Finite Group Databasebeta

← #17043426

Subgroups — #17043426

Order 4352.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763380C211yesmaximal, characteristic
2C2×#11762776C211yesmaximal, characteristic
3#14529215C212yesmaximal, complemented
4#14529205C212yesmaximal, complemented
5#14529096C212yesmaximal, complemented
6#14529091C212yesmaximal, complemented
7#14528876C212yesmaximal, complemented
8#14528718C212yesmaximal, complemented
9C17×#2311C211yesmaximal, characteristic
10C23×Dic34C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C23×#3622C2211yescharacteristic
13#11763380—24no—
14#11763380C418yescomplemented
15C2×#10587630—21no—
16C2×#10587622—21no—
17C2×#10587621—21no—
18C2×#10587621—21no—
19C2×#10587610C2212yes—
20C2×#10587610C2212yes—
21C2×#10587612C2211yescharacteristic
22C2×#10587481C2212yes—
23C2×#10587481C2212yes—
24C2×#10587481C2212yes—
25C2×#10587481C2212yes—
26#11762776—24no—
27C2×C22⋊#3622C2212yes—
28C2×C22⋊#3622C2212yes—
29#11762571C2212yes—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33C22×#41×C17C2211yescharacteristic
34C2×C17×#117C2211yescharacteristic
35C22×C4⋊C4×C17C2211yescharacteristic
36C17×#391—21no—
37C17×#390—21no—
38C17×#386—21no—
39C17×#386—21no—
40C17×#380C2212yes—
41C17×#380C2212yes—
42C2×#41×C17—41no—
43C2×#41×C17—41no—
44C2×#41×C17—41no—
45C2×#41×C17—41no—
46C2×#41×C17C2×C414yes—
47C2×#41×C17D414yes—
48C22×Q8×C17D411yescharacteristic
49C22×D4×C17D411yescharacteristic
50C22×D4×C17D411yescharacteristic
51C22×D4×C17D411yescharacteristic
52C23×C4×C17—21no—
53C23×C4×C17C2311yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C23×C4×C17C2311yescharacteristic
56C23×Dic17—21no—
57C22×Dic34—21no—
58C22×Dic34—22no—
59C22×Dic34—41no—
60C22×Dic34—41no—
61C22×Dic34—41no—
62C22×Dic34—41no—
63C22×Dic34C2×C414yescomplemented
64C22×Dic34C2312yes—
65C17×#117—41no—
66C17×#117—41no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—41no—
69C2×C4⋊C4×C17—41no—
70C2×C4⋊C4×C17C2312yes—
71C2×C4⋊C4×C17C2312yes—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—22no—
74C2×C22⋊C4×C17—22no—
75C2×C42×C17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—22no—
78C2×C22⋊Dic17—22no—
79#10587630—28no—
80#10587622—28no—
81#10587621—28no—
82#10587621—28no—
83#10587612—22no—
84#10587612C2×C414yescomplemented
85C2×C4⋊Dic17C2312yes—
86#10587610—24no—
87#10587610—24no—
88#10587610C2×C418yes—
89#10587610C2×C418yescomplemented
90C2×#63225—22no—
91C2×#63225—22no—
92C2×C4×Dic17—21no—
93C2×#63223—22no—
94C2×#63223—41no—
95C2×#63223—41no—
96C22×#3622—22no—
97C22×#3622—41no—
98C22×#3622—41no—
99C22×#3622C2312yes—
100C22×#3622C2312yes—
101C17×#95—22no—
102C17×#95—22no—
103C22⋊#3622—24no—
104C22⋊#3622—24no—
105#10587481—42no—
106#10587481—42no—
107#10587481—42no—
108#10587481—42no—
109C24×C17C2×D411yescharacteristic
110C24×C17C2×D411yescharacteristic
111C24×C17C2×D411yescharacteristic
112#41×C17—44no—
113#41×C17—44no—
114#41×C17—44no—
115#41×C17—44no—
116C2×Q8×C17—41no—
117C2×Q8×C17—41no—
118C2×Q8×C17C22⋊C414yes—
119C2×D4×C17—41no—
120C2×D4×C17—41no—
121C2×D4×C17—41no—
122C2×D4×C17—41no—
123C2×D4×C17—41no—
124C2×D4×C17—41no—
125C2×D4×C17C22⋊C414yes—
126C2×D4×C17C22⋊C414yes—
127C2×D4×C17C22⋊C414yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—22no—
141C22×C4×C17—22no—
142C22×C4×C17—42no—
143C22×C4×C17—82no—
144C22×C4×C17C22×C412yes—
145C22×C4×C17C22×C412yes—
146C22×C4×C17C22×C412yes—
147C22×C4×C17C2×D412yes—
148C22×C4×C17C2×D412yes—
149C22×C4×C17C2×D412yes—
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—24no—
155C22×Dic17—22no—
156C22×Dic17—22no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C2×Dic34—22no—
160C2×Dic34—22no—
161C2×Dic34—22no—
162C2×Dic34—22no—
163C2×Dic34—24no—
164C2×Dic34—22no—
165C2×Dic34—44no—
166C2×Dic34—44no—
167C2×Dic34—42no—
168C2×Dic34—42no—
169C2×Dic34—44no—
170C2×Dic34—44no—
171C2×Dic34C22×C414yes—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—22no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17—42no—
177C22⋊C4×C17—42no—
178C22⋊C4×C17—82no—
179C22⋊C4×C17—82no—
180C42×C17—42no—
181C22⋊Dic17—28no—
182C22⋊Dic17—28no—
183C22⋊Dic17—24no—
184C4⋊Dic17—22no—
185C4⋊Dic17C22×C414yes—
186#63225—28no—
187#63225—28no—
188C4×Dic17—24no—
189#63223—44no—
190C2×#3622—22no—
191C2×#3622—22no—
192C2×#3622—22no—
193C2×#3622—22no—
194C2×#3622—22no—
195C2×#3622—22no—
196C2×#3622—44no—
197#21771—171nomaximal, Sylow
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17—21no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—22no—
207C23×C17—22no—
208C23×C17—22no—
209C23×C17—22no—
210C23×C17C2×C22⋊C412yes—
211C23×C17C2×C22⋊C412yes—
212C23×C17C2×C22⋊C412yes—
213C23×C17C22×D411yescharacteristic
214Q8×C17—44no—
215D4×C17—44no—
216D4×C17—44no—
217D4×C17—44no—
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—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—22no—
230C2×C4×C17—22no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—42no—
239C2×C4×C17—42no—
240C2×C4×C17—42no—
241C2×C4×C17—42no—
242C2×C4×C17—84no—
243C2×C4×C17C2×C22⋊C412yes—
244C2×C4×C17C2×C22⋊C412yes—
245C2×C4×C17C2×C22⋊C412yes—
246C2×C4×C17C23×C411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17H(5,𝔽2)11yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—48no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×Dic17—42no—
259Dic34—24no—
260Dic34—24no—
261Dic34—48no—
262Dic34—42no—
263Dic34—42no—
264#3622—42no—
265#3622—42no—
266C2×#536—171no—
267C2×#408—171no—
268#2311D1711yescharacteristic
269#1894—172no—
270#1892—172no—
271#1873—172no—
272#1872—172no—
273#1825—172no—
274#1787—172no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—24no—
279C22×C17—24no—
280C22×C17—24no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17—41no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—41no—
288C22×C17—41no—
289C22×C17—41no—
290C22×C17C22×C22⋊C411yescharacteristic
291C22×C17#53411yescharacteristic
292C22×C17#53411yescharacteristic
293C22×C17#57712yes—
294C22×C17#57812yes—
295C4×C17—22no—
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—82no—
299C4×C17—82no—
300C4×C17#51811yescharacteristic
301C4×C17#51811yescharacteristic
302C4×C17#53411yescharacteristic
303C4×C17#53411yescharacteristic
304Dic17—44no—
305Dic17—44no—
306C22×#41C2×D1711yescharacteristic
307C23×Q8—171no—
308C22×M16—171no—
309C23×C8—171no—
310#536—178no—
311#536—344no—
312C2×C4⋊Q8—341no—
313C2×#124—341no—
314C2×#123—341no—
315C2×#123—341no—
316C2×C22⋊Q8—172no—
317C2×C22⋊Q8—172no—
318C2×#117—171no—
319C2×#117C2×D1711yescharacteristic
320C22×C4⋊C4C2×D1711yescharacteristic
321C2×#103—172no—
322C2×#103—172no—
323C2×#103—172no—
324C2×#103—172no—
325#408—344no—
326C2×C22⋊C8—172no—
327C2×C22⋊C8—172no—
328#391—21no—
329#390—21no—
330#386—21no—
331#386—21no—
332#380C2×D1712yes—
333#380C2×D1712yes—
334#340—172no—
335#340—172no—
336#340—172no—
337#340—172no—
338C2×C17—42no—
339C2×C17—42no—
340C2×C17—42no—
341C2×C17#230811yescharacteristic
342C2×C17#289812yes—
343C2×C17#320011yescharacteristic
344C2×C17#320112yes—
345C2×C17#320211yescharacteristic
346C2×#41—41no—
347C2×#41—41no—
348C2×#41—41no—
349C2×#41—41no—
350C2×#41C4×D1714yes—
351C2×#41D6814yes—
352C22×Q8—174no—
353C22×Q8—172no—
354C22×Q8—342no—
355C22×Q8—341no—
356C22×Q8—681no—
357C22×Q8—681no—
358C22×Q8—681no—
359C22×Q8—681no—
360C22×Q8C22⋊D1711yescharacteristic
361C22×D4C22⋊D1711yescharacteristic
362C22×D4C22⋊D1711yescharacteristic
363C22×D4C22⋊D1711yescharacteristic
364C23×C4—21no—
365C23×C4—341no—
366C23×C4C22×D1711yescharacteristic
367C23×C4C22×D1711yescharacteristic
368C23×C4C22×D1711yescharacteristic
369C2×M16—342no—
370C2×M16—681no—
371C2×M16—681no—
372C22×C8—172no—
373C22×C8—172no—
374C22×C8—342no—
375C22×C8—681no—
376C22×C8—681no—
377C4⋊Q8—348no—
378#124—348no—
379#123—348no—
380#123—348no—
381C22⋊Q8—178no—
382C22⋊Q8—178no—
383C22⋊Q8—344no—
384C22⋊Q8—344no—
385#117—41no—
386#117—41no—
387#117—174no—
388#117—342no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—172no—
393C2×C4⋊C4—342no—
394C2×C4⋊C4—342no—
395C2×C4⋊C4C22×D1712yes—
396C2×C4⋊C4C22×D1712yes—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—342no—
402C2×C22⋊C4—342no—
403C2×C42—21no—
404C2×C42—341no—
405#103—682no—
406#103—682no—
407#103—682no—
408#103—682no—
409C22⋊C8—344no—
410C22⋊C8—344no—
411#95—22no—
412#95—22no—
413C17#2177111yescharacteristic, complemented, Sylow
414C24C2×C22⋊D1711yescharacteristic
415C24C2×C22⋊D1711yescharacteristic
416C24C2×C22⋊D1711yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—174no—
424C2×Q8—342no—
425C2×Q8—344no—
426C2×Q8—342no—
427C2×Q8—342no—
428C2×Q8—342no—
429C2×Q8—342no—
430C2×Q8—684no—
431C2×Q8—684no—
432C2×Q8—682no—
433C2×Q8—682no—
434C2×Q8—684no—
435C2×Q8—684no—
436C2×Q8#6322714yes—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4#6322714yes—
444C2×D4#6322714yes—
445C2×D4#6322714yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—82no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—344no—
465C22×C4—682no—
466C22×C4—682no—
467C22×C4C2×C4×D1712yes—
468C22×C4C2×C4×D1712yes—
469C22×C4C2×C4×D1712yes—
470C22×C4C2×D6812yes—
471C22×C4C2×D6812yes—
472C22×C4C2×D6812yes—
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C2×C22⋊D1711yescharacteristic
476C22×C4C23×D1711yescharacteristic
477M16—684no—
478C2×C8—342no—
479C2×C8—342no—
480C2×C8—342no—
481C2×C8—342no—
482C2×C8—342no—
483C2×C8—342no—
484C2×C8—684no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—174no—
491C4⋊C4—342no—
492C4⋊C4—348no—
493C4⋊C4—348no—
494C22⋊C4—42no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—344no—
498C22⋊C4—348no—
499C22⋊C4—348no—
500C42—42no—
501C42—344no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23C2×#6322712yes—
515C23C2×#6322712yes—
516C23C2×#6322712yes—
517C23C22×C22⋊D1711yescharacteristic
518Q8—44no—
519Q8—344no—
520Q8—344no—
521Q8—688no—
522Q8—682no—
523Q8—682no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—344no—
553C2×C4—344no—
554C2×C4—688no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4C2×#6322712yes—
560C2×C4C2×#6322712yes—
561C2×C4C2×#6322712yes—
562C2×C4C22×C4×D1711yescharacteristic
563C2×C4C22×D6811yescharacteristic
564C2×C4#1058769111yescharacteristic
565C2×C4#1058769111yescharacteristic
566C2×C4#1058769111yescharacteristic
567C2×C4#1058769111yescharacteristic
568C8—682no—
569C8—682no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×#6322711yescharacteristic
586C22#1176336811yescharacteristic
587C22#1176336811yescharacteristic
588C22#1176362012yes—
589C22#1176362412yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—82no—
594C4—82no—
595C4—684no—
596C4—684no—
597C4#1176328211yescharacteristic
598C4#1176328211yescharacteristic
599C4#1176337111yescharacteristic
600C4#1176337111yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#1453135911yescharacteristic
605C2#1453731412yes—
606C2#1453965611yescharacteristic
607C2#1453966012yes—
608C2#1453967511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2311C2
socleC23×C17C22×D4
prime radicalC22×#41×C17C22
cosocleC22×C4×C17C24
hypercenter#2311D17
hypocenterC17#21771
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #14529215C2semidirect
4. #14529205C2semidirect
5. #14529096C2semidirect
6. #14529091C2semidirect
7. #14528876C2semidirect
8. #14528718C2semidirect
14. #11763380C4semidirect
63. C22×Dic34C2×C4semidirect
84. #10587612C2×C4semidirect
89. #10587610C2×C4semidirect
413. C17#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#117633802
C2×#117627762
#145292152
#145292052
#145290962
#145290912
#145288762
#145287182
C17×#23112
#2177117