Finite Group Databasebeta

← #17005356

Subgroups — #17005356

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538827C211yesmaximal, characteristic
2M16×D68C211yesmaximal, characteristic, complemented
3#14527696C211yesmaximal, characteristic, complemented
4#14527674C211yesmaximal, characteristic, complemented
5#14527662C211yesmaximal, characteristic, complemented
6#14527646C211yesmaximal, characteristic, complemented
7#14527618C211yesmaximal, characteristic, complemented
8#14527613C211yesmaximal, characteristic, complemented
9#14527594C211yesmaximal, characteristic, complemented
10C8×#63228C211yesmaximal, characteristic, complemented
11#14527572C211yesmaximal, characteristic, complemented
12#14527249C211yesmaximal, characteristic, complemented
13#14527174C211yesmaximal, characteristic, complemented
14#14527081C211yesmaximal, characteristic, complemented
15C17×#2969C211yesmaximal, characteristic, complemented
16C2×M16×D17—21no—
17C2×C4×D68C2211yescharacteristic
18#11763142C411yescharacteristic
19#11763125C411yescharacteristic
20#11763106C411yescharacteristic
21#11763106C411yescharacteristic
22#11763021C411yescharacteristic
23#11763021C411yescharacteristic
24#11762959C411yescharacteristic
25#11762942C411yescharacteristic
26C4×#63230C2211yescharacteristic
27C4×#63230C2211yescharacteristic
28C4×#63228C2211yescharacteristic
29C4×#63228C2211yescharacteristic
30#11762888C2211yescharacteristic
31#11762888C2211yescharacteristic
32#11762882—21no—
33#11762882—21no—
34C8×D68C2211yescharacteristic
35C8×D68C2211yescharacteristic
36#11762862C2211yescharacteristic
37#11762862C2211yescharacteristic
38#11762833C2211yescharacteristic
39#11762772—21no—
40#11762561C2211yescharacteristic, complemented
41#11762561C2211yescharacteristic, complemented
42#11762559C2211yescharacteristic, complemented
43#11762559C2211yescharacteristic, complemented
44Q8⋊#3622C2211yescharacteristic, complemented
45Q8⋊#3622C2211yescharacteristic, complemented
46D4⋊#3622C2211yescharacteristic, complemented
47D4⋊#3622C2211yescharacteristic, complemented
48#11762536C2211yescharacteristic
49#11762536C2211yescharacteristic
50#11762536C2211yescharacteristic
51#11762536C2211yescharacteristic
52#11762531C2211yescharacteristic, complemented
53#11762531C2211yescharacteristic, complemented
54#11762530C2211yescharacteristic, complemented
55C8×#3622C2211yescharacteristic, complemented
56C17×#529C2211yescharacteristic
57C17×#446C2211yescharacteristic, complemented
58C8×Q8×C17C2211yescharacteristic, complemented
59C17×#436—21no—
60C17×#436C2211yescharacteristic, complemented
61C17×#435—21no—
62C8×D4×C17C2211yescharacteristic, complemented
63C17×#433—21no—
64C17×#431C2211yescharacteristic
65C17×#405—21no—
66C4×M16×C17C2211yescharacteristic
67C22×D68C2×C411yescharacteristic
68C22×C4×D17—21no—
69C2×M16×C17—21no—
70C2×M16×C17D411yescharacteristic
71C2×M16×C17D411yescharacteristic
72C22×C8×C17—21no—
73C17×#126—21no—
74C17×#125C2×C411yescharacteristic
75C17×#124C2×C411yescharacteristic
76C17×#123—21no—
77C17×C22⋊Q8C2×C411yescharacteristic
78C17×#121C2×C411yescharacteristic
79C4×Q8×C17C2311yescharacteristic
80C4×D4×C17—21no—
81C4×D4×C17C2311yescharacteristic
82C17×#117—21no—
83C2×C42×C17C2311yescharacteristic
84C2×#63230C2×C411yescharacteristic
85C2×#63230C2×C411yescharacteristic
86C2×#63228C2×C411yescharacteristic
87C2×#63228C2×C411yescharacteristic
88C2×#63227—21no—
89C2×C4⋊Dic17C2×C411yescharacteristic
90C2×#63223—21no—
91M16×D17—44no—
92C2×#63218—21no—
93C2×#63218—21no—
94C2×C8×D17—21no—
95C2×C8×D17—21no—
96C4×D68—21no—
97C4×D68—21no—
98C4×D68C2×C411yescharacteristic
99C4×D68C2×C411yescharacteristic
100C4×D68C2311yescharacteristic
101C4×D68C2311yescharacteristic
102C17×C4⋊C8—21no—
103C17×C4⋊C8—21no—
104C17×C4⋊C8C2311yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×C22⋊C8—21no—
107C17×C22⋊C8—21no—
108C17×C22⋊C8—21no—
109C17×#97—21no—
110C17×#97D411yescharacteristic
111C17×#97D411yescharacteristic
112C4×C8×C17—21no—
113C4×C8×C17—21no—
114C4×C8×C17C2311yescharacteristic
115C4×C8×C17C2311yescharacteristic
116#10587506C2×C411yescharacteristic
117#10587506C2×C411yescharacteristic
118#10587503C2×C411yescharacteristic
119#10587503C2×C411yescharacteristic
120C22⋊#3622—21no—
121C22⋊#3622C2×C411yescharacteristic
122C22⋊#3622C2×C411yescharacteristic
123#10587491—21no—
124#10587491—21no—
125#10587491—21no—
126#10587491—21no—
127#10587480C2×C411yescharacteristic
128#10587480C2×C411yescharacteristic
129#10587480C2×C411yescharacteristic
130#10587480C2×C411yescharacteristic
131#10587479C2×C411yescharacteristic
132#10587479C2×C411yescharacteristic
133#10587478C2×C411yescharacteristic
134#10587478C2×C411yescharacteristic
135C4⋊#3622—21no—
136C4⋊#3622—21no—
137C4⋊#3622C2×C411yescharacteristic
138C4⋊#3622C2×C411yescharacteristic
139C4⋊#3622C2311yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C23×D17—21no—
146C2×Q8×C17C22×C411yescharacteristic
147C2×D4×C17—21no—
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17C22×C411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17#4111yescharacteristic
153C22×Dic17—21no—
154C2×D68—22no—
155C2×D68—21no—
156C2×D68—41no—
157C2×D68C22×C411yescharacteristic
158C2×D68C22×C411yescharacteristic
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—21no—
163C2×C4×D17—42no—
164C2×C4×D17—42no—
165M16×C17—22no—
166M16×C17—41no—
167M16×C17—42no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—42no—
171C2×C8×C17C2×D411yescharacteristic
172C2×C8×C17C2×D411yescharacteristic
173C2×C8×C17C2×D411yescharacteristic
174C2×C8×C17C2×D411yescharacteristic
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17C22×C411yescharacteristic
180C4⋊C4×C17C22×C411yescharacteristic
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C22⋊C4×C17—21no—
185C22⋊C4×C17—21no—
186C42×C17—21no—
187C42×C17C22×C411yescharacteristic
188C42×C17C2×D411yescharacteristic
189C42×C17#4111yescharacteristic
190C42×C17C2411yescharacteristic
191#63230—22no—
192#63230—41no—
193#63228—22no—
194#63228—41no—
195#63227—21no—
196#63227—21no—
197#63227—21no—
198#63227—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17C22×C411yescharacteristic
201C4⋊Dic17C22×C411yescharacteristic
202#63223—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622C22×C411yescharacteristic
208C2×#3622C22×C411yescharacteristic
209C2×#3622C22×C411yescharacteristic
210C2×#3622C22×C411yescharacteristic
211#63218—42no—
212#63218—42no—
213C8×D17—42no—
214C8×D17—42no—
215#18006—171nomaximal, Sylow
216C23×C17—21no—
217C23×C17C4×D411yescharacteristic
218C22×D17—21no—
219C22×D17—21no—
220C22×D17—42no—
221C22×D17—42no—
222C22×D17—81no—
223Q8×C17—21no—
224Q8×C17—21no—
225D4×C17—21no—
226D4×C17—21no—
227D4×C17—81no—
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—21no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17C4×D411yescharacteristic
239C2×C4×C17C4×D411yescharacteristic
240C2×C4×C17C4×D411yescharacteristic
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—42no—
247D68—21no—
248D68—21no—
249D68—21no—
250D68—42no—
251D68#13112yes—
252C4×D17—42no—
253C4×D17—42no—
254C4×D17—41no—
255C4×D17—41no—
256C4×D17—41no—
257C4×D17—41no—
258C8×C17—21no—
259C8×C17—21no—
260C8×C17—21no—
261C8×C17—41no—
262C8×C17—41no—
263C8×C17AGL(1,ℤ/8)12yes—
264#3622—21no—
265#3622—21no—
266#3622—21no—
267#3622—21no—
268#3622—41no—
269#3622—41no—
270#3622—41no—
271#3622—41no—
272#3105—171no—
273#2969D1711yescharacteristic, complemented
274D4×M16—171no—
275#1594—171no—
276#1590—171no—
277#1588—171no—
278#1586—171no—
279#1582—171no—
280#1581—171no—
281SD16⋊C8—171no—
282C8×D8—171no—
283#1574—171no—
284#1495—171no—
285#1478—171no—
286#1453—171no—
287C22×C17—21no—
288C22×C17—21no—
289C22×C17—21no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17C2×C4×D411yescharacteristic
294C2×D17—22no—
295C2×D17—41no—
296C2×D17—41no—
297C2×D17—41no—
298C2×D17—41no—
299C2×D17—41no—
300C2×D17—84no—
301C4×C17—21no—
302C4×C17—21no—
303C4×C17—21no—
304C4×C17—21no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17—41no—
308C4×C17C2×#13111yescharacteristic
309C4×C17#56811yescharacteristic
310C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
311C4×C17#57511yescharacteristic
312Dic17—41no—
313Dic17—41no—
314C22×M16—341no—
315#529C2×D1711yescharacteristic
316C2×C4×D4—171no—
317#484—171no—
318#481—171no—
319#478—171no—
320#478—171no—
321#463—171no—
322#463—171no—
323#450—171no—
324#447—171no—
325#446C2×D1711yescharacteristic, complemented
326C8×Q8C2×D1711yescharacteristic, complemented
327C4×SD16—171no—
328C4×SD16—171no—
329C4×D8—171no—
330C4×D8—171no—
331#436—21no—
332#436—171no—
333#436—171no—
334#436C2×D1711yescharacteristic, complemented
335#435—21no—
336#435—341no—
337#435—341no—
338C8×D4—171no—
339C8×D4—171no—
340C8×D4C2×D1711yescharacteristic, complemented
341#433—21no—
342#431—171no—
343#431—171no—
344#431C2×D1711yescharacteristic
345#423—171no—
346#407—341no—
347#405—21no—
348C4×M16C2×D1711yescharacteristic
349#335—171no—
350#335—171no—
351#334—171no—
352#334—171no—
353Q8⋊C8—171no—
354Q8⋊C8—171no—
355D4⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8—171no—
358D4⋊C8—171no—
359D4⋊C8—171no—
360D4⋊C8—171no—
361#322—171no—
362#322—171no—
363#322—171no—
364C82—171no—
365C2×C17—41no—
366C2×C17—81no—
367C2×C17#293311yescharacteristic
368C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
369C2×C17#295711yescharacteristic
370D17—42no—
371D17—81no—
372C22×D4—171no—
373C23×C4—341no—
374C2×SD16—171no—
375C2×SD16—171no—
376C2×D8—171no—
377C2×D8—171no—
378C2×M16—21no—
379C2×M16—341no—
380C2×M16—341no—
381C2×M16—341no—
382C2×M16—684no—
383C2×M16C22⋊D1711yescharacteristic
384C2×M16C22⋊D1711yescharacteristic
385C22×C8—21no—
386C22×C8—341no—
387C22×C8—341no—
388#126—21no—
389#125C4×D1711yescharacteristic
390#124C4×D1711yescharacteristic
391#123—21no—
392C22⋊Q8C4×D1711yescharacteristic
393#121C4×D1711yescharacteristic
394C4×Q8C22×D1711yescharacteristic
395C4×D4—21no—
396C4×D4—171no—
397C4×D4—171no—
398C4×D4—171no—
399C4×D4—171no—
400C4×D4—341no—
401C4×D4—341no—
402C4×D4C22×D1711yescharacteristic
403#117—21no—
404C2×C4⋊C4—171no—
405C2×C22⋊C4—341no—
406C2×C42C22×D1711yescharacteristic
407#107—171no—
408#107—171no—
409#106—171no—
410#106—171no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—171no—
414C4⋊C8—171no—
415C4⋊C8—171no—
416C4⋊C8—171no—
417C4⋊C8—341no—
418C4⋊C8—341no—
419C4⋊C8C22×D1711yescharacteristic
420C4⋊C8C22×D1711yescharacteristic
421#103—171no—
422#103—171no—
423#102—171no—
424#102—171no—
425#102—171no—
426#102—171no—
427#102—171no—
428#102—171no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—171no—
433C22⋊C8—171no—
434C22⋊C8—341no—
435C22⋊C8—341no—
436C22⋊C8—341no—
437C22⋊C8—341no—
438C22⋊C8—341no—
439#97—21no—
440#97C22⋊D1711yescharacteristic
441#97C22⋊D1711yescharacteristic
442C4×C8—21no—
443C4×C8—21no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—171no—
447C4×C8—171no—
448C4×C8C22×D1711yescharacteristic
449C4×C8C22×D1711yescharacteristic
450C17#1800611yescharacteristic, complemented, Sylow
451C24—341no—
452C2×Q8C2×C4×D1711yescharacteristic
453C2×D4—21no—
454C2×D4—171no—
455C2×D4—171no—
456C2×D4—342no—
457C2×D4—341no—
458C2×D4—681no—
459C2×D4C2×C4×D1711yescharacteristic
460C22×C4—21no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4C2×C4×D1711yescharacteristic
469C22×C4#6325211yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471SD16—342no—
472SD16—681no—
473D8—342no—
474D8—681no—
475M16—22no—
476M16—41no—
477M16—42no—
478M16—682no—
479M16—682no—
480M16—682no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—171no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—171no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—682no—
493C2×C8—682no—
494C2×C8C2×C22⋊D1711yescharacteristic
495C2×C8C2×C22⋊D1711yescharacteristic
496C2×C8C2×C22⋊D1711yescharacteristic
497C2×C8C2×C22⋊D1711yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—171no—
503C4⋊C4—171no—
504C4⋊C4—341no—
505C4⋊C4C2×C4×D1711yescharacteristic
506C4⋊C4C2×C4×D1711yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—341no—
513C22⋊C4—341no—
514C22⋊C4—341no—
515C22⋊C4—341no—
516C42—21no—
517C42C2×C4×D1711yescharacteristic
518C42#6325211yescharacteristic
519C42C2×C22⋊D1711yescharacteristic
520C42C23×D1711yescharacteristic
521C23—21no—
522C23—341no—
523C23—341no—
524C23—682no—
525C23—682no—
526C23—1361no—
527C23C4×C22⋊D1711yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—172no—
534D4—341no—
535D4—341no—
536D4—341no—
537D4—682no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4C4×C22⋊D1711yescharacteristic
558C2×C4C4×C22⋊D1711yescharacteristic
559C2×C4C4×C22⋊D1711yescharacteristic
560C2×C4C22×C4×D1711yescharacteristic
561C2×C4C2×#6325211yescharacteristic
562C2×C4C22×C22⋊D1711yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—341no—
569C8—341no—
570C8—341no—
571C8—341no—
572C8—681no—
573C8—681no—
574C8—681no—
575C8—681no—
576C8#1058763512yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—342no—
584C22—681no—
585C22—681no—
586C22—681no—
587C22—681no—
588C22—681no—
589C22—1364no—
590C22C2×C4×C22⋊D1711yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—681no—
599C4—681no—
600C4#131×D1711yescharacteristic
601C4#1176357411yescharacteristic
602C4C2×#1058763511yescharacteristic
603C4#1176360711yescharacteristic
604C2—41no—
605C2—81no—
606C2—682no—
607C2—1361no—
608C2#1453750611yescharacteristic
609C2C4×#1058763511yescharacteristic
610C2#1453769911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#2969C2
socleC22×C17C2×C4×D4
prime radical#11762942C4
cosocleC42×C17C24
hypercenter#2969D17
hypocenterC17#18006
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D68C2semidirect
3. #14527696C2semidirect
4. #14527674C2semidirect
5. #14527662C2semidirect
6. #14527646C2semidirect
7. #14527618C2semidirect
8. #14527613C2semidirect
9. #14527594C2semidirect
10. C8×#63228C2semidirect
11. #14527572C2semidirect
12. #14527249C2semidirect
13. #14527174C2semidirect
14. #14527081C2semidirect
15. C17×#2969C2semidirect
40. #11762561C22semidirect
41. #11762561C22semidirect
42. #11762559C22semidirect
43. #11762559C22semidirect
44. Q8⋊#3622C22semidirect
45. Q8⋊#3622C22semidirect
46. D4⋊#3622C22semidirect
47. D4⋊#3622C22semidirect
52. #11762531C22semidirect
53. #11762531C22semidirect
54. #11762530C22semidirect
55. C8×#3622C22semidirect
57. C17×#446C22semidirect
58. C8×Q8×C17C22semidirect
60. C17×#436C22semidirect
62. C8×D4×C17C22semidirect
273. #2969D17semidirect
325. #446C2×D17semidirect
326. C8×Q8C2×D17semidirect
334. #436C2×D17semidirect
340. C8×D4C2×D17semidirect
450. C17#18006semidirect

Maximal subgroups

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

subgroupindex
#145388272
M16×D682
#145276962
#145276742
#145276622
#145276462
#145276182
#145276132
#145275942
C8×#632282
#145275722
#145272492
#145271742
#145270812
C17×#29692
#1800617