Finite Group Databasebeta

← C2×#22571377 #31727861

Subgroups — C2×#22571377 #31727861

Order 17152.

1822 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863930C211yesmaximal, characteristic
2#22571377C2116yesmaximal, direct product factor
3C2×#16863795C212yesmaximal, complemented
4C2×#16863753C212yesmaximal, complemented
5C2×#16863731C211yesmaximal, characteristic, complemented
6C2×#16863714C211yesmaximal, characteristic
7C2×#16863726C211yesmaximal, characteristic, complemented
8C2×#16863709C211yesmaximal, characteristic
9C2×#16863700C212yesmaximal, complemented
10C2×#16863693C212yesmaximal, complemented
11C2×#16863449C211yesmaximal, characteristic, complemented
12C22×#125×C67C211yesmaximal, characteristic, complemented
13#16863930—24no—
14#16863930C2218yes—
15C2×#14524037—21no—
16C2×#14524035C2211yescharacteristic
17C2×#14524031C2211yescharacteristic
18C2×#14524026—21no—
19C2×#14524022C2211yescharacteristic
20C2×#14524017C2211yescharacteristic
21C2×C4×Dic134C2211yescharacteristic
22C2×C4×D268C2211yescharacteristic
23C2×C4×C22⋊D67—21no—
24C2×#14524000—21no—
25#16863795C22116yescomplemented
26#16863753C22116yescomplemented
27#16863731C2218yescomplemented
28#16863726C2218yescomplemented
29#16863714C2218yes—
30#16863709C2218yes—
31#16863700C22116yescomplemented
32#16863693C22116yescomplemented
33#16863449C2218yescomplemented
34C2×#14523976C2212yes—
35C2×#14523976C2212yes—
36C2×#14523974C2212yes—
37C2×#14523974C2212yes—
38C2×C4⋊(C67⋊C8)C2212yes—
39C2×#14523966C2212yes—
40C2×#14523966C2212yes—
41C2×#14523963C2212yes—
42C2×#14523963C2212yes—
43C2×C22⋊(C67⋊C8)C2211yescharacteristic
44C2×C22⋊(C67⋊C8)C2211yescharacteristic
45C2×C4×C67⋊C8C2212yes—
46C2×#125×C67—21no—
47C2×#125×C67—21no—
48C2×#125×C67—24no—
49C2×#125×C67C2212yes—
50C2×#125×C67C2212yes—
51C2×#125×C67C2218yescomplemented
52C22×C4⋊C4×C67—21no—
53C22×C4⋊C4×C67—21no—
54C22×C4⋊C4×C67C2211yescharacteristic
55C22×C4⋊C4×C67C2211yescharacteristic
56C22×C42×C67C2211yescharacteristic
57C22×Dic134C2311yescharacteristic
58C22×D268C2311yescharacteristic
59C22×C22⋊D67—21no—
60C22×C4×D67—21no—
61C22×C67⋊C8C2312yes—
62C22×C67⋊C8C2312yes—
63#14524037—28no—
64#14524035—22no—
65#14524035C2314yes—
66#14524031—22no—
67#14524031C2314yes—
68#14524026—28no—
69#14524022—22no—
70#14524022C2314yes—
71#14524017—22no—
72#14524017C2314yes—
73C4×Dic134—22no—
74C4×Dic134C2314yes—
75C4×D268—22no—
76C4×D268C2314yes—
77C4×C22⋊D67—28no—
78#14524000—28no—
79C2×C4⋊Dic67C2311yescharacteristic
80C2×C4⋊Dic67C2311yescharacteristic
81C2×Dic67⋊C4—21no—
82C2×Dic67⋊C4—21no—
83C2×Dic67⋊C4—21no—
84C2×Dic67⋊C4—21no—
85C2×C22⋊Dic67—21no—
86C2×(C2×D67)⋊C4—21no—
87C2×(C2×D67)⋊C4—21no—
88C2×(C2×D67)⋊C4—21no—
89C2×(C2×D67)⋊C4—21no—
90C2×C4×Dic67—21no—
91#14523976C2318yes—
92#14523976C2318yes—
93#14523974C2318yes—
94#14523974C2318yes—
95C4⋊(C67⋊C8)C2318yes—
96#14523966C2318yes—
97#14523966C2318yes—
98#14523963C2318yes—
99#14523963C2318yes—
100C22⋊(C67⋊C8)C2314yes—
101C22⋊(C67⋊C8)C2314yes—
102C4×C67⋊C8C2318yes—
103C23×C4×C67—21no—
104C23×C4×C67—21no—
105C23×C4×C67D411yescharacteristic
106C23×C4×C67D411yescharacteristic
107C23×C4×C67C2311yescharacteristic
108#125×C67—24no—
109#125×C67—24no—
110#125×C67—28no—
111#125×C67—28no—
112#125×C67C2318yes—
113#125×C67C2318yes—
114C2×C4⋊C4×C67—22no—
115C2×C4⋊C4×C67—22no—
116C2×C4⋊C4×C67—22no—
117C2×C4⋊C4×C67—22no—
118C2×C4⋊C4×C67—28no—
119C2×C4⋊C4×C67—28no—
120C2×C4⋊C4×C67—21no—
121C2×C4⋊C4×C67—21no—
122C2×C4⋊C4×C67—22no—
123C2×C4⋊C4×C67—22no—
124C2×C4⋊C4×C67C2312yes—
125C2×C4⋊C4×C67C2312yes—
126C2×C4⋊C4×C67C2314yes—
127C2×C4⋊C4×C67C2314yes—
128C2×C42×C67D411yescharacteristic
129C2×C42×C67D411yescharacteristic
130C2×C42×C67D414yes—
131C2×C42×C67C2311yescharacteristic
132C2×C42×C67C2311yescharacteristic
133C2×C42×C67C2314yes—
134C23×D67—21no—
135C2×Dic134—24no—
136C2×Dic134—21no—
137C2×Dic134C2412yes—
138C2×D268—24no—
139C2×D268—21no—
140C2×D268C2412yes—
141C2×C22⋊D67—24no—
142C2×C22⋊D67—44no—
143C22×Dic67—21no—
144C22×Dic67—21no—
145C22×Dic67—21no—
146C2×C4×D67—24no—
147C2×C4×D67—44no—
148C2×C67⋊C8—24no—
149C2×C67⋊C8—24no—
150C2×C67⋊C8C2414yes—
151C2×C67⋊C8C2414yes—
152C4⋊Dic67—21no—
153C4⋊Dic67—21no—
154C4⋊Dic67C2412yes—
155C4⋊Dic67C2412yes—
156Dic67⋊C4—24no—
157Dic67⋊C4—24no—
158Dic67⋊C4—24no—
159Dic67⋊C4—24no—
160C22⋊Dic67—24no—
161(C2×D67)⋊C4—24no—
162(C2×D67)⋊C4—24no—
163(C2×D67)⋊C4—24no—
164(C2×D67)⋊C4—24no—
165C4×Dic67—24no—
166C24×C67C2×D411yescharacteristic
167C22×C4×C67—24no—
168C22×C4×C67—24no—
169C22×C4×C67—24no—
170C22×C4×C67—24no—
171C22×C4×C67—22no—
172C22×C4×C67—22no—
173C22×C4×C67—42no—
174C22×C4×C67—44no—
175C22×C4×C67—44no—
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D412yes—
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67C2×D414yes—
183C22×C4×C67C2×D414yes—
184C22×C4×C67C2412yes—
185C22×C4×C67C2411yescharacteristic
186C4⋊C4×C67—28no—
187C4⋊C4×C67—28no—
188C4⋊C4×C67—28no—
189C4⋊C4×C67—28no—
190C4⋊C4×C67—24no—
191C4⋊C4×C67—24no—
192C4⋊C4×C67—22no—
193C4⋊C4×C67—22no—
194C4⋊C4×C67C2414yes—
195C4⋊C4×C67C2414yes—
196C42×C67C2×D414yes—
197C42×C67C2×D412yes—
198C42×C67C2×D412yes—
199C42×C67C2×D414yes—
200C42×C67C2412yes—
201C42×C67C2412yes—
202C23×C67—22no—
203C23×C67—22no—
204C23×C67#13514yes—
205C23×C67C22×D412yes—
206C23×C67C22×D411yescharacteristic
207C22×D67—22no—
208C22×D67—42no—
209C22×D67—42no—
210C22×D67—42no—
211C2×C4×C67—22no—
212C2×C4×C67—28no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—22no—
216C2×C4×C67—22no—
217C2×C4×C67—24no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—24no—
221C2×C4×C67—44no—
222C2×C4×C67—48no—
223C2×C4×C67—48no—
224C2×C4×C67—44no—
225C2×C4×C67—44no—
226C2×C4×C67C22×D412yes—
227C2×C4×C67C22×D412yes—
228C2×C4×C67C22×D412yes—
229C2×C4×C67C22×D412yes—
230C2×C4×C67C22×D412yes—
231C2×C4×C67C22×D411yescharacteristic
232C2×C4×C672−1+414yes—
233C2×C4×C67C2511yescharacteristic
234C22⋊D67—48no—
235C2×Dic67—22no—
236C2×Dic67—22no—
237C2×Dic67—22no—
238C2×Dic67—42no—
239C2×Dic67—42no—
240C2×Dic67—42no—
241D268—24no—
242D268—42no—
243C4×D67—48no—
244Dic134—24no—
245Dic134—42no—
246C67⋊C8—24no—
247C67⋊C8—24no—
248C22×C67—22no—
249C22×C67—28no—
250C22×C67—22no—
251C22×C67#53612yes—
252C22×C67C2×#13514yes—
253C22×C67C2×#13512yes—
254C22×C67#57512yes—
255C22×C67C23×D411yescharacteristic
256C2×D67—42no—
257C2×D67—42no—
258C2×D67—42no—
259C2×D67—84no—
260C4×C67—22no—
261C4×C67—22no—
262C4×C67—22no—
263C4×C67—22no—
264C4×C67—22no—
265C4×C67—44no—
266C4×C67—44no—
267C4×C67#53612yes—
268C4×C67C2×2−1+412yes—
269Dic67—42no—
270Dic67—42no—
271Dic67—42no—
272C2×#3232—671nomaximal, Sylow
273C2×C67—24no—
274C2×C67#323214yescomplemented
275C2×C67C2×#53611yescharacteristic
276C2×C67C22×#13511yescharacteristic
277C2×C67C2×#57511yescharacteristic
278D67—82no—
279C2×#529—671no—
280C22×#125D6711yescharacteristic, complemented
281#3232—6716no—
282C2×#499—672no—
283C2×#488—672no—
284C2×#484—671no—
285C2×#481—671no—
286C2×#483—671no—
287C2×#480—671no—
288C2×#479—672no—
289C2×#478—672no—
290C2×#431—671no—
291C67C2×#323211yescharacteristic, complemented, Sylow
292#529—678no—
293#529—1344no—
294C2×#126—1341no—
295C2×#125—21no—
296C2×#125—21no—
297C2×#125—24no—
298C2×#125—671no—
299C2×#125C2×D6712yes—
300C2×#125C2×D6712yes—
301C2×#125C2×D6718yescomplemented
302C2×#124—671no—
303C2×#123—1341no—
304C2×C22⋊Q8—671no—
305C2×#121—671no—
306C2×C4×Q8—671no—
307C2×C4×D4—671no—
308C2×C4×D4—1341no—
309C2×#117—1341no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4—21no—
312C22×C4⋊C4C2×D6711yescharacteristic
313C22×C4⋊C4C2×D6711yescharacteristic
314C22×C42C2×D6711yescharacteristic
315#499—6716no—
316#488—6716no—
317#484—678no—
318#483—678no—
319#481—678no—
320#480—678no—
321#479—6716no—
322#478—6716no—
323#431—678no—
324C2×#107—672no—
325C2×#107—672no—
326C2×#106—672no—
327C2×#106—672no—
328C2×C4⋊C8—672no—
329C2×#103—672no—
330C2×#103—672no—
331C2×#102—672no—
332C2×#102—672no—
333C2×C22⋊C8—671no—
334C2×C22⋊C8—671no—
335C2×C4×C8—672no—
336C22×Q8—671no—
337C22×D4—671no—
338C22×D4—1341no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1341no—
342C23×C4C22⋊D6711yescharacteristic
343C23×C4C22⋊D6711yescharacteristic
344C23×C4C22×D6711yescharacteristic
345C22×C8—672no—
346C22×C8—672no—
347#126—1348no—
348#125—24no—
349#125—24no—
350#125—28no—
351#125—28no—
352#125—674no—
353#125—1342no—
354#125C22×D6718yes—
355#125C22×D6718yes—
356#124—674no—
357#124—1342no—
358#123—1348no—
359C22⋊Q8—674no—
360C22⋊Q8—1342no—
361#121—674no—
362#121—1342no—
363C4×Q8—674no—
364C4×Q8—1342no—
365C4×D4—674no—
366C4×D4—1348no—
367C4×D4—1342no—
368#117—1348no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—671no—
380C2×C4⋊C4—671no—
381C2×C4⋊C4—1341no—
382C2×C4⋊C4—1341no—
383C2×C4⋊C4—1341no—
384C2×C4⋊C4—1341no—
385C2×C4⋊C4C22×D6712yes—
386C2×C4⋊C4C22×D6712yes—
387C2×C4⋊C4C22×D6714yes—
388C2×C4⋊C4C22×D6714yes—
389C2×C22⋊C4—1341no—
390C2×C22⋊C4—1341no—
391C2×C22⋊C4—1341no—
392C2×C22⋊C4—1341no—
393C2×C22⋊C4—1341no—
394C2×C42—1341no—
395C2×C42C22⋊D6711yescharacteristic
396C2×C42C22⋊D6711yescharacteristic
397C2×C42C22⋊D6714yes—
398C2×C42C22×D6711yescharacteristic
399C2×C42C22×D6711yescharacteristic
400C2×C42C22×D6714yes—
401#107—678no—
402#107—678no—
403#106—678no—
404#106—678no—
405C4⋊C8—678no—
406#103—678no—
407#103—678no—
408#102—678no—
409#102—678no—
410C22⋊C8—674no—
411C22⋊C8—674no—
412C4×C8—678no—
413C24—1341no—
414C24C2×C22⋊D6711yescharacteristic
415C2×Q8—672no—
416C2×Q8—1344no—
417C2×Q8—1341no—
418C2×D4—2684no—
419C2×D4—672no—
420C2×D4—1344no—
421C2×D4—1341no—
422C2×D4—1344no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—2684no—
433C22×C4—1341no—
434C22×C4—1341no—
435C22×C4—1341no—
436C22×C4—1344no—
437C22×C4C2×C22⋊D6712yes—
438C22×C4C2×C22⋊D6714yes—
439C22×C4C2×C22⋊D6711yescharacteristic
440C22×C4C2×C22⋊D6711yescharacteristic
441C22×C4C2×C22⋊D6711yescharacteristic
442C22×C4C2×C22⋊D6711yescharacteristic
443C22×C4C2×C22⋊D6711yescharacteristic
444C22×C4C2×C22⋊D6714yes—
445C22×C4C23×D6712yes—
446C22×C4C23×D6711yescharacteristic
447C2×C8—674no—
448C2×C8—674no—
449C2×C8—1344no—
450C2×C8—1344no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—672no—
460C4⋊C4—672no—
461C4⋊C4—1341no—
462C4⋊C4—1341no—
463C4⋊C4—1344no—
464C4⋊C4—1344no—
465C4⋊C4—1344no—
466C4⋊C4—1344no—
467C4⋊C4C23×D6714yes—
468C4⋊C4C23×D6714yes—
469C22⋊C4—1344no—
470C22⋊C4—1344no—
471C22⋊C4—1344no—
472C22⋊C4—1344no—
473C22⋊C4—1344no—
474C42—1344no—
475C42C2×C22⋊D6714yes—
476C42C2×C22⋊D6712yes—
477C42C2×C22⋊D6712yes—
478C42C2×C22⋊D6714yes—
479C42C23×D6712yes—
480C42C23×D6712yes—
481C23—22no—
482C23—22no—
483C23—2682no—
484C23—2682no—
485C23—2682no—
486C23—1342no—
487C23#1452406314yes—
488C23C22×C22⋊D6712yes—
489C23C22×C22⋊D6711yescharacteristic
490Q8—2682no—
491Q8—1344no—
492D4—2688no—
493D4—2682no—
494D4—1344no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—2688no—
511C2×C4—2682no—
512C2×C4—2682no—
513C2×C4—2682no—
514C2×C4—1342no—
515C2×C4—1342no—
516C2×C4—1342no—
517C2×C4C22×C22⋊D6712yes—
518C2×C4C22×C22⋊D6712yes—
519C2×C4C22×C22⋊D6712yes—
520C2×C4C22×C22⋊D6712yes—
521C2×C4C22×C22⋊D6712yes—
522C2×C4C22×C22⋊D6711yescharacteristic
523C2×C4#1452409114yes—
524C2×C4C24×D6711yescharacteristic
525C8—1344no—
526C8—1344no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—2682no—
531C22—2682no—
532C22—2682no—
533C22—5364no—
534C22#1686397012yes—
535C22C2×#1452406314yes—
536C22C2×#1452406312yes—
537C22#1686419412yes—
538C22C23×C22⋊D6711yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—2682no—
547C4—2682no—
548C4—2682no—
549C4#1686397012yes—
550C4C2×#1452409112yes—
551C2—24no—
552C2—5362no—
553C2#2257137714yesdirect product factor
554C2C2×#1686397011yescharacteristic
555C2C22×#1452406311yescharacteristic
556C2C2×#1686419411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC22×#125×C67C2
socleC23×C67C22×D4
prime radicalC2×#14524017C22
hypercenterC22×#125D67
hypocenterC67C2×#3232
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #22571377C2direct (N × H)
3. C2×#16863795C2semidirect
4. C2×#16863753C2semidirect
5. C2×#16863731C2semidirect
7. C2×#16863726C2semidirect
9. C2×#16863700C2semidirect
10. C2×#16863693C2semidirect
11. C2×#16863449C2semidirect
12. C22×#125×C67C2semidirect
25. #16863795C22semidirect
26. #16863753C22semidirect
27. #16863731C22semidirect
28. #16863726C22semidirect
31. #16863700C22semidirect
32. #16863693C22semidirect
33. #16863449C22semidirect
51. C2×#125×C67C22semidirect
274. C2×C67#3232semidirect
280. C22×#125D67semidirect
291. C67C2×#3232semidirect
301. C2×#125C2×D67semidirect
553. C2#22571377direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168639302
#225713772
C2×#168637952
C2×#168637532
C2×#168637312
C2×#168637142
C2×#168637262
C2×#168637092
C2×#168637002
C2×#168636932
C2×#168634492
C22×#125×C672
C2×#323267