Finite Group Databasebeta

← #17063019

Subgroups — #17063019

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×#11762852C211yesmaximal, characteristic, complemented
2C2×#11762833C211yesmaximal, characteristic, complemented
3C2×#11762810C211yesmaximal, characteristic
4C2×#11762810C211yesmaximal, characteristic
5C2×#11762707C211yesmaximal, characteristic, complemented
6#14530179C212yesmaximal, complemented
7#14530172C212yesmaximal, complemented
8#14530165C212yesmaximal, complemented
9#14530158C212yesmaximal, complemented
10#14528505C211yesmaximal, characteristic, complemented
11C17×#2297C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C22×#63223C2211yescharacteristic
14C22×C4⋊Dic17C2211yescharacteristic
15C22×C4⋊Dic17C2211yescharacteristic
16C22×#63225—21no—
17#11762852—24no—
18C2×#10587478C2212yes—
19C2×#10587479C2212yes—
20#11762833—24no—
21C2×C4⋊#3622C2212yes—
22C2×C4⋊#3622C2212yes—
23#11762810—24no—
24#11762810—24no—
25C2×#10587506C2212yes—
26C2×#10587506C2212yes—
27C2×#10587503C2212yes—
28C2×#10587503C2212yes—
29#11762707—21no—
30#11762707—21no—
31#11762707C2212yes—
32#11762707C2212yes—
33C2×#10587502—21no—
34#11762571C2212yes—
35#11762571C2212yes—
36C22×#41×C17C2211yescharacteristic
37C2×C17×#117C2211yescharacteristic
38C22×C42×C17C2211yescharacteristic
39C17×#391C2211yescharacteristic
40C17×#391C2211yescharacteristic
41C17×#390C2211yescharacteristic
42C17×#390C2211yescharacteristic
43C17×#386—21no—
44C17×#386—21no—
45C4×C22⋊C4×C17—21no—
46C4×C22⋊C4×C17—21no—
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×C17D411yescharacteristic
58C23×C4×C17D411yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C23×Dic17—21no—
62C17×#117—22no—
63C17×#117—22no—
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—22no—
72C2×C42×C17—22no—
73C2×C42×C17—21no—
74C2×C42×C17D411yescharacteristic
75C2×C42×C17D411yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—22no—
81C2×C4⋊Dic17—22no—
82C2×C4⋊Dic17—22no—
83C2×C4⋊Dic17D414yes—
84C2×C4⋊Dic17C2312yes—
85C2×C4⋊Dic17C2312yes—
86C2×#63225—22no—
87C2×#63225—22no—
88C2×#63225—44no—
89C2×#63223—22no—
90C2×#63223—22no—
91C2×#63223—22no—
92C2×#63223—22no—
93C2×#63223—22no—
94C2×#63223D414yes—
95C22×#3622C2312yes—
96C22×#3622C2312yes—
97C17×#95—21no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101#10587506—24no—
102#10587506—24no—
103#10587503—24no—
104#10587503—24no—
105#10587502—22no—
106#10587502—22no—
107#10587479—24no—
108#10587478—24no—
109C4⋊#3622—24no—
110C4⋊#3622—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—22no—
133C22×C4×C17—22no—
134C22×C4×C17—21no—
135C22×C4×C17—24no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—41no—
141C22×C4×C17—41no—
142C22×C4×C17—42no—
143C22×C4×C17—42no—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×Q812yes—
152C22×C4×C17#4112yes—
153C22×C4×C17#4112yes—
154C22×C4×C17#4112yes—
155C22×C4×C17C2411yescharacteristic
156C22×Dic17—24no—
157C22×Dic17—22no—
158C22×Dic17—22no—
159C22×Dic17—42no—
160C22×Dic17—42no—
161C22×Dic17—42no—
162C22×Dic17—42no—
163C22×Dic17—42no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C22⋊C4×C17—42no—
169C22⋊C4×C17—42no—
170C22⋊C4×C17—42no—
171C22⋊C4×C17—42no—
172C22⋊C4×C17—42no—
173C42×C17—22no—
174C42×C17—22no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—42no—
180C4⋊Dic17—24no—
181C4⋊Dic17—22no—
182C4⋊Dic17—22no—
183C4⋊Dic17—22no—
184C4⋊Dic17—42no—
185C4⋊Dic17C2×D414yes—
186#63225—44no—
187#63225—44no—
188#63223—28no—
189#63223—44no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622C2×D414yes—
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—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
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—48no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17#12114yes—
244C2×C4×C17#12112yes—
245C2×C4×C17#12112yes—
246C2×C4×C17C22⋊Q814yes—
247C2×C4×C17C22⋊Q814yes—
248C2×C4×C17C22⋊Q812yes—
249C2×C4×C17#12614yes—
250C2×C4×C17AGL(1,ℤ/8)12yes—
251C2×C4×C17#13712yes—
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C22×Q811yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×Dic17—24no—
259C2×Dic17—48no—
260C2×Dic17—42no—
261C2×Dic17—42no—
262C2×Dic17—42no—
263C2×Dic17—42no—
264C2×Dic17—42no—
265C2×Dic17—84no—
266#3622—24no—
267#3622—42no—
268C2×#428—171no—
269C2×#423—171no—
270C2×#417—171no—
271C2×#417—171no—
272#2297D1711yescharacteristic
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—22no—
301C4×C17—22no—
302C4×C17—42no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—81no—
306C4×C17—81no—
307C4×C17#39614yes—
308C4×C17C2×#12111yescharacteristic
309C4×C17C2×C22⋊Q811yescharacteristic
310C4×C17C2×C22⋊Q811yescharacteristic
311C4×C17C2×#12611yescharacteristic
312Dic17—44no—
313Dic17—82no—
314C22×#41C2×D1711yescharacteristic
315C22×M16—171no—
316C22×M16—171no—
317C2×#117C2×D1711yescharacteristic
318C22×C4⋊C4—171no—
319C22×C4⋊C4—171no—
320C22×C4⋊C4—341no—
321C22×C42C2×D1711yescharacteristic
322#428—344no—
323C2×#107—172no—
324C2×#106—172no—
325#423—344no—
326C2×C4⋊C8—172no—
327C2×C4⋊C8—172no—
328#417—344no—
329#417—344no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#102—172no—
333C2×#102—172no—
334#391C2×D1711yescharacteristic
335#391C2×D1711yescharacteristic
336#390C2×D1711yescharacteristic
337#390C2×D1711yescharacteristic
338#389—172no—
339#389—172no—
340#389—341no—
341#389—341no—
342#386—21no—
343#386—21no—
344C4×C22⋊C4—21no—
345C4×C22⋊C4—21no—
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—341no—
369C23×C4—341no—
370C23×C4C22⋊D1711yescharacteristic
371C23×C4C22⋊D1711yescharacteristic
372C23×C4C22×D1711yescharacteristic
373C2×M16—174no—
374C2×M16—342no—
375C2×M16—342no—
376C2×M16—342no—
377C2×M16—342no—
378C2×M16—342no—
379C22×C8—172no—
380C22×C8—172no—
381#117—22no—
382#117—22no—
383C2×C4⋊C4—174no—
384C2×C4⋊C4—172no—
385C2×C4⋊C4—172no—
386C2×C4⋊C4—342no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—342no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—341no—
391C2×C4⋊C4—342no—
392C2×C4⋊C4—342no—
393C2×C4⋊C4—684no—
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22⋊D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—344no—
409#106—344no—
410C4⋊C8—344no—
411C4⋊C8—344no—
412#103—344no—
413#103—344no—
414#102—344no—
415#102—344no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—342no—
421#95—342no—
422C17#2361711yescharacteristic, complemented, Sylow
423C24—21no—
424C24C2×C22⋊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—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—342no—
457C22×C4—344no—
458C22×C4—342no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4C2×Dic3412yes—
465C22×C4#6325212yes—
466C22×C4D4×D1712yes—
467C22×C4D4×D1712yes—
468C22×C4#6325412yes—
469C22×C4#6325412yes—
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C23×D1711yescharacteristic
476M16—348no—
477M16—684no—
478C2×C8—174no—
479C2×C8—342no—
480C2×C8—342no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—174no—
489C4⋊C4—342no—
490C4⋊C4—344no—
491C4⋊C4—342no—
492C4⋊C4—342no—
493C4⋊C4—684no—
494C4⋊C4—682no—
495C4⋊C4—684no—
496C22⋊C4—42no—
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—42no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#1058761012yes—
520C23#1058762412yes—
521C23#1058762512yes—
522C23C22×C22⋊D1711yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
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—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—344no—
555C2×C4—688no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—1364no—
562C2×C4#1058756414yes—
563C2×C4#1058756414yes—
564C2×C4#1058756514yes—
565C2×C4#1058757014yes—
566C2×C4#1058761012yes—
567C2×C4#1058762412yes—
568C2×C4#1058762512yes—
569C2×C4#1058763512yes—
570C2×C4#1058763712yes—
571C2×C4C22×Dic3411yescharacteristic
572C2×C4C2×#6325211yescharacteristic
573C2×C4C2×D4×D1711yescharacteristic
574C2×C4C2×D4×D1711yescharacteristic
575C2×C4C2×#6325411yescharacteristic
576C2×C4C2×#6325411yescharacteristic
577C8—344no—
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×#1058762511yescharacteristic
595C22C2×#1058761011yescharacteristic
596C22C2×#1058763511yescharacteristic
597C22C2×#1058763711yescharacteristic
598C22#1176360711yescharacteristic
599C22#1176360711yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—684no—
608C4—1362no—
609C4#1176273214yes—
610C4C2×#1058757011yescharacteristic
611C4C2×#1058756411yescharacteristic
612C4C2×#1058756411yescharacteristic
613C4C2×#1058756511yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#1176273211yescharacteristic
617C2#1453824411yescharacteristic
618C2#1453825211yescharacteristic
619C2#1453840511yescharacteristic
620C2#1453841611yescharacteristic
621C2#1453859911yescharacteristic
622C2#1453860611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2297C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
hypercenter#2297D17
hypocenterC17#23617
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C2×#11762852C2semidirect
2. C2×#11762833C2semidirect
5. C2×#11762707C2semidirect
6. #14530179C2semidirect
7. #14530172C2semidirect
8. #14530165C2semidirect
9. #14530158C2semidirect
10. #14528505C2semidirect
422. C17#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#117628522
C2×#117628332
C2×#117628102
C2×#117627072
#145301792
#145301722
#145301652
#145301582
#145285052
C17×#22972
#2361717