Finite Group Databasebeta

← C2×#13131787 #16808902

Subgroups — C2×#13131787 #16808902

Order 3328.

1982 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131787C2116yesmaximal, direct product factor
2C2×#11719158C211yesmaximal, characteristic, complemented
3C2×Q8×#1057C211yesmaximal, characteristic, complemented
4C2×#11719076C211yesmaximal, characteristic, complemented
5C2×#11719064C212yesmaximal, complemented
6C2×#11719073C211yesmaximal, characteristic, complemented
7C2×#11719063C212yesmaximal, complemented
8C2×#11719077C212yesmaximal, complemented
9C2×#11719125C211yesmaximal, characteristic, complemented
10C2×#11719079C212yesmaximal, complemented
11C22×C4⋊#1057C211yesmaximal, characteristic, complemented
12C2×#11719329C211yesmaximal, characteristic
13C23×#1057C2212yes—
14C22×#6118C2212yes—
15C22×D13⋊C8C2212yes—
16C2×C13×C22⋊Q8C411yescharacteristic
17C2×#88201C411yescharacteristic
18C2×Q8×Dic13C2211yescharacteristic
19C2×#88184C411yescharacteristic
20C22×C4×Dic13C2211yescharacteristic
21#11719329C418yes—
22#11719329C2218yes—
23C2×#88152C412yes—
24C2×#88147C2211yescharacteristic
25C2×#88146C2212yes—
26C2×#88144C411yescharacteristic
27C2×#88140C412yes—
28C2×#88135C2212yes—
29C2×C22⋊#1057C2211yescharacteristic
30C2×C22⋊#1057C2211yescharacteristic
31#11719158C2218yescomplemented
32Q8×#1057C2218yescomplemented
33#11719125C2218yescomplemented
34C2×#88112C2212yes—
35C2×#88112C2211yescharacteristic
36C2×#88112C2211yescharacteristic
37C2×#88111C2212yes—
38C2×#88111C2212yes—
39C2×#88110C2212yes—
40C2×C4⋊#1057C2211yescharacteristic
41C2×C4⋊#1057C2212yes—
42C2×C4⋊#1057C2211yescharacteristic
43C2×C4⋊#1057C2212yes—
44C2×C4⋊#1057C2218yescomplemented
45C2×C4×#1057C2212yes—
46C2×C4×#1057C2211yescharacteristic
47C2×C4×#1057C2211yescharacteristic
48#11719079C22116yescomplemented
49#11719077C22116yescomplemented
50#11719076C2218yescomplemented
51#11719073C2218yescomplemented
52#11719064C22116yescomplemented
53#11719063C22116yescomplemented
54C22×Q8×C13C2×C411yescharacteristic
55C23×C4×C13C2×C411yescharacteristic
56C22×C22⋊D13C2×C412yes—
57C23×Dic13C2×C411yescharacteristic
58C23×Dic13C2311yescharacteristic
59C22×D52C2×C411yescharacteristic
60C22×C4×D13C2312yes—
61C22×#1057—44no—
62C22×#1057C2311yescharacteristic
63C22×#1057C2311yescharacteristic
64C22×#1057C2311yescharacteristic
65C22×#1057C2311yescharacteristic
66C22×#1057C2312yes—
67C22×#1057C2312yes—
68C22×#1057C2318yes—
69C2×#6118—44no—
70C2×#6118C2318yes—
71C2×D13⋊C8—44no—
72C2×D13⋊C8C2318yes—
73C13×C22⋊Q8C2×C418yes—
74C2×C13×C4⋊C4C2×C411yescharacteristic
75C2×C13×C4⋊C4C2×C412yes—
76C2×C13×C22⋊C4C2×C412yes—
77#88201C2×C418yes—
78Q8×Dic13C2×C414yes—
79Q8×Dic13C2314yes—
80#88184C2×C418yes—
81C2×#6103C2×C412yes—
82C2×#6103C2×C412yes—
83C2×#6103C2×C412yes—
84C2×#6103C2×C412yes—
85C2×C4⋊Dic13C2×C412yes—
86C2×C4⋊Dic13C2×C412yes—
87C2×C4⋊Dic13C2×C411yescharacteristic
88C2×#6101C2×C412yes—
89C2×C4×Dic13C2×C411yescharacteristic
90C2×C4×Dic13C2×C411yescharacteristic
91C2×C4×Dic13C2×C414yes—
92C2×C4×Dic13C2312yes—
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2314yes—
96#88152C2×C4116yes—
97#88147C2×C414yes—
98#88147C2314yes—
99#88146C2×C418yes—
100#88146C2318yes—
101#88144C2×C418yes—
102#88140C2×C4116yes—
103#88135C2×C418yes—
104#88135C2318yes—
105C22⋊#1057C2314yes—
106C22⋊#1057C2314yes—
107#88112C2318yes—
108#88112C2314yes—
109#88112C2314yes—
110#88111C2318yes—
111#88111C2318yes—
112#88110C2318yes—
113C4⋊#1057C2318yes—
114C4⋊#1057C2314yes—
115C4⋊#1057C2314yes—
116C4⋊#1057C2318yes—
117C4×#1057C2318yes—
118C4×#1057C2314yes—
119C4×#1057C2314yes—
120C2×#2994—131nomaximal, Sylow
121C24×C13C22×C411yescharacteristic
122C23×D13C22×C412yes—
123C2×Q8×C13—24no—
124C2×Q8×C13C22×C414yes—
125C22×C4×C13—24no—
126C22×C4×C13C22×C411yescharacteristic
127C22×C4×C13C22×C414yes—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C22×C411yescharacteristic
131C2×C22⋊D13—28no—
132C2×C22⋊D13C22×C418yes—
133C22×Dic13—24no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13C22×C412yes—
137C22×Dic13C22×C414yes—
138C22×Dic13C22×C411yescharacteristic
139C22×Dic13C22×C411yescharacteristic
140C22×Dic13C22×C411yescharacteristic
141C22×Dic13C22×C412yes—
142C22×Dic13C2412yes—
143C22×Dic13C2411yescharacteristic
144C2×D52—24no—
145C2×D52C22×C414yes—
146C2×C4×D13—24no—
147C2×C4×D13—24no—
148C2×C4×D13C22×C414yes—
149C2×C4×D13C2414yes—
150C2×#1057—22no—
151C2×#1057—24no—
152C2×#1057—22no—
153C2×#1057—24no—
154C2×#1057—22no—
155C2×#1057—22no—
156C2×#1057—48no—
157C2×#1057C2412yes—
158C2×#1057C2414yes—
159C2×#1057C2412yes—
160C2×#1057C2412yes—
161C2×#1057C2412yes—
162C2×#1057C2414yes—
163#6118—48no—
164D13⋊C8—48no—
165C13×C4⋊C4C22×C414yes—
166C13×C4⋊C4C22×C418yes—
167C13×C22⋊C4C22×C418yes—
168#6103C22×C418yes—
169#6103C22×C418yes—
170#6103C22×C418yes—
171#6103C22×C418yes—
172C4⋊Dic13C22×C418yes—
173C4⋊Dic13C22×C418yes—
174C4⋊Dic13C22×C414yes—
175#6101C22×C418yes—
176C4×Dic13C22×C414yes—
177C4×Dic13C22×C414yes—
178C4×Dic13C22×C412yes—
179C4×Dic13C22×C412yes—
180C4×Dic13C22×C414yes—
181C4×Dic13C2412yes—
182C4×Dic13C2414yes—
183C4×Dic13C2412yes—
184C2×#529—131no—
185#2994—1316no—
186C2×#446—131no—
187C2×C8×Q8—131no—
188C2×#436—131no—
189C2×#435—132no—
190C2×C8×D4—132no—
191C2×C8×D4—131no—
192C2×#433—132no—
193C2×#431—131no—
194C2×#424—132no—
195C22×C4⋊C8—131no—
196C23×C13—22no—
197C23×C13—22no—
198C23×C13#13114yes—
199C23×C13C23×C412yes—
200C23×C13C23×C411yescharacteristic
201C22×D13—24no—
202C22×D13—24no—
203C22×D13—24no—
204C22×D13C23×C414yes—
205Q8×C13—28no—
206C2×C4×C13—24no—
207C2×C4×C13—22no—
208C2×C4×C13—24no—
209C2×C4×C13—28no—
210C2×C4×C13#13114yes—
211C2×C4×C13C23×C412yes—
212C2×C4×C13C23×C414yes—
213C2×C4×C13C23×C414yes—
214C2×C4×C13C23×C412yes—
215C22⋊D13—216no—
216C2×Dic13—22no—
217C2×Dic13—28no—
218C2×Dic13—22no—
219C2×Dic13—22no—
220C2×Dic13—24no—
221C2×Dic13—24no—
222C2×Dic13—22no—
223C2×Dic13—22no—
224C2×Dic13C23×C412yes—
225C2×Dic13C23×C412yes—
226C2×Dic13C23×C412yes—
227C2×Dic13C23×C414yes—
228C2×Dic13C23×C411yescharacteristic
229C2×Dic13H(5,𝔽2)12yes—
230C2×Dic132−1+412yes—
231C2×Dic13C2511yescharacteristic
232D52—28no—
233C4×D13—28no—
234C4×D13—24no—
235C4×D13—24no—
236#1057—22no—
237#1057—24no—
238#1057—22no—
239#1057—24no—
240#1057—22no—
241#1057—22no—
242C22×M16—132no—
243C23×C8—132no—
244C23×C8—132no—
245#529—138no—
246#529—138no—
247C2×#126—132no—
248C2×#125—131no—
249C2×#124—131no—
250C2×#123—132no—
251C2×C22⋊Q8#26111yescharacteristic
252C2×#121—131no—
253C2×C4×Q8—131no—
254C2×C4×D4—131no—
255C2×C4×D4—132no—
256C2×#117—132no—
257C22×C42—131no—
258#446—138no—
259C8×Q8—138no—
260#436—138no—
261#435—1316no—
262C8×D4—1316no—
263C8×D4—138no—
264#433—1316no—
265#431—138no—
266#424—1316no—
267C2×C4⋊C8—132no—
268C2×C4⋊C8—131no—
269C2×C4⋊C8—131no—
270C2×C4⋊C8—138no—
271C2×C4⋊C8—131no—
272C2×C4⋊C8—132no—
273C2×C4⋊C8—132no—
274C2×C4⋊C8—131no—
275C2×C22⋊C8—131no—
276C2×C22⋊C8—131no—
277C2×C22⋊C8—132no—
278C2×C22⋊C8—132no—
279C2×#97—132no—
280C2×C4×C8—131no—
281C2×C4×C8—131no—
282C2×C4×C8—132no—
283C22×C13—22no—
284C22×C13—28no—
285C22×C13—22no—
286C22×C13#52012yes—
287C22×C13C2×#13112yes—
288C22×C13C2×#13114yes—
289C22×C13C2×#13112yes—
290C22×C13C24×C411yescharacteristic
291C2×D13—24no—
292C2×D13—24no—
293C2×D13—24no—
294C2×D13—48no—
295C4×C13—24no—
296C4×C13—22no—
297C4×C13—24no—
298C4×C13C2×#13114yes—
299Dic13—22no—
300Dic13—22no—
301Dic13—22no—
302Dic13—24no—
303Dic13#52012yes—
304Dic13C2×H(5,𝔽2)11yescharacteristic
305Dic13C2×2−1+411yescharacteristic
306C22×Q8C2×#26111yescharacteristic
307C22×D4—132no—
308C22×D4—131no—
309C23×C4—131no—
310C23×C4—131no—
311C23×C4—132no—
312C23×C4C2×#26111yescharacteristic
313C2×M16—138no—
314C2×M16—524no—
315C22×C8—132no—
316C22×C8—131no—
317C22×C8—132no—
318C22×C8—131no—
319C22×C8—131no—
320C22×C8—131no—
321C22×C8—138no—
322C22×C8—138no—
323C22×C8—524no—
324C22×C8—524no—
325#126—1316no—
326#125—138no—
327#124—138no—
328#123—1316no—
329C22⋊Q8C2×#26118yes—
330#121—138no—
331C4×Q8—134no—
332C4×Q8—134no—
333C4×D4—134no—
334C4×D4—134no—
335C4×D4—138no—
336C4×D4—138no—
337#117—138no—
338#117—138no—
339C2×C4⋊C4—132no—
340C2×C4⋊C4—132no—
341C2×C4⋊C4—131no—
342C2×C4⋊C4—132no—
343C2×C4⋊C4C2×#26112yes—
344C2×C4⋊C4C2×#26111yescharacteristic
345C2×C22⋊C4—132no—
346C2×C22⋊C4—132no—
347C2×C22⋊C4—132no—
348C2×C22⋊C4—132no—
349C2×C22⋊C4C2×#26112yes—
350C2×C42—131no—
351C2×C42—132no—
352C2×C42—131no—
353C2×C42—131no—
354C2×C42—131no—
355C2×C42—134no—
356C2×C42—134no—
357C4⋊C8—138no—
358C4⋊C8—134no—
359C4⋊C8—134no—
360C4⋊C8—138no—
361C4⋊C8—134no—
362C4⋊C8—138no—
363C4⋊C8—134no—
364C22⋊C8—138no—
365C22⋊C8—138no—
366C22⋊C8—134no—
367C22⋊C8—134no—
368#97—138no—
369C4×C8—134no—
370C4×C8—134no—
371C4×C8—138no—
372C2×C13—24no—
373C2×C13#299414yescomplemented
374C2×C13C2×#52011yescharacteristic
375C2×C13C22×#13111yescharacteristic
376C2×C13C22×#13111yescharacteristic
377D13—44no—
378C24—132no—
379C24C22×#26111yescharacteristic
380C2×Q8—24no—
381C2×Q8C22×#26114yes—
382C2×D4—138no—
383C2×D4—134no—
384C2×D4—268no—
385C2×D4—264no—
386C22×C4—24no—
387C22×C4—131no—
388C22×C4—132no—
389C22×C4—132no—
390C22×C4—134no—
391C22×C4—131no—
392C22×C4—131no—
393C22×C4—131no—
394C22×C4—132no—
395C22×C4—134no—
396C22×C4—134no—
397C22×C4—264no—
398C22×C4—264no—
399C22×C4—262no—
400C22×C4—262no—
401C22×C4—264no—
402C22×C4C22×#26114yes—
403C22×C4C22×#26111yescharacteristic
404C22×C4C22×#26112yes—
405C22×C4C22×#26112yes—
406C22×C4C22×#26111yescharacteristic
407M16—528no—
408C2×C8—132no—
409C2×C8—132no—
410C2×C8—132no—
411C2×C8—134no—
412C2×C8—132no—
413C2×C8—134no—
414C2×C8—262no—
415C2×C8—264no—
416C2×C8—264no—
417C2×C8—262no—
418C2×C8—262no—
419C2×C8—262no—
420C2×C8—528no—
421C2×C8—528no—
422C4⋊C4—138no—
423C4⋊C4—138no—
424C4⋊C4—134no—
425C4⋊C4—138no—
426C4⋊C4C22×#26118yes—
427C4⋊C4C22×#26114yes—
428C22⋊C4—138no—
429C22⋊C4—138no—
430C22⋊C4—138no—
431C22⋊C4—138no—
432C22⋊C4C22×#26118yes—
433C42—132no—
434C42—134no—
435C42—132no—
436C42—132no—
437C42—132no—
438C42—134no—
439C42—134no—
440C42—134no—
441C13C2×#299411yescharacteristic, complemented, Sylow
442C23—22no—
443C23—22no—
444C23—134no—
445C23—264no—
446C23—264no—
447C23—264no—
448C23#8824014yes—
449C23C23×#26112yes—
450C23C23×#26111yescharacteristic
451Q8—28no—
452D4—2616no—
453D4—268no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—132no—
459C2×C4—132no—
460C2×C4—131no—
461C2×C4—131no—
462C2×C4—132no—
463C2×C4—132no—
464C2×C4—132no—
465C2×C4—134no—
466C2×C4—264no—
467C2×C4—268no—
468C2×C4—264no—
469C2×C4—262no—
470C2×C4—264no—
471C2×C4—262no—
472C2×C4—268no—
473C2×C4—262no—
474C2×C4—262no—
475C2×C4—264no—
476C2×C4—262no—
477C2×C4#8823814yes—
478C2×C4C23×#26112yes—
479C2×C4C23×#26114yes—
480C2×C4C23×#26114yes—
481C2×C4C23×#26112yes—
482C8—262no—
483C8—264no—
484C8—264no—
485C8—262no—
486C8—262no—
487C8—262no—
488C22—28no—
489C22—22no—
490C22—22no—
491C22—264no—
492C22—264no—
493C22—264no—
494C22—528no—
495C22C2×#8823812yes—
496C22C2×#8824014yes—
497C22C2×#8824012yes—
498C22#1171963112yes—
499C22C24×#26111yescharacteristic
500C4—24no—
501C4—22no—
502C4—24no—
503C4—132no—
504C4—131no—
505C4—131no—
506C4—262no—
507C4—262no—
508C4—262no—
509C4—264no—
510C4C2×#8823814yes—
511C2—24no—
512C2—524no—
513C2#1313178714yesdirect product factor
514C2C2×#1171963111yescharacteristic
515C2C22×#8823811yescharacteristic
516C2C22×#8824011yescharacteristic
subgroupG/H
centerC23C23×#261
commutatorC22×C13C24×C4
FrattiniC22C24×#261
FittingC2×C13×C22⋊Q8C4
socleC23×C13C23×C4
prime radicalC2×#88184C4
cosocleC2×Dic13C25
hypercenterC2×C22⋊Q8#261
hypocenterC13C2×#2994
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #13131787C2direct (N × H)
2. C2×#11719158C2semidirect
3. C2×Q8×#1057C2semidirect
4. C2×#11719076C2semidirect
5. C2×#11719064C2semidirect
6. C2×#11719073C2semidirect
7. C2×#11719063C2semidirect
8. C2×#11719077C2semidirect
9. C2×#11719125C2semidirect
10. C2×#11719079C2semidirect
11. C22×C4⋊#1057C2semidirect
31. #11719158C22semidirect
32. Q8×#1057C22semidirect
33. #11719125C22semidirect
44. C2×C4⋊#1057C22semidirect
48. #11719079C22semidirect
49. #11719077C22semidirect
50. #11719076C22semidirect
51. #11719073C22semidirect
52. #11719064C22semidirect
53. #11719063C22semidirect
373. C2×C13#2994semidirect
441. C13C2×#2994semidirect
513. C2#13131787direct (N × H)

Maximal subgroups

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

subgroupindex
#131317872
C2×#117191582
C2×Q8×#10572
C2×#117190762
C2×#117190642
C2×#117190732
C2×#117190632
C2×#117190772
C2×#117191252
C2×#117190792
C22×C4⋊#10572
C2×#117193292
C2×#299413