Finite Group Databasebeta

← #16955715

Subgroups — #16955715

Order 4352.

1958 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63227C211yesmaximal, characteristic
2C2×#11762656C211yesmaximal, characteristic
3C2×#95×D17C211yesmaximal, characteristic
4#14527011C211yesmaximal, characteristic, complemented
5#14527010C212yesmaximal, complemented
6#14527009C211yesmaximal, characteristic, complemented
7#14527009C211yesmaximal, characteristic, complemented
8#14527008C212yesmaximal, complemented
9#14527005C212yesmaximal
10#14527005C212yesmaximal
11C17×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D17C2212yes—
13C23×C4×D17C2211yescharacteristic
14C22×#63227C2212yes—
15C22×#63227C2211yescharacteristic
16C22×#63227C2211yescharacteristic
17C22×C4×Dic17C2212yes—
18C22×C4×Dic17C2211yescharacteristic
19C4×#63227C418yescomplemented
20#11762656C418yescomplemented
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C2×#10587502C2211yescharacteristic
30C2×#10587502C2211yescharacteristic
31C2×#10587502C2211yescharacteristic
32#95×D17C418yescomplemented
33C22×C42×C17C2211yescharacteristic
34C2×C17×#95C2212yes—
35C2×C17×#95C2212yes—
36C2×C17×#95C2211yescharacteristic
37C2×C17×#95C2211yescharacteristic
38C24×D17C2311yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×C4×C17C2312yes—
42C23×C4×C17C2312yes—
43C23×C4×C17C2311yescharacteristic
44C23×Dic17C2312yes—
45C23×Dic17C2311yescharacteristic
46C23×Dic17C2311yescharacteristic
47C23×Dic17C2312yes—
48C23×Dic17C2311yescharacteristic
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—42no—
52C22×C4×D17—42no—
53C22×C4×D17—42no—
54C22×C4×D17—42no—
55C22×C4×D17—42no—
56C22×C4×D17—42no—
57C22×C4×D17C2×C418yes—
58C22×C4×D17C2×C414yes—
59C22×C4×D17C2×C414yes—
60C22×C4×D17C2×C414yes—
61C2×C42×C17—42no—
62C2×C42×C17C2×C414yescomplemented
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227C2×C418yes—
68C2×#63227C2×C414yes—
69C2×#63227C2×C414yes—
70C2×C4×Dic17—42no—
71C2×C4×Dic17—42no—
72C2×C4×Dic17—42no—
73C2×C4×Dic17C2×C418yescomplemented
74C2×C4×Dic17C2×C414yescomplemented
75C17×#95—42no—
76C17×#95—42no—
77C17×#95C2×C414yes—
78C17×#95C2×C414yescomplemented
79#10587502—42no—
80#10587502—42no—
81#10587502—42no—
82#10587502—42no—
83#10587502—42no—
84#10587502—42no—
85#10587502C2×C414yes—
86#10587502C2×C418yes—
87#10587502C2×C414yescomplemented
88#10587502C2×C414yescomplemented
89#10587502C2×C414yescomplemented
90C24×C17C2411yescharacteristic
91C23×D17—22no—
92C23×D17—21no—
93C23×D17—22no—
94C23×D17—21no—
95C23×D17—21no—
96C23×D17—21no—
97C23×D17C22×C412yes—
98C23×D17C22×C412yes—
99C23×D17C22×C412yes—
100C23×D17#4114yes—
101C23×D17#4114yes—
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—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17C22×C414yes—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C22×C412yes—
127C22×C4×C17C22×C414yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C22×C412yes—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
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—21no—
144C22×Dic17—21no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—21no—
151C22×Dic17—22no—
152C22×Dic17—21no—
153C22×Dic17C22×C414yes—
154C22×Dic17C22×C414yes—
155C22×Dic17C22×C412yes—
156C22×Dic17C22×C412yes—
157C22×Dic17C22×C412yes—
158C22×Dic17C22×C412yes—
159C22×Dic17C22×C412yes—
160C2×C4×D17—216no—
161C2×C4×D17—44no—
162C2×C4×D17—48no—
163C2×C4×D17—44no—
164C2×C4×D17—48no—
165C2×C4×D17—48no—
166C2×C4×D17—48no—
167C2×C4×D17—48no—
168C2×C4×D17—48no—
169C2×C4×D17—48no—
170C2×C4×D17—48no—
171C2×C4×D17C42116yes—
172C42×C17—44no—
173#63227—216no—
174#63227—28no—
175#63227—28no—
176C4×Dic17—48no—
177C4×Dic17—44no—
178#13796—171nomaximal, Sylow
179C23×C17C23×C411yescharacteristic
180C23×C17C23×C411yescharacteristic
181C23×C17C23×C411yescharacteristic
182C23×C17C2×#4112yes—
183C23×C17C2×#4112yes—
184C23×C17H(5,𝔽2)12yes—
185C23×C17H(5,𝔽2)11yescharacteristic
186C23×C17H(5,𝔽2)12yes—
187C23×C17H(5,𝔽2)11yescharacteristic
188C23×C172−1+411yescharacteristic
189C23×C172−1+411yescharacteristic
190C22×D17—22no—
191C22×D17—24no—
192C22×D17—24no—
193C22×D17—22no—
194C22×D17—22no—
195C22×D17—22no—
196C22×D17—22no—
197C22×D17—22no—
198C22×D17—22no—
199C22×D17—22no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—24no—
207C22×D17—24no—
208C22×D17C2×C4214yes—
209C22×D17#11718yes—
210C22×D17#11714yes—
211C22×D17#11718yes—
212C22×D17#11714yes—
213C2×C4×C17—22no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—22no—
217C2×C4×C17—24no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—24no—
224C2×C4×C17—24no—
225C2×C4×C17—24no—
226C2×C4×C17—24no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
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×C17C2×C4214yes—
242C2×Dic17—24no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—24no—
246C2×Dic17—24no—
247C2×Dic17—22no—
248C2×Dic17—24no—
249C2×Dic17—22no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—24no—
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×Dic17C2×C4214yes—
271C4×D17—416no—
272C4×D17—432no—
273C2×C4×C22⋊C4—171no—
274C2×C4×C22⋊C4—171no—
275C22×#95—171no—
276#1434—171no—
277#1434—172no—
278#1434—171no—
279#1434—171no—
280#1434—172no—
281#1434D1711yescharacteristic, complemented
282C22⋊#95—172no—
283C22⋊#95—172no—
284C22×C17C22×C4211yescharacteristic
285C22×C17C2×#11712yes—
286C22×C17C2×#11711yescharacteristic
287C22×C17C2×#11712yes—
288C22×C17C2×#11711yescharacteristic
289C22×C17#51811yescharacteristic
290C22×C17#51812yes—
291C22×C17#51812yes—
292C22×C17#51811yescharacteristic
293C22×C17#51811yescharacteristic
294C22×C17#51911yescharacteristic
295C22×C17#52011yescharacteristic
296C22×C17#52011yescharacteristic
297C22×C17#52011yescharacteristic
298C22×C17#52011yescharacteristic
299C22×C17#53812yes—
300C22×C17#53812yes—
301C22×C17#53812yes—
302C22×C17#53812yes—
303C22×C17#53912yes—
304C22×C17#53912yes—
305C22×C17#53912yes—
306C22×C17#53912yes—
307C2×D17—28no—
308C2×D17—28no—
309C2×D17—24no—
310C2×D17—28no—
311C2×D17—24no—
312C2×D17—28no—
313C2×D17—24no—
314C2×D17—24no—
315C2×D17—24no—
316C2×D17—24no—
317C2×D17#37618yes—
318C4×C17—24no—
319C4×C17—44no—
320C4×C17—42no—
321C4×C17—44no—
322C4×C17—42no—
323Dic17—24no—
324Dic17—44no—
325Dic17—42no—
326Dic17—44no—
327Dic17—42no—
328C24×C4—171no—
329C24×C4—172no—
330C22×C22⋊C4—171no—
331C22×C22⋊C4—171no—
332C22×C22⋊C4—172no—
333C22×C42—172no—
334C22×C42—171no—
335C22×C42C2×D1711yescharacteristic
336C4×C22⋊C4—178no—
337C4×C22⋊C4—178no—
338C2×#95—172no—
339C2×#95—171no—
340C2×#95—171no—
341C2×#95—171no—
342C2×#95—171no—
343C2×#95—178no—
344C2×#95—172no—
345C2×#95—172no—
346C2×#95—172no—
347C2×#95—172no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95C2×D1712yes—
351C2×#95C2×D1712yes—
352C2×#95C2×D1711yescharacteristic
353C2×#95C2×D1711yescharacteristic
354C2×C17C2×#37611yescharacteristic
355C2×C17#227311yescharacteristic
356C2×C17#227411yescharacteristic
357C2×C17#231212yes—
358C2×C17#231312yes—
359C2×C17#231811yescharacteristic
360C2×C17#231812yes—
361C2×C17#231811yescharacteristic
362C2×C17#231812yes—
363C2×C17#231811yescharacteristic
364C2×C17#231811yescharacteristic
365D17—28no—
366C25—171no—
367C23×C4—171no—
368C23×C4—174no—
369C23×C4—174no—
370C23×C4—174no—
371C23×C4—171no—
372C23×C4—172no—
373C23×C4—171no—
374C23×C4—172no—
375C23×C4—178no—
376C23×C4—344no—
377C23×C4—344no—
378C23×C4—682no—
379C23×C4—682no—
380C23×C4—682no—
381C23×C4—682no—
382C23×C4—682no—
383C23×C4—682no—
384C23×C4C22×D1711yescharacteristic
385C23×C4C22×D1711yescharacteristic
386C23×C4C22×D1712yes—
387C23×C4C22×D1712yes—
388C23×C4C22×D1711yescharacteristic
389C2×C22⋊C4—174no—
390C2×C22⋊C4—174no—
391C2×C22⋊C4—178no—
392C2×C22⋊C4—344no—
393C2×C22⋊C4—344no—
394C2×C22⋊C4—344no—
395C2×C22⋊C4—344no—
396C2×C42—42no—
397C2×C42—178no—
398C2×C42—174no—
399C2×C42—682no—
400C2×C42—682no—
401C2×C42—682no—
402C2×C42C4×D1714yescomplemented
403#95—42no—
404#95—42no—
405#95—178no—
406#95—174no—
407#95—174no—
408#95—174no—
409#95—174no—
410#95—682no—
411#95—682no—
412#95—682no—
413#95—682no—
414#95—682no—
415#95—682no—
416#95C4×D1714yes—
417#95C4×D1714yescomplemented
418C17#1379611yescharacteristic, complemented, Sylow
419C24—172no—
420C24—172no—
421C24—172no—
422C24—174no—
423C24—174no—
424C24—342no—
425C24—341no—
426C24—341no—
427C24—342no—
428C24—341no—
429C24—341no—
430C24C23×D1711yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—172no—
454C22×C4—172no—
455C22×C4—172no—
456C22×C4—1716no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—172no—
460C22×C4—172no—
461C22×C4—3416no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—341no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—341no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—341no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—341no—
484C22×C4—688no—
485C22×C4—688no—
486C22×C4—684no—
487C22×C4—684no—
488C22×C4—688no—
489C22×C4—688no—
490C22×C4—688no—
491C22×C4—688no—
492C22×C4—688no—
493C22×C4—688no—
494C22×C4C2×C4×D1714yes—
495C22×C4C2×C4×D1712yes—
496C22×C4C2×C4×D1712yes—
497C22×C4C2×C4×D1714yes—
498C22×C4C2×C4×D1712yes—
499C22×C4C2×C4×D1712yes—
500C22×C4C2×C4×D1712yes—
501C22⋊C4—348no—
502C22⋊C4—3416no—
503C22⋊C4—348no—
504C42—44no—
505C42—684no—
506C42—688no—
507C23—174no—
508C23—174no—
509C23—178no—
510C23—174no—
511C23—178no—
512C23—342no—
513C23—344no—
514C23—344no—
515C23—342no—
516C23—342no—
517C23—342no—
518C23—342no—
519C23—342no—
520C23—342no—
521C23—342no—
522C23—344no—
523C23—344no—
524C23—344no—
525C23—344no—
526C23—344no—
527C23—344no—
528C23—344no—
529C23—344no—
530C23C22×C4×D1711yescharacteristic
531C23C22×C4×D1711yescharacteristic
532C23C22×C4×D1711yescharacteristic
533C23#1058769112yes—
534C23#1058769111yescharacteristic
535C23#1058769411yescharacteristic
536C23#41×D1712yes—
537C23#41×D1712yes—
538C23#1058769612yes—
539C23#1058769611yescharacteristic
540C23#1058769711yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
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—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—174no—
570C2×C4—344no—
571C2×C4—344no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—344no—
576C2×C4—342no—
577C2×C4—344no—
578C2×C4—344no—
579C2×C4—344no—
580C2×C4—344no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—344no—
586C2×C4—6816no—
587C2×C4—6832no—
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×C4C42×D1714yes—
601C22—178no—
602C22—348no—
603C22—348no—
604C22—344no—
605C22—344no—
606C22—344no—
607C22—348no—
608C22—344no—
609C22—348no—
610C22—344no—
611C22—344no—
612C22C2×C42×D1711yescharacteristic
613C22#117×D1712yes—
614C22#117×D1711yescharacteristic
615C22#117×D1712yes—
616C22#117×D1711yescharacteristic
617C22#1176328111yescharacteristic
618C22#1176328112yes—
619C22#1176328112yes—
620C22#1176328211yescharacteristic
621C22#1176328311yescharacteristic
622C22#1176328511yescharacteristic
623C22#1176328811yescharacteristic
624C22#1176328911yescharacteristic
625C22#1176329011yescharacteristic
626C22#1176329111yescharacteristic
627C22#1176339112yes—
628C22#1176339312yes—
629C22#1176339512yes—
630C22#1176339812yes—
631C22#1176340012yes—
632C22#1176340212yes—
633C22#1176340512yes—
634C22#1176340712yes—
635C4—24no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—344no—
641C4—684no—
642C4—682no—
643C4—684no—
644C4—682no—
645C2—348no—
646C2D17×#37611yescharacteristic
647C2#1453113611yescharacteristic
648C2#1453113811yescharacteristic
649C2#1453137812yes—
650C2#1453138312yes—
651C2#1453141311yescharacteristic
652C2#1453141412yes—
653C2#1453141511yescharacteristic
654C2#1453141612yes—
655C2#1453141911yescharacteristic
656C2#1453142011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC22×C17C22×C42
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC24×D17C23
hypercenter#1434D17
hypocenterC17#13796
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #14527011C2semidirect
5. #14527010C2semidirect
6. #14527009C2semidirect
7. #14527009C2semidirect
8. #14527008C2semidirect
11. C17×#1434C2semidirect
19. C4×#63227C4semidirect
20. #11762656C4semidirect
32. #95×D17C4semidirect
62. C2×C42×C17C2×C4semidirect
73. C2×C4×Dic17C2×C4semidirect
74. C2×C4×Dic17C2×C4semidirect
78. C17×#95C2×C4semidirect
87. #10587502C2×C4semidirect
88. #10587502C2×C4semidirect
89. #10587502C2×C4semidirect
281. #1434D17semidirect
402. C2×C42C4×D17semidirect
417. #95C4×D17semidirect
418. C17#13796semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632272
C2×#117626562
C2×#95×D172
#145270112
#145270102
#145270092
#145270082
#145270052
C17×#14342
#1379617