Finite Group Databasebeta

← C2×#22572834 #31734580

Subgroups — C2×#22572834 #31734580

Order 17152.

1930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864112C211yesmaximal, characteristic
2C2×#16864107C211yesmaximal, characteristic, complemented
3#22572834C2116yesmaximal, direct product factor
4C2×#16863739C212yesmaximal, complemented
5C2×#16863711C212yesmaximal, complemented
6C2×#16863688C211yesmaximal, characteristic, complemented
7C2×#16863683C211yesmaximal, characteristic, complemented
8C2×#16863643C212yesmaximal, complemented
9C2×C4×#11761617C211yesmaximal, characteristic, complemented
10C2×#16863475C211yesmaximal, characteristic, complemented
11C2×#16863437C212yesmaximal, complemented
12C2×#500×C67C211yesmaximal, characteristic, complemented
13C22×#11761617C2211yescharacteristic
14C22×#11761611C2212yes—
15#16864112—24no—
16#16864112C2218yes—
17#16864107—24no—
18#16864107C2218yescomplemented
19C2×#14524034—21no—
20C2×#14524032—22no—
21C2×#14524023C2212yes—
22C2×#14524019—22no—
23C2×#14524018—21no—
24C2×#14524018C2212yes—
25C2×C4×Dic134C2211yescharacteristic
26C2×C4×D268C2211yescharacteristic
27C2×C4×D268C2211yescharacteristic
28C2×#14524005—21no—
29C2×#14524001—21no—
30C2×#14524001C2212yes—
31C2×C4⋊C4×D67C2212yes—
32#16863739C22116yescomplemented
33#16863711C22116yescomplemented
34#16863688C2218yescomplemented
35#16863683C2218yescomplemented
36#16863643C22116yescomplemented
37C4×#11761617C2218yescomplemented
38#16863475C2218yescomplemented
39#16863437C22116yescomplemented
40C2×#14523976C2212yes—
41C2×#14523975C2211yescharacteristic
42C2×#14523974C2212yes—
43C2×C4⋊(C67⋊C8)C2211yescharacteristic
44C2×#14523968C2211yescharacteristic
45C2×#14523966C2212yes—
46C2×#14523965C2211yescharacteristic
47C2×#14523963C2212yes—
48C2×(C2×D67)⋊C8C2212yes—
49C2×C4⋊Q8×C67C2211yescharacteristic
50C2×C4⋊Q8×C67C2211yescharacteristic
51#500×C67C2218yescomplemented
52C2×#107×C67C2212yes—
53C2×#107×C67C2212yes—
54C2×C4×C8×C67C2211yescharacteristic
55C22×Dic134C2311yescharacteristic
56C22×D268—22no—
57C22×D268D411yescharacteristic
58C22×D268C2311yescharacteristic
59C22×C4×D67—22no—
60C22×C4×D67—21no—
61C22×C4×D67C2312yes—
62C2×#11761617—24no—
63C2×#11761617C2314yes—
64C2×#11761611—28no—
65C2×#11761611C2318yes—
66C22×C67⋊C8C2312yes—
67#14524034—28no—
68#14524032—216no—
69#14524023—24no—
70#14524023C2318yes—
71#14524019—216no—
72#14524018—28no—
73#14524018—24no—
74#14524018C2318yes—
75C4×Dic134—22no—
76C4×Dic134C2314yes—
77C4×D268—22no—
78C4×D268D414yes—
79C4×D268C2314yes—
80C4×D268C2314yes—
81#14524005—28no—
82#14524001—24no—
83#14524001—28no—
84#14524001C2318yes—
85C2×C4⋊Dic67—22no—
86C2×C4⋊Dic67D411yescharacteristic
87C2×C4⋊Dic67C2311yescharacteristic
88C2×C4⋊Dic67C2311yescharacteristic
89C2×Dic67⋊C4—22no—
90C2×Dic67⋊C4—21no—
91C4⋊C4×D67—24no—
92C4⋊C4×D67C2318yes—
93C2×(C2×D67)⋊C4—22no—
94C2×(C2×D67)⋊C4—22no—
95C2×(C2×D67)⋊C4—22no—
96C2×(C2×D67)⋊C4—21no—
97C2×(C2×D67)⋊C4D412yes—
98C2×C4×Dic67—22no—
99C2×C4×Dic67—21no—
100#14523976C2318yes—
101#14523975C2314yes—
102#14523974C2318yes—
103C4⋊(C67⋊C8)C2314yes—
104#14523968C2314yes—
105#14523966C2318yes—
106#14523965C2314yes—
107#14523963C2318yes—
108(C2×D67)⋊C8C2318yes—
109C22×Q8×C67—21no—
110C22×Q8×C67—21no—
111C22×C8×C67C2311yescharacteristic
112C22×C8×C67C2311yescharacteristic
113C4⋊Q8×C67—22no—
114C4⋊Q8×C67—22no—
115C4⋊Q8×C67C2314yes—
116C4⋊Q8×C67C2314yes—
117C2×C4⋊C4×C67—21no—
118C2×C4⋊C4×C67—21no—
119C2×C4⋊C4×C67C2312yes—
120C2×C4⋊C4×C67C2312yes—
121C2×C42×C67C2311yescharacteristic
122#107×C67C2318yes—
123#107×C67C2318yes—
124C4×C8×C67C2314yes—
125C23×D67—21no—
126C23×D67C2×D412yes—
127C2×Dic134—22no—
128C2×Dic134—22no—
129C2×Dic134—21no—
130C2×Dic134C2412yes—
131C2×D268—24no—
132C2×D268—22no—
133C2×D268—22no—
134C2×D268—21no—
135C2×D268—28no—
136C2×D268—84no—
137C2×D268C2×D414yes—
138C2×D268C2412yes—
139C22×Dic67—21no—
140C22×Dic67—21no—
141C22×Dic67—21no—
142C22×Dic67C2×D412yes—
143C2×C4×D67—24no—
144C2×C4×D67—24no—
145C2×C4×D67—24no—
146C2×C4×D67—28no—
147C2×C4×D67—44no—
148C2×C4×D67—84no—
149C2×C4×D67C2×D414yes—
150C2×C4×D67C2414yes—
151#11761617—28no—
152#11761611—216no—
153C2×C67⋊C8—24no—
154C2×C67⋊C8C2414yes—
155C4⋊Dic67—21no—
156C4⋊Dic67—21no—
157C4⋊Dic67—28no—
158C4⋊Dic67C2×D414yes—
159C4⋊Dic67C2412yes—
160C4⋊Dic67C2412yes—
161Dic67⋊C4—24no—
162Dic67⋊C4—28no—
163(C2×D67)⋊C4—28no—
164(C2×D67)⋊C4—28no—
165(C2×D67)⋊C4—24no—
166(C2×D67)⋊C4—28no—
167(C2×D67)⋊C4C2×D418yes—
168C4×Dic67—24no—
169C4×Dic67—28no—
170C2×Q8×C67—24no—
171C2×Q8×C67—24no—
172C2×Q8×C67—82no—
173C2×Q8×C67—82no—
174C22×C4×C67—22no—
175C22×C4×C67—22no—
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2411yescharacteristic
179C2×C8×C67—22no—
180C2×C8×C67#4114yes—
181C2×C8×C67C2412yes—
182C2×C8×C67C2412yes—
183C4⋊C4×C67—24no—
184C4⋊C4×C67—24no—
185C4⋊C4×C67—22no—
186C4⋊C4×C67—22no—
187C4⋊C4×C67C2414yes—
188C4⋊C4×C67C2414yes—
189C42×C67C2×D412yes—
190C42×C67C2412yes—
191C23×C67C22×D411yescharacteristic
192C22×D67—24no—
193C22×D67—24no—
194C22×D67—24no—
195C22×D67—22no—
196C22×D67—42no—
197C22×D67—42no—
198C22×D67—42no—
199C22×D67C22×D414yes—
200Q8×C67—84no—
201Q8×C67—84no—
202C2×C4×C67—22no—
203C2×C4×C67—24no—
204C2×C4×C67—24no—
205C2×C4×C67—44no—
206C2×C4×C67—44no—
207C2×C4×C67#13714yes—
208C2×C4×C67C22×D412yes—
209C2×C4×C67C22×D411yescharacteristic
210C2×C4×C67C22×D412yes—
211C2×C4×C67C2×#4112yes—
212C2×C4×C672−1+412yes—
213C2×C4×C67C2511yescharacteristic
214C2×Dic67—24no—
215C2×Dic67—22no—
216C2×Dic67—22no—
217C2×Dic67—22no—
218C2×Dic67—42no—
219C2×Dic67—42no—
220C2×Dic67—42no—
221C2×Dic67C22×D414yes—
222D268—28no—
223D268—22no—
224D268—22no—
225D268—42no—
226D268—88no—
227C4×D67—28no—
228C4×D67—24no—
229C4×D67—24no—
230C4×D67—48no—
231C4×D67—88no—
232Dic134—22no—
233Dic134—22no—
234Dic134—42no—
235C8×C67—22no—
236C8×C67C2×#4114yes—
237C67⋊C8—24no—
238C22×C67#54712yes—
239C22×C67C2×#13712yes—
240C22×C67#57612yes—
241C22×C67C23×D411yescharacteristic
242C2×D67—24no—
243C2×D67—24no—
244C2×D67—24no—
245C2×D67—48no—
246C2×D67—42no—
247C2×D67—42no—
248C2×D67—42no—
249C2×D67—84no—
250C4×C67—22no—
251C4×C67—44no—
252C4×C67—44no—
253C4×C67#54712yes—
254C4×C67C2×#13714yes—
255C4×C67C22×#4111yescharacteristic
256C4×C67C2×2−1+411yescharacteristic
257Dic67—24no—
258Dic67—42no—
259Dic67—42no—
260Dic67—42no—
261C2×#3357—671nomaximal, Sylow
262C2×C67#335714yescomplemented
263C2×C67C2×#54711yescharacteristic
264C2×C67C22×#13711yescharacteristic
265C2×C67C2×#57611yescharacteristic
266D67—44no—
267D67—82no—
268C2×#555—671no—
269C2×#554—671no—
270#3357—6716no—
271C2×#500D6711yescharacteristic, complemented
272C2×#485—672no—
273C2×#480—672no—
274C2×#477—671no—
275C2×#476—671no—
276C2×#468—672no—
277C2×C4×SD16—671no—
278C2×#436—671no—
279C2×#428—672no—
280C67C2×#335711yescharacteristic, complemented, Sylow
281C22×SD16—671no—
282C22×M16—672no—
283#555—678no—
284#555—1344no—
285#554—678no—
286#554—1344no—
287C2×C4⋊Q8C2×D6711yescharacteristic
288C2×C4⋊Q8C2×D6711yescharacteristic
289C2×#125—1341no—
290C2×#124—1342no—
291C2×C22⋊Q8—672no—
292C2×C22⋊Q8—1342no—
293C2×#121—672no—
294C2×#121—1341no—
295C2×C4×Q8—671no—
296C2×C4×D4—671no—
297C2×C4×D4—671no—
298C2×C4×D4—1341no—
299C2×#117—672no—
300C2×#117—1341no—
301C22×C4⋊C4—672no—
302#500C2×D6718yescomplemented
303#485—6716no—
304#480—6716no—
305#477—678no—
306#476—678no—
307#468—6716no—
308C4×SD16—678no—
309#436—678no—
310#428—6716no—
311C2×#107—672no—
312C2×#107C2×D6712yes—
313C2×#107C2×D6712yes—
314C2×#106—671no—
315C2×#106—672no—
316C2×C4⋊C8—671no—
317C2×#103—671no—
318C2×#103—672no—
319C2×#102—671no—
320C2×#102—672no—
321C2×C22⋊C8—672no—
322C2×C4×C8C2×D6711yescharacteristic
323C22×Q8—21no—
324C22×Q8—21no—
325C22×Q8—671no—
326C22×D4—671no—
327C22×D4—671no—
328C22×D4—1342no—
329C23×C4—672no—
330C23×C4—1342no—
331C23×C4—1341no—
332C2×SD16—674no—
333C2×SD16—1344no—
334C2×M16—678no—
335C2×M16—1348no—
336C22×C8—672no—
337C22×C8C22×D6711yescharacteristic
338C22×C8C22×D6711yescharacteristic
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8C22×D6714yes—
342C4⋊Q8C22×D6714yes—
343#125—1348no—
344#124—13416no—
345C22⋊Q8—678no—
346C22⋊Q8—13416no—
347C22⋊Q8—1344no—
348#121—678no—
349#121—1344no—
350#121—1348no—
351C4×Q8—674no—
352C4×Q8—1342no—
353C4×D4—674no—
354C4×D4—674no—
355C4×D4—674no—
356C4×D4—1348no—
357C4×D4—1342no—
358#117—678no—
359#117—1348no—
360#117—1344no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—671no—
364C2×C4⋊C4—671no—
365C2×C4⋊C4—671no—
366C2×C4⋊C4—678no—
367C2×C4⋊C4—1344no—
368C2×C4⋊C4—1342no—
369C2×C4⋊C4—1342no—
370C2×C4⋊C4—1341no—
371C2×C4⋊C4C22×D6712yes—
372C2×C4⋊C4C22×D6712yes—
373C2×C22⋊C4—672no—
374C2×C22⋊C4—1342no—
375C2×C22⋊C4—1342no—
376C2×C22⋊C4—1342no—
377C2×C22⋊C4—1341no—
378C2×C42—1342no—
379C2×C42—1341no—
380C2×C42C22×D6711yescharacteristic
381#107—678no—
382#107C22×D6718yes—
383#107C22×D6718yes—
384#106—674no—
385#106—678no—
386C4⋊C8—674no—
387#103—674no—
388#103—678no—
389#102—674no—
390#102—678no—
391C22⋊C8—678no—
392C4×C8C22×D6714yes—
393C24—672no—
394C24—1341no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—672no—
400C2×Q8—1342no—
401C2×Q8—1342no—
402C2×Q8—1341no—
403C2×D4—5364no—
404C2×D4—674no—
405C2×D4—672no—
406C2×D4—1348no—
407C2×D4—1344no—
408C2×D4—1341no—
409C2×D4—1342no—
410C2×D4—1342no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—2684no—
414C22×C4—5364no—
415C22×C4—672no—
416C22×C4—674no—
417C22×C4—674no—
418C22×C4—1348no—
419C22×C4—1341no—
420C22×C4—1341no—
421C22×C4—1341no—
422C22×C4—1344no—
423C22×C4—1344no—
424C22×C4—1344no—
425C22×C4D4×D6711yescharacteristic
426C22×C4D4×D6711yescharacteristic
427C22×C4C23×D6711yescharacteristic
428SD16—1348no—
429M16—13416no—
430C2×C8—22no—
431C2×C8—674no—
432C2×C8—1344no—
433C2×C8#1176162914yes—
434C2×C8C23×D6712yes—
435C2×C8C23×D6712yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—674no—
441C4⋊C4—672no—
442C4⋊C4—672no—
443C4⋊C4—1348no—
444C4⋊C4—1348no—
445C4⋊C4—1341no—
446C4⋊C4—1341no—
447C4⋊C4—1344no—
448C4⋊C4C23×D6714yes—
449C4⋊C4C23×D6714yes—
450C22⋊C4—678no—
451C22⋊C4—1348no—
452C22⋊C4—1348no—
453C22⋊C4—1348no—
454C22⋊C4—1344no—
455C42—1348no—
456C42—1344no—
457C42D4×D6712yes—
458C42C23×D6712yes—
459C23—2682no—
460C23—2682no—
461C23—2682no—
462C23—674no—
463C23—1342no—
464C23—1344no—
465C23—1344no—
466C23—1344no—
467C23C2×D4×D6711yescharacteristic
468Q8—84no—
469Q8—84no—
470Q8—2682no—
471Q8—1342no—
472Q8—1342no—
473D4—2682no—
474D4—5368no—
475D4—1348no—
476D4—1342no—
477D4—1342no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—2682no—
484C2×C4—2682no—
485C2×C4—2682no—
486C2×C4—2688no—
487C2×C4—5368no—
488C2×C4—674no—
489C2×C4—1342no—
490C2×C4—1342no—
491C2×C4—1342no—
492C2×C4—1344no—
493C2×C4—1348no—
494C2×C4—1344no—
495C2×C4—1344no—
496C2×C4#1452407514yes—
497C2×C4C2×D4×D6712yes—
498C2×C4C2×D4×D6711yescharacteristic
499C2×C4C2×D4×D6712yes—
500C2×C4C2×#1176162912yes—
501C2×C4#1452409112yes—
502C2×C4C24×D6711yescharacteristic
503C8—22no—
504C8—1344no—
505C8C2×#1176162914yes—
506C22—2688no—
507C22—2682no—
508C22—2682no—
509C22—2682no—
510C22—5364no—
511C22—1344no—
512C22—1344no—
513C22—1344no—
514C22#1686404912yes—
515C22C2×#1452407512yes—
516C22#1686419712yes—
517C22C22×D4×D6711yescharacteristic
518C4—22no—
519C4—44no—
520C4—44no—
521C4—2682no—
522C4—2682no—
523C4—2682no—
524C4—1344no—
525C4#1686404912yes—
526C4C2×#1452407514yes—
527C4C22×#1176162911yescharacteristic
528C4C2×#1452409111yescharacteristic
529C2—2684no—
530C2—5362no—
531C2#2257283414yesdirect product factor
532C2C2×#1686404911yescharacteristic
533C2C22×#1452407511yescharacteristic
534C2C2×#1686419711yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#500×C67C2
socleC23×C67C22×D4
prime radicalC2×#16864112C2
hypercenterC2×#500D67
hypocenterC67C2×#3357
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
2. C2×#16864107C2semidirect
3. #22572834C2direct (N × H)
4. C2×#16863739C2semidirect
5. C2×#16863711C2semidirect
6. C2×#16863688C2semidirect
7. C2×#16863683C2semidirect
8. C2×#16863643C2semidirect
9. C2×C4×#11761617C2semidirect
10. C2×#16863475C2semidirect
11. C2×#16863437C2semidirect
12. C2×#500×C67C2semidirect
18. #16864107C22semidirect
32. #16863739C22semidirect
33. #16863711C22semidirect
34. #16863688C22semidirect
35. #16863683C22semidirect
36. #16863643C22semidirect
37. C4×#11761617C22semidirect
38. #16863475C22semidirect
39. #16863437C22semidirect
51. #500×C67C22semidirect
262. C2×C67#3357semidirect
271. C2×#500D67semidirect
280. C67C2×#3357semidirect
302. #500C2×D67semidirect
531. C2#22572834direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168641122
C2×#168641072
#225728342
C2×#168637392
C2×#168637112
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C2×#168636432
C2×C4×#117616172
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C2×#500×C672
C2×#335767