Finite Group Databasebeta

← C2×#20275152 #27385779

Subgroups — C2×#20275152 #27385779

Order 13568.

2310 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719912C211yesmaximal, characteristic, complemented
2C22×#13133192C211yesmaximal, characteristic
3C22×#13133139C211yesmaximal, characteristic, complemented
4C22×#13133137C211yesmaximal, characteristic, complemented
5C2×#16810480C211yesmaximal, characteristic, complemented
6#20275152C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C2×C53×#340C211yesmaximal, characteristic, complemented
9C23×D212—21no—
10C23×D212C2211yescharacteristic
11C23×C4×D53C2211yescharacteristic
12C22×#11719912—24no—
13C22×#11719912—24no—
14C22×#11719912—24no—
15C22×#11719912C2214yescomplemented
16C23×#88322C2211yescharacteristic
17C2×#13133192—24no—
18C2×#13133192—22no—
19C2×#13133192C418yescomplemented
20C2×#13133192C2214yes—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719921—21no—
23C22×#11719921—21no—
24C2×#13133139—24no—
25C2×#13133139C2214yescomplemented
26C2×#13133137—24no—
27C2×#13133137C2214yescomplemented
28#16810480—22no—
29#16810480C2214yescomplemented
30C2×#13133124—21no—
31#16810385C2214yescomplemented
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#340C2214yescomplemented
36C24×D53—21no—
37C24×D53D411yescharacteristic
38C22×D212—22no—
39C22×D212—22no—
40C22×D212—22no—
41C22×D212—22no—
42C22×D212—28no—
43C22×D212—24no—
44C22×D212—21no—
45C22×D212—22no—
46C22×D212—44no—
47C22×D212—44no—
48C22×D212C2×C414yes—
49C22×D212C2312yes—
50C23×Dic53—21no—
51C23×Dic53D411yescharacteristic
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53C2×C414yes—
57C22×C4×D53D412yes—
58C22×C4×D53D414yes—
59C22×C4×D53C2312yes—
60C2×#11719912—232no—
61C2×#11719912—28no—
62C2×#11719912—28no—
63C2×#11719912—28no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322D414yescomplemented
67C22×#88322C2312yes—
68#13133192—216no—
69#13133192—28no—
70#13133192C2×C4116yescomplemented
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53C2312yes—
75C2×#11719921—24no—
76C2×#11719921—28no—
77C2×#11719921—24no—
78C2×#11719921—44no—
79#13133139—28no—
80#13133137—28no—
81#13133124—24no—
82C23×C4×C53—21no—
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85C22×C8×C53—22no—
86C22×C8×C53—22no—
87C22×C8×C53D414yescomplemented
88C22×C8×C53C2312yes—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53C2×C414yes—
95C2×C4⋊C4×C53C2312yes—
96C2×C4⋊C4×C53C2312yes—
97C24×C53C2×D411yescharacteristic
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—24no—
105C23×D53—22no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C23×D53C22⋊C414yes—
113C23×D53C2×D412yes—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53—42no—
136C22×Dic53C22⋊C414yes—
137C22×Dic53C2×D412yes—
138C2×D212—28no—
139C2×D212—28no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—22no—
143C2×D212—22no—
144C2×D212—216no—
145C2×D212—24no—
146C2×D212—22no—
147C2×D212—48no—
148C2×D212—48no—
149C2×D212—432no—
150C2×D212—44no—
151C2×D212—42no—
152C2×D212—42no—
153C2×D212—44no—
154C2×D212C22×C414yes—
155C2×C4×D53—28no—
156C2×C4×D53—28no—
157C2×C4×D53—28no—
158C2×C4×D53—24no—
159C2×C4×D53—24no—
160C2×C4×D53—28no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53C22⋊C418yes—
166C2×C4×D53C22×C414yes—
167C2×C4×D53C2×D414yes—
168C2×C8×C53—28no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53C2×D414yescomplemented
172C4⋊C4×C53—24no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53C22×C414yes—
178#11719921—216no—
179#11719921—48no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—42no—
183C2×#88322—28no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322C2×D414yescomplemented
187#11719912—232no—
188C23×C53C2×C22⋊C412yes—
189C23×C53#12112yes—
190C23×C53#12112yes—
191C23×C53AGL(1,ℤ/8)12yes—
192C23×C53AGL(1,ℤ/8)12yes—
193C23×C53AGL(1,ℤ/8)12yes—
194C23×C53#13712yes—
195C23×C53C22×D411yescharacteristic
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—28no—
203C22×D53—28no—
204C22×D53—28no—
205C22×D53—24no—
206C22×D53—44no—
207C22×D53—44no—
208C22×D53—44no—
209C22×D53—44no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—48no—
217C22×D53—48no—
218C22×D53—48no—
219C22×D53—84no—
220C22×D53—84no—
221C22×D53—84no—
222C22×D53—84no—
223C22×D53C2×C22⋊C414yes—
224C2×C4×C53—24no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—48no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—84no—
232C2×C4×C53C2×C22⋊C412yes—
233C2×C4×C53C4×D414yes—
234C2×C4×C53C4×D414yes—
235C2×C4×C53#12014yes—
236C2×C4×C53#12112yes—
237C2×C4×C53#12112yes—
238C2×C4×C53#12314yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—28no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—84no—
252C2×Dic53C2×C22⋊C414yes—
253D212—24no—
254D212—24no—
255D212—432no—
256D212—42no—
257D212—42no—
258D212—48no—
259C4×D53—24no—
260C4×D53—24no—
261C4×D53—24no—
262C4×D53—24no—
263C4×D53—216no—
264C8×C53—24no—
265#88322—24no—
266C2×#1895—531nomaximal, Sylow
267C22×C53#38614yes—
268C22×C53#41714yes—
269C22×C53#41814yes—
270C22×C53#46814yes—
271C22×C53#46814yes—
272C22×C53#46914yes—
273C22×C53#46914yes—
274C22×C53C22×C22⋊C411yescharacteristic
275C22×C53C2×#12111yescharacteristic
276C22×C53C2×#12111yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C2×D53—24no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53—416no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—816no—
293C2×D53—82no—
294C2×D53—82no—
295C2×D53—82no—
296C2×D53—82no—
297C4×C53—44no—
298C4×C53—82no—
299C4×C53#38614yes—
300C4×C53C2×C4×D411yescharacteristic
301C4×C53C2×C4×D411yescharacteristic
302C4×C53C2×#12011yescharacteristic
303C4×C53C2×#12311yescharacteristic
304Dic53—24no—
305Dic53—82no—
306C23×M16—531no—
307C22×#121—531no—
308C22×#102—531no—
309C22×#102—531no—
310C2×#380—531no—
311#1895—538no—
312C2×#340—531no—
313C2×#340D5311yescharacteristic, complemented
314C2×C53#189518yescomplemented
315C2×C53C2×#38611yescharacteristic
316C2×C53C2×#41811yescharacteristic
317C2×C53C2×#41711yescharacteristic
318C2×C53C2×#46811yescharacteristic
319C2×C53C2×#46811yescharacteristic
320C2×C53C2×#46911yescharacteristic
321C2×C53C2×#46911yescharacteristic
322D53—44no—
323D53—84no—
324C23×D4—531no—
325C23×D4—1061no—
326C24×C4—531no—
327C22×M16—534no—
328C22×M16—1064no—
329C22×M16—1064no—
330C22×M16—1064no—
331C23×C8—531no—
332C23×C8C2×D5311yescharacteristic
333C2×#121—538no—
334C2×#121—534no—
335C2×#121—1064no—
336C2×#121—1062no—
337C22×C4⋊C4—531no—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C4⋊C4C2×D5311yescharacteristic
340C22×C22⋊C4—1061no—
341C22×C22⋊C4—1061no—
342C2×#102—534no—
343C2×#102—534no—
344C2×#102—1064no—
345C2×#102—1064no—
346#380—534no—
347#380—1062no—
348C2×#95—1061no—
349#340—534no—
350#340C2×D5314yescomplemented
351C53C2×#189511yescharacteristic, complemented, Sylow
352C25—531no—
353C25—1061no—
354C22×D4—534no—
355C22×D4—532no—
356C22×D4—1068no—
357C22×D4—1064no—
358C22×D4—1062no—
359C22×D4—1062no—
360C22×D4—1062no—
361C22×D4—1061no—
362C22×D4—1062no—
363C22×D4—1062no—
364C22×D4—2124no—
365C22×D4—2124no—
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—531no—
369C23×C4—534no—
370C23×C4—534no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—1062no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×M16—1068no—
380C2×M16—1068no—
381C2×M16—10632no—
382C2×M16—1068no—
383C22×C8—22no—
384C22×C8—22no—
385C22×C8—534no—
386C22×C8—532no—
387C22×C8—1062no—
388C22×C8—1062no—
389C22×C8D21214yescomplemented
390C22×C8C22×D5312yes—
391#121—5316no—
392#121—10616no—
393#121—1068no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—532no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C4×D5314yes—
404C2×C4⋊C4C22×D5312yes—
405C2×C4⋊C4C22×D5312yes—
406C2×C22⋊C4—1068no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—2124no—
410#102—1068no—
411#102—1068no—
412#95—1064no—
413C24—534no—
414C24—532no—
415C24—1062no—
416C24—1062no—
417C24—1062no—
418C24—1062no—
419C24—1062no—
420C24—1062no—
421C24—1064no—
422C24—1062no—
423C24—2122no—
424C24—2122no—
425C24—2122no—
426C24—2122no—
427C24—2122no—
428C24—2122no—
429C24D4×D5311yescharacteristic
430C2×D4—534no—
431C2×D4—10616no—
432C2×D4—1064no—
433C2×D4—1068no—
434C2×D4—1068no—
435C2×D4—1062no—
436C2×D4—1062no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4—21232no—
441C2×D4—2128no—
442C2×D4—2128no—
443C2×D4—2124no—
444C2×D4—2122no—
445C2×D4—2122no—
446C2×D4—2124no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—538no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1068no—
463C22×C4—1068no—
464C22×C4—1068no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1068no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4C2×C4×D5312yes—
477C22×C4C2×D21212yes—
478C22×C4#1171994512yes—
479C22×C4D4×D5311yescharacteristic
480C22×C4D4×D5312yes—
481C22×C4D4×D5312yes—
482C22×C4#1171994912yes—
483C22×C4C23×D5311yescharacteristic
484M16—10632no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—534no—
489C2×C8—1068no—
490C2×C8—1062no—
491C2×C8—1062no—
492C2×C8C2×D21214yescomplemented
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—2122no—
501C4⋊C4C2×C4×D5314yes—
502C22⋊C4—10616no—
503C22⋊C4—2128no—
504C23—534no—
505C23—1068no—
506C23—1062no—
507C23—1062no—
508C23—1068no—
509C23—1068no—
510C23—1062no—
511C23—1062no—
512C23—1062no—
513C23—1062no—
514C23—1064no—
515C23—4244no—
516C23—4244no—
517C23—4244no—
518C23—4244no—
519C23—2128no—
520C23—2122no—
521C23—2122no—
522C23—2128no—
523C23—2128no—
524C23—2122no—
525C23—2122no—
526C23—2122no—
527C23—2122no—
528C23—2124no—
529C23—2124no—
530C23—2124no—
531C23—2124no—
532C23C22⋊C4×D5312yes—
533C23#1313319012yes—
534C23#1313319212yes—
535C23#1313324112yes—
536C23#1313324412yes—
537C23#1313324412yes—
538C23#1313324912yes—
539C23C2×D4×D5311yescharacteristic
540D4—1064no—
541D4—1064no—
542D4—21232no—
543D4—2128no—
544D4—2122no—
545D4—2122no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—534no—
555C2×C4—1068no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—10616no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1064no—
563C2×C4—4244no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4C22⋊C4×D5312yes—
568C2×C4#1313317914yes—
569C2×C4C4×D21214yes—
570C2×C4#1313318714yes—
571C2×C4#1313319012yes—
572C2×C4#1313319212yes—
573C2×C4#1313320214yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4C2×D4×D5311yescharacteristic
576C2×C4C2×D4×D5311yescharacteristic
577C2×C4C22×D21211yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C2×C4C2×#1171994911yescharacteristic
580C8—24no—
581C8—1064no—
582C22—1064no—
583C22—1064no—
584C22—1064no—
585C22—4242no—
586C22—4242no—
587C22—4242no—
588C22—4242no—
589C22—42416no—
590C22—2124no—
591C22—2124no—
592C22—2124no—
593C22—2122no—
594C22—2122no—
595C22—2122no—
596C22—2122no—
597C22—21216no—
598C22#1681050114yes—
599C22#1681062314yes—
600C22#1681062614yes—
601C22#1681087014yes—
602C22#1681087214yes—
603C22#1681087514yes—
604C22#1681087714yes—
605C22C2×C22⋊C4×D5311yescharacteristic
606C22C2×#1313319011yescharacteristic
607C22C2×#1313319211yescharacteristic
608C22C2×#1313324111yescharacteristic
609C22C2×#1313324411yescharacteristic
610C22C2×#1313324411yescharacteristic
611C22C2×#1313324911yescharacteristic
612C4—44no—
613C4—82no—
614C4—1064no—
615C4—4242no—
616C4#1681050114yes—
617C4C2×#1313317911yescharacteristic
618C4C2×C4×D21211yescharacteristic
619C4C2×#1313318711yescharacteristic
620C4C2×#1313320211yescharacteristic
621C2—4244no—
622C2—2124no—
623C2#2027515218yesdirect product factor
624C2C2×#1681050111yescharacteristic
625C2C2×#1681062611yescharacteristic
626C2C2×#1681062311yescharacteristic
627C2C2×#1681087011yescharacteristic
628C2C2×#1681087211yescharacteristic
629C2C2×#1681087511yescharacteristic
630C2C2×#1681087711yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC24×C53C2×D4
prime radicalC22×#13133192C2
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53C2×#1895
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
1. C23×#11719912C2semidirect
3. C22×#13133139C2semidirect
4. C22×#13133137C2semidirect
5. C2×#16810480C2semidirect
6. #20275152C2direct (N × H)
7. C2×#16810385C2semidirect
8. C2×C53×#340C2semidirect
15. C22×#11719912C22semidirect
19. C2×#13133192C4semidirect
25. C2×#13133139C22semidirect
27. C2×#13133137C22semidirect
29. #16810480C22semidirect
31. #16810385C22semidirect
35. C53×#340C22semidirect
66. C22×#88322D4semidirect
70. #13133192C2×C4semidirect
87. C22×C8×C53D4semidirect
171. C2×C8×C53C2×D4semidirect
186. C2×#88322C2×D4semidirect
313. C2×#340D53semidirect
314. C2×C53#1895semidirect
350. #340C2×D53semidirect
351. C53C2×#1895semidirect
389. C22×C8D212semidirect
492. C2×C8C2×D212semidirect
623. C2#20275152direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199122
C22×#131331922
C22×#131331392
C22×#131331372
C2×#168104802
#202751522
C2×#168103852
C2×C53×#3402
C2×#189553