Finite Group Databasebeta

← #16962726

Subgroups — #16962726

Order 4352.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762711C211yesmaximal, characteristic
2C2×#11762687C212yesmaximal
3C2×#11762680C212yesmaximal, complemented
4#14527036C212yesmaximal, complemented
5#14527028C212yesmaximal, complemented
6#14527026C212yesmaximal, complemented
7#14527011C211yesmaximal, characteristic, complemented
8#14527005C212yesmaximal, complemented
9C17×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C23×C4×D17C2212yes—
12C22×C4⋊Dic17C2211yescharacteristic
13C22×#63225C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19#11762711C418yescomplemented
20#11762687—28no—
21#11762680—28no—
22C2×#10587502C2212yes—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C2×#10587502C2211yescharacteristic
29C22×C4⋊C4×C17C2212yes—
30C22×C42×C17C2211yescharacteristic
31C2×C17×#95C2212yes—
32C2×C17×#95C2211yescharacteristic
33C2×C17×#95C2211yescharacteristic
34C24×D17C2312yes—
35C23×C4×C17C2312yes—
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2312yes—
38C23×C4×C17C2311yescharacteristic
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2312yes—
41C23×Dic17C2312yes—
42C23×Dic17C2312yes—
43C22×D68—42no—
44C22×D68—42no—
45C22×D68C2×C414yes—
46C22×D68D414yes—
47C22×D68Q814yes—
48C22×C4×D17—24no—
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—42no—
52C22×C4×D17—42no—
53C22×C4×D17—42no—
54C22×C4×D17—42no—
55C2×C4⋊C4×C17—24no—
56C2×C4⋊C4×C17—24no—
57C2×C4⋊C4×C17—42no—
58C2×C42×C17—42no—
59C2×C42×C17C2×C414yescomplemented
60C2×#63227—24no—
61C2×#63227—24no—
62C2×#63227—24no—
63C2×#63227—24no—
64C2×#63227—24no—
65C2×#63227—24no—
66C2×#63227—24no—
67C2×#63227—42no—
68C2×#63227—42no—
69C2×#63227—42no—
70C2×#63227—42no—
71C2×#63227—42no—
72C2×#63227—42no—
73C2×#63227C2×C418yescomplemented
74C2×#63227C2×C418yescomplemented
75C2×C4⋊Dic17—42no—
76C2×C4⋊Dic17C2×C414yescomplemented
77C2×#63225—24no—
78C2×#63225—42no—
79C2×#63225—42no—
80C17×#95—24no—
81C17×#95—41no—
82C17×#95—41no—
83#10587502—24no—
84#10587502—22no—
85#10587502—24no—
86#10587502—42no—
87#10587502—42no—
88#10587502—42no—
89#10587502D414yescomplemented
90C24×C17C2411yescharacteristic
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—
104C23×D17C2×D414yes—
105C23×D17C2×Q814yes—
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—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
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×C17C22×C412yes—
128C22×C4×C17C22×C412yes—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C2×D412yes—
131C22×C4×C17C2×D412yes—
132C22×C4×C17C2×Q812yes—
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×Dic17—22no—
154C22×Dic17C22×C414yes—
155C22×Dic17C2×D414yes—
156C2×D68—44no—
157C2×D68—44no—
158C2×D68—48no—
159C2×D68—48no—
160C2×D68C4⋊C4116yescomplemented
161C2×C4×D17—48no—
162C2×C4×D17—48no—
163C2×C4×D17—48no—
164C2×C4×D17—48no—
165C2×C4×D17—48no—
166C2×C4×D17—48no—
167C2×C4×D17—48no—
168C4⋊C4×C17—48no—
169C42×C17—44no—
170#63227—216no—
171#63227—48no—
172#63227—48no—
173#63227—48no—
174#63227—48no—
175C4⋊Dic17—44no—
176#63225—48no—
177#14438—171nomaximal, Sylow
178C23×C17C23×C411yescharacteristic
179C23×C17C22×D411yescharacteristic
180C23×C17C22×D411yescharacteristic
181C23×C17C22×Q811yescharacteristic
182C23×C17C2×#4112yes—
183C23×C17C2×#4111yescharacteristic
184C23×C17H(5,𝔽2)12yes—
185C23×C17H(5,𝔽2)11yescharacteristic
186C23×C17H(5,𝔽2)12yes—
187C23×C17H(5,𝔽2)12yes—
188C23×C172−1+411yescharacteristic
189C22×D17—24no—
190C22×D17—24no—
191C22×D17—24no—
192C22×D17—24no—
193C22×D17—24no—
194C22×D17—24no—
195C22×D17—24no—
196C22×D17—24no—
197C22×D17—24no—
198C22×D17—24no—
199C22×D17—24no—
200C22×D17—24no—
201C22×D17—24no—
202C22×D17—24no—
203C22×D17—24no—
204C22×D17—24no—
205C22×D17—24no—
206C22×D17—24no—
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—
219C22×D17—42no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—42no—
223C22×D17C2×C4⋊C418yes—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—24no—
229C2×C4×C17—24no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
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×C17—42no—
248C2×C4×C17—42no—
249C2×C4×C17—42no—
250C2×C4×C17—42no—
251C2×C4×C17C2×C4⋊C414yes—
252C2×C4×C17C4×D414yes—
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—
271C2×Dic17—42no—
272C2×Dic17—42no—
273C2×Dic17—42no—
274D68—416no—
275C4×D17—432no—
276C2×#390—171no—
277C2×#386—172no—
278C2×#385—172no—
279#1441—172no—
280#1439—172no—
281#1439—172no—
282C4⋊#95D1711yescharacteristic, complemented
283#1434—171no—
284C22⋊#95—172no—
285C22×C17C22×C4⋊C411yescharacteristic
286C22×C17C2×C4×D411yescharacteristic
287C22×C17C4×#4111yescharacteristic
288C22×C17#51811yescharacteristic
289C22×C17#51811yescharacteristic
290C22×C17#51811yescharacteristic
291C22×C17#52011yescharacteristic
292C22×C17#53412yes—
293C22×C17#53512yes—
294C22×C17#53812yes—
295C22×C17#54012yes—
296C22×C17#54212yes—
297C22×C17#54312yes—
298C22×C17D4211yescharacteristic
299C22×C17#54612yes—
300C22×C17#54612yes—
301C22×C17#54711yescharacteristic
302C22×C17D4×Q811yescharacteristic
303C22×C17#55312yes—
304C22×C17#55412yes—
305C22×C17#55411yescharacteristic
306C22×C17#56012yes—
307C22×C17#56212yes—
308C2×D17—28no—
309C2×D17—28no—
310C2×D17—28no—
311C2×D17—44no—
312C2×D17—44no—
313C2×D17—44no—
314C2×D17—44no—
315C2×D17—44no—
316C2×D17—44no—
317C2×D17—44no—
318C2×D17—44no—
319C2×D17—44no—
320C2×D17—44no—
321C2×D17—44no—
322C2×D17—44no—
323C4×C17—24no—
324C4×C17—24no—
325C4×C17—42no—
326C4×C17—44no—
327C4×C17—82no—
328Dic17—44no—
329Dic17—44no—
330Dic17—82no—
331C23×D4—171no—
332C24×C4—172no—
333C22×C4⋊C4—171no—
334C22×C4⋊C4—172no—
335C22×C4⋊C4C2×D1712yes—
336C22×C22⋊C4—172no—
337C22×C22⋊C4—172no—
338C22×C22⋊C4—172no—
339C22×C22⋊C4—172no—
340C22×C22⋊C4—172no—
341C22×C42C2×D1711yescharacteristic
342#390—178no—
343#386—348no—
344#385—348no—
345C2×#95—171no—
346C2×#95—172no—
347C2×#95—171no—
348C2×#95—172no—
349C2×#95—172no—
350C2×#95—172no—
351C2×#95—172no—
352C2×#95C2×D1712yes—
353C2×#95C2×D1711yescharacteristic
354C2×#95C2×D1711yescharacteristic
355C2×C17#231511yescharacteristic
356C2×C17#232811yescharacteristic
357C2×C17#233711yescharacteristic
358C2×C17D4×C4⋊C411yescharacteristic
359C2×C17#234911yescharacteristic
360C2×C17#236411yescharacteristic
361C2×C17#237711yescharacteristic
362C2×C17#267812yes—
363C2×C17#269212yes—
364C2×C17#270112yes—
365C2×C17#278012yes—
366D17—48no—
367C25—172no—
368C22×D4—174no—
369C22×D4—174no—
370C22×D4—174no—
371C22×D4—682no—
372C22×D4—682no—
373C23×C4—172no—
374C23×C4—172no—
375C23×C4—172no—
376C23×C4—344no—
377C23×C4—344no—
378C23×C4—344no—
379C23×C4—682no—
380C23×C4—682no—
381C23×C4—682no—
382C23×C4—682no—
383C23×C4C22×D1712yes—
384C23×C4C22×D1711yescharacteristic
385C23×C4C22×D1712yes—
386C23×C4C22×D1711yescharacteristic
387C23×C4C22×D1711yescharacteristic
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—42no—
391C2×C4⋊C4—174no—
392C2×C4⋊C4—344no—
393C2×C4⋊C4—682no—
394C2×C4⋊C4—682no—
395C2×C4⋊C4—682no—
396C2×C22⋊C4—178no—
397C2×C22⋊C4—178no—
398C2×C22⋊C4—344no—
399C2×C22⋊C4—344no—
400C2×C22⋊C4—344no—
401C2×C22⋊C4—344no—
402C2×C22⋊C4—344no—
403C2×C22⋊C4—344no—
404C2×C22⋊C4—344no—
405C2×C22⋊C4—682no—
406C2×C22⋊C4—682no—
407C2×C22⋊C4—682no—
408C2×C22⋊C4—682no—
409C2×C22⋊C4—682no—
410C2×C22⋊C4—682no—
411C2×C42—42no—
412C2×C42C4×D1714yescomplemented
413#95—24no—
414#95—41no—
415#95—41no—
416#95—174no—
417#95—344no—
418#95—344no—
419#95—342no—
420#95—682no—
421#95—682no—
422#95—682no—
423C17#1443811yescharacteristic, complemented, Sylow
424C24—174no—
425C24—174no—
426C24—174no—
427C24—342no—
428C24—342no—
429C24—342no—
430C24—342no—
431C24—342no—
432C24—342no—
433C24—342no—
434C24—342no—
435C24—342no—
436C24—342no—
437C24—342no—
438C24—342no—
439C24C23×D1711yescharacteristic
440C2×D4—1716no—
441C2×D4—684no—
442C2×D4—684no—
443C2×D4—688no—
444C2×D4—688no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
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—22no—
465C22×C4—22no—
466C22×C4—174no—
467C22×C4—174no—
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—688no—
488C22×C4—688no—
489C22×C4—688no—
490C22×C4—688no—
491C22×C4—688no—
492C22×C4—688no—
493C22×C4—688no—
494C22×C4C2×C4×D1712yes—
495C22×C4C2×C4×D1712yes—
496C22×C4C2×C4×D1712yes—
497C22×C4#6325214yes—
498C22×C4D4×D1712yes—
499C22×C4D4×D1712yes—
500C22×C4Q8×D1712yes—
501C22×C4#6325612yes—
502C4⋊C4—48no—
503C4⋊C4—684no—
504C4⋊C4—688no—
505C22⋊C4—3416no—
506C22⋊C4—688no—
507C22⋊C4—688no—
508C22⋊C4—688no—
509C22⋊C4—688no—
510C42—44no—
511C23—178no—
512C23—344no—
513C23—344no—
514C23—344no—
515C23—344no—
516C23—344no—
517C23—344no—
518C23—344no—
519C23—344no—
520C23—344no—
521C23—344no—
522C23—344no—
523C23—344no—
524C23—344no—
525C23—344no—
526C23—344no—
527C23—344no—
528C23—344no—
529C23—344no—
530C23—682no—
531C23—682no—
532C23—682no—
533C23—682no—
534C23—682no—
535C23—682no—
536C23—682no—
537C23—682no—
538C23—682no—
539C23—682no—
540C23—682no—
541C23—682no—
542C23—682no—
543C23—682no—
544C23—682no—
545C23—682no—
546C23C22×C4×D1711yescharacteristic
547C23C2×#6325212yes—
548C23C2×D4×D1711yescharacteristic
549C23C2×D4×D1711yescharacteristic
550C23#1058769112yes—
551C23#1058769111yescharacteristic
552C23C2×Q8×D1711yescharacteristic
553C23C2×#6325611yescharacteristic
554C23#1058769411yescharacteristic
555C23#1058769612yes—
556C23#1058769612yes—
557D4—6816no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—344no—
586C2×C4—344no—
587C2×C4—344no—
588C2×C4—344no—
589C2×C4—344no—
590C2×C4—344no—
591C2×C4—6832no—
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×C4C4⋊C4×D1714yes—
608C2×C4#1058757914yes—
609C22—348no—
610C22—348no—
611C22—348no—
612C22—684no—
613C22—684no—
614C22—684no—
615C22—684no—
616C22—684no—
617C22—684no—
618C22—684no—
619C22—684no—
620C22—684no—
621C22—684no—
622C22—684no—
623C22—684no—
624C22C2×C4⋊C4×D1711yescharacteristic
625C22C2×#1058757911yescharacteristic
626C22C4×#6325211yescharacteristic
627C22#1176328211yescharacteristic
628C22#1176328311yescharacteristic
629C22#1176328311yescharacteristic
630C22#1176329011yescharacteristic
631C22#1176336712yes—
632C22#1176337712yes—
633C22#1176339212yes—
634C22#1176341512yes—
635C22#1176343112yes—
636C22#1176343412yes—
637C22#1176344411yescharacteristic
638C22#1176344912yes—
639C22#1176345512yes—
640C22#1176346211yescharacteristic
641C22#1176347411yescharacteristic
642C22#1176350712yes—
643C22#1176351912yes—
644C22#1176352011yescharacteristic
645C22#1176354712yes—
646C22#1176355512yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—82no—
652C4—684no—
653C4—684no—
654C4—1362no—
655C2—688no—
656C2#1453139711yescharacteristic
657C2#1453150711yescharacteristic
658C2#1453159711yescharacteristic
659C2#1453168411yescharacteristic
660C2#1453169511yescharacteristic
661C2#1453183111yescharacteristic
662C2#1453193811yescharacteristic
663C2#1453510612yes—
664C2#1453527512yes—
665C2#1453539212yes—
666C2#1453642612yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×C4⋊#95C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenterC4⋊#95D17
hypocenterC17#14438
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#11762680C2semidirect
4. #14527036C2semidirect
5. #14527028C2semidirect
6. #14527026C2semidirect
7. #14527011C2semidirect
8. #14527005C2semidirect
9. C17×C4⋊#95C2semidirect
19. #11762711C4semidirect
59. C2×C42×C17C2×C4semidirect
73. C2×#63227C2×C4semidirect
74. C2×#63227C2×C4semidirect
76. C2×C4⋊Dic17C2×C4semidirect
89. #10587502D4semidirect
160. C2×D68C4⋊C4semidirect
282. C4⋊#95D17semidirect
412. C2×C42C4×D17semidirect
423. C17#14438semidirect

Maximal subgroups

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

subgroupindex
C2×#117627112
C2×#117626872
C2×#117626802
#145270362
#145270282
#145270262
#145270112
#145270052
C17×C4⋊#952
#1443817