Finite Group Databasebeta

← C2×#20274906 #27384801

Subgroups — C2×#20274906 #27384801

Order 13568.

1966 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
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C22×#13133148C211yesmaximal, characteristic, complemented
3C22×#13133129C211yesmaximal, characteristic
4C2×#16810480C211yesmaximal, characteristic
5#20274906C218yesmaximal, direct product factor
6C2×#16810386C211yesmaximal, characteristic, complemented
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#106×C53C211yesmaximal, characteristic, complemented
9C23×C4×D53—21no—
10C23×C4×D53C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919—21no—
15C2×C4⋊C4×D53—22no—
16C2×C4⋊C4×D53—24no—
17C2×C4⋊C4×D53—24no—
18C2×C4⋊C4×D53C2214yes—
19C22×#11719921—21no—
20C2×#13133148—24no—
21C2×#13133148C2214yescomplemented
22C2×#13133129C418yescomplemented
23C2×#13133129C2214yes—
24#16810480—22no—
25#16810480C2214yes—
26C2×#13133124—21no—
27#16810386C2214yescomplemented
28#16810385C2214yescomplemented
29C23×C8×C53C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C2×#106×C53—24no—
33C2×#106×C53C2214yescomplemented
34C24×D53D411yescharacteristic
35C23×Dic53—21no—
36C23×Dic53—21no—
37C23×Dic53D411yescharacteristic
38C22×C4×D53—22no—
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—28no—
43C22×C4×D53—24no—
44C22×C4×D53—44no—
45C22×C4×D53—44no—
46C22×C4×D53C2×C414yes—
47C22×C4×D53D412yes—
48C22×C4×D53Q814yes—
49C22×C4×D53C2312yes—
50C22×#88322—22no—
51C22×#88322—22no—
52C22×#88322C2×C414yescomplemented
53C22×#88322C2312yes—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53D414yes—
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—28no—
63C2×#11719919—24no—
64C4⋊C4×D53—216no—
65C4⋊C4×D53—28no—
66C4⋊C4×D53—28no—
67C2×#11719921—24no—
68C2×#11719921—44no—
69#13133148—28no—
70#13133129C2×C4116yescomplemented
71#13133124—24no—
72C23×C4×C53—21no—
73C23×C4×C53—21no—
74C23×C4×C53C2311yescharacteristic
75C22×C8×C53—22no—
76C22×C8×C53—22no—
77C22×C8×C53C2×C414yescomplemented
78C22×C8×C53C2312yes—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53D414yes—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2312yes—
87#106×C53—28no—
88C24×C53C2×D411yescharacteristic
89C23×D53—22no—
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53C4⋊C414yes—
94C23×D53SD1614yes—
95C23×D53SD1614yes—
96C23×D53C2×D412yes—
97C22×C4×C53—24no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—42no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C22×C412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2×Q812yes—
110C22×C4×C53#4112yes—
111C22×C4×C53#4112yes—
112C22×C4×C53C2411yescharacteristic
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—24no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C22×Dic53—42no—
122C22×Dic53—42no—
123C22×Dic53C4⋊C414yes—
124C22×Dic53C2×D412yes—
125C2×C4×D53—28no—
126C2×C4×D53—28no—
127C2×C4×D53—24no—
128C2×C4×D53—28no—
129C2×C4×D53—28no—
130C2×C4×D53—24no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—216no—
136C2×C4×D53—48no—
137C2×C4×D53—432no—
138C2×C4×D53—48no—
139C2×C4×D53C4⋊C418yes—
140C2×C4×D53C22×C414yes—
141C2×C4×D53C2×Q814yes—
142C2×C8×C53—28no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C2×C8×C53C22×C414yescomplemented
146C4⋊C4×C53—24no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53C2×D414yes—
152#11719921—48no—
153C4⋊Dic53—24no—
154C4⋊Dic53—22no—
155C4⋊Dic53—22no—
156C4⋊Dic53—22no—
157C4⋊Dic53—42no—
158C4⋊Dic53C2×D414yes—
159#11719919—216no—
160C2×#88322—28no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322C22×C414yescomplemented
164C23×C53C2×C4⋊C412yes—
165C23×C53#12012yes—
166C23×C53#12312yes—
167C23×C53C2×SD1612yes—
168C23×C53C2×SD1612yes—
169C23×C53AGL(1,ℤ/8)12yes—
170C23×C53#13712yes—
171C23×C53C22×D411yescharacteristic
172C22×D53—22no—
173C22×D53—22no—
174C22×D53—22no—
175C22×D53—22no—
176C22×D53—28no—
177C22×D53—28no—
178C22×D53—28no—
179C22×D53—28no—
180C22×D53—28no—
181C22×D53—28no—
182C22×D53#106116yescomplemented
183C22×D53C2×C4⋊C414yes—
184C22×D53C2×SD1614yes—
185C22×D53C2×SD1614yes—
186C2×C4×C53—24no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—48no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—84no—
194C2×C4×C53C2×C4⋊C412yes—
195C2×C4×C53C4×D414yes—
196C2×C4×C53C4×D414yes—
197C2×C4×C53#12012yes—
198C2×C4×C53C22⋊Q814yes—
199C2×C4×C53C22⋊Q814yes—
200C2×C4×C53#12312yes—
201C2×C4×C53C23×C411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C22×Q811yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C2×Dic53—22no—
208C2×Dic53—22no—
209C2×Dic53—28no—
210C2×Dic53—24no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—48no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—84no—
218C2×Dic53C2×C4⋊C414yes—
219C4×D53—24no—
220C4×D53—24no—
221C4×D53—24no—
222C4×D53—24no—
223C4×D53—216no—
224C4×D53—432no—
225C8×C53—24no—
226#88322—24no—
227C2×#1853—531nomaximal, Sylow
228C22×C53#38514yes—
229C22×C53C2×#10614yes—
230C22×C53#42814yes—
231C22×C53#44814yes—
232C22×C53#45014yes—
233C22×C53#48114yes—
234C22×C53#48314yes—
235C22×C53C22×C4⋊C411yescharacteristic
236C22×C53C2×#12011yescharacteristic
237C22×C53C2×#12311yescharacteristic
238C22×C53C22×SD1611yescharacteristic
239C22×C53C22×SD1611yescharacteristic
240C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C53C2×#13711yescharacteristic
242C2×D53—24no—
243C2×D53—24no—
244C2×D53—232no—
245C2×D53—24no—
246C2×D53—24no—
247C2×D53—24no—
248C2×D53—24no—
249C2×D53C2×#10618yescomplemented
250C4×C53—44no—
251C4×C53—82no—
252C4×C53#38514yes—
253C4×C53C2×C4×D411yescharacteristic
254C4×C53C2×C4×D411yescharacteristic
255C4×C53C2×C22⋊Q811yescharacteristic
256C4×C53C2×C22⋊Q811yescharacteristic
257Dic53—24no—
258Dic53—44no—
259Dic53—82no—
260C23×C4⋊C4—531no—
261C22×#106—531no—
262C22×#106D5311yescharacteristic, complemented
263C22×C22⋊C8—531no—
264C2×#380—531no—
265#1853—538no—
266C2×#340—531no—
267C2×#340—531no—
268C2×C53#185318yescomplemented
269C2×C53C2×#38511yescharacteristic
270C2×C53C22×#10611yescharacteristic
271C2×C53C2×#42811yescharacteristic
272C2×C53C2×#45011yescharacteristic
273C2×C53C2×#44811yescharacteristic
274C2×C53C2×#48311yescharacteristic
275C2×C53C2×#48111yescharacteristic
276D53—28no—
277C24×C4—531no—
278C24×C4—1061no—
279C23×C8—531no—
280C23×C8C2×D5311yescharacteristic
281C22×C4⋊C4—531no—
282C22×C4⋊C4—531no—
283C22×C4⋊C4—534no—
284C22×C4⋊C4—1064no—
285C22×C4⋊C4—1062no—
286C22×C4⋊C4—1061no—
287C22×C4⋊C4—1064no—
288C22×C4⋊C4C2×D5311yescharacteristic
289C22×C4⋊C4C2×D5311yescharacteristic
290C22×C22⋊C4—1061no—
291C2×#106—24no—
292C2×#106—534no—
293C2×#106—1064no—
294C2×#106C2×D5314yescomplemented
295C2×C22⋊C8—538no—
296C2×C22⋊C8—534no—
297#380—534no—
298#380—1062no—
299C2×#95—1061no—
300#340—534no—
301#340—534no—
302C53C2×#185311yescharacteristic, complemented, Sylow
303C25—531no—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4—531no—
307C23×C4—534no—
308C23×C4—532no—
309C23×C4—534no—
310C23×C4—532no—
311C23×C4—1064no—
312C23×C4—1068no—
313C23×C4—1062no—
314C23×C4—1062no—
315C23×C4—1062no—
316C23×C4—1062no—
317C23×C4—1061no—
318C23×C4—1061no—
319C23×C4—2124no—
320C23×C4—2124no—
321C23×C4C22×D5311yescharacteristic
322C22×C8—22no—
323C22×C8—22no—
324C22×C8—534no—
325C22×C8—532no—
326C22×C8—1062no—
327C22×C8—1062no—
328C22×C8C4×D5314yescomplemented
329C22×C8C22×D5312yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—534no—
336C2×C4⋊C4—532no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—1068no—
339C2×C4⋊C4—10616no—
340C2×C4⋊C4—1064no—
341C2×C4⋊C4—1068no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4—1062no—
344C2×C4⋊C4—1068no—
345C2×C4⋊C4—1062no—
346C2×C4⋊C4—1062no—
347C2×C4⋊C4—1061no—
348C2×C4⋊C4C22⋊D5314yes—
349C2×C4⋊C4C22×D5312yes—
350C2×C4⋊C4C22×D5312yes—
351C2×C22⋊C4—1064no—
352C2×C22⋊C4—2124no—
353#106—28no—
354#106—1068no—
355C22⋊C8—5316no—
356#95—1064no—
357C24—534no—
358C24—534no—
359C24—534no—
360C24—532no—
361C24—1062no—
362C24—1062no—
363C24—1062no—
364C24—1062no—
365C24D4×D5311yescharacteristic
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—534no—
375C22×C4—532no—
376C22×C4—538no—
377C22×C4—534no—
378C22×C4—534no—
379C22×C4—10616no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—1068no—
383C22×C4—1068no—
384C22×C4—1064no—
385C22×C4—1068no—
386C22×C4—1068no—
387C22×C4—1064no—
388C22×C4—1062no—
389C22×C4—1062no—
390C22×C4—1062no—
391C22×C4—1062no—
392C22×C4—1064no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—2128no—
396C22×C4—21232no—
397C22×C4—2128no—
398C22×C4—2122no—
399C22×C4—2122no—
400C22×C4—2122no—
401C22×C4—2122no—
402C22×C4—2122no—
403C22×C4C2×C4×D5312yes—
404C22×C4#1171994512yes—
405C22×C4D4×D5311yescharacteristic
406C22×C4D4×D5312yes—
407C22×C4Q8×D5312yes—
408C22×C4#1171994912yes—
409C22×C4C2×C22⋊D5312yes—
410C22×C4C23×D5311yescharacteristic
411C2×C8—28no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—534no—
415C2×C8—1068no—
416C2×C8—1062no—
417C2×C8—1062no—
418C2×C8C2×C4×D5314yescomplemented
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—42no—
424C4⋊C4—534no—
425C4⋊C4—10616no—
426C4⋊C4—1064no—
427C4⋊C4—1062no—
428C4⋊C4—1062no—
429C4⋊C4—1062no—
430C4⋊C4—2122no—
431C4⋊C4C2×C22⋊D5314yes—
432C22⋊C4—2128no—
433C23—534no—
434C23—534no—
435C23—534no—
436C23—5316no—
437C23—1068no—
438C23—1068no—
439C23—1062no—
440C23—1062no—
441C23—1062no—
442C23—1062no—
443C23—1068no—
444C23—1068no—
445C23—1068no—
446C23—1068no—
447C23C4⋊C4×D5312yes—
448C23#1313318512yes—
449C23#1313320212yes—
450C23SD16×D5312yes—
451C23SD16×D5312yes—
452C23#1313324412yes—
453C23#1313324612yes—
454C23C2×D4×D5311yescharacteristic
455C2×C4—24no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—534no—
464C2×C4—1062no—
465C2×C4—1062no—
466C2×C4—1068no—
467C2×C4—1064no—
468C2×C4—1064no—
469C2×C4—1064no—
470C2×C4—1064no—
471C2×C4—10616no—
472C2×C4—1064no—
473C2×C4—4244no—
474C2×C4—21232no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4—2122no—
478C2×C4—2128no—
479C2×C4—2122no—
480C2×C4—2122no—
481C2×C4C4⋊C4×D5312yes—
482C2×C4C4×C22⋊D5314yes—
483C2×C4#1313317914yes—
484C2×C4#1313318512yes—
485C2×C4#1313319314yes—
486C2×C4#1313319514yes—
487C2×C4#1313320212yes—
488C2×C4C22×C4×D5311yescharacteristic
489C2×C4C2×D4×D5311yescharacteristic
490C2×C4C22×C22⋊D5311yescharacteristic
491C2×C4C2×Q8×D5311yescharacteristic
492C2×C4C2×#1171994511yescharacteristic
493C2×C4C2×#1171994911yescharacteristic
494C8—24no—
495C8—1064no—
496C22—538no—
497C22—1064no—
498C22—1064no—
499C22—1064no—
500C22—1064no—
501C22—10632no—
502C22—1064no—
503C22—1064no—
504C22#1681049414yes—
505C22#106×D5314yes—
506C22#1681066214yes—
507C22#1681076014yes—
508C22#1681077214yes—
509C22#1681094314yes—
510C22#1681095514yes—
511C22C2×C4⋊C4×D5311yescharacteristic
512C22C2×#1313318511yescharacteristic
513C22C2×#1313320211yescharacteristic
514C22C2×SD16×D5311yescharacteristic
515C22C2×SD16×D5311yescharacteristic
516C22C2×#1313324411yescharacteristic
517C22C2×#1313324611yescharacteristic
518C4—44no—
519C4—82no—
520C4—1064no—
521C4—4242no—
522C4—2124no—
523C4#1681049414yes—
524C4C2×C4×C22⋊D5311yescharacteristic
525C4C2×#1313317911yescharacteristic
526C4C2×#1313319311yescharacteristic
527C4C2×#1313319511yescharacteristic
528C2—1068no—
529C2#2027490618yesdirect product factor
530C2C2×#1681049411yescharacteristic
531C2C2×#106×D5311yescharacteristic
532C2C2×#1681066211yescharacteristic
533C2C2×#1681077211yescharacteristic
534C2C2×#1681076011yescharacteristic
535C2C2×#1681095511yescharacteristic
536C2C2×#1681094311yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#106×C53C2
socleC24×C53C2×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC22×#106D53
hypocenterC53C2×#1853
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×#13133148C2semidirect
5. #20274906C2direct (N × H)
6. C2×#16810386C2semidirect
7. C2×#16810385C2semidirect
8. C22×#106×C53C2semidirect
21. C2×#13133148C22semidirect
22. C2×#13133129C4semidirect
27. #16810386C22semidirect
28. #16810385C22semidirect
33. C2×#106×C53C22semidirect
52. C22×#88322C2×C4semidirect
70. #13133129C2×C4semidirect
77. C22×C8×C53C2×C4semidirect
145. C2×C8×C53C22×C4semidirect
163. C2×#88322C22×C4semidirect
182. C22×D53#106semidirect
249. C2×D53C2×#106semidirect
262. C22×#106D53semidirect
268. C2×C53#1853semidirect
294. C2×#106C2×D53semidirect
302. C53C2×#1853semidirect
328. C22×C8C4×D53semidirect
418. C2×C8C2×C4×D53semidirect
529. C2#20274906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C22×#131331482
C22×#131331292
C2×#168104802
#202749062
C2×#168103862
C2×#168103852
C22×#106×C532
C2×#185353