Finite Group Databasebeta

← #17068669

Subgroups — #17068669

Order 4352.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763091C211yesmaximal, characteristic, complemented
2C2×#11763006C211yesmaximal, characteristic, complemented
3C4×#10587610C211yesmaximal, characteristic, complemented
4#14529970C212yesmaximal, complemented
5#14529787C212yesmaximal, complemented
6#14529737C211yesmaximal, characteristic, complemented
7#14529712C211yesmaximal, characteristic, complemented
8#14529118C212yesmaximal, complemented
9#14529045C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C2×C17×#532C211yesmaximal, characteristic
12C22×#63229C2212yes—
13C2×#10587610C2211yescharacteristic
14C2×C4×Dic34C2211yescharacteristic
15#11763091—24no—
16#11763006—24no—
17C2×#10587479C2212yes—
18C2×C4⋊#3622C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587481C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2212yes—
24C2×C22⋊#3622C2212yes—
25#11762715C2211yescharacteristic
26#11762707C2211yescharacteristic
27#11762693—21no—
28#11762682—21no—
29#11762669C2211yescharacteristic
30C4×#63225—21no—
31C4×C4⋊Dic17C2211yescharacteristic
32C4×C22⋊Dic17—21no—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C22×#41×C17—21no—
37C17×#532—24no—
38C17×#532—24no—
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×#127C2211yescharacteristic
41C2×C17×#124—21no—
42C2×C17×#121—21no—
43C2×C17×#121C2212yes—
44C2×C4×D4×C17—21no—
45C2×C4×D4×C17C2212yes—
46C22×C42×C17C2211yescharacteristic
47C2×#41×C17—28no—
48C2×#41×C17—44no—
49C2×#41×C17—44no—
50C22×Q8×C17—21no—
51C22×D4×C17—22no—
52C22×D4×C17—21no—
53C22×D4×C17—21no—
54C22×D4×C17C2312yes—
55C23×C4×C17—22no—
56C23×C4×C17D411yescharacteristic
57C23×C4×C17D411yescharacteristic
58C23×C4×C17C2311yescharacteristic
59C22×Dic34C2311yescharacteristic
60C17×C4⋊Q8—22no—
61C17×C4⋊Q8D414yes—
62C17×#127—22no—
63C17×#127D414yes—
64C17×#124—28no—
65C17×#121—28no—
66C17×#121—24no—
67C17×#121—24no—
68C4×D4×C17—28no—
69C4×D4×C17—24no—
70C4×D4×C17—24no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2312yes—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17—22no—
75C2×C42×C17—22no—
76C2×C42×C17—22no—
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×#63229—24no—
84C2×#63229—24no—
85C2×#63229—42no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89#10587610—22no—
90#10587610—22no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C22×#3622C2312yes—
97C22×#3622C2312yes—
98C4×Dic34—22no—
99C4×Dic34—22no—
100#10587503—24no—
101#10587503—24no—
102#10587503—42no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108C22⋊#3622—42no—
109#10587481—24no—
110#10587479—24no—
111C4⋊#3622—22no—
112C4⋊#3622—22no—
113C24×C17—22no—
114C24×C17C2×D411yescharacteristic
115#41×C17—432no—
116C2×Q8×C17—24no—
117C2×Q8×C17—42no—
118C2×Q8×C17—42no—
119C2×D4×C17—28no—
120C2×D4×C17—24no—
121C2×D4×C17—24no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17—42no—
128C2×D4×C17—42no—
129C2×D4×C17—44no—
130C2×D4×C17—44no—
131C2×D4×C17—42no—
132C2×D4×C17—42no—
133C2×D4×C17C2×D414yes—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—28no—
137C22×C4×C17—22no—
138C22×C4×C17—22no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—41no—
146C22×C4×C17—41no—
147C22×C4×C17—44no—
148C22×C4×C17—44no—
149C22×C4×C17SD1614yescomplemented
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2411yescharacteristic
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17—21no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34#4112yescomplemented
166C4⋊C4×C17—24no—
167C4⋊C4×C17—22no—
168C4⋊C4×C17C2×D414yes—
169C22⋊C4×C17—28no—
170C22⋊C4×C17—28no—
171C42×C17—22no—
172C42×C17—22no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17C2×D412yes—
176C42×C17C2×D412yes—
177C22⋊Dic17—24no—
178C22⋊Dic17—42no—
179#63229—48no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17#4112yescomplemented
186#63225—24no—
187#63225—42no—
188#63225—42no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—22no—
197#24154—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—24no—
205C23×C17—42no—
206C23×C17—42no—
207C23×C17—42no—
208C23×C17—42no—
209C23×C17—42no—
210C23×C17—42no—
211C23×C17C2×SD1612yes—
212C23×C17C22×D411yescharacteristic
213Q8×C17—48no—
214D4×C17—24no—
215D4×C17—24no—
216D4×C17—48no—
217D4×C17—416no—
218D4×C17—42no—
219D4×C17—42no—
220D4×C17—48no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—28no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—24no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—44no—
234C2×C4×C17—416no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17#12012yes—
242C2×C4×C17#12012yes—
243C2×C4×C17C2×SD1612yes—
244C2×C4×C17C2×SD1612yes—
245C2×C4×C17AGL(1,ℤ/8)12yes—
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C172−1+411yescharacteristic
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264Dic34—22no—
265Dic34—22no—
266Dic34—41no—
267Dic34—41no—
268#3622—42no—
269#3622—42no—
270C2×#532D1711yescharacteristic
271C2×#475—171no—
272C2×#460—171no—
273C4×C22⋊Q8—171no—
274#2034—172no—
275#2000—172no—
276#1988—171no—
277#1982—171no—
278#1877—172no—
279#1864—172no—
280#1797—171no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—41no—
285C22×C17—41no—
286C22×C17—41no—
287C22×C17—41no—
288C22×C17—44no—
289C22×C17—44no—
290C22×C17—44no—
291C22×C17—82no—
292C22×C17—82no—
293C22×C17—82no—
294C22×C17—82no—
295C22×C17C2×#12011yescharacteristic
296C22×C17#52511yescharacteristic
297C22×C17#53711yescharacteristic
298C22×C17C22×SD1611yescharacteristic
299C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C17#57511yescharacteristic
301C22×C17#57511yescharacteristic
302C4×C17—22no—
303C4×C17—22no—
304C4×C17—41no—
305C4×C17—44no—
306C4×C17—41no—
307C4×C17#45014yes—
308C4×C17#54611yescharacteristic
309C4×C17#54711yescharacteristic
310C4×C17#54811yescharacteristic
311C4×C17#55011yescharacteristic
312Dic17—42no—
313Dic17—42no—
314Dic17—81no—
315Dic17—81no—
316C22×#41—21no—
317C22×SD16—172no—
318#532—24no—
319#532—24no—
320C2×C4⋊Q8C2×D1711yescharacteristic
321C2×#127C2×D1711yescharacteristic
322C2×#124—21no—
323C2×C22⋊Q8—171no—
324C2×#121—21no—
325C2×#121C2×D1712yes—
326C2×C4×Q8—171no—
327C2×C4×D4—21no—
328C2×C4×D4C2×D1712yes—
329C22×C42C2×D1711yescharacteristic
330#475—344no—
331#460—344no—
332C2×#106—172no—
333C2×C4⋊C8—171no—
334C2×C4⋊C8—171no—
335C2×#103—172no—
336C2×#102—172no—
337C2×#102—172no—
338C2×#102—172no—
339C2×C22⋊C8—172no—
340#391—171no—
341#389—171no—
342#387—341no—
343#385—341no—
344#380—171no—
345C4×C4⋊C4—171no—
346C4×C4⋊C4—341no—
347C4×C22⋊C4—341no—
348#340—172no—
349#340—172no—
350#336—172no—
351C2×C17—42no—
352C2×C17—84no—
353C2×C17#240711yescharacteristic
354C2×C17C2×#45011yescharacteristic
355C2×C17#300511yescharacteristic
356C2×C17#310311yescharacteristic
357C2×C17#311511yescharacteristic
358C2×C17#324611yescharacteristic
359C2×C17#325111yescharacteristic
360C2×#41—28no—
361C2×#41—44no—
362C2×#41—44no—
363C22×Q8—21no—
364C22×Q8—171no—
365C22×D4—22no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4C22×D1712yes—
369C23×C4—22no—
370C23×C4C22⋊D1711yescharacteristic
371C23×C4C22⋊D1711yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C2×SD16—344no—
374C2×SD16—344no—
375C2×SD16—682no—
376C22×C8—172no—
377C22×C8—172no—
378C4⋊Q8—22no—
379C4⋊Q8C22⋊D1714yes—
380#127—22no—
381#127C22⋊D1714yes—
382#124—28no—
383C22⋊Q8—342no—
384C22⋊Q8—342no—
385#121—28no—
386#121—24no—
387#121—24no—
388C4×Q8—342no—
389C4×Q8—342no—
390C4×D4—28no—
391C4×D4—24no—
392C4×D4—24no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4C22×D1712yes—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—341no—
404C2×C22⋊C4—341no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42C22⋊D1711yescharacteristic
410C2×C42C22⋊D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412C2×C42C22×D1711yescharacteristic
413#106—344no—
414C4⋊C8—342no—
415C4⋊C8—342no—
416#103—344no—
417#102—344no—
418#102—344no—
419#102—682no—
420C22⋊C8—682no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426C17#2415411yescharacteristic, complemented, Sylow
427C24—22no—
428C24C2×C22⋊D1711yescharacteristic
429#41—432no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—172no—
434C2×Q8—341no—
435C2×Q8—341no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×C22⋊D1714yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—341no—
470C22×C4—341no—
471C22×C4—341no—
472C22×C4—341no—
473C22×C4#6322914yescomplemented
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C2×C22⋊D1711yescharacteristic
477C22×C4C2×C22⋊D1711yescharacteristic
478C22×C4C2×C22⋊D1711yescharacteristic
479C22×C4C23×D1711yescharacteristic
480SD16—688no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C2×C8—342no—
485C2×C8—342no—
486C2×C8—342no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—172no—
490C4⋊C4—344no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—682no—
497C4⋊C4—682no—
498C4⋊C4C2×C22⋊D1714yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—344no—
502C22⋊C4—682no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—682no—
508C42—682no—
509C42C2×C22⋊D1712yes—
510C42C2×C22⋊D1712yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×#6322912yes—
525C23C22×C22⋊D1711yescharacteristic
526Q8—48no—
527Q8—342no—
528Q8—342no—
529Q8—681no—
530Q8—681no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—416no—
535D4—42no—
536D4—48no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4#1058761812yes—
571C2×C4C2×#6322912yes—
572C2×C4C2×#6322912yes—
573C2×C4#1058763912yes—
574C2×C4#1058763912yes—
575C2×C4#41×D1711yescharacteristic
576C2×C4#41×D1711yescharacteristic
577C2×C4#1058769611yescharacteristic
578C2×C4#1058769711yescharacteristic
579C2×C4C22×C22⋊D1711yescharacteristic
580C2×C4C22×C22⋊D1711yescharacteristic
581C8—682no—
582C8—682no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#1058763911yescharacteristic
598C22#1176331611yescharacteristic
599C22#1176338711yescharacteristic
600C22C22×#6322911yescharacteristic
601C22C2×#1058761811yescharacteristic
602C22#1176360711yescharacteristic
603C22#1176360711yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—44no—
609C4—682no—
610C4—682no—
611C4—1361no—
612C4—1361no—
613C4#1176296014yes—
614C4#1176344711yescharacteristic
615C4#1176345811yescharacteristic
616C4#1176346611yescharacteristic
617C4#1176347811yescharacteristic
618C2—42no—
619C2—84no—
620C2#1453212311yescharacteristic
621C2C2×#1176296011yescharacteristic
622C2#1453802611yescharacteristic
623C2#1453881111yescharacteristic
624C2#1453892911yescharacteristic
625C2#1454010011yescharacteristic
626C2#1454014011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#532C2
socleC23×C17C22×D4
hypercenterC2×#532D17
hypocenterC17#24154
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#11763091C2semidirect
2. C2×#11763006C2semidirect
3. C4×#10587610C2semidirect
4. #14529970C2semidirect
5. #14529787C2semidirect
6. #14529737C2semidirect
7. #14529712C2semidirect
8. #14529118C2semidirect
9. #14529045C2semidirect
10. #14528764C2semidirect
149. C22×C4×C17SD16semidirect
165. C2×Dic34#41semidirect
185. C4⋊Dic17#41semidirect
426. C17#24154semidirect
473. C22×C4#63229semidirect

Maximal subgroups

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

subgroupindex
C2×#117630912
C2×#117630062
C4×#105876102
#145299702
#145297872
#145297372
#145297122
#145291182
#145290452
#145287642
C2×C17×#5322
#2415417