Finite Group Databasebeta

← #17005486

Subgroups — #17005486

Order 4352.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538827C211yesmaximal, characteristic
2#14537944C211yesmaximal, characteristic, complemented
3#14537821C211yesmaximal, characteristic, complemented
4#14527671C211yesmaximal, characteristic, complemented
5#14527666C211yesmaximal, characteristic, complemented
6#14527659C211yesmaximal, characteristic, complemented
7#14527651C211yesmaximal, characteristic, complemented
8#14527645C211yesmaximal, characteristic, complemented
9#14527644C211yesmaximal, characteristic, complemented
10#14527581C211yesmaximal, characteristic, complemented
11#14527576C211yesmaximal, characteristic, complemented
12#14527570C211yesmaximal, characteristic, complemented
13C2×#11762538C211yesmaximal, characteristic, complemented
14#14527067C211yesmaximal, characteristic, complemented
15C17×#1494C211yesmaximal, characteristic, complemented
16C2×C4×D68C2211yescharacteristic
17#11763142C411yescharacteristic
18#11763125C411yescharacteristic
19#11763106C411yescharacteristic
20#11763106C411yescharacteristic
21#11763021C411yescharacteristic
22#11763021C411yescharacteristic
23#11762959C411yescharacteristic
24#11762942C411yescharacteristic
25#11762940C2211yescharacteristic, complemented
26Q8×#3622C2211yescharacteristic, complemented
27C4×#63230C2211yescharacteristic
28C4×#63230C2211yescharacteristic
29C4×#63228C2211yescharacteristic
30C4×#63228C2211yescharacteristic
31#11762887C2211yescharacteristic, complemented
32#11762886C2211yescharacteristic
33#11762886C2211yescharacteristic
34#11762882—21no—
35#11762882—21no—
36#11762881—21no—
37D4×#3622—21no—
38D4×#3622C2211yescharacteristic, complemented
39#11762874C2211yescharacteristic
40#11762874C2211yescharacteristic
41#11762870—21no—
42#11762862C2211yescharacteristic
43#11762862C2211yescharacteristic
44#11762836—21no—
45C2×C4⋊#3622C2211yescharacteristic
46#11762771—21no—
47C22⋊C8×D17—21no—
48#11762561C2211yescharacteristic, complemented
49#11762561C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762559C2211yescharacteristic, complemented
52#11762538C2211yescharacteristic
53#11762538C2211yescharacteristic
54#11762538C2211yescharacteristic
55#11762538C2211yescharacteristic
56#11762530C2211yescharacteristic, complemented
57#11762530C2211yescharacteristic, complemented
58C8×#3622C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C17×#529C2211yescharacteristic
61C17×#432C2211yescharacteristic
62C2×C4×C8×C17C2211yescharacteristic
63C17×Q8⋊C8C2211yescharacteristic, complemented
64C17×Q8⋊C8C2211yescharacteristic, complemented
65C17×D4⋊C8C2211yescharacteristic, complemented
66C17×D4⋊C8C2211yescharacteristic, complemented
67C22×D68C2×C411yescharacteristic
68C22×C4×D17—21no—
69C22×C8×C17—21no—
70C17×#126—21no—
71C17×#125C2×C411yescharacteristic
72C17×#124C2×C411yescharacteristic
73C17×#123—21no—
74C17×C22⋊Q8C2×C411yescharacteristic
75C17×#121C2×C411yescharacteristic
76C4×Q8×C17C2311yescharacteristic
77C4×D4×C17—21no—
78C4×D4×C17C2311yescharacteristic
79C17×#117—21no—
80C2×C42×C17C2311yescharacteristic
81C2×#63230C2×C411yescharacteristic
82C2×#63230C2×C411yescharacteristic
83C2×#63228C2×C411yescharacteristic
84C2×#63228C2×C411yescharacteristic
85C2×#63227—21no—
86C2×C4⋊Dic17C2×C411yescharacteristic
87C2×#63223—21no—
88C22×#3622—21no—
89C22×#3622—21no—
90C2×#63218—21no—
91C2×#63218—21no—
92C2×C8×D17—21no—
93C2×C8×D17—21no—
94C4×D68—21no—
95C4×D68—21no—
96C4×D68C2×C411yescharacteristic
97C4×D68C2×C411yescharacteristic
98C4×D68C2311yescharacteristic
99C4×D68C2311yescharacteristic
100C17×C4⋊C8C2311yescharacteristic
101C17×C4⋊C8C2311yescharacteristic
102C17×C22⋊C8—21no—
103C17×#97—21no—
104C4×C8×C17—21no—
105C4×C8×C17C2311yescharacteristic
106C4×C8×C17C2311yescharacteristic
107#10587506C2×C411yescharacteristic
108#10587506C2×C411yescharacteristic
109#10587503C2×C411yescharacteristic
110#10587503C2×C411yescharacteristic
111C22⋊#3622—21no—
112C22⋊#3622—21no—
113C22⋊#3622C2×C411yescharacteristic
114C22⋊#3622C2×C411yescharacteristic
115C22⋊#3622D411yescharacteristic
116C22⋊#3622D411yescharacteristic
117#10587491—21no—
118#10587491—21no—
119#10587491—21no—
120#10587491—21no—
121#10587480C2×C411yescharacteristic
122#10587480C2×C411yescharacteristic
123#10587480C2×C411yescharacteristic
124#10587480C2×C411yescharacteristic
125#10587479C2×C411yescharacteristic
126#10587479C2×C411yescharacteristic
127#10587478C2×C411yescharacteristic
128#10587478C2×C411yescharacteristic
129C4⋊#3622—21no—
130C4⋊#3622—21no—
131C4⋊#3622—21no—
132C4⋊#3622C2×C411yescharacteristic
133C4⋊#3622C2×C411yescharacteristic
134C4⋊#3622D411yescharacteristic
135C4⋊#3622D411yescharacteristic
136C4⋊#3622C2311yescharacteristic
137C4⋊#3622C2311yescharacteristic
138#10587474—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C23×D17—21no—
145C2×Q8×C17C22×C411yescharacteristic
146C2×D4×C17—21no—
147C2×D4×C17C22×C411yescharacteristic
148C22×C4×C17—21no—
149C22×C4×C17C22×C411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17#4111yescharacteristic
152C22×Dic17—21no—
153C2×D68—22no—
154C2×D68—21no—
155C2×D68—41no—
156C2×D68C22×C411yescharacteristic
157C2×D68C22×C411yescharacteristic
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×C8×C17—24no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17C22×C411yescharacteristic
174C4⋊C4×C17C22×C411yescharacteristic
175C22⋊C4×C17—21no—
176C22⋊C4×C17—21no—
177C22⋊C4×C17—21no—
178C22⋊C4×C17—21no—
179C22⋊C4×C17—21no—
180C42×C17—21no—
181C42×C17C22×C411yescharacteristic
182C42×C17C2×D411yescharacteristic
183C42×C17#4111yescharacteristic
184C42×C17C2411yescharacteristic
185#63230—22no—
186#63230—41no—
187#63228—22no—
188#63228—41no—
189#63227—21no—
190#63227—21no—
191#63227—21no—
192#63227—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17C22×C411yescharacteristic
195C4⋊Dic17C22×C411yescharacteristic
196#63223—81no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—81no—
200C2×#3622—81no—
201C2×#3622C22×C411yescharacteristic
202C2×#3622C22×C411yescharacteristic
203C2×#3622C22×C411yescharacteristic
204C2×#3622C22×C411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206C2×#3622C2×D411yescharacteristic
207C2×#3622C2×D411yescharacteristic
208C2×#3622C2×D411yescharacteristic
209#63218—81no—
210#63218—81no—
211C8×D17—81no—
212C8×D17—81no—
213#18014—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17C4×D411yescharacteristic
216C22×D17—21no—
217C22×D17—21no—
218C22×D17—42no—
219C22×D17—42no—
220C22×D17—81no—
221Q8×C17—21no—
222Q8×C17—21no—
223D4×C17—21no—
224D4×C17—21no—
225D4×C17—42no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17C4×D411yescharacteristic
237C2×C4×C17C4×D411yescharacteristic
238C2×C4×C17C4×D411yescharacteristic
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—81no—
245D68—21no—
246D68—21no—
247D68—21no—
248D68—42no—
249D68#13112yes—
250C4×D17—41no—
251C4×D17—41no—
252C4×D17—41no—
253C4×D17—41no—
254C4×D17—81no—
255C4×D17—81no—
256C8×C17—22no—
257C8×C17—22no—
258C8×C17—41no—
259C8×C17—41no—
260#3622—21no—
261#3622—21no—
262#3622—21no—
263#3622—21no—
264#3622—21no—
265#3622—21no—
266#3622—21no—
267#3622—21no—
268#3622—41no—
269#3622—41no—
270#3105—171no—
271#2994—171no—
272#2974—171no—
273#1590—171no—
274#1589—171no—
275#1588—171no—
276#1586—171no—
277#1586—171no—
278#1585—171no—
279C8×SD16—171no—
280C8×D8—171no—
281#1573—171no—
282#1494D1711yescharacteristic, complemented
283C2×D4⋊C8—171no—
284#1450—171no—
285C22×C17—21no—
286C22×C17—21no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17#44312yes—
291C22×C17C2×C4×D411yescharacteristic
292C2×D17—22no—
293C2×D17—41no—
294C2×D17—41no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C2×D17—84no—
299C4×C17—21no—
300C4×C17—21no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17C2×#13111yescharacteristic
307C4×C17#56811yescharacteristic
308C4×C17#57611yescharacteristic
309C4×C17#57711yescharacteristic
310Dic17—41no—
311Dic17—41no—
312#529C2×D1711yescharacteristic
313C2×C4×D4—171no—
314#484—171no—
315#481—171no—
316#478—171no—
317#478—171no—
318#463—171no—
319#463—171no—
320#450—171no—
321#447—171no—
322#446—171no—
323C8×Q8—171no—
324C4×SD16—171no—
325C4×SD16—171no—
326C4×D8—171no—
327C4×D8—171no—
328#436—171no—
329#436—171no—
330#436—171no—
331#435—341no—
332#435—341no—
333#435—341no—
334C8×D4—171no—
335C8×D4—171no—
336C8×D4—171no—
337C8×D4—341no—
338#433—341no—
339#432C2×D1711yescharacteristic
340#431—171no—
341#431—171no—
342#424—341no—
343C2×C4⋊C8—171no—
344#407—341no—
345C2×C22⋊C8—341no—
346C2×C4×C8C2×D1711yescharacteristic
347#335—171no—
348#335—171no—
349#334—171no—
350#334—171no—
351Q8⋊C8C2×D1711yescharacteristic, complemented
352Q8⋊C8C2×D1711yescharacteristic, complemented
353D4⋊C8—171no—
354D4⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8C2×D1711yescharacteristic, complemented
358D4⋊C8C2×D1711yescharacteristic, complemented
359#322—171no—
360#322—171no—
361C82—171no—
362C82—171no—
363C2×C17—22no—
364C2×C17—81no—
365C2×C17#293311yescharacteristic
366C2×C17#294611yescharacteristic
367C2×C17C2×#44311yescharacteristic
368D17—42no—
369D17—81no—
370C22×D4—171no—
371C23×C4—341no—
372C2×SD16—171no—
373C2×SD16—171no—
374C2×D8—171no—
375C2×D8—171no—
376C2×M16—341no—
377C2×M16—341no—
378C2×M16—341no—
379C22×C8—21no—
380C22×C8—341no—
381C22×C8—341no—
382C22×C8—341no—
383C22×C8—341no—
384#126—21no—
385#125C4×D1711yescharacteristic
386#124C4×D1711yescharacteristic
387#123—21no—
388C22⋊Q8C4×D1711yescharacteristic
389#121C4×D1711yescharacteristic
390C4×Q8C22×D1711yescharacteristic
391C4×D4—21no—
392C4×D4—171no—
393C4×D4—171no—
394C4×D4—171no—
395C4×D4—171no—
396C4×D4—341no—
397C4×D4—341no—
398C4×D4C22×D1711yescharacteristic
399#117—21no—
400C2×C4⋊C4—171no—
401C2×C22⋊C4—341no—
402C2×C42C22×D1711yescharacteristic
403#107—171no—
404#107—171no—
405#106—171no—
406#106—171no—
407C4⋊C8—171no—
408C4⋊C8—171no—
409C4⋊C8—171no—
410C4⋊C8—171no—
411C4⋊C8—171no—
412C4⋊C8—171no—
413C4⋊C8—341no—
414C4⋊C8—341no—
415C4⋊C8—341no—
416C4⋊C8C22×D1711yescharacteristic
417C4⋊C8C22×D1711yescharacteristic
418#103—171no—
419#103—171no—
420#102—171no—
421#102—171no—
422#102—171no—
423#102—171no—
424#102—171no—
425#102—171no—
426C22⋊C8—21no—
427C22⋊C8—171no—
428C22⋊C8—171no—
429C22⋊C8—171no—
430C22⋊C8—171no—
431C22⋊C8—341no—
432C22⋊C8—341no—
433C22⋊C8—341no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437#97—21no—
438#97—341no—
439C4×C8—21no—
440C4×C8—171no—
441C4×C8—171no—
442C4×C8—171no—
443C4×C8—171no—
444C4×C8—341no—
445C4×C8C22×D1711yescharacteristic
446C4×C8C22×D1711yescharacteristic
447C17#1801411yescharacteristic, complemented, Sylow
448C24—341no—
449C2×Q8C2×C4×D1711yescharacteristic
450C2×D4—21no—
451C2×D4—171no—
452C2×D4—171no—
453C2×D4—342no—
454C2×D4—341no—
455C2×D4—681no—
456C2×D4C2×C4×D1711yescharacteristic
457C22×C4—21no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—1361no—
464C22×C4—1361no—
465C22×C4C2×C4×D1711yescharacteristic
466C22×C4D4×D1711yescharacteristic
467C22×C4#6325411yescharacteristic
468SD16—342no—
469SD16—681no—
470D8—342no—
471D8—681no—
472M16—1361no—
473M16—1361no—
474M16—1361no—
475C2×C8—24no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—171no—
481C2×C8—171no—
482C2×C8—171no—
483C2×C8—171no—
484C2×C8—171no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—171no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—1361no—
491C2×C8—1361no—
492C2×C8—1361no—
493C2×C8—1361no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—171no—
499C4⋊C4—171no—
500C4⋊C4—341no—
501C4⋊C4C2×C4×D1711yescharacteristic
502C4⋊C4C2×C4×D1711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—341no—
509C22⋊C4—341no—
510C22⋊C4—341no—
511C22⋊C4—341no—
512C42—21no—
513C42C2×C4×D1711yescharacteristic
514C42D4×D1711yescharacteristic
515C42#6325411yescharacteristic
516C42C23×D1711yescharacteristic
517C23—21no—
518C23—341no—
519C23—341no—
520C23—682no—
521C23—682no—
522C23—1361no—
523C23#1058756711yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—42no—
529D4—172no—
530D4—341no—
531D4—341no—
532D4—341no—
533D4—682no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—1361no—
551C2×C4—1361no—
552C2×C4—1361no—
553C2×C4#1058756711yescharacteristic
554C2×C4#1058756711yescharacteristic
555C2×C4#1058756711yescharacteristic
556C2×C4C22×C4×D1711yescharacteristic
557C2×C4C2×D4×D1711yescharacteristic
558C2×C4C2×#6325411yescharacteristic
559C8—22no—
560C8—22no—
561C8—41no—
562C8—41no—
563C8—341no—
564C8—341no—
565C8—341no—
566C8—341no—
567C8—341no—
568C8—341no—
569C8—341no—
570C8—341no—
571C8—681no—
572C8—681no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—342no—
579C22—681no—
580C22—681no—
581C22—681no—
582C22—681no—
583C22—681no—
584C22—1364no—
585C22#1176292312yes—
586C22C2×#1058756711yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—681no—
595C4—681no—
596C4#131×D1711yescharacteristic
597C4#1176357411yescharacteristic
598C4#1176361111yescharacteristic
599C4#1176361611yescharacteristic
600C2—22no—
601C2—81no—
602C2—682no—
603C2—1361no—
604C2#1453751411yescharacteristic
605C2#1453761911yescharacteristic
606C2C2×#1176292311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1494C2
socleC22×C17C2×C4×D4
prime radical#11762942C4
cosocleC42×C17C24
hypercenter#1494D17
hypocenterC17#18014
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #14537944C2semidirect
3. #14537821C2semidirect
4. #14527671C2semidirect
5. #14527666C2semidirect
6. #14527659C2semidirect
7. #14527651C2semidirect
8. #14527645C2semidirect
9. #14527644C2semidirect
10. #14527581C2semidirect
11. #14527576C2semidirect
12. #14527570C2semidirect
13. C2×#11762538C2semidirect
14. #14527067C2semidirect
15. C17×#1494C2semidirect
25. #11762940C22semidirect
26. Q8×#3622C22semidirect
31. #11762887C22semidirect
38. D4×#3622C22semidirect
48. #11762561C22semidirect
49. #11762561C22semidirect
50. #11762559C22semidirect
51. #11762559C22semidirect
56. #11762530C22semidirect
57. #11762530C22semidirect
58. C8×#3622C22semidirect
59. C8×#3622C22semidirect
63. C17×Q8⋊C8C22semidirect
64. C17×Q8⋊C8C22semidirect
65. C17×D4⋊C8C22semidirect
66. C17×D4⋊C8C22semidirect
282. #1494D17semidirect
351. Q8⋊C8C2×D17semidirect
352. Q8⋊C8C2×D17semidirect
357. D4⋊C8C2×D17semidirect
358. D4⋊C8C2×D17semidirect
447. C17#18014semidirect

Maximal subgroups

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

subgroupindex
#145388272
#145379442
#145378212
#145276712
#145276662
#145276592
#145276512
#145276452
#145276442
#145275812
#145275762
#145275702
C2×#117625382
#145270672
C17×#14942
#1801417