Finite Group Databasebeta

← C2×#20273182 #27350709

Subgroups — C2×#20273182 #27350709

Order 13568.

1270 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811170C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic, complemented
3C2×#16810674C211yesmaximal, characteristic, complemented
4#20273182C218yesmaximal, direct product factor
5C2×#16810352C211yesmaximal, characteristic, complemented
6C2×D4⋊#88322C211yesmaximal, characteristic, complemented
7C2×#16810350C211yesmaximal, characteristic, complemented
8C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#11719945—21no—
10#16811170—22no—
11#16811170—22no—
12#16811170C414yes—
13#16811170C2214yes—
14C2×#13133216C411yescharacteristic
15C2×#13133213C411yescharacteristic
16C2×#13133203—21no—
17C2×#13133192C411yescharacteristic
18C2×#13133190—21no—
19C2×#13133188C411yescharacteristic
20C2×C4×D212C2211yescharacteristic
21C2×C4×C22⋊D53—21no—
22C2×C42×D53C2211yescharacteristic
23#16810679—22no—
24#16810679C2214yescomplemented
25#16810674—22no—
26#16810674C2214yescomplemented
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133146—21no—
29C2×#13133129—21no—
30C2×#13133129—21no—
31C2×#13133127—21no—
32C2×C4×#88322C2211yescharacteristic
33#16810352C2214yescomplemented
34D4⋊#88322C2214yescomplemented
35#16810350C2214yescomplemented
36C2×C4×D4×C53C2211yescharacteristic
37C2×C4⋊C8×C53C2211yescharacteristic
38C2×C4×C8×C53C2211yescharacteristic
39C53×D4⋊C8C2214yescomplemented
40C2×#11719945—24no—
41C2×#11719945—24no—
42C2×#11719945—44no—
43C2×#11719945—44no—
44C22×Dic106—21no—
45C22×D212—21no—
46C22×D212C2×C411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C22⋊D53—21no—
49C22×C22⋊D53—21no—
50C22×C4×D53—21no—
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C22×#88322—21no—
55C22×#88322—21no—
56#13133216—22no—
57#13133216C2×C414yes—
58#13133213—22no—
59#13133213C2×C414yes—
60#13133203—24no—
61#13133203—24no—
62#13133192—22no—
63#13133192C2×C414yes—
64#13133190—24no—
65#13133190—24no—
66#13133188—22no—
67#13133188C2×C414yes—
68C4×D212—21no—
69C4×D212—21no—
70C4×D212C2×C412yes—
71C4×D212C2312yes—
72C4×C22⋊D53—24no—
73C4×C22⋊D53—24no—
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D411yescharacteristic
82C42×D53C2×C412yes—
83C42×D53D412yes—
84C42×D53D412yes—
85C42×D53C2312yes—
86C4⋊#88322—21no—
87C4⋊#88322C2312yes—
88#13133146—24no—
89#13133129—24no—
90#13133129—24no—
91#13133127—24no—
92C4×#88322—21no—
93C4×#88322C2312yes—
94C22×D4×C53C2×C411yescharacteristic
95C23×C4×C53—21no—
96C22×C8×C53—21no—
97C22×C8×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53C2×C412yes—
101C4×D4×C53C2312yes—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107C4×C8×C53—21no—
108C4×C8×C53C2312yes—
109C24×C53—21no—
110C23×D53—21no—
111C23×D53C22⋊C411yescharacteristic
112C2×D4×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53C22×C412yes—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C2×C22⋊D53—24no—
125C2×C22⋊D53—24no—
126C2×C22⋊D53—24no—
127C2×C22⋊D53—44no—
128C2×C22⋊D53—44no—
129C2×C22⋊D53—44no—
130C22×Dic53—21no—
131C22×Dic53C22⋊C411yescharacteristic
132C22×Dic53C22⋊C411yescharacteristic
133C22×Dic53C22⋊C411yescharacteristic
134#11719945—48no—
135#11719945—48no—
136#11719945—416no—
137C2×D212—22no—
138C2×D212—22no—
139C2×D212—24no—
140C2×D212—21no—
141C2×D212—44no—
142C2×D212C22×C412yes—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53C22⋊C412yes—
153C2×C4×D53C22⋊C412yes—
154C2×C4×D53C22⋊C412yes—
155C2×C4×D53C22×C412yes—
156C2×C4×D53C2×D412yes—
157C2×C4×D53C2×D412yes—
158C2×Dic106—24no—
159C2×Dic106—44no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—42no—
163C2×C8×C53—42no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C412yes—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C42×C53C22×C411yescharacteristic
169C42×C53C2×D411yescharacteristic
170C42×C53C2×D411yescharacteristic
171C42×C53C2411yescharacteristic
172C22⋊Dic53—24no—
173#11719921—22no—
174#11719921—22no—
175#11719921—24no—
176C4⋊Dic53—21no—
177C4⋊Dic53C22×C412yes—
178#11719919—24no—
179C4×Dic53C22⋊C412yes—
180C4×Dic53C22⋊C412yes—
181C4×Dic53C22⋊C412yes—
182C4×Dic53C22×C412yes—
183C4×Dic53C2×D412yes—
184C4×Dic53C2×D412yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—42no—
188C2×#88322—42no—
189C23×C53—22no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53C2×C22⋊C411yescharacteristic
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53C2×C22⋊C412yes—
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—42no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—44no—
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C22⋊C411yescharacteristic
214C2×C4×C53#13112yes—
215C2×C4×C53#13112yes—
216C2×C4×C53AGL(1,ℤ/8)12yes—
217C2×C4×C53AGL(1,ℤ/8)12yes—
218C2×C4×C53C23×C411yescharacteristic
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C22×D411yescharacteristic
221C22⋊D53—48no—
222C22⋊D53—48no—
223C22⋊D53—48no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—42no—
229C2×Dic53C2×C22⋊C412yes—
230C2×Dic53C2×C22⋊C412yes—
231C2×Dic53C2×C22⋊C412yes—
232D212—22no—
233D212—22no—
234D212—48no—
235D212—42no—
236C4×D53—22no—
237C4×D53—22no—
238C4×D53—22no—
239C4×D53—22no—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—42no—
243C4×D53—42no—
244C4×D53—42no—
245C4×D53—42no—
246C4×D53—44no—
247Dic106—48no—
248C8×C53—42no—
249C8×C53—42no—
250#88322—42no—
251#88322—42no—
252C2×#1496—531nomaximal, Sylow
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—84no—
257C22×C53#40812yes—
258C22×C53#41812yes—
259C22×C53#42112yes—
260C22×C53C22×C22⋊C411yescharacteristic
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—44no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—84no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—42no—
272C4×C53#40812yes—
273C4×C53#41812yes—
274C4×C53C2×#13111yescharacteristic
275C4×C53C2×#13111yescharacteristic
276C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
278Dic53—22no—
279Dic53—22no—
280Dic53—22no—
281Dic53—42no—
282C2×#532—531no—
283C2×#432—531no—
284C2×#431—531no—
285#1496—538no—
286C2×D4⋊C8—531no—
287C2×D4⋊C8—531no—
288C2×D4⋊C8—531no—
289C2×D4⋊C8D5311yescharacteristic, complemented
290C2×C53—82no—
291C2×C53#149614yescomplemented
292C2×C53C2×#40811yescharacteristic
293C2×C53C2×#41811yescharacteristic
294C2×C53C2×#42111yescharacteristic
295D53—42no—
296D53—82no—
297C22×#41—1061no—
298#532—534no—
299#532—534no—
300#532—1062no—
301#532—1062no—
302C2×C4⋊Q8—531no—
303C2×#127—531no—
304C2×#124—1061no—
305C2×#121—531no—
306C2×#121—531no—
307C2×#121—1061no—
308C2×C4×D4—531no—
309C2×C4×D4—1061no—
310C2×C4×D4C2×D5311yescharacteristic
311C22×C42—531no—
312#432—534no—
313#432—1062no—
314#431—534no—
315#431—1062no—
316C2×C4⋊C8—531no—
317C2×C4⋊C8—1061no—
318C2×C4⋊C8C2×D5311yescharacteristic
319C2×C22⋊C8—1061no—
320C2×C22⋊C8—1061no—
321C2×#97—1061no—
322C2×C4×C8—531no—
323C2×C4×C8C2×D5311yescharacteristic
324D4⋊C8—534no—
325D4⋊C8—534no—
326D4⋊C8—534no—
327D4⋊C8C2×D5314yescomplemented
328C53C2×#149611yescharacteristic, complemented, Sylow
329C2×#41—1064no—
330C2×#41—1064no—
331C2×#41—2124no—
332C2×#41—2124no—
333C22×Q8—1061no—
334C22×D4—531no—
335C22×D4—1061no—
336C22×D4—1061no—
337C22×D4—1061no—
338C22×D4—1061no—
339C22×D4C4×D5311yescharacteristic
340C23×C4—21no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—1061no—
345C22×C8—21no—
346C22×C8—21no—
347C22×C8—1061no—
348C22×C8—1061no—
349C4⋊Q8—534no—
350C4⋊Q8—1062no—
351#127—534no—
352#127—1062no—
353#124—1064no—
354#124—1064no—
355#121—534no—
356#121—534no—
357#121—1064no—
358#121—1064no—
359#121—1062no—
360#121—1062no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—532no—
364C4×D4—532no—
365C4×D4—1064no—
366C4×D4—1064no—
367C4×D4—1061no—
368C4×D4—1061no—
369C4×D4C4×D5312yes—
370C4×D4C22×D5312yes—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4C4×D5311yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C42—531no—
379C2×C42—531no—
380C2×C42—531no—
381C2×C42—532no—
382C2×C42—532no—
383C2×C42—532no—
384C2×C42—532no—
385C2×C42C22×D5311yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—532no—
388C4⋊C8—1064no—
389C4⋊C8—1061no—
390C4⋊C8C22×D5312yes—
391C22⋊C8—1064no—
392C22⋊C8—1064no—
393#97—1064no—
394C4×C8—21no—
395C4×C8—532no—
396C4×C8—1061no—
397C4×C8C22×D5312yes—
398C24—21no—
399C24—531no—
400C24—1061no—
401#41—21216no—
402#41—2128no—
403#41—2128no—
404C2×Q8—1064no—
405C2×Q8—2124no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—532no—
410C2×D4—1064no—
411C2×D4—1064no—
412C2×D4—1064no—
413C2×D4—1064no—
414C2×D4—1061no—
415C2×D4—1062no—
416C2×D4—1062no—
417C2×D4—2124no—
418C2×D4—2124no—
419C2×D4—2124no—
420C2×D4—2124no—
421C2×D4C2×C4×D5312yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—532no—
431C22×C4—532no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1061no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—2122no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4C2×C4×D5311yescharacteristic
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5311yescharacteristic
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—1062no—
454C2×C8—1062no—
455C2×C8—2122no—
456C2×C8—2122no—
457C4⋊C4—21no—
458C4⋊C4—532no—
459C4⋊C4—1064no—
460C4⋊C4—1061no—
461C4⋊C4C2×C4×D5312yes—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—1064no—
465C22⋊C4—1064no—
466C22⋊C4—1062no—
467C22⋊C4—1062no—
468C42—532no—
469C42—532no—
470C42—532no—
471C42—532no—
472C42—532no—
473C42—532no—
474C42C2×C4×D5311yescharacteristic
475C42D4×D5311yescharacteristic
476C42D4×D5311yescharacteristic
477C42C23×D5311yescharacteristic
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—532no—
483C23—1062no—
484C23—1062no—
485C23—1062no—
486C23—1062no—
487C23—2122no—
488C23—2122no—
489C23—2122no—
490C23C22⋊C4×D5311yescharacteristic
491Q8—2128no—
492D4—22no—
493D4—22no—
494D4—42no—
495D4—1062no—
496D4—1062no—
497D4—2128no—
498D4—2128no—
499D4—2128no—
500D4—2128no—
501D4—2122no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—532no—
509C2×C4—532no—
510C2×C4—532no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1064no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2124no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4C22⋊C4×D5311yescharacteristic
528C2×C4C22⋊C4×D5311yescharacteristic
529C2×C4C22⋊C4×D5311yescharacteristic
530C2×C4#1313322312yes—
531C2×C4#1313322512yes—
532C2×C4#1313324112yes—
533C2×C4#1313324412yes—
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C2×D4×D5311yescharacteristic
536C2×C4C2×D4×D5311yescharacteristic
537C8—42no—
538C8—42no—
539C8—2122no—
540C8—2122no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—84no—
545C22—1062no—
546C22—1062no—
547C22—1062no—
548C22—4244no—
549C22—2122no—
550C22—2122no—
551C22—2122no—
552C22—2124no—
553C22#1681058812yes—
554C22#1681062612yes—
555C22#1681063812yes—
556C22C2×C22⋊C4×D5311yescharacteristic
557C4—21no—
558C4—21no—
559C4—42no—
560C4—1062no—
561C4—1062no—
562C4—1062no—
563C4—2122no—
564C4#1681058812yes—
565C4#1681062612yes—
566C4C2×#1313322311yescharacteristic
567C4C2×#1313322511yescharacteristic
568C4C2×#1313324111yescharacteristic
569C4C2×#1313324411yescharacteristic
570C2—82no—
571C2—4242no—
572C2—2122no—
573C2#2027318214yesdirect product factor
574C2C2×#1681058811yescharacteristic
575C2C2×#1681062611yescharacteristic
576C2C2×#1681063811yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811170C2
cosocleC42×C53C24
hypercenterC2×D4⋊C8D53
hypocenterC53C2×#1496
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810679C2semidirect
3. C2×#16810674C2semidirect
4. #20273182C2direct (N × H)
5. C2×#16810352C2semidirect
6. C2×D4⋊#88322C2semidirect
7. C2×#16810350C2semidirect
8. C2×C53×D4⋊C8C2semidirect
24. #16810679C22semidirect
26. #16810674C22semidirect
33. #16810352C22semidirect
34. D4⋊#88322C22semidirect
35. #16810350C22semidirect
39. C53×D4⋊C8C22semidirect
289. C2×D4⋊C8D53semidirect
291. C2×C53#1496semidirect
327. D4⋊C8C2×D53semidirect
328. C53C2×#1496semidirect
573. C2#20273182direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111702
C2×#168106792
C2×#168106742
#202731822
C2×#168103522
C2×D4⋊#883222
C2×#168103502
C2×C53×D4⋊C82
C2×#149653