Finite Group Databasebeta

← C4×#16810847 #27390331

Subgroups — C4×#16810847 #27390331

Order 13568.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810847C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic
3C4×#13133206C211yesmaximal, characteristic
4C4×Q8×Dic53C211yesmaximal, characteristic, complemented
5#20275644C211yesmaximal, characteristic
6#20275480C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20275334C211yesmaximal, characteristic
9#20275068C211yesmaximal, characteristic
10#20275058C211yesmaximal, characteristic, complemented
11#20274728C211yesmaximal, characteristic, complemented
12C4×#13133146C211yesmaximal, characteristic, complemented
13C4×#13133141C211yesmaximal, characteristic, complemented
14C4×#13133139C211yesmaximal, characteristic
15C2×C4×Q16×C53C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×#13133206C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×C4×D212C2211yescharacteristic
21#16810847C418yesdirect product factor
22C4×#11719924—24no—
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133139C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810524C2211yescharacteristic
35#16810513—21no—
36#16810513—21no—
37#16810488C2211yescharacteristic
38C4⋊C4×Dic53—21no—
39C4⋊C4×Dic53—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C4⋊Dic53—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×#11719921—21no—
45C4×#11719921—21no—
46C42×Dic53—21no—
47C42×Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×Q16×C53C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×C4×Q8×C53C2211yescharacteristic
57C4×Q16×C53—24no—
58C2×#107×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C2×C4×C8×C53C2211yescharacteristic
62C22×D212C2311yescharacteristic
63C22×C4×D53—21no—
64C2×#11719924—22no—
65C2×#11719924—22no—
66C2×#11719924C2×C414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133206—22no—
70#13133206C2×C414yes—
71Q8×Dic53—22no—
72Q8×Dic53—22no—
73Q8×Dic53—22no—
74Q8×Dic53C2×C414yescomplemented
75C4×D212—22no—
76C4×D212—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98#13133148—22no—
99#13133146—22no—
100#13133146C2×C414yescomplemented
101#13133141—22no—
102#13133141—22no—
103#13133141C2×C414yescomplemented
104#13133139—22no—
105#13133139C2×C414yes—
106#13133137—22no—
107C4×#88322—22no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115C22×Q8×C53C2311yescharacteristic
116C22×Q8×C53C2311yescharacteristic
117C2×Q16×C53—22no—
118C2×Q16×C53—22no—
119C2×Q16×C53C2×C414yescomplemented
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C4×Q8×C53—22no—
123C4×Q8×C53—22no—
124C4×Q8×C53—22no—
125C4×Q8×C53D414yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C42×C53—21no—
131C2×C42×C53—21no—
132C2×C42×C53C2311yescharacteristic
133#107×C53—22no—
134#103×C53—22no—
135#103×C53—22no—
136C4×C8×C53—22no—
137C23×D53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C22×C412yes—
147C2×Q8×C53C22×C412yes—
148C2×Q8×C53C2×D412yes—
149C2×Q8×C53#4112yes—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C2×D212—21no—
165C2×D212—21no—
166C2×D212—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212C22×C412yes—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173Q16×C53—28no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53C22×C412yes—
180C4⋊C4×C53—24no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—42no—
185C4⋊C4×C53C2×D412yes—
186C42×C53—24no—
187C42×C53—42no—
188C42×C53C2×D412yes—
189C42×C53#4112yes—
190#11719924—28no—
191#11719921—24no—
192#11719921—24no—
193#11719921—42no—
194#11719921—42no—
195#11719921—42no—
196C4⋊Dic53—24no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—24no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53C22×C412yes—
203C4⋊Dic53#4112yes—
204C4×Dic53—24no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—24no—
210C4×Dic53—22no—
211C4×Dic53—41no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215C4×Dic53C22×C412yes—
216C4×Dic53#4112yes—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322C22×C412yes—
223C23×C53C22×D411yescharacteristic
224C22×D53—22no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—41no—
231Q8×C53—22no—
232Q8×C53—22no—
233Q8×C53—22no—
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238Q8×C53C4×D414yes—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53#12112yes—
254C2×C4×C53#12112yes—
255C2×C4×C53#13512yes—
256C2×C4×C53#13712yes—
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53C4×D412yes—
282C2×Dic53C4×D412yes—
283C2×Dic53#13512yes—
284D212—22no—
285D212—22no—
286D212—41no—
287D212—41no—
288C4×D53—84no—
289C8×C53—22no—
290C8×C53—41no—
291#88322—22no—
292#88322—41no—
293C4×#464—531nomaximal, Sylow
294C22×C53C2×C4×D411yescharacteristic
295C22×C53C2×#12111yescharacteristic
296C22×C53#52511yescharacteristic
297C22×C53C2×#13511yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C2×D53—81no—
308C4×C53—22no—
309C4×C53—24no—
310C4×C53—42no—
311C4×C53—42no—
312C4×C53—81no—
313C4×C53#46414yescomplemented
314C4×C53C2×C4×D411yescharacteristic
315C4×C53C4×#4111yescharacteristic
316C4×C53#52911yescharacteristic
317C4×C53#53211yescharacteristic
318Dic53—22no—
319Dic53—22no—
320Dic53—22no—
321Dic53—41no—
322Dic53—42no—
323Dic53—42no—
324Dic53—81no—
325C2×#464—531no—
326C2×C4×Q16D5311yescharacteristic, complemented
327C2×C4×SD16—531no—
328C4×#124—531no—
329C42×Q8—531no—
330#1983—531no—
331#1949—531no—
332#1927—531no—
333#1925—531no—
334#1880—531no—
335#1878—531no—
336#1805—531no—
337C4×C4⋊C8—531no—
338C4×#103—531no—
339C4×#102—531no—
340C2×C53C4×#12111yescharacteristic
341C2×C53C4×#13511yescharacteristic
342C2×C53C4×#13711yescharacteristic
343C2×C53C2×#46411yescharacteristic
344C2×C53#303511yescharacteristic
345C2×C53#310211yescharacteristic
346C2×C53#310811yescharacteristic
347D53—82no—
348C22×Q16C2×D5311yescharacteristic
349C22×SD16—531no—
350C2×#124—531no—
351C2×C4×Q8—531no—
352C2×C4×Q8—531no—
353C2×C4×Q8C2×D5311yescharacteristic
354C2×C4×Q8C2×D5311yescharacteristic
355C2×C4×D4—531no—
356#464—538no—
357C4×Q16—24no—
358C4×SD16—1064no—
359C2×#107C2×D5311yescharacteristic
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#103—531no—
366C2×#103C2×D5311yescharacteristic
367C2×#103C2×D5311yescharacteristic
368C2×#102—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×C4×C8—531no—
372C2×C4×C8C2×D5311yescharacteristic
373#391—531no—
374#390—531no—
375#388—1061no—
376#388—1061no—
377#382—531no—
378C4×C4⋊C4—531no—
379C4×C4⋊C4—531no—
380C4×C4⋊C4—531no—
381C4×C4⋊C4—1061no—
382C4×C4⋊C4—1061no—
383C4×C4⋊C4—1061no—
384C4×C22⋊C4—1061no—
385C4×C22⋊C4—1061no—
386C43—531no—
387C43—1061no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394C53C4×#46411yescharacteristic, complemented, Sylow
395C22×Q8C22×D5311yescharacteristic
396C22×Q8C22×D5311yescharacteristic
397C22×D4—531no—
398C23×C4—1061no—
399C2×Q16—22no—
400C2×Q16—22no—
401C2×Q16C4×D5314yescomplemented
402C2×SD16—534no—
403C2×SD16—1062no—
404C2×SD16—1062no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409#124—534no—
410#124—1062no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—534no—
415C4×Q8—1062no—
416C4×Q8—1062no—
417C4×Q8—1062no—
418C4×Q8C22⋊D5314yes—
419C4×D4—1062no—
420C4×D4—1062no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42C22×D5311yescharacteristic
449#107—22no—
450#106—1062no—
451C4⋊C8—534no—
452C4⋊C8—1062no—
453#103—22no—
454#103—22no—
455#103—534no—
456#103—1062no—
457#103—1062no—
458#102—534no—
459#102—1062no—
460#102—1062no—
461C4×C8—22no—
462C4×C8—1062no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470C24—1061no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8C2×C4×D5312yes—
480C2×Q8C2×C4×D5312yes—
481C2×Q8#1171994512yes—
482C2×Q8C2×C22⋊D5312yes—
483C2×D4—532no—
484C2×D4—1061no—
485C2×D4—1061no—
486C2×D4—1061no—
487C2×D4—1061no—
488C2×D4—1061no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—2122no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4D4×D5311yescharacteristic
504C22×C4D4×D5311yescharacteristic
505C22×C4C23×D5311yescharacteristic
506Q16—28no—
507SD16—1068no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—532no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8C2×C4×D5312yes—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—532no—
526C4⋊C4—532no—
527C4⋊C4—1064no—
528C4⋊C4—1064no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4C2×C22⋊D5312yes—
534C22⋊C4—1064no—
535C22⋊C4—1064no—
536C22⋊C4—2122no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C42—24no—
540C42—42no—
541C42—532no—
542C42—532no—
543C42—1062no—
544C42—1064no—
545C42—1064no—
546C42—1062no—
547C42—1062no—
548C42—1061no—
549C42—1061no—
550C42—2121no—
551C42—2122no—
552C42—2122no—
553C42—2122no—
554C42#1171994712yes—
555C42C2×C22⋊D5312yes—
556C23—1062no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23C2×D4×D5311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8C4×C22⋊D5314yes—
572D4—1062no—
573D4—1062no—
574D4—2121no—
575D4—2121no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—4244no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313318912yes—
613C2×C4#1313318912yes—
614C2×C4#1313324012yes—
615C2×C4#1313324912yes—
616C2×C4C22×C4×D5311yescharacteristic
617C2×C4C22×C22⋊D5311yescharacteristic
618C2×C4#41×D5311yescharacteristic
619C2×C4#41×D5311yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4C2×#1171994511yescharacteristic
622C8—22no—
623C8—41no—
624C8—1062no—
625C8—2121no—
626C22—4241no—
627C22—4241no—
628C22—4241no—
629C22—4241no—
630C22—2122no—
631C22—2122no—
632C22—2122no—
633C22C4×D4×D5311yescharacteristic
634C22C2×#1313318911yescharacteristic
635C22#1681113211yescharacteristic
636C22#135×D5311yescharacteristic
637C22C2×#1313324011yescharacteristic
638C22C2×#1313324911yescharacteristic
639C22#1681142011yescharacteristic
640C4—22no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—81no—
645C4—1062no—
646C4—1062no—
647C4—1062no—
648C4—4241no—
649C4—2121no—
650C4—2122no—
651C4—2122no—
652C4#1681084714yesdirect product factor
653C4C2×C4×C22⋊D5311yescharacteristic
654C4C4×#1171994711yescharacteristic
655C4#1681115311yescharacteristic
656C4#1681116911yescharacteristic
657C2—4242no—
658C2C4×#1313318911yescharacteristic
659C2C4×#1313324011yescharacteristic
660C2C4×#1313324911yescharacteristic
661C2C2×#1681084711yescharacteristic
662C2#2028416611yescharacteristic
663C2#2028473911yescharacteristic
664C2#2028479911yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×Q16×C53C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenterC2×C4×Q16D53
hypocenterC53C4×#464
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
4. C4×Q8×Dic53C2semidirect
6. #20275480C2semidirect
7. #20275352C2semidirect
10. #20275058C2semidirect
11. #20274728C2semidirect
12. C4×#13133146C2semidirect
13. C4×#13133141C2semidirect
15. C2×C4×Q16×C53C2semidirect
21. #16810847C4direct (N × H)
74. Q8×Dic53C2×C4semidirect
100. #13133146C2×C4semidirect
103. #13133141C2×C4semidirect
119. C2×Q16×C53C2×C4semidirect
313. C4×C53#464semidirect
326. C2×C4×Q16D53semidirect
394. C53C4×#464semidirect
401. C2×Q16C4×D53semidirect
652. C4#16810847direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168108472
C2×C4×#117199242
C4×#131332062
C4×Q8×Dic532
#202756442
#202754802
#202753522
#202753342
#202750682
#202750582
#202747282
C4×#131331462
C4×#131331412
C4×#131331392
C2×C4×Q16×C532
C4×#46453