Finite Group Databasebeta

← #17086216

Subgroups — #17086216

Order 4352.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic
2C2×Q8×#3622C211yesmaximal, characteristic, complemented
3#14534172C211yesmaximal, characteristic, complemented
4#14530286C211yesmaximal, characteristic, complemented
5#14530280C211yesmaximal, characteristic, complemented
6#14530172C211yesmaximal, characteristic
7#14530158C211yesmaximal, characteristic
8#14529748C211yesmaximal, characteristic, complemented
9#14529550C211yesmaximal, characteristic, complemented
10#14529514C211yesmaximal, characteristic, complemented
11#14529509C211yesmaximal, characteristic, complemented
12#14529470C211yesmaximal, characteristic
13#14529470C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C17×#2580C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17#11763109—24no—
18Q8×#3622—24no—
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33#11762740—21no—
34#11762728—21no—
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762704—21no—
40#11762704—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#399—21no—
53C17×#393—21no—
54C17×#390C2211yescharacteristic
55C17×#390C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#385—21no—
59C17×#385—21no—
60C17×#382C2211yescharacteristic
61C22×Q8×C17D411yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17D411yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99#10587556—22no—
100#10587556—22no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105#10587503—22no—
106#10587503—22no—
107#10587503—22no—
108#10587503Q814yescomplemented
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587479—22no—
115#10587479—22no—
116#10587478—22no—
117#10587478—22no—
118C4⋊#3622—22no—
119C4⋊#3622—22no—
120C4⋊#3622D414yes—
121C4×#3622—22no—
122C4×#3622—22no—
123C4×#3622—22no—
124C24×C17—21no—
125C2×Q8×C17—22no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17C2×Q812yes—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—42no—
138C22×C4×C17—81no—
139C22×C4×C17—81no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2411yescharacteristic
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—22no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—81no—
161C4⋊C4×C17—81no—
162C22⋊C4×C17—42no—
163C22⋊C4×C17—42no—
164C22⋊C4×C17—42no—
165C22⋊C4×C17—81no—
166C22⋊C4×C17—81no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17C2×D412yes—
172C42×C17#4112yes—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17C2×Q812yes—
180C4⋊Dic17#4112yes—
181#63225—42no—
182#63225—42no—
183#63225—42no—
184#63225—42no—
185#63225—81no—
186#63225—81no—
187C4×Dic17—81no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622C2×D412yes—
198C2×#3622C2×D412yes—
199C2×#3622C2×Q812yes—
200#25473—171nomaximal, Sylow
201C23×C17—22no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17C22×D411yescharacteristic
209Q8×C17—28no—
210D4×C17—22no—
211D4×C17—22no—
212D4×C17—41no—
213D4×C17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
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—21no—
227C2×C4×C17—21no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—84no—
238C2×C4×C17#12112yes—
239C2×C4×C17#12112yes—
240C2×C4×C17#13512yes—
241C2×C4×C17#13512yes—
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C22×Q811yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260#3622—22no—
261#3622—22no—
262#3622—22no—
263#3622—41no—
264C2×#478—171no—
265C2×C8×Q8—171no—
266#2584—171no—
267#2580D1711yescharacteristic
268#2088—171no—
269#2087—171no—
270#2067—171no—
271#2065—171no—
272#1991—171no—
273#1949—171no—
274#1942—171no—
275#1941—171no—
276#1934—171no—
277#1934—171no—
278C4×#102—171no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17—81no—
286C22×C17C2×#12111yescharacteristic
287C22×C17#54811yescharacteristic
288C22×C17D4×Q811yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17#57611yescharacteristic
292C22×C17#57711yescharacteristic
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17—81no—
300C4×C17—81no—
301C4×C17—81no—
302C4×C17#53311yescharacteristic
303C4×C17#54111yescharacteristic
304C4×C17#54711yescharacteristic
305C4×C17D4×Q811yescharacteristic
306Dic17—42no—
307Dic17—42no—
308Dic17—81no—
309Dic17—81no—
310C2×#125—171no—
311C2×C4×Q8C2×D1711yescharacteristic
312C2×C4×D4C2×D1711yescharacteristic
313#478—344no—
314C8×Q8—344no—
315C2×#107—171no—
316C2×#107—171no—
317C2×#106—171no—
318C2×#106—171no—
319C2×C4⋊C8—171no—
320C2×C4⋊C8—171no—
321C2×C4⋊C8—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×#102—171no—
325C2×#102—171no—
326C2×C4×C8—171no—
327C2×C4×C8—171no—
328C2×C4×C8—171no—
329#399—21no—
330#398—341no—
331#395—341no—
332#393—21no—
333#390C2×D1711yescharacteristic
334#390C2×D1711yescharacteristic
335#389—171no—
336#389—171no—
337#389—171no—
338#389—171no—
339#389—341no—
340#389—341no—
341#389C2×D1711yescharacteristic
342#389C2×D1711yescharacteristic
343#385—21no—
344#385—21no—
345#382C2×D1711yescharacteristic
346C4×C4⋊C4—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355C2×C17—82no—
356C2×C17#242911yescharacteristic
357C2×C17#305011yescharacteristic
358C2×C17#306211yescharacteristic
359C2×C17#336111yescharacteristic
360C2×C17#336511yescharacteristic
361C2×C17#338011yescharacteristic
362C2×C17#338111yescharacteristic
363C22×Q8C22⋊D1711yescharacteristic
364C22×D4C22×D1711yescharacteristic
365C23×C4—21no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8—171no—
369C22×C8—171no—
370#125—342no—
371#125—342no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×D4—22no—
375C4×D4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4C22⋊D1711yescharacteristic
389C2×C4⋊C4C22⋊D1711yescharacteristic
390C2×C4⋊C4C22⋊D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C42—341no—
400C2×C42C22×D1711yescharacteristic
401C2×C42C22×D1711yescharacteristic
402C2×C42C22×D1711yescharacteristic
403#107—342no—
404#107—342no—
405#106—342no—
406#106—342no—
407C4⋊C8—174no—
408C4⋊C8—342no—
409C4⋊C8—342no—
410#102—174no—
411#102—342no—
412#102—342no—
413#102—342no—
414C4×C8—342no—
415C4×C8—342no—
416C4×C8—342no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426C17#2547311yescharacteristic, complemented, Sylow
427C24—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D1712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C2×C22⋊D1711yescharacteristic
449C22×C4C2×C22⋊D1711yescharacteristic
450C22×C4C2×C22⋊D1711yescharacteristic
451C22×C4C2×C22⋊D1711yescharacteristic
452C22×C4C2×C22⋊D1711yescharacteristic
453C22×C4C23×D1711yescharacteristic
454C2×C8—172no—
455C2×C8—172no—
456C2×C8—172no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—172no—
478C4⋊C4—172no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C4⋊C4—682no—
488C4⋊C4—682no—
489C4⋊C4—1361no—
490C4⋊C4—1361no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—1361no—
501C42D4×D1712yes—
502C42#6325412yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D1711yescharacteristic
511Q8—28no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4#1058762512yes—
553C2×C4#1058762512yes—
554C2×C4#1058763612yes—
555C2×C4#1058763612yes—
556C2×C4C2×D4×D1711yescharacteristic
557C2×C4C2×#6325411yescharacteristic
558C2×C4C2×Q8×D1711yescharacteristic
559C2×C4#1058769411yescharacteristic
560C2×C4#41×D1711yescharacteristic
561C2×C4#1058769711yescharacteristic
562C8—342no—
563C8—342no—
564C8—342no—
565C8—681no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1058762511yescharacteristic
574C22#1176346611yescharacteristic
575C22Q8×C22⋊D1711yescharacteristic
576C22C2×#1058763611yescharacteristic
577C22C2×#1058763611yescharacteristic
578C22#1176361211yescharacteristic
579C22#1176362011yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—682no—
590C4—682no—
591C4—1361no—
592C4—1361no—
593C4#1176336111yescharacteristic
594C4#1176341911yescharacteristic
595C4#1176346111yescharacteristic
596C4#1176347211yescharacteristic
597C2—82no—
598C2#1453238511yescharacteristic
599C2#1453836211yescharacteristic
600C2#1453846011yescharacteristic
601C2#1454144011yescharacteristic
602C2#1454147611yescharacteristic
603C2#1454164711yescharacteristic
604C2#1454165911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2580C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2580D17
hypocenterC17#25473
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#3622C2semidirect
3. #14534172C2semidirect
4. #14530286C2semidirect
5. #14530280C2semidirect
8. #14529748C2semidirect
9. #14529550C2semidirect
10. #14529514C2semidirect
11. #14529509C2semidirect
108. #10587503Q8semidirect
426. C17#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
C2×Q8×#36222
#145341722
#145302862
#145302802
#145301722
#145301582
#145297482
#145295502
#145295142
#145295092
#145294702
C4×#105875032
C17×#25802
#2547317