Finite Group Databasebeta

← C2×#20275889 #27411518

Subgroups — C2×#20275889 #27411518

Order 13568.

1234 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811176C211yesmaximal, characteristic, complemented
2C2×#16810640C211yesmaximal, characteristic
3C22×#13133144C211yesmaximal, characteristic
4#20275889C218yesmaximal, direct product factor
5C2×#16810393C211yesmaximal, characteristic, complemented
6C2×#16810382C211yesmaximal, characteristic, complemented
7C2×#16810379C211yesmaximal, characteristic, complemented
8C2×C53×#410C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811176—22no—
12#16811176—22no—
13#16811176—22no—
14#16811176C2214yescomplemented
15C2×#13133216—21no—
16C2×#13133208—21no—
17C2×#13133194—22no—
18C2×C4×Dic106—22no—
19C2×#13133176C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21#16810640—22no—
22#16810640C2214yes—
23C2×#13133144—21no—
24C2×#13133144—21no—
25C2×#13133144—22no—
26C2×#13133144—22no—
27C2×#13133144C2214yes—
28#16810393C2214yescomplemented
29#16810382C2214yescomplemented
30#16810379C2214yescomplemented
31C22×#41×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C53×#410—22no—
35C53×#410C2214yes—
36C2×#99×C53—22no—
37C22×Dic106—22no—
38C23×Dic53—21no—
39C2×#11719917—21no—
40C2×#11719917—21no—
41C2×#11719917—21no—
42C2×#11719917—21no—
43C2×#11719917—21no—
44C2×#11719917—21no—
45C2×#11719917—21no—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917C2312yes—
50C2×#11719917C2312yes—
51C22×#88322—21no—
52C22×#88322—21no—
53#13133216—24no—
54#13133216—24no—
55#13133208—24no—
56#13133208—24no—
57#13133194—28no—
58#13133194—28no—
59C4×Dic106—28no—
60C4×Dic106—28no—
61#13133176—21no—
62#13133176—21no—
63#13133176D412yes—
64#13133176C2312yes—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—21no—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×C22⋊Dic53—21no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73C2×C4×Dic53D411yescharacteristic
74C2×C4×Dic53D411yescharacteristic
75C2×C4×Dic53D412yes—
76C2×C4×Dic53D412yes—
77C2×C4×Dic53C2312yes—
78#13133144—24no—
79#13133144—24no—
80#13133144—28no—
81C2×#41×C53—21no—
82C2×#41×C53—21no—
83C2×#41×C53—21no—
84C2×#41×C53—21no—
85C2×#41×C53—21no—
86C2×#41×C53—21no—
87C2×#41×C53—21no—
88C2×#41×C53—21no—
89C2×#41×C53—21no—
90C2×#41×C53—21no—
91C2×#41×C53Q812yescomplemented
92C2×#41×C53C2312yes—
93C22×Q8×C53Q811yescharacteristic, complemented
94C22×D4×C53—21no—
95C22×D4×C53Q811yescharacteristic, complemented
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98#117×C53—21no—
99#117×C53—21no—
100#117×C53—21no—
101#117×C53—21no—
102#117×C53—21no—
103#117×C53D412yes—
104#117×C53C2312yes—
105#117×C53C2312yes—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53D412yes—
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112#99×C53—28no—
113C24×C53—21no—
114C24×C53C2×D411yescharacteristic
115#41×C53—22no—
116#41×C53—22no—
117#41×C53—22no—
118#41×C53—22no—
119#41×C53—22no—
120#41×C53—22no—
121#41×C53—22no—
122#41×C53—22no—
123#41×C53—41no—
124#41×C53—41no—
125#41×C53—41no—
126#41×C53—41no—
127#41×C53—41no—
128#41×C53—41no—
129#41×C53—41no—
130#41×C53—41no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53C2×Q812yescomplemented
137C2×D4×C53—22no—
138C2×D4×C53—22no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—41no—
145C2×D4×C53—41no—
146C2×D4×C53—41no—
147C2×D4×C53—41no—
148C2×D4×C53C2×Q812yescomplemented
149C22×C4×C53—22no—
150C22×C4×C53—21no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53—41no—
156C22×C4×C53—41no—
157C22×C4×C53—41no—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×Q811yescharacteristic
164C22×C4×C53C2×Q811yescharacteristic
165C22×C4×C53#4111yescharacteristic
166C22×C4×C53#4111yescharacteristic
167C22×C4×C53#4111yescharacteristic
168C22×C4×C53#4111yescharacteristic
169C22×C4×C53#4111yescharacteristic
170C22×C4×C53#4111yescharacteristic
171C22×C4×C53C2411yescharacteristic
172C22×Dic53—21no—
173C22×Dic53—21no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—44no—
178C2×Dic106—28no—
179C2×Dic106—48no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—41no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—41no—
187C22⋊C4×C53C2×D414yes—
188C42×C53—22no—
189C42×C53—22no—
190C42×C53—22no—
191C42×C53—41no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C4⋊Dic53—28no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197#11719919—28no—
198#11719919—28no—
199C4×Dic53—22no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—22no—
203C4×Dic53—22no—
204C4×Dic53C2×D412yes—
205C4×Dic53C2×D412yes—
206C4×Dic53C2×D414yes—
207#11719917—22no—
208#11719917—22no—
209#11719917—22no—
210#11719917—22no—
211#11719917—41no—
212#11719917—41no—
213#11719917—41no—
214#11719917—41no—
215C2×#88322—22no—
216C2×#88322—22no—
217C2×#88322—41no—
218C2×#88322—41no—
219C2×#88322—41no—
220C2×#88322—41no—
221C23×C53—22no—
222C23×C53—42no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—82no—
226C23×C53#12011yescharacteristic
227C23×C53#12011yescharacteristic
228C23×C53C22⋊Q811yescharacteristic
229C23×C53C22⋊Q811yescharacteristic
230C23×C53#12411yescharacteristic
231C23×C53#12411yescharacteristic
232C23×C53C22×D411yescharacteristic
233Q8×C53—22no—
234Q8×C53—22no—
235Q8×C53—41no—
236Q8×C53—41no—
237D4×C53—22no—
238D4×C53—22no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—42no—
242D4×C53—42no—
243D4×C53—41no—
244D4×C53—41no—
245C2×C4×C53—24no—
246C2×C4×C53—22no—
247C2×C4×C53—44no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—82no—
256C2×C4×C53#12011yescharacteristic
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53C22⋊Q811yescharacteristic
259C2×C4×C53C22⋊Q811yescharacteristic
260C2×C4×C53C22⋊Q811yescharacteristic
261C2×C4×C53C22⋊Q811yescharacteristic
262C2×C4×C53C22⋊Q811yescharacteristic
263C2×C4×C53C22⋊Q811yescharacteristic
264C2×C4×C53C22⋊Q811yescharacteristic
265C2×C4×C53C22⋊Q811yescharacteristic
266C2×C4×C53C22⋊Q811yescharacteristic
267C2×C4×C53C22⋊Q811yescharacteristic
268C2×C4×C53#12411yescharacteristic
269C2×C4×C53#12411yescharacteristic
270C2×C4×C53#12411yescharacteristic
271C2×C4×C53#12411yescharacteristic
272C2×C4×C53#12411yescharacteristic
273C2×C4×C53#12411yescharacteristic
274C2×C4×C53#12411yescharacteristic
275C2×C4×C53#12411yescharacteristic
276C2×C4×C53#12411yescharacteristic
277C2×C4×C53#12411yescharacteristic
278C2×C4×C53C22×D411yescharacteristic
279C2×C4×C53C22×D411yescharacteristic
280C2×C4×C53C22×Q811yescharacteristic
281C2×C4×C53C2×#4111yescharacteristic
282C2×C4×C53C2×#4111yescharacteristic
283C2×C4×C53C2×#4111yescharacteristic
284C2×Dic53—24no—
285C2×Dic53—22no—
286C2×Dic53—24no—
287C2×Dic53—22no—
288C2×Dic53—44no—
289C2×Dic53—44no—
290C2×Dic53—44no—
291C2×Dic53—42no—
292Dic106—416no—
293#88322—42no—
294#88322—42no—
295C2×#2031—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—42no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53—82no—
307C22×C53#39311yescharacteristic
308C22×C53#39311yescharacteristic
309C22×C53#39311yescharacteristic
310C22×C53#39311yescharacteristic
311C22×C53C2×#12011yescharacteristic
312C22×C53C2×C22⋊Q811yescharacteristic
313C22×C53C2×#12411yescharacteristic
314C4×C53—44no—
315C4×C53—42no—
316C4×C53—82no—
317C4×C53#39311yescharacteristic
318C4×C53#39311yescharacteristic
319C4×C53#39311yescharacteristic
320C4×C53#39311yescharacteristic
321C4×C53C2×C22⋊Q811yescharacteristic
322C4×C53C2×C22⋊Q811yescharacteristic
323C4×C53C2×#12411yescharacteristic
324C4×C53C2×#12411yescharacteristic
325Dic53—24no—
326Dic53—44no—
327Dic53—42no—
328C2×#533—531no—
329C2×#421—531no—
330C22×#104—531no—
331C2×#410D5311yescharacteristic
332#2031—538no—
333C2×#344—531no—
334C2×#339—531no—
335C2×H(3,ℤ/4)—531no—
336C2×C53—21no—
337C2×C53—21no—
338C2×C53—21no—
339C2×C53—21no—
340C2×C53—21no—
341C2×C53—21no—
342C2×C53—82no—
343C2×C53#203112yescomplemented
344C2×C53C2×#39311yescharacteristic
345C22×#41C2×D5311yescharacteristic
346C22×M16—531no—
347C22×M16—531no—
348#533—534no—
349#533—1062no—
350#533—1062no—
351#533—1062no—
352C2×C4⋊Q8—1061no—
353C2×#125—1061no—
354C2×C22⋊Q8—1062no—
355C2×C4×Q8—1062no—
356C2×#117—531no—
357C2×#117C2×D5311yescharacteristic
358C2×#117C2×D5311yescharacteristic
359C22×C42—531no—
360#421—534no—
361#421—1062no—
362C2×#104—534no—
363C2×#104—1061no—
364C2×#104—1061no—
365C2×#104—1062no—
366C2×#104—1062no—
367#410—22no—
368#410C2×D5314yes—
369C2×#99—22no—
370#344—534no—
371#339—534no—
372H(3,ℤ/4)—534no—
373C53C2×#203111yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—21no—
377C2×#41—21no—
378C2×#41—21no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41Dic10612yescomplemented
385C2×#41C22×D5312yes—
386C22×Q8—1062no—
387C22×Q8Dic10611yescharacteristic, complemented
388C22×D4—21no—
389C22×D4Dic10611yescharacteristic, complemented
390C23×C4—21no—
391C23×C4—1061no—
392C23×C4C22×D5311yescharacteristic
393C2×M16—532no—
394C2×M16—532no—
395C2×M16—1061no—
396C2×M16—1061no—
397C2×M16—1061no—
398C2×M16—1061no—
399C2×M16—1061no—
400C2×M16—1061no—
401C2×M16—1061no—
402C2×M16—1061no—
403C2×M16—1061no—
404C2×M16—1061no—
405C22×C8—1061no—
406C22×C8—1061no—
407C4⋊Q8—1064no—
408C4⋊Q8—1064no—
409#125—1064no—
410#125—1064no—
411C22⋊Q8—1068no—
412C22⋊Q8—1068no—
413C4×Q8—1068no—
414C4×Q8—1068no—
415#117—21no—
416#117—21no—
417#117—21no—
418#117—21no—
419#117—21no—
420#117—532no—
421#117—532no—
422#117—1061no—
423#117—1061no—
424#117C22⋊D5312yes—
425#117C22×D5312yes—
426#117C22×D5312yes—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1061no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4C22⋊D5312yes—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—532no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—532no—
442C2×C42—532no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446#104—1064no—
447#104—1064no—
448#104—1068no—
449#99—28no—
450C24—21no—
451C24C2×C22⋊D5311yescharacteristic
452#41—22no—
453#41—22no—
454#41—22no—
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—22no—
460#41—41no—
461#41—41no—
462#41—41no—
463#41—41no—
464#41—41no—
465#41—41no—
466#41—41no—
467#41—41no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—1068no—
474C2×Q8—2128no—
475C2×Q8C2×Dic10612yescomplemented
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—41no—
487C2×D4C2×Dic10612yescomplemented
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—2124no—
503C22×C4C2×Dic10611yescharacteristic
504C22×C4C2×Dic10611yescharacteristic
505C22×C4#1171994511yescharacteristic
506C22×C4#1171994511yescharacteristic
507C22×C4D4×D5311yescharacteristic
508C22×C4D4×D5311yescharacteristic
509C22×C4D4×D5311yescharacteristic
510C22×C4D4×D5311yescharacteristic
511C22×C4#1171994711yescharacteristic
512C22×C4#1171994711yescharacteristic
513C22×C4#1171994711yescharacteristic
514C22×C4#1171994711yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C23×D5311yescharacteristic
517M16—1062no—
518M16—1062no—
519M16—1062no—
520M16—1062no—
521M16—2121no—
522M16—2121no—
523M16—2121no—
524M16—2121no—
525C2×C8—1062no—
526C2×C8—1062no—
527C2×C8—2121no—
528C2×C8—2121no—
529C2×C8—2121no—
530C2×C8—2121no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—41no—
535C4⋊C4—1068no—
536C4⋊C4—1068no—
537C4⋊C4—1068no—
538C4⋊C4—1062no—
539C4⋊C4—1062no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—41no—
543C22⋊C4—1062no—
544C22⋊C4—1062no—
545C22⋊C4C2×C22⋊D5314yes—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—41no—
550C42—532no—
551C42—532no—
552C42—534no—
553C42—1062no—
554C42—1061no—
555C42—1061no—
556C42—1062no—
557C42—1062no—
558C23—22no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—82no—
563C23#1313318511yescharacteristic
564C23#1313318511yescharacteristic
565C23#1313319611yescharacteristic
566C23#1313319611yescharacteristic
567C23#1313320411yescharacteristic
568C23#1313320411yescharacteristic
569C23C22×C22⋊D5311yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—21216no—
575D4—22no—
576D4—22no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—41no—
582D4—41no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—1064no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1064no—
598C2×C4—2124no—
599C2×C4—2124no—
600C2×C4—2122no—
601C2×C4—2124no—
602C2×C4#1313318511yescharacteristic
603C2×C4#1313318511yescharacteristic
604C2×C4#1313319411yescharacteristic
605C2×C4#1313319411yescharacteristic
606C2×C4#1313319411yescharacteristic
607C2×C4#1313319411yescharacteristic
608C2×C4#1313319411yescharacteristic
609C2×C4#1313319411yescharacteristic
610C2×C4#1313319411yescharacteristic
611C2×C4#1313319411yescharacteristic
612C2×C4#1313319611yescharacteristic
613C2×C4#1313319611yescharacteristic
614C2×C4#1313320311yescharacteristic
615C2×C4#1313320311yescharacteristic
616C2×C4#1313320311yescharacteristic
617C2×C4#1313320311yescharacteristic
618C2×C4#1313320311yescharacteristic
619C2×C4#1313320311yescharacteristic
620C2×C4#1313320311yescharacteristic
621C2×C4#1313320311yescharacteristic
622C2×C4#1313320411yescharacteristic
623C2×C4#1313320411yescharacteristic
624C2×C4C2×D4×D5311yescharacteristic
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4C22×Dic10611yescharacteristic
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994711yescharacteristic
629C2×C4C2×#1171994511yescharacteristic
630C8—2122no—
631C8—2122no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22#1681053511yescharacteristic
644C22#1681053511yescharacteristic
645C22#1681053511yescharacteristic
646C22#1681053511yescharacteristic
647C22C2×#1313318511yescharacteristic
648C22C2×#1313319611yescharacteristic
649C22C2×#1313320411yescharacteristic
650C4—44no—
651C4—42no—
652C4—82no—
653C4—1064no—
654C4—2122no—
655C4—2124no—
656C4#1681053511yescharacteristic
657C4#1681053511yescharacteristic
658C4#1681053511yescharacteristic
659C4#1681053511yescharacteristic
660C4C2×#1313319411yescharacteristic
661C4C2×#1313319411yescharacteristic
662C4C2×#1313320311yescharacteristic
663C4C2×#1313320311yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2#2027588912yesdirect product factor
672C2C2×#1681053511yescharacteristic
subgroupG/H
centerC2×C4#13133203
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#410C2
socleC22×C53#393
prime radicalC22×#41×C53C22
hypercenterC2×#410D53
hypocenterC53C2×#2031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. C2×#16811176C2semidirect
4. #20275889C2direct (N × H)
5. C2×#16810393C2semidirect
6. C2×#16810382C2semidirect
7. C2×#16810379C2semidirect
14. #16811176C22semidirect
28. #16810393C22semidirect
29. #16810382C22semidirect
30. #16810379C22semidirect
91. C2×#41×C53Q8semidirect
93. C22×Q8×C53Q8semidirect
95. C22×D4×C53Q8semidirect
136. C2×Q8×C53C2×Q8semidirect
148. C2×D4×C53C2×Q8semidirect
343. C2×C53#2031semidirect
373. C53C2×#2031semidirect
384. C2×#41Dic106semidirect
387. C22×Q8Dic106semidirect
389. C22×D4Dic106semidirect
475. C2×Q8C2×Dic106semidirect
487. C2×D4C2×Dic106semidirect
671. C2#20275889direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111762
C2×#168106402
C22×#131331442
#202758892
C2×#168103932
C2×#168103822
C2×#168103792
C2×C53×#4102
C2×#203153