Finite Group Databasebeta

← #17063017

Subgroups — #17063017

Order 4352.

1178 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×#11762851C211yesmaximal, characteristic, complemented
2C2×#11762832C211yesmaximal, characteristic, complemented
3C2×#11762810C211yesmaximal, characteristic
4#14531275C211yesmaximal, characteristic
5C2×#11762705C211yesmaximal, characteristic, complemented
6#14530177C212yesmaximal, complemented
7#14530170C212yesmaximal, complemented
8#14530163C212yesmaximal, complemented
9#14530156C212yesmaximal, complemented
10#14528503C211yesmaximal, characteristic, complemented
11C2×C17×#417C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×#63225—21no—
16C22×C4×Dic17C2211yescharacteristic
17#11762851—24no—
18C2×#10587489C2212yes—
19C2×#10587488C2212yes—
20#11762832—24no—
21C2×#10587485C2212yes—
22C2×#10587485C2212yes—
23#11762810—24no—
24C2×#10587506C2212yes—
25C2×#10587503C2212yes—
26#11762716C2211yescharacteristic
27#11762716C2211yescharacteristic
28#11762712C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762705—21no—
31#11762705—21no—
32#11762705C2212yes—
33#11762705C2212yes—
34#11762689—21no—
35#11762689—21no—
36C4×C22⋊Dic17—21no—
37C4×C22⋊Dic17—21no—
38C2×#10587502—21no—
39#11762571C2212yes—
40#11762571C2212yes—
41C22×#41×C17C2211yescharacteristic
42C22×M16×C17C2211yescharacteristic
43C22×C4⋊C4×C17C2211yescharacteristic
44C17×#417—24no—
45C2×C17×#103C2212yes—
46C2×C17×#102C2212yes—
47C2×#41×C17—22no—
48C2×#41×C17—22no—
49C2×#41×C17—22no—
50C2×#41×C17—22no—
51C2×#41×C17—22no—
52C2×#41×C17Q814yes—
53C22×Q8×C17C2311yescharacteristic
54C22×D4×C17—21no—
55C22×D4×C17C2311yescharacteristic
56C23×C4×C17—21no—
57C23×C4×C17—21no—
58C23×C4×C17C2311yescharacteristic
59C23×Dic17—21no—
60C23×Dic17D411yescharacteristic
61C23×Dic17D411yescharacteristic
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C2×M16×C17D414yes—
65C22×C8×C17C2312yes—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17C2312yes—
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75#10587612—22no—
76#10587612—22no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17—22no—
79C2×C4⋊Dic17D414yes—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2312yes—
83C2×#63225—22no—
84C2×#63225—22no—
85C2×#63225—44no—
86C2×C4×Dic17—22no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—22no—
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C2×#63223—22no—
94C2×#63223—22no—
95C2×#63223—22no—
96C22×#3622C2312yes—
97C17×#103—24no—
98C17×#102—24no—
99#10587506—24no—
100#10587503—24no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—22no—
106#10587502—22no—
107#10587489—24no—
108#10587488—24no—
109#10587485—24no—
110#10587485—24no—
111C24×C17—21no—
112C24×C17C2×D411yescharacteristic
113#41×C17—28no—
114#41×C17—28no—
115#41×C17—44no—
116#41×C17—44no—
117C2×Q8×C17—21no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17—21no—
121C2×Q8×C17—21no—
122C2×Q8×C17C2×Q812yes—
123C2×D4×C17—24no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—42no—
130C2×D4×C17—42no—
131C2×D4×C17C2×Q812yes—
132C22×C4×C17—21no—
133C22×C4×C17—24no—
134C22×C4×C17—22no—
135C22×C4×C17—42no—
136C22×C4×C17—42no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17—42no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D412yes—
142C22×C4×C17C2×D412yes—
143C22×C4×C17C2×Q812yes—
144C22×C4×C17#4112yes—
145C22×C4×C17#4112yes—
146C22×C4×C17#4112yes—
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—24no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—41no—
157C22×Dic17—41no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164M16×C17—28no—
165C2×C8×C17—22no—
166C2×C8×C17—22no—
167C2×C8×C17C2×D414yes—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—42no—
171C22⋊Dic17—42no—
172C22⋊Dic17—42no—
173C22⋊Dic17—42no—
174C22⋊Dic17—42no—
175C22⋊Dic17—42no—
176C4⋊Dic17—24no—
177C4⋊Dic17—22no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17C2×D414yes—
183#63225—44no—
184#63225—44no—
185C4×Dic17—22no—
186C4×Dic17—22no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—42no—
192#63223—44no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196#23617—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—22no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17#12112yes—
209C23×C17#12112yes—
210C23×C17C22⋊Q812yes—
211C23×C17C22×D411yescharacteristic
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17—41no—
215Q8×C17—41no—
216D4×C17—22no—
217D4×C17—22no—
218D4×C17—48no—
219D4×C17—41no—
220D4×C17—41no—
221C2×C4×C17—22no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—48no—
225C2×C4×C17—42no—
226C2×C4×C17—42no—
227C2×C4×C17—42no—
228C2×C4×C17—84no—
229C2×C4×C17#12112yes—
230C2×C4×C17#12112yes—
231C2×C4×C17#12114yes—
232C2×C4×C17C22⋊Q814yes—
233C2×C4×C17C22⋊Q812yes—
234C2×C4×C17C22⋊Q814yes—
235C2×C4×C17#12614yes—
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C22×Q811yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—24no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—44no—
254C2×Dic17—44no—
255C2×Dic17—48no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17AGL(1,ℤ/8)12yes—
265C2×Dic17#13712yes—
266C8×C17—24no—
267#3622—42no—
268C2×#428—171no—
269C2×#423—171no—
270C2×#417—171no—
271C2×#417D1711yescharacteristic
272#2297—171no—
273C2×#389—171no—
274#2068—172no—
275#2067—172no—
276#2066—172no—
277#2065—172no—
278#1736—171no—
279C22×C17—22no—
280C22×C17—22no—
281C22×C17—22no—
282C22×C17—22no—
283C22×C17—22no—
284C22×C17—22no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17#39614yes—
293C22×C17C2×#12111yescharacteristic
294C22×C17C2×#12111yescharacteristic
295C22×C17C2×C22⋊Q811yescharacteristic
296C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C17C2×#13711yescharacteristic
298C22×C17#57511yescharacteristic
299C22×C17#57511yescharacteristic
300C4×C17—42no—
301C4×C17—82no—
302C4×C17#39614yes—
303C4×C17C2×#12111yescharacteristic
304C4×C17C2×C22⋊Q811yescharacteristic
305C4×C17C2×C22⋊Q811yescharacteristic
306C4×C17C2×#12611yescharacteristic
307Dic17—22no—
308Dic17—22no—
309Dic17—44no—
310Dic17—41no—
311Dic17—41no—
312Dic17—81no—
313Dic17—81no—
314C22×#41C2×D1711yescharacteristic
315C22×M16—171no—
316C22×M16C2×D1711yescharacteristic
317C2×#117—171no—
318C22×C4⋊C4—171no—
319C22×C4⋊C4—341no—
320C22×C4⋊C4C2×D1711yescharacteristic
321C22×C42—171no—
322#428—344no—
323C2×#107—172no—
324C2×#106—172no—
325#423—344no—
326C2×C4⋊C8—172no—
327C2×C4⋊C8—172no—
328#417—24no—
329#417—344no—
330C2×#103—172no—
331C2×#103C2×D1712yes—
332C2×#102—172no—
333C2×#102C2×D1712yes—
334#391—171no—
335#391—171no—
336#390—171no—
337#390—171no—
338#389—172no—
339#389—172no—
340#389—341no—
341#389—341no—
342#386—341no—
343#386—341no—
344C4×C22⋊C4—341no—
345C4×C22⋊C4—341no—
346C2×#95—341no—
347#340—172no—
348#340—172no—
349C2×C17—24no—
350C2×C17—82no—
351C2×C17C2×#39611yescharacteristic
352C2×C17#303611yescharacteristic
353C2×C17#303711yescharacteristic
354C2×C17#305511yescharacteristic
355C2×C17#305611yescharacteristic
356C2×C17#307911yescharacteristic
357C2×C17#308011yescharacteristic
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41Dic3414yes—
364C22×Q8C22×D1711yescharacteristic
365C22×D4—21no—
366C22×D4C22×D1711yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—171no—
370C23×C4—171no—
371C23×C4—341no—
372C23×C4C22×D1711yescharacteristic
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—342no—
376C2×M16—342no—
377C2×M16—342no—
378C2×M16C22⋊D1714yes—
379C22×C8—172no—
380C22×C8C22×D1712yes—
381#117—342no—
382#117—342no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—174no—
387C2×C4⋊C4—172no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—342no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4—684no—
395C2×C4⋊C4C22×D1712yes—
396C2×C22⋊C4—341no—
397C2×C22⋊C4—341no—
398C2×C22⋊C4—341no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C42—171no—
402C2×C42—171no—
403C2×C42—171no—
404C2×C42—171no—
405C2×C42—342no—
406C2×C42—342no—
407C2×C42—341no—
408#107—344no—
409#106—344no—
410C4⋊C8—344no—
411C4⋊C8—344no—
412#103—24no—
413#103—344no—
414#102—24no—
415#102—344no—
416#95—342no—
417#95—342no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2361711yescharacteristic, complemented, Sylow
423C24—21no—
424C24D4×D1711yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic3412yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C2×Dic3412yes—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—171no—
455C22×C4—171no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—344no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—681no—
465C22×C4—681no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4C2×Dic3412yes—
469C22×C4#6325212yes—
470C22×C4D4×D1712yes—
471C22×C4D4×D1711yescharacteristic
472C22×C4#6325412yes—
473C22×C4#6325412yes—
474C22×C4C2×C22⋊D1712yes—
475C22×C4C23×D1711yescharacteristic
476M16—28no—
477M16—684no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—342no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8C2×C22⋊D1714yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—174no—
488C4⋊C4—342no—
489C4⋊C4—344no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—684no—
495C4⋊C4—684no—
496C22⋊C4—682no—
497C22⋊C4—682no—
498C22⋊C4—682no—
499C22⋊C4—682no—
500C22⋊C4—682no—
501C42—341no—
502C42—341no—
503C42—341no—
504C42—341no—
505C42—342no—
506C42—342no—
507C42—682no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#1058756412yes—
520C23#1058757012yes—
521C23#1058762412yes—
522C23C2×D4×D1711yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—172no—
541C2×C4—172no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—344no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—684no—
554C2×C4—684no—
555C2×C4—688no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4#1058756412yes—
565C2×C4#1058756414yes—
566C2×C4#1058756514yes—
567C2×C4#1058757012yes—
568C2×C4#1058761014yes—
569C2×C4#1058762412yes—
570C2×C4#1058762514yes—
571C2×C4C22×Dic3411yescharacteristic
572C2×C4C2×#6325211yescharacteristic
573C2×C4C2×D4×D1711yescharacteristic
574C2×C4C2×#6325411yescharacteristic
575C2×C4C2×#6325411yescharacteristic
576C2×C4C22×C22⋊D1711yescharacteristic
577C8—24no—
578C8—682no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#1176273214yes—
593C22C2×#1058762411yescharacteristic
594C22C2×#1058757011yescharacteristic
595C22C2×#1058756411yescharacteristic
596C22AGL(1,ℤ/8)×D1711yescharacteristic
597C22#137×D1711yescharacteristic
598C22#1176360511yescharacteristic
599C22#1176360811yescharacteristic
600C4—42no—
601C4—82no—
602C4—342no—
603C4—342no—
604C4—684no—
605C4—681no—
606C4—681no—
607C4—1361no—
608C4—1361no—
609C4#1176273214yes—
610C4C2×#1058762511yescharacteristic
611C4C2×#1058756411yescharacteristic
612C4C2×#1058761011yescharacteristic
613C4C2×#1058756511yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#1176273211yescharacteristic
617C2#1453824711yescharacteristic
618C2#1453825511yescharacteristic
619C2#1453839711yescharacteristic
620C2#1453840811yescharacteristic
621C2#1453859511yescharacteristic
622C2#1453860411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#417C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
hypercenterC2×#417D17
hypocenterC17#23617
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C2×#11762851C2semidirect
2. C2×#11762832C2semidirect
5. C2×#11762705C2semidirect
6. #14530177C2semidirect
7. #14530170C2semidirect
8. #14530163C2semidirect
9. #14530156C2semidirect
10. #14528503C2semidirect
422. C17#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#117628512
C2×#117628322
C2×#117628102
#145312752
C2×#117627052
#145301772
#145301702
#145301632
#145301562
#145285032
C2×C17×#4172
#2361717