Finite Group Databasebeta

← C2×#20276357 #27536814

Subgroups — C2×#20276357 #27536814

Order 13568.

1398 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811130C211yesmaximal, characteristic
2C2×#16810597C211yesmaximal, characteristic, complemented
3#20276357C218yesmaximal, direct product factor
4C2×#16810414C212yesmaximal, complemented
5C2×#16810411C212yesmaximal, complemented
6C2×C53×#413C211yesmaximal, characteristic, complemented
7C23×C4×D53C411yescharacteristic, complemented
8#16811130—22no—
9#16811130—22no—
10#16811130C414yescomplemented
11#16811130C2214yes—
12C2×#13133198C411yescharacteristic, complemented
13C2×#13133198C2211yescharacteristic
14C2×#13133193—21no—
15C2×#13133188—21no—
16C2×#13133185C411yescharacteristic, complemented
17C2×#13133185C2211yescharacteristic
18C2×C4×C22⋊D53—21no—
19C2×C4×C22⋊D53—21no—
20C2×#13133176—21no—
21C2×#13133176C411yescharacteristic, complemented
22#16810597—22no—
23#16810597C2214yescomplemented
24C2×#13133132—21no—
25C2×#13133132C2212yes—
26#16810414C2218yescomplemented
27#16810411C2218yescomplemented
28C22×#41×C53C2211yescharacteristic
29C22×M16×C53—21no—
30C53×#413—22no—
31C53×#413C2214yescomplemented
32C2×#101×C53—21no—
33C2×#100×C53C2212yes—
34C24×D53C2×C411yescharacteristic
35C22×C22⋊D53—21no—
36C22×C22⋊D53—21no—
37C23×Dic53C2×C411yescharacteristic
38C22×C4×D53—21no—
39C22×C4×D53—21no—
40C22×C4×D53—22no—
41C22×C4×D53—42no—
42C22×C4×D53—41no—
43C22×C4×D53—41no—
44C22×C4×D53C2×C414yescomplemented
45#13133198—22no—
46#13133198—21no—
47#13133198—21no—
48#13133198C2×C412yes—
49#13133198C2×C414yescomplemented
50#13133198C2312yes—
51#13133193—24no—
52#13133193—24no—
53#13133188—24no—
54#13133188—24no—
55#13133185—22no—
56#13133185—21no—
57#13133185—21no—
58#13133185C2×C412yes—
59#13133185C2×C414yescomplemented
60#13133185C2312yes—
61C4×C22⋊D53—24no—
62C4×C22⋊D53—24no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65#13133176—22no—
66#13133176—24no—
67#13133176—42no—
68#13133176C2×C414yescomplemented
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—41no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—22no—
76C2×C22⋊Dic53C2×C411yescharacteristic
77C2×C22⋊Dic53C2×C411yescharacteristic
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—41no—
82#13133132—24no—
83#13133132—22no—
84#13133132C2314yes—
85C2×#41×C53—41no—
86C2×#41×C53—41no—
87C2×#41×C53—42no—
88C2×#41×C53C2×C412yes—
89C2×#41×C53D412yes—
90C2×#41×C53D412yes—
91C2×#41×C53C2312yes—
92C22×Q8×C53D411yescharacteristic
93C22×D4×C53C2×C411yescharacteristic
94C22×D4×C53D411yescharacteristic
95C22×D4×C53C2311yescharacteristic
96C23×C4×C53C2×C411yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99C2×M16×C53—22no—
100C2×M16×C53—24no—
101C2×M16×C53—41no—
102C2×M16×C53—42no—
103C22×C8×C53—41no—
104#101×C53—24no—
105#100×C53—22no—
106#100×C53C2314yes—
107C24×C53C22⋊C411yescharacteristic
108C24×C53C22×C411yescharacteristic
109C24×C53C2×D411yescharacteristic
110C23×D53—21no—
111C23×D53—21no—
112C23×D53—44no—
113C23×D53—41no—
114C23×D53—41no—
115C23×D53C22×C412yes—
116#41×C53—42no—
117#41×C53—42no—
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—42no—
123#41×C53—42no—
124C2×Q8×C53—41no—
125C2×Q8×C53—41no—
126C2×Q8×C53C22⋊C412yes—
127C2×Q8×C53C2×D412yes—
128C2×D4×C53—41no—
129C2×D4×C53—41no—
130C2×D4×C53—41no—
131C2×D4×C53—41no—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53C22⋊C412yes—
135C2×D4×C53C22⋊C412yes—
136C2×D4×C53C22×C412yes—
137C2×D4×C53C22×C411yescharacteristic
138C2×D4×C53C2×D412yes—
139C2×D4×C53C2×D411yescharacteristic
140C2×D4×C53C2×D411yescharacteristic
141C2×D4×C53C2411yescharacteristic
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—22no—
151C22×C4×C53C22⋊C411yescharacteristic
152C22×C4×C53C22⋊C412yes—
153C22×C4×C53C22⋊C411yescharacteristic
154C22×C4×C53C22⋊C412yes—
155C22×C4×C53C22⋊C412yes—
156C22×C4×C53C22⋊C411yescharacteristic
157C22×C4×C53C22×C412yes—
158C22×C4×C53C2×D412yes—
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D412yes—
161C2×C22⋊D53—22no—
162C2×C22⋊D53—24no—
163C2×C22⋊D53—22no—
164C2×C22⋊D53—44no—
165C2×C22⋊D53—44no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—41no—
171C22×Dic53—41no—
172C22×Dic53—42no—
173C22×Dic53C22×C412yes—
174C2×C4×D53—24no—
175C2×C4×D53—24no—
176C2×C4×D53—42no—
177C2×C4×D53—42no—
178C2×C4×D53—44no—
179C2×C4×D53—44no—
180C2×C4×D53—42no—
181C2×C4×D53—41no—
182C2×C4×D53—41no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185M16×C53—24no—
186M16×C53—42no—
187M16×C53—44no—
188C2×C8×C53—42no—
189C2×C8×C53—44no—
190C22⋊Dic53—22no—
191C22⋊Dic53—24no—
192C22⋊Dic53—21no—
193C22⋊Dic53—21no—
194C22⋊Dic53—22no—
195C22⋊Dic53—24no—
196C22⋊Dic53—42no—
197C22⋊Dic53C22×C412yes—
198C22⋊Dic53C22×C412yes—
199#11719921—22no—
200#11719921—24no—
201#11719921—22no—
202C4⋊Dic53—24no—
203C4⋊Dic53—44no—
204#11719919—22no—
205#11719919—24no—
206#11719919—22no—
207C4×Dic53—24no—
208C4×Dic53—44no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53C2×C22⋊C411yescharacteristic
219C23×C53C2×C22⋊C411yescharacteristic
220C23×C53C2×C22⋊C411yescharacteristic
221C23×C53C2×C22⋊C412yes—
222C23×C53C23×C411yescharacteristic
223C23×C53C22×D411yescharacteristic
224C22×D53—24no—
225C22×D53—22no—
226C22×D53—44no—
227C22×D53—44no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—48no—
231C22×D53—44no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—42no—
235C22×D53—44no—
236Q8×C53—42no—
237Q8×C53—42no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—41no—
242D4×C53—42no—
243D4×C53—41no—
244D4×C53—41no—
245D4×C53—41no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—22no—
261C2×C4×C53—22no—
262C2×C4×C53—22no—
263C2×C4×C53—22no—
264C2×C4×C53#9912yes—
265C2×C4×C53#9911yescharacteristic
266C2×C4×C53#9911yescharacteristic
267C2×C4×C53C2×C22⋊C412yes—
268C2×C4×C53C2×C22⋊C412yes—
269C2×C4×C53C2×C22⋊C411yescharacteristic
270C2×C4×C53C2×C22⋊C412yes—
271C2×C4×C53C22×D411yescharacteristic
272C22⋊D53—44no—
273C22⋊D53—48no—
274C22⋊D53—44no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—42no—
284C2×Dic53—44no—
285C2×Dic53—84no—
286C4×D53—48no—
287C4×D53—44no—
288C4×D53—44no—
289C8×C53—44no—
290C2×#2120—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—22no—
297C22×C53—42no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53C2×#9911yescharacteristic
309C22×C53C2×#9911yescharacteristic
310C22×C53C22×C22⋊C411yescharacteristic
311C2×D53—24no—
312C2×D53—42no—
313C2×D53—44no—
314C2×D53—48no—
315C2×D53—44no—
316C2×D53—42no—
317C2×D53—44no—
318C2×D53—44no—
319C4×C53—21no—
320C4×C53—22no—
321C4×C53—22no—
322C4×C53—21no—
323C4×C53C2×#9912yes—
324C4×C53C2×#9912yes—
325Dic53—42no—
326Dic53—42no—
327Dic53—42no—
328Dic53—84no—
329C2×#525—531no—
330C2×#413D5311yescharacteristic, complemented
331C2×#410—531no—
332#2120—538no—
333C2×#352—532no—
334C2×AΣL(1,𝔽16)—532no—
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—41no—
338C2×C53—41no—
339C2×C53—41no—
340C2×C53—41no—
341C2×C53—42no—
342C2×C53#212012yescomplemented
343C2×C53C22×#9911yescharacteristic
344D53—44no—
345C22×#41C2×D5311yescharacteristic
346C24×C4—531no—
347C22×M16—21no—
348#525—534no—
349#525—534no—
350#525—1062no—
351#525—1062no—
352C2×#123—531no—
353C2×#123—531no—
354C2×C22⋊Q8—1061no—
355C2×#121—1061no—
356C2×#120—531no—
357C2×#120—531no—
358C2×C4×D4—1061no—
359C2×C4×D4—1061no—
360C2×#117—531no—
361C2×#117—1061no—
362#413—22no—
363#413C2×D5314yescomplemented
364C2×#101—21no—
365C2×#100C2×D5312yes—
366#410—534no—
367#410—1062no—
368C2×#99—532no—
369C2×#99—1061no—
370#352—538no—
371AΣL(1,𝔽16)—538no—
372C53C2×#212011yescharacteristic, complemented, Sylow
373C25—531no—
374C2×#41—41no—
375C2×#41—42no—
376C2×#41—41no—
377C2×#41C4×D5312yes—
378C2×#41D21212yes—
379C2×#41C22⋊D5312yes—
380C2×#41C22×D5312yes—
381C22×Q8C22⋊D5311yescharacteristic
382C22×D4—1061no—
383C22×D4—1061no—
384C22×D4C4×D5311yescharacteristic
385C22×D4D21211yescharacteristic
386C22×D4C22×D5311yescharacteristic
387C23×C4—531no—
388C23×C4—534no—
389C23×C4—1061no—
390C23×C4—1062no—
391C23×C4—1061no—
392C23×C4—2121no—
393C23×C4—2122no—
394C23×C4—2121no—
395C23×C4C4×D5311yescharacteristic
396C23×C4D21211yescharacteristic
397C23×C4C22⋊D5311yescharacteristic
398C2×M16—22no—
399C2×M16—24no—
400C2×M16—41no—
401C2×M16—42no—
402C22×C8—41no—
403#123—532no—
404#123—534no—
405#123—532no—
406#123—1061no—
407#123—1061no—
408#123—1062no—
409C22⋊Q8—1064no—
410C22⋊Q8—1064no—
411#121—1064no—
412#121—1064no—
413#120—532no—
414#120—534no—
415#120—532no—
416#120—1061no—
417#120—1061no—
418#120—1062no—
419C4×D4—1064no—
420C4×D4—1064no—
421C4×D4—1064no—
422C4×D4—1064no—
423#117—534no—
424#117—1062no—
425#117—1064no—
426#117—2122no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—2121no—
431C2×C22⋊C4—531no—
432C2×C22⋊C4—531no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1062no—
438C2×C42—1061no—
439C2×C42—2121no—
440#101—24no—
441#100—22no—
442#100C22×D5314yes—
443#99—534no—
444#99—1062no—
445#99—1064no—
446C24—532no—
447C24—1061no—
448C24—1061no—
449C24—2121no—
450C24—2121no—
451C24—2124no—
452C24#1171992111yescharacteristic
453C24C2×C4×D5311yescharacteristic
454C24C2×D21211yescharacteristic
455#41—42no—
456#41—42no—
457#41—42no—
458#41—42no—
459#41—42no—
460#41—42no—
461#41—42no—
462#41—42no—
463C2×Q8—41no—
464C2×Q8—41no—
465C2×Q8#1171992112yes—
466C2×Q8C2×C22⋊D5312yes—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—1064no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—2124no—
477C2×D4—2124no—
478C2×D4#1171992112yes—
479C2×D4#1171992112yes—
480C2×D4C2×C4×D5312yes—
481C2×D4C2×C4×D5311yescharacteristic
482C2×D4C2×D21211yescharacteristic
483C2×D4C2×D21212yes—
484C2×D4C2×C22⋊D5311yescharacteristic
485C2×D4C23×D5311yescharacteristic
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—532no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1064no—
501C22×C4—1064no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4—2124no—
505C22×C4—2124no—
506C22×C4—2124no—
507C22×C4—2124no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4—2121no—
511C22×C4—2121no—
512C22×C4—2121no—
513C22×C4—2121no—
514C22×C4#1171992111yescharacteristic
515C22×C4#1171992111yescharacteristic
516C22×C4#1171992112yes—
517C22×C4#1171992112yes—
518C22×C4#1171992112yes—
519C22×C4#1171992111yescharacteristic
520C22×C4C2×C4×D5312yes—
521C22×C4C2×D21212yes—
522C22×C4C2×C22⋊D5311yescharacteristic
523C22×C4C2×C22⋊D5312yes—
524M16—24no—
525M16—42no—
526M16—44no—
527C2×C8—42no—
528C2×C8—44no—
529C4⋊C4—1064no—
530C4⋊C4—1064no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—2124no—
534C22⋊C4—532no—
535C22⋊C4—532no—
536C22⋊C4—1061no—
537C22⋊C4—1061no—
538C22⋊C4—1064no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1064no—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C22⋊C4—1064no—
545C22⋊C4—2122no—
546C42—1064no—
547C42—2124no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—1064no—
558C23—1062no—
559C23—2122no—
560C23—2122no—
561C23—2122no—
562C23—2124no—
563C23—2124no—
564C23—2124no—
565C23—2128no—
566C23—2124no—
567C23—2121no—
568C23—2121no—
569C23C2×#1171992111yescharacteristic
570C23C2×#1171992111yescharacteristic
571C23C2×#1171992112yes—
572C23C2×#1171992111yescharacteristic
573C23C22×C4×D5311yescharacteristic
574C23C22×D21211yescharacteristic
575Q8—42no—
576Q8—42no—
577D4—42no—
578D4—41no—
579D4—42no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—2124no—
586D4—2128no—
587D4—2124no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—4244no—
610C2×C4—2128no—
611C2×C4—2124no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2124no—
618C2×C4—2124no—
619C2×C4—2122no—
620C2×C4#1313313111yescharacteristic
621C2×C4#1313313112yes—
622C2×C4#1313313111yescharacteristic
623C2×C4C2×#1171992112yes—
624C2×C4C2×#1171992111yescharacteristic
625C2×C4C2×#1171992112yes—
626C2×C4C2×#1171992112yes—
627C2×C4C22×C22⋊D5311yescharacteristic
628C8—44no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—1064no—
647C22—2122no—
648C22—2124no—
649C22—2124no—
650C22—2124no—
651C22—2128no—
652C22—2122no—
653C22—2124no—
654C22C2×#1313313111yescharacteristic
655C22C2×#1313313111yescharacteristic
656C22C22×#1171992111yescharacteristic
657C4—21no—
658C4—22no—
659C4—21no—
660C4—22no—
661C4—4244no—
662C4—2122no—
663C4—2122no—
664C4—2122no—
665C4C2×#1313313112yes—
666C4C2×#1313313112yes—
667C2—21no—
668C2—21no—
669C2—41no—
670C2—41no—
671C2—41no—
672C2—41no—
673C2—42no—
674C2—2124no—
675C2#2027635712yesdirect product factor
676C2C22×#1313313111yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC23×C53C23×C4
FrattiniC2×D4C23×D53
FittingC2×C53×#413C2
socleC22×C53C2×#99
prime radicalC2×#16811130C2
cosocleC2×D4×C53C24
hypercenterC2×#413D53
hypocenterC53C2×#2120
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16810597C2semidirect
3. #20276357C2direct (N × H)
4. C2×#16810414C2semidirect
5. C2×#16810411C2semidirect
6. C2×C53×#413C2semidirect
7. C23×C4×D53C4semidirect
10. #16811130C4semidirect
12. C2×#13133198C4semidirect
16. C2×#13133185C4semidirect
21. C2×#13133176C4semidirect
23. #16810597C22semidirect
26. #16810414C22semidirect
27. #16810411C22semidirect
31. C53×#413C22semidirect
44. C22×C4×D53C2×C4semidirect
49. #13133198C2×C4semidirect
59. #13133185C2×C4semidirect
68. #13133176C2×C4semidirect
330. C2×#413D53semidirect
342. C2×C53#2120semidirect
363. #413C2×D53semidirect
372. C53C2×#2120semidirect
675. C2#20276357direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111302
C2×#168105972
#202763572
C2×#168104142
C2×#168104112
C2×C53×#4132
C2×#212053