Finite Group Databasebeta

← C2×#20290906 #28320569

Subgroups — C2×#20290906 #28320569

Order 13568.

1698 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811565C211yesmaximal, characteristic
2C2×C4×C22⋊#6157C211yesmaximal, characteristic
3#20290906C218yesmaximal, direct product factor
4C2×#16811501C212yesmaximal
5C2×#16811501C212yesmaximal
6C22×#13133195C211yesmaximal, characteristic
7C22×C4⋊#6157—21no—
8C22×C22⋊#6157C2212yes—
9C22×C22⋊#6157C2212yes—
10C22×C4×#6157—21no—
11#16811565—22no—
12#16811565C2214yes—
13C4×C22⋊#6157—22no—
14C4×C22⋊#6157C2214yes—
15C2×(C2×#6157)⋊C4—21no—
16C2×(C2×#6157)⋊C4—21no—
17#16811501C2218yes—
18#16811501C2218yes—
19C23×Dic106C411yescharacteristic, complemented
20C23×C4×D53C2211yescharacteristic
21C2×#13133195—21no—
22C2×#13133195—21no—
23C2×#13133195—21no—
24C2×#13133195—21no—
25C2×#13133195—22no—
26C2×#13133195—22no—
27C2×#13133195C414yescomplemented
28C2×#13133195C2214yes—
29C22×#11719919C411yescharacteristic, complemented
30C22×#11719919C411yescharacteristic, complemented
31C22×#11719921C2211yescharacteristic
32C22×#11719921C2211yescharacteristic
33C22×C4⋊C4×C53C411yescharacteristic, complemented
34C23×#6157—22no—
35C23×#6157—22no—
36C2×C4⋊#6157—24no—
37C2×C4⋊#6157—44no—
38C2×C22⋊#6157—22no—
39C2×C22⋊#6157—24no—
40C2×C22⋊#6157—28no—
41C2×C22⋊#6157—22no—
42C2×C22⋊#6157Q818yescomplemented
43C2×C22⋊#6157C2314yes—
44C2×C22⋊#6157C2314yes—
45C2×C4×#6157—24no—
46C2×C4×#6157—44no—
47(C2×#6157)⋊C4—24no—
48(C2×#6157)⋊C4—24no—
49C24×D53C2311yescharacteristic
50C22×Dic106—21no—
51C22×Dic106—22no—
52C22×Dic106—21no—
53C22×Dic106—44no—
54C22×Dic106C2×C414yescomplemented
55C23×Dic53C2×C411yescharacteristic
56C23×Dic53C2×C411yescharacteristic
57C23×Dic53C2×C411yescharacteristic
58C22×C4×D53—44no—
59C22×C4×D53C2×C412yes—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53D412yes—
65C22×C4×D53D412yes—
66C22×C4×D53C2312yes—
67#13133195—24no—
68#13133195—24no—
69#13133195—24no—
70#13133195—24no—
71#13133195—24no—
72#13133195—24no—
73#13133195—24no—
74#13133195—24no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919C2×C414yescomplemented
82C2×#11719919C2×C414yescomplemented
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921C2×C412yes—
92C2×#11719921C2×C412yes—
93C2×#11719921C2312yes—
94C2×#11719921C2312yes—
95C23×C4×C53C2×C411yescharacteristic
96C23×C4×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53C2×C414yescomplemented
102C24×C53C22×C411yescharacteristic
103C23×D53—42no—
104C23×D53—42no—
105C23×D53C22×C411yescharacteristic
106C23×D53C2×D411yescharacteristic
107C23×D53C2×D411yescharacteristic
108C23×D53C2×D411yescharacteristic
109C23×D53C2×D411yescharacteristic
110C23×D53C2×Q814yes—
111C23×D53#4114yes—
112C23×D53C2411yescharacteristic
113C22×#6157—28no—
114C22×#6157—24no—
115C22×#6157—24no—
116C22×#6157—44no—
117C22×#6157—44no—
118C22×#6157—44no—
119C22×#6157—44no—
120C22×#6157—44no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53C22⋊C411yescharacteristic
129C22×C4×C53C22⋊C411yescharacteristic
130C22×C4×C53C22⋊C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C412yes—
143C22×Dic53C22⋊C411yescharacteristic
144C22×Dic53C22×C412yes—
145C22×Dic53C22×C412yes—
146C22×Dic53C22×C412yes—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—44no—
150C2×C4×D53—44no—
151C2×C4×D53—44no—
152C2×C4×D53—44no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53C22⋊C412yes—
156C2×C4×D53C22⋊C412yes—
157C2×C4×D53C22⋊C412yes—
158C2×C4×D53C22⋊C412yes—
159C2×C4×D53C2×D412yes—
160C2×C4×D53C2×D412yes—
161C2×C4×D53C2×D412yes—
162C2×C4×D53C2×D412yes—
163C2×Dic106—24no—
164C2×Dic106—24no—
165C2×Dic106—44no—
166C2×Dic106—48no—
167C2×Dic106—44no—
168C2×Dic106—48no—
169C22⋊#6157—28no—
170C22⋊#6157—24no—
171C22⋊#6157—28no—
172C22⋊#6157—44no—
173C22⋊#6157C2×Q818yescomplemented
174C4⋊#6157—48no—
175C4×#6157—48no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—24no—
178#11719921—22no—
179#11719921—22no—
180#11719921—22no—
181#11719921—22no—
182#11719921—22no—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719919—24no—
187#11719919—24no—
188#11719919—24no—
189#11719919—24no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53C2×C22⋊C411yescharacteristic
193C23×C53C2×C22⋊C411yescharacteristic
194C23×C53C4×Q814yes—
195C23×C53C23×C411yescharacteristic
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—48no—
208C22×D53—48no—
209C22×D53—41no—
210C22×D53C2×C22⋊C411yescharacteristic
211C22×D53C2×C22⋊C411yescharacteristic
212C22×D53C4×Q814yes—
213C22×D53C22⋊Q814yes—
214C22×D53C22⋊Q814yes—
215C22×D53#12412yes—
216C22×D53#12412yes—
217C22×D53C4⋊Q812yes—
218C22×D53C4⋊Q812yes—
219C22×D53C22×D411yescharacteristic
220C22×D53C22×D411yescharacteristic
221C22×D53C22×Q812yes—
222C22×D53C2×#4112yes—
223C2×#6157—28no—
224C2×#6157—44no—
225C2×#6157—44no—
226C2×#6157—416no—
227C2×#6157—44no—
228C2×#6157—44no—
229C2×#6157—44no—
230C2×#6157—88no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
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×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C22⋊C412yes—
245C2×Dic53—22no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53#9914yes—
257C2×Dic53C2×C22⋊C412yes—
258C2×Dic53C2×C22⋊C412yes—
259C4×D53—24no—
260C4×D53—24no—
261C4×D53—24no—
262C4×D53—24no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266C4×D53—42no—
267Dic106—48no—
268Dic106—48no—
269C2×#1996—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—28no—
275C22×C53#39112yes—
276C22×C53#39112yes—
277C22×C53C2×#9912yes—
278C22×C53#41012yes—
279C22×C53C22×C22⋊C411yescharacteristic
280C22×C53C2×C4×Q812yes—
281C2×D53—21no—
282C2×D53—21no—
283C2×D53—21no—
284C2×D53—21no—
285C2×D53—21no—
286C2×D53—21no—
287C2×D53—21no—
288C2×D53—21no—
289C2×D53—28no—
290C2×D53—24no—
291C2×D53—24no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C2×D53#39112yes—
299C2×D53#39112yes—
300C2×D53C2×C22⋊Q812yes—
301C2×D53C2×#12411yescharacteristic
302C2×D53C2×C4⋊Q811yescharacteristic
303#6157—48no—
304#6157—84no—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—42no—
308C4×C53—42no—
309Dic53—22no—
310Dic53—42no—
311Dic53—42no—
312Dic53C2×#9914yes—
313C22×C22⋊Q8—531no—
314C2×#380—531no—
315C2×C4×C22⋊C4—531no—
316#1996—538no—
317C2×#342—532no—
318C2×#342—532no—
319C2×C53—22no—
320C2×C53—22no—
321C2×C53#199614yescomplemented
322C2×C53C2×#39111yescharacteristic
323C2×C53C22×#9911yescharacteristic
324C2×C53C2×#41011yescharacteristic
325D53—22no—
326D53—21no—
327D53—21no—
328D53—22no—
329D53—22no—
330C23×Q8—531no—
331C24×C4—531no—
332C2×C22⋊Q8—534no—
333C2×C22⋊Q8—534no—
334C2×C22⋊Q8—1061no—
335C2×C22⋊Q8—1061no—
336C2×C22⋊Q8—1061no—
337C2×C22⋊Q8—1061no—
338C2×C22⋊Q8—1062no—
339C2×C22⋊Q8—1062no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—1061no—
343C22×C4⋊C4#615711yescharacteristic, complemented
344C22×C22⋊C4—532no—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—532no—
348C22×C42—1061no—
349#380—534no—
350#380—1062no—
351C4×C22⋊C4—534no—
352C4×C22⋊C4—1062no—
353C2×#95—1061no—
354C2×#95—1061no—
355#342—538no—
356#342—538no—
357C53C2×#199611yescharacteristic, complemented, Sylow
358C25—531no—
359C22×Q8—534no—
360C22×Q8—1062no—
361C22×Q8—1061no—
362C22×Q8—1061no—
363C22×Q8—2124no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—1062no—
376C23×C4—1062no—
377C23×C4—2124no—
378C23×C4C2×#615711yescharacteristic
379C23×C4C2×#615711yescharacteristic
380C23×C4C2×#615711yescharacteristic
381C22⋊Q8—1064no—
382C22⋊Q8—1064no—
383C22⋊Q8—1064no—
384C22⋊Q8—1064no—
385C22⋊Q8—1064no—
386C22⋊Q8—1064no—
387C22⋊Q8—1064no—
388C22⋊Q8—1064no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—534no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—2124no—
402C2×C4⋊C4C2×#615714yescomplemented
403C2×C22⋊C4—534no—
404C2×C22⋊C4—538no—
405C2×C22⋊C4—532no—
406C2×C22⋊C4—532no—
407C2×C22⋊C4—532no—
408C2×C22⋊C4—532no—
409C2×C22⋊C4—534no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1062no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1068no—
421C2×C22⋊C4—1062no—
422C2×C42—1064no—
423C2×C42—2124no—
424#95—1064no—
425#95—1064no—
426C24—531no—
427C24—531no—
428C24—531no—
429C24—531no—
430C24—534no—
431C24—534no—
432C24—531no—
433C24—531no—
434C24—2122no—
435C24—2122no—
436C24C22×#615711yescharacteristic
437C2×Q8—1064no—
438C2×Q8—1064no—
439C2×Q8—2124no—
440C2×Q8—2128no—
441C2×Q8—2128no—
442C2×Q8—2124no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—531no—
451C22×C4—532no—
452C22×C4—531no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—532no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1064no—
468C22×C4—1068no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1064no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2124no—
477C22×C4—2124no—
478C22×C4—2124no—
479C22×C4—2124no—
480C22×C4—2124no—
481C22×C4—2124no—
482C22×C4—2124no—
483C22×C4—2124no—
484C22×C4—2124no—
485C22×C4—2122no—
486C22×C4—2122no—
487C22×C4C22⋊#615711yescharacteristic
488C22×C4C22⋊#615711yescharacteristic
489C22×C4C22⋊#615712yes—
490C22×C4C22×#615712yes—
491C22×C4C22×#615712yes—
492C22×C4C22×#615712yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—1064no—
496C4⋊C4—1064no—
497C4⋊C4—1064no—
498C4⋊C4—1064no—
499C4⋊C4—2128no—
500C22⋊C4—538no—
501C22⋊C4—1062no—
502C22⋊C4—1062no—
503C22⋊C4—1062no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—1068no—
510C22⋊C4—1064no—
511C22⋊C4—1068no—
512C22⋊C4—2124no—
513C42—2128no—
514C23—22no—
515C23—22no—
516C23—534no—
517C23—532no—
518C23—532no—
519C23—531no—
520C23—531no—
521C23—531no—
522C23—531no—
523C23—534no—
524C23—532no—
525C23—532no—
526C23—534no—
527C23—532no—
528C23—532no—
529C23—1062no—
530C23—1062no—
531C23—1062no—
532C23—1062no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2128no—
538C23—2128no—
539C23—2121no—
540C23—2121no—
541C23—2121no—
542C23—2121no—
543C23C2×C22⋊#615711yescharacteristic
544C23C2×C22⋊#615711yescharacteristic
545C23Q8×#615714yes—
546C23C23×#615711yescharacteristic
547Q8—2128no—
548Q8—2128no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—534no—
562C2×C4—532no—
563C2×C4—532no—
564C2×C4—1064no—
565C2×C4—1064no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1062no—
571C2×C4—1068no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—4248no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2124no—
580C2×C4—2124no—
581C2×C4—2124no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2124no—
585C2×C4—21216no—
586C2×C4—2124no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4C2×C22⋊#615712yes—
592C2×C4C2×C22⋊#615712yes—
593C22—28no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—532no—
599C22—531no—
600C22—531no—
601C22—532no—
602C22—532no—
603C22—1064no—
604C22—1068no—
605C22—1064no—
606C22—1061no—
607C22—1061no—
608C22—1061no—
609C22—1061no—
610C22—1061no—
611C22—1061no—
612C22—1061no—
613C22—1061no—
614C22—2122no—
615C22—2124no—
616C22—2122no—
617C22—2122no—
618C22—2122no—
619C22—2124no—
620C22(C2×#6157)⋊Q812yes—
621C22(C2×#6157)⋊Q812yes—
622C22D53⋊#9912yes—
623C22#1681160212yes—
624C22C22×C22⋊#615711yescharacteristic
625C22C2×Q8×#615712yes—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—534no—
631C4—1062no—
632C4—4244no—
633C4—2128no—
634C4—2122no—
635C4—2122no—
636C2—22no—
637C2—22no—
638C2—1061no—
639C2—1062no—
640C2—1061no—
641C2—1062no—
642C2—1062no—
643C2#2029090614yesdirect product factor
644C2C2×(C2×#6157)⋊Q811yescharacteristic
645C2C2×D53⋊#9911yescharacteristic
646C2C2×#1681160211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1996
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20290906C2direct (N × H)
19. C23×Dic106C4semidirect
27. C2×#13133195C4semidirect
29. C22×#11719919C4semidirect
30. C22×#11719919C4semidirect
33. C22×C4⋊C4×C53C4semidirect
42. C2×C22⋊#6157Q8semidirect
54. C22×Dic106C2×C4semidirect
81. C2×#11719919C2×C4semidirect
82. C2×#11719919C2×C4semidirect
101. C2×C4⋊C4×C53C2×C4semidirect
173. C22⋊#6157C2×Q8semidirect
321. C2×C53#1996semidirect
343. C22×C4⋊C4#6157semidirect
357. C53C2×#1996semidirect
402. C2×C4⋊C4C2×#6157semidirect
643. C2#20290906direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115652
C2×C4×C22⋊#61572
#202909062
C2×#168115012
C22×#131331952
C2×#199653