Finite Group Databasebeta

← #16965392

Subgroups — #16965392

Order 4352.

1494 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762711C211yesmaximal, characteristic
2#14527036C212yesmaximal, complemented
3#14527032C212yesmaximal, complemented
4#14527032C212yesmaximal, complemented
5#14527028C212yesmaximal, complemented
6#14527028C212yesmaximal, complemented
7#14527024C212yesmaximal
8#14527011C211yesmaximal, characteristic, complemented
9C17×#1434C211yesmaximal, characteristic, complemented
10C23×D68C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×#63225C2212yes—
13C22×#63227C2212yes—
14C22×#63227C2212yes—
15C22×#63227C2212yes—
16C22×#63227C2212yes—
17C22×#63227C2212yes—
18C22×#63227C2212yes—
19#11762711C418yescomplemented
20C2×#10587502C2212yes—
21C2×#10587502C2211yescharacteristic
22C2×#10587502C2212yes—
23C2×#10587502C2211yescharacteristic
24C2×#10587502C2212yes—
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C22×C42×C17C2211yescharacteristic
28C2×C17×#95C2212yes—
29C2×C17×#95C2211yescharacteristic
30C2×C17×#95C2212yes—
31C2×C17×#95C2211yescharacteristic
32C24×D17C2312yes—
33C23×C4×C17C2312yes—
34C23×C4×C17C2312yes—
35C23×C4×C17C2311yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2311yescharacteristic
38C23×Dic17C2312yes—
39C23×Dic17C2312yes—
40C23×Dic17C2312yes—
41C22×D68—24no—
42C22×D68—41no—
43C22×D68—41no—
44C22×D68—42no—
45C22×D68C2×C414yes—
46C2×C42×C17—42no—
47C2×C42×C17C2×C414yes—
48C2×#63227—42no—
49C2×#63227—42no—
50C2×#63227—42no—
51C2×#63227—42no—
52C2×#63227—42no—
53C2×#63227—42no—
54C2×#63227—42no—
55C2×#63227—42no—
56C2×#63227—42no—
57C2×#63227—42no—
58C2×#63227—42no—
59C2×#63227—42no—
60C2×#63227—42no—
61C2×#63227—42no—
62C2×#63227—42no—
63C2×#63227—42no—
64C2×#63227C2×C418yes—
65C2×#63227C2×C418yes—
66C2×C4⋊Dic17—41no—
67C2×C4⋊Dic17—41no—
68C2×C4⋊Dic17C2×C414yes—
69C2×#63225—42no—
70C2×#63225—42no—
71C2×#63225—42no—
72C17×#95—24no—
73C17×#95—41no—
74C17×#95—41no—
75C17×#95—42no—
76#10587502—24no—
77#10587502—42no—
78#10587502—42no—
79#10587502—42no—
80#10587502—41no—
81#10587502—41no—
82#10587502—42no—
83C24×C17C2411yescharacteristic
84C23×D17—22no—
85C23×D17—22no—
86C23×D17—22no—
87C23×D17—22no—
88C23×D17—22no—
89C23×D17—22no—
90C23×D17—22no—
91C23×D17—22no—
92C23×D17—22no—
93C23×D17—22no—
94C23×D17—22no—
95C23×D17—22no—
96C23×D17—22no—
97C23×D17—22no—
98C23×D17C22×C414yes—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—22no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—21no—
109C22×C4×C17—21no—
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—22no—
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×C17C22×C412yes—
125C22×C4×C17C22×C412yes—
126C22×C4×C17C22×C412yes—
127C22×C4×C17#4114yes—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
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—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17C22×C414yes—
149C2×D68—28no—
150C2×D68—44no—
151C2×D68—44no—
152C2×D68—48no—
153C2×D68—48no—
154C42×C17—44no—
155#63227—48no—
156#63227—48no—
157#63227—48no—
158#63227—48no—
159#63227—48no—
160#63227—48no—
161C4⋊Dic17—44no—
162#63225—48no—
163#14667—171nomaximal, Sylow
164C23×C17C23×C411yescharacteristic
165C23×C17C2×#4112yes—
166C23×C17H(5,𝔽2)12yes—
167C23×C17H(5,𝔽2)11yescharacteristic
168C23×C17H(5,𝔽2)11yescharacteristic
169C23×C17H(5,𝔽2)11yescharacteristic
170C23×C17H(5,𝔽2)12yes—
171C23×C17H(5,𝔽2)12yes—
172C23×C17H(5,𝔽2)11yescharacteristic
173C23×C172−1+411yescharacteristic
174C23×C172−1+411yescharacteristic
175C22×D17—24no—
176C22×D17—24no—
177C22×D17—24no—
178C22×D17—24no—
179C22×D17—24no—
180C22×D17—24no—
181C22×D17—24no—
182C22×D17—42no—
183C22×D17—42no—
184C22×D17—42no—
185C22×D17—42no—
186C22×D17—42no—
187C22×D17—42no—
188C22×D17—42no—
189C22×D17—42no—
190C22×D17—42no—
191C22×D17—42no—
192C22×D17—42no—
193C22×D17—42no—
194C22×D17—42no—
195C22×D17—42no—
196C22×D17—42no—
197C22×D17—42no—
198C22×D17—42no—
199C22×D17—42no—
200C22×D17—42no—
201C22×D17—42no—
202C22×D17—42no—
203C22×D17—42no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—24no—
213C2×C4×C17—24no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
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#11714yes—
239C2×Dic17—24no—
240C2×Dic17—24no—
241C2×Dic17—24no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259C2×Dic17—42no—
260D68—416no—
261C2×#390—171no—
262#1441—172no—
263#1440—172no—
264#1440—172no—
265#1439—172no—
266#1439—172no—
267#1438—172no—
268#1434—171no—
269#1434D1711yescharacteristic, complemented
270C22×C17C2×#11711yescharacteristic
271C22×C17#51811yescharacteristic
272C22×C17#51811yescharacteristic
273C22×C17#51811yescharacteristic
274C22×C17#51811yescharacteristic
275C22×C17#52011yescharacteristic
276C22×C17#52011yescharacteristic
277C22×C17#53812yes—
278C22×C17#53812yes—
279C22×C17#54012yes—
280C22×C17#54012yes—
281C22×C17#54212yes—
282C22×C17#54212yes—
283C22×C17#56012yes—
284C22×C17#56012yes—
285C22×C17#56012yes—
286C22×C17#56012yes—
287C22×C17H(3,𝔽4)11yescharacteristic
288C22×C17#56211yescharacteristic
289C22×C17#56212yes—
290C22×C17#56212yes—
291C22×C17#56211yescharacteristic
292C22×C17#56311yescharacteristic
293C2×D17—44no—
294C2×D17—44no—
295C2×D17—44no—
296C2×D17—44no—
297C2×D17—44no—
298C2×D17—44no—
299C2×D17—44no—
300C2×D17—82no—
301C2×D17—82no—
302C2×D17—82no—
303C2×D17—82no—
304C2×D17—82no—
305C2×D17—82no—
306C2×D17—82no—
307C2×D17—82no—
308C4×C17—24no—
309C4×C17—42no—
310C4×C17—42no—
311C4×C17—82no—
312C4×C17—82no—
313Dic17—44no—
314Dic17—82no—
315Dic17—82no—
316C23×D4—171no—
317C22×C4⋊C4—171no—
318C22×C4⋊C4—172no—
319C22×C22⋊C4—172no—
320C22×C22⋊C4—172no—
321C22×C22⋊C4—172no—
322C22×C22⋊C4—172no—
323C22×C22⋊C4—172no—
324C22×C22⋊C4—172no—
325C22×C42C2×D1711yescharacteristic
326#390—178no—
327C2×#95—172no—
328C2×#95—171no—
329C2×#95—172no—
330C2×#95—171no—
331C2×#95—172no—
332C2×#95—172no—
333C2×#95—172no—
334C2×#95C2×D1712yes—
335C2×#95C2×D1711yescharacteristic
336C2×#95C2×D1712yes—
337C2×#95C2×D1711yescharacteristic
338C2×C17#231211yescharacteristic
339C2×C17#231911yescharacteristic
340C2×C17#231911yescharacteristic
341C2×C17#237711yescharacteristic
342C2×C17#237711yescharacteristic
343C2×C17#237911yescharacteristic
344C2×C17#238011yescharacteristic
345C2×C17#282812yes—
346C2×C17#282912yes—
347C2×C17#282912yes—
348C2×C17#283012yes—
349D17—84no—
350C25—172no—
351C22×D4—174no—
352C22×D4—344no—
353C22×D4—682no—
354C22×D4—681no—
355C22×D4—681no—
356C23×C4—172no—
357C23×C4—172no—
358C23×C4—172no—
359C23×C4C22×D1712yes—
360C23×C4C22×D1712yes—
361C23×C4C22×D1711yescharacteristic
362C23×C4C22×D1711yescharacteristic
363C23×C4C22×D1711yescharacteristic
364C2×C4⋊C4—174no—
365C2×C4⋊C4—682no—
366C2×C4⋊C4—682no—
367C2×C4⋊C4—682no—
368C2×C4⋊C4—681no—
369C2×C4⋊C4—681no—
370C2×C22⋊C4—178no—
371C2×C22⋊C4—178no—
372C2×C22⋊C4—682no—
373C2×C22⋊C4—682no—
374C2×C22⋊C4—682no—
375C2×C22⋊C4—682no—
376C2×C22⋊C4—682no—
377C2×C22⋊C4—682no—
378C2×C22⋊C4—682no—
379C2×C22⋊C4—682no—
380C2×C22⋊C4—682no—
381C2×C22⋊C4—682no—
382C2×C22⋊C4—682no—
383C2×C22⋊C4—682no—
384C2×C22⋊C4—682no—
385C2×C22⋊C4—682no—
386C2×C22⋊C4—682no—
387C2×C22⋊C4—682no—
388C2×C42—42no—
389C2×C42C4×D1714yes—
390#95—24no—
391#95—41no—
392#95—41no—
393#95—42no—
394#95—344no—
395#95—682no—
396#95—681no—
397#95—681no—
398#95—682no—
399#95—682no—
400#95—682no—
401C17#1466711yescharacteristic, complemented, Sylow
402C24—174no—
403C24—342no—
404C24—342no—
405C24—342no—
406C24—342no—
407C24—342no—
408C24—342no—
409C24—342no—
410C24—342no—
411C24—342no—
412C24—342no—
413C24—342no—
414C24—342no—
415C24—342no—
416C24—342no—
417C24C23×D1711yescharacteristic
418C2×D4—348no—
419C2×D4—684no—
420C2×D4—684no—
421C2×D4—688no—
422C2×D4—688no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
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—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—174no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—342no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4—342no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4C2×C4×D1712yes—
470C22×C4C2×C4×D1712yes—
471C22×C4C2×C4×D1712yes—
472C22×C4#6325214yes—
473C4⋊C4—684no—
474C4⋊C4—688no—
475C22⋊C4—688no—
476C22⋊C4—688no—
477C22⋊C4—688no—
478C22⋊C4—688no—
479C22⋊C4—688no—
480C22⋊C4—688no—
481C42—44no—
482C23—344no—
483C23—344no—
484C23—344no—
485C23—344no—
486C23—344no—
487C23—344no—
488C23—344no—
489C23—682no—
490C23—682no—
491C23—682no—
492C23—682no—
493C23—682no—
494C23—682no—
495C23—682no—
496C23—682no—
497C23—682no—
498C23—682no—
499C23—682no—
500C23—682no—
501C23—682no—
502C23—682no—
503C23—682no—
504C23—682no—
505C23—682no—
506C23—682no—
507C23—682no—
508C23—682no—
509C23—682no—
510C23—682no—
511C23—682no—
512C23—682no—
513C23—682no—
514C23—682no—
515C23—682no—
516C23—682no—
517C23C22×C4×D1711yescharacteristic
518C23C2×#6325212yes—
519C23#1058769112yes—
520C23#1058769111yescharacteristic
521C23#1058769411yescharacteristic
522C23#1058769612yes—
523C23#1058769612yes—
524C23#1058769611yescharacteristic
525C23#1058769611yescharacteristic
526C23#1058769611yescharacteristic
527C23#1058769711yescharacteristic
528D4—6816no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—344no—
558C2×C4—344no—
559C2×C4—344no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4—682no—
568C2×C4—682no—
569C2×C4—682no—
570C2×C4—682no—
571C2×C4—682no—
572C2×C4—682no—
573C2×C4—682no—
574C2×C4—682no—
575C2×C4—682no—
576C2×C4—682no—
577C2×C4—682no—
578C2×C4#1058755814yes—
579C22—684no—
580C22—684no—
581C22—684no—
582C22—684no—
583C22—684no—
584C22—684no—
585C22—684no—
586C22—1362no—
587C22—1362no—
588C22—1362no—
589C22—1362no—
590C22—1362no—
591C22—1362no—
592C22—1362no—
593C22—1362no—
594C22C2×#1058755811yescharacteristic
595C22#1176328211yescharacteristic
596C22#1176328311yescharacteristic
597C22#1176328311yescharacteristic
598C22#1176328311yescharacteristic
599C22#1176328911yescharacteristic
600C22#1176329011yescharacteristic
601C22#1176339212yes—
602C22#1176339212yes—
603C22#1176341012yes—
604C22#1176341412yes—
605C22#1176343012yes—
606C22#1176343112yes—
607C22#1176354712yes—
608C22#1176354912yes—
609C22#1176355012yes—
610C22#1176355012yes—
611C22#1176355111yescharacteristic
612C22#1176355311yescharacteristic
613C22#1176355512yes—
614C22#1176355612yes—
615C22#1176355711yescharacteristic
616C22#1176356311yescharacteristic
617C4—24no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—684no—
623C4—1362no—
624C4—1362no—
625C2—1364no—
626C2#1453137911yescharacteristic
627C2#1453142211yescharacteristic
628C2#1453142811yescharacteristic
629C2#1453193811yescharacteristic
630C2#1453194311yescharacteristic
631C2#1453195511yescharacteristic
632C2#1453196011yescharacteristic
633C2#1453690112yes—
634C2#1453691012yes—
635C2#1453691412yes—
636C2#1453692512yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1434C2
socle, cosocleC24×C17C24
prime radicalC23×D68C22
hypercenter#1434D17
hypocenterC17#14667
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #14527036C2semidirect
3. #14527032C2semidirect
4. #14527032C2semidirect
5. #14527028C2semidirect
6. #14527028C2semidirect
8. #14527011C2semidirect
9. C17×#1434C2semidirect
19. #11762711C4semidirect
269. #1434D17semidirect
401. C17#14667semidirect

Maximal subgroups

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

subgroupindex
C2×#117627112
#145270362
#145270322
#145270282
#145270242
#145270112
C17×#14342
#1466717