Finite Group Databasebeta

← C2×#14540048 #17906525

Subgroups — C2×#14540048 #17906525

Order 4352.

1678 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763440C211yesmaximal, characteristic
2C2×#11763419C211yesmaximal, characteristic
3#14540048C2116yesmaximal, direct product factor
4C2×#11763216C211yesmaximal, characteristic, complemented
5C2×#11763209C211yesmaximal, characteristic, complemented
6C2×#11763176C211yesmaximal, characteristic, complemented
7C2×#11763158C211yesmaximal, characteristic, complemented
8C2×#11763144C211yesmaximal, characteristic
9C2×#11763151C211yesmaximal, characteristic
10C2×#11763147C211yesmaximal, characteristic
11C2×#11763119C211yesmaximal, characteristic, complemented
12C2×#11763114C211yesmaximal, characteristic, complemented
13C2×#11763102C211yesmaximal, characteristic, complemented
14C2×#11763093C211yesmaximal, characteristic, complemented
15C2×#11762867C211yesmaximal, characteristic
16C2×C17×#483C211yesmaximal, characteristic
17#11763440—24no—
18#11763440C2218yes—
19#11763419—24no—
20#11763419C2218yes—
21C2×#10587574—21no—
22C2×#10587574C2211yescharacteristic
23C2×#10587574C2211yescharacteristic
24C2×#10587630C2211yescharacteristic
25C2×#10587565—21no—
26C2×#10587565—21no—
27C2×#10587576—21no—
28C2×#10587576—21no—
29C2×#10587576C2211yescharacteristic
30C2×#10587576C2211yescharacteristic
31C2×#10587576C2211yescharacteristic
32C2×#10587575—21no—
33C2×#10587610—21no—
34C2×#10587610—21no—
35C2×#10587610C2211yescharacteristic
36C2×Q8×Dic17C2211yescharacteristic
37C2×#10587573C2211yescharacteristic
38C2×#10587612C2211yescharacteristic
39C2×#10587612C2211yescharacteristic
40#11763216C2218yescomplemented
41#11763209C2218yescomplemented
42#11763176C2218yescomplemented
43#11763158C2218yescomplemented
44#11763151C2218yes—
45#11763147C2218yes—
46#11763144C2218yes—
47#11763119C2218yescomplemented
48#11763114C2218yescomplemented
49#11763102C2218yescomplemented
50#11763093C2218yescomplemented
51#11762867C2218yes—
52C2×#10587489C2211yescharacteristic
53C2×#10587478C2211yescharacteristic
54C2×#10587478C2211yescharacteristic
55C2×#10587478C2211yescharacteristic
56C2×#10587488C2211yescharacteristic
57C2×#10587479C2211yescharacteristic
58C2×#10587485C2211yescharacteristic
59C2×#10587485C2211yescharacteristic
60C2×#10587487C2211yescharacteristic
61C2×#10587487C2211yescharacteristic
62C2×#10587506C2211yescharacteristic
63C2×#10587506C2211yescharacteristic
64C2×#10587481C2211yescharacteristic
65C2×#10587481C2211yescharacteristic
66C2×C22⋊#3622C2211yescharacteristic
67C2×#10587486C2211yescharacteristic
68C2×C8×Dic17C2211yescharacteristic
69C2×C17×C22⋊Q8C2211yescharacteristic
70C22×C4⋊C4×C17C2211yescharacteristic
71C17×#483C2218yes—
72C2×C17×#106C2211yescharacteristic
73C2×C17×#106C2211yescharacteristic
74C2×C17×#103C2211yescharacteristic
75C2×C17×#103C2211yescharacteristic
76C2×C17×C22⋊C8C2211yescharacteristic
77C22×Q8×C17C2311yescharacteristic
78C23×C4×C17—21no—
79C23×C4×C17C2311yescharacteristic
80C22×Dic34—21no—
81C22×Dic34—21no—
82C22×Dic34C2311yescharacteristic
83C22×C8×C17C2311yescharacteristic
84C22×C8×C17C2311yescharacteristic
85C17×C22⋊Q8—22no—
86C17×C22⋊Q8C2314yes—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—22no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17C2311yescharacteristic
91C2×C4⋊C4×C17C2311yescharacteristic
92C2×C4⋊C4×C17C2314yes—
93C2×C4⋊C4×C17C2311yescharacteristic
94C2×C22⋊C4×C17—21no—
95C2×C22⋊Dic17—21no—
96C2×C22⋊Dic17—21no—
97C2×C22⋊Dic17D411yescharacteristic
98C2×C22⋊Dic17D411yescharacteristic
99Q8×Dic17—22no—
100Q8×Dic17C2314yes—
101#10587630—22no—
102#10587630C2314yes—
103#10587612—22no—
104#10587612D414yes—
105#10587612C2314yes—
106#10587612C2314yes—
107C2×C4⋊Dic17—21no—
108C2×C4⋊Dic17—21no—
109C2×C4⋊Dic17—21no—
110C2×C4⋊Dic17—21no—
111C2×C4⋊Dic17D411yescharacteristic
112C2×C4⋊Dic17D411yescharacteristic
113C2×C4⋊Dic17C2311yescharacteristic
114C2×C4⋊Dic17C2311yescharacteristic
115C2×C4⋊Dic17C2311yescharacteristic
116#10587610—22no—
117#10587610—28no—
118#10587610—28no—
119#10587610C2314yes—
120C2×#63225—21no—
121C2×#63225—21no—
122C2×#63225—21no—
123C2×#63225—21no—
124C2×#63225—21no—
125C2×#63225—21no—
126C2×#63225—21no—
127C2×#63225—21no—
128C2×#63225—21no—
129C2×C4×Dic17—21no—
130C2×C4×Dic17—21no—
131C2×C4×Dic17—21no—
132C2×C4×Dic17C2311yescharacteristic
133C2×C4×Dic17C2311yescharacteristic
134C22×#3622C2311yescharacteristic
135C22×#3622C2311yescharacteristic
136#10587576—28no—
137#10587576—28no—
138#10587576—22no—
139#10587576—22no—
140#10587576—22no—
141#10587576C2314yes—
142#10587576C2314yes—
143#10587576C2314yes—
144#10587575—28no—
145#10587574—28no—
146#10587574—22no—
147#10587574—22no—
148#10587574C2314yes—
149#10587574C2314yes—
150#10587573—22no—
151#10587573C2314yes—
152#10587565—28no—
153#10587565—28no—
154C17×#106C2314yes—
155C17×#106C2314yes—
156C17×#103C2314yes—
157C17×#103C2314yes—
158C17×C22⋊C8C2314yes—
159#10587506C2314yes—
160#10587506C2314yes—
161C22⋊#3622C2314yes—
162#10587489C2314yes—
163#10587488C2314yes—
164#10587487C2314yes—
165#10587487C2314yes—
166#10587486C2314yes—
167#10587485C2314yes—
168#10587485C2314yes—
169C8×Dic17C2314yes—
170#10587481C2314yes—
171#10587481C2314yes—
172#10587479C2314yes—
173#10587478C2314yes—
174#10587478C2314yes—
175#10587478C2314yes—
176C24×C17C2×D411yescharacteristic
177C2×Q8×C17—21no—
178C2×Q8×C17—22no—
179C2×Q8×C17—22no—
180C2×Q8×C17C2412yes—
181C22×C4×C17—22no—
182C22×C4×C17—22no—
183C22×C4×C17—21no—
184C22×C4×C17—21no—
185C22×C4×C17—21no—
186C22×C4×C17—21no—
187C22×C4×C17—24no—
188C22×C4×C17—82no—
189C22×C4×C17C2×D412yes—
190C22×C4×C17C2×D411yescharacteristic
191C22×C4×C17C2412yes—
192C22×C4×C17C2411yescharacteristic
193C22×Dic17—21no—
194C22×Dic17—21no—
195C22×Dic17—21no—
196C22×Dic17—21no—
197C22×Dic17C2×D411yescharacteristic
198C22×Dic17C2×D411yescharacteristic
199C22×Dic17C2×D411yescharacteristic
200C22×Dic17C2×D411yescharacteristic
201C2×Dic34—22no—
202C2×Dic34—22no—
203C2×Dic34—21no—
204C2×Dic34—24no—
205C2×Dic34—24no—
206C2×Dic34—82no—
207C2×Dic34—82no—
208C2×Dic34C2412yes—
209C2×C8×C17—22no—
210C2×C8×C17—22no—
211C2×C8×C17C2412yes—
212C2×C8×C17C2412yes—
213C4⋊C4×C17—21no—
214C4⋊C4×C17—24no—
215C4⋊C4×C17—24no—
216C4⋊C4×C17—21no—
217C4⋊C4×C17—21no—
218C4⋊C4×C17C2412yes—
219C4⋊C4×C17C2412yes—
220C4⋊C4×C17C2412yes—
221C22⋊C4×C17—24no—
222C22⋊Dic17—24no—
223C22⋊Dic17—24no—
224C22⋊Dic17C2×D414yes—
225C22⋊Dic17C2×D414yes—
226C4⋊Dic17—21no—
227C4⋊Dic17—21no—
228C4⋊Dic17—21no—
229C4⋊Dic17—24no—
230C4⋊Dic17—24no—
231C4⋊Dic17—24no—
232C4⋊Dic17—24no—
233C4⋊Dic17C2×D414yes—
234C4⋊Dic17C2×D414yes—
235C4⋊Dic17C2412yes—
236C4⋊Dic17C2412yes—
237C4⋊Dic17C2412yes—
238#63225—24no—
239#63225—24no—
240#63225—24no—
241#63225—24no—
242#63225—24no—
243#63225—24no—
244#63225—24no—
245#63225—24no—
246#63225—24no—
247C4×Dic17—24no—
248C4×Dic17—24no—
249C4×Dic17—24no—
250C4×Dic17C2×D412yes—
251C4×Dic17C2×D412yes—
252C4×Dic17C2412yes—
253C4×Dic17C2412yes—
254C2×#3622—22no—
255C2×#3622—22no—
256C2×#3622C2412yes—
257C2×#3622C2412yes—
258C2×#3242—171nomaximal, Sylow
259C23×C17—22no—
260C23×C17—22no—
261C23×C17—22no—
262C23×C17C22×D412yes—
263C23×C17C22×D411yescharacteristic
264Q8×C17—22no—
265Q8×C17—22no—
266Q8×C17—42no—
267C2×C4×C17—22no—
268C2×C4×C17—24no—
269C2×C4×C17—22no—
270C2×C4×C17—22no—
271C2×C4×C17—22no—
272C2×C4×C17—22no—
273C2×C4×C17—22no—
274C2×C4×C17—22no—
275C2×C4×C17—42no—
276C2×C4×C17—42no—
277C2×C4×C17—42no—
278C2×C4×C17—42no—
279C2×C4×C17—84no—
280C2×C4×C17C22×D412yes—
281C2×C4×C17C22×D411yescharacteristic
282C2×C4×C172−1+412yes—
283C2×C4×C172−1+412yes—
284C2×C4×C17C2511yescharacteristic
285C2×Dic17—22no—
286C2×Dic17—22no—
287C2×Dic17—22no—
288C2×Dic17—22no—
289C2×Dic17—22no—
290C2×Dic17—22no—
291C2×Dic17—22no—
292C2×Dic17—22no—
293C2×Dic17—42no—
294C2×Dic17—42no—
295C2×Dic17—42no—
296C2×Dic17—42no—
297C2×Dic17C22×D412yes—
298C2×Dic17C22×D412yes—
299C2×Dic17C22×D412yes—
300C2×Dic17C22×D412yes—
301Dic34—22no—
302Dic34—22no—
303Dic34—42no—
304Dic34—84no—
305Dic34—84no—
306C8×C17—22no—
307C8×C17—22no—
308#3622—22no—
309#3622—22no—
310C2×#544—171no—
311C2×#541—171no—
312#3242—1716no—
313C2×#501—171no—
314C2×#499—171no—
315C2×#491—171no—
316C2×#487—171no—
317C2×#484—171no—
318C2×#485—171no—
319C2×#485—171no—
320C2×#483D1711yescharacteristic
321C2×#479—171no—
322C2×#479—171no—
323C2×#477—171no—
324C2×#475—171no—
325C2×#433—171no—
326C22×C17—22no—
327C22×C17—22no—
328C22×C17—22no—
329C22×C17—44no—
330C22×C17#53612yes—
331C22×C17#57712yes—
332C22×C17#57812yes—
333C22×C17C23×D411yescharacteristic
334C4×C17—22no—
335C4×C17—42no—
336C4×C17—42no—
337C4×C17—42no—
338C4×C17—42no—
339C4×C17#53612yes—
340C4×C17C2×2−1+411yescharacteristic
341C4×C17C2×2−1+411yescharacteristic
342Dic17—22no—
343Dic17—22no—
344Dic17—22no—
345Dic17—22no—
346Dic17—42no—
347Dic17—42no—
348Dic17—42no—
349Dic17—42no—
350#544—178no—
351#544—344no—
352#541—178no—
353#541—344no—
354C2×C4⋊Q8—171no—
355C2×C4⋊Q8—171no—
356C2×C4⋊Q8—171no—
357C2×C4⋊Q8—341no—
358C2×#126—341no—
359C2×#126—341no—
360C2×#125—171no—
361C2×#125—171no—
362C2×#125—171no—
363C2×#125—341no—
364C2×#125—341no—
365C2×#125—341no—
366C2×C22⋊Q8—171no—
367C2×C22⋊Q8—341no—
368C2×C22⋊Q8—341no—
369C2×C22⋊Q8C2×D1711yescharacteristic
370C2×C4×Q8—171no—
371C2×C4×Q8—171no—
372C2×#117—171no—
373C2×#117—171no—
374C22×C4⋊C4C2×D1711yescharacteristic
375#501—178no—
376#499—178no—
377#491—178no—
378#487—178no—
379#485—178no—
380#485—178no—
381#484—178no—
382#483C2×D1718yes—
383#479—178no—
384#479—178no—
385#477—178no—
386#475—178no—
387#433—178no—
388C2×#107—171no—
389C2×#107—171no—
390C2×#107—171no—
391C2×#107—171no—
392C2×#106—171no—
393C2×#106—171no—
394C2×#106C2×D1711yescharacteristic
395C2×#106C2×D1711yescharacteristic
396C2×C4⋊C8—171no—
397C2×C4⋊C8—171no—
398C2×#103—171no—
399C2×#103—171no—
400C2×#103—171no—
401C2×#103—171no—
402C2×#103—171no—
403C2×#103—171no—
404C2×#103C2×D1711yescharacteristic
405C2×#103C2×D1711yescharacteristic
406C2×C22⋊C8—171no—
407C2×C22⋊C8C2×D1711yescharacteristic
408C2×#97—171no—
409C2×C4×C8—171no—
410C2×C17—42no—
411C2×C17#324214yescomplemented
412C2×C17C2×#53611yescharacteristic
413C2×C17C2×#57711yescharacteristic
414C2×C17C2×#57811yescharacteristic
415C22×Q8—171no—
416C22×Q8—341no—
417C22×Q8—341no—
418C22×Q8C22×D1711yescharacteristic
419C23×C4—21no—
420C23×C4C22×D1711yescharacteristic
421C22×C8—171no—
422C22×C8—171no—
423C22×C8C22×D1711yescharacteristic
424C22×C8C22×D1711yescharacteristic
425C4⋊Q8—174no—
426C4⋊Q8—174no—
427C4⋊Q8—174no—
428C4⋊Q8—342no—
429C4⋊Q8—342no—
430C4⋊Q8—342no—
431C4⋊Q8—348no—
432#126—348no—
433#126—348no—
434#125—174no—
435#125—174no—
436#125—174no—
437#125—342no—
438#125—342no—
439#125—342no—
440#125—348no—
441#125—348no—
442#125—348no—
443C22⋊Q8—22no—
444C22⋊Q8—174no—
445C22⋊Q8—348no—
446C22⋊Q8—348no—
447C22⋊Q8—342no—
448C22⋊Q8C22×D1714yes—
449C4×Q8—174no—
450C4×Q8—174no—
451C4×Q8—342no—
452C4×Q8—342no—
453#117—174no—
454#117—174no—
455#117—174no—
456#117—342no—
457C2×C4⋊C4—21no—
458C2×C4⋊C4—22no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—171no—
461C2×C4⋊C4—171no—
462C2×C4⋊C4—171no—
463C2×C4⋊C4—171no—
464C2×C4⋊C4—171no—
465C2×C4⋊C4—341no—
466C2×C4⋊C4—341no—
467C2×C4⋊C4—341no—
468C2×C4⋊C4—341no—
469C2×C4⋊C4—341no—
470C2×C4⋊C4—341no—
471C2×C4⋊C4—341no—
472C2×C4⋊C4—341no—
473C2×C4⋊C4—341no—
474C2×C4⋊C4—341no—
475C2×C4⋊C4—341no—
476C2×C4⋊C4—341no—
477C2×C4⋊C4—341no—
478C2×C4⋊C4C22×D1711yescharacteristic
479C2×C4⋊C4C22×D1711yescharacteristic
480C2×C4⋊C4C22×D1714yes—
481C2×C4⋊C4C22×D1711yescharacteristic
482C2×C22⋊C4—21no—
483C2×C22⋊C4—171no—
484C2×C22⋊C4—171no—
485C2×C22⋊C4—341no—
486C2×C22⋊C4—341no—
487C2×C42—171no—
488C2×C42—171no—
489C2×C42—341no—
490C2×C42—341no—
491C2×C42—341no—
492#107—174no—
493#107—174no—
494#107—174no—
495#107—174no—
496#106—174no—
497#106—174no—
498#106C22×D1714yes—
499#106C22×D1714yes—
500C4⋊C8—174no—
501C4⋊C8—174no—
502#103—174no—
503#103—174no—
504#103—174no—
505#103—174no—
506#103—174no—
507#103—174no—
508#103C22×D1714yes—
509#103C22×D1714yes—
510C22⋊C8—174no—
511C22⋊C8C22×D1714yes—
512#97—174no—
513C4×C8—174no—
514C17C2×#324211yescharacteristic, complemented, Sylow
515C24D4×D1711yescharacteristic
516C2×Q8—21no—
517C2×Q8—22no—
518C2×Q8—22no—
519C2×Q8—172no—
520C2×Q8—344no—
521C2×Q8—344no—
522C2×Q8—341no—
523C2×Q8—342no—
524C2×Q8—342no—
525C2×Q8—1362no—
526C2×Q8—1362no—
527C2×Q8C23×D1712yes—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—24no—
535C22×C4—82no—
536C22×C4—171no—
537C22×C4—171no—
538C22×C4—171no—
539C22×C4—171no—
540C22×C4—341no—
541C22×C4—341no—
542C22×C4—341no—
543C22×C4—341no—
544C22×C4D4×D1712yes—
545C22×C4D4×D1711yescharacteristic
546C22×C4C23×D1712yes—
547C22×C4C23×D1711yescharacteristic
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—172no—
551C2×C8—172no—
552C2×C8—342no—
553C2×C8—342no—
554C2×C8C23×D1712yes—
555C2×C8C23×D1712yes—
556C4⋊C4—24no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—24no—
560C4⋊C4—21no—
561C4⋊C4—174no—
562C4⋊C4—174no—
563C4⋊C4—172no—
564C4⋊C4—172no—
565C4⋊C4—172no—
566C4⋊C4—344no—
567C4⋊C4—344no—
568C4⋊C4—344no—
569C4⋊C4—344no—
570C4⋊C4—344no—
571C4⋊C4—344no—
572C4⋊C4—344no—
573C4⋊C4—344no—
574C4⋊C4—344no—
575C4⋊C4—344no—
576C4⋊C4—344no—
577C4⋊C4—344no—
578C4⋊C4—341no—
579C4⋊C4—341no—
580C4⋊C4—341no—
581C4⋊C4—344no—
582C4⋊C4C23×D1712yes—
583C4⋊C4C23×D1712yes—
584C4⋊C4C23×D1712yes—
585C22⋊C4—24no—
586C22⋊C4—174no—
587C22⋊C4—174no—
588C22⋊C4—344no—
589C22⋊C4—344no—
590C42—172no—
591C42—172no—
592C42—172no—
593C42—172no—
594C42—344no—
595C42—344no—
596C42—344no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23C2×D4×D1712yes—
601C23C2×D4×D1711yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—42no—
605Q8—342no—
606Q8—342no—
607Q8—682no—
608Q8—1364no—
609Q8—1364no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—84no—
623C2×C4—172no—
624C2×C4—172no—
625C2×C4—172no—
626C2×C4—172no—
627C2×C4—342no—
628C2×C4—342no—
629C2×C4—342no—
630C2×C4—342no—
631C2×C4—342no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—682no—
636C2×C4—682no—
637C2×C4—682no—
638C2×C4—682no—
639C2×C4C2×D4×D1712yes—
640C2×C4C2×D4×D1711yescharacteristic
641C2×C4#1058769712yes—
642C2×C4#1058769712yes—
643C2×C4C24×D1711yescharacteristic
644C8—22no—
645C8—22no—
646C8—342no—
647C8—342no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—44no—
652C22#1176338112yes—
653C22#1176361512yes—
654C22#1176362512yes—
655C22C22×D4×D1711yescharacteristic
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—342no—
662C4—342no—
663C4—342no—
664C4—342no—
665C4—682no—
666C4—682no—
667C4—682no—
668C4—682no—
669C4#1176338112yes—
670C4C2×#1058769711yescharacteristic
671C4C2×#1058769711yescharacteristic
672C2—42no—
673C2#1454004814yesdirect product factor
674C2C2×#1176338111yescharacteristic
675C2C2×#1176361511yescharacteristic
676C2C2×#1176362511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC2×C4×C17C25
FrattiniC2×C4C24×D17
FittingC2×C17×#483C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#483D17
hypocenterC17C2×#3242
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #14540048C2direct (N × H)
4. C2×#11763216C2semidirect
5. C2×#11763209C2semidirect
6. C2×#11763176C2semidirect
7. C2×#11763158C2semidirect
11. C2×#11763119C2semidirect
12. C2×#11763114C2semidirect
13. C2×#11763102C2semidirect
14. C2×#11763093C2semidirect
40. #11763216C22semidirect
41. #11763209C22semidirect
42. #11763176C22semidirect
43. #11763158C22semidirect
47. #11763119C22semidirect
48. #11763114C22semidirect
49. #11763102C22semidirect
50. #11763093C22semidirect
411. C2×C17#3242semidirect
514. C17C2×#3242semidirect
673. C2#14540048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117634402
C2×#117634192
#145400482
C2×#117632162
C2×#117632092
C2×#117631762
C2×#117631582
C2×#117631442
C2×#117631512
C2×#117631472
C2×#117631192
C2×#117631142
C2×#117631022
C2×#117630932
C2×#117628672
C2×C17×#4832
C2×#324217