Finite Group Databasebeta

← C2×#22570154 #31723634

Subgroups — C2×#22570154 #31723634

Order 17152.

2326 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D67C211yesmaximal, characteristic
2C22×#14524001C211yesmaximal, characteristic
3#22570154C2116yesmaximal, direct product factor
4C2×#16863731C211yesmaximal, characteristic, complemented
5C2×#16863739C212yesmaximal, complemented
6C2×#16863735C212yesmaximal, complemented
7C2×#16863727C211yesmaximal, characteristic, complemented
8C2×#16863724C211yesmaximal, characteristic, complemented
9C2×#16863437C212yesmaximal, complemented
10C2×#16863403C212yesmaximal, complemented
11C2×#16863358C211yesmaximal, characteristic
12C2×#484×C67C211yesmaximal, characteristic, complemented
13C22×Q8×D67C2211yescharacteristic
14C23×C4×D67C2211yescharacteristic
15C22×#11761611C2212yes—
16C2×#14524023C2211yescharacteristic
17C2×#14524022C2211yescharacteristic
18C2×#14524021—21no—
19C2×#14524020—21no—
20C2×#14524019C2211yescharacteristic
21C22⋊Q8×D67—24no—
22C22⋊Q8×D67C2218yes—
23C2×#14524001—21no—
24C2×#14524001—24no—
25C2×#14524001—21no—
26C2×#14524001C2212yes—
27C2×#14524001C2212yes—
28C2×#14524001C2218yes—
29C22×C4⋊Dic67C2211yescharacteristic
30C2×C4⋊C4×D67—21no—
31C2×C4⋊C4×D67C2211yescharacteristic
32C22×(C2×D67)⋊C4—21no—
33C2×C22⋊C4×D67—21no—
34C22×C4×Dic67—21no—
35#16863739C22116yescomplemented
36#16863735C22116yescomplemented
37#16863731C2218yescomplemented
38#16863727C2218yescomplemented
39#16863724C2218yescomplemented
40#16863437C22116yescomplemented
41C2×#14523976C2212yescomplemented
42C2×#14523975C2212yescomplemented
43C2×#14523974C2212yescomplemented
44#16863403C22116yescomplemented
45C2×#14523968C2212yescomplemented
46C2×#14523967C2212yescomplemented
47C2×#14523966C2212yescomplemented
48#16863358C2218yes—
49C2×C22⋊(C67⋊C8)C2211yescharacteristic
50C2×(C2×D67)⋊C8C2212yes—
51C2×C22⋊Q8×C67C2211yescharacteristic
52C22×C4⋊C4×C67C2211yescharacteristic
53#484×C67C2218yescomplemented
54C2×#107×C67C2212yescomplemented
55C2×#103×C67C2212yescomplemented
56C2×C22⋊C8×C67C2211yescharacteristic
57C24×D67D411yescharacteristic
58C22×Dic134—21no—
59C22×Dic134C2311yescharacteristic
60C2×Q8×D67—22no—
61C2×Q8×D67—28no—
62C2×Q8×D67C2314yes—
63C23×Dic67—21no—
64C23×Dic67D411yescharacteristic
65C22×C4×D67—28no—
66C22×C4×D67—21no—
67C22×C4×D67—21no—
68C22×C4×D67D411yescharacteristic
69C22×C4×D67D414yes—
70C22×C4×D67D411yescharacteristic
71C22×C4×D67C2311yescharacteristic
72C22×C4×D67C2314yes—
73C22×C4×D67C2311yescharacteristic
74C2×#11761611—28no—
75C2×#11761611C2318yes—
76C22×C67⋊C8C2312yes—
77#14524023—22no—
78#14524023C2314yes—
79#14524022—22no—
80#14524022C2314yes—
81#14524021—28no—
82#14524020—28no—
83#14524019—22no—
84#14524019C2314yes—
85#14524001—28no—
86#14524001—24no—
87#14524001—28no—
88#14524001—24no—
89#14524001C2318yes—
90#14524001C2318yes—
91C2×C4⋊Dic67—21no—
92C2×C4⋊Dic67—21no—
93C2×C4⋊Dic67—22no—
94C2×C4⋊Dic67C2311yescharacteristic
95C2×C4⋊Dic67C2312yes—
96C2×C4⋊Dic67C2314yes—
97C2×Dic67⋊C4—21no—
98C2×Dic67⋊C4—21no—
99C2×Dic67⋊C4—21no—
100C4⋊C4×D67—28no—
101C4⋊C4×D67—22no—
102C4⋊C4×D67C2314yes—
103C2×C22⋊Dic67—21no—
104C2×(C2×D67)⋊C4—21no—
105C2×(C2×D67)⋊C4—21no—
106C2×(C2×D67)⋊C4—22no—
107C2×(C2×D67)⋊C4—28no—
108C2×(C2×D67)⋊C4—22no—
109C22⋊C4×D67—28no—
110C2×C4×Dic67—22no—
111C2×C4×Dic67—28no—
112C2×C4×Dic67—22no—
113#14523976C2318yescomplemented
114#14523975C2318yescomplemented
115#14523974C2318yescomplemented
116#14523968C2318yescomplemented
117#14523967C2318yescomplemented
118#14523966C2318yescomplemented
119C22⋊(C67⋊C8)C2314yes—
120(C2×D67)⋊C8C2318yes—
121C22×Q8×C67C2311yescharacteristic
122C23×C4×C67—21no—
123C23×C4×C67C2311yescharacteristic
124C22×C8×C67C2312yes—
125C22⋊Q8×C67—22no—
126C22⋊Q8×C67C2314yes—
127C2×C4⋊C4×C67—21no—
128C2×C4⋊C4×C67—22no—
129C2×C4⋊C4×C67—21no—
130C2×C4⋊C4×C67C2314yes—
131C2×C4⋊C4×C67C2312yes—
132C2×C4⋊C4×C67C2311yescharacteristic
133C2×C22⋊C4×C67—21no—
134#107×C67C2318yescomplemented
135#103×C67C2318yescomplemented
136C22⋊C8×C67C2314yes—
137C23×D67—24no—
138C23×D67—24no—
139C23×D67—42no—
140C23×D67C2×D411yescharacteristic
141C23×D67C2×D414yes—
142C23×D67C2×D411yescharacteristic
143C2×Dic134—24no—
144C2×Dic134—21no—
145C2×Dic134—24no—
146C2×Dic134—44no—
147C2×Dic134C2412yes—
148Q8×D67—216no—
149Q8×D67—48no—
150C22×Dic67—24no—
151C22×Dic67—24no—
152C22×Dic67—22no—
153C22×Dic67—21no—
154C22×Dic67—21no—
155C22×Dic67—44no—
156C22×Dic67C2×D411yescharacteristic
157C22×Dic67C2×D414yes—
158C22×Dic67C2×D411yescharacteristic
159C2×C4×D67—24no—
160C2×C4×D67—24no—
161C2×C4×D67—24no—
162C2×C4×D67—216no—
163C2×C4×D67—216no—
164C2×C4×D67—24no—
165C2×C4×D67—24no—
166C2×C4×D67—44no—
167C2×C4×D67—44no—
168C2×C4×D67C2×D414yes—
169C2×C4×D67C2×D412yes—
170C2×C4×D67C2×D414yes—
171C2×C4×D67C2×D412yes—
172C2×C4×D67#4118yes—
173C2×C4×D67C2412yes—
174C2×C4×D67C2412yes—
175#11761611—216no—
176C2×C67⋊C8—24no—
177C2×C67⋊C8C2414yes—
178C4⋊Dic67—21no—
179C4⋊Dic67—24no—
180C4⋊Dic67—22no—
181C4⋊Dic67—24no—
182C4⋊Dic67C2414yes—
183C4⋊Dic67C2412yes—
184Dic67⋊C4—24no—
185Dic67⋊C4—24no—
186Dic67⋊C4—24no—
187C22⋊Dic67—24no—
188(C2×D67)⋊C4—24no—
189(C2×D67)⋊C4—24no—
190(C2×D67)⋊C4—28no—
191(C2×D67)⋊C4—28no—
192C4×Dic67—28no—
193C4×Dic67—28no—
194C24×C67C2×D411yescharacteristic
195C2×Q8×C67—21no—
196C2×Q8×C67—24no—
197C2×Q8×C67C2412yes—
198C22×C4×C67—24no—
199C22×C4×C67—21no—
200C22×C4×C67—21no—
201C22×C4×C67—24no—
202C22×C4×C67—22no—
203C22×C4×C67—44no—
204C22×C4×C67C2×D411yescharacteristic
205C22×C4×C67C2×D412yes—
206C22×C4×C67C2412yes—
207C22×C4×C67C2411yescharacteristic
208C2×C8×C67—24no—
209C2×C8×C67C2414yes—
210C4⋊C4×C67—24no—
211C4⋊C4×C67—22no—
212C4⋊C4×C67—24no—
213C4⋊C4×C67—21no—
214C4⋊C4×C67C2414yes—
215C4⋊C4×C67C2412yes—
216C22⋊C4×C67—24no—
217C23×C67—22no—
218C23×C67—22no—
219C23×C67#13714yes—
220C23×C67C22×D412yes—
221C23×C67C22×D411yescharacteristic
222C22×D67—22no—
223C22×D67—22no—
224C22×D67—22no—
225C22×D67—28no—
226C22×D67—22no—
227C22×D67—22no—
228C22×D67—28no—
229C22×D67—44no—
230C22×D67—416no—
231C22×D67#12314yes—
232C22×D67C22×D412yes—
233C22×D67C22×D412yes—
234Q8×C67—24no—
235Q8×C67—42no—
236C2×C4×C67—22no—
237C2×C4×C67—24no—
238C2×C4×C67—24no—
239C2×C4×C67—22no—
240C2×C4×C67—22no—
241C2×C4×C67—24no—
242C2×C4×C67—48no—
243C2×C4×C67—42no—
244C2×C4×C67—44no—
245C2×C4×C67—42no—
246C2×C4×C67C22×D412yes—
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67C2×#4114yes—
249C2×C4×C67C2511yescharacteristic
250C2×Dic67—22no—
251C2×Dic67—28no—
252C2×Dic67—24no—
253C2×Dic67—22no—
254C2×Dic67—22no—
255C2×Dic67—42no—
256C2×Dic67—48no—
257C2×Dic67—44no—
258C2×Dic67—42no—
259C2×Dic67#12314yes—
260C2×Dic67C22×D412yes—
261C2×Dic67C22×D412yes—
262C4×D67—28no—
263C4×D67—24no—
264C4×D67—24no—
265C4×D67—28no—
266C4×D67—48no—
267C4×D67—48no—
268C4×D67#12318yes—
269C4×D67C2×#4118yes—
270Dic134—24no—
271Dic134—42no—
272Dic134—48no—
273C8×C67—24no—
274C67⋊C8—24no—
275C22×C67—28no—
276C22×C67—22no—
277C22×C67—22no—
278C22×C67C2×#12312yes—
279C22×C67C2×#13714yes—
280C22×C67C2×#13712yes—
281C22×C67#57512yes—
282C22×C67C23×D411yescharacteristic
283C2×D67—28no—
284C2×D67—22no—
285C2×D67—22no—
286C2×D67—22no—
287C2×D67—22no—
288C2×D67—22no—
289C2×D67—416no—
290C2×D67—44no—
291C2×D67C2×#12314yes—
292C4×C67—22no—
293C4×C67—42no—
294C4×C67—42no—
295C4×C67—44no—
296C4×C67C2×#12312yes—
297C4×C67C22×#4112yes—
298Dic67—22no—
299Dic67—42no—
300Dic67—44no—
301Dic67—42no—
302Dic67C2×#12314yes—
303C2×#3095—671nomaximal, Sylow
304C2×C67—24no—
305C2×C67#309514yescomplemented
306C2×C67C22×#12311yescharacteristic
307C2×C67C22×#13711yescharacteristic
308C2×C67C2×#57511yescharacteristic
309D67—24no—
310D67—42no—
311C22×C22⋊Q8—671no—
312C22×#117—671no—
313#3095—6716no—
314C2×#484—671no—
315C2×#484D6711yescharacteristic, complemented
316C2×#485—672no—
317C2×#485—672no—
318C2×#483—671no—
319C2×#483—671no—
320C2×#428—672no—
321C2×#419—672no—
322C2×#407—671no—
323C67C2×#309511yescharacteristic, complemented, Sylow
324C23×Q8—671no—
325C24×C4—671no—
326C22×M16—672no—
327C2×C22⋊Q8—671no—
328C2×C22⋊Q8—671no—
329C2×C22⋊Q8—678no—
330C2×C22⋊Q8—671no—
331C2×C22⋊Q8—1341no—
332C2×C22⋊Q8—1341no—
333C2×C22⋊Q8—1344no—
334C2×C22⋊Q8C2×D6711yescharacteristic
335C2×#117—672no—
336C2×#117—678no—
337C2×#117—672no—
338C2×#117—1341no—
339C2×#117—1344no—
340C2×#117—1341no—
341C22×C4⋊C4—671no—
342C22×C4⋊C4—671no—
343C22×C4⋊C4—1341no—
344C22×C4⋊C4C2×D6711yescharacteristic
345C22×C22⋊C4—1341no—
346C22×C22⋊C4—1341no—
347C22×C42—1341no—
348#485—6716no—
349#485—6716no—
350#484—678no—
351#484C2×D6718yescomplemented
352#483—678no—
353#483—678no—
354#428—6716no—
355C2×#107—672no—
356C2×#107C2×D6712yescomplemented
357C2×#106—672no—
358C2×#106—672no—
359#419—6716no—
360C2×#103—672no—
361C2×#103—672no—
362C2×#103—672no—
363C2×#103C2×D6712yescomplemented
364#407—678no—
365C2×C22⋊C8—671no—
366C2×C22⋊C8—672no—
367C2×C22⋊C8C2×D6711yescharacteristic
368C25—671no—
369C22×Q8—674no—
370C22×Q8—671no—
371C22×Q8—1341no—
372C22×Q8—1348no—
373C22×Q8—1342no—
374C22×Q8C22×D6711yescharacteristic
375C23×C4—21no—
376C23×C4—671no—
377C23×C4—671no—
378C23×C4—671no—
379C23×C4—671no—
380C23×C4—674no—
381C23×C4—671no—
382C23×C4—674no—
383C23×C4—1341no—
384C23×C4—1341no—
385C23×C4—1348no—
386C23×C4—1341no—
387C23×C4C22×D6711yescharacteristic
388C2×M16—678no—
389C2×M16—1348no—
390C22×C8—672no—
391C22×C8C22×D6712yes—
392C22⋊Q8—22no—
393C22⋊Q8—674no—
394C22⋊Q8—674no—
395C22⋊Q8—674no—
396C22⋊Q8—1348no—
397C22⋊Q8—1348no—
398C22⋊Q8—1342no—
399C22⋊Q8—1342no—
400C22⋊Q8—1342no—
401C22⋊Q8C22×D6714yes—
402#117—678no—
403#117—678no—
404#117—1344no—
405#117—1348no—
406#117—1344no—
407#117—1348no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—672no—
412C2×C4⋊C4—674no—
413C2×C4⋊C4—671no—
414C2×C4⋊C4—674no—
415C2×C4⋊C4—1342no—
416C2×C4⋊C4—1348no—
417C2×C4⋊C4—1341no—
418C2×C4⋊C4—1341no—
419C2×C4⋊C4—1341no—
420C2×C4⋊C4—1342no—
421C2×C4⋊C4—1341no—
422C2×C4⋊C4—1341no—
423C2×C4⋊C4C22×D6711yescharacteristic
424C2×C4⋊C4C22×D6712yes—
425C2×C4⋊C4C22×D6714yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1348no—
428C2×C22⋊C4—1348no—
429C2×C22⋊C4—1342no—
430C2×C22⋊C4—1342no—
431C2×C22⋊C4—1341no—
432C2×C22⋊C4—1341no—
433C2×C22⋊C4—1341no—
434C2×C42—1348no—
435C2×C42—1342no—
436C2×C42—1342no—
437#107—678no—
438#107C22×D6718yescomplemented
439#106—678no—
440#106—678no—
441#103—678no—
442#103—678no—
443#103—678no—
444#103C22×D6718yescomplemented
445C22⋊C8—674no—
446C22⋊C8—678no—
447C22⋊C8C22×D6714yes—
448C24—2682no—
449C24—674no—
450C24—671no—
451C24—671no—
452C24—1344no—
453C24—1344no—
454C24D4×D6711yescharacteristic
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—2684no—
458C2×Q8—2688no—
459C2×Q8—672no—
460C2×Q8—1344no—
461C2×Q8—13416no—
462C2×Q8—1344no—
463C2×Q8—1341no—
464C2×Q8C23×D6712yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—2684no—
472C22×C4—2684no—
473C22×C4—2684no—
474C22×C4—671no—
475C22×C4—672no—
476C22×C4—678no—
477C22×C4—672no—
478C22×C4—674no—
479C22×C4—671no—
480C22×C4—674no—
481C22×C4—674no—
482C22×C4—672no—
483C22×C4—672no—
484C22×C4—1344no—
485C22×C4—1344no—
486C22×C4—1344no—
487C22×C4—13416no—
488C22×C4—13416no—
489C22×C4—1344no—
490C22×C4—1344no—
491C22×C4—1342no—
492C22×C4—1341no—
493C22×C4—1341no—
494C22×C4—1344no—
495C22×C4—1344no—
496C22×C4D4×D6712yes—
497C22×C4D4×D6711yescharacteristic
498C22×C4C23×D6712yes—
499C22×C4C23×D6711yescharacteristic
500M16—13416no—
501C2×C8—24no—
502C2×C8—674no—
503C2×C8—1344no—
504C2×C8C23×D6714yes—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—674no—
510C4⋊C4—672no—
511C4⋊C4—1344no—
512C4⋊C4—1344no—
513C4⋊C4—1344no—
514C4⋊C4—1344no—
515C4⋊C4—1342no—
516C4⋊C4—1341no—
517C4⋊C4—1344no—
518C4⋊C4C23×D6714yes—
519C4⋊C4C23×D6712yes—
520C22⋊C4—24no—
521C22⋊C4—1348no—
522C22⋊C4—1348no—
523C22⋊C4—1344no—
524C22⋊C4—1344no—
525C22⋊C4—1344no—
526C42—1348no—
527C42—1348no—
528C23—22no—
529C23—22no—
530C23—26816no—
531C23—2684no—
532C23—674no—
533C23—672no—
534C23—672no—
535C23—1348no—
536C23—1342no—
537C23—1348no—
538C23—1342no—
539C23—1342no—
540C23—1342no—
541C23—1342no—
542C23#1452407514yes—
543C23C2×D4×D6712yes—
544C23C2×D4×D6711yescharacteristic
545Q8—24no—
546Q8—42no—
547Q8—2688no—
548Q8—2682no—
549Q8—1344no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—2688no—
561C2×C4—2688no—
562C2×C4—2682no—
563C2×C4—2688no—
564C2×C4—2684no—
565C2×C4—2682no—
566C2×C4—674no—
567C2×C4—672no—
568C2×C4—678no—
569C2×C4—678no—
570C2×C4—672no—
571C2×C4—1348no—
572C2×C4—1348no—
573C2×C4—1344no—
574C2×C4—1342no—
575C2×C4—1342no—
576C2×C4—1348no—
577C2×C4—1342no—
578C2×C4—1344no—
579C2×C4—1344no—
580C2×C4C2×D4×D6712yes—
581C2×C4C2×D4×D6711yescharacteristic
582C2×C4#41×D6714yes—
583C2×C4C24×D6711yescharacteristic
584C8—24no—
585C8—1344no—
586C22—22no—
587C22—22no—
588C22—28no—
589C22—26816no—
590C22—2684no—
591C22—674no—
592C22—1342no—
593C22—1342no—
594C22—1342no—
595C22—1342no—
596C22—1342no—
597C22—1348no—
598C22#123×D6712yes—
599C22C2×#1452407514yes—
600C22C2×#1452407512yes—
601C22#1686419212yes—
602C22C22×D4×D6711yescharacteristic
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—2684no—
608C4—2682no—
609C4—2682no—
610C4—674no—
611C4—1342no—
612C4#123×D6712yes—
613C4C2×#41×D6712yes—
614C2—24no—
615C2—2682no—
616C2—1344no—
617C2#2257015414yesdirect product factor
618C2C2×#123×D6711yescharacteristic
619C2C22×#1452407511yescharacteristic
620C2C2×#1686419211yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#484×C67C2
socleC23×C67C22×D4
prime radicalC24×D67D4
hypercenterC2×#484D67
hypocenterC67C2×#3095
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
3. #22570154C2direct (N × H)
4. C2×#16863731C2semidirect
5. C2×#16863739C2semidirect
6. C2×#16863735C2semidirect
7. C2×#16863727C2semidirect
8. C2×#16863724C2semidirect
9. C2×#16863437C2semidirect
10. C2×#16863403C2semidirect
12. C2×#484×C67C2semidirect
35. #16863739C22semidirect
36. #16863735C22semidirect
37. #16863731C22semidirect
38. #16863727C22semidirect
39. #16863724C22semidirect
40. #16863437C22semidirect
41. C2×#14523976C22semidirect
42. C2×#14523975C22semidirect
43. C2×#14523974C22semidirect
44. #16863403C22semidirect
45. C2×#14523968C22semidirect
46. C2×#14523967C22semidirect
47. C2×#14523966C22semidirect
53. #484×C67C22semidirect
54. C2×#107×C67C22semidirect
55. C2×#103×C67C22semidirect
113. #14523976C23semidirect
114. #14523975C23semidirect
115. #14523974C23semidirect
116. #14523968C23semidirect
117. #14523967C23semidirect
118. #14523966C23semidirect
134. #107×C67C23semidirect
135. #103×C67C23semidirect
305. C2×C67#3095semidirect
315. C2×#484D67semidirect
323. C67C2×#3095semidirect
351. #484C2×D67semidirect
356. C2×#107C2×D67semidirect
363. C2×#103C2×D67semidirect
438. #107C22×D67semidirect
444. #103C22×D67semidirect
617. C2#22570154direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D672
C22×#145240012
#225701542
C2×#168637312
C2×#168637392
C2×#168637352
C2×#168637272
C2×#168637242
C2×#168634372
C2×#168634032
C2×#168633582
C2×#484×C672
C2×#309567