Finite Group Databasebeta

← C4×#16810559 #27305921

Subgroups — C4×#16810559 #27305921

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2C2×C4×#11719919C212yesmaximal, complemented
3C2×C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Dic53C212yesmaximal
5#20272984C211yesmaximal, characteristic, complemented
6#20272981C212yesmaximal, complemented
7#20272979C211yesmaximal, characteristic, complemented
8#20272960C211yesmaximal, characteristic
9#20272958C212yesmaximal
10C4×#13133124C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C4×Dic53C2212yes—
19C22×C4×Dic53C2212yes—
20#16810559C418yesdirect product factor
21C4×#11719919—28no—
22C4×C4⋊Dic53—24no—
23C4×C22⋊Dic53—28no—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C24×C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C42×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C4×C22⋊C4×C53—24no—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—24no—
48C2×C4⋊Dic53C2×C414yescomplemented
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919C2×C418yescomplemented
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—24no—
57C2×C22⋊Dic53—24no—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53C2×C418yes—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—24no—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53D418yes—
67#13133124—24no—
68#13133124—22no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124C2×C414yescomplemented
75C25×C53C2311yescharacteristic
76C23×C4×C53—22no—
77C23×C4×C53—24no—
78C23×C4×C53—22no—
79C23×C4×C53—24no—
80C23×C4×C53Q814yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—22no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53C2×C414yes—
93C2×C42×C53—22no—
94C2×C42×C53—24no—
95C2×C42×C53—24no—
96C2×C42×C53D414yes—
97#95×C53—22no—
98#95×C53—22no—
99C24×C53—21no—
100C24×C53—22no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—22no—
104C24×C53—21no—
105C24×C53—22no—
106C24×C53—22no—
107C24×C53C22×C412yes—
108C24×C53C2×Q812yes—
109C24×C53#4112yes—
110C24×C53C2411yescharacteristic
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—216no—
130C22×C4×C53—28no—
131C22×C4×C53—44no—
132C22×C4×C53—48no—
133C22×C4×C53—44no—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D414yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×Q812yes—
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53#4112yes—
144C22×C4×C53#4112yes—
145C22×C4×C53#4114yes—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53C22×C414yes—
169C22×Dic53C22×C414yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D414yes—
172C22×Dic53#4114yes—
173C22×Dic53#4114yes—
174C22⋊C4×C53—28no—
175C22⋊C4×C53—28no—
176C42×C53—28no—
177C42×C53—44no—
178C22⋊Dic53—216no—
179C22⋊Dic53—216no—
180C4⋊Dic53—28no—
181C4⋊Dic53—28no—
182#11719919—216no—
183#11719919—216no—
184C4×Dic53—216no—
185C4×Dic53—48no—
186C23×C53—24no—
187C23×C53—22no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—24no—
191C23×C53—22no—
192C23×C53—24no—
193C23×C53—22no—
194C23×C53—24no—
195C23×C53—22no—
196C23×C53—24no—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—42no—
208C23×C53C4×Q814yes—
209C23×C53C23×C411yescharacteristic
210C23×C53C22×D412yes—
211C23×C53C22×D411yescharacteristic
212C23×C53C22×Q811yescharacteristic
213C23×C53C2×#4112yes—
214C23×C53C2×#4112yes—
215C23×C53C2×#4112yes—
216C23×C53C2×#4111yescharacteristic
217C23×C53C2×#4111yescharacteristic
218C23×C53C2×#4111yescharacteristic
219C23×C53C2×#4111yescharacteristic
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—416no—
245C2×C4×C53C4×D414yes—
246C2×C4×C53C22⋊Q818yes—
247C2×C4×C53C22⋊Q814yes—
248C2×C4×C53#12314yes—
249C2×C4×C53#12318yes—
250C2×C4×C53#12714yes—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53C4×D418yes—
279C4×#399—531nomaximal, Sylow
280C22×C53—24no—
281C22×C53—28no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—44no—
286C22×C53—42no—
287C22×C53—44no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—44no—
291C22×C53C2×C4×D412yes—
292C22×C53C2×C4×D411yescharacteristic
293C22×C53C2×C4×Q811yescharacteristic
294C22×C53C4×#4112yes—
295C22×C53C4×#4111yescharacteristic
296C22×C53C2×C22⋊Q812yes—
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53C2×#12311yescharacteristic
299C22×C53C2×#12312yes—
300C22×C53#52511yescharacteristic
301C22×C53#52512yes—
302C22×C53#52512yes—
303C22×C53#52511yescharacteristic
304C22×C53#52912yes—
305C22×C53#52912yes—
306C22×C53#52912yes—
307C22×C53#52912yes—
308C22×C53#52911yescharacteristic
309C22×C53C2×#12711yescharacteristic
310C22×C53#53211yescharacteristic
311C22×C53#53212yes—
312C22×C53#53312yes—
313C22×C53#53311yescharacteristic
314C4×C53—24no—
315C4×C53—42no—
316C4×C53—42no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53#39918yescomplemented
320Dic53—28no—
321Dic53—44no—
322Dic53—44no—
323Dic53—44no—
324C2×#399—531no—
325C2×C4×C4⋊C4—531no—
326C2×C4×C4⋊C4—532no—
327C2×C4×C22⋊C4—532no—
328C2×C4×C22⋊C4D5311yescharacteristic
329#1445—532no—
330#1445—531no—
331#1444—531no—
332#1439—532no—
333#1439—531no—
334C4×#95—531no—
335C2×C53—44no—
336C2×C53C4×#12311yescharacteristic
337C2×C53C4×#12312yes—
338C2×C53C4×C22⋊Q811yescharacteristic
339C2×C53C4×C22⋊Q812yes—
340C2×C53C4×#12711yescharacteristic
341C2×C53C2×#39911yescharacteristic
342C2×C53#239511yescharacteristic
343C2×C53#239512yes—
344C2×C53#239611yescharacteristic
345C2×C53#239612yes—
346C2×C53#239711yescharacteristic
347C24×C4C2×D5311yescharacteristic
348C22×C4⋊C4—532no—
349C22×C4⋊C4—531no—
350C22×C4⋊C4—532no—
351C22×C4⋊C4—531no—
352C22×C22⋊C4—532no—
353C22×C22⋊C4—532no—
354C22×C22⋊C4C2×D5311yescharacteristic
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C42—532no—
357C22×C42—532no—
358C22×C42C2×D5311yescharacteristic
359C22×C42C2×D5311yescharacteristic
360#399—538no—
361C4×C4⋊C4—1068no—
362C4×C4⋊C4—1064no—
363C4×C22⋊C4—24no—
364C4×C22⋊C4—1068no—
365C2×#95—531no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95C2×D5311yescharacteristic
374C2×#95C2×D5311yescharacteristic
375C53C4×#39911yescharacteristic, complemented, Sylow
376C25C22×D5311yescharacteristic
377C23×C4—22no—
378C23×C4—24no—
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—532no—
382C23×C4—532no—
383C23×C4—532no—
384C23×C4—532no—
385C23×C4Dic10614yes—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5311yescharacteristic
392C2×C4⋊C4—538no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—538no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1064no—
413C2×C22⋊C4—1064no—
414C2×C22⋊C4C4×D5314yes—
415C2×C42—24no—
416C2×C42—24no—
417C2×C42—22no—
418C2×C42—538no—
419C2×C42—1064no—
420C2×C42—1064no—
421C2×C42—1064no—
422C2×C42—1064no—
423C2×C42—1064no—
424C2×C42C22⋊D5314yes—
425#95—22no—
426#95—22no—
427#95—534no—
428#95—1064no—
429#95—1064no—
430#95—1064no—
431#95—1064no—
432#95—1064no—
433#95—1062no—
434#95—1064no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—21no—
443C24C2×Dic10612yes—
444C24C2×C4×D5312yes—
445C24#1171994512yes—
446C24C23×D5311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—534no—
474C22×C4—534no—
475C22×C4—534no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4C2×Dic10612yes—
499C22×C4C2×Dic10612yes—
500C22×C4C2×C4×D5312yes—
501C22×C4C2×C4×D5312yes—
502C22×C4#1171994512yes—
503C22×C4#1171994512yes—
504C22×C4#1171994512yes—
505C22×C4D4×D5314yes—
506C22×C4#1171994714yes—
507C22×C4C2×C22⋊D5312yes—
508C22×C4C2×C22⋊D5312yes—
509C22×C4C2×C22⋊D5312yes—
510C4⋊C4—10616no—
511C4⋊C4—10616no—
512C4⋊C4—1068no—
513C4⋊C4—1068no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—10616no—
517C22⋊C4—10616no—
518C42—28no—
519C42—44no—
520C42—10616no—
521C42—2128no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23C4×Dic10614yes—
545C23C22×C4×D5311yescharacteristic
546C23C2×D4×D5312yes—
547C23C22×C22⋊D5311yescharacteristic
548C23C22×Dic10611yescharacteristic
549C23#41×D5312yes—
550C23#41×D5312yes—
551C23C2×#1171994712yes—
552C23C2×#1171994511yescharacteristic
553C23C2×#1171994511yescharacteristic
554C23C2×#1171994511yescharacteristic
555C23C2×#1171994511yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—416no—
581C2×C4—538no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4C4×C22⋊D5314yes—
610C2×C4#1313319418yes—
611C2×C4#1313319614yes—
612C2×C4#1313319814yes—
613C2×C4#1313319918yes—
614C2×C4#1313321314yes—
615C22—24no—
616C22—28no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22C4×D4×D5312yes—
627C22C2×C4×C22⋊D5311yescharacteristic
628C22C2×C4×Dic10611yescharacteristic
629C22C4×#1171994712yes—
630C22C4×#1171994511yescharacteristic
631C22C2×#1313319412yes—
632C22C2×#1313319611yescharacteristic
633C22C2×#1313319811yescharacteristic
634C22C2×#1313319912yes—
635C22#1681113011yescharacteristic
636C22#1681113212yes—
637C22#1681113212yes—
638C22#1681113411yescharacteristic
639C22#1681115112yes—
640C22#1681115312yes—
641C22#1681115312yes—
642C22#1681115512yes—
643C22#1681115711yescharacteristic
644C22C2×#1313321311yescharacteristic
645C22#1681116911yescharacteristic
646C22#1681117012yes—
647C22#1681117612yes—
648C22#1681118011yescharacteristic
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—1068no—
655C4—2124no—
656C4—2124no—
657C4—2124no—
658C4#1681055918yesdirect product factor
659C2—44no—
660C2C4×#1313319811yescharacteristic
661C2C4×#1313319912yes—
662C2C4×#1313319412yes—
663C2C4×#1313319611yescharacteristic
664C2C4×#1313321311yescharacteristic
665C2C2×#1681055911yescharacteristic
666C2#2027797411yescharacteristic
667C2#2027797612yes—
668C2#2027798311yescharacteristic
669C2#2027798912yes—
670C2#2027799411yescharacteristic
subgroupG/H
centerC23×C4C22×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53C4×#399
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719919C2semidirect
3. C2×C4×C4⋊Dic53C2semidirect
5. #20272984C2semidirect
6. #20272981C2semidirect
7. #20272979C2semidirect
10. C4×#13133124C2semidirect
20. #16810559C4direct (N × H)
48. C2×C4⋊Dic53C2×C4semidirect
54. C2×#11719919C2×C4semidirect
74. #13133124C2×C4semidirect
319. C4×C53#399semidirect
375. C53C4×#399semidirect
658. C4#16810559direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105592
C2×C4×#117199192
C2×C4×C4⋊Dic532
C2×C4×C22⋊Dic532
#202729842
#202729812
#202729792
#202729602
#202729582
C4×#131331242
C2×C4×C22⋊C4×C532
C4×#39953