Finite Group Databasebeta

← #17054201

Subgroups — #17054201

Order 4352.

838 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762910C211yesmaximal, characteristic
2#14531748C211yesmaximal, characteristic, complemented
3#14531732C211yesmaximal, characteristic
4#14529667C211yesmaximal, characteristic
5#14529428C211yesmaximal, characteristic, complemented
6#14529415C211yesmaximal, characteristic, complemented
7#14529414C211yesmaximal, characteristic
8#14529401C211yesmaximal, characteristic
9#14528949C211yesmaximal, characteristic, complemented
10#14528792C211yesmaximal, characteristic
11#14528787C211yesmaximal, characteristic, complemented
12#14528684C211yesmaximal, characteristic, complemented
13C4×#10587489C211yesmaximal, characteristic, complemented
14#14528658C211yesmaximal, characteristic, complemented
15C17×#1968C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×#10587563—21no—
19#11762910C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×#10587486C2211yescharacteristic
31#11762716C2211yescharacteristic
32#11762712C2211yescharacteristic
33#11762703—21no—
34#11762697—21no—
35#11762694—21no—
36#11762692—21no—
37#11762678C2211yescharacteristic
38#11762676—21no—
39#11762674C2211yescharacteristic
40#11762667—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46#11762654—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C17×#106C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C2×C4×C8×C17C2211yescharacteristic
58C17×#391C2211yescharacteristic
59C17×#390C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C2×SD16×C17—22no—
64C2×SD16×C17—22no—
65C2×SD16×C17C2×C414yes—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76Q8×Dic17—22no—
77Q8×Dic17C2×C414yes—
78D4×Dic17—22no—
79D4×Dic17C2×C414yes—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225D411yescharacteristic
88C2×#63225D411yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98#10587563—44no—
99C17×#106—22no—
100C17×#103—41no—
101C17×#102—41no—
102C4×C8×C17—22no—
103C17×#95—21no—
104C17×#95—21no—
105#10587506—41no—
106#10587506C2×C414yes—
107#10587503—41no—
108#10587503C2×C414yes—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502D411yescharacteristic
118#10587502D411yescharacteristic
119#10587489—22no—
120#10587489C2×C414yes—
121#10587486—22no—
122#10587486C2×C414yes—
123#10587485—22no—
124#10587479—41no—
125#10587478—41no—
126C24×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17C22×C412yes—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17C22×C412yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—24no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—42no—
151C22×Dic17—42no—
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156SD16×C17—28no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17C22×C412yes—
161C2×C8×C17#4112yes—
162C2×C8×C17#4112yes—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—81no—
169C42×C17—21no—
170C42×C17—21no—
171C22⋊Dic17—24no—
172C22⋊Dic17—42no—
173C22⋊Dic17—81no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—24no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17C22×C412yes—
182C4⋊Dic17C22×C412yes—
183#63225—22no—
184#63225—41no—
185#63225—42no—
186#63225—81no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—24no—
190C4×Dic17—42no—
191C4×Dic17—42no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C4×Dic17C22×C412yes—
195C4×Dic17#4112yes—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622C22×C412yes—
202#22820—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17C22×D411yescharacteristic
211Q8×C17—22no—
212Q8×C17—22no—
213Q8×C17—41no—
214Q8×C17—41no—
215D4×C17—22no—
216D4×C17—22no—
217D4×C17—41no—
218D4×C17—41no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—22no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17AGL(1,ℤ/8)12yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17H(5,𝔽2)11yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—48no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17C4×D412yes—
264C2×Dic17C4×D412yes—
265C8×C17—41no—
266C8×C17#11714yescomplemented
267#3622—22no—
268#3622—41no—
269C2×#440—171no—
270#2356—171no—
271#2354—171no—
272#1973—171no—
273#1968D1711yescharacteristic
274#1928—171no—
275#1927—171no—
276#1926—171no—
277#1925—171no—
278#1847—171no—
279#1804—171no—
280#1803—171no—
281#1776—171no—
282C4×#107—171no—
283#1771—171no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—81no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17C2×C4×D411yescharacteristic
292C22×C17#54711yescharacteristic
293C22×C17#55011yescharacteristic
294C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C17#57511yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57811yescharacteristic
298C4×C17—22no—
299C4×C17—41no—
300C4×C17—42no—
301C4×C17—81no—
302C4×C17—81no—
303C4×C17C2×#11711yescharacteristic
304C4×C17#52011yescharacteristic
305C4×C17#55511yescharacteristic
306C4×C17#55611yescharacteristic
307Dic17—22no—
308Dic17—22no—
309Dic17—41no—
310Dic17—41no—
311Dic17—42no—
312Dic17—42no—
313Dic17—81no—
314Dic17—81no—
315C22×SD16C2×D1711yescharacteristic
316C2×C4×Q8—171no—
317C2×C4×D4—171no—
318C2×#117—341no—
319#440—178no—
320C2×#107—171no—
321C2×#107—171no—
322C2×#107—171no—
323C2×#106—171no—
324C2×#106C2×D1711yescharacteristic
325C2×C4⋊C8—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#103C2×D1711yescharacteristic
329C2×#102—171no—
330C2×#102—171no—
331C2×#102C2×D1711yescharacteristic
332C2×#97—171no—
333C2×#97—171no—
334C2×C4×C8C2×D1711yescharacteristic
335#391—171no—
336#391C2×D1711yescharacteristic
337#390—171no—
338#390C2×D1711yescharacteristic
339#388—341no—
340#388—341no—
341#387—341no—
342#387—341no—
343#384—171no—
344#383—341no—
345#382—171no—
346#380—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4—171no—
352#376—341no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#336—171no—
358#336—171no—
359C2×C17—82no—
360C2×C17#235411yescharacteristic
361C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
362C2×C17#294711yescharacteristic
363C2×C17#334311yescharacteristic
364C2×C17#334811yescharacteristic
365C2×C17#339711yescharacteristic
366C2×C17#340211yescharacteristic
367C22×Q8C22×D1711yescharacteristic
368C22×D4C22×D1711yescharacteristic
369C23×C4—341no—
370C2×SD16—22no—
371C2×SD16—22no—
372C2×SD16C4×D1714yes—
373C22×C8—171no—
374C22×C8—171no—
375C22×C8C22×D1711yescharacteristic
376C22×C8C22×D1711yescharacteristic
377C4×Q8—174no—
378C4×Q8—342no—
379C4×D4—174no—
380C4×D4—342no—
381#117—684no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—341no—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42C22×D1711yescharacteristic
406#107—174no—
407#107—342no—
408#107—681no—
409#106—22no—
410#106—681no—
411C4⋊C8—342no—
412#103—41no—
413#103—174no—
414#103—681no—
415#102—41no—
416#102—174no—
417#102—681no—
418#97—174no—
419#97—342no—
420C4×C8—22no—
421#95—21no—
422#95—21no—
423#95—171no—
424#95—171no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433C17#2282011yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D1712yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D1712yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—344no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4D4×D1711yescharacteristic
462C22×C4D4×D1711yescharacteristic
463C22×C4C23×D1711yescharacteristic
464SD16—28no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—172no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8C2×C4×D1712yes—
475C2×C8#6325412yes—
476C2×C8#6325412yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—172no—
482C4⋊C4—172no—
483C4⋊C4—344no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—342no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—681no—
492C4⋊C4—682no—
493C4⋊C4—1361no—
494C22⋊C4—42no—
495C22⋊C4—81no—
496C22⋊C4—344no—
497C22⋊C4—682no—
498C22⋊C4—1361no—
499C42—21no—
500C42—21no—
501C42—172no—
502C42—172no—
503C42—344no—
504C42—341no—
505C42—341no—
506C42—682no—
507C42—682no—
508C42—682no—
509C42—682no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D1711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—172no—
541C2×C4—172no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—342no—
549C2×C4—342no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—688no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4#1058760012yes—
566C2×C4C22×C4×D1711yescharacteristic
567C2×C4C2×#6325411yescharacteristic
568C2×C4C2×#6325411yescharacteristic
569C2×C4#1058769111yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#1058769711yescharacteristic
572C8—41no—
573C8—342no—
574C8—681no—
575C8#1058756314yescomplemented
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×D4×D1711yescharacteristic
584C22#1176345911yescharacteristic
585C22#1176347911yescharacteristic
586C22C2×#1058760011yescharacteristic
587C22#1176360611yescharacteristic
588C22#1176361511yescharacteristic
589C22#1176362311yescharacteristic
590C4—22no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—342no—
596C4—342no—
597C4—682no—
598C4—682no—
599C4—681no—
600C4—681no—
601C4—1361no—
602C4—1361no—
603C4C2×#1058756311yescharacteristic
604C4#1176328811yescharacteristic
605C4#1176352311yescharacteristic
606C4#1176352811yescharacteristic
607C2—82no—
608C2#1453173211yescharacteristic
609C2C4×#1058760011yescharacteristic
610C2#1453762011yescharacteristic
611C2#1454123011yescharacteristic
612C2#1454130511yescharacteristic
613C2#1454180911yescharacteristic
614C2#1454186611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1968C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1968D17
hypocenterC17#22820
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #14531748C2semidirect
5. #14529428C2semidirect
6. #14529415C2semidirect
9. #14528949C2semidirect
11. #14528787C2semidirect
12. #14528684C2semidirect
13. C4×#10587489C2semidirect
14. #14528658C2semidirect
19. #11762910C4semidirect
266. C8×C17#117semidirect
433. C17#22820semidirect
575. C8#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×#117629102
#145317482
#145317322
#145296672
#145294282
#145294152
#145294142
#145294012
#145289492
#145287922
#145287872
#145286842
C4×#105874892
#145286582
C17×#19682
#2282017