Finite Group Databasebeta

← #17038657

Subgroups — #17038657

Order 4352.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2#14531261C211yesmaximal, characteristic, complemented
3#14529664C211yesmaximal, characteristic
4#14529637C211yesmaximal, characteristic
5#14529519C211yesmaximal, characteristic, complemented
6#14529514C211yesmaximal, characteristic, complemented
7#14529484C211yesmaximal, characteristic
8#14529473C211yesmaximal, characteristic, complemented
9#14529470C211yesmaximal, characteristic
10#14529459C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12#14528617C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14C4×#10587474C211yesmaximal, characteristic, complemented
15C17×#2293C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11762909C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587481C2211yescharacteristic
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762693—21no—
39#11762693—21no—
40C4×#63225—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42#11762655—21no—
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C2×C17×#117—21no—
51C17×#390C2211yescharacteristic
52C17×#390C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C17×#388—21no—
56C17×#388—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C43×C17C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×Dic34C2311yescharacteristic
63C4×D4×C17—22no—
64C4×D4×C17C2×C414yes—
65C17×#117—44no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17D411yescharacteristic
80C2×C42×C17D411yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×#63229—22no—
85C2×#63229—22no—
86C2×#63229C2×C414yes—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C4×Dic34—22no—
100C4×Dic34C2×C414yes—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104#10587503—22no—
105#10587503—22no—
106#10587503C2×C414yes—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587481—22no—
115#10587481—22no—
116#10587481C2×C414yes—
117#10587479—22no—
118#10587478—22no—
119#10587478C2×C414yes—
120#10587474—22no—
121#10587474—22no—
122#10587474—22no—
123#10587474C2×C414yes—
124C4×#3622—22no—
125C4×#3622—22no—
126C24×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17C22×C412yes—
133C22×C4×C17—21no—
134C22×C4×C17—24no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34—21no—
153C2×Dic34—21no—
154C2×Dic34—21no—
155C2×Dic34C22×C412yes—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—42no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17C22×C412yes—
164C22⋊C4×C17—24no—
165C22⋊C4×C17—42no—
166C22⋊C4×C17—42no—
167C22⋊C4×C17—42no—
168C22⋊C4×C17—42no—
169C42×C17—22no—
170C42×C17—22no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—41no—
176C42×C17—41no—
177C42×C17—42no—
178C42×C17C22×C412yes—
179C42×C17#4112yes—
180#63229—28no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17C22×C412yes—
187#63225—24no—
188#63225—42no—
189#63225—42no—
190C4×Dic17—24no—
191C4×Dic17—42no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199C2×#3622C22×C412yes—
200C2×#3622#4112yes—
201C2×#3622#4112yes—
202C2×#3622#4112yes—
203C2×#3622#4112yes—
204#21288—171nomaximal, Sylow
205C23×C17—22no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17—41no—
212C23×C17C22×D411yescharacteristic
213D4×C17—22no—
214D4×C17—22no—
215D4×C17—41no—
216D4×C17—41no—
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—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—48no—
233C2×C4×C17—41no—
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×C17C4×D412yes—
242C2×C4×C17C4×D412yes—
243C2×C4×C17AGL(1,ℤ/8)12yes—
244C2×C4×C17#13712yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—41no—
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—
263Dic34—22no—
264Dic34—22no—
265Dic34—41no—
266Dic34—41no—
267#3622—22no—
268#3622—22no—
269#3622—22no—
270#3622#11714yescomplemented
271C2×#440—171no—
272#2295—171no—
273#2293D1711yescharacteristic
274#1973—171no—
275#1968—171no—
276#1943—171no—
277#1942—171no—
278#1936—171no—
279#1935—171no—
280#1934—171no—
281#1933—171no—
282#1776—171no—
283#1763—171no—
284#1762—171no—
285C4×#97—171no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17—81no—
293C22×C17C2×C4×D411yescharacteristic
294C22×C17#53211yescharacteristic
295C22×C17#53211yescharacteristic
296C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C17C2×#13711yescharacteristic
298C22×C17#57511yescharacteristic
299C22×C17#57511yescharacteristic
300C4×C17—22no—
301C4×C17—22no—
302C4×C17—22no—
303C4×C17—22no—
304C4×C17—41no—
305C4×C17—41no—
306C4×C17—42no—
307C4×C17—81no—
308C4×C17—81no—
309C4×C17C2×#11711yescharacteristic
310C4×C17C4×#4111yescharacteristic
311C4×C17#52911yescharacteristic
312C4×C17#52911yescharacteristic
313Dic17—42no—
314Dic17—42no—
315Dic17—81no—
316Dic17—81no—
317C22×SD16—171no—
318C2×C4×Q8—171no—
319C2×C4×D4C2×D1711yescharacteristic
320C2×#117—21no—
321#440—178no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#106—171no—
325C2×#103—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#97—171no—
332C2×#97—171no—
333C2×#97—171no—
334C2×#97—171no—
335C2×C4×C8—171no—
336C2×C4×C8—171no—
337#391—171no—
338#391—171no—
339#390C2×D1711yescharacteristic
340#390C2×D1711yescharacteristic
341#389—171no—
342#389—171no—
343#389C2×D1711yescharacteristic
344#389C2×D1711yescharacteristic
345#388—21no—
346#388—21no—
347#387—341no—
348#387—341no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—341no—
351C4×C4⋊C4C2×D1711yescharacteristic
352C4×C22⋊C4—21no—
353#376—341no—
354C43C2×D1711yescharacteristic
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361C2×C17—82no—
362C2×C17#229311yescharacteristic
363C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
364C2×C17C4×#13711yescharacteristic
365C2×C17#318011yescharacteristic
366C2×C17#318111yescharacteristic
367C2×C17#318211yescharacteristic
368C2×C17#318311yescharacteristic
369C22×Q8—171no—
370C22×D4C22×D1711yescharacteristic
371C23×C4—21no—
372C2×SD16—174no—
373C2×SD16—342no—
374C2×SD16—342no—
375C22×C8—171no—
376C22×C8—171no—
377C22×C8—171no—
378C22×C8—171no—
379C4×Q8—174no—
380C4×Q8—342no—
381C4×D4—22no—
382C4×D4C4×D1714yes—
383#117—44no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C4⋊C4C22×D1711yescharacteristic
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—21no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42C22⋊D1711yescharacteristic
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22⋊D1711yescharacteristic
406C2×C42C22⋊D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408C2×C42C22×D1711yescharacteristic
409C2×C42C22×D1711yescharacteristic
410#107—174no—
411#107—342no—
412#106—342no—
413#103—174no—
414#103—342no—
415#103—342no—
416#102—174no—
417#102—342no—
418#102—342no—
419#97—174no—
420#97—342no—
421#97—342no—
422#97—342no—
423C4×C8—342no—
424C4×C8—342no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434#95—341no—
435C17#2128811yescharacteristic, complemented, Sylow
436C24—21no—
437C2×Q8—172no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×Q8—341no—
442C2×Q8—341no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D1712yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4C2×C22⋊D1711yescharacteristic
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C23×D1711yescharacteristic
466SD16—348no—
467C2×C8—172no—
468C2×C8—172no—
469C2×C8—172no—
470C2×C8—172no—
471C2×C8—172no—
472C2×C8—172no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—172no—
487C4⋊C4—341no—
488C4⋊C4—344no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4C2×C4×D1712yes—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—344no—
511C42—682no—
512C42C2×C4×D1712yes—
513C42#6325212yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊D1711yescharacteristic
522Q8—342no—
523Q8—342no—
524Q8—681no—
525Q8—681no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4C4×C22⋊D1712yes—
567C2×C4C4×C22⋊D1712yes—
568C2×C4#1058761812yes—
569C2×C4#1058763712yes—
570C2×C4C22×C4×D1711yescharacteristic
571C2×C4C2×#6325211yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C2×C4#41×D1711yescharacteristic
575C2×C4#41×D1711yescharacteristic
576C8—174no—
577C8—342no—
578C8—342no—
579C8—342no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×C4×C22⋊D1711yescharacteristic
588C22#1176335511yescharacteristic
589C22#1176335511yescharacteristic
590C22C2×#1058761811yescharacteristic
591C22C2×#1058763711yescharacteristic
592C22#1176360711yescharacteristic
593C22#1176360711yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—682no—
604C4—682no—
605C4—1361no—
606C4—1361no—
607C4#117×D1711yescharacteristic
608C4#1176327711yescharacteristic
609C4#1176333711yescharacteristic
610C4#1176333711yescharacteristic
611C2—82no—
612C2#1453124311yescharacteristic
613C2C4×#1058761811yescharacteristic
614C2C4×#1058763711yescharacteristic
615C2#1453947611yescharacteristic
616C2#1453949411yescharacteristic
617C2#1453949811yescharacteristic
618C2#1453950711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2293C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2293D17
hypocenterC17#21288
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #14531261C2semidirect
5. #14529519C2semidirect
6. #14529514C2semidirect
8. #14529473C2semidirect
10. #14529459C2semidirect
11. #14528682C2semidirect
12. #14528617C2semidirect
14. C4×#10587474C2semidirect
18. #11762909C4semidirect
270. #3622#117semidirect
435. C17#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
#145312612
#145296642
#145296372
#145295192
#145295142
#145294842
#145294732
#145294702
#145294592
#145286822
#145286172
#145286092
C4×#105874742
C17×#22932
#2128817