Finite Group Databasebeta

← #17036508

Subgroups — #17036508

Order 4352.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762848C211yesmaximal, characteristic
2D17×#389C211yesmaximal, characteristic
3C4×#10587578C211yesmaximal, characteristic
4#14529377C211yesmaximal, characteristic, complemented
5#14529373C211yesmaximal, characteristic, complemented
6#14529369C211yesmaximal, characteristic, complemented
7#14529364C211yesmaximal, characteristic, complemented
8#14529360C211yesmaximal, characteristic, complemented
9#14529358C211yesmaximal, characteristic, complemented
10#14529354C211yesmaximal, characteristic, complemented
11#14528502C211yesmaximal, characteristic
12#14528502C211yesmaximal, characteristic
13#14528502C211yesmaximal, characteristic
14#14528476C211yesmaximal, characteristic
15C17×#1920C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C4⋊C4×D17C2211yescharacteristic
23C2×C42×D17C2211yescharacteristic
24#11762848C418yescomplemented
25C2×#10587489C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762707C2211yescharacteristic
30#11762706—21no—
31#11762705C2211yescharacteristic
32#11762704—21no—
33C4⋊C4×Dic17C2211yescharacteristic
34C4⋊C4×Dic17C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36C4×#63227—21no—
37#11762656—21no—
38#11762654—21no—
39#95×D17—21no—
40C42×Dic17—21no—
41#11762572C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×#106C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×C4×D17—21no—
61C22×C4×D17—21no—
62C22×C4×D17D411yescharacteristic
63C22×C4×D17D411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C42×C17—21no—
73C2×C42×C17C2311yescharacteristic
74C2×#63227—21no—
75C2×#63227—21no—
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×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C2×#63218—22no—
96C2×#63218—22no—
97C2×#63218—22no—
98C2×#63218—22no—
99C2×#63218—22no—
100C2×#63218C2×C414yes—
101#10587578—22no—
102#10587578—22no—
103#10587578C2×C414yes—
104#10587578Q814yes—
105C4⋊C4×D17—22no—
106C4⋊C4×D17—22no—
107C4⋊C4×D17—22no—
108C4⋊C4×D17—44no—
109C4⋊C4×D17C2×C414yes—
110C42×D17—22no—
111C42×D17—22no—
112C17×#106C2×C414yescomplemented
113C17×#95—21no—
114C17×#95—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587489C2×C414yescomplemented
122#10587479C2×C414yescomplemented
123#10587478C2×C414yescomplemented
124C23×D17C2×D411yescharacteristic
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C2×C4×D17—24no—
140C2×C4×D17—22no—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—41no—
145C2×C4×D17—41no—
146C2×C4×D17—42no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C2×C4×D17C22×C412yes—
152C2×C4×D17C2×Q812yes—
153C2×C4×D17#4112yes—
154C2×C4×D17#4112yes—
155C2×C4×D17#4112yes—
156C2×C4×D17#4112yes—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17C22×C412yes—
170C4⋊C4×C17C22×C412yes—
171C4⋊C4×C17C2×Q812yes—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—42no—
175#63227—24no—
176#63227—24no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—42no—
183C4⋊Dic17C22×C412yes—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17C2×Q812yes—
186#63225—24no—
187#63225—42no—
188#63225—42no—
189#63225—42no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—24no—
193C4×Dic17—24no—
194C4×Dic17—22no—
195C4×Dic17—22no—
196C4×Dic17—42no—
197C4×Dic17—42no—
198C4×Dic17—42no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622C22×C412yes—
205#63218—28no—
206#63218—44no—
207#21034—171nomaximal, Sylow
208C23×C17C22×D411yescharacteristic
209C22×D17—21no—
210C22×D17—21no—
211C22×D17—21no—
212C22×D17—21no—
213C22×D17C4×D412yes—
214C22×D17C22⋊Q812yes—
215C22×D17#12312yes—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—41no—
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×C17#13712yes—
234C2×C4×C17C23×C411yescharacteristic
235C2×C4×C17C22×Q811yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17C4×D412yes—
258C2×Dic17C22⋊Q812yes—
259C2×Dic17#12312yes—
260C2×Dic17AGL(1,ℤ/8)12yes—
261C4×D17—44no—
262C4×D17—44no—
263C4×D17—48no—
264C4×D17—84no—
265C4×D17#11714yes—
266C4×D17C4×Q814yes—
267C4×D17#12514yes—
268C4×D17#12614yes—
269C8×C17—22no—
270C8×C17—41no—
271#3622—22no—
272#3622—41no—
273C2×#428—171no—
274C2×#389—171no—
275C4×#117—171no—
276#1921—171no—
277#1921—171no—
278#1920—171no—
279#1920D1711yescharacteristic, complemented
280#1919—171no—
281#1919—171no—
282#1918—171no—
283#1918—171no—
284#1736—171no—
285#1736—171no—
286#1736—171no—
287#1728—171no—
288C22×C17C2×C4×D411yescharacteristic
289C22×C17C2×C22⋊Q811yescharacteristic
290C22×C17C2×#12311yescharacteristic
291C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C17C2×#13711yescharacteristic
293C22×C17#57511yescharacteristic
294C22×C17#57511yescharacteristic
295C2×D17—22no—
296C2×D17—22no—
297C2×D17—22no—
298C2×D17—22no—
299C2×D17—22no—
300C2×D17—22no—
301C2×D17#38714yes—
302C4×C17—22no—
303C4×C17—42no—
304C4×C17—41no—
305C4×C17—42no—
306C4×C17—42no—
307C4×C17—81no—
308C4×C17C2×#11711yescharacteristic
309C4×C17C2×C4×Q811yescharacteristic
310C4×C17C2×#12511yescharacteristic
311C4×C17C2×#12611yescharacteristic
312Dic17—22no—
313Dic17—42no—
314Dic17—41no—
315Dic17—42no—
316Dic17—42no—
317Dic17—81no—
318Dic17#38714yes—
319C22×M16—171no—
320C22×M16—171no—
321C2×#117—171no—
322C2×#117—171no—
323C22×C4⋊C4—171no—
324C22×C4⋊C4—171no—
325C22×C4⋊C4—341no—
326C22×C42—171no—
327#428—178no—
328C2×#107—171no—
329C2×#107—171no—
330C2×#106—171no—
331C2×#106C2×D1711yescharacteristic
332#389—171no—
333#389—171no—
334#389—171no—
335#389—341no—
336#389—341no—
337#389C2×D1711yescharacteristic
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4C2×D1711yescharacteristic
342C4×C22⋊C4—341no—
343C4×C22⋊C4—341no—
344#376—341no—
345C2×#95—341no—
346C43—341no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340C2×D1711yescharacteristic
357#340C2×D1711yescharacteristic
358#340C2×D1711yescharacteristic
359#336—171no—
360#336—171no—
361#336—171no—
362#336C2×D1711yescharacteristic
363C2×C17C2×#38711yescharacteristic
364C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
365C2×C17C4×#13711yescharacteristic
366C2×C17#307911yescharacteristic
367C2×C17#308011yescharacteristic
368C2×C17#309411yescharacteristic
369C2×C17#309511yescharacteristic
370D17—24no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4—171no—
374C23×C4—341no—
375C23×C4—341no—
376C2×M16—174no—
377C2×M16—342no—
378C2×M16—342no—
379C2×M16—342no—
380C2×M16—342no—
381C2×M16—342no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8C22×D1711yescharacteristic
385C22×C8C22×D1711yescharacteristic
386#117—174no—
387#117—174no—
388#117—342no—
389#117—342no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—174no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—342no—
397C2×C4⋊C4—342no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—684no—
405C2×C4⋊C4C22×D1711yescharacteristic
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C4⋊C4C22×D1711yescharacteristic
409C2×C22⋊C4—341no—
410C2×C22⋊C4—341no—
411C2×C42—21no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—171no—
415C2×C42—342no—
416C2×C42—342no—
417C2×C42—341no—
418C2×C42—341no—
419C2×C42—341no—
420C2×C42—341no—
421C2×C42—341no—
422C2×C42C22×D1711yescharacteristic
423#107—174no—
424#107—174no—
425#106—174no—
426#106C4×D1714yescomplemented
427#95—21no—
428#95—21no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435C17#2103411yescharacteristic, complemented, Sylow
436C24—171no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—172no—
443C22×C4—172no—
444C22×C4—172no—
445C22×C4—172no—
446C22×C4—172no—
447C22×C4—172no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—344no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—342no—
458C22×C4—341no—
459C22×C4—681no—
460C22×C4—681no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4D4×D1711yescharacteristic
467C22×C4D4×D1711yescharacteristic
468C22×C4C23×D1711yescharacteristic
469M16—348no—
470M16—684no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—172no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8C2×C4×D1712yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—172no—
490C4⋊C4—172no—
491C4⋊C4—172no—
492C4⋊C4—344no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—682no—
499C4⋊C4—682no—
500C4⋊C4—682no—
501C4⋊C4—682no—
502C4⋊C4C2×C4×D1712yes—
503C4⋊C4C2×C4×D1712yes—
504C4⋊C4Q8×D1712yes—
505C22⋊C4—344no—
506C22⋊C4—344no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—342no—
511C42—342no—
512C42—344no—
513C42—344no—
514C42—341no—
515C42—341no—
516C42—682no—
517C42—682no—
518C42—682no—
519C23—172no—
520C23—172no—
521C23—172no—
522C23—341no—
523C23—341no—
524C23—341no—
525C23—341no—
526C23C2×D4×D1711yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—172no—
545C2×C4—172no—
546C2×C4—172no—
547C2×C4—174no—
548C2×C4—174no—
549C2×C4—174no—
550C2×C4—174no—
551C2×C4—172no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—684no—
561C2×C4—684no—
562C2×C4—688no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—1364no—
573C2×C4#1058760112yes—
574C2×C4C22×C4×D1711yescharacteristic
575C2×C4C2×Q8×D1711yescharacteristic
576C2×C4#41×D1711yescharacteristic
577C2×C4#41×D1711yescharacteristic
578C2×C4#41×D1711yescharacteristic
579C2×C4#41×D1711yescharacteristic
580C8—22no—
581C8—41no—
582C8—342no—
583C8—681no—
584C22—174no—
585C22—342no—
586C22—342no—
587C22—342no—
588C22—342no—
589C22—342no—
590C22—342no—
591C22C4×D4×D1711yescharacteristic
592C22C22⋊Q8×D1711yescharacteristic
593C22#123×D1711yescharacteristic
594C22AGL(1,ℤ/8)×D1711yescharacteristic
595C22C2×#1058760111yescharacteristic
596C22#1176360611yescharacteristic
597C22#1176360811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—174no—
605C4—342no—
606C4—682no—
607C4—682no—
608C4—682no—
609C4—681no—
610C4—1361no—
611C4#117×D1711yescharacteristic
612C4C4×Q8×D1711yescharacteristic
613C4#125×D1711yescharacteristic
614C4#126×D1711yescharacteristic
615C2—344no—
616C2D17×#38711yescharacteristic
617C2#1453758911yescharacteristic
618C2C4×#1058760111yescharacteristic
619C2#1453859611yescharacteristic
620C2#1453860311yescharacteristic
621C2#1453872311yescharacteristic
622C2#1453872711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1920C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1920D17
hypocenterC17#21034
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #14529377C2semidirect
5. #14529373C2semidirect
6. #14529369C2semidirect
7. #14529364C2semidirect
8. #14529360C2semidirect
9. #14529358C2semidirect
10. #14529354C2semidirect
15. C17×#1920C2semidirect
24. #11762848C4semidirect
112. C17×#106C2×C4semidirect
121. #10587489C2×C4semidirect
122. #10587479C2×C4semidirect
123. #10587478C2×C4semidirect
279. #1920D17semidirect
426. #106C4×D17semidirect
435. C17#21034semidirect

Maximal subgroups

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

subgroupindex
C2×#117628482
D17×#3892
C4×#105875782
#145293772
#145293732
#145293692
#145293642
#145293602
#145293582
#145293542
#145285022
#145284762
C17×#19202
#2103417