Finite Group Databasebeta

← C2×#20275024 #27385295

Subgroups — C2×#20275024 #27385295

Order 13568.

1966 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
1C22×#13133193C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic
3C22×#13133141C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic
5#20275024C218yesmaximal, direct product factor
6C2×#16810386C211yesmaximal, characteristic, complemented
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#103×C53C211yesmaximal, characteristic, complemented
9C23×C4×D53C2211yescharacteristic
10C23×#88322C2211yescharacteristic
11C2×#13133193—24no—
12C2×#13133193—22no—
13C2×#13133193C418yescomplemented
14C2×#13133193C2214yes—
15C2×#13133175—22no—
16C2×#13133175—24no—
17C2×#13133175—24no—
18C2×#13133175C2214yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×#11719919—21no—
22C22×#11719921—21no—
23C22×#11719921—21no—
24C22×C4×Dic53—21no—
25C2×#13133141—24no—
26C2×#13133141C2214yescomplemented
27C2×#13133129—24no—
28C2×#13133129C2214yes—
29#16810386C2214yescomplemented
30#16810385C2214yescomplemented
31C23×Q8×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#103×C53—24no—
35C2×#103×C53C2214yescomplemented
36C24×D53D411yescharacteristic
37C23×Dic53—21no—
38C23×Dic53—21no—
39C23×Dic53D411yescharacteristic
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53C2×C414yes—
45C22×C4×D53D412yes—
46C22×C4×D53D414yes—
47C22×C4×D53C2312yes—
48C22×#88322—22no—
49C22×#88322—22no—
50C22×#88322—22no—
51C22×#88322C2312yes—
52#13133193—216no—
53#13133193—28no—
54#13133193C2×C4116yescomplemented
55#13133175—216no—
56#13133175—28no—
57#13133175—28no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2×C414yes—
63C2×C4⋊Dic53D414yes—
64C2×C4⋊Dic53C2312yes—
65C2×C4⋊Dic53C2312yes—
66C2×#11719919—28no—
67C2×#11719919—24no—
68C2×#11719921—28no—
69C2×#11719921—28no—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×C4×Dic53—28no—
73C2×C4×Dic53—24no—
74#13133141—28no—
75#13133129—28no—
76C22×Q8×C53—22no—
77C22×Q8×C53—21no—
78C22×Q8×C53—22no—
79C22×Q8×C53—22no—
80C22×Q8×C53—22no—
81C22×Q8×C53—22no—
82C22×Q8×C53C2×C414yes—
83C22×Q8×C53C2312yes—
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53C2311yescharacteristic
87C22×C8×C53—22no—
88C22×C8×C53—22no—
89C22×C8×C53—22no—
90C22×C8×C53C2312yes—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53D414yes—
94C2×C4⋊C4×C53C2312yes—
95#103×C53—28no—
96C24×C53C2×D411yescharacteristic
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53C22⋊C414yes—
104C23×D53C2×D412yes—
105C2×Q8×C53—24no—
106C2×Q8×C53—22no—
107C2×Q8×C53—28no—
108C2×Q8×C53—28no—
109C2×Q8×C53—22no—
110C2×Q8×C53—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—42no—
114C2×Q8×C53—44no—
115C2×Q8×C53—44no—
116C2×Q8×C53—42no—
117C2×Q8×C53C22×C414yes—
118C22×C4×C53—24no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—24no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53C22⋊C414yes—
145C22×Dic53C2×D412yes—
146C2×C4×D53—28no—
147C2×C4×D53—28no—
148C2×C4×D53—24no—
149C2×C4×D53—28no—
150C2×C4×D53—28no—
151C2×C4×D53—24no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53C22⋊C418yes—
157C2×C4×D53C22×C414yes—
158C2×C4×D53C2×D414yes—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—44no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53C2×D414yes—
166#11719921—216no—
167#11719921—216no—
168C4⋊Dic53—24no—
169C4⋊Dic53—24no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53C22×C414yes—
173C4⋊Dic53C2×D414yes—
174#11719919—216no—
175C4×Dic53—216no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—44no—
180C23×C53C2×C22⋊C412yes—
181C23×C53#12012yes—
182C23×C53#12312yes—
183C23×C53#13512yes—
184C23×C53#13512yes—
185C23×C53#13712yes—
186C23×C53#13712yes—
187C23×C53C22×D411yescharacteristic
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—28no—
195C22×D53—28no—
196C22×D53—28no—
197C22×D53—44no—
198C22×D53—44no—
199C22×D53—44no—
200C22×D53—44no—
201C22×D53C2×C22⋊C414yes—
202Q8×C53—24no—
203Q8×C53—24no—
204Q8×C53—42no—
205Q8×C53—42no—
206Q8×C53—48no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—48no—
214C2×C4×C53—48no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C4×D414yes—
217C2×C4×C53C4×D414yes—
218C2×C4×C53#12012yes—
219C2×C4×C53#12114yes—
220C2×C4×C53#12114yes—
221C2×C4×C53#12312yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—28no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—48no—
238C2×Dic53—48no—
239C2×Dic53C2×C22⋊C414yes—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—24no—
243C4×D53—24no—
244C4×D53—216no—
245C8×C53—42no—
246#88322—42no—
247C2×#1873—531nomaximal, Sylow
248C22×C53#38614yes—
249C22×C53#41614yes—
250C22×C53#41914yes—
251C22×C53#45214yes—
252C22×C53#45214yes—
253C22×C53#48514yes—
254C22×C53#48514yes—
255C22×C53C22×C22⋊C411yescharacteristic
256C22×C53C2×#12011yescharacteristic
257C22×C53C2×#12311yescharacteristic
258C22×C53C2×#13511yescharacteristic
259C22×C53C2×#13511yescharacteristic
260C22×C53C2×#13711yescharacteristic
261C22×C53C2×#13711yescharacteristic
262C2×D53—24no—
263C2×D53—24no—
264C2×D53—24no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—416no—
268C2×D53—42no—
269C2×D53—42no—
270C4×C53—44no—
271C4×C53—44no—
272C4×C53#38614yes—
273C4×C53C2×C4×D411yescharacteristic
274C4×C53C2×C4×D411yescharacteristic
275C4×C53C2×#12111yescharacteristic
276C4×C53C2×#12111yescharacteristic
277Dic53—24no—
278Dic53—44no—
279Dic53—44no—
280C22×C22⋊Q8—531no—
281C22×#117—531no—
282C22×#103—531no—
283C22×#103D5311yescharacteristic, complemented
284C22×C22⋊C8—531no—
285#1873—538no—
286C2×#340—531no—
287C2×#340—531no—
288C2×C53#187318yescomplemented
289C2×C53C2×#38611yescharacteristic
290C2×C53C2×#41611yescharacteristic
291C2×C53C2×#41911yescharacteristic
292C2×C53C2×#45211yescharacteristic
293C2×C53C2×#45211yescharacteristic
294C2×C53C2×#48511yescharacteristic
295C2×C53C2×#48511yescharacteristic
296D53—44no—
297C23×Q8C2×D5311yescharacteristic
298C24×C4—531no—
299C23×C8—531no—
300C23×C8C2×D5311yescharacteristic
301C2×C22⋊Q8—538no—
302C2×C22⋊Q8—534no—
303C2×C22⋊Q8—1062no—
304C2×C22⋊Q8—1064no—
305C2×#117—534no—
306C2×#117—1064no—
307C2×#117—1062no—
308C2×#117—1064no—
309C22×C4⋊C4—531no—
310C22×C4⋊C4—531no—
311C22×C4⋊C4—1061no—
312C22×C4⋊C4C2×D5311yescharacteristic
313C22×C22⋊C4—1061no—
314C22×C22⋊C4—1061no—
315C22×C42—1061no—
316C2×#103—24no—
317C2×#103—534no—
318C2×#103—1064no—
319C2×#103C2×D5314yescomplemented
320C2×C22⋊C8—534no—
321C2×C22⋊C8—1064no—
322#340—534no—
323#340—534no—
324C53C2×#187311yescharacteristic, complemented, Sylow
325C25—531no—
326C22×Q8—22no—
327C22×Q8—21no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8C4×D5314yes—
333C22×Q8C22×D5312yes—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—531no—
337C23×C4—534no—
338C23×C4—532no—
339C23×C4—534no—
340C23×C4—532no—
341C23×C4—1062no—
342C23×C4—1062no—
343C23×C4—1062no—
344C23×C4—1062no—
345C23×C4—1061no—
346C23×C4—1061no—
347C23×C4C22×D5311yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—532no—
352C22×C8—1062no—
353C22×C8—1062no—
354C22×C8—1062no—
355C22×C8C22×D5312yes—
356C22⋊Q8—5316no—
357C22⋊Q8—10616no—
358C22⋊Q8—1068no—
359#117—1068no—
360#117—10616no—
361#117—1068no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—534no—
365C2×C4⋊C4—534no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—532no—
368C2×C4⋊C4—1068no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4C22⋊D5314yes—
375C2×C4⋊C4C22×D5312yes—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1068no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1068no—
380C2×C42—1064no—
381C2×C42—1068no—
382#103—28no—
383#103—1068no—
384C22⋊C8—1068no—
385C24—534no—
386C24—532no—
387C24—1062no—
388C24—1062no—
389C24—1062no—
390C24—1062no—
391C24—1062no—
392C24—1062no—
393C24D4×D5311yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8C2×C4×D5314yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—534no—
416C22×C4—532no—
417C22×C4—538no—
418C22×C4—534no—
419C22×C4—534no—
420C22×C4—1062no—
421C22×C4—1062no—
422C22×C4—1068no—
423C22×C4—1068no—
424C22×C4—1064no—
425C22×C4—1068no—
426C22×C4—1068no—
427C22×C4—1064no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1064no—
433C22×C4—1064no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—2122no—
437C22×C4—2122no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4C2×C4×D5312yes—
441C22×C4#1171994512yes—
442C22×C4D4×D5311yescharacteristic
443C22×C4D4×D5312yes—
444C22×C4D4×D5312yes—
445C22×C4#1171994712yes—
446C22×C4C2×C22⋊D5312yes—
447C22×C4C23×D5311yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—1062no—
453C2×C8—1062no—
454C2×C8—1062no—
455C2×C8—2124no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—534no—
459C4⋊C4—534no—
460C4⋊C4—10616no—
461C4⋊C4—1064no—
462C4⋊C4—1064no—
463C4⋊C4—1062no—
464C4⋊C4—1062no—
465C4⋊C4C2×C22⋊D5314yes—
466C22⋊C4—10616no—
467C22⋊C4—10616no—
468C42—10616no—
469C23—534no—
470C23—1068no—
471C23—1062no—
472C23—1062no—
473C23—1068no—
474C23—1068no—
475C23—1062no—
476C23—1062no—
477C23—1062no—
478C23—1062no—
479C23—2124no—
480C23—2124no—
481C23—2124no—
482C23—2124no—
483C23C22⋊C4×D5312yes—
484C23#1313318512yes—
485C23#1313319912yes—
486C23#1313323812yes—
487C23#1313324012yes—
488C23#1313324612yes—
489C23#1313324912yes—
490C23C2×D4×D5311yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—42no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—534no—
505C2×C4—1068no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—10616no—
509C2×C4—1064no—
510C2×C4—1064no—
511C2×C4—1064no—
512C2×C4—1064no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—2128no—
516C2×C4—2128no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4C22⋊C4×D5312yes—
522C2×C4#1313317714yes—
523C2×C4C4×C22⋊D5314yes—
524C2×C4#1313318512yes—
525C2×C4#1313318814yes—
526C2×C4#1313319014yes—
527C2×C4#1313319912yes—
528C2×C4C22×C4×D5311yescharacteristic
529C2×C4C2×D4×D5311yescharacteristic
530C2×C4C2×D4×D5311yescharacteristic
531C2×C4C22×C22⋊D5311yescharacteristic
532C2×C4C2×#1171994711yescharacteristic
533C2×C4C2×#1171994511yescharacteristic
534C8—42no—
535C8—2122no—
536C22—1064no—
537C22—1064no—
538C22—1064no—
539C22—2122no—
540C22—2122no—
541C22—21216no—
542C22—2122no—
543C22—2122no—
544C22#1681049914yes—
545C22#1681061814yes—
546C22#1681063014yes—
547C22#1681078214yes—
548C22#1681078414yes—
549C22#1681096214yes—
550C22#1681096314yes—
551C22C2×C22⋊C4×D5311yescharacteristic
552C22C2×#1313318511yescharacteristic
553C22C2×#1313319911yescharacteristic
554C22C2×#1313323811yescharacteristic
555C22C2×#1313324011yescharacteristic
556C22C2×#1313324611yescharacteristic
557C22C2×#1313324911yescharacteristic
558C4—44no—
559C4—44no—
560C4—1064no—
561C4—2124no—
562C4—2124no—
563C4#1681049914yes—
564C4C2×#1313317711yescharacteristic
565C4C2×C4×C22⋊D5311yescharacteristic
566C4C2×#1313318811yescharacteristic
567C4C2×#1313319011yescharacteristic
568C2—2124no—
569C2#2027502418yesdirect product factor
570C2C2×#1681049911yescharacteristic
571C2C2×#1681061811yescharacteristic
572C2C2×#1681063011yescharacteristic
573C2C2×#1681078211yescharacteristic
574C2C2×#1681078411yescharacteristic
575C2C2×#1681096211yescharacteristic
576C2C2×#1681096311yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#103×C53C2
socleC24×C53C2×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC22×#103D53
hypocenterC53C2×#1873
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#13133141C2semidirect
5. #20275024C2direct (N × H)
6. C2×#16810386C2semidirect
7. C2×#16810385C2semidirect
8. C22×#103×C53C2semidirect
13. C2×#13133193C4semidirect
26. C2×#13133141C22semidirect
29. #16810386C22semidirect
30. #16810385C22semidirect
35. C2×#103×C53C22semidirect
54. #13133193C2×C4semidirect
283. C22×#103D53semidirect
288. C2×C53#1873semidirect
319. C2×#103C2×D53semidirect
324. C53C2×#1873semidirect
569. C2#20275024direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331932
C22×#131331752
C22×#131331412
C22×#131331292
#202750242
C2×#168103862
C2×#168103852
C22×#103×C532
C2×#187353