527021C2semidirect
8. #14527018C2semidirect
32. C17×#386C4semidirect
88. #10587502D4semidirect
344. #386Dic17semidirect
421. C17#14410semidirect

Maximal subgroups

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

subgroupindex
C2×#117627022
C2×#117626692
#145270412
#145270292
#145270252
#145270212
#145270182
C2×C17×#3862
#1441017