Finite Group Databasebeta

← #17052019

Subgroups — #17052019

Order 4352.

866 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D17C211yesmaximal, characteristic
2#14531652C211yesmaximal, characteristic
3#14531649C211yesmaximal, characteristic
4#14529521C211yesmaximal, characteristic, complemented
5#14529519C211yesmaximal, characteristic, complemented
6#14529378C211yesmaximal, characteristic, complemented
7#14529377C211yesmaximal, characteristic, complemented
8#14529375C211yesmaximal, characteristic, complemented
9#14528869C211yesmaximal, characteristic
10#14528869C211yesmaximal, characteristic
11#14528808C211yesmaximal, characteristic
12#14528682C211yesmaximal, characteristic, complemented
13#14528680C211yesmaximal, characteristic, complemented
14#14528559C211yesmaximal, characteristic
15C17×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D17C2211yescharacteristic
17C2×#10587578—21no—
18C2×#10587558—21no—
19C2×C4⋊C4×D17C2211yescharacteristic
20C2×C4⋊C4×D17C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24#107×D17C418yescomplemented
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762705C2211yescharacteristic
33#11762696—21no—
34#11762693—21no—
35#11762691—21no—
36#11762690—21no—
37#11762671—21no—
38#11762671—21no—
39#11762666C2211yescharacteristic
40#11762666C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×C4⋊Dic17C2211yescharacteristic
44C4×#63227C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×C4×D17—21no—
61C22×C4×D17—21no—
62C22×C4×D17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227D411yescharacteristic
75C2×#63227D411yescharacteristic
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C2×C8×D17—22no—
92C2×C8×D17—22no—
93C2×C8×D17C2×C414yes—
94#10587578—44no—
95C4⋊C4×D17—22no—
96C4⋊C4×D17—22no—
97C4⋊C4×D17C2×C414yes—
98C4⋊C4×D17C2×C414yes—
99#10587558—44no—
100C17×#107C2×C414yes—
101C17×#95—21no—
102C17×#95—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587491—22no—
114#10587489C2×C414yes—
115#10587486—22no—
116#10587479—22no—
117#10587479—22no—
118#10587478C2×C414yes—
119#10587478C2×C414yes—
120#10587474—22no—
121C23×D17C2×D411yescharacteristic
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C22×Dic17C2×D411yescharacteristic
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—24no—
141C2×C4×D17—24no—
142C2×C4×D17—42no—
143C2×C4×D17—42no—
144C2×C4×D17—42no—
145C2×C4×D17—42no—
146C2×C4×D17C22×C412yes—
147C2×C4×D17#4112yes—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17C22×C412yes—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17C22×C412yes—
162C4⋊C4×C17C22×C412yes—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—42no—
166#63227—22no—
167#63227—22no—
168#63227—42no—
169#63227—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—42no—
177C4⋊Dic17C22×C412yes—
178C4⋊Dic17C22×C412yes—
179#63225—24no—
180#63225—24no—
181#63225—42no—
182#63225—42no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—42no—
186C4×Dic17—42no—
187C4×Dic17—42no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622C22×C412yes—
192C2×#3622#4112yes—
193C2×#3622#4112yes—
194C8×D17—28no—
195#22617—171nomaximal, Sylow
196C23×C17C22×D411yescharacteristic
197C22×D17—21no—
198C22×D17—21no—
199C22×D17—21no—
200C22×D17—21no—
201C22×D17C4×D412yes—
202C22×D17AGL(1,ℤ/8)12yes—
203C22×D17AGL(1,ℤ/8)12yes—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17C23×C411yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C2×C4×C172−1+411yescharacteristic
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17C4×D412yes—
249C4×D17—22no—
250C4×D17—22no—
251C4×D17—22no—
252C4×D17—22no—
253C4×D17—48no—
254C4×D17—48no—
255C8×C17—22no—
256C8×C17—41no—
257#3622—41no—
258#3622#11714yescomplemented
259C22×#107—171no—
260#2344—171no—
261#2344—171no—
262#1943—171no—
263#1943—171no—
264#1921—171no—
265#1921—171no—
266#1921—171no—
267#1921D1711yescharacteristic, complemented
268#1823—171no—
269#1823—171no—
270#1807—171no—
271#1776—171no—
272#1776—171no—
273#1752—171no—
274C22×C17C2×C4×D411yescharacteristic
275C22×C17#54711yescharacteristic
276C22×C17#54711yescharacteristic
277C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C17#57811yescharacteristic
280C22×C17#57811yescharacteristic
281C2×D17—22no—
282C2×D17—22no—
283C2×D17—22no—
284C2×D17—22no—
285C2×D17—22no—
286C2×D17—22no—
287C2×D17#44214yes—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—42no—
291C4×C17—42no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17C2×#11711yescharacteristic
295C4×C17#51911yescharacteristic
296C4×C17#55211yescharacteristic
297C4×C17#55211yescharacteristic
298Dic17—22no—
299Dic17—41no—
300Dic17—41no—
301Dic17—42no—
302Dic17—42no—
303Dic17—81no—
304Dic17—81no—
305C23×C8—171no—
306C2×#117—341no—
307C2×#117—341no—
308C22×C4⋊C4—171no—
309C22×C4⋊C4—171no—
310C2×#107—171no—
311C2×#107—171no—
312C2×#107—171no—
313C2×#107—178no—
314C2×#107C2×D1711yescharacteristic
315C2×#106—171no—
316C2×#106—171no—
317C2×C22⋊C8—171no—
318C2×#97—171no—
319C2×#97—171no—
320#389—171no—
321#389—171no—
322#389—171no—
323#389C2×D1711yescharacteristic
324#387—341no—
325#387—341no—
326#387—341no—
327#387—341no—
328#381—341no—
329#381—341no—
330#380—171no—
331#380—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C4⋊C4—171no—
335C4×C4⋊C4C2×D1711yescharacteristic
336C4×C22⋊C4—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340C2×D1711yescharacteristic
345#340C2×D1711yescharacteristic
346#340C2×D1711yescharacteristic
347#336—171no—
348#336C2×D1711yescharacteristic
349C2×C17#234411yescharacteristic
350C2×C17C2×#44211yescharacteristic
351C2×C17#295011yescharacteristic
352C2×C17#334311yescharacteristic
353C2×C17#334311yescharacteristic
354C2×C17#334311yescharacteristic
355C2×C17#334311yescharacteristic
356D17—24no—
357C23×C4—171no—
358C23×C4—341no—
359C23×C4—341no—
360C22×C8—171no—
361C22×C8—174no—
362C22×C8—171no—
363C22×C8—342no—
364C22×C8—342no—
365C22×C8C22×D1711yescharacteristic
366C22×C8C22×D1711yescharacteristic
367#117—684no—
368#117—684no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—174no—
373C2×C4⋊C4—174no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—342no—
377C2×C4⋊C4—342no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C22⋊C4—171no—
388C2×C22⋊C4—171no—
389C2×C22⋊C4—341no—
390C2×C22⋊C4—341no—
391C2×C42—21no—
392C2×C42—171no—
393C2×C42—341no—
394C2×C42—341no—
395C2×C42—341no—
396C2×C42C22×D1711yescharacteristic
397#107—174no—
398#107—174no—
399#107—174no—
400#107C4×D1714yes—
401#106—342no—
402#106—342no—
403C22⋊C8—342no—
404#97—342no—
405#97—342no—
406#95—21no—
407#95—21no—
408#95—171no—
409#95—171no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418C17#2261711yescharacteristic, complemented, Sylow
419C24—171no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—171no—
425C22×C4—172no—
426C22×C4—172no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—344no—
430C22×C4—344no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—682no—
440C22×C4—682no—
441C22×C4—682no—
442C22×C4—682no—
443C22×C4D4×D1711yescharacteristic
444C22×C4D4×D1711yescharacteristic
445C22×C4C23×D1711yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—172no—
452C2×C8—172no—
453C2×C8—172no—
454C2×C8—348no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8C2×C4×D1712yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—172no—
467C4⋊C4—172no—
468C4⋊C4—344no—
469C4⋊C4—344no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4C2×C4×D1712yes—
480C4⋊C4C2×C4×D1712yes—
481C22⋊C4—342no—
482C22⋊C4—342no—
483C22⋊C4—682no—
484C22⋊C4—682no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—341no—
489C42—341no—
490C42—682no—
491C42—682no—
492C42—682no—
493C23—172no—
494C23—172no—
495C23—172no—
496C23—341no—
497C23—341no—
498C23—341no—
499C23—341no—
500C23C2×D4×D1711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—172no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—342no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—341no—
527C2×C4—341no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—688no—
535C2×C4—688no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4C22×C4×D1711yescharacteristic
547C2×C4#1058769411yescharacteristic
548C2×C4#41×D1711yescharacteristic
549C2×C4#41×D1711yescharacteristic
550C2×C4#41×D1711yescharacteristic
551C2×C4#1058769711yescharacteristic
552C8—22no—
553C8—41no—
554C8—174no—
555C8—681no—
556C22—174no—
557C22—342no—
558C22—342no—
559C22—342no—
560C22—342no—
561C22—342no—
562C22—342no—
563C22C4×D4×D1711yescharacteristic
564C22#1176345711yescharacteristic
565C22#1176346111yescharacteristic
566C22AGL(1,ℤ/8)×D1711yescharacteristic
567C22AGL(1,ℤ/8)×D1711yescharacteristic
568C22#1176362211yescharacteristic
569C22#1176362311yescharacteristic
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—342no—
577C4—682no—
578C4—682no—
579C4—681no—
580C4—681no—
581C4—1361no—
582C4—1361no—
583C4#117×D1711yescharacteristic
584C4#1176328511yescharacteristic
585C4#1176349111yescharacteristic
586C4#1176349511yescharacteristic
587C2—344no—
588C2#1453165011yescharacteristic
589C2D17×#44211yescharacteristic
590C2#1453764711yescharacteristic
591C2#1454121711yescharacteristic
592C2#1454122111yescharacteristic
593C2#1454122711yescharacteristic
594C2#1454123111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1921C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1921D17
hypocenterC17#22617
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #14529521C2semidirect
5. #14529519C2semidirect
6. #14529378C2semidirect
7. #14529377C2semidirect
8. #14529375C2semidirect
12. #14528682C2semidirect
13. #14528680C2semidirect
15. C17×#1921C2semidirect
24. #107×D17C4semidirect
258. #3622#117semidirect
267. #1921D17semidirect
418. C17#22617semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D172
#145316522
#145316492
#145295212
#145295192
#145293782
#145293772
#145293752
#145288692
#145288082
#145286822
#145286802
#145285592
C17×#19212
#2261717