Finite Group Databasebeta

← #17070528

Subgroups — #17070528

Order 4352.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×#11763101C211yesmaximal, characteristic, complemented
3#14531466C211yesmaximal, characteristic
4#14530119C211yesmaximal, characteristic, complemented
5#14530109C211yesmaximal, characteristic, complemented
6#14529932C211yesmaximal, characteristic, complemented
7#14529922C211yesmaximal, characteristic, complemented
8#14529742C211yesmaximal, characteristic, complemented
9#14529737C211yesmaximal, characteristic, complemented
10#14529096C211yesmaximal, characteristic
11#14529091C211yesmaximal, characteristic
12#14528983C211yesmaximal, characteristic
13#14528972C211yesmaximal, characteristic
14#14528768C211yesmaximal, characteristic
15C2×C17×#544C211yesmaximal, characteristic
16C2×#10587610C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18#11763113—24no—
19#11763101—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32#11762715C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762704—21no—
35#11762703—21no—
36#11762697—21no—
37#11762690—21no—
38#11762669C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C17×#544—24no—
51C17×#544—24no—
52C2×C17×C4⋊Q8—21no—
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×C4⋊Q8C2211yescharacteristic
55C2×C17×#125—21no—
56C2×C17×#125C2211yescharacteristic
57C2×C17×#125C2211yescharacteristic
58C2×C17×C22⋊Q8—21no—
59C2×C17×C22⋊Q8—21no—
60C2×C17×#117C2211yescharacteristic
61C2×C17×#117C2211yescharacteristic
62C22×C4⋊C4×C17C2211yescharacteristic
63C22×Q8×C17—21no—
64C22×Q8×C17—21no—
65C23×C4×C17—21no—
66C23×C4×C17C2311yescharacteristic
67C22×Dic34C2311yescharacteristic
68C17×C4⋊Q8—28no—
69C17×C4⋊Q8—22no—
70C17×C4⋊Q8—22no—
71C17×C4⋊Q8—22no—
72C17×C4⋊Q8D414yes—
73C17×#125—28no—
74C17×#125—22no—
75C17×#125—22no—
76C17×#125—22no—
77C17×#125D414yes—
78C17×C22⋊Q8—28no—
79C17×C22⋊Q8—28no—
80C17×#117—22no—
81C17×#117—22no—
82C17×#117—22no—
83C17×#117—22no—
84C2×C4⋊C4×C17—21no—
85C2×C4⋊C4×C17—21no—
86C2×C4⋊C4×C17—21no—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17—21no—
91C2×C4⋊C4×C17—22no—
92C2×C4⋊C4×C17—22no—
93C2×C4⋊C4×C17D411yescharacteristic
94C2×C4⋊C4×C17D411yescharacteristic
95C2×C4⋊C4×C17C2311yescharacteristic
96C2×C4⋊C4×C17C2311yescharacteristic
97C2×C4⋊C4×C17C2311yescharacteristic
98C2×C4⋊C4×C17C2311yescharacteristic
99C2×C22⋊C4×C17—21no—
100C2×C22⋊C4×C17D411yescharacteristic
101C2×C22⋊C4×C17D411yescharacteristic
102C2×C42×C17—21no—
103C2×C42×C17C2311yescharacteristic
104C2×C42×C17C2311yescharacteristic
105C2×C22⋊Dic17—21no—
106C2×C22⋊Dic17—21no—
107C2×C4⋊Dic17C2311yescharacteristic
108C2×C4⋊Dic17C2311yescharacteristic
109C2×C4⋊Dic17C2311yescharacteristic
110#10587610—22no—
111#10587610Q814yescomplemented
112C2×#63225—21no—
113C2×#63225—21no—
114C2×#63225—21no—
115C2×C4×Dic17—21no—
116C2×C4×Dic17—21no—
117C22×#3622C2311yescharacteristic
118C22×#3622C2311yescharacteristic
119C22×#3622C2311yescharacteristic
120C22×#3622C2311yescharacteristic
121C4×Dic34—22no—
122C4×Dic34—22no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128C22⋊#3622—22no—
129C22⋊#3622—22no—
130#10587481—22no—
131#10587481—22no—
132#10587481—22no—
133#10587481—22no—
134#10587479—22no—
135#10587478—22no—
136#10587478—22no—
137#10587478—22no—
138C4⋊#3622—22no—
139C4⋊#3622—22no—
140C24×C17C2×D411yescharacteristic
141C2×Q8×C17—24no—
142C2×Q8×C17—24no—
143C2×Q8×C17—81no—
144C2×Q8×C17—81no—
145C2×Q8×C17—81no—
146C2×Q8×C17—81no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—24no—
156C22×C4×C17—81no—
157C22×C4×C17—81no—
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×Q812yes—
164C22×C4×C17#4112yes—
165C22×C4×C17C2411yescharacteristic
166C22×Dic17—21no—
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34C2×Q812yes—
176C4⋊C4×C17—24no—
177C4⋊C4×C17—24no—
178C4⋊C4×C17—24no—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—24no—
182C4⋊C4×C17—24no—
183C4⋊C4×C17—22no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17C2×D412yes—
190C4⋊C4×C17C2×D412yes—
191C4⋊C4×C17C2×D412yes—
192C4⋊C4×C17C2×D412yes—
193C22⋊C4×C17—24no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—22no—
196C42×C17—24no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17C2×D412yes—
200C42×C17C2×D412yes—
201C22⋊Dic17—24no—
202C22⋊Dic17—81no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17C2×Q812yes—
209#63225—24no—
210#63225—42no—
211#63225—81no—
212C4×Dic17—42no—
213C4×Dic17—81no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226#24311—171nomaximal, Sylow
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17C22⋊Q812yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—84no—
236Q8×C17—84no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—84no—
266C2×C4×C17#12012yes—
267C2×C4×C17#12012yes—
268C2×C4×C17C22⋊Q812yes—
269C2×C4×C17C22×D411yescharacteristic
270C2×C4×C17C22×D411yescharacteristic
271C2×C4×C17C22×Q811yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C172−1+411yescharacteristic
274C2×C4×C172−1+411yescharacteristic
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287Dic34—22no—
288Dic34—22no—
289Dic34—41no—
290Dic34—41no—
291#3622—41no—
292#3622—41no—
293#3622—41no—
294#3622—41no—
295C2×#544D1711yescharacteristic
296C2×#479—171no—
297C2×#477—171no—
298#2323—171no—
299#2059—171no—
300#2057—171no—
301#2026—171no—
302#2024—171no—
303#1989—171no—
304#1988—171no—
305#1873—171no—
306#1872—171no—
307#1854—171no—
308#1852—171no—
309#1798—171no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—41no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17C2×#12011yescharacteristic
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17#53611yescharacteristic
320C22×C17#57611yescharacteristic
321C22×C17#57711yescharacteristic
322C22×C17#57811yescharacteristic
323C22×C17#57811yescharacteristic
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—42no—
329C4×C17—41no—
330C4×C17—42no—
331C4×C17—42no—
332C4×C17—42no—
333C4×C17#54811yescharacteristic
334C4×C17#54811yescharacteristic
335C4×C17D4×Q811yescharacteristic
336C4×C17D4×Q811yescharacteristic
337Dic17—42no—
338Dic17—42no—
339Dic17—81no—
340Dic17—81no—
341#544—24no—
342#544—24no—
343C2×C4⋊Q8—21no—
344C2×C4⋊Q8C2×D1711yescharacteristic
345C2×C4⋊Q8C2×D1711yescharacteristic
346C2×#125—21no—
347C2×#125C2×D1711yescharacteristic
348C2×#125C2×D1711yescharacteristic
349C2×C22⋊Q8—21no—
350C2×C22⋊Q8—21no—
351C2×C22⋊Q8—171no—
352C2×C4×Q8—171no—
353C2×#117C2×D1711yescharacteristic
354C2×#117C2×D1711yescharacteristic
355C22×C4⋊C4C2×D1711yescharacteristic
356#479—344no—
357#477—344no—
358C2×#107—171no—
359C2×#107—171no—
360C2×#107—171no—
361C2×#106—171no—
362C2×C4⋊C8—171no—
363C2×C4⋊C8—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×C22⋊C8—171no—
369C2×C22⋊C8—171no—
370#391—171no—
371#389—171no—
372#389—341no—
373#388—341no—
374#388—341no—
375#387—341no—
376#380—171no—
377C4×C4⋊C4—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340—171no—
386#336—171no—
387#336—171no—
388C2×C17—42no—
389C2×C17#240911yescharacteristic
390C2×C17#301111yescharacteristic
391C2×C17#301211yescharacteristic
392C2×C17#308211yescharacteristic
393C2×C17#308311yescharacteristic
394C2×C17#324211yescharacteristic
395C2×C17#324311yescharacteristic
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—171no—
399C23×C4—21no—
400C23×C4C22×D1711yescharacteristic
401C22×C8—171no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C4⋊Q8—28no—
406C4⋊Q8—22no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8C22⋊D1714yes—
410#125—28no—
411#125—22no—
412#125—22no—
413#125—22no—
414#125C22⋊D1714yes—
415C22⋊Q8—28no—
416C22⋊Q8—28no—
417C22⋊Q8—174no—
418C22⋊Q8—342no—
419C4×Q8—342no—
420C4×Q8—342no—
421#117—22no—
422#117—22no—
423#117—22no—
424#117—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—171no—
435C2×C4⋊C4—171no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4C22⋊D1711yescharacteristic
441C2×C4⋊C4C22⋊D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C4⋊C4C22×D1711yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4C22⋊D1711yescharacteristic
450C2×C22⋊C4C22⋊D1711yescharacteristic
451C2×C42—21no—
452C2×C42—341no—
453C2×C42—341no—
454C2×C42C22×D1711yescharacteristic
455C2×C42C22×D1711yescharacteristic
456#107—342no—
457#107—342no—
458#107—342no—
459#106—342no—
460C4⋊C8—342no—
461C4⋊C8—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#103—342no—
466C22⋊C8—342no—
467C22⋊C8—342no—
468#95—341no—
469#95—341no—
470#95—341no—
471#95—341no—
472#95—341no—
473C17#2431111yescharacteristic, complemented, Sylow
474C24C2×C22⋊D1711yescharacteristic
475C2×Q8—24no—
476C2×Q8—24no—
477C2×Q8—81no—
478C2×Q8—81no—
479C2×Q8—81no—
480C2×Q8—81no—
481C2×Q8—172no—
482C2×Q8—341no—
483C2×Q8—341no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4Q8×D1712yes—
503C22×C4#6325612yes—
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C2×C22⋊D1711yescharacteristic
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C23×D1711yescharacteristic
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—172no—
536C4⋊C4—341no—
537C4⋊C4—344no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—682no—
543C4⋊C4—1361no—
544C4⋊C4C2×C22⋊D1712yes—
545C4⋊C4C2×C22⋊D1712yes—
546C4⋊C4C2×C22⋊D1712yes—
547C4⋊C4C2×C22⋊D1712yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—344no—
552C22⋊C4—1361no—
553C42—24no—
554C42—21no—
555C42—21no—
556C42—682no—
557C42—1361no—
558C42C2×C22⋊D1712yes—
559C42C2×C22⋊D1712yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1058763212yes—
567C23C22×C22⋊D1711yescharacteristic
568Q8—84no—
569Q8—84no—
570Q8—342no—
571Q8—342no—
572Q8—681no—
573Q8—681no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—342no—
604C2×C4—342no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4#1058763212yes—
616C2×C4#1058763912yes—
617C2×C4#1058763912yes—
618C2×C4C2×Q8×D1711yescharacteristic
619C2×C4C2×#6325611yescharacteristic
620C2×C4#1058769411yescharacteristic
621C2×C4#1058769411yescharacteristic
622C2×C4C22×C22⋊D1711yescharacteristic
623C2×C4C22×C22⋊D1711yescharacteristic
624C8—681no—
625C8—681no—
626C8—681no—
627C8—681no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1058763911yescharacteristic
636C22C2×#1058763211yescharacteristic
637C22#1176338311yescharacteristic
638C22#1176361211yescharacteristic
639C22#1176362011yescharacteristic
640C22#1176362411yescharacteristic
641C22#1176362411yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—682no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4#1176347011yescharacteristic
656C4#1176347011yescharacteristic
657C4Q8×C22⋊D1711yescharacteristic
658C4Q8×C22⋊D1711yescharacteristic
659C2—42no—
660C2#1453214311yescharacteristic
661C2#1453807911yescharacteristic
662C2#1453808811yescharacteristic
663C2#1453862011yescharacteristic
664C2#1453863111yescharacteristic
665C2#1454005211yescharacteristic
666C2#1454006711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#544C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#544D17
hypocenterC17#24311
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
2. C2×#11763101C2semidirect
4. #14530119C2semidirect
5. #14530109C2semidirect
6. #14529932C2semidirect
7. #14529922C2semidirect
8. #14529742C2semidirect
9. #14529737C2semidirect
111. #10587610Q8semidirect
473. C17#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117631012
#145314662
#145301192
#145301092
#145299322
#145299222
#145297422
#145297372
#145290962
#145290912
#145289832
#145289722
#145287682
C2×C17×#5442
#2431117