Finite Group Databasebeta

← C2×#22570730 #31725444

Subgroups — C2×#22570730 #31725444

Order 17152.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863952C211yesmaximal, characteristic
2C2×#16863951C211yesmaximal, characteristic
3#22570730C2116yesmaximal, direct product factor
4C2×#16863806C211yesmaximal, characteristic, complemented
5C2×#16863805C211yesmaximal, characteristic, complemented
6C2×#16863804C212yesmaximal, complemented
7C2×#16863803C212yesmaximal, complemented
8C2×#16863802C211yesmaximal, characteristic, complemented
9C2×#16863433C212yesmaximal
10C2×#16863431C212yesmaximal
11C2×#97×D67C211yesmaximal, characteristic
12C2×#501×C67C211yesmaximal, characteristic, complemented
13C22×C8×D67C2212yes—
14#16863952—24no—
15#16863952C2218yes—
16#16863951—24no—
17#16863951C2218yes—
18C2×#14524043C2211yescharacteristic
19C2×#14524042C2211yescharacteristic
20C2×#14524042C2211yescharacteristic
21C2×#14524035C2211yescharacteristic
22C2×#14524034C2212yes—
23C2×#14524023—21no—
24C2×#14524021—21no—
25C2×#14524021—21no—
26C2×#14524019—21no—
27C2×Q8×Dic67—21no—
28C2×#14524008—21no—
29C2×#14524008—21no—
30C2×#14524008—21no—
31C2×#14524001C2211yescharacteristic
32C2×#14524001C2212yes—
33C2×#14524001C2211yescharacteristic
34C2×C42×D67C2211yescharacteristic
35#16863806C2218yescomplemented
36#16863805C2218yescomplemented
37#16863804C22116yescomplemented
38#16863803C22116yescomplemented
39#16863802C2218yescomplemented
40#16863433C22116yes—
41#16863431C22116yes—
42C2×#14523977C2212yes—
43C2×#14523976C2212yes—
44C2×#14523976C2212yes—
45C2×#14523975C2212yes—
46C2×#14523974C2212yes—
47C2×#14523974C2212yes—
48C2×#14523954C2211yescharacteristic
49C2×#14523953C2212yes—
50#97×D67C2218yes—
51C2×C4⋊Q8×C67C2211yescharacteristic
52C2×#125×C67C2211yescharacteristic
53#501×C67C2218yescomplemented
54C2×#107×C67C2212yes—
55C2×#106×C67C2212yes—
56C2×#97×C67C2211yescharacteristic
57C22×Dic134—21no—
58C22×Dic134—21no—
59C22×Dic134—21no—
60C22×C4×D67D411yescharacteristic
61C22×C4×D67D411yescharacteristic
62C22×C4×D67C2311yescharacteristic
63C22×C67⋊C8C2312yes—
64C2×C8×D67Q8116yescomplemented
65C2×C8×D67C2318yes—
66#14524043—22no—
67#14524043C2314yes—
68#14524042—22no—
69#14524042—22no—
70#14524042C2314yes—
71#14524042C2314yes—
72#14524035—22no—
73#14524035C2314yes—
74#14524034—24no—
75#14524034C2318yes—
76#14524023—28no—
77#14524021—28no—
78#14524021—28no—
79#14524019—28no—
80Q8×Dic67—28no—
81#14524008—28no—
82#14524008—28no—
83#14524008—28no—
84#14524001—24no—
85#14524001—22no—
86#14524001—22no—
87#14524001C2314yes—
88#14524001C2318yes—
89#14524001C2314yes—
90C2×C4⋊Dic67—22no—
91C2×C4⋊Dic67—21no—
92C2×C4⋊Dic67C2312yes—
93C2×C4⋊Dic67C2311yescharacteristic
94C2×C4⋊Dic67C2311yescharacteristic
95C2×Dic67⋊C4—22no—
96C2×Dic67⋊C4—21no—
97C2×Dic67⋊C4—21no—
98C2×Dic67⋊C4—21no—
99C2×Dic67⋊C4—21no—
100C2×(C2×D67)⋊C4—22no—
101C2×(C2×D67)⋊C4—21no—
102C2×(C2×D67)⋊C4—21no—
103C2×C4×Dic67—21no—
104C2×C4×Dic67—21no—
105C2×C4×Dic67—22no—
106C2×C4×Dic67D411yescharacteristic
107C2×C4×Dic67D411yescharacteristic
108C2×C4×Dic67C2311yescharacteristic
109C42×D67D414yes—
110C42×D67C2314yes—
111#14523977C2318yes—
112#14523976C2318yes—
113#14523976C2318yes—
114#14523975C2318yes—
115#14523974C2318yes—
116#14523974C2318yes—
117#14523954C2314yes—
118#14523953C2318yes—
119C22×Q8×C67—21no—
120C22×C8×C67C2312yes—
121C4⋊Q8×C67—22no—
122C4⋊Q8×C67C2314yes—
123#125×C67—22no—
124#125×C67C2314yes—
125C2×C4⋊C4×C67—21no—
126C2×C4⋊C4×C67—21no—
127C2×C4⋊C4×C67—21no—
128C2×C4⋊C4×C67C2311yescharacteristic
129C2×C4⋊C4×C67C2311yescharacteristic
130C2×C4⋊C4×C67C2312yes—
131C2×C42×C67C2311yescharacteristic
132#107×C67C2318yes—
133#106×C67C2318yes—
134#97×C67C2314yes—
135C23×D67C2×D411yescharacteristic
136C2×Dic134—24no—
137C2×Dic134—24no—
138C2×Dic134—24no—
139C2×Dic134—44no—
140C2×Dic134—44no—
141C2×Dic134—44no—
142C22×Dic67—21no—
143C22×Dic67—21no—
144C22×Dic67—22no—
145C22×Dic67C2×D411yescharacteristic
146C22×Dic67C2×D411yescharacteristic
147C22×Dic67C2×D411yescharacteristic
148C2×C4×D67—42no—
149C2×C4×D67—42no—
150C2×C4×D67C2×D414yes—
151C2×C4×D67C2×D414yes—
152C2×C4×D67C2×D412yes—
153C2×C4×D67C2×Q818yes—
154C2×C4×D67C2412yes—
155C2×C67⋊C8C2×Q818yes—
156C2×C67⋊C8C2414yes—
157C8×D67C2×Q8132yescomplemented
158C4⋊Dic67—24no—
159C4⋊Dic67—22no—
160C4⋊Dic67—21no—
161C4⋊Dic67—21no—
162C4⋊Dic67—28no—
163C4⋊Dic67C2412yes—
164C4⋊Dic67C2414yes—
165C4⋊Dic67C2412yes—
166Dic67⋊C4—24no—
167Dic67⋊C4—24no—
168Dic67⋊C4—28no—
169Dic67⋊C4—24no—
170Dic67⋊C4—24no—
171(C2×D67)⋊C4—28no—
172(C2×D67)⋊C4—24no—
173(C2×D67)⋊C4—24no—
174C4×Dic67—24no—
175C4×Dic67—24no—
176C4×Dic67—28no—
177C4×Dic67C2×D412yes—
178C4×Dic67C2×D414yes—
179C4×Dic67C2×D414yes—
180C4×Dic67C2412yes—
181C2×Q8×C67—24no—
182C2×Q8×C67—44no—
183C22×C4×C67—21no—
184C22×C4×C67—21no—
185C22×C4×C67—22no—
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67C2×D411yescharacteristic
188C22×C4×C67C2411yescharacteristic
189C2×C8×C67C2×Q818yes—
190C2×C8×C67C2414yes—
191C4⋊C4×C67—24no—
192C4⋊C4×C67—24no—
193C4⋊C4×C67—24no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—22no—
197C4⋊C4×C67C2412yes—
198C4⋊C4×C67C2414yes—
199C4⋊C4×C67C2412yes—
200C42×C67C2×D412yes—
201C42×C67C2412yes—
202C23×C67C22×D411yescharacteristic
203C22×D67—22no—
204C22×D67—22no—
205C22×D67C4⋊Q814yes—
206C22×D67C22×D412yes—
207Q8×C67—48no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—22no—
211C2×C4×C67—22no—
212C2×C4×C67—24no—
213C2×C4×C67—44no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67C22×D412yes—
217C2×C4×C67C22×D412yes—
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C22×Q814yes—
220C2×C4×C67C2511yescharacteristic
221C2×Dic67—22no—
222C2×Dic67—22no—
223C2×Dic67—22no—
224C2×Dic67—22no—
225C2×Dic67—24no—
226C2×Dic67—42no—
227C2×Dic67—42no—
228C2×Dic67—44no—
229C2×Dic67C4⋊Q814yes—
230C2×Dic67C22×D412yes—
231C2×Dic67C22×D412yes—
232C2×Dic67C22×D412yes—
233C4×D67—44no—
234C4×D67—44no—
235C4×D67C4⋊Q818yes—
236C4×D67C22×Q818yes—
237Dic134—48no—
238Dic134—48no—
239Dic134—48no—
240C8×C67C22×Q818yes—
241C67⋊C8C22×Q818yes—
242C22×C67C2×C4⋊Q812yes—
243C22×C67#57512yes—
244C22×C67#57512yes—
245C22×C67C23×D411yescharacteristic
246C2×D67—28no—
247C2×D67—22no—
248C2×D67—22no—
249C2×D67C2×C4⋊Q814yes—
250C4×C67—22no—
251C4×C67—22no—
252C4×C67—42no—
253C4×C67—42no—
254C4×C67—44no—
255C4×C67C2×C4⋊Q812yes—
256C4×C67C23×Q812yes—
257Dic67—22no—
258Dic67—22no—
259Dic67—42no—
260Dic67—42no—
261Dic67—44no—
262Dic67C2×C4⋊Q814yes—
263C2×#3162—671nomaximal, Sylow
264C2×C67#316214yescomplemented
265C2×C67C22×C4⋊Q811yescharacteristic
266C2×C67C2×#57511yescharacteristic
267C2×C67C2×#57511yescharacteristic
268D67—24no—
269C2×#533—671no—
270C2×#533—671no—
271#3162—6716no—
272C2×#501—672no—
273C2×#501—672no—
274C2×#501—671no—
275C2×#501—671no—
276C2×#501—671no—
277C2×#501D6711yescharacteristic, complemented
278C2×#427—672no—
279C2×#427—672no—
280C22×#97—671no—
281C67C2×#316211yescharacteristic, complemented, Sylow
282C23×C8—672no—
283#533—678no—
284#533—678no—
285#533—1344no—
286#533—1344no—
287C2×C4⋊Q8—671no—
288C2×C4⋊Q8—671no—
289C2×C4⋊Q8—671no—
290C2×C4⋊Q8C2×D6711yescharacteristic
291C2×#125—672no—
292C2×#125—671no—
293C2×#125C2×D6711yescharacteristic
294C2×C22⋊Q8—1341no—
295C2×C22⋊Q8—1341no—
296C2×C22⋊Q8—1341no—
297C2×C22⋊Q8—1341no—
298C2×C4×Q8—1341no—
299C2×C4×Q8—1341no—
300C2×C4×Q8—1341no—
301C2×C4×Q8—1341no—
302C2×#117—672no—
303C2×#117—671no—
304C2×#117—671no—
305C22×C42—671no—
306#501—678no—
307#501—678no—
308#501—6716no—
309#501—6716no—
310#501—678no—
311#501C2×D6718yescomplemented
312#427—6716no—
313#427—6716no—
314C2×#107—672no—
315C2×#107—672no—
316C2×#107—672no—
317C2×#107C2×D6712yes—
318C2×#106—672no—
319C2×#106—672no—
320C2×#106—672no—
321C2×#106C2×D6712yes—
322C2×#97—672no—
323C2×#97—671no—
324C2×#97—678no—
325C2×#97C2×D6711yescharacteristic
326C22×Q8—21no—
327C22×Q8—1341no—
328C22×Q8—1341no—
329C22×Q8—1341no—
330C23×C4—671no—
331C23×C4—671no—
332C23×C4—671no—
333C22×C8—672no—
334C22×C8—6716no—
335C22×C8—678no—
336C22×C8C22×D6712yes—
337C4⋊Q8—22no—
338C4⋊Q8—674no—
339C4⋊Q8—674no—
340C4⋊Q8—674no—
341C4⋊Q8—1342no—
342C4⋊Q8—1342no—
343C4⋊Q8—1342no—
344C4⋊Q8C22×D6714yes—
345#125—22no—
346#125—678no—
347#125—674no—
348#125—1344no—
349#125—1342no—
350#125C22×D6714yes—
351C22⋊Q8—1348no—
352C22⋊Q8—1348no—
353C22⋊Q8—1348no—
354C22⋊Q8—1348no—
355C4×Q8—1348no—
356C4×Q8—1348no—
357C4×Q8—1348no—
358C4×Q8—1348no—
359#117—674no—
360#117—678no—
361#117—674no—
362#117—1342no—
363#117—1342no—
364#117—1344no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—672no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—1342no—
372C2×C4⋊C4—1342no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4—1341no—
375C2×C4⋊C4—1341no—
376C2×C4⋊C4—1341no—
377C2×C4⋊C4—1341no—
378C2×C4⋊C4C22×D6712yes—
379C2×C4⋊C4C22×D6711yescharacteristic
380C2×C4⋊C4C22×D6711yescharacteristic
381C2×C22⋊C4—1341no—
382C2×C22⋊C4—1341no—
383C2×C22⋊C4—1342no—
384C2×C42—671no—
385C2×C42—671no—
386C2×C42—671no—
387C2×C42—674no—
388C2×C42—674no—
389C2×C42—1341no—
390C2×C42—1341no—
391C2×C42—1342no—
392C2×C42C22×D6711yescharacteristic
393#107—678no—
394#107—678no—
395#107—678no—
396#107C22×D6718yes—
397#106—678no—
398#106—678no—
399#106—678no—
400#106C22×D6718yes—
401#97—678no—
402#97—674no—
403#97C22×D6714yes—
404C24—671no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—2684no—
408C2×Q8—2684no—
409C2×Q8—2684no—
410C2×Q8—1344no—
411C2×Q8—1344no—
412C2×Q8—1344no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—2682no—
417C22×C4—2682no—
418C22×C4—674no—
419C22×C4—674no—
420C22×C4—671no—
421C22×C4—672no—
422C22×C4—678no—
423C22×C4—672no—
424C22×C4—671no—
425C22×C4—671no—
426C22×C4—1341no—
427C22×C4—1342no—
428C22×C4—1341no—
429C22×C4D4×D6711yescharacteristic
430C22×C4D4×D6711yescharacteristic
431C22×C4C23×D6711yescharacteristic
432C2×C8—678no—
433C2×C8—674no—
434C2×C8—6732no—
435C2×C8Q8×D6718yes—
436C2×C8C23×D6714yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—672no—
444C4⋊C4—674no—
445C4⋊C4—672no—
446C4⋊C4—1348no—
447C4⋊C4—1348no—
448C4⋊C4—1344no—
449C4⋊C4—1344no—
450C4⋊C4—1342no—
451C4⋊C4—1341no—
452C4⋊C4—1341no—
453C4⋊C4—1344no—
454C4⋊C4—1344no—
455C4⋊C4—1344no—
456C4⋊C4C23×D6712yes—
457C4⋊C4C23×D6714yes—
458C4⋊C4C23×D6712yes—
459C22⋊C4—1344no—
460C22⋊C4—1344no—
461C22⋊C4—1348no—
462C42—674no—
463C42—674no—
464C42—672no—
465C42—672no—
466C42—1344no—
467C42—1344no—
468C42—1348no—
469C42D4×D6712yes—
470C42C23×D6712yes—
471C23—674no—
472C23—672no—
473C23—1342no—
474C23—1342no—
475C23C2×D4×D6711yescharacteristic
476Q8—48no—
477Q8—2688no—
478Q8—2688no—
479Q8—2688no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—2684no—
489C2×C4—2684no—
490C2×C4—2682no—
491C2×C4—2684no—
492C2×C4—2682no—
493C2×C4—674no—
494C2×C4—672no—
495C2×C4—678no—
496C2×C4—678no—
497C2×C4—672no—
498C2×C4—672no—
499C2×C4—1342no—
500C2×C4—1344no—
501C2×C4—1342no—
502C2×C4—1342no—
503C2×C4—1342no—
504C2×C4C2×D4×D6712yes—
505C2×C4C2×D4×D6712yes—
506C2×C4C2×D4×D6711yescharacteristic
507C2×C4C2×Q8×D6714yes—
508C2×C4C24×D6711yescharacteristic
509C8—678no—
510C8C2×Q8×D6718yes—
511C22—674no—
512C22—1342no—
513C22—1342no—
514C22—1348no—
515C22C4⋊Q8×D6712yes—
516C22#1686419212yes—
517C22#1686419512yes—
518C22C22×D4×D6711yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—2682no—
525C4—2684no—
526C4—2682no—
527C4—674no—
528C4—1342no—
529C4—1342no—
530C4C4⋊Q8×D6712yes—
531C4C22×Q8×D6712yes—
532C2—1344no—
533C2#2257073014yesdirect product factor
534C2C2×C4⋊Q8×D6711yescharacteristic
535C2C2×#1686419211yescharacteristic
536C2C2×#1686419511yescharacteristic
subgroupG/H
centerC23C2×D4×D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#501×C67C2
socleC23×C67C22×D4
prime radicalC23×D67C2×D4
hypercenterC2×#501D67
hypocenterC67C2×#3162
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #22570730C2direct (N × H)
4. C2×#16863806C2semidirect
5. C2×#16863805C2semidirect
6. C2×#16863804C2semidirect
7. C2×#16863803C2semidirect
8. C2×#16863802C2semidirect
12. C2×#501×C67C2semidirect
35. #16863806C22semidirect
36. #16863805C22semidirect
37. #16863804C22semidirect
38. #16863803C22semidirect
39. #16863802C22semidirect
53. #501×C67C22semidirect
64. C2×C8×D67Q8semidirect
157. C8×D67C2×Q8semidirect
264. C2×C67#3162semidirect
277. C2×#501D67semidirect
281. C67C2×#3162semidirect
311. #501C2×D67semidirect
533. C2#22570730direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168639522
C2×#168639512
#225707302
C2×#168638062
C2×#168638052
C2×#168638042
C2×#168638032
C2×#168638022
C2×#168634332
C2×#168634312
C2×#97×D672
C2×#501×C672
C2×#316267