Finite Group Databasebeta

← C2×#20290113 #28315008

Subgroups — C2×#20290113 #28315008

Order 13568.

1554 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811595C211yesmaximal, characteristic
2C2×C4×C22⋊#6157C211yesmaximal, characteristic
3#20290113C218yesmaximal, direct product factor
4C2×#16811477C212yesmaximal, complemented
5C2×#16811476C212yesmaximal, complemented
6C2×C22⋊Q8×D53C211yesmaximal, characteristic
7C22×#11719940—21no—
8C22×C22⋊#6157—21no—
9C22×C4×#6157—21no—
10#16811595—22no—
11#16811595C2214yes—
12C2×#13133274—21no—
13C2×#13133274C2212yes—
14C4×C22⋊#6157—22no—
15C4×C22⋊#6157C2214yes—
16C2×(C2×#6157)⋊C4C2212yes—
17#16811477C2218yescomplemented
18#16811476C2218yescomplemented
19C22×Q8×D53C411yescharacteristic, complemented
20C23×C4×D53C2211yescharacteristic
21C2×#13133197C2211yescharacteristic
22C2×#13133196C411yescharacteristic, complemented
23C2×#13133195—21no—
24C2×#13133194—21no—
25C2×#13133193C2211yescharacteristic
26C22⋊Q8×D53—22no—
27C22⋊Q8×D53—22no—
28C22⋊Q8×D53C414yescomplemented
29C22⋊Q8×D53C2214yes—
30C2×C4⋊C4×D53—21no—
31C2×C4⋊C4×D53C411yescharacteristic, complemented
32C2×C22⋊C4×D53—21no—
33C2×C22⋊Q8×C53C411yescharacteristic, complemented
34C23×#6157—22no—
35C2×#11719940—24no—
36C2×#11719940—44no—
37C22×#88324—22no—
38C2×C22⋊#6157—28no—
39C2×C22⋊#6157—24no—
40C2×C4×#6157—24no—
41C2×C4×#6157—44no—
42#13133274—24no—
43#13133274—22no—
44#13133274C2314yes—
45(C2×#6157)⋊C4—22no—
46(C2×#6157)⋊C4C2314yes—
47C24×D53D411yescharacteristic
48C22×Dic106—21no—
49C22×Dic106C2×C411yescharacteristic
50C2×Q8×D53—22no—
51C2×Q8×D53—44no—
52C2×Q8×D53C2×C414yescomplemented
53C23×Dic53D411yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53—44no—
57C22×C4×D53C2×C411yescharacteristic
58C22×C4×D53C2×C412yes—
59C22×C4×D53D411yescharacteristic
60C22×C4×D53D412yes—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D412yes—
63C22×C4×D53C2311yescharacteristic
64C22×C4×D53C2312yes—
65#13133197—21no—
66#13133197—21no—
67#13133197C2×C412yes—
68#13133197C2312yes—
69#13133196—22no—
70#13133196C2×C414yescomplemented
71#13133195—24no—
72#13133195—24no—
73#13133194—24no—
74#13133194—24no—
75#13133193—21no—
76#13133193—21no—
77#13133193C2×C412yes—
78#13133193C2312yes—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2×C411yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C4⋊C4×D53—24no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—24no—
87C4⋊C4×D53C2×C414yescomplemented
88C2×C22⋊Dic53—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C22⋊C4×D53—24no—
92C22⋊C4×D53—24no—
93C22×Q8×C53C2×C411yescharacteristic
94C23×C4×C53C2×C411yescharacteristic
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53C2×C414yescomplemented
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2×C411yescharacteristic
99C2×C22⋊C4×C53—21no—
100C24×C53C22⋊C411yescharacteristic
101C23×D53—24no—
102C23×D53—24no—
103C23×D53—42no—
104C23×D53C22⋊C412yes—
105C23×D53C22⋊C411yescharacteristic
106C23×D53C2×D412yes—
107C23×D53C2×D411yescharacteristic
108C22×#6157—28no—
109C22×#6157—24no—
110C22×#6157—44no—
111C22×#6157—44no—
112C2×Q8×C53—21no—
113C2×Q8×C53—42no—
114C2×Q8×C53C22×C412yes—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—42no—
118C22×C4×C53C22⋊C411yescharacteristic
119C22×C4×C53C22⋊C412yes—
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C412yes—
122C22×Dic53—24no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53C22⋊C412yes—
126C22×Dic53C22⋊C411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D412yes—
129Q8×D53—48no—
130Q8×D53—48no—
131C2×C4×D53—24no—
132C2×C4×D53—24no—
133C2×C4×D53—24no—
134C2×C4×D53—24no—
135C2×C4×D53—44no—
136C2×C4×D53—44no—
137C2×C4×D53—44no—
138C2×C4×D53—44no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53—44no—
142C2×C4×D53—44no—
143C2×C4×D53C22⋊C412yes—
144C2×C4×D53C22⋊C412yes—
145C2×C4×D53C22⋊C412yes—
146C2×C4×D53C22×C412yes—
147C2×C4×D53C22×C411yescharacteristic
148C2×C4×D53C2×D412yes—
149C2×C4×D53C2×D411yescharacteristic
150C2×C4×D53C2×D412yes—
151C2×C4×D53C2×D411yescharacteristic
152C2×C4×D53C2411yescharacteristic
153C2×Dic106—21no—
154C2×Dic106—24no—
155C2×Dic106—44no—
156C2×Dic106—42no—
157C2×Dic106C22×C412yes—
158C22⋊#6157—216no—
159#11719940—48no—
160C2×#88324—24no—
161C2×#88324—44no—
162C4×#6157—48no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53C22×C412yes—
166C22⋊C4×C53—24no—
167C22⋊Dic53—24no—
168#11719921—22no—
169#11719921—22no—
170#11719921—22no—
171#11719921—22no—
172C4⋊Dic53—21no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53C22×C412yes—
176#11719919—24no—
177#11719919—22no—
178#11719919—24no—
179#11719919—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53C2×C22⋊C412yes—
184C23×C53C2×C22⋊C411yescharacteristic
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—416no—
196C22×D53#10414yes—
197C22×D53C2×C22⋊C411yescharacteristic
198C22×D53C2×C22⋊C412yes—
199C22×D53#13714yes—
200C22×D53C22×D411yescharacteristic
201C2×#6157—28no—
202C2×#6157—44no—
203C2×#6157—416no—
204C2×#6157—44no—
205Q8×C53—42no—
206Q8×C53—42no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53C23×C411yescharacteristic
218C2×Dic53—22no—
219C2×Dic53—24no—
220C2×Dic53—24no—
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53#9914yes—
226C2×Dic53C2×C22⋊C411yescharacteristic
227C2×Dic53C2×C22⋊C412yes—
228C2×Dic53C22×D411yescharacteristic
229C4×D53—24no—
230C4×D53—24no—
231C4×D53—24no—
232C4×D53—24no—
233C4×D53—42no—
234C4×D53—41no—
235C4×D53—41no—
236C4×D53—42no—
237C4×D53—41no—
238C4×D53—41no—
239C4×D53—48no—
240C4×D53—48no—
241Dic106—42no—
242Dic106—48no—
243Dic106—42no—
244#88324—44no—
245C2×#1508—531nomaximal, Sylow
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—44no—
250C22×C53C2×#9912yes—
251C22×C53#41912yes—
252C22×C53C2×#10412yes—
253C22×C53C22×C22⋊C411yescharacteristic
254C2×D53—22no—
255C2×D53—21no—
256C2×D53—21no—
257C2×D53—22no—
258C2×D53—21no—
259C2×D53—21no—
260C2×D53—28no—
261C2×D53—28no—
262C2×D53—48no—
263C2×D53—48no—
264C2×D53#41912yes—
265C2×D53C2×#10414yes—
266C2×D53C2×#13712yes—
267#6157—48no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—22no—
271C4×C53—42no—
272C4×C53—42no—
273Dic53—21no—
274Dic53—21no—
275Dic53—42no—
276Dic53—42no—
277Dic53C2×#9914yes—
278C22×C22⋊Q8—531no—
279C2×#407—531no—
280C2×C4×C22⋊C4—531no—
281#1508—538no—
282C2×#328—532no—
283C2×#328—532no—
284C2×C53—42no—
285C2×C53#150814yescomplemented
286C2×C53C22×#9911yescharacteristic
287C2×C53C2×#41911yescharacteristic
288C2×C53C22×#10411yescharacteristic
289D53—22no—
290D53—22no—
291D53—24no—
292C23×Q8—531no—
293C24×C4—531no—
294C22×M16—1061no—
295C2×C22⋊Q8—531no—
296C2×C22⋊Q8—531no—
297C2×C22⋊Q8—534no—
298C2×C22⋊Q8—534no—
299C2×C22⋊Q8—531no—
300C2×C22⋊Q8—1061no—
301C2×C22⋊Q8—1061no—
302C2×C22⋊Q8—1062no—
303C2×C22⋊Q8—1062no—
304C2×C22⋊Q8#615711yescharacteristic, complemented
305C22×C4⋊C4—531no—
306C22×C4⋊C4—1061no—
307C22×C22⋊C4—1061no—
308C22×C22⋊C4—1061no—
309C22×C42—1061no—
310#407—534no—
311#407—1062no—
312C2×C22⋊C8—532no—
313C2×C22⋊C8—1061no—
314C4×C22⋊C4—534no—
315C4×C22⋊C4—1062no—
316C2×#95—532no—
317#328—538no—
318#328—538no—
319C53C2×#150811yescharacteristic, complemented, Sylow
320C25—531no—
321C22×Q8—534no—
322C22×Q8—531no—
323C22×Q8—1061no—
324C22×Q8—1062no—
325C22×Q8—2124no—
326C22×Q8C2×#615711yescharacteristic
327C23×C4—531no—
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—532no—
332C23×C4—532no—
333C23×C4—531no—
334C23×C4—532no—
335C23×C4—532no—
336C23×C4—1061no—
337C23×C4—1061no—
338C23×C4—1062no—
339C23×C4—2124no—
340C23×C4C2×#615711yescharacteristic
341C2×M16—1064no—
342C2×M16—2124no—
343C22×C8—1062no—
344C22⋊Q8—22no—
345C22⋊Q8—532no—
346C22⋊Q8—532no—
347C22⋊Q8—534no—
348C22⋊Q8—532no—
349C22⋊Q8—532no—
350C22⋊Q8—1064no—
351C22⋊Q8—1064no—
352C22⋊Q8—1064no—
353C22⋊Q8—1064no—
354C22⋊Q8—1061no—
355C22⋊Q8—1061no—
356C22⋊Q8—1061no—
357C22⋊Q8—1061no—
358C22⋊Q8—1062no—
359C22⋊Q8C2×#615714yescomplemented
360C2×C4⋊C4—21no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—534no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4C2×#615711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1064no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—1061no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1068no—
378C2×C22⋊C4—1061no—
379C2×C42—1064no—
380C2×C42—2124no—
381C22⋊C8—534no—
382C22⋊C8—1062no—
383C22⋊C8—1064no—
384#95—534no—
385#95—1062no—
386C24—531no—
387C24—531no—
388C24—532no—
389C24—532no—
390C24—1064no—
391C24—1064no—
392C24—2122no—
393C24C22⋊#615711yescharacteristic
394C2×Q8—21no—
395C2×Q8—42no—
396C2×Q8—532no—
397C2×Q8—1064no—
398C2×Q8—1061no—
399C2×Q8—2122no—
400C2×Q8—2124no—
401C2×Q8—2128no—
402C2×Q8—2128no—
403C2×Q8C22×#615712yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—531no—
408C22×C4—532no—
409C22×C4—532no—
410C22×C4—531no—
411C22×C4—532no—
412C22×C4—532no—
413C22×C4—532no—
414C22×C4—532no—
415C22×C4—532no—
416C22×C4—531no—
417C22×C4—532no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—1064no—
422C22×C4—1064no—
423C22×C4—1064no—
424C22×C4—1064no—
425C22×C4—1064no—
426C22×C4—1064no—
427C22×C4—1068no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—2122no—
431C22×C4—2122no—
432C22×C4—2124no—
433C22×C4—2124no—
434C22×C4—2124no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4—2124no—
438C22×C4—2124no—
439C22×C4—2124no—
440C22×C4C22⋊#615711yescharacteristic
441C22×C4C22⋊#615712yes—
442C22×C4C22×#615711yescharacteristic
443C22×C4C22×#615712yes—
444M16—2128no—
445C2×C8—1064no—
446C2×C8—2124no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—532no—
450C4⋊C4—1064no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C4⋊C4—1061no—
456C4⋊C4—1064no—
457C4⋊C4C22×#615712yes—
458C22⋊C4—24no—
459C22⋊C4—1062no—
460C22⋊C4—1062no—
461C22⋊C4—1062no—
462C22⋊C4—1062no—
463C22⋊C4—10616no—
464C22⋊C4—1064no—
465C42—2128no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—534no—
470C23—534no—
471C23—531no—
472C23—532no—
473C23—531no—
474C23—1064no—
475C23—1064no—
476C23—1064no—
477C23—1064no—
478C23—1062no—
479C23—1062no—
480C23—1062no—
481C23—1062no—
482C23—2122no—
483C23—2122no—
484C23—21216no—
485C23C2×C22⋊#615712yes—
486C23C2×C22⋊#615711yescharacteristic
487Q8—42no—
488Q8—42no—
489Q8—2122no—
490Q8—2122no—
491Q8—2128no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—534no—
501C2×C4—532no—
502C2×C4—531no—
503C2×C4—531no—
504C2×C4—1064no—
505C2×C4—1064no—
506C2×C4—1064no—
507C2×C4—1064no—
508C2×C4—1064no—
509C2×C4—1064no—
510C2×C4—1062no—
511C2×C4—1068no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2128no—
521C2×C4—2128no—
522C2×C4—2124no—
523C2×C4—2124no—
524C2×C4—21216no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4C2×C22⋊#615712yes—
528C2×C4C2×C22⋊#615711yescharacteristic
529C2×C4C23×#615711yescharacteristic
530C8—2124no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—44no—
535C22—532no—
536C22—532no—
537C22—534no—
538C22—1062no—
539C22—1061no—
540C22—1061no—
541C22—1068no—
542C22—1068no—
543C22—1061no—
544C22—1062no—
545C22—1061no—
546C22—2128no—
547C22—2128no—
548C22D53⋊#9912yes—
549C22#1681161612yes—
550C22D53⋊#10412yes—
551C22C22×C22⋊#615711yescharacteristic
552C4—22no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—534no—
558C4—1061no—
559C4—1061no—
560C4—2128no—
561C4—2122no—
562C4—2122no—
563C2—42no—
564C2—1062no—
565C2—1062no—
566C2—1064no—
567C2#2029011314yesdirect product factor
568C2C2×D53⋊#9911yescharacteristic
569C2C2×#1681161611yescharacteristic
570C2C2×D53⋊#10411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C22⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC2×C4×D53C24
hypercenterC2×C22⋊Q8#6157
hypocenterC53C2×#1508
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20290113C2direct (N × H)
4. C2×#16811477C2semidirect
5. C2×#16811476C2semidirect
17. #16811477C22semidirect
18. #16811476C22semidirect
19. C22×Q8×D53C4semidirect
22. C2×#13133196C4semidirect
28. C22⋊Q8×D53C4semidirect
31. C2×C4⋊C4×D53C4semidirect
33. C2×C22⋊Q8×C53C4semidirect
52. C2×Q8×D53C2×C4semidirect
70. #13133196C2×C4semidirect
87. C4⋊C4×D53C2×C4semidirect
96. C22⋊Q8×C53C2×C4semidirect
285. C2×C53#1508semidirect
304. C2×C22⋊Q8#6157semidirect
319. C53C2×#1508semidirect
359. C22⋊Q8C2×#6157semidirect
567. C2#20290113direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115952
C2×C4×C22⋊#61572
#202901132
C2×#168114772
C2×#168114762
C2×C22⋊Q8×D532
C2×#150853