Finite Group Databasebeta

← C2×#22570065 #31723417

Subgroups — C2×#22570065 #31723417

Order 17152.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864027C211yesmaximal, characteristic, complemented
2C2×#16863877C211yesmaximal, characteristic
3#22570065C2116yesmaximal, direct product factor
4C2×#16863702C212yesmaximal, complemented
5C2×#16863690C212yesmaximal, complemented
6C2×#16863699C212yesmaximal, complemented
7C2×#16863675C212yesmaximal, complemented
8C2×#16863436C211yesmaximal, characteristic, complemented
9C2×#16863422C211yesmaximal, characteristic, complemented
10C2×#16863395C211yesmaximal, characteristic
11C2×#16863393C211yesmaximal, characteristic
12C22×#107×C67C211yesmaximal, characteristic, complemented
13C22×#11761628C2211yescharacteristic
14C22×#11761612C2211yescharacteristic
15#16864027—24no—
16#16864027C2218yescomplemented
17C2×#14524042—21no—
18C2×#14524042C2212yes—
19C2×#14524034—21no—
20C2×#14524034C2212yes—
21C2×#14524022—22no—
22#16863877—24no—
23#16863877C2218yes—
24C2×#14524008C2212yes—
25C2×#14524005C2212yes—
26C2×C4×C22⋊D67—22no—
27C2×#14524002C2211yescharacteristic
28C2×#14524002C2211yescharacteristic
29C2×#14524001—21no—
30C2×C4⋊C4×D67—21no—
31#16863702C22116yescomplemented
32#16863699C22116yescomplemented
33#16863690C22116yescomplemented
34#16863675C22116yescomplemented
35#16863436C2218yescomplemented
36C2×#14523976C2212yes—
37C2×#14523974C2212yes—
38#16863422C2218yescomplemented
39C2×Dic67⋊C8C2212yes—
40C2×Dic67⋊C8C2212yes—
41#16863395C2218yes—
42#16863393C2218yes—
43C2×#14523968C2211yescharacteristic
44C2×#14523968C2211yescharacteristic
45C2×#14523966C2212yes—
46C2×#14523965C2211yescharacteristic
47C2×#14523965C2211yescharacteristic
48C2×#14523963C2212yes—
49C23×C8×C67C2211yescharacteristic
50C22×C4⋊C4×C67C2211yescharacteristic
51C22×C4⋊C4×C67C2211yescharacteristic
52C2×#107×C67C2212yes—
53C2×#107×C67C2212yes—
54C2×#107×C67C2218yescomplemented
55C2×#11761628—22no—
56C2×#11761628—24no—
57C2×#11761628Q818yescomplemented
58C2×#11761628C2314yes—
59C22×Dic134—22no—
60C22×Dic134C2311yescharacteristic
61C22×D268C2311yescharacteristic
62C22×C22⋊D67—21no—
63C22×C4×D67—22no—
64C22×C4×D67—21no—
65C2×#11761612—24no—
66C2×#11761612C2314yes—
67C22×C67⋊C8C2312yes—
68#14524042—28no—
69#14524042—24no—
70#14524042C2318yes—
71#14524034—28no—
72#14524034—24no—
73#14524034C2318yes—
74#14524022—216no—
75#14524008—24no—
76#14524008C2318yes—
77#14524005—24no—
78#14524005C2318yes—
79C4×C22⋊D67—216no—
80#14524002—22no—
81#14524002D414yes—
82#14524002C2314yes—
83#14524002C2314yes—
84#14524001—28no—
85C2×C4⋊Dic67—22no—
86C2×C4⋊Dic67D411yescharacteristic
87C2×C4⋊Dic67D411yescharacteristic
88C2×C4⋊Dic67C2311yescharacteristic
89C2×C4⋊Dic67C2311yescharacteristic
90C2×Dic67⋊C4—22no—
91C2×Dic67⋊C4—22no—
92C2×Dic67⋊C4—22no—
93C4⋊C4×D67—28no—
94C2×C22⋊Dic67—21no—
95C2×C22⋊Dic67D412yes—
96C2×(C2×D67)⋊C4—22no—
97C2×C4×Dic67—22no—
98C2×C4×Dic67—21no—
99C2×C4×Dic67C2312yes—
100#14523976C2318yes—
101#14523974C2318yes—
102Dic67⋊C8C2318yes—
103Dic67⋊C8C2318yes—
104#14523968C2314yes—
105#14523968C2314yes—
106#14523966C2318yes—
107#14523965C2314yes—
108#14523965C2314yes—
109#14523963C2318yes—
110C23×C4×C67—21no—
111C23×C4×C67—21no—
112C23×C4×C67C2311yescharacteristic
113C22×C8×C67—24no—
114C22×C8×C67C2311yescharacteristic
115C22×C8×C67C2311yescharacteristic
116C22×C8×C67C2314yes—
117C2×C4⋊C4×C67—22no—
118C2×C4⋊C4×C67—21no—
119C2×C4⋊C4×C67—22no—
120C2×C4⋊C4×C67—21no—
121C2×C4⋊C4×C67C2314yes—
122C2×C4⋊C4×C67C2312yes—
123C2×C4⋊C4×C67C2314yes—
124C2×C4⋊C4×C67C2312yes—
125#107×C67C2318yes—
126#107×C67C2318yes—
127C23×D67—21no—
128#11761628—216no—
129#11761628—28no—
130#11761628C2×Q8116yescomplemented
131C2×Dic134—21no—
132C2×Dic134—22no—
133C2×Dic134—28no—
134C2×Dic134—84no—
135C2×Dic134C2×Q814yes—
136C2×Dic134C2412yes—
137C2×D268—21no—
138C2×D268—22no—
139C2×D268C2×Q814yes—
140C2×D268C2412yes—
141C2×C22⋊D67—28no—
142C2×C22⋊D67—24no—
143C22×Dic67—21no—
144C22×Dic67—21no—
145C22×Dic67—21no—
146C22×Dic67C2×D412yes—
147C22×Dic67C2×D412yes—
148C2×C4×D67—28no—
149C2×C4×D67—24no—
150C2×C4×D67—28no—
151C2×C4×D67—84no—
152#11761612—28no—
153C2×C67⋊C8—24no—
154C2×C67⋊C8C2414yes—
155C4⋊Dic67—21no—
156C4⋊Dic67—21no—
157C4⋊Dic67—28no—
158C4⋊Dic67C2×D414yes—
159C4⋊Dic67C2×D414yes—
160C4⋊Dic67C2412yes—
161C4⋊Dic67C2412yes—
162Dic67⋊C4—28no—
163Dic67⋊C4—28no—
164Dic67⋊C4—28no—
165C22⋊Dic67—24no—
166C22⋊Dic67C2×D418yes—
167(C2×D67)⋊C4—28no—
168C4×Dic67—24no—
169C4×Dic67—28no—
170C4×Dic67C2×D414yes—
171C4×Dic67C2414yes—
172C24×C67C2×D411yescharacteristic
173C22×C4×C67—22no—
174C22×C4×C67—22no—
175C22×C4×C67—24no—
176C22×C4×C67—24no—
177C22×C4×C67—82no—
178C22×C4×C67—82no—
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2×D412yes—
181C22×C4×C67C2×Q814yes—
182C22×C4×C67#4114yes—
183C22×C4×C67C2411yescharacteristic
184C22×C4×C67C2412yes—
185C2×C8×C67—22no—
186C2×C8×C67—22no—
187C2×C8×C67—28no—
188C2×C8×C67C2412yes—
189C2×C8×C67C2412yes—
190C4⋊C4×C67—24no—
191C4⋊C4×C67—22no—
192C4⋊C4×C67—24no—
193C4⋊C4×C67—22no—
194C4⋊C4×C67C2414yes—
195C4⋊C4×C67C2414yes—
196C23×C67—22no—
197C23×C67—22no—
198C23×C67C22⋊Q814yes—
199C23×C67C22×D412yes—
200C23×C67C22×D411yescharacteristic
201C22×D67—24no—
202C22×D67—22no—
203C22×D67—42no—
204C22×D67—42no—
205C2×C4×C67—24no—
206C2×C4×C67—24no—
207C2×C4×C67—44no—
208C2×C4×C67—44no—
209C2×C4×C67—84no—
210C2×C4×C67—84no—
211C2×C4×C67C22⋊Q814yes—
212C2×C4×C67C22⋊Q818yes—
213C2×C4×C67C22×D412yes—
214C2×C4×C67C22×D411yescharacteristic
215C2×C4×C67C22×Q812yes—
216C2×C4×C67C22×Q814yes—
217C2×C4×C67C2×#4112yes—
218C2×C4×C67C2×#4114yes—
219C2×C4×C67C2511yescharacteristic
220C22⋊D67—216no—
221C2×Dic67—24no—
222C2×Dic67—24no—
223C2×Dic67—24no—
224C2×Dic67—22no—
225C2×Dic67—22no—
226C2×Dic67—22no—
227C2×Dic67—42no—
228C2×Dic67—42no—
229C2×Dic67C22×D414yes—
230C2×Dic67C22×D414yes—
231D268—24no—
232D268—22no—
233D268C22×Q814yes—
234C4×D67—216no—
235C4×D67—88no—
236Dic134—24no—
237Dic134—22no—
238Dic134—88no—
239Dic134C22×Q814yes—
240C8×C67—22no—
241C8×C67—22no—
242C67⋊C8—24no—
243C22×C67—28no—
244C22×C67—22no—
245C22×C67—22no—
246C22×C67C2×C22⋊Q814yes—
247C22×C67C2×C22⋊Q812yes—
248C22×C67#57612yes—
249C22×C67#57812yes—
250C22×C67C23×D411yescharacteristic
251C2×D67—24no—
252C2×D67—48no—
253C2×D67—42no—
254C2×D67—42no—
255C4×C67—44no—
256C4×C67—44no—
257C4×C67C2×C22⋊Q812yes—
258C4×C67C2×C22⋊Q814yes—
259C4×C67C23×Q811yescharacteristic
260C4×C67C22×#4111yescharacteristic
261Dic67—24no—
262Dic67—24no—
263Dic67—24no—
264Dic67—42no—
265Dic67—42no—
266C2×#3083—671nomaximal, Sylow
267C2×C67—24no—
268C2×C67#308314yescomplemented
269C2×C67C22×C22⋊Q811yescharacteristic
270C2×C67C2×#57611yescharacteristic
271C2×C67C2×#57811yescharacteristic
272D67—44no—
273C2×#544—671no—
274C2×#520—671no—
275#3083—6716no—
276C2×#479—672no—
277C2×#477—672no—
278C2×#478—672no—
279C2×#474—672no—
280C2×#428—671no—
281C22×#107D6711yescharacteristic, complemented
282C2×#424—671no—
283C2×#417—671no—
284C2×#416—671no—
285C67C2×#308311yescharacteristic, complemented, Sylow
286C22×#41—671no—
287C22×M16—671no—
288C23×C8C2×D6711yescharacteristic
289#544—678no—
290#544—1344no—
291C2×C4⋊Q8—672no—
292C2×C4⋊Q8—1341no—
293C2×#125—672no—
294C2×#125—1341no—
295C2×C22⋊Q8—1342no—
296#520—678no—
297#520—1344no—
298C2×C4×Q8—672no—
299C2×C4×D4—672no—
300C2×C4×D4—1342no—
301C2×#117—671no—
302C2×#117—671no—
303C2×#117—1341no—
304C22×C4⋊C4—1341no—
305C22×C4⋊C4C2×D6711yescharacteristic
306C22×C4⋊C4C2×D6711yescharacteristic
307#479—6716no—
308#478—6716no—
309#477—6716no—
310#474—6716no—
311#428—678no—
312C2×#107—672no—
313C2×#107C2×D6712yes—
314C2×#107C2×D6712yes—
315C2×#107C2×D6718yescomplemented
316C2×#106—672no—
317#424—678no—
318C2×C4⋊C8—672no—
319C2×C4⋊C8—672no—
320#417—678no—
321#416—678no—
322C2×#103—671no—
323C2×#103—671no—
324C2×#103—672no—
325C2×#102—671no—
326C2×#102—671no—
327C2×#102—672no—
328C2×#41—678no—
329C2×#41—674no—
330C2×#41—1342no—
331C2×#41—1344no—
332C22×Q8—671no—
333C22×Q8—1342no—
334C22×D4—671no—
335C22×D4—1341no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—1342no—
339C23×C4—1341no—
340C23×C4C22×D6711yescharacteristic
341C2×M16—674no—
342C2×M16—1344no—
343C22×C8—24no—
344C22×C8—672no—
345C22×C8C22×D6711yescharacteristic
346C22×C8C22×D6711yescharacteristic
347C22×C8C22×D6714yes—
348C4⋊Q8—678no—
349C4⋊Q8—1344no—
350C4⋊Q8—1348no—
351#125—678no—
352#125—1344no—
353#125—1348no—
354C22⋊Q8—13416no—
355C4×Q8—678no—
356C4×Q8—1344no—
357C4×D4—678no—
358C4×D4—1344no—
359C4×D4—13416no—
360#117—674no—
361#117—674no—
362#117—674no—
363#117—1348no—
364#117—1342no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—671no—
373C2×C4⋊C4—1348no—
374C2×C4⋊C4—1342no—
375C2×C4⋊C4—1342no—
376C2×C4⋊C4—1342no—
377C2×C4⋊C4—1342no—
378C2×C4⋊C4C22×D6712yes—
379C2×C4⋊C4C22×D6714yes—
380C2×C4⋊C4C22×D6712yes—
381C2×C4⋊C4C22×D6714yes—
382C2×C22⋊C4—672no—
383C2×C22⋊C4—1342no—
384C2×C22⋊C4—1341no—
385C2×C42—672no—
386C2×C42—1342no—
387C2×C42—1341no—
388#107—678no—
389#107C22×D6718yes—
390#107C22×D6718yes—
391#106—678no—
392C4⋊C8—678no—
393C4⋊C8—678no—
394#103—674no—
395#103—674no—
396#103—678no—
397#102—674no—
398#102—674no—
399#102—678no—
400C24—1341no—
401C24D4×D6711yescharacteristic
402#41—6716no—
403#41—13416no—
404#41—1348no—
405C2×Q8—5364no—
406C2×Q8—674no—
407C2×Q8—672no—
408C2×Q8—1348no—
409C2×Q8—1341no—
410C2×Q8—1342no—
411C2×D4—674no—
412C2×D4—672no—
413C2×D4—1348no—
414C2×D4—1344no—
415C2×D4—1341no—
416C2×D4—1342no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—5364no—
424C22×C4—672no—
425C22×C4—672no—
426C22×C4—1348no—
427C22×C4—1341no—
428C22×C4—1341no—
429C22×C4—1341no—
430C22×C4—1348no—
431C22×C4—1344no—
432C22×C4D4×D6712yes—
433C22×C4D4×D6711yescharacteristic
434C22×C4Q8×D6714yes—
435C22×C4#1176162814yes—
436C22×C4C23×D6712yes—
437C22×C4C23×D6711yescharacteristic
438M16—1348no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—28no—
442C2×C8—674no—
443C2×C8—1344no—
444C2×C8C23×D6712yes—
445C2×C8C23×D6712yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—674no—
451C4⋊C4—674no—
452C4⋊C4—672no—
453C4⋊C4—672no—
454C4⋊C4—1348no—
455C4⋊C4—1348no—
456C4⋊C4—1348no—
457C4⋊C4—1348no—
458C4⋊C4—1341no—
459C4⋊C4—1341no—
460C4⋊C4C23×D6714yes—
461C4⋊C4C23×D6714yes—
462C22⋊C4—678no—
463C22⋊C4—1348no—
464C22⋊C4—1344no—
465C42—674no—
466C42—674no—
467C42—1348no—
468C42—1344no—
469C23—22no—
470C23—22no—
471C23—2682no—
472C23—2682no—
473C23—1344no—
474C23—1342no—
475C23#1452402114yes—
476C23C2×D4×D6712yes—
477C23C2×D4×D6711yescharacteristic
478Q8—5368no—
479Q8—674no—
480Q8—1344no—
481Q8—1342no—
482D4—674no—
483D4—13416no—
484D4—1344no—
485D4—1342no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—2682no—
493C2×C4—2682no—
494C2×C4—5368no—
495C2×C4—674no—
496C2×C4—674no—
497C2×C4—1342no—
498C2×C4—1342no—
499C2×C4—1344no—
500C2×C4—1344no—
501C2×C4—1344no—
502C2×C4—1342no—
503C2×C4—13416no—
504C2×C4#1452402114yes—
505C2×C4#1452402118yes—
506C2×C4C2×D4×D6711yescharacteristic
507C2×C4C2×D4×D6712yes—
508C2×C4C2×Q8×D6712yes—
509C2×C4C2×Q8×D6714yes—
510C2×C4C2×#1176162812yes—
511C2×C4C2×#1176162814yes—
512C2×C4C24×D6711yescharacteristic
513C8—22no—
514C8—22no—
515C8—1344no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—2688no—
520C22—2682no—
521C22—2682no—
522C22—1344no—
523C22C2×#1452402114yes—
524C22C2×#1452402112yes—
525C22#1686419712yes—
526C22#1686420912yes—
527C22C22×D4×D6711yescharacteristic
528C4—44no—
529C4—44no—
530C4—2682no—
531C4—2682no—
532C4—1344no—
533C4—1344no—
534C4—1344no—
535C4C2×#1452402114yes—
536C4C2×#1452402112yes—
537C4C22×Q8×D6711yescharacteristic
538C4C22×#1176162811yescharacteristic
539C2—24no—
540C2—2684no—
541C2#2257006514yesdirect product factor
542C2C22×#1452402111yescharacteristic
543C2C2×#1686419711yescharacteristic
544C2C2×#1686420911yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC22×#107×C67C2
socleC23×C67C22×D4
prime radicalC22×#11761628C22
hypercenterC22×#107D67
hypocenterC67C2×#3083
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
1. C2×#16864027C2semidirect
3. #22570065C2direct (N × H)
4. C2×#16863702C2semidirect
5. C2×#16863690C2semidirect
6. C2×#16863699C2semidirect
7. C2×#16863675C2semidirect
8. C2×#16863436C2semidirect
9. C2×#16863422C2semidirect
12. C22×#107×C67C2semidirect
16. #16864027C22semidirect
31. #16863702C22semidirect
32. #16863699C22semidirect
33. #16863690C22semidirect
34. #16863675C22semidirect
35. #16863436C22semidirect
38. #16863422C22semidirect
54. C2×#107×C67C22semidirect
57. C2×#11761628Q8semidirect
130. #11761628C2×Q8semidirect
268. C2×C67#3083semidirect
281. C22×#107D67semidirect
285. C67C2×#3083semidirect
315. C2×#107C2×D67semidirect
541. C2#22570065direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168640272
C2×#168638772
#225700652
C2×#168637022
C2×#168636902
C2×#168636992
C2×#168636752
C2×#168634362
C2×#168634222
C2×#168633952
C2×#168633932
C22×#107×C672
C2×#308367