Finite Group Databasebeta

← C2×#20273246 #27351326

Subgroups — C2×#20273246 #27351326

Order 13568.

1446 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D53C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3C2×#16810485C211yesmaximal, characteristic
4#20273246C218yesmaximal, direct product factor
5C2×#16810361C211yesmaximal, characteristic, complemented
6C2×#16810360C211yesmaximal, characteristic, complemented
7C2×#16810359C211yesmaximal, characteristic, complemented
8C2×C53×#328C211yesmaximal, characteristic, complemented
9C22×Q8×D53C411yescharacteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C22×#11719912—21no—
12C2×#13133197C411yescharacteristic, complemented
13C2×#13133196C2211yescharacteristic
14C2×#13133195—21no—
15C2×#13133194—21no—
16C2×#13133193C411yescharacteristic, complemented
17C22⋊Q8×D53—22no—
18C22⋊Q8×D53—22no—
19C22⋊Q8×D53C414yescomplemented
20C22⋊Q8×D53C2214yes—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53C411yescharacteristic, complemented
23C22×#11719921—21no—
24C2×C22⋊C4×D53—21no—
25#16810585—22no—
26#16810585C2214yes—
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133129—21no—
29#16810485—22no—
30#16810485C2214yes—
31C2×#13133124—21no—
32C2×#13133124C2211yescharacteristic
33#16810361C2214yescomplemented
34#16810360C2214yescomplemented
35#16810359C2214yescomplemented
36C2×C22⋊Q8×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#328C2214yescomplemented
40C24×D53D411yescharacteristic
41C22×Dic106—21no—
42C22×Dic106C2×C411yescharacteristic
43C2×Q8×D53—22no—
44C2×Q8×D53—44no—
45C2×Q8×D53C2×C414yescomplemented
46C23×Dic53—21no—
47C23×Dic53D411yescharacteristic
48C22×C4×D53—21no—
49C22×C4×D53—21no—
50C22×C4×D53—44no—
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C412yes—
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53D411yescharacteristic
55C22×C4×D53D412yes—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D412yes—
58C22×C4×D53C2312yes—
59C2×#11719912—24no—
60C2×#11719912—44no—
61C22×#88322—21no—
62#13133197—22no—
63#13133197C2×C414yescomplemented
64#13133196—21no—
65#13133196—21no—
66#13133196C2×C412yes—
67#13133196C2312yes—
68#13133195—24no—
69#13133195—24no—
70#13133194—24no—
71#13133194—24no—
72#13133193—22no—
73#13133193C2×C414yescomplemented
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2×C411yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C4⋊C4×D53—24no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—24no—
82C4⋊C4×D53C2×C414yescomplemented
83C2×C22⋊Dic53—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—24no—
87C2×#11719921—44no—
88C22⋊C4×D53—24no—
89C22⋊C4×D53—24no—
90C22⋊#88322—21no—
91C22⋊#88322C2312yes—
92#13133129—24no—
93#13133124—21no—
94#13133124—24no—
95#13133124C2312yes—
96C22×Q8×C53C2×C411yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—21no—
100C22⋊Q8×C53—21no—
101C22⋊Q8×C53—21no—
102C22⋊Q8×C53C2×C412yes—
103C22⋊Q8×C53C2312yes—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2×C411yescharacteristic
106C2×C22⋊C4×C53—21no—
107C22⋊C8×C53—21no—
108C22⋊C8×C53C2312yes—
109#95×C53—21no—
110#95×C53C2312yes—
111C24×C53C2×D411yescharacteristic
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—24no—
115C23×D53—42no—
116C23×D53C22⋊C411yescharacteristic
117C23×D53C22⋊C412yes—
118C23×D53C22⋊C411yescharacteristic
119C23×D53C2×D412yes—
120C2×Q8×C53—21no—
121C2×Q8×C53—42no—
122C2×Q8×C53C22×C412yes—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53C22⋊C411yescharacteristic
144C22×Dic53C22⋊C411yescharacteristic
145C22×Dic53C22⋊C412yes—
146C22×Dic53C2×D412yes—
147Q8×D53—48no—
148Q8×D53—48no—
149C2×C4×D53—24no—
150C2×C4×D53—24no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—44no—
154C2×C4×D53—44no—
155C2×C4×D53—44no—
156C2×C4×D53—44no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—44no—
160C2×C4×D53—44no—
161C2×C4×D53C22⋊C412yes—
162C2×C4×D53C22⋊C412yes—
163C2×C4×D53C22⋊C412yes—
164C2×C4×D53C22⋊C412yes—
165C2×C4×D53C22×C412yes—
166C2×C4×D53C22×C412yes—
167C2×C4×D53C2×D412yes—
168C2×C4×D53C2×D412yes—
169C2×Dic106—21no—
170C2×Dic106—24no—
171C2×Dic106—44no—
172C2×Dic106—42no—
173C2×Dic106C22×C412yes—
174C2×C8×C53—22no—
175C2×C8×C53—42no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53C22×C412yes—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊Dic53—22no—
183C22⋊Dic53—22no—
184#11719921—24no—
185#11719921—24no—
186#11719921—48no—
187C4⋊Dic53—21no—
188C4⋊Dic53—24no—
189C4⋊Dic53C22×C412yes—
190#11719919—22no—
191#11719919—22no—
192#11719919—24no—
193#11719919—24no—
194C2×#88322—22no—
195C2×#88322—42no—
196#11719912—48no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C2×C22⋊C411yescharacteristic
201C23×C53#13712yes—
202C23×C53#13712yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—48no—
217C22×D53—48no—
218C22×D53#9914yes—
219C22×D53C2×C22⋊C412yes—
220C22×D53C2×C22⋊C412yes—
221Q8×C53—42no—
222Q8×C53—42no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—84no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—84no—
253C2×Dic53C2×C22⋊C412yes—
254C2×Dic53C2×C22⋊C412yes—
255C4×D53—24no—
256C4×D53—24no—
257C4×D53—24no—
258C4×D53—24no—
259C4×D53—42no—
260C4×D53—42no—
261C4×D53—42no—
262C4×D53—42no—
263C4×D53—48no—
264C4×D53—48no—
265Dic106—48no—
266Dic106—42no—
267Dic106—42no—
268C8×C53—42no—
269#88322—42no—
270C2×#1511—531nomaximal, Sylow
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—24no—
276C22×C53—24no—
277C22×C53C2×#9912yes—
278C22×C53#41912yes—
279C22×C53#41912yes—
280C22×C53#42112yes—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—28no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—48no—
293C2×D53—44no—
294C2×D53C2×#9914yes—
295C4×C53—21no—
296C4×C53—21no—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53—42no—
300C4×C53—42no—
301C4×C53—82no—
302Dic53—22no—
303Dic53—22no—
304Dic53—42no—
305Dic53—42no—
306Dic53—82no—
307C22×C22⋊Q8—531no—
308C2×#407—531no—
309C2×#381—531no—
310#1511—538no—
311C2×#328—531no—
312C2×#328—531no—
313C2×#328—531no—
314C2×#328D5311yescharacteristic, complemented
315C2×C53—21no—
316C2×C53—21no—
317C2×C53—22no—
318C2×C53#151114yescomplemented
319C2×C53C22×#9911yescharacteristic
320C2×C53C2×#41911yescharacteristic
321C2×C53C2×#42111yescharacteristic
322D53—24no—
323D53—42no—
324C23×Q8—531no—
325C24×C4—531no—
326C22×M16—1061no—
327C2×C22⋊Q8—531no—
328C2×C22⋊Q8—531no—
329C2×C22⋊Q8—534no—
330C2×C22⋊Q8—534no—
331C2×C22⋊Q8—531no—
332C2×C22⋊Q8—1061no—
333C2×C22⋊Q8—1061no—
334C2×C22⋊Q8—1062no—
335C2×C22⋊Q8—1062no—
336C2×C22⋊Q8C2×D5311yescharacteristic
337C22×C4⋊C4—531no—
338C22×C4⋊C4—1061no—
339C22×C22⋊C4—1061no—
340C22×C22⋊C4—1061no—
341#407—534no—
342#407—1062no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—1061no—
345C2×C22⋊C8C2×D5311yescharacteristic
346#381—534no—
347#381—1062no—
348C2×#95—531no—
349C2×#95—1061no—
350C2×#95C2×D5311yescharacteristic
351#328—534no—
352#328—534no—
353#328—534no—
354#328C2×D5314yescomplemented
355C53C2×#151111yescharacteristic, complemented, Sylow
356C25—531no—
357C22×Q8—534no—
358C22×Q8—531no—
359C22×Q8—1061no—
360C22×Q8—1062no—
361C22×Q8—2124no—
362C22×Q8C4×D5311yescharacteristic
363C23×C4—21no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—1061no—
374C23×C4—1061no—
375C23×C4—1061no—
376C23×C4—2124no—
377C23×C4C22×D5311yescharacteristic
378C2×M16—1064no—
379C2×M16—2124no—
380C22×C8—21no—
381C22×C8—1061no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—534no—
385C22⋊Q8—532no—
386C22⋊Q8—532no—
387C22⋊Q8—534no—
388C22⋊Q8—1064no—
389C22⋊Q8—1064no—
390C22⋊Q8—1064no—
391C22⋊Q8—1064no—
392C22⋊Q8—1062no—
393C22⋊Q8—1062no—
394C22⋊Q8—1061no—
395C22⋊Q8—1061no—
396C22⋊Q8C4×D5312yes—
397C22⋊Q8C22×D5312yes—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—534no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4C4×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1064no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1064no—
416C2×C22⋊C4—2124no—
417C22⋊C8—21no—
418C22⋊C8—532no—
419C22⋊C8—1064no—
420C22⋊C8—1061no—
421C22⋊C8C22×D5312yes—
422#95—21no—
423#95—532no—
424#95—1064no—
425#95—1061no—
426#95C22×D5312yes—
427C24—531no—
428C24—531no—
429C24—532no—
430C24—532no—
431C24—1064no—
432C24—1062no—
433C24—1062no—
434C24—2122no—
435C24D4×D5311yescharacteristic
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8—532no—
439C2×Q8—1064no—
440C2×Q8—1061no—
441C2×Q8—2122no—
442C2×Q8—2124no—
443C2×Q8—2128no—
444C2×Q8—2128no—
445C2×Q8C2×C4×D5312yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—531no—
454C22×C4—531no—
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—532no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1064no—
468C22×C4—1064no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1062no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2124no—
475C22×C4—2124no—
476C22×C4—2124no—
477C22×C4—2124no—
478C22×C4—2124no—
479C22×C4—2124no—
480C22×C4—2122no—
481C22×C4—2122no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4C2×C4×D5311yescharacteristic
485C22×C4C2×C4×D5311yescharacteristic
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4D4×D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490M16—2128no—
491C2×C8—22no—
492C2×C8—42no—
493C2×C8—1062no—
494C2×C8—2122no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1064no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1062no—
504C4⋊C4—1062no—
505C4⋊C4C2×C4×D5312yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—1064no—
509C22⋊C4—1064no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—2128no—
513C23—22no—
514C23—22no—
515C23—532no—
516C23—534no—
517C23—532no—
518C23—1064no—
519C23—1064no—
520C23—1062no—
521C23—1062no—
522C23—1062no—
523C23—1064no—
524C23—1062no—
525C23—1062no—
526C23—1062no—
527C23—1062no—
528C23—2122no—
529C23—2122no—
530C23—2128no—
531C23—2128no—
532C23C22⋊C4×D5311yescharacteristic
533C23C22⋊C4×D5311yescharacteristic
534C23#1313324612yes—
535C23#1313324912yes—
536C23C2×D4×D5311yescharacteristic
537Q8—42no—
538Q8—42no—
539Q8—2122no—
540Q8—2122no—
541Q8—2128no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—4244no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2128no—
572C2×C4—2128no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4C22⋊C4×D5311yescharacteristic
581C2×C4C22⋊C4×D5311yescharacteristic
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4C2×D4×D5311yescharacteristic
584C8—42no—
585C8—2122no—
586C22—21no—
587C22—21no—
588C22—24no—
589C22—24no—
590C22—21no—
591C22—21no—
592C22—534no—
593C22—1062no—
594C22—1068no—
595C22—1062no—
596C22—1062no—
597C22—1062no—
598C22—1062no—
599C22—2124no—
600C22—2124no—
601C22—2128no—
602C22—2124no—
603C22#99×D5312yes—
604C22#1681063012yes—
605C22#1681063012yes—
606C22#1681063812yes—
607C22C2×C22⋊C4×D5311yescharacteristic
608C22C2×#1313324611yescharacteristic
609C22C2×#1313324911yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—1062no—
618C4—1062no—
619C4—4242no—
620C4—2122no—
621C4—2122no—
622C2—21no—
623C2—21no—
624C2—22no—
625C2—1064no—
626C2—2122no—
627C2#2027324614yesdirect product factor
628C2C2×#99×D5311yescharacteristic
629C2C2×#1681063011yescharacteristic
630C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#328C2
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC2×#328D53
hypocenterC53C2×#1511
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
4. #20273246C2direct (N × H)
5. C2×#16810361C2semidirect
6. C2×#16810360C2semidirect
7. C2×#16810359C2semidirect
8. C2×C53×#328C2semidirect
9. C22×Q8×D53C4semidirect
12. C2×#13133197C4semidirect
16. C2×#13133193C4semidirect
19. C22⋊Q8×D53C4semidirect
22. C2×C4⋊C4×D53C4semidirect
33. #16810361C22semidirect
34. #16810360C22semidirect
35. #16810359C22semidirect
39. C53×#328C22semidirect
45. C2×Q8×D53C2×C4semidirect
63. #13133197C2×C4semidirect
73. #13133193C2×C4semidirect
82. C4⋊C4×D53C2×C4semidirect
314. C2×#328D53semidirect
318. C2×C53#1511semidirect
354. #328C2×D53semidirect
355. C53C2×#1511semidirect
627. C2#20273246direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D532
C2×#168105852
C2×#168104852
#202732462
C2×#168103612
C2×#168103602
C2×#168103592
C2×C53×#3282
C2×#151153