Finite Group Databasebeta

← C2×#22560442 #30856693

Subgroups — C2×#22560442 #30856693

Order 17152.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863439C211yesmaximal, characteristic, complemented
2C22×#14523970C211yesmaximal, characteristic
3C2×#16863397C211yesmaximal, characteristic
4C2×#16863261C211yesmaximal, characteristic, complemented
5#22560442C218yesmaximal, direct product factor
6C2×#16863165C211yesmaximal, characteristic, complemented
7C2×#16863155C211yesmaximal, characteristic, complemented
8C2×#520×C67C211yesmaximal, characteristic
9C22×#11761612C2211yescharacteristic
10C22×#11761612C2211yescharacteristic
11C22×C4⋊Dic67C2211yescharacteristic
12C22×C4×Dic67C2211yescharacteristic
13#16863439—22no—
14#16863439C2214yescomplemented
15C2×#14523976—21no—
16C2×#14523974—21no—
17C2×#14523970C418yescomplemented
18C2×#14523970C2214yes—
19#16863397—22no—
20#16863397C2214yes—
21C2×#14523967—21no—
22C2×#14523964—21no—
23#16863261—22no—
24#16863261C2214yescomplemented
25C2×C67⋊#95—22no—
26#16863165C2214yescomplemented
27#16863155C2214yescomplemented
28C22×#41×C67C2211yescharacteristic
29#520×C67—22no—
30#520×C67—22no—
31#520×C67—22no—
32#520×C67C2214yes—
33C2×C4×Q8×C67—21no—
34C2×C4×D4×C67—21no—
35C2×C4×D4×C67—22no—
36C2×#117×C67—21no—
37C2×#117×C67C2211yescharacteristic
38C22×C4⋊C4×C67—21no—
39C22×C4⋊C4×C67C2211yescharacteristic
40C23×Dic67—21no—
41C23×Dic67—21no—
42C2×#11761612—21no—
43C2×#11761612—21no—
44C2×#11761612—22no—
45C2×#11761612—22no—
46C2×#11761612—22no—
47C2×#11761612C2×C414yescomplemented
48C2×#11761612C2312yes—
49C2×#11761612C2312yes—
50C22×C67⋊C8—21no—
51C22×C67⋊C8—21no—
52C2×C4⋊Dic67—21no—
53C2×C4⋊Dic67—21no—
54C2×C4⋊Dic67—21no—
55C2×C4⋊Dic67—21no—
56C2×C4⋊Dic67—21no—
57C2×C4⋊Dic67C2312yes—
58C2×C4×Dic67—21no—
59C2×C4×Dic67—22no—
60C2×C4×Dic67C2×C414yescomplemented
61C2×C4×Dic67C2312yes—
62#14523976—24no—
63#14523974—24no—
64#14523970C2×C4116yescomplemented
65#14523967—24no—
66#14523964—24no—
67C67⋊#95—28no—
68C2×#41×C67—21no—
69C2×#41×C67—21no—
70C2×#41×C67—22no—
71C2×#41×C67—22no—
72C2×#41×C67C2×C414yes—
73C2×#41×C67D412yes—
74C2×#41×C67Q814yes—
75C2×#41×C67C2312yes—
76C22×Q8×C67D411yescharacteristic
77C22×D4×C67—21no—
78C22×D4×C67D411yescharacteristic
79C23×C4×C67—21no—
80C23×C4×C67—22no—
81C23×C4×C67D412yes—
82C23×C4×C67C2311yescharacteristic
83C4×Q8×C67—24no—
84C4×Q8×C67—24no—
85C4×D4×C67—28no—
86C4×D4×C67—24no—
87C4×D4×C67—24no—
88C4×D4×C67—28no—
89#117×C67—24no—
90#117×C67—21no—
91#117×C67—21no—
92#117×C67—24no—
93#117×C67D412yes—
94#117×C67C2312yes—
95C2×C4⋊C4×C67—22no—
96C2×C4⋊C4×C67—24no—
97C2×C4⋊C4×C67—21no—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—24no—
102C2×C4⋊C4×C67D412yes—
103C2×C4⋊C4×C67D411yescharacteristic
104C2×C4⋊C4×C67D411yescharacteristic
105C2×C4⋊C4×C67C2312yes—
106C2×C22⋊C4×C67—22no—
107C2×C22⋊C4×C67—21no—
108C2×C42×C67—22no—
109C2×C42×C67—21no—
110C22×Dic67—24no—
111C22×Dic67—22no—
112C22×Dic67—22no—
113C22×Dic67—42no—
114C22×Dic67—42no—
115C22×Dic67—44no—
116C22×Dic67—41no—
117C22×Dic67—41no—
118#11761612—24no—
119#11761612—22no—
120#11761612—22no—
121#11761612—22no—
122#11761612—42no—
123#11761612C22×C414yescomplemented
124C2×C67⋊C8—24no—
125C2×C67⋊C8—22no—
126C2×C67⋊C8—22no—
127C2×C67⋊C8—42no—
128C4⋊Dic67—22no—
129C4⋊Dic67—22no—
130C4⋊Dic67—41no—
131C4⋊Dic67—41no—
132C4×Dic67—24no—
133C4×Dic67—22no—
134C4×Dic67C22×C414yescomplemented
135C24×C67—21no—
136C24×C67C2×D411yescharacteristic
137#41×C67—24no—
138#41×C67—24no—
139#41×C67—24no—
140#41×C67—24no—
141#41×C67—24no—
142#41×C67—22no—
143#41×C67—22no—
144#41×C67C4⋊C418yes—
145#41×C67C22×C414yes—
146#41×C67C2×Q814yes—
147C2×Q8×C67—21no—
148C2×Q8×C67—22no—
149C2×Q8×C67C4⋊C414yes—
150C2×Q8×C67C2×D412yes—
151C2×D4×C67—24no—
152C2×D4×C67—22no—
153C2×D4×C67—24no—
154C2×D4×C67—22no—
155C2×D4×C67—21no—
156C2×D4×C67—22no—
157C2×D4×C67C4⋊C414yes—
158C2×D4×C67C2×D412yes—
159C22×C4×C67—21no—
160C22×C4×C67—24no—
161C22×C4×C67—22no—
162C22×C4×C67—24no—
163C22×C4×C67—22no—
164C22×C4×C67—21no—
165C22×C4×C67—22no—
166C22×C4×C67—21no—
167C22×C4×C67—28no—
168C22×C4×C67—22no—
169C22×C4×C67—44no—
170C22×C4×C67—84no—
171C22×C4×C67C22×C412yes—
172C22×C4×C67C2×D414yes—
173C22×C4×C67C2×D411yescharacteristic
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×Q812yes—
179C22×C4×C67#4112yes—
180C22×C4×C67#4112yes—
181C22×C4×C67C2411yescharacteristic
182C4⋊C4×C67—28no—
183C4⋊C4×C67—22no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—22no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—22no—
188C4⋊C4×C67—22no—
189C4⋊C4×C67C2×D412yes—
190C4⋊C4×C67C2×D412yes—
191C22⋊C4×C67—28no—
192C22⋊C4×C67—22no—
193C22⋊C4×C67—22no—
194C42×C67—28no—
195C42×C67—22no—
196C42×C67—22no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—22no—
200C23×C67—24no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67C4×D412yes—
204C23×C67#12011yescharacteristic
205C23×C67#12011yescharacteristic
206C23×C67C22⋊Q812yes—
207C23×C67C22×D411yescharacteristic
208Q8×C67—22no—
209Q8×C67—24no—
210Q8×C67C2×C4⋊C414yes—
211D4×C67—24no—
212D4×C67—24no—
213D4×C67—28no—
214D4×C67—22no—
215D4×C67—24no—
216D4×C67C2×C4⋊C414yes—
217C2×C4×C67—22no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—24no—
221C2×C4×C67—28no—
222C2×C4×C67—22no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—42no—
226C2×C4×C67—44no—
227C2×C4×C67—42no—
228C2×C4×C67—44no—
229C2×C4×C67—82no—
230C2×C4×C67—88no—
231C2×C4×C67C2×C4⋊C412yes—
232C2×C4×C67C2×C4⋊C412yes—
233C2×C4×C67C4×D412yes—
234C2×C4×C67C4×D412yes—
235C2×C4×C67C4×D412yes—
236C2×C4×C67#12011yescharacteristic
237C2×C4×C67#12011yescharacteristic
238C2×C4×C67C22⋊Q812yes—
239C2×C4×C67C22⋊Q812yes—
240C2×C4×C67C22⋊Q812yes—
241C2×C4×C67#12312yes—
242C2×C4×C67#12312yes—
243C2×C4×C67C23×C411yescharacteristic
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×Q811yescharacteristic
247C2×C4×C67C2×#4111yescharacteristic
248C2×C4×C67C2×#4111yescharacteristic
249C2×Dic67—24no—
250C2×Dic67—48no—
251C2×Dic67—42no—
252C2×Dic67—42no—
253C2×Dic67—44no—
254C2×Dic67—42no—
255C2×Dic67—82no—
256C2×Dic67—82no—
257C67⋊C8—24no—
258C67⋊C8—42no—
259C22×C67—24no—
260C22×C67—24no—
261C22×C67—21no—
262C22×C67—21no—
263C22×C67—21no—
264C22×C67—21no—
265C22×C67—24no—
266C22×C67—48no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67#38512yes—
270C22×C67#38512yes—
271C22×C67#45412yes—
272C22×C67#45512yes—
273C22×C67C2×C4×D411yescharacteristic
274C22×C67C2×#12011yescharacteristic
275C22×C67C2×C22⋊Q811yescharacteristic
276C4×C67—24no—
277C4×C67—44no—
278C4×C67—42no—
279C4×C67—42no—
280C4×C67#38512yes—
281C4×C67#38512yes—
282C4×C67C22×C4⋊C411yescharacteristic
283C4×C67C2×C4×D411yescharacteristic
284C4×C67C2×C22⋊Q811yescharacteristic
285C4×C67C2×#12311yescharacteristic
286Dic67—44no—
287Dic67—82no—
288C2×#1860—671nomaximal, Sylow
289C2×C67—21no—
290C2×C67—21no—
291C2×C67—22no—
292C2×C67—44no—
293C2×C67#186014yescomplemented
294C2×C67C2×#38511yescharacteristic
295C2×C67C2×#45511yescharacteristic
296C2×C67C2×#45411yescharacteristic
297C2×#520D6711yescharacteristic
298C2×#428—671no—
299C22×#104—671no—
300C2×#417—671no—
301C2×#382—671no—
302#1860—678no—
303C2×#343—671no—
304C2×#339—671no—
305C67C2×#186011yescharacteristic, complemented, Sylow
306C22×#41C2×D6711yescharacteristic
307C22×M16—671no—
308C22×M16—671no—
309#520—22no—
310#520—22no—
311#520—22no—
312#520C2×D6714yes—
313C2×C4×Q8—21no—
314C2×C4×D4—21no—
315C2×C4×D4—22no—
316C2×#117—21no—
317C2×#117C2×D6711yescharacteristic
318C22×C4⋊C4—21no—
319C22×C4⋊C4—671no—
320C22×C4⋊C4C2×D6711yescharacteristic
321C22×C42—671no—
322#428—674no—
323#428—1342no—
324C2×#107—1341no—
325C2×#106—1341no—
326C2×#104—678no—
327C2×#104—674no—
328#417—674no—
329#417—1342no—
330C2×#103—1341no—
331C2×#102—1341no—
332#382—674no—
333#382—1342no—
334C2×#95—1342no—
335#343—674no—
336#339—674no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic13414yes—
342C2×#41C4×D6714yes—
343C2×#41C22⋊D6712yes—
344C2×#41C22×D6712yes—
345C22×Q8C22⋊D6711yescharacteristic
346C22×D4—21no—
347C22×D4C22⋊D6711yescharacteristic
348C23×C4—21no—
349C23×C4—22no—
350C23×C4—1341no—
351C23×C4—1341no—
352C23×C4C22⋊D6712yes—
353C23×C4C22×D6711yescharacteristic
354C2×M16—674no—
355C2×M16—672no—
356C2×M16—672no—
357C2×M16—1342no—
358C2×M16—1342no—
359C2×M16—1341no—
360C2×M16—1341no—
361C2×M16—1342no—
362C22×C8—1341no—
363C22×C8—1341no—
364C4×Q8—24no—
365C4×Q8—24no—
366C4×D4—28no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—28no—
370#117—24no—
371#117—21no—
372#117—21no—
373#117—24no—
374#117C22⋊D6712yes—
375#117C22×D6712yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—672no—
384C2×C4⋊C4—1341no—
385C2×C4⋊C4—1341no—
386C2×C4⋊C4—1341no—
387C2×C4⋊C4—1341no—
388C2×C4⋊C4—1341no—
389C2×C4⋊C4C22⋊D6712yes—
390C2×C4⋊C4C22⋊D6711yescharacteristic
391C2×C4⋊C4C22⋊D6711yescharacteristic
392C2×C4⋊C4C22×D6712yes—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—674no—
398C2×C42—672no—
399C2×C42—1341no—
400C2×C42—1342no—
401#107—1344no—
402#106—1344no—
403#104—6716no—
404#103—1344no—
405#102—1344no—
406#95—1348no—
407C24—21no—
408C24C2×C22⋊D6711yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41Dic67⋊C418yes—
417#41C2×C4×D6714yes—
418#41C2×Dic13414yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8Dic67⋊C414yes—
422C2×Q8C2×C22⋊D6712yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4Dic67⋊C414yes—
430C2×D4C2×C22⋊D6712yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—2682no—
444C22×C4—2682no—
445C22×C4—2684no—
446C22×C4—2681no—
447C22×C4—2681no—
448C22×C4—1342no—
449C22×C4—1344no—
450C22×C4—1342no—
451C22×C4C2×C4×D6712yes—
452C22×C4C2×C22⋊D6714yes—
453C22×C4C2×C22⋊D6711yescharacteristic
454C22×C4C2×C22⋊D6711yescharacteristic
455C22×C4C2×C22⋊D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4C2×C22⋊D6711yescharacteristic
458C22×C4C2×Dic13412yes—
459C22×C4#1176162812yes—
460C22×C4#1176162812yes—
461C22×C4C23×D6711yescharacteristic
462M16—2682no—
463M16—674no—
464M16—1342no—
465M16—1342no—
466M16—1344no—
467M16—1342no—
468C2×C8—2682no—
469C2×C8—1342no—
470C2×C8—1344no—
471C2×C8—1342no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—2681no—
480C4⋊C4—2681no—
481C4⋊C4—1342no—
482C4⋊C4—1342no—
483C4⋊C4C2×C22⋊D6712yes—
484C4⋊C4C2×C22⋊D6712yes—
485C22⋊C4—28no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—674no—
492C42—1344no—
493C42—1342no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23C4×C22⋊D6712yes—
501C23#1452401211yescharacteristic
502C23#1452401211yescharacteristic
503C23#1452402212yes—
504C23C22×C22⋊D6711yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×Dic67⋊C414yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×Dic67⋊C414yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—82no—
527C2×C4—88no—
528C2×C4—2682no—
529C2×C4—2682no—
530C2×C4—2688no—
531C2×C4—2684no—
532C2×C4—2682no—
533C2×C4—5362no—
534C2×C4—5362no—
535C2×C4—1344no—
536C2×C4C2×Dic67⋊C412yes—
537C2×C4C2×Dic67⋊C412yes—
538C2×C4C4×C22⋊D6712yes—
539C2×C4C4×C22⋊D6712yes—
540C2×C4C4×C22⋊D6712yes—
541C2×C4#1452401211yescharacteristic
542C2×C4#1452401211yescharacteristic
543C2×C4#1452402212yes—
544C2×C4#1452402212yes—
545C2×C4#1452402212yes—
546C2×C4#1452402612yes—
547C2×C4#1452402612yes—
548C2×C4C22×C4×D6711yescharacteristic
549C2×C4C22×C22⋊D6711yescharacteristic
550C2×C4C22×C22⋊D6711yescharacteristic
551C2×C4C22×Dic13411yescharacteristic
552C2×C4C2×#1176162811yescharacteristic
553C2×C4C2×#1176162811yescharacteristic
554C8—2682no—
555C8—1344no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1686326912yes—
567C22#1686326912yes—
568C22#1686356912yes—
569C22#1686357412yes—
570C22C2×C4×C22⋊D6711yescharacteristic
571C22C2×#1452401211yescharacteristic
572C22C2×#1452402211yescharacteristic
573C4—24no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—2684no—
578C4—5362no—
579C4#1686326912yes—
580C4#1686326912yes—
581C4C22×Dic67⋊C411yescharacteristic
582C4C2×C4×C22⋊D6711yescharacteristic
583C4C2×#1452402211yescharacteristic
584C4C2×#1452402611yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2256044214yesdirect product factor
590C2C2×#1686326911yescharacteristic
591C2C2×#1686357411yescharacteristic
592C2C2×#1686356911yescharacteristic
subgroupG/H
centerC23#14524012
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#520×C67C2
socleC23×C67#120
prime radicalC22×#41×C67C22
cosocleC22×C4×C67C24
hypercenterC2×#520D67
hypocenterC67C2×#1860
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16863439C2semidirect
4. C2×#16863261C2semidirect
5. #22560442C2direct (N × H)
6. C2×#16863165C2semidirect
7. C2×#16863155C2semidirect
14. #16863439C22semidirect
17. C2×#14523970C4semidirect
24. #16863261C22semidirect
26. #16863165C22semidirect
27. #16863155C22semidirect
47. C2×#11761612C2×C4semidirect
60. C2×C4×Dic67C2×C4semidirect
64. #14523970C2×C4semidirect
123. #11761612C22×C4semidirect
134. C4×Dic67C22×C4semidirect
293. C2×C67#1860semidirect
305. C67C2×#1860semidirect
589. C2#22560442direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168634392
C22×#145239702
C2×#168633972
C2×#168632612
#225604422
C2×#168631652
C2×#168631552
C2×#520×C672
C2×#186067