Finite Group Databasebeta

← #17004339

Subgroups — #17004339

Order 4352.

712 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538829C211yesmaximal, characteristic
2#14537756C211yesmaximal, characteristic, complemented
3C8×D4×D17C211yesmaximal, characteristic, complemented
4#14527619C211yesmaximal, characteristic, complemented
5#14527617C211yesmaximal, characteristic, complemented
6#14527616C211yesmaximal, characteristic, complemented
7#14527614C211yesmaximal, characteristic, complemented
8#14527606C211yesmaximal, characteristic, complemented
9#14527598C211yesmaximal, characteristic, complemented
10#14527596C211yesmaximal, characteristic, complemented
11#14527566C211yesmaximal, characteristic, complemented
12#14527225C211yesmaximal, characteristic, complemented
13#14527194C211yesmaximal, characteristic, complemented
14#14527072C211yesmaximal, characteristic, complemented
15C17×#1580C211yesmaximal, characteristic, complemented
16C22×C8×D17—21no—
17#11763341C2211yescharacteristic
18C4×D4×D17C2211yescharacteristic
19#11763146C411yescharacteristic
20#11763126C411yescharacteristic
21#11763112C411yescharacteristic
22#11763111C411yescharacteristic
23#11763023C411yescharacteristic
24#11763022C411yescharacteristic
25#11762961C411yescharacteristic
26#11762941C411yescharacteristic
27#11762936C2211yescharacteristic, complemented
28C8×Dic34C2211yescharacteristic, complemented
29C4×#63219C2211yescharacteristic
30C4×#63229C2211yescharacteristic
31C4×#63228C2211yescharacteristic
32#11762887C2211yescharacteristic, complemented
33#11762883—21no—
34C8×D68C2211yescharacteristic
35D4×#3622C2211yescharacteristic
36#11762874C2211yescharacteristic
37C8×C22⋊D17—21no—
38C8×C22⋊D17—21no—
39#11762872—21no—
40#11762867—21no—
41#11762865C2211yescharacteristic
42#11762860C2211yescharacteristic
43C4⋊C8×D17C2211yescharacteristic
44C22⋊C8×D17—21no—
45#11762764—21no—
46C4×#63218C2211yescharacteristic
47C4×C8×D17C2211yescharacteristic
48#11762562C2211yescharacteristic, complemented
49#11762561C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762541C2211yescharacteristic, complemented
52#11762539C2211yescharacteristic, complemented
53#11762538C2211yescharacteristic
54D4⋊#3622C2211yescharacteristic
55D4⋊#3622C2211yescharacteristic, complemented
56#11762536C2211yescharacteristic
57#11762531C2211yescharacteristic, complemented
58#11762531C2211yescharacteristic, complemented
59#11762531C2211yescharacteristic, complemented
60C4×D8×C17C2211yescharacteristic
61C8×D4×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic, complemented
63C17×#334C2211yescharacteristic, complemented
64C17×D4⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic, complemented
66C17×#322C2211yescharacteristic, complemented
67C2×D4×D17C2×C411yescharacteristic
68C22×C4×D17—21no—
69C2×D8×C17C2×C411yescharacteristic
70C22×C8×C17—21no—
71C22×C8×C17—21no—
72C4×D4×C17C2311yescharacteristic
73C4×D4×C17C2311yescharacteristic
74#10587624C2×C411yescharacteristic
75#10587622C2×C411yescharacteristic
76D4×Dic17C2×C411yescharacteristic
77C2×#63229C2×C411yescharacteristic
78C2×#63228C2×C411yescharacteristic
79C4×C22⋊D17—21no—
80C4×C22⋊D17—21no—
81#10587612—21no—
82C2×#63223—21no—
83C22×#3622—21no—
84C2×#63219C2×C411yescharacteristic
85C2×#63218—21no—
86C2×C8×D17—21no—
87C2×C8×D17—21no—
88C2×C8×D17—21no—
89C2×C8×D17—44no—
90C2×C8×D17D411yescharacteristic
91C2×C8×D17D411yescharacteristic
92#10587583C2×C411yescharacteristic
93#10587579C2×C411yescharacteristic
94C4⋊C4×D17C2×C411yescharacteristic
95#10587576C2×C411yescharacteristic
96#10587569—21no—
97#10587567—21no—
98C22⋊C4×D17—21no—
99#10587565—21no—
100C4×D68C2311yescharacteristic
101C42×D17C2311yescharacteristic
102C4×Dic34C2311yescharacteristic
103C17×#107C2×C411yescharacteristic
104C17×C4⋊C8C2311yescharacteristic
105C17×C4⋊C8C2311yescharacteristic
106C17×#102C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×C22⋊C8—21no—
109C17×C22⋊C8—21no—
110C4×C8×C17C2311yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C4×C8×C17C2311yescharacteristic
113#10587503C2×C411yescharacteristic
114#10587503C2×C411yescharacteristic
115C22⋊#3622—21no—
116C22⋊#3622—21no—
117#10587492C2×C411yescharacteristic
118#10587491—21no—
119#10587491—21no—
120#10587491—21no—
121#10587491C2×C411yescharacteristic
122#10587491C2×C411yescharacteristic
123#10587488C2×C411yescharacteristic
124#10587487C2×C411yescharacteristic
125#10587486—21no—
126#10587486—21no—
127#10587486—21no—
128#10587485—21no—
129#10587485—21no—
130#10587485—21no—
131#10587485C2×C411yescharacteristic
132#10587485C2×C411yescharacteristic
133C8×Dic17—21no—
134C8×Dic17D411yescharacteristic
135C8×Dic17D411yescharacteristic
136#10587481C2×C411yescharacteristic
137#10587480C2×C411yescharacteristic
138#10587479C2×C411yescharacteristic
139#10587478C2×C411yescharacteristic
140C4⋊#3622C2311yescharacteristic
141C4⋊#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C23×D17—21no—
146C2×D4×C17C22×C411yescharacteristic
147C2×D4×C17C22×C411yescharacteristic
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C2×C22⋊D17—21no—
151C2×C22⋊D17—21no—
152C22×Dic17—21no—
153D4×D17—22no—
154D4×D17—22no—
155C2×D68C22×C411yescharacteristic
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—42no—
160C2×C4×D17—42no—
161C2×C4×D17C22×C411yescharacteristic
162C2×C4×D17C2×D411yescharacteristic
163C2×C4×D17#4111yescharacteristic
164C2×Dic34C22×C411yescharacteristic
165D8×C17—22no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—42no—
170C2×C8×C17—42no—
171C2×C8×C17C22×C411yescharacteristic
172C2×C8×C17C22×C411yescharacteristic
173C2×C8×C17C2×D411yescharacteristic
174C2×C8×C17C2×D411yescharacteristic
175C4⋊C4×C17C22×C411yescharacteristic
176C4⋊C4×C17C22×C411yescharacteristic
177C22⋊C4×C17—21no—
178C22⋊C4×C17—21no—
179C42×C17C2411yescharacteristic
180C22⋊Dic17—21no—
181C22⋊Dic17—21no—
182C22⋊Dic17—21no—
183C22⋊Dic17—21no—
184#63229—22no—
185#63228—22no—
186#63227—21no—
187#63227—21no—
188#63227—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17C22×C411yescharacteristic
192C4⋊Dic17C22×C411yescharacteristic
193#63225—21no—
194#63225—21no—
195#63225—21no—
196C4×Dic17—21no—
197C4×Dic17C22×C411yescharacteristic
198C4×Dic17C2×D411yescharacteristic
199C4×Dic17#4111yescharacteristic
200#63223—42no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—42no—
205C2×#3622C22×C411yescharacteristic
206C2×#3622C22×C411yescharacteristic
207C2×#3622C2×D411yescharacteristic
208C2×#3622C2×D411yescharacteristic
209#63219—22no—
210#63218—42no—
211C8×D17—22no—
212C8×D17—41no—
213C8×D17—42no—
214C8×D17—42no—
215C8×D17—42no—
216#17914—171nomaximal, Sylow
217C23×C17—21no—
218C23×C17—21no—
219C22×D17—24no—
220C22×D17—21no—
221C22×D17—42no—
222C22×D17—42no—
223C22×D17C4×D411yescharacteristic
224D4×C17—21no—
225D4×C17—21no—
226D4×C17—21no—
227D4×C17#13112yes—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C22⋊D17—24no—
238C22⋊D17—42no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—42no—
243C2×Dic17C4×D411yescharacteristic
244C2×Dic17C4×D411yescharacteristic
245C2×Dic17C4×D411yescharacteristic
246D68—21no—
247D68#13112yes—
248C4×D17—22no—
249C4×D17—21no—
250C4×D17—21no—
251C4×D17—21no—
252C4×D17—21no—
253C4×D17—42no—
254C4×D17—42no—
255C4×D17—41no—
256C4×D17—41no—
257Dic34—21no—
258Dic34—21no—
259C8×C17—21no—
260C8×C17—21no—
261C8×C17—21no—
262C8×C17—21no—
263C8×C17—41no—
264C8×C17—41no—
265C8×C17—41no—
266#3622—21no—
267#3622—21no—
268#3622—21no—
269#3622—41no—
270#3622—41no—
271#3622—41no—
272#3622AGL(1,ℤ/8)12yes—
273#3105—171no—
274#2966—171no—
275C2×C8×D4—171no—
276#1582—171no—
277#1582—171no—
278#1581—171no—
279#1581—171no—
280#1580—171no—
281#1580D1711yescharacteristic, complemented
282SD16⋊C8—171no—
283SD16⋊C8—171no—
284#1573—171no—
285#1490—171no—
286#1483—171no—
287#1451—171no—
288C22×C17—22no—
289C22×C17—41no—
290C22×C17—41no—
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17C2×C4×D411yescharacteristic
295C2×D17—21no—
296C2×D17—21no—
297C2×D17—22no—
298C2×D17—48no—
299C2×D17—41no—
300C2×D17—41no—
301C2×D17#44412yes—
302C4×C17—21no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17C2×#13111yescharacteristic
306C4×C17C2×#13111yescharacteristic
307C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C17#57511yescharacteristic
309Dic17—21no—
310Dic17—21no—
311Dic17—21no—
312Dic17—41no—
313Dic17—41no—
314Dic17—41no—
315C23×C8—341no—
316#529—171no—
317C2×C4×D4—171no—
318#484—171no—
319#481—171no—
320#478—171no—
321#478—171no—
322#463—171no—
323#463—171no—
324#450—171no—
325#447—171no—
326#446—171no—
327C8×Q8—171no—
328C4×SD16—171no—
329C4×SD16—171no—
330C4×D8—171no—
331C4×D8C2×D1711yescharacteristic
332#436—171no—
333#435—341no—
334C8×D4—171no—
335C8×D4—171no—
336C8×D4—171no—
337C8×D4—341no—
338C8×D4—341no—
339C8×D4—341no—
340C8×D4C2×D1711yescharacteristic
341C8×D4C2×D1711yescharacteristic, complemented
342#433—341no—
343#432—171no—
344#431—171no—
345C2×C4⋊C8—171no—
346C2×C22⋊C8—341no—
347#405—341no—
348C4×M16—171no—
349C2×C4×C8—171no—
350#335—171no—
351#335—171no—
352#334—171no—
353#334C2×D1711yescharacteristic, complemented
354Q8⋊C8—171no—
355Q8⋊C8—171no—
356D4⋊C8—171no—
357D4⋊C8—171no—
358D4⋊C8—171no—
359D4⋊C8—171no—
360D4⋊C8C2×D1711yescharacteristic
361D4⋊C8C2×D1711yescharacteristic, complemented
362#322—171no—
363#322—171no—
364#322—171no—
365#322C2×D1711yescharacteristic, complemented
366C2×C17—42no—
367C2×C17—81no—
368C2×C17#292911yescharacteristic
369C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
370C2×C17C2×#44411yescharacteristic
371D17—22no—
372D17—42no—
373C22×D4—171no—
374C23×C4—341no—
375C2×SD16—171no—
376C2×SD16—171no—
377C2×D8—171no—
378C2×D8C4×D1711yescharacteristic
379C2×M16—341no—
380C2×M16—341no—
381C22×C8—21no—
382C22×C8—21no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—341no—
386C22×C8—341no—
387C22×C8—341no—
388C22×C8—341no—
389C22×C8—684no—
390#126—341no—
391#125—171no—
392#124—171no—
393#123—341no—
394C22⋊Q8—171no—
395#121—171no—
396C4×Q8—171no—
397C4×D4—171no—
398C4×D4—171no—
399C4×D4—171no—
400C4×D4—341no—
401C4×D4—341no—
402C4×D4—341no—
403C4×D4C22×D1711yescharacteristic
404C4×D4C22×D1711yescharacteristic
405#117—341no—
406C2×C4⋊C4—171no—
407C2×C22⋊C4—341no—
408C2×C42—171no—
409#107—171no—
410#107C4×D1711yescharacteristic
411#106—171no—
412#106—171no—
413C4⋊C8—171no—
414C4⋊C8—171no—
415C4⋊C8—171no—
416C4⋊C8—171no—
417C4⋊C8—341no—
418C4⋊C8—341no—
419C4⋊C8—341no—
420C4⋊C8C22×D1711yescharacteristic
421C4⋊C8C22×D1711yescharacteristic
422#103—171no—
423#103—171no—
424#102—171no—
425#102—171no—
426#102—171no—
427#102—171no—
428#102C4×D1711yescharacteristic
429#102C4×D1711yescharacteristic
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—341no—
440#97—341no—
441#97—341no—
442C4×C8—171no—
443C4×C8—171no—
444C4×C8—171no—
445C4×C8—171no—
446C4×C8—171no—
447C4×C8—341no—
448C4×C8C22×D1711yescharacteristic
449C4×C8C22×D1711yescharacteristic
450C4×C8C22×D1711yescharacteristic
451C17#1791411yescharacteristic, complemented, Sylow
452C24—341no—
453C2×Q8—171no—
454C2×D4—171no—
455C2×D4—341no—
456C2×D4—342no—
457C2×D4—342no—
458C2×D4—341no—
459C2×D4C2×C4×D1711yescharacteristic
460C2×D4C2×C4×D1711yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—171no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4—341no—
470C22×C4—682no—
471C22×C4—682no—
472SD16—342no—
473SD16—342no—
474D8—22no—
475D8—342no—
476M16—682no—
477M16—682no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—171no—
484C2×C8—171no—
485C2×C8—171no—
486C2×C8—171no—
487C2×C8—342no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—681no—
492C2×C8—682no—
493C2×C8—682no—
494C2×C8—682no—
495C2×C8—682no—
496C2×C8C2×C4×D1711yescharacteristic
497C2×C8C2×C4×D1711yescharacteristic
498C2×C8D4×D1711yescharacteristic
499C2×C8D4×D1711yescharacteristic
500C4⋊C4—171no—
501C4⋊C4—171no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4C2×C4×D1711yescharacteristic
508C4⋊C4C2×C4×D1711yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—341no—
512C22⋊C4—341no—
513C22⋊C4—341no—
514C22⋊C4—341no—
515C22⋊C4—341no—
516C22⋊C4—341no—
517C22⋊C4—341no—
518C42—171no—
519C42—171no—
520C42—171no—
521C42—341no—
522C42C23×D1711yescharacteristic
523C23—21no—
524C23—21no—
525C23—171no—
526C23—344no—
527C23—341no—
528C23—682no—
529C23—682no—
530Q8—341no—
531Q8—341no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—172no—
536D4—344no—
537D4—341no—
538D4—682no—
539D4#1058758712yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—171no—
547C2×C4—171no—
548C2×C4—171no—
549C2×C4—342no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4C22×C4×D1711yescharacteristic
563C2×C4C2×D4×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—172no—
573C8—341no—
574C8—341no—
575C8—341no—
576C8—681no—
577C8—681no—
578C8—681no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—172no—
586C22—341no—
587C22—341no—
588C22—342no—
589C22—688no—
590C22—681no—
591C22—681no—
592C22C4×D4×D1711yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—341no—
597C4—341no—
598C4—341no—
599C4—681no—
600C4—681no—
601C4—681no—
602C4#131×D1711yescharacteristic
603C4C2×#1058758711yescharacteristic
604C4AGL(1,ℤ/8)×D1711yescharacteristic
605C4#1176360511yescharacteristic
606C2—42no—
607C2—81no—
608C2—342no—
609C2—682no—
610C2#1453747311yescharacteristic
611C2#1453759511yescharacteristic
612C2D17×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1580C2
socleC22×C17C2×C4×D4
prime radical#11762941C4
cosocleC42×C17C24
hypercenter#1580D17
hypocenterC17#17914
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #14537756C2semidirect
3. C8×D4×D17C2semidirect
4. #14527619C2semidirect
5. #14527617C2semidirect
6. #14527616C2semidirect
7. #14527614C2semidirect
8. #14527606C2semidirect
9. #14527598C2semidirect
10. #14527596C2semidirect
11. #14527566C2semidirect
12. #14527225C2semidirect
13. #14527194C2semidirect
14. #14527072C2semidirect
15. C17×#1580C2semidirect
27. #11762936C22semidirect
28. C8×Dic34C22semidirect
32. #11762887C22semidirect
48. #11762562C22semidirect
49. #11762561C22semidirect
50. #11762559C22semidirect
51. #11762541C22semidirect
52. #11762539C22semidirect
55. D4⋊#3622C22semidirect
57. #11762531C22semidirect
58. #11762531C22semidirect
59. #11762531C22semidirect
62. C8×D4×C17C22semidirect
63. C17×#334C22semidirect
65. C17×D4⋊C8C22semidirect
66. C17×#322C22semidirect
281. #1580D17semidirect
341. C8×D4C2×D17semidirect
353. #334C2×D17semidirect
361. D4⋊C8C2×D17semidirect
365. #322C2×D17semidirect
451. C17#17914semidirect

Maximal subgroups

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

subgroupindex
#145388292
#145377562
C8×D4×D172
#145276192
#145276172
#145276162
#145276142
#145276062
#145275982
#145275962
#145275662
#145272252
#145271942
#145270722
C17×#15802
#1791417