Finite Group Databasebeta

← C4×#16810558 #27305919

Subgroups — C4×#16810558 #27305919

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×#16810558C211yesmaximal, characteristic
2C2×C4×#11719919C212yesmaximal, complemented
3C2×C4×#11719921C212yesmaximal
4C2×C4×#11719921C211yesmaximal, characteristic
5#20272983C211yesmaximal, characteristic, complemented
6#20272981C212yesmaximal, complemented
7#20272978C211yesmaximal, characteristic, complemented
8#20272959C212yesmaximal
9#20272957C211yesmaximal, characteristic
10C4×#13133124C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719919C2212yes—
14C22×#11719919C2212yes—
15C22×#11719921C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×#11719921C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21#16810558C418yesdirect product factor
22C4×#11719919—28no—
23C4×#11719921—28no—
24C4×#11719921—24no—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C42×C53C2212yes—
37C22×C42×C53C2211yescharacteristic
38C4×C4⋊C4×C53—24no—
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2312yes—
46C22×C4×D53—24no—
47C22×C4×D53—22no—
48C22×C4×D53—24no—
49C22×C4×D53—22no—
50C22×C4×D53Q814yes—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—24no—
56C2×#11719919C2×C418yescomplemented
57C2×#11719921—24no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921—22no—
61C2×#11719921—24no—
62C2×#11719921—22no—
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—22no—
66C2×#11719921—24no—
67C2×#11719921C2×C414yes—
68C2×#11719921C2×C418yes—
69C2×C4×Dic53—22no—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53D418yes—
74#13133124—22no—
75#13133124—22no—
76#13133124—22no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—24no—
81#13133124—24no—
82#13133124C2×C414yescomplemented
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—24no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53C2×C414yescomplemented
93C2×C42×C53—24no—
94C2×C42×C53—24no—
95C2×C42×C53—24no—
96C2×C42×C53—24no—
97C2×C42×C53D414yes—
98#95×C53—24no—
99C24×C53C2411yescharacteristic
100C23×D53—22no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—21no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—21no—
108C23×D53C22×C412yes—
109C23×D53C2×Q812yes—
110C23×D53#4112yes—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53C22×C414yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D414yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×Q812yes—
136C22×C4×C53#4114yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4114yes—
139C22×C4×C53#4112yes—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—21no—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C414yes—
164C22×Dic53C2×D414yes—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×Q812yes—
167C22×Dic53#4112yes—
168C22×Dic53#4114yes—
169C2×C4×D53—28no—
170C2×C4×D53—216no—
171C2×C4×D53—44no—
172C2×C4×D53—48no—
173C2×C4×D53—44no—
174C4⋊C4×C53—28no—
175C4⋊C4×C53—28no—
176C42×C53—28no—
177C42×C53—48no—
178#11719921—216no—
179#11719921—28no—
180#11719921—28no—
181#11719921—216no—
182#11719919—216no—
183#11719919—216no—
184C4×Dic53—216no—
185C4×Dic53—44no—
186C23×C53C23×C411yescharacteristic
187C23×C53C22×D412yes—
188C23×C53C22×D411yescharacteristic
189C23×C53C22×Q811yescharacteristic
190C23×C53C2×#4112yes—
191C23×C53C2×#4111yescharacteristic
192C23×C53C2×#4111yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C23×C53C2×#4112yes—
195C23×C53C2×#4112yes—
196C23×C53C2×#4111yescharacteristic
197C22×D53—24no—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—24no—
203C22×D53—22no—
204C22×D53—24no—
205C22×D53—24no—
206C22×D53—22no—
207C22×D53—24no—
208C22×D53—22no—
209C22×D53—41no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—41no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—42no—
219C22×D53C4×Q814yes—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—22no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53C4×D414yes—
244C2×C4×C53C22⋊Q814yes—
245C2×C4×C53C22⋊Q818yes—
246C2×C4×C53#12314yes—
247C2×C4×C53#12318yes—
248C2×C4×C53#12714yes—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—41no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—41no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53C4×D418yes—
278C4×D53—416no—
279C4×#399—531nomaximal, Sylow
280C22×C53C2×C4×D412yes—
281C22×C53C2×C4×D411yescharacteristic
282C22×C53C2×C4×Q811yescharacteristic
283C22×C53C4×#4112yes—
284C22×C53C4×#4111yescharacteristic
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53C2×C22⋊Q812yes—
287C22×C53C2×#12311yescharacteristic
288C22×C53C2×#12312yes—
289C22×C53#52511yescharacteristic
290C22×C53#52512yes—
291C22×C53#52512yes—
292C22×C53#52511yescharacteristic
293C22×C53#52912yes—
294C22×C53#52912yes—
295C22×C53#52912yes—
296C22×C53#52911yescharacteristic
297C22×C53#52912yes—
298C22×C53C2×#12711yescharacteristic
299C22×C53#53211yescharacteristic
300C22×C53#53212yes—
301C22×C53#53312yes—
302C22×C53#53311yescharacteristic
303C2×D53—28no—
304C2×D53—24no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—44no—
309C2×D53—42no—
310C2×D53—44no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—44no—
314C4×C53—24no—
315C4×C53—44no—
316C4×C53—42no—
317C4×C53—44no—
318C4×C53#39918yescomplemented
319Dic53—28no—
320Dic53—42no—
321Dic53—42no—
322Dic53—44no—
323Dic53—42no—
324C2×#399—531no—
325C2×C4×C4⋊C4—532no—
326C2×C4×C4⋊C4D5311yescharacteristic, complemented
327C2×C4×C22⋊C4—532no—
328C2×C4×C22⋊C4—531no—
329#1445—532no—
330#1445—531no—
331#1444—531no—
332#1439—532no—
333#1439—531no—
334C4×#95—531no—
335C2×C53C4×#12311yescharacteristic
336C2×C53C4×#12312yes—
337C2×C53C4×C22⋊Q812yes—
338C2×C53C4×C22⋊Q811yescharacteristic
339C2×C53C4×#12711yescharacteristic
340C2×C53C2×#39911yescharacteristic
341C2×C53#239511yescharacteristic
342C2×C53#239512yes—
343C2×C53#239611yescharacteristic
344C2×C53#239612yes—
345C2×C53#239711yescharacteristic
346D53—44no—
347C24×C4—531no—
348C22×C4⋊C4—532no—
349C22×C4⋊C4—532no—
350C22×C4⋊C4C2×D5311yescharacteristic
351C22×C4⋊C4C2×D5311yescharacteristic
352C22×C22⋊C4—532no—
353C22×C22⋊C4—531no—
354C22×C22⋊C4—531no—
355C22×C22⋊C4—532no—
356C22×C42—532no—
357C22×C42—531no—
358C22×C42C2×D5312yes—
359C22×C42C2×D5311yescharacteristic
360#399—538no—
361C4×C4⋊C4—24no—
362C4×C4⋊C4—1068no—
363C4×C22⋊C4—1068no—
364C4×C22⋊C4—1064no—
365C2×#95—531no—
366C2×#95—532no—
367C2×#95—531no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—531no—
372C2×#95—532no—
373C2×#95—531no—
374C2×#95C2×D5312yes—
375C53C4×#39911yescharacteristic, complemented, Sylow
376C25—531no—
377C23×C4—532no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—532no—
381C23×C4—531no—
382C23×C4—534no—
383C23×C4—1064no—
384C23×C4—1064no—
385C23×C4—1062no—
386C23×C4—1062no—
387C23×C4C22×D5312yes—
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390C23×C4C22×D5311yescharacteristic
391C23×C4C22×D5311yescharacteristic
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—538no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4C4×D5314yescomplemented
403C2×C22⋊C4—538no—
404C2×C22⋊C4—534no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1064no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1062no—
415C2×C42—24no—
416C2×C42—24no—
417C2×C42—24no—
418C2×C42—24no—
419C2×C42—538no—
420C2×C42—1064no—
421C2×C42—1064no—
422C2×C42—1064no—
423C2×C42—1062no—
424C2×C42C22⋊D5314yes—
425#95—24no—
426#95—534no—
427#95—1064no—
428#95—1064no—
429#95—1064no—
430#95—1064no—
431#95—1064no—
432#95—1062no—
433#95—1062no—
434#95—1062no—
435C24—532no—
436C24—532no—
437C24—532no—
438C24—1062no—
439C24—1061no—
440C24—1062no—
441C24—1061no—
442C24—1061no—
443C24—1062no—
444C24—1062no—
445C24—1061no—
446C24C23×D5311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—534no—
466C22×C4—534no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—534no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1068no—
495C22×C4—10616no—
496C22×C4—2124no—
497C22×C4—2128no—
498C22×C4—2124no—
499C22×C4C2×C4×D5314yes—
500C22×C4C2×C4×D5312yes—
501C22×C4#1171994514yes—
502C22×C4#1171994512yes—
503C22×C4#1171994514yes—
504C22×C4D4×D5314yes—
505C22×C4Q8×D5312yes—
506C22×C4#1171994912yes—
507C22×C4C2×C22⋊D5312yes—
508C22×C4C2×C22⋊D5312yes—
509C22×C4C2×C22⋊D5312yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—10616no—
513C4⋊C4—10616no—
514C22⋊C4—10616no—
515C22⋊C4—1068no—
516C22⋊C4—10616no—
517C22⋊C4—1068no—
518C42—28no—
519C42—48no—
520C42—10616no—
521C42—2124no—
522C23—534no—
523C23—1064no—
524C23—1062no—
525C23—1064no—
526C23—1062no—
527C23—1062no—
528C23—1064no—
529C23—1062no—
530C23—1064no—
531C23—1064no—
532C23—1062no—
533C23—1064no—
534C23—1062no—
535C23—2121no—
536C23—2122no—
537C23—2122no—
538C23—2121no—
539C23—2122no—
540C23—2122no—
541C23—2122no—
542C23—2121no—
543C23—2121no—
544C23—2122no—
545C23C22×C4×D5311yescharacteristic
546C23C2×D4×D5312yes—
547C23C22×C22⋊D5311yescharacteristic
548C23C2×Q8×D5311yescharacteristic
549C23#41×D5312yes—
550C23#41×D5311yescharacteristic
551C23#41×D5311yescharacteristic
552C23C2×#1171994511yescharacteristic
553C23C2×#1171994512yes—
554C23C2×#1171994512yes—
555C23C2×#1171994911yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—538no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1062no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—2121no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2122no—
608C2×C4—21216no—
609C2×C4C4×C22⋊D5314yes—
610C2×C4#1313319314yes—
611C2×C4#1313319518yes—
612C2×C4#1313320014yes—
613C2×C4#1313320218yes—
614C2×C4#1313321314yes—
615C22—1068no—
616C22—1064no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22—2124no—
621C22—2122no—
622C22—2124no—
623C22—2122no—
624C22—2122no—
625C22—2124no—
626C22C4×D4×D5312yes—
627C22C2×C4×C22⋊D5311yescharacteristic
628C22C4×Q8×D5311yescharacteristic
629C22C4×#1171994512yes—
630C22C4×#1171994911yescharacteristic
631C22C2×#1313319311yescharacteristic
632C22C2×#1313319512yes—
633C22C2×#1313320011yescharacteristic
634C22C2×#1313320212yes—
635C22#1681113011yescharacteristic
636C22#1681113212yes—
637C22#1681113212yes—
638C22#1681113411yescharacteristic
639C22#1681115112yes—
640C22#1681115112yes—
641C22#1681115712yes—
642C22#1681116011yescharacteristic
643C22#1681116012yes—
644C22C2×#1313321311yescharacteristic
645C22#1681116911yescharacteristic
646C22#1681117012yes—
647C22#1681117712yes—
648C22#1681117711yescharacteristic
649C4—24no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—1068no—
654C4—2122no—
655C4—2122no—
656C4—2124no—
657C4—2122no—
658C4#1681055818yesdirect product factor
659C2—2124no—
660C2C4×#1313320011yescharacteristic
661C2C4×#1313320212yes—
662C2C4×#1313319311yescharacteristic
663C2C4×#1313319512yes—
664C2C4×#1313321311yescharacteristic
665C2C2×#1681055811yescharacteristic
666C2#2027797211yescharacteristic
667C2#2027797812yes—
668C2#2027798511yescharacteristic
669C2#2027799112yes—
670C2#2027799411yescharacteristic
subgroupG/H
centerC23×C4C22×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C4×C4⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×C4×C4⋊C4D53
hypocenterC53C4×#399
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719919C2semidirect
5. #20272983C2semidirect
6. #20272981C2semidirect
7. #20272978C2semidirect
10. C4×#13133124C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
21. #16810558C4direct (N × H)
56. C2×#11719919C2×C4semidirect
82. #13133124C2×C4semidirect
92. C2×C4⋊C4×C53C2×C4semidirect
318. C4×C53#399semidirect
326. C2×C4×C4⋊C4D53semidirect
375. C53C4×#399semidirect
402. C2×C4⋊C4C4×D53semidirect
658. C4#16810558direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105582
C2×C4×#117199192
C2×C4×#117199212
#202729832
#202729812
#202729782
#202729592
#202729572
C4×#131331242
C2×C4×C4⋊C4×C532
C4×#39953