Finite Group Databasebeta

← #17045451

Subgroups — #17045451

Order 4352.

1308 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×#11762942C211yesmaximal, characteristic
2C2×#11762711C211yesmaximal, characteristic, complemented
3#14529649C212yesmaximal, complemented
4#14529466C212yesmaximal, complemented
5#14529404C212yesmaximal, complemented
6#14529132C212yesmaximal, complemented
7#14529102C211yesmaximal, characteristic, complemented
8#14528971C211yesmaximal, characteristic, complemented
9#14528714C211yesmaximal, characteristic, complemented
10C4×C22⋊#3622C211yesmaximal, characteristic, complemented
11C4×C17×#121C211yesmaximal, characteristic, complemented
12C23×D68C2211yescharacteristic
13C22×#63228C2212yes—
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63227—21no—
16C22×#63227—21no—
17#11762942C418yescomplemented
18C2×#10587478C2212yescomplemented
19C2×#10587503C2212yescomplemented
20C2×#10587480C2212yes—
21C2×#10587480C2212yes—
22C2×#10587480C2212yes—
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C4×#3622C2212yescomplemented
26#11762711—21no—
27#11762711—21no—
28#11762711C2212yes—
29#11762711C2212yes—
30#11762571C2212yescomplemented
31#11762571C2212yescomplemented
32#11762564C2212yescomplemented
33C2×C17×#121C2211yescharacteristic
34C2×C4×D4×C17—21no—
35C2×C4×D4×C17C2211yescharacteristic, complemented
36C22×C42×C17C2211yescharacteristic
37C17×#390C2211yescharacteristic, complemented
38C17×#389C2211yescharacteristic, complemented
39C17×#388—21no—
40C17×#386—21no—
41C17×#380C2211yescharacteristic
42C4×C4⋊C4×C17C2211yescharacteristic, complemented
43C4×C22⋊C4×C17—21no—
44C24×D17—21no—
45C22×D4×C17—21no—
46C22×D4×C17C2311yescharacteristic
47C23×C4×C17—21no—
48C23×C4×C17D411yescharacteristic
49C23×C4×C17D411yescharacteristic
50C23×C4×C17C2311yescharacteristic
51C23×Dic17—21no—
52C22×D68—21no—
53C22×D68—22no—
54C22×D68—42no—
55C22×D68—42no—
56C22×D68C2×C414yescomplemented
57C22×D68C2312yes—
58C17×#121—22no—
59C17×#121C2×C414yes—
60C4×D4×C17—42no—
61C4×D4×C17—44no—
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C42×C17—21no—
72C2×C42×C17—42no—
73C2×C42×C17D411yescharacteristic
74C2×C42×C17D411yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×#63228—42no—
78C2×#63228—42no—
79C2×#63228C2×C418yescomplemented
80C2×#63227—22no—
81C2×#63227—22no—
82C2×#63227—22no—
83C2×#63227—22no—
84C2×#63227—44no—
85C2×#63227—44no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—42no—
88C2×C4⋊Dic17C2312yes—
89C22×#3622C2312yes—
90C22×#3622C2312yes—
91C17×#95—21no—
92C17×#95—21no—
93C17×#95—21no—
94#10587503—42no—
95C22⋊#3622—22no—
96C22⋊#3622C2×C414yescomplemented
97#10587480—42no—
98#10587480—42no—
99#10587480C2×C418yescomplemented
100#10587478—42no—
101C4×#3622—42no—
102C24×C17—21no—
103C24×C17C2×D411yescharacteristic
104C23×D17—24no—
105C23×D17—22no—
106C23×D17—42no—
107C23×D17—42no—
108C23×D17—42no—
109C23×D17—42no—
110C23×D17—42no—
111C23×D17—42no—
112C2×D4×C17—24no—
113C2×D4×C17—21no—
114C2×D4×C17—22no—
115C2×D4×C17—22no—
116C2×D4×C17—42no—
117C2×D4×C17—42no—
118C2×D4×C17C22×C412yes—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—41no—
127C22×C4×C17—41no—
128C22×C4×C17—41no—
129C22×C4×C17—42no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17D814yescomplemented
133C22×C4×C17C22×C412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17#4112yes—
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—22no—
142C22×Dic17—42no—
143C22×Dic17—42no—
144C22×Dic17—42no—
145C2×D68—22no—
146C2×D68—22no—
147C2×D68—22no—
148C2×D68—22no—
149C2×D68—24no—
150C2×D68—22no—
151C2×D68—48no—
152C2×D68—42no—
153C2×D68—42no—
154C2×D68—48no—
155C2×D68C22×C414yes—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—42no—
160C4⋊C4×C17C22×C412yes—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—42no—
163C22⋊C4×C17—42no—
164C22⋊C4×C17—42no—
165C22⋊C4×C17—42no—
166C42×C17—22no—
167C42×C17—22no—
168C42×C17—41no—
169C42×C17—41no—
170C42×C17—42no—
171#63228—48no—
172#63227—44no—
173#63227—44no—
174#63227—44no—
175#63227—44no—
176C4⋊Dic17—22no—
177C4⋊Dic17—22no—
178C4⋊Dic17—42no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622C22×C414yes—
185#22040—171nomaximal, Sylow
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17C4×D412yes—
198C23×C17C2×D812yes—
199C23×C17#13512yes—
200C23×C17C22×D411yescharacteristic
201C22×D17—24no—
202C22×D17—48no—
203C22×D17—48no—
204C22×D17—42no—
205C22×D17—42no—
206C22×D17—42no—
207C22×D17—42no—
208C22×D17—42no—
209C22×D17—42no—
210C22×D17—48no—
211C22×D17—84no—
212C22×D17—84no—
213C22×D17—84no—
214C22×D17—84no—
215D4×C17—24no—
216D4×C17—48no—
217D4×C17—42no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—24no—
227C2×C4×C17—24no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—44no—
233C2×C4×C17—44no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—84no—
243C2×C4×C17C4×D412yes—
244C2×C4×C17C2×D812yes—
245C2×C4×C17C2×D812yes—
246C2×C4×C17AGL(1,ℤ/8)12yes—
247C2×C4×C17C23×C411yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17H(5,𝔽2)12yes—
250C2×C4×C17H(5,𝔽2)12yes—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—84no—
255D68—24no—
256D68—24no—
257D68—48no—
258D68—42no—
259D68—42no—
260#3622—24no—
261#3622—42no—
262C2×#447—171no—
263C4×#121D1711yescharacteristic, complemented
264C2×#390—171no—
265#1970—172no—
266#1934—172no—
267#1925—172no—
268#1879—172no—
269#1874—171no—
270#1852—171no—
271#1786—171no—
272C4×C22⋊C8—171no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—22no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17C4×D814yes—
287C22×C17C2×C4×D411yescharacteristic
288C22×C17#53411yescharacteristic
289C22×C17#53511yescharacteristic
290C22×C17C22×D811yescharacteristic
291C22×C17C2×#13511yescharacteristic
292C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C17#57511yescharacteristic
294C2×D17—44no—
295C2×D17—44no—
296C2×D17—44no—
297C2×D17—816no—
298C2×D17—82no—
299C2×D17—82no—
300C2×D17—82no—
301C2×D17—82no—
302C4×C17—22no—
303C4×C17—22no—
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—41no—
307C4×C17—42no—
308C4×C17—81no—
309C4×C17—81no—
310C4×C17#51812yes—
311C4×C17#53812yes—
312Dic17—82no—
313C23×D4—171no—
314C22×D8—172no—
315C2×#121C2×D1711yescharacteristic
316C2×C4×D4—21no—
317C2×C4×D4C2×D1711yescharacteristic, complemented
318C22×C4⋊C4—171no—
319C22×C22⋊C4—341no—
320C22×C22⋊C4—341no—
321C22×C42C2×D1711yescharacteristic
322#447—178no—
323C2×#107—172no—
324C2×#102—172no—
325C2×#102—172no—
326C2×#102—172no—
327C2×#102—172no—
328C2×C22⋊C8—171no—
329C2×C22⋊C8—171no—
330C2×C4×C8—172no—
331#390—172no—
332#390—172no—
333#390—341no—
334#390—341no—
335#390C2×D1711yescharacteristic, complemented
336#389C2×D1711yescharacteristic, complemented
337#388—21no—
338#386—21no—
339#380C2×D1711yescharacteristic
340C4×C4⋊C4C2×D1711yescharacteristic, complemented
341C4×C22⋊C4—21no—
342#340—172no—
343#340—172no—
344#336—172no—
345C2×C17—24no—
346C2×C17—82no—
347C2×C17#232111yescharacteristic
348C2×C17C2×C4×D811yescharacteristic
349C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
350C2×C17#318611yescharacteristic
351C2×C17#318811yescharacteristic
352C2×C17#320511yescharacteristic
353C2×C17#321111yescharacteristic
354D17—84no—
355C25—341no—
356C22×D4—21no—
357C22×D4—174no—
358C22×D4—172no—
359C22×D4—342no—
360C22×D4—341no—
361C22×D4—682no—
362C22×D4—682no—
363C22×D4C22×D1711yescharacteristic
364C23×C4—21no—
365C23×C4—341no—
366C23×C4C22⋊D1711yescharacteristic
367C23×C4C22⋊D1711yescharacteristic
368C23×C4C22×D1711yescharacteristic
369C2×D8—178no—
370C2×D8—682no—
371C2×D8—682no—
372C22×C8—172no—
373C22×C8—172no—
374#121—22no—
375#121C4×D1714yes—
376C4×D4—42no—
377C4×D4—44no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—172no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—682no—
382C2×C4⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—342no—
391C2×C22⋊C4—342no—
392C2×C22⋊C4—342no—
393C2×C22⋊C4—342no—
394C2×C22⋊C4—684no—
395C2×C22⋊C4—684no—
396C2×C42—21no—
397C2×C42—42no—
398C2×C42C22⋊D1711yescharacteristic
399C2×C42C22⋊D1711yescharacteristic
400C2×C42C22×D1711yescharacteristic
401C2×C42C22×D1711yescharacteristic
402#107—682no—
403#102—178no—
404#102—682no—
405#102—682no—
406#102—682no—
407C22⋊C8—174no—
408C22⋊C8—342no—
409C4×C8—682no—
410#95—21no—
411#95—21no—
412#95—21no—
413C17#2204011yescharacteristic, complemented, Sylow
414C24—21no—
415C24—342no—
416C24—344no—
417C24—682no—
418C24—682no—
419C24—682no—
420C24—682no—
421C24—682no—
422C24—682no—
423C24C2×C22⋊D1711yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—174no—
431C2×D4—342no—
432C2×D4—342no—
433C2×D4—342no—
434C2×D4—342no—
435C2×D4—342no—
436C2×D4—344no—
437C2×D4—688no—
438C2×D4—682no—
439C2×D4—682no—
440C2×D4—688no—
441C2×D4C2×C4×D1712yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—342no—
456C22×C4—682no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4#6322814yescomplemented
460C22×C4C2×C4×D1712yes—
461C22×C4#6325212yes—
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468D8—688no—
469C2×C8—174no—
470C2×C8—342no—
471C2×C8—342no—
472C2×C8—342no—
473C2×C8—342no—
474C2×C8—342no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—342no—
480C4⋊C4—342no—
481C4⋊C4—682no—
482C4⋊C4C2×C4×D1712yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—684no—
489C22⋊C4—684no—
490C22⋊C4—684no—
491C22⋊C4—684no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—344no—
509C23—688no—
510C23—688no—
511C23—682no—
512C23—682no—
513C23—682no—
514C23—682no—
515C23—682no—
516C23—682no—
517C23—688no—
518C23—1364no—
519C23—1364no—
520C23—1364no—
521C23—1364no—
522C23C4×C22⋊D1712yes—
523C23C2×#6322812yes—
524C23#1058763612yes—
525C23C22×C22⋊D1711yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—344no—
530D4—344no—
531D4—688no—
532D4—682no—
533D4—682no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—1364no—
563C2×C4C4×C22⋊D1712yes—
564C2×C4C2×#6322812yes—
565C2×C4C2×#6322812yes—
566C2×C4#1058763512yes—
567C2×C4C22×C4×D1711yescharacteristic
568C2×C4C2×#6325211yescharacteristic
569C2×C4#1058769112yes—
570C2×C4#1058769612yes—
571C8—344no—
572C8—682no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—684no—
587C22—684no—
588C22—684no—
589C22—13616no—
590C22—1362no—
591C22—1362no—
592C22—1362no—
593C22—1362no—
594C22C4×#6322814yes—
595C22C2×C4×C22⋊D1711yescharacteristic
596C22#1176336811yescharacteristic
597C22#1176337311yescharacteristic
598C22C22×#6322811yescharacteristic
599C22C2×#1058763611yescharacteristic
600C22C2×#1058763511yescharacteristic
601C22#1176360711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—1362no—
611C4#1176328112yes—
612C4#1176339212yes—
613C2—24no—
614C2—82no—
615C2—1364no—
616C2#1453143811yescharacteristic
617C2C2×C4×#6322811yescharacteristic
618C2C4×#1058763511yescharacteristic
619C2#1453953211yescharacteristic
620C2#1453954511yescharacteristic
621C2#1453969911yescharacteristic
622C2#1453974511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#121C2
socleC23×C17C22×D4
prime radicalC2×#11762942C2
cosocleC22×C4×C17C24
hypercenterC4×#121D17
hypocenterC17#22040
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#11762711C2semidirect
3. #14529649C2semidirect
4. #14529466C2semidirect
5. #14529404C2semidirect
6. #14529132C2semidirect
7. #14529102C2semidirect
8. #14528971C2semidirect
9. #14528714C2semidirect
10. C4×C22⋊#3622C2semidirect
11. C4×C17×#121C2semidirect
17. #11762942C4semidirect
18. C2×#10587478C22semidirect
19. C2×#10587503C22semidirect
25. C2×C4×#3622C22semidirect
30. #11762571C22semidirect
31. #11762571C22semidirect
32. #11762564C22semidirect
35. C2×C4×D4×C17C22semidirect
37. C17×#390C22semidirect
38. C17×#389C22semidirect
42. C4×C4⋊C4×C17C22semidirect
56. C22×D68C2×C4semidirect
79. C2×#63228C2×C4semidirect
96. C22⋊#3622C2×C4semidirect
99. #10587480C2×C4semidirect
132. C22×C4×C17D8semidirect
263. C4×#121D17semidirect
317. C2×C4×D4C2×D17semidirect
335. #390C2×D17semidirect
336. #389C2×D17semidirect
340. C4×C4⋊C4C2×D17semidirect
413. C17#22040semidirect
459. C22×C4#63228semidirect

Maximal subgroups

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

subgroupindex
C2×#117629422
C2×#117627112
#145296492
#145294662
#145294042
#145291322
#145291022
#145289712
#145287142
C4×C22⋊#36222
C4×C17×#1212
#2204017