Finite Group Databasebeta

← #17039392

Subgroups — #17039392

Order 4352.

1290 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763345C211yesmaximal, characteristic
2C2×#11762859C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
4#14529738C212yesmaximal, complemented
5#14529713C212yesmaximal, complemented
6#14529101C211yesmaximal, characteristic
7#14529099C211yesmaximal, characteristic
8#14529092C211yesmaximal, characteristic, complemented
9#103×Dic17C212yesmaximal, complemented
10#14528598C212yesmaximal, complemented
11C17×#1872C211yesmaximal, characteristic, complemented
12#11763345—24no—
13#11763345C418yescomplemented
14C2×#10587584—21no—
15C2×#10587576C2211yescharacteristic
16C2×#10587633C2211yescharacteristic
17C2×#10587572—21no—
18C2×#10587616C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×#10587579C2211yescharacteristic
21C2×#10587567—21no—
22C2×#10587578—21no—
23C22×C4⋊Dic17C2211yescharacteristic
24C22×C4×Dic17—21no—
25C22×C4×Dic17C2211yescharacteristic
26#11762859—24no—
27C2×#10587485C2212yes—
28C2×#10587506C2212yes—
29C2×#10587492C2212yes—
30C2×#10587480C2212yes—
31C2×C22⋊#3622C2211yescharacteristic
32C2×C8×Dic17C2212yes—
33C4⋊C4×Dic17—21no—
34C4⋊C4×Dic17—21no—
35C4⋊C4×Dic17C2212yes—
36C4⋊C4×Dic17C2212yes—
37C2×#10587502—21no—
38#11762572C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C2×C17×C22⋊Q8C2211yescharacteristic
42C22×C4⋊C4×C17C2211yescharacteristic
43C2×C17×#103C2212yes—
44C2×C17×C22⋊C8C2211yescharacteristic
45C17×#340C2212yes—
46C22×Q8×C17C2311yescharacteristic
47C23×C4×C17—21no—
48C23×C4×C17C2311yescharacteristic
49C22×C22⋊D17—21no—
50C23×Dic17—21no—
51C23×Dic17D411yescharacteristic
52C23×Dic17D411yescharacteristic
53C22×D68C2311yescharacteristic
54C22×C4×D17—21no—
55C22×C8×C17C2312yes—
56C17×C22⋊Q8—22no—
57C17×C22⋊Q8C2×C414yes—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—21no—
60C2×C4⋊C4×C17—24no—
61C2×C4⋊C4×C17C2311yescharacteristic
62C2×C4⋊C4×C17C2312yes—
63C2×C22⋊C4×C17—21no—
64#10587633—22no—
65#10587633C2×C414yescomplemented
66Q8×Dic17—22no—
67Q8×Dic17C2×C414yescomplemented
68#10587616—22no—
69#10587616C2×C414yes—
70C2×#63227—21no—
71C2×#63227—21no—
72C2×#63227—21no—
73C2×#63227—21no—
74C2×C4⋊Dic17—24no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2312yes—
80C2×#63225—21no—
81C2×C4×Dic17—21no—
82C2×C4×Dic17—22no—
83C2×C4×Dic17—22no—
84C2×C4×Dic17—44no—
85C2×C4×Dic17C2×C414yes—
86C2×C4×Dic17D414yes—
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2312yes—
92#10587584—28no—
93#10587579—22no—
94#10587579C2×C414yescomplemented
95#10587578—28no—
96#10587576—22no—
97#10587576C2×C414yescomplemented
98#10587572—28no—
99#10587567—28no—
100C17×#103—24no—
101C17×C22⋊C8—22no—
102#10587506—24no—
103#10587502—22no—
104#10587502—22no—
105C22⋊#3622—22no—
106#10587492—24no—
107#10587485—24no—
108C8×Dic17—24no—
109#10587480—24no—
110C24×C17C2×D411yescharacteristic
111C23×D17—21no—
112C2×Q8×C17—21no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17C22×C412yes—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—24no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—21no—
122C22×C4×C17—42no—
123C22×C4×C17—42no—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D412yes—
127C22×C4×C17C2411yescharacteristic
128C2×C22⋊D17—24no—
129C2×C22⋊D17—42no—
130C2×C22⋊D17—42no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—24no—
136C22×Dic17—22no—
137C22×Dic17—21no—
138C22×Dic17—42no—
139C22×Dic17—42no—
140C22×Dic17C22⋊C414yes—
141C22×Dic17C22⋊C414yes—
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C22×Dic17C2×D411yescharacteristic
146C2×D68—22no—
147C2×D68—22no—
148C2×D68—21no—
149C2×D68C22×C412yes—
150C2×C4×D17—24no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C8×C17—22no—
154C2×C8×C17—22no—
155C2×C8×C17—22no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—22no—
160C4⋊C4×C17C22×C412yes—
161C4⋊C4×C17#4114yes—
162C22⋊C4×C17—24no—
163#63227—24no—
164#63227—24no—
165#63227—24no—
166#63227—24no—
167C4⋊Dic17—24no—
168C4⋊Dic17—21no—
169C4⋊Dic17—22no—
170C4⋊Dic17—24no—
171C4⋊Dic17—24no—
172C4⋊Dic17C22×C412yes—
173C4⋊Dic17#4114yes—
174#63225—24no—
175C4×Dic17—24no—
176C4×Dic17—44no—
177C4×Dic17—44no—
178C4×Dic17C22⋊C414yes—
179C4×Dic17C22⋊C414yes—
180C4×Dic17C22×C412yes—
181C4×Dic17C22×C412yes—
182C4×Dic17C2×D412yes—
183C4×Dic17C2×D412yes—
184C2×#3622—22no—
185C2×#3622—22no—
186C2×#3622—22no—
187#21388—171nomaximal, Sylow
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—21no—
191C23×C17—21no—
192C23×C17C2×C22⋊C412yes—
193C23×C17#13512yes—
194C23×C17#13512yes—
195C23×C17C22×D411yescharacteristic
196C22×D17—22no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—41no—
203Q8×C17—24no—
204Q8×C17—42no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—24no—
208C2×C4×C17—24no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—24no—
212C2×C4×C17—48no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—42no—
216C2×C4×C17—41no—
217C2×C4×C17—42no—
218C2×C4×C17—41no—
219C2×C4×C17C2×C22⋊C412yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C22×D411yescharacteristic
222C2×C4×C17C2×#4112yes—
223C2×C4×C17C2×#4112yes—
224C22⋊D17—48no—
225C2×Dic17—21no—
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—28no—
232C2×Dic17—28no—
233C2×Dic17—22no—
234C2×Dic17—24no—
235C2×Dic17—48no—
236C2×Dic17—42no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—42no—
240C2×Dic17C2×C22⋊C412yes—
241C2×Dic17C2×C22⋊C412yes—
242C2×Dic17C2×C22⋊C412yes—
243C2×Dic17C2×C22⋊C412yes—
244C2×Dic17AGL(1,ℤ/8)12yes—
245C2×Dic17#13712yes—
246D68—24no—
247D68—42no—
248C4×D17—48no—
249C8×C17—42no—
250#3622—42no—
251C2×#529—171no—
252C2×#431—171no—
253C2×C4×C4⋊C4—171no—
254#1989—172no—
255#1983—172no—
256#1874—171no—
257#1874—171no—
258#1872—171no—
259#1872D1711yescharacteristic, complemented
260C4×#103—172no—
261C4×#102—172no—
262C22×C17—22no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17#41614yes—
269C22×C17C22×C22⋊C411yescharacteristic
270C22×C17#52511yescharacteristic
271C22×C17#52511yescharacteristic
272C22×C17C2×#13511yescharacteristic
273C22×C17C2×#13511yescharacteristic
274C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C17C2×#13711yescharacteristic
276C2×D17—42no—
277C2×D17—42no—
278C2×D17—42no—
279C2×D17—81no—
280C2×D17—81no—
281C2×D17—81no—
282C2×D17—81no—
283C4×C17—22no—
284C4×C17—44no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17C4×#4112yes—
288C4×C17#53212yes—
289Dic17—22no—
290Dic17—22no—
291Dic17—22no—
292Dic17—44no—
293Dic17—42no—
294Dic17#41714yes—
295#529—178no—
296#529—344no—
297C2×#126—341no—
298C2×#125—171no—
299C2×#124—171no—
300C2×#123—341no—
301C2×C22⋊Q8C2×D1711yescharacteristic
302C2×#121—171no—
303C2×C4×Q8—171no—
304C2×C4×D4—171no—
305C2×C4×D4—341no—
306C2×#117—341no—
307C22×C4⋊C4—171no—
308C22×C4⋊C4C2×D1711yescharacteristic
309C22×C42—171no—
310C22×C42—341no—
311#431—344no—
312C2×C4⋊C8—172no—
313C2×#103—172no—
314C2×#103C2×D1712yes—
315C2×#102—172no—
316C2×#102—172no—
317C2×C22⋊C8—171no—
318C2×C22⋊C8C2×D1711yescharacteristic
319C2×C4×C8—172no—
320C4×C4⋊C4—172no—
321C4×C4⋊C4—172no—
322C4×C4⋊C4—341no—
323C4×C4⋊C4—341no—
324C2×#95—341no—
325#340—172no—
326#340—172no—
327#340—172no—
328#340C2×D1712yes—
329C2×C17—24no—
330C2×C17#229711yescharacteristic
331C2×C17C2×#41611yescharacteristic
332C2×C17C2×#41711yescharacteristic
333C2×C17#310511yescharacteristic
334C2×C17#310511yescharacteristic
335C2×C17#310811yescharacteristic
336C2×C17#310811yescharacteristic
337D17—82no—
338C22×Q8C22×D1711yescharacteristic
339C22×D4—171no—
340C22×D4—341no—
341C23×C4—21no—
342C23×C4—171no—
343C23×C4—171no—
344C23×C4—341no—
345C23×C4—341no—
346C23×C4C22×D1711yescharacteristic
347C22×C8—172no—
348C22×C8C22×D1712yes—
349#126—348no—
350#125—174no—
351#125—342no—
352#124—174no—
353#124—342no—
354#123—348no—
355C22⋊Q8—22no—
356C22⋊Q8C4×D1714yes—
357#121—174no—
358#121—342no—
359C4×Q8—174no—
360C4×Q8—342no—
361C4×D4—174no—
362C4×D4—342no—
363C4×D4—348no—
364#117—348no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—172no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—344no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1712yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—341no—
382C2×C42—174no—
383C2×C42—171no—
384C2×C42—171no—
385C2×C42—174no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—341no—
389C2×C42—342no—
390C2×C42—342no—
391C2×C42—684no—
392C4⋊C8—344no—
393#103—24no—
394#103—344no—
395#102—344no—
396#102—344no—
397C22⋊C8—22no—
398C22⋊C8—342no—
399C4×C8—344no—
400#95—342no—
401#95—342no—
402C17#2138811yescharacteristic, complemented, Sylow
403C24—341no—
404C24D4×D1711yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×C4×D1712yes—
409C2×D4—172no—
410C2×D4—344no—
411C2×D4—341no—
412C2×D4—342no—
413C2×D4—342no—
414C2×D4—682no—
415C2×D4—682no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—171no—
427C22×C4—171no—
428C22×C4—174no—
429C22×C4—174no—
430C22×C4—344no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—344no—
436C22×C4—342no—
437C22×C4—341no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4—682no—
441C22×C4—682no—
442C22×C4C2×C4×D1712yes—
443C22×C4D4×D1712yes—
444C22×C4D4×D1711yescharacteristic
445C22×C4C23×D1711yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—342no—
450C2×C8—342no—
451C2×C8—342no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—172no—
457C4⋊C4—174no—
458C4⋊C4—344no—
459C4⋊C4—344no—
460C4⋊C4—344no—
461C4⋊C4—344no—
462C4⋊C4—341no—
463C4⋊C4—342no—
464C4⋊C4C2×C4×D1712yes—
465C4⋊C4#6325414yes—
466C22⋊C4—24no—
467C22⋊C4—344no—
468C22⋊C4—344no—
469C22⋊C4—344no—
470C22⋊C4—344no—
471C42—172no—
472C42—172no—
473C42—172no—
474C42—172no—
475C42—174no—
476C42—174no—
477C42—344no—
478C42—684no—
479C42—684no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—342no—
485C23—681no—
486C23—681no—
487C23—681no—
488C23—681no—
489C23—681no—
490C23—681no—
491C23C22⋊C4×D1712yes—
492C23#1058760212yes—
493C23#1058760512yes—
494C23C2×D4×D1711yescharacteristic
495Q8—24no—
496Q8—42no—
497D4—344no—
498D4—688no—
499D4—682no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—172no—
515C2×C4—172no—
516C2×C4—172no—
517C2×C4—172no—
518C2×C4—172no—
519C2×C4—172no—
520C2×C4—348no—
521C2×C4—348no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—342no—
529C2×C4—344no—
530C2×C4—688no—
531C2×C4—688no—
532C2×C4—681no—
533C2×C4—682no—
534C2×C4—681no—
535C2×C4—682no—
536C2×C4C22⋊C4×D1712yes—
537C2×C4C22×C4×D1711yescharacteristic
538C2×C4C2×D4×D1711yescharacteristic
539C2×C4C2×#6325412yes—
540C2×C4#41×D1712yes—
541C8—42no—
542C8—682no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—682no—
550C22—682no—
551C22—682no—
552C22—1361no—
553C22—1361no—
554C22—1361no—
555C22—1361no—
556C22#1176280414yes—
557C22C2×C22⋊C4×D1711yescharacteristic
558C22#1176331511yescharacteristic
559C22#1176331711yescharacteristic
560C22C2×#1058760211yescharacteristic
561C22C2×#1058760511yescharacteristic
562C22AGL(1,ℤ/8)×D1711yescharacteristic
563C22#137×D1711yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—174no—
569C4—342no—
570C4—342no—
571C4—342no—
572C4—682no—
573C4—684no—
574C4C4×#6325412yes—
575C4#1176335412yes—
576C2—24no—
577C2—1362no—
578C2#1453126911yescharacteristic
579C2C2×#1176280411yescharacteristic
580C2D17×#41711yescharacteristic
581C2#1453883211yescharacteristic
582C2#1453883411yescharacteristic
583C2#1453886511yescharacteristic
584C2#1453886711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1872C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
cosocleC22×C4×C17C24
hypercenter#1872D17
hypocenterC17#21388
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#11762859C2semidirect
3. C2×C4⋊C4×Dic17C2semidirect
4. #14529738C2semidirect
5. #14529713C2semidirect
8. #14529092C2semidirect
9. #103×Dic17C2semidirect
10. #14528598C2semidirect
11. C17×#1872C2semidirect
13. #11763345C4semidirect
65. #10587633C2×C4semidirect
67. Q8×Dic17C2×C4semidirect
94. #10587579C2×C4semidirect
97. #10587576C2×C4semidirect
259. #1872D17semidirect
402. C17#21388semidirect

Maximal subgroups

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

subgroupindex
C2×#117633452
C2×#117628592
C2×C4⋊C4×Dic172
#145297382
#145297132
#145291012
#145290992
#145290922
#103×Dic172
#145285982
C17×#18722
#2138817