Finite Group Databasebeta

← C2×#14543239 #17942088

Subgroups — C2×#14543239 #17942088

Order 4352.

6566 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14543239C2132yesmaximal, direct product factor
2C2×C17⋊#564C211yesmaximal, characteristic, complemented
3C2×#11763559C212yesmaximal, complemented
4C2×#11763513C214yesmaximal, complemented
5C2×#11763506C214yesmaximal, complemented
6C2×#11763496C214yesmaximal, complemented
7C2×#11763419C214yesmaximal, complemented
8C2×#11763404C214yesmaximal, complemented
9C2×#11763402C214yesmaximal, complemented
10C2×#11763338C212yesmaximal, complemented
11C2×#125×D17C211yesmaximal, characteristic, complemented
12C2×C17×#562C211yesmaximal, characteristic, complemented
13C17⋊#564C22116yescomplemented
14#11763559C22132yescomplemented
15#11763513C22164yescomplemented
16#11763506C22164yescomplemented
17#11763496C22164yescomplemented
18#11763419C22164yescomplemented
19#11763404C22164yescomplemented
20#11763402C22164yescomplemented
21C2×#10587630C2214yes—
22#11763338C22132yescomplemented
23C2×#10587565C2214yes—
24C2×#10587565C2214yes—
25C2×#10587565C2214yes—
26C2×#10587565C2214yes—
27C2×#10587565C2214yes—
28C2×#10587556C2211yescharacteristic
29C2×#10587556C2212yes—
30C2×#10587576C2214yes—
31C2×#10587576C2214yes—
32C2×#10587576C2212yes—
33C2×#10587575C2214yes—
34C2×#10587575C2214yes—
35C2×#10587575C2214yes—
36C2×#10587575C2214yes—
37#125×D17C22116yescomplemented
38C2×#10587569C2214yes—
39C2×#10587621C2214yes—
40C2×#10587621C2214yes—
41C2×#10587621C2214yes—
42C2×#10587610C2214yes—
43C2×#10587582C2214yes—
44C2×#10587564C2214yes—
45C2×#10587564C2214yes—
46C2×#10587564C2214yes—
47C2×#10587632C2212yes—
48C2×#10587624C2214yes—
49C2×#10587624C2212yes—
50C2×Q8×Dic17C2212yes—
51C2×#10587573C2214yes—
52C2×D4×Dic17C2212yes—
53C2×D4×Dic17C2214yes—
54C2×C4×C22⋊D17C2214yes—
55C2×#10587567C2214yes—
56C2×#10587567C2214yes—
57C2×#10587612C2214yes—
58C2×#10587563C2214yes—
59C22×#63225C2214yes—
60C2×C4⋊C4×D17C2212yes—
61C2×C4⋊C4×D17C2214yes—
62C2×C42×D17C2211yescharacteristic
63C17×#562C22116yescomplemented
64C2×C17×#125C2211yescharacteristic
65C2×C17×#124C2212yes—
66C2×C17×#123C2214yes—
67C2×C17×C22⋊Q8C2214yes—
68C2×C17×#121C2214yes—
69C22×Q8×C17C2312yes—
70C22×D4×C17C2314yes—
71C22×D4×C17C2312yes—
72C23×C4×C17C2314yes—
73C22×C22⋊D17C2314yes—
74C23×Dic17C2314yes—
75C23×Dic17C2314yes—
76C22×C4×D17C2311yescharacteristic
77C22×C4×D17C2312yes—
78C22×C4×D17C2314yes—
79C22×Dic34C2314yes—
80C17×#125C2318yes—
81C17×#124C23116yes—
82C17×#123C23132yes—
83C17×C22⋊Q8C23132yes—
84C17×#121C23132yes—
85C2×C4⋊C4×C17C2314yes—
86C2×C4⋊C4×C17C2314yes—
87C2×C4⋊C4×C17C2312yes—
88C2×C22⋊C4×C17C2314yes—
89C2×C22⋊C4×C17C2314yes—
90C2×C22⋊C4×C17C2314yes—
91C2×C42×C17C2311yescharacteristic
92C2×C22⋊Dic17C2314yes—
93C2×C22⋊Dic17C2314yes—
94C2×C22⋊Dic17C2314yes—
95C2×C22⋊Dic17C2314yes—
96C2×C22⋊Dic17C2314yes—
97#10587632C23116yes—
98Q8×Dic17C23116yes—
99#10587630C23132yes—
100#10587624C23116yes—
101#10587624C23132yes—
102#10587621C23132yes—
103#10587621C23132yes—
104#10587621C23132yes—
105D4×Dic17C23132yes—
106D4×Dic17C23116yes—
107C4×C22⋊D17C23132yes—
108C2×#63227C2314yes—
109#10587612C23132yes—
110C2×C4⋊Dic17C2314yes—
111C2×C4⋊Dic17C2312yes—
112C2×C4⋊Dic17C2314yes—
113C2×C4⋊Dic17C2312yes—
114C2×C4⋊Dic17C2312yes—
115C2×C4⋊Dic17C2314yes—
116#10587610C23132yes—
117C2×#63225C23132yes—
118C2×#63225C2314yes—
119C2×#63225C2314yes—
120C2×#63225C2314yes—
121C2×#63225C2314yes—
122C2×#63225C2314yes—
123C2×#63225C2314yes—
124C2×#63225C2314yes—
125C2×#63225C2314yes—
126C2×#63225C2314yes—
127C2×#63225C2314yes—
128C2×C4×Dic17C2311yescharacteristic
129C2×C4×Dic17C2312yes—
130C2×C4×Dic17C2314yes—
131C2×C4×Dic17C2314yes—
132C2×C4×Dic17C2314yes—
133#10587582C23132yes—
134C4⋊C4×D17C23132yes—
135C4⋊C4×D17C23116yes—
136#10587576C23116yes—
137#10587576C23132yes—
138#10587576C23132yes—
139#10587575C23132yes—
140#10587575C23132yes—
141#10587575C23132yes—
142#10587575C23132yes—
143#10587573C23132yes—
144#10587569C23132yes—
145#10587567C23132yes—
146#10587567C23132yes—
147#10587565C23132yes—
148#10587565C23132yes—
149#10587565C23132yes—
150#10587565C23132yes—
151#10587565C23132yes—
152#10587564C23132yes—
153#10587564C23132yes—
154#10587564C23132yes—
155#10587563C23132yes—
156C42×D17C2318yes—
157#10587556C2318yes—
158#10587556C23116yes—
159C24×C17C2414yes—
160C23×D17C2411yescharacteristic
161C2×Q8×C17—44no—
162C2×Q8×C17C2418yes—
163C2×D4×C17—48no—
164C2×D4×C17—44no—
165C2×D4×C17C24116yes—
166C2×D4×C17C2418yes—
167C22×C4×C17—48no—
168C22×C4×C17C2414yes—
169C22×C4×C17C2414yes—
170C22×C4×C17C2412yes—
171C22×C4×C17C2411yescharacteristic
172C22×C4×C17C24116yes—
173C2×C22⋊D17—216no—
174C2×C22⋊D17C24116yes—
175C22×Dic17—48no—
176C22×Dic17—48no—
177C22×Dic17C2414yes—
178C22×Dic17C2414yes—
179C22×Dic17C2412yes—
180C22×Dic17C2411yescharacteristic
181C22×Dic17C2414yes—
182C22×Dic17C24116yes—
183C22×Dic17C24116yes—
184C2×C4×D17—24no—
185C2×C4×D17—48no—
186C2×C4×D17—44no—
187C2×C4×D17C2414yes—
188C2×C4×D17C2418yes—
189C2×C4×D17C24116yes—
190C2×Dic34—48no—
191C2×Dic34C24116yes—
192C4⋊C4×C17C24116yes—
193C4⋊C4×C17C24116yes—
194C4⋊C4×C17C2418yes—
195C22⋊C4×C17C24116yes—
196C22⋊C4×C17C24116yes—
197C22⋊C4×C17C24116yes—
198C42×C17C2414yes—
199C22⋊Dic17C24116yes—
200C22⋊Dic17C24116yes—
201C22⋊Dic17C24116yes—
202C22⋊Dic17C24116yes—
203C22⋊Dic17C24116yes—
204#63227C24116yes—
205C4⋊Dic17C2418yes—
206C4⋊Dic17C2418yes—
207C4⋊Dic17C2418yes—
208C4⋊Dic17C24116yes—
209C4⋊Dic17C24116yes—
210C4⋊Dic17C24116yes—
211#63225C24116yes—
212#63225C24116yes—
213#63225C24116yes—
214#63225C24116yes—
215#63225C24116yes—
216#63225C24116yes—
217#63225C24116yes—
218#63225C24116yes—
219#63225C24116yes—
220#63225C24116yes—
221C4×Dic17C2414yes—
222C4×Dic17C2418yes—
223C4×Dic17C24116yes—
224C4×Dic17C24116yes—
225C4×Dic17C24116yes—
226C2×#3553—171nomaximal, Sylow
227C23×C17—28no—
228C23×C17—28no—
229C23×C17—28no—
230C23×C17C2518yes—
231C23×C17C2511yescharacteristic
232C22×D17—24no—
233C22×D172−1+414yes—
234C22×D17C2512yes—
235Q8×C17—48no—
236D4×C17—416no—
237D4×C17—48no—
238C2×C4×C17—22no—
239C2×C4×C17—28no—
240C2×C4×C17—28no—
241C2×C4×C17—24no—
242C2×C4×C17—416no—
243C2×C4×C17C2512yes—
244C2×C4×C17C2514yes—
245C2×C4×C17C2518yes—
246C2×C4×C17C2518yes—
247C22⋊D17—232no—
248C2×Dic17—22no—
249C2×Dic17—28no—
250C2×Dic17—28no—
251C2×Dic17—28no—
252C2×Dic17—24no—
253C2×Dic17—416no—
254C2×Dic17—416no—
255C2×Dic17C2512yes—
256C2×Dic17C2518yes—
257C2×Dic17C2518yes—
258C2×Dic17C2518yes—
259C2×Dic17C2514yes—
260C4×D17—28no—
261C4×D17—416no—
262C4×D17—48no—
263Dic34—416no—
264#3553—1732no—
265C2×#564—171no—
266C2×#563—172no—
267C2×#562D1711yescharacteristic, complemented
268C2×#554—174no—
269C2×#553—174no—
270C2×#552—174no—
271C2×#541—174no—
272C2×#539—174no—
273C2×#539—174no—
274C2×#529—172no—
275C22×#125—171no—
276C22×C17—28no—
277C22×C17—28no—
278C22×C17—28no—
279C22×C17—416no—
280C22×C17C2×2−1+412yes—
281C22×C17#58514yes—
282C22×C17C2611yescharacteristic
283C2×D17—24no—
284C2×D17—28no—
285C2×D17C2×2−1+414yes—
286C4×C17—22no—
287C4×C17—28no—
288C4×C17—28no—
289C4×C17—24no—
290Dic17—22no—
291Dic17—28no—
292Dic17—28no—
293Dic17—28no—
294Dic17—24no—
295#564—1716no—
296#563—1732no—
297#562C2×D17116yescomplemented
298#554—1764no—
299#553—1764no—
300#552—1764no—
301#541—1764no—
302#539—1764no—
303#539—1764no—
304C2×C4⋊Q8—174no—
305#529—1732no—
306C2×#126—174no—
307C2×#126—174no—
308C2×#126—174no—
309C2×#126—174no—
310C2×#126—174no—
311C2×#125—1716no—
312C2×#125—174no—
313C2×#125—172no—
314C2×#125—174no—
315C2×#125—174no—
316C2×#125—174no—
317C2×#125—174no—
318C2×#125—174no—
319C2×#125—171no—
320C2×#125—172no—
321C2×#125C2×D1711yescharacteristic
322C2×#124C2×D1712yes—
323C2×#123—174no—
324C2×#123—174no—
325C2×#123—174no—
326C2×#123—174no—
327C2×#123C2×D1714yes—
328C2×C22⋊Q8—172no—
329C2×C22⋊Q8—174no—
330C2×C22⋊Q8—174no—
331C2×C22⋊Q8—174no—
332C2×C22⋊Q8—174no—
333C2×C22⋊Q8—174no—
334C2×C22⋊Q8C2×D1714yes—
335C2×#121—172no—
336C2×#121—174no—
337C2×#121C2×D1714yes—
338C2×C4×Q8—172no—
339C2×C4×Q8—174no—
340C2×C4×D4—172no—
341C2×C4×D4—174no—
342C2×C4×D4—174no—
343C2×C4×D4—174no—
344C2×C4×D4—174no—
345C2×#117—174no—
346C2×#117—174no—
347C22×C4⋊C4—174no—
348C22×C4⋊C4—172no—
349C22×C4⋊C4—174no—
350C22×C42—171no—
351C2×C17—48no—
352C2×C17#355314yescomplemented
353C2×C17C22×2−1+411yescharacteristic
354C2×C17C2×#58512yes—
355D17—24no—
356C22×Q8—174no—
357C22×Q8C22×D1712yes—
358C22×D4—174no—
359C22×D4C22×D1714yes—
360C22×D4C22×D1712yes—
361C23×C4—174no—
362C23×C4—174no—
363C23×C4—171no—
364C23×C4—174no—
365C23×C4—172no—
366C23×C4C22×D1714yes—
367C4⋊Q8—1732no—
368#126—1732no—
369#126—1732no—
370#126—1732no—
371#126—1732no—
372#126—1732no—
373#125—1732no—
374#125—1716no—
375#125—1732no—
376#125—1732no—
377#125—1732no—
378#125—1732no—
379#125—1732no—
380#125—178no—
381#125—1716no—
382#125C22×D1718yes—
383#124C22×D17116yes—
384#123—1732no—
385#123—1732no—
386#123—1732no—
387#123—1732no—
388#123C22×D17132yes—
389C22⋊Q8—1716no—
390C22⋊Q8—1732no—
391C22⋊Q8—1732no—
392C22⋊Q8—1732no—
393C22⋊Q8—1732no—
394C22⋊Q8—1732no—
395C22⋊Q8C22×D17132yes—
396#121—1716no—
397#121—1732no—
398#121C22×D17132yes—
399C4×Q8—1716no—
400C4×Q8—1732no—
401C4×D4—1716no—
402C4×D4—1732no—
403C4×D4—1732no—
404C4×D4—1732no—
405C4×D4—1732no—
406#117—1732no—
407#117—1732no—
408C2×C4⋊C4—174no—
409C2×C4⋊C4—174no—
410C2×C4⋊C4—174no—
411C2×C4⋊C4—174no—
412C2×C4⋊C4—172no—
413C2×C4⋊C4—174no—
414C2×C4⋊C4—174no—
415C2×C4⋊C4—174no—
416C2×C4⋊C4—1732no—
417C2×C4⋊C4—174no—
418C2×C4⋊C4—174no—
419C2×C4⋊C4—172no—
420C2×C4⋊C4—174no—
421C2×C4⋊C4—174no—
422C2×C4⋊C4—174no—
423C2×C4⋊C4—172no—
424C2×C4⋊C4—174no—
425C2×C4⋊C4—1716no—
426C2×C4⋊C4—1732no—
427C2×C4⋊C4C22×D1714yes—
428C2×C4⋊C4C22×D1712yes—
429C2×C4⋊C4C22×D1714yes—
430C2×C22⋊C4—174no—
431C2×C22⋊C4—174no—
432C2×C22⋊C4—174no—
433C2×C22⋊C4—174no—
434C2×C22⋊C4—174no—
435C2×C22⋊C4—174no—
436C2×C22⋊C4C22×D1714yes—
437C2×C22⋊C4C22×D1714yes—
438C2×C22⋊C4C22×D1714yes—
439C2×C42—171no—
440C2×C42—174no—
441C2×C42—174no—
442C2×C42—174no—
443C2×C42—172no—
444C2×C42—178no—
445C2×C42C22×D1711yescharacteristic
446C17C2×#355311yescharacteristic, complemented, Sylow
447C24—171no—
448C24C23×D1714yes—
449C2×Q8—44no—
450C2×Q8—1716no—
451C2×Q8—688no—
452C2×Q8C23×D1718yes—
453C2×D4—48no—
454C2×D4—44no—
455C2×D4—1716no—
456C2×D4—3416no—
457C2×D4C23×D17116yes—
458C2×D4C23×D1718yes—
459C22×C4—48no—
460C22×C4—172no—
461C22×C4—174no—
462C22×C4—171no—
463C22×C4—174no—
464C22×C4—174no—
465C22×C4—1716no—
466C22×C4—1716no—
467C22×C4—174no—
468C22×C4—178no—
469C22×C4—1716no—
470C22×C4—344no—
471C22×C4—688no—
472C22×C4—684no—
473C22×C4—688no—
474C22×C4—688no—
475C22×C4C23×D1714yes—
476C22×C4C23×D1711yescharacteristic
477C22×C4C23×D1712yes—
478C22×C4C23×D1714yes—
479C22×C4C23×D17116yes—
480C4⋊C4—1716no—
481C4⋊C4—1716no—
482C4⋊C4—1716no—
483C4⋊C4—1716no—
484C4⋊C4—178no—
485C4⋊C4—1716no—
486C4⋊C4—1716no—
487C4⋊C4—1716no—
488C4⋊C4—1716no—
489C4⋊C4—1716no—
490C4⋊C4—178no—
491C4⋊C4—1716no—
492C4⋊C4—1716no—
493C4⋊C4—1716no—
494C4⋊C4—178no—
495C4⋊C4—1716no—
496C4⋊C4C23×D17116yes—
497C4⋊C4C23×D1718yes—
498C4⋊C4C23×D17116yes—
499C22⋊C4—1716no—
500C22⋊C4—1716no—
501C22⋊C4—1716no—
502C22⋊C4—1716no—
503C22⋊C4—1716no—
504C22⋊C4—1716no—
505C22⋊C4C23×D17116yes—
506C22⋊C4C23×D17116yes—
507C22⋊C4C23×D17116yes—
508C42—174no—
509C42—178no—
510C42—1716no—
511C42—1716no—
512C42—1716no—
513C42C23×D1714yes—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—174no—
518C23—172no—
519C23—344no—
520C23C24×D1718yes—
521C23C24×D1711yescharacteristic
522Q8—48no—
523Q8—6816no—
524D4—416no—
525D4—48no—
526D4—3432no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—416no—
532C2×C4—174no—
533C2×C4—178no—
534C2×C4—172no—
535C2×C4—178no—
536C2×C4—178no—
537C2×C4—348no—
538C2×C4—342no—
539C2×C4—348no—
540C2×C4—344no—
541C2×C4—348no—
542C2×C4—348no—
543C2×C4—6816no—
544C2×C4—688no—
545C2×C4—6816no—
546C2×C4—6816no—
547C2×C4C24×D1718yes—
548C2×C4C24×D1712yes—
549C2×C4C24×D1714yes—
550C2×C4C24×D1718yes—
551C22—28no—
552C22—28no—
553C22—28no—
554C22—416no—
555C22—174no—
556C22—348no—
557C22—344no—
558C222−1+4×D1712yes—
559C22#1176364414yes—
560C22C25×D1711yescharacteristic
561C4—22no—
562C4—28no—
563C4—28no—
564C4—24no—
565C4—342no—
566C4—348no—
567C4—344no—
568C4—348no—
569C4—348no—
570C2—48no—
571C2—344no—
572C2#1454323914yesdirect product factor
573C2C2×2−1+4×D1711yescharacteristic
574C2C2×#1176364412yes—
subgroupG/H
centerC23C24×D17
commutator, cosocleC22×C17C26
FrattiniC22C25×D17
FittingC2×C17×#562C2
socleC23×C17C25
hypercenterC2×#562D17
hypocenterC17C2×#3553
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #14543239C2direct (N × H)
2. C2×C17⋊#564C2semidirect
3. C2×#11763559C2semidirect
4. C2×#11763513C2semidirect
5. C2×#11763506C2semidirect
6. C2×#11763496C2semidirect
7. C2×#11763419C2semidirect
8. C2×#11763404C2semidirect
9. C2×#11763402C2semidirect
10. C2×#11763338C2semidirect
11. C2×#125×D17C2semidirect
12. C2×C17×#562C2semidirect
13. C17⋊#564C22semidirect
14. #11763559C22semidirect
15. #11763513C22semidirect
16. #11763506C22semidirect
17. #11763496C22semidirect
18. #11763419C22semidirect
19. #11763404C22semidirect
20. #11763402C22semidirect
22. #11763338C22semidirect
37. #125×D17C22semidirect
63. C17×#562C22semidirect
267. C2×#562D17semidirect
297. #562C2×D17semidirect
352. C2×C17#3553semidirect
446. C17C2×#3553semidirect
572. C2#14543239direct (N × H)

Maximal subgroups

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

subgroupindex
#145432392
C2×C17⋊#5642
C2×#117635592
C2×#117635132
C2×#117635062
C2×#117634962
C2×#117634192
C2×#117634042
C2×#117634022
C2×#117633382
C2×#125×D172
C2×C17×#5622
C2×#355317