Finite Group Databasebeta

← C2×#20289900 #28312857

Subgroups — C2×#20289900 #28312857

Order 13568.

3246 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊#6157C211yesmaximal, characteristic
2C22×(C2×#6157)⋊C4C212yesmaximal
3C22×(C2×#6157)⋊C4C211yesmaximal, characteristic
4C22×(C2×#6157)⋊C4C212yesmaximal
5#20289900C218yesmaximal, direct product factor
6C22×C4⋊C4×D53C211yesmaximal, characteristic
7C24×#6157C2212yes—
8C24×#6157C2212yes—
9C22×C4⋊#6157—28no—
10C22×C4⋊#6157—24no—
11C22×C4⋊#6157C2214yes—
12C2×(C2×#6157)⋊C4—28no—
13C2×(C2×#6157)⋊C4—24no—
14C2×(C2×#6157)⋊C4—28no—
15C2×(C2×#6157)⋊C4C2218yes—
16C2×(C2×#6157)⋊C4C2214yes—
17C2×(C2×#6157)⋊C4C2218yes—
18C23×C4×D53C2211yescharacteristic
19C23×C4×D53C2211yescharacteristic
20C23×C4×D53C2211yescharacteristic
21C22×C4⋊Dic53C411yescharacteristic, complemented
22C22×#11719919C411yescharacteristic, complemented
23C22×#11719919C411yescharacteristic, complemented
24C2×C4⋊C4×D53—24no—
25C2×C4⋊C4×D53—24no—
26C2×C4⋊C4×D53C414yescomplemented
27C2×C4⋊C4×D53C2214yes—
28C22×C4⋊C4×C53C411yescharacteristic, complemented
29C23×#6157—24no—
30C23×#6157—24no—
31C23×#6157—24no—
32C23×#6157—24no—
33C23×#6157—24no—
34C23×#6157—24no—
35C23×#6157—24no—
36C23×#6157—24no—
37C23×#6157—24no—
38C23×#6157—24no—
39C23×#6157—24no—
40C23×#6157—24no—
41C23×#6157D418yes—
42C23×#6157Q818yes—
43C23×#6157C2314yes—
44C23×#6157C2314yes—
45C2×C4⋊#6157—232no—
46C2×C4⋊#6157—216no—
47C2×C4⋊#6157—28no—
48(C2×#6157)⋊C4—216no—
49(C2×#6157)⋊C4—28no—
50(C2×#6157)⋊C4—216no—
51C24×D53C2311yescharacteristic
52C23×Dic53C2×C411yescharacteristic
53C23×Dic53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53—24no—
58C22×C4×D53—22no—
59C22×C4×D53—22no—
60C22×C4×D53—24no—
61C22×C4×D53—24no—
62C22×C4×D53—22no—
63C22×C4×D53—22no—
64C22×C4×D53—24no—
65C22×C4×D53C2×C412yes—
66C22×C4×D53C2×C412yes—
67C22×C4×D53C2×C412yes—
68C22×C4×D53D414yes—
69C22×C4×D53D414yes—
70C22×C4×D53D414yes—
71C22×C4×D53Q814yes—
72C22×C4×D53C2312yes—
73C22×C4×D53C2312yes—
74C22×C4×D53C2312yes—
75C2×C4⋊Dic53—24no—
76C2×C4⋊Dic53C2×C414yescomplemented
77C2×#11719919—24no—
78C2×#11719919—24no—
79C2×#11719919C2×C414yescomplemented
80C2×#11719919C2×C414yescomplemented
81C4⋊C4×D53—216no—
82C4⋊C4×D53—28no—
83C4⋊C4×D53—28no—
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53C2×C414yescomplemented
89C24×C53C22×C411yescharacteristic
90C23×D53C22×C411yescharacteristic
91C23×D53C2×D412yes—
92C23×D53C2×D412yes—
93C23×D53C2×D412yes—
94C23×D53C2×D414yes—
95C23×D53C2×Q812yes—
96C23×D53C2×Q814yes—
97C23×D53#4112yes—
98C23×D53#4112yes—
99C23×D53#4114yes—
100C23×D53#4114yes—
101C23×D53C2411yescharacteristic
102C22×#6157—216no—
103C22×#6157—216no—
104C22×#6157—216no—
105C22×#6157—216no—
106C22×#6157—216no—
107C22×#6157—216no—
108C22×#6157—24no—
109C22×#6157—24no—
110C22×#6157—24no—
111C22×#6157—24no—
112C22×#6157—24no—
113C22×#6157—24no—
114C22×#6157—24no—
115C22×#6157—24no—
116C22×#6157—24no—
117C22×#6157—24no—
118C22×#6157—24no—
119C22×#6157—24no—
120C22×#6157—48no—
121C22×#6157—48no—
122C22×#6157—48no—
123C22×#6157—48no—
124C22×#6157—48no—
125C22×#6157—48no—
126C22×#6157—48no—
127C22×#6157—48no—
128C22×#6157C2×D418yes—
129C22×#6157C2×Q818yes—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—24no—
133C22×C4×C53—22no—
134C22×C4×C53—24no—
135C22×C4×C53C22⋊C414yes—
136C22×C4×C53C4⋊C414yescomplemented
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C22×C412yes—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—24no—
143C22×Dic53—22no—
144C22×Dic53—24no—
145C22×Dic53C22⋊C414yes—
146C22×Dic53C4⋊C414yescomplemented
147C22×Dic53C22×C412yes—
148C22×Dic53C22×C412yes—
149C22×Dic53C22×C412yes—
150C2×C4×D53—24no—
151C2×C4×D53—24no—
152C2×C4×D53—28no—
153C2×C4×D53—28no—
154C2×C4×D53—28no—
155C2×C4×D53—28no—
156C2×C4×D53—216no—
157C2×C4×D53—28no—
158C2×C4×D53—28no—
159C2×C4×D53—28no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—24no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—24no—
166C2×C4×D53—24no—
167C2×C4×D53—28no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—24no—
171C2×C4×D53—24no—
172C2×C4×D53—44no—
173C2×C4×D53—48no—
174C2×C4×D53—44no—
175C2×C4×D53C22⋊C418yes—
176C2×C4×D53C4⋊C418yescomplemented
177C2×C4×D53C2×D414yes—
178C2×C4×D53C2×D414yes—
179C2×C4×D53C2×D414yes—
180C2×C4×D53C2×Q814yes—
181C4⋊#6157—232no—
182C4⋊C4×C53—28no—
183C4⋊Dic53—28no—
184#11719919—28no—
185#11719919—28no—
186C23×C53C2×C22⋊C412yes—
187C23×C53C2×C4⋊C412yes—
188C23×C53#11712yes—
189C23×C53C4×D414yes—
190C23×C53C4×Q814yes—
191C23×C53C23×C411yescharacteristic
192C22×D53C2×C22⋊C412yes—
193C22×D53C2×C4⋊C412yes—
194C22×D53#11712yes—
195C22×D53C4×D414yes—
196C22×D53C4×Q814yes—
197C22×D53#12014yes—
198C22×D53#12114yes—
199C22×D53#12114yes—
200C22×D53#12118yes—
201C22×D53#12118yes—
202C22×D53C22⋊Q814yes—
203C22×D53C22⋊Q814yes—
204C22×D53C22⋊Q818yes—
205C22×D53C22⋊Q818yes—
206C22×D53C22⋊Q818yes—
207C22×D53#12314yes—
208C22×D53#12318yes—
209C22×D53#12414yes—
210C22×D53#12518yes—
211C22×D53#12518yes—
212C22×D53#12618yes—
213C22×D53C4⋊Q818yes—
214C22×D53C4⋊Q814yes—
215C22×D53C22×D412yes—
216C22×D53C22×D411yescharacteristic
217C22×D53C22×D411yescharacteristic
218C22×D53C22×D411yescharacteristic
219C22×D53C22×Q812yes—
220C22×D53C22×Q811yescharacteristic
221C22×D53C2×#4111yescharacteristic
222C22×D53C2×#4111yescharacteristic
223C22×D53C2×#4112yes—
224C22×D53C2×#4112yes—
225C2×#6157—28no—
226C2×#6157—28no—
227C2×#6157—28no—
228C2×#6157—28no—
229C2×#6157—28no—
230C2×#6157—28no—
231C2×#6157—44no—
232C2×#6157—44no—
233C2×#6157—44no—
234C2×#6157—44no—
235C2×#6157—44no—
236C2×#6157—44no—
237C2×#6157—44no—
238C2×#6157—44no—
239C2×#6157—432no—
240C2×#6157—432no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—24no—
244C2×C4×C53—28no—
245C2×C4×C53—28no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—44no—
249C2×C4×C53C2×C22⋊C414yes—
250C2×C4×C53C2×C4⋊C414yescomplemented
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—28no—
255C2×Dic53—28no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—44no—
259C2×Dic53C2×C22⋊C414yes—
260C2×Dic53C2×C4⋊C414yescomplemented
261C4×D53—24no—
262C4×D53—24no—
263C4×D53—24no—
264C4×D53—28no—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—24no—
268C4×D53—216no—
269C4×D53—216no—
270C4×D53—42no—
271C4×D53—44no—
272C4×D53—42no—
273C4×D53—48no—
274C2×#1441—531nomaximal, Sylow
275C22×C53#38014yes—
276C22×C53#38614yes—
277C22×C53#38718yes—
278C22×C53#38918yes—
279C22×C53#39114yes—
280C22×C53C22×C22⋊C411yescharacteristic
281C22×C53C22×C4⋊C411yescharacteristic
282C22×C53C2×#11711yescharacteristic
283C22×C53C2×C4×D412yes—
284C22×C53C2×C4×Q812yes—
285C2×D53#38014yes—
286C2×D53#38614yes—
287C2×D53#38718yes—
288C2×D53#38918yes—
289C2×D53#39114yes—
290C2×D53#39418yes—
291C2×D53#39518yes—
292C2×D53#396116yes—
293C2×D53#39818yes—
294C2×D53#398116yes—
295C2×D53#40018yes—
296C2×D53C2×#12011yescharacteristic
297C2×D53C2×#12112yes—
298C2×D53C2×#12111yescharacteristic
299C2×D53C2×#12111yescharacteristic
300C2×D53C2×#12112yes—
301C2×D53C2×C22⋊Q812yes—
302C2×D53C2×C22⋊Q811yescharacteristic
303C2×D53C2×C22⋊Q812yes—
304C2×D53C2×C22⋊Q812yes—
305C2×D53C2×C22⋊Q811yescharacteristic
306C2×D53C2×#12312yes—
307C2×D53C2×#12311yescharacteristic
308C2×D53C2×#12411yescharacteristic
309C2×D53C2×#12512yes—
310C2×D53C2×#12512yes—
311C2×D53C2×#12612yes—
312C2×D53C2×C4⋊Q812yes—
313C2×D53C2×C4⋊Q811yescharacteristic
314#6157—48no—
315#6157—48no—
316C4×C53—24no—
317C4×C53—24no—
318C4×C53—42no—
319Dic53—24no—
320Dic53—24no—
321Dic53—42no—
322C23×C4⋊C4—531no—
323C23×C4⋊C4—531no—
324C22×#95—531no—
325C22×#95—532no—
326C22×#95—532no—
327#1441—538no—
328C2×C53#144118yescomplemented
329C2×C53C2×#38011yescharacteristic
330C2×C53C2×#38611yescharacteristic
331C2×C53C2×#38712yes—
332C2×C53C2×#38912yes—
333C2×C53C2×#39111yescharacteristic
334D53#144118yescomplemented
335D53C2×#39411yescharacteristic
336D53C2×#39511yescharacteristic
337D53C2×#39612yes—
338D53C2×#39811yescharacteristic
339D53C2×#39812yes—
340D53C2×#40011yescharacteristic
341C24×C4—531no—
342C24×C4—531no—
343C24×C4—532no—
344C24×C4—532no—
345C24×C4—531no—
346C22×C4⋊C4—531no—
347C22×C4⋊C4—531no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4—534no—
350C22×C4⋊C4—534no—
351C22×C4⋊C4—534no—
352C22×C4⋊C4—1064no—
353C22×C4⋊C4—1064no—
354C22×C4⋊C4—1064no—
355C22×C4⋊C4—1068no—
356C22×C4⋊C4#615711yescharacteristic, complemented
357C2×#95—534no—
358C2×#95—538no—
359C2×#95—538no—
360C2×#95—1068no—
361C2×#95—1064no—
362C2×#95—1068no—
363C53C2×#144111yescharacteristic, complemented, Sylow
364C25—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—534no—
368C23×C4—534no—
369C23×C4—534no—
370C23×C4—534no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4—532no—
375C23×C4—534no—
376C23×C4—538no—
377C23×C4—534no—
378C23×C4—538no—
379C23×C4—531no—
380C23×C4—532no—
381C23×C4—532no—
382C23×C4—1062no—
383C23×C4—1062no—
384C23×C4—1064no—
385C23×C4—1062no—
386C23×C4—1062no—
387C23×C4—1064no—
388C23×C4—1062no—
389C23×C4—1062no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—1064no—
393C23×C4—1064no—
394C23×C4—1064no—
395C23×C4—1064no—
396C23×C4—1064no—
397C23×C4—1064no—
398C23×C4—1064no—
399C23×C4—1064no—
400C23×C4—1064no—
401C23×C4—1064no—
402C23×C4—1064no—
403C23×C4—1064no—
404C23×C4C2×#615711yescharacteristic
405C23×C4C2×#615711yescharacteristic
406C23×C4C2×#615711yescharacteristic
407C2×C4⋊C4—24no—
408C2×C4⋊C4—534no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—10616no—
412C2×C4⋊C4—1068no—
413C2×C4⋊C4—1068no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1068no—
417C2×C4⋊C4—10616no—
418C2×C4⋊C4—10632no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4C2×#615714yescomplemented
421#95—1068no—
422#95—10616no—
423#95—10616no—
424C24—532no—
425C24—532no—
426C24—532no—
427C24—532no—
428C24—532no—
429C24—532no—
430C24—534no—
431C24—534no—
432C24—534no—
433C24—534no—
434C24—531no—
435C24—531no—
436C24C22×#615711yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—534no—
443C22×C4—534no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—538no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—538no—
450C22×C4—534no—
451C22×C4—534no—
452C22×C4—538no—
453C22×C4—538no—
454C22×C4—532no—
455C22×C4—1068no—
456C22×C4—1064no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1064no—
460C22×C4—1068no—
461C22×C4—1068no—
462C22×C4—1068no—
463C22×C4—1068no—
464C22×C4—10616no—
465C22×C4—1068no—
466C22×C4—1064no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1068no—
470C22×C4—1064no—
471C22×C4—1064no—
472C22×C4—1064no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1068no—
476C22×C4—1064no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—1064no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—10616no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—10616no—
488C22×C4—10616no—
489C22×C4—10616no—
490C22×C4—10616no—
491C22×C4—10616no—
492C22×C4—1064no—
493C22×C4—1064no—
494C22×C4—1064no—
495C22×C4—1064no—
496C22×C4—1064no—
497C22×C4—1064no—
498C22×C4—1062no—
499C22×C4—1064no—
500C22×C4—2124no—
501C22×C4—2128no—
502C22×C4—2124no—
503C22×C4—2128no—
504C22×C4—2128no—
505C22×C4—2128no—
506C22×C4—2128no—
507C22×C4—2128no—
508C22×C4—2128no—
509C22×C4—2128no—
510C22×C4—2128no—
511C22×C4C4⋊#615714yescomplemented
512C22×C4C22⋊#615714yes—
513C22×C4C22×#615712yes—
514C22×C4C22×#615712yes—
515C22×C4C22×#615712yes—
516C4⋊C4—28no—
517C4⋊C4—1068no—
518C4⋊C4—1068no—
519C4⋊C4—10632no—
520C4⋊C4—1068no—
521C23—534no—
522C23—534no—
523C23—534no—
524C23—534no—
525C23—534no—
526C23—538no—
527C23—534no—
528C23—538no—
529C23—538no—
530C23—538no—
531C23—538no—
532C23—538no—
533C23—538no—
534C23—538no—
535C23—538no—
536C23—538no—
537C23—534no—
538C23—534no—
539C23—532no—
540C23—531no—
541C23—531no—
542C23—532no—
543C23—531no—
544C23—531no—
545C23—532no—
546C23—531no—
547C23—531no—
548C23—534no—
549C23—532no—
550C23—532no—
551C23—534no—
552C23—532no—
553C23—532no—
554C23C2×C22⋊#615712yes—
555C23C2×C4⋊#615712yes—
556C23#1313329612yes—
557C23D4×#615714yes—
558C23Q8×#615714yes—
559C23C23×#615711yescharacteristic
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—534no—
569C2×C4—534no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—10616no—
576C2×C4—1068no—
577C2×C4—1064no—
578C2×C4—10616no—
579C2×C4—1068no—
580C2×C4—1068no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1064no—
584C2×C4—1068no—
585C2×C4—1068no—
586C2×C4—1068no—
587C2×C4—1068no—
588C2×C4—1068no—
589C2×C4—1068no—
590C2×C4—1062no—
591C2×C4—1064no—
592C2×C4—2122no—
593C2×C4—2124no—
594C2×C4—2122no—
595C2×C4—2128no—
596C2×C4—2124no—
597C2×C4—2124no—
598C2×C4—2124no—
599C2×C4—2124no—
600C2×C4—2124no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2124no—
604C2×C4—21232no—
605C2×C4—21232no—
606C2×C4—2124no—
607C2×C4C2×C22⋊#615714yes—
608C2×C4C2×C4⋊#615714yescomplemented
609C22—531no—
610C22—531no—
611C22—531no—
612C22—531no—
613C22—534no—
614C22—531no—
615C22—532no—
616C22—531no—
617C22—534no—
618C22—532no—
619C22—532no—
620C22—532no—
621C22—532no—
622C22—538no—
623C22—532no—
624C22—532no—
625C22—532no—
626C22—532no—
627C22—538no—
628C22—532no—
629C22—531no—
630C22—531no—
631C22—534no—
632C22—538no—
633C22—5316no—
634C22—538no—
635C22—538no—
636C22—5316no—
637C22—538no—
638C22#1681156514yes—
639C22#1681157114yes—
640C22#1681157218yes—
641C22#1681157418yes—
642C22(C2×#6157)⋊Q814yes—
643C22C22×C22⋊#615711yescharacteristic
644C22C22×C4⋊#615711yescharacteristic
645C22C2×#1313329611yescharacteristic
646C22C2×D4×#615712yes—
647C22C2×Q8×#615712yes—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—1064no—
652C4—1064no—
653C4—2128no—
654C4—2128no—
655C4—2122no—
656C2—531no—
657C2—532no—
658C2—531no—
659C2—531no—
660C2—532no—
661C2—531no—
662C2—538no—
663C2#2028990018yesdirect product factor
664C2C2×#1681156511yescharacteristic
665C2C2×#1681157111yescharacteristic
666C2C2×#1681157212yes—
667C2C2×#1681157412yes—
668C2C2×(C2×#6157)⋊Q811yescharacteristic
subgroupG/H
centerC24C22×#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC24×C53C22×C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1441
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #20289900C2direct (N × H)
21. C22×C4⋊Dic53C4semidirect
22. C22×#11719919C4semidirect
23. C22×#11719919C4semidirect
26. C2×C4⋊C4×D53C4semidirect
28. C22×C4⋊C4×C53C4semidirect
76. C2×C4⋊Dic53C2×C4semidirect
79. C2×#11719919C2×C4semidirect
80. C2×#11719919C2×C4semidirect
88. C2×C4⋊C4×C53C2×C4semidirect
136. C22×C4×C53C4⋊C4semidirect
146. C22×Dic53C4⋊C4semidirect
176. C2×C4×D53C4⋊C4semidirect
250. C2×C4×C53C2×C4⋊C4semidirect
260. C2×Dic53C2×C4⋊C4semidirect
328. C2×C53#1441semidirect
334. D53#1441semidirect
356. C22×C4⋊C4#6157semidirect
363. C53C2×#1441semidirect
420. C2×C4⋊C4C2×#6157semidirect
511. C22×C4C4⋊#6157semidirect
608. C2×C4C2×C4⋊#6157semidirect
663. C2#20289900direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊#61572
C22×(C2×#6157)⋊C42
#202899002
C22×C4⋊C4×D532
C2×#144153