Finite Group Databasebeta

← C2×#14544193 #17965326

Subgroups — C2×#14544193 #17965326

Order 4352.

1446 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763780C211yesmaximal, characteristic
2C2×#11763752C211yesmaximal, characteristic
3#14544193C218yesmaximal, direct product factor
4C2×#11763663C212yesmaximal, complemented
5C2×#11763662C212yesmaximal, complemented
6C2×C22⋊Q8×D17C211yesmaximal, characteristic
7C22×#63243—21no—
8C22×C22⋊#595—21no—
9#11763780—22no—
10#11763780C2214yes—
11C2×#10587543C2212yes—
12C2×C22⋊#3624—21no—
13#11763752—22no—
14#11763752C2214yes—
15C2×(C2×#595)⋊C4—21no—
16C2×(C2×#595)⋊C4C2212yes—
17#11763663C2218yescomplemented
18#11763662C2218yescomplemented
19C22×Q8×D17C411yescharacteristic, complemented
20C23×C4×D17C2211yescharacteristic
21C2×#10587583C2211yescharacteristic
22C2×#10587610C411yescharacteristic, complemented
23C2×#10587582—21no—
24C2×#10587564—21no—
25C2×#10587632C2211yescharacteristic
26C22⋊Q8×D17—22no—
27C22⋊Q8×D17—22no—
28C22⋊Q8×D17C414yescomplemented
29C22⋊Q8×D17C2214yes—
30C2×C4⋊C4×D17—21no—
31C2×C4⋊C4×D17C411yescharacteristic, complemented
32C2×C22⋊C4×D17—21no—
33C2×C17×C22⋊Q8C411yescharacteristic, complemented
34C24×D17D411yescharacteristic
35C23×#595—22no—
36C22×Q8×C17C2×C411yescharacteristic
37C23×C4×C17C2×C411yescharacteristic
38C23×Dic17D411yescharacteristic
39C2×Q8×D17—22no—
40C2×Q8×D17—44no—
41C2×Q8×D17C2×C414yescomplemented
42C22×C4×D17—21no—
43C22×C4×D17—21no—
44C22×C4×D17—44no—
45C22×C4×D17C2×C411yescharacteristic
46C22×C4×D17C2×C412yes—
47C22×C4×D17D411yescharacteristic
48C22×C4×D17D411yescharacteristic
49C22×C4×D17D412yes—
50C22×C4×D17D412yes—
51C22×C4×D17C2311yescharacteristic
52C22×C4×D17C2312yes—
53C22×Dic34—21no—
54C22×Dic34C2×C411yescharacteristic
55C2×C22⋊#595—24no—
56C2×C22⋊#595—44no—
57C22×#3624—22no—
58C2×#63243—24no—
59C2×#63243—44no—
60C17×C22⋊Q8—22no—
61C17×C22⋊Q8C2×C414yescomplemented
62C2×C4⋊C4×C17—21no—
63C2×C4⋊C4×C17C2×C411yescharacteristic
64C2×C22⋊C4×C17—21no—
65C2×C22⋊Dic17—21no—
66#10587632—21no—
67#10587632—21no—
68#10587632C2×C412yes—
69#10587632C2312yes—
70C2×#63227—21no—
71C2×#63227—21no—
72C2×C4⋊Dic17—21no—
73C2×C4⋊Dic17C2×C411yescharacteristic
74#10587610—22no—
75#10587610C2×C414yescomplemented
76C2×#63225—21no—
77C2×#63225—21no—
78C2×#63225—21no—
79#10587583—21no—
80#10587583—21no—
81#10587583C2×C412yes—
82#10587583C2312yes—
83#10587582—24no—
84#10587582—24no—
85C4⋊C4×D17—24no—
86C4⋊C4×D17—22no—
87C4⋊C4×D17—24no—
88C4⋊C4×D17C2×C414yescomplemented
89C22⋊C4×D17—24no—
90C22⋊C4×D17—24no—
91#10587564—24no—
92#10587564—24no—
93C22⋊#3624—24no—
94(C2×#595)⋊C4—24no—
95(C2×#595)⋊C4—22no—
96(C2×#595)⋊C4C2314yes—
97#10587543—22no—
98#10587543C2314yes—
99C24×C17C22⋊C411yescharacteristic
100C23×D17—24no—
101C23×D17—24no—
102C23×D17—42no—
103C23×D17C22⋊C412yes—
104C23×D17C22⋊C411yescharacteristic
105C23×D17C2×D412yes—
106C23×D17C2×D411yescharacteristic
107C22×#595—24no—
108C22×#595—44no—
109C22×#595—44no—
110C22×#595—44no—
111C2×Q8×C17—21no—
112C2×Q8×C17—42no—
113C2×Q8×C17C22×C412yes—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—42no—
117C22×C4×C17C22⋊C411yescharacteristic
118C22×C4×C17C22⋊C412yes—
119C22×C4×C17C22×C411yescharacteristic
120C22×C4×C17C22×C412yes—
121C22×Dic17—21no—
122C22×Dic17—21no—
123C22×Dic17—42no—
124C22×Dic17C22⋊C411yescharacteristic
125C22×Dic17C22⋊C412yes—
126C22×Dic17C2×D411yescharacteristic
127C22×Dic17C2×D412yes—
128Q8×D17—48no—
129Q8×D17—48no—
130C2×C4×D17—24no—
131C2×C4×D17—24no—
132C2×C4×D17—24no—
133C2×C4×D17—24no—
134C2×C4×D17—44no—
135C2×C4×D17—44no—
136C2×C4×D17—44no—
137C2×C4×D17—44no—
138C2×C4×D17—42no—
139C2×C4×D17—42no—
140C2×C4×D17—44no—
141C2×C4×D17—44no—
142C2×C4×D17C22⋊C412yes—
143C2×C4×D17C22⋊C412yes—
144C2×C4×D17C22⋊C412yes—
145C2×C4×D17C22×C412yes—
146C2×C4×D17C22×C411yescharacteristic
147C2×C4×D17C2×D412yes—
148C2×C4×D17C2×D412yes—
149C2×C4×D17C2×D411yescharacteristic
150C2×C4×D17C2×D411yescharacteristic
151C2×C4×D17C2411yescharacteristic
152C2×Dic34—21no—
153C2×Dic34—24no—
154C2×Dic34—44no—
155C2×Dic34—42no—
156C2×Dic34C22×C412yes—
157C22⋊#595—48no—
158C2×#3624—24no—
159C2×#3624—44no—
160#63243—48no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—24no—
163C4⋊C4×C17C22×C412yes—
164C22⋊C4×C17—24no—
165C22⋊Dic17—24no—
166#63227—22no—
167#63227—22no—
168#63227—22no—
169#63227—22no—
170C4⋊Dic17—21no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17C22×C412yes—
174#63225—24no—
175#63225—22no—
176#63225—24no—
177#63225—22no—
178C2×#1511—171nomaximal, Sylow
179C23×C17—22no—
180C23×C17—22no—
181C23×C17—22no—
182C23×C17C2×C22⋊C412yes—
183C23×C17C2×C22⋊C411yescharacteristic
184C22×D17—22no—
185C22×D17—22no—
186C22×D17—22no—
187C22×D17—22no—
188C22×D17—24no—
189C22×D17—24no—
190C22×D17—24no—
191C22×D17—24no—
192C22×D17—42no—
193C22×D17—42no—
194C22×D17—416no—
195C22×D17#9914yes—
196C22×D17C2×C22⋊C412yes—
197C22×D17C2×C22⋊C411yescharacteristic
198C22×D17#13714yes—
199C22×D17C22×D411yescharacteristic
200C2×#595—44no—
201C2×#595—44no—
202C2×#595—44no—
203C2×#595—88no—
204Q8×C17—42no—
205Q8×C17—42no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17C2×C22⋊C412yes—
215C2×C4×C17C2×C22⋊C411yescharacteristic
216C2×C4×C17C23×C411yescharacteristic
217C2×Dic17—22no—
218C2×Dic17—22no—
219C2×Dic17—22no—
220C2×Dic17—22no—
221C2×Dic17—42no—
222C2×Dic17—42no—
223C2×Dic17—42no—
224C2×Dic17—42no—
225C2×Dic17C2×C22⋊C412yes—
226C2×Dic17C2×C22⋊C411yescharacteristic
227C2×Dic17C22×D411yescharacteristic
228C4×D17—24no—
229C4×D17—24no—
230C4×D17—24no—
231C4×D17—24no—
232C4×D17—42no—
233C4×D17—41no—
234C4×D17—41no—
235C4×D17—42no—
236C4×D17—41no—
237C4×D17—41no—
238C4×D17—48no—
239C4×D17—48no—
240Dic34—48no—
241Dic34—42no—
242Dic34—42no—
243#3624—44no—
244C22×C22⋊Q8—171no—
245C2×#407—171no—
246C2×#381—171no—
247#1511—178no—
248C2×#328—172no—
249C2×#328—172no—
250C22×C17—22no—
251C22×C17—22no—
252C22×C17—22no—
253C22×C17—44no—
254C22×C17C2×#9912yes—
255C22×C17#41912yes—
256C22×C17#42112yes—
257C22×C17C22×C22⋊C411yescharacteristic
258C2×D17—21no—
259C2×D17—21no—
260C2×D17—22no—
261C2×D17—22no—
262C2×D17—21no—
263C2×D17—21no—
264C2×D17—28no—
265C2×D17—28no—
266C2×D17—48no—
267C2×D17—48no—
268C2×D17C2×#9914yes—
269C2×D17#41912yes—
270C2×D17C2×#13712yes—
271#595—84no—
272C4×C17—21no—
273C4×C17—21no—
274C4×C17—22no—
275C4×C17—42no—
276C4×C17—42no—
277Dic17—21no—
278Dic17—22no—
279Dic17—21no—
280Dic17—42no—
281Dic17—42no—
282C23×Q8—171no—
283C24×C4—171no—
284C22×M16—341no—
285C2×C22⋊Q8—171no—
286C2×C22⋊Q8—171no—
287C2×C22⋊Q8—174no—
288C2×C22⋊Q8—174no—
289C2×C22⋊Q8—171no—
290C2×C22⋊Q8—341no—
291C2×C22⋊Q8—341no—
292C2×C22⋊Q8—342no—
293C2×C22⋊Q8—342no—
294C2×C22⋊Q8#59511yescharacteristic, complemented
295C22×C4⋊C4—171no—
296C22×C4⋊C4—341no—
297C22×C22⋊C4—341no—
298C22×C22⋊C4—341no—
299#407—174no—
300#407—342no—
301C2×C22⋊C8—172no—
302C2×C22⋊C8—341no—
303#381—174no—
304#381—342no—
305C2×#95—172no—
306C2×#95—341no—
307#328—178no—
308#328—178no—
309C2×C17—42no—
310C2×C17#151114yescomplemented
311C2×C17C22×#9911yescharacteristic
312C2×C17C2×#41911yescharacteristic
313C2×C17C2×#42111yescharacteristic
314D17—22no—
315D17—24no—
316D17—22no—
317C25—171no—
318C22×Q8—174no—
319C22×Q8—171no—
320C22×Q8—341no—
321C22×Q8—342no—
322C22×Q8—684no—
323C22×Q8C2×#59511yescharacteristic
324C23×C4—171no—
325C23×C4—171no—
326C23×C4—171no—
327C23×C4—171no—
328C23×C4—172no—
329C23×C4—172no—
330C23×C4—172no—
331C23×C4—171no—
332C23×C4—172no—
333C23×C4—341no—
334C23×C4—341no—
335C23×C4—342no—
336C23×C4—684no—
337C23×C4C2×#59511yescharacteristic
338C2×M16—344no—
339C2×M16—684no—
340C22×C8—342no—
341C22⋊Q8—22no—
342C22⋊Q8—172no—
343C22⋊Q8—172no—
344C22⋊Q8—174no—
345C22⋊Q8—172no—
346C22⋊Q8—172no—
347C22⋊Q8—344no—
348C22⋊Q8—344no—
349C22⋊Q8—344no—
350C22⋊Q8—344no—
351C22⋊Q8—341no—
352C22⋊Q8—341no—
353C22⋊Q8—341no—
354C22⋊Q8—341no—
355C22⋊Q8—342no—
356C22⋊Q8C2×#59514yescomplemented
357C2×C4⋊C4—21no—
358C2×C4⋊C4—171no—
359C2×C4⋊C4—174no—
360C2×C4⋊C4—342no—
361C2×C4⋊C4—344no—
362C2×C4⋊C4—344no—
363C2×C4⋊C4—341no—
364C2×C4⋊C4—341no—
365C2×C4⋊C4—341no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4C2×#59511yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—344no—
370C2×C22⋊C4—344no—
371C2×C22⋊C4—341no—
372C2×C22⋊C4—341no—
373C2×C22⋊C4—341no—
374C2×C22⋊C4—344no—
375C2×C22⋊C4—684no—
376C22⋊C8—174no—
377C22⋊C8—342no—
378C22⋊C8—344no—
379#95—174no—
380#95—342no—
381#95—344no—
382C17C2×#151111yescharacteristic, complemented, Sylow
383C24—171no—
384C24—171no—
385C24—172no—
386C24—172no—
387C24—344no—
388C24—344no—
389C24—682no—
390C24C22⋊#59511yescharacteristic
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8—172no—
394C2×Q8—344no—
395C2×Q8—341no—
396C2×Q8—682no—
397C2×Q8—684no—
398C2×Q8—688no—
399C2×Q8—688no—
400C2×Q8C22×#59512yes—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—42no—
404C22×C4—171no—
405C22×C4—171no—
406C22×C4—172no—
407C22×C4—172no—
408C22×C4—172no—
409C22×C4—172no—
410C22×C4—172no—
411C22×C4—172no—
412C22×C4—172no—
413C22×C4—171no—
414C22×C4—172no—
415C22×C4—171no—
416C22×C4—171no—
417C22×C4—171no—
418C22×C4—344no—
419C22×C4—344no—
420C22×C4—344no—
421C22×C4—344no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—344no—
425C22×C4—682no—
426C22×C4—682no—
427C22×C4—684no—
428C22×C4—684no—
429C22×C4—684no—
430C22×C4—684no—
431C22×C4—684no—
432C22×C4—684no—
433C22×C4—682no—
434C22×C4—684no—
435C22×C4—684no—
436C22×C4—684no—
437C22×C4C22⋊#59511yescharacteristic
438C22×C4C22⋊#59512yes—
439C22×C4C22×#59511yescharacteristic
440C22×C4C22×#59512yes—
441M16—688no—
442C2×C8—344no—
443C2×C8—684no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—172no—
447C4⋊C4—344no—
448C4⋊C4—342no—
449C4⋊C4—342no—
450C4⋊C4—342no—
451C4⋊C4—342no—
452C4⋊C4—341no—
453C4⋊C4—344no—
454C4⋊C4C22×#59512yes—
455C22⋊C4—24no—
456C22⋊C4—342no—
457C22⋊C4—342no—
458C22⋊C4—342no—
459C22⋊C4—342no—
460C22⋊C4—344no—
461C22⋊C4—688no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—174no—
466C23—174no—
467C23—171no—
468C23—171no—
469C23—172no—
470C23—344no—
471C23—344no—
472C23—344no—
473C23—344no—
474C23—342no—
475C23—342no—
476C23—342no—
477C23—342no—
478C23—682no—
479C23—682no—
480C23—6816no—
481C23C2×C22⋊#59512yes—
482C23C2×C22⋊#59511yescharacteristic
483Q8—42no—
484Q8—42no—
485Q8—682no—
486Q8—682no—
487Q8—688no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—171no—
497C2×C4—171no—
498C2×C4—172no—
499C2×C4—344no—
500C2×C4—344no—
501C2×C4—344no—
502C2×C4—344no—
503C2×C4—342no—
504C2×C4—342no—
505C2×C4—342no—
506C2×C4—342no—
507C2×C4—681no—
508C2×C4—681no—
509C2×C4—682no—
510C2×C4—682no—
511C2×C4—681no—
512C2×C4—681no—
513C2×C4—688no—
514C2×C4—688no—
515C2×C4—682no—
516C2×C4—682no—
517C2×C4—682no—
518C2×C4—682no—
519C2×C4—684no—
520C2×C4—684no—
521C2×C4—684no—
522C2×C4—1368no—
523C2×C4C2×C22⋊#59512yes—
524C2×C4C2×C22⋊#59511yescharacteristic
525C2×C4C23×#59511yescharacteristic
526C8—684no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—44no—
531C22—174no—
532C22—172no—
533C22—172no—
534C22—341no—
535C22—342no—
536C22—341no—
537C22—348no—
538C22—348no—
539C22—341no—
540C22—341no—
541C22—342no—
542C22—688no—
543C22—688no—
544C22D17⋊#9912yes—
545C22#1176380212yes—
546C22#1176380612yes—
547C22C22×C22⋊#59511yescharacteristic
548C4—21no—
549C4—22no—
550C4—21no—
551C4—42no—
552C4—42no—
553C4—341no—
554C4—342no—
555C4—341no—
556C4—682no—
557C4—682no—
558C4—1364no—
559C2—42no—
560C2—342no—
561C2—344no—
562C2—342no—
563C2#1454419314yesdirect product factor
564C2C2×D17⋊#9911yescharacteristic
565C2C2×#1176380211yescharacteristic
566C2C2×#1176380611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C23×C4
FrattiniC2×C4C23×#595
FittingC2×C17×C22⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17D4
cosocleC2×C4×D17C24
hypercenterC2×C22⋊Q8#595
hypocenterC17C2×#1511
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14544193C2direct (N × H)
4. C2×#11763663C2semidirect
5. C2×#11763662C2semidirect
17. #11763663C22semidirect
18. #11763662C22semidirect
19. C22×Q8×D17C4semidirect
22. C2×#10587610C4semidirect
28. C22⋊Q8×D17C4semidirect
31. C2×C4⋊C4×D17C4semidirect
33. C2×C17×C22⋊Q8C4semidirect
41. C2×Q8×D17C2×C4semidirect
61. C17×C22⋊Q8C2×C4semidirect
75. #10587610C2×C4semidirect
88. C4⋊C4×D17C2×C4semidirect
294. C2×C22⋊Q8#595semidirect
310. C2×C17#1511semidirect
356. C22⋊Q8C2×#595semidirect
382. C17C2×#1511semidirect
563. C2#14544193direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117637802
C2×#117637522
#145441932
C2×#117636632
C2×#117636622
C2×C22⋊Q8×D172
C2×#151117