Finite Group Databasebeta

← C2×#20275929 #27411682

Subgroups — C2×#20275929 #27411682

Order 13568.

1434 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic, complemented
2C22×#13133144C211yesmaximal, characteristic
3C2×#16810624C211yesmaximal, characteristic
4C22×#13133136C211yesmaximal, characteristic, complemented
5C2×#16810530C211yesmaximal, characteristic, complemented
6#20275929C218yesmaximal, direct product factor
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#536C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13#16810666—22no—
14#16810666C2214yescomplemented
15C2×#13133150—21no—
16C2×#13133148—21no—
17C2×#13133144—24no—
18C2×#13133144C2214yes—
19#16810624—22no—
20#16810624C2214yes—
21C2×#13133141—21no—
22C2×#13133138—21no—
23C2×#13133136—24no—
24C2×#13133136C2214yescomplemented
25#16810530—22no—
26#16810530C2214yescomplemented
27C2×#13133124—22no—
28#16810382C2214yescomplemented
29C22×#41×C53C2211yescharacteristic
30C23×Q8×C53C2211yescharacteristic
31C53×#536—22no—
32C53×#536—22no—
33C53×#536—22no—
34C53×#536C2214yes—
35C2×C4⋊Q8×C53—21no—
36C2×#124×C53—21no—
37C2×#123×C53—22no—
38C2×C22⋊Q8×C53—21no—
39C2×C22⋊Q8×C53—21no—
40C2×#117×C53C2211yescharacteristic
41C23×Dic53—21no—
42C23×Dic53—21no—
43C2×#11719917—21no—
44C2×#11719917—21no—
45C2×#11719917—22no—
46C2×#11719917—22no—
47C2×#11719917—22no—
48C2×#11719917D414yes—
49C2×#11719917C2312yes—
50C2×#11719917C2312yes—
51C22×#88322—21no—
52C22×#88322—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2312yes—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53C2312yes—
63#13133150—24no—
64#13133148—24no—
65#13133144—28no—
66#13133141—24no—
67#13133138—24no—
68#13133136—28no—
69#13133124—28no—
70C2×#41×C53—21no—
71C2×#41×C53—21no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—22no—
76C2×#41×C53D412yes—
77C2×#41×C53C2312yes—
78C22×Q8×C53—21no—
79C22×Q8×C53—21no—
80C22×Q8×C53—21no—
81C22×Q8×C53—21no—
82C22×Q8×C53—22no—
83C22×Q8×C53—22no—
84C22×Q8×C53—22no—
85C22×Q8×C53—22no—
86C22×Q8×C53D411yescharacteristic
87C22×Q8×C53D412yes—
88C22×Q8×C53C2312yes—
89C22×D4×C53—21no—
90C22×D4×C53D411yescharacteristic
91C23×C4×C53—21no—
92C23×C4×C53D412yes—
93C23×C4×C53C2311yescharacteristic
94C4⋊Q8×C53—24no—
95C4⋊Q8×C53—24no—
96#124×C53—24no—
97#124×C53—24no—
98#123×C53—28no—
99#123×C53—28no—
100C22⋊Q8×C53—24no—
101C22⋊Q8×C53—24no—
102C22⋊Q8×C53—24no—
103C22⋊Q8×C53—24no—
104#117×C53—21no—
105#117×C53—21no—
106#117×C53D412yes—
107#117×C53C2312yes—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53—22no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53—22no—
113C2×C22⋊C4×C53—22no—
114C2×C22⋊C4×C53—21no—
115C2×C42×C53—21no—
116C24×C53—21no—
117C24×C53C2×D411yescharacteristic
118#41×C53—24no—
119#41×C53—24no—
120#41×C53—22no—
121#41×C53—22no—
122#41×C53—22no—
123#41×C53—22no—
124#41×C53—42no—
125#41×C53—42no—
126#41×C53—42no—
127#41×C53—42no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—21no—
131C2×Q8×C53—24no—
132C2×Q8×C53—24no—
133C2×Q8×C53—22no—
134C2×Q8×C53—22no—
135C2×Q8×C53—22no—
136C2×Q8×C53—22no—
137C2×Q8×C53—22no—
138C2×Q8×C53—22no—
139C2×Q8×C53—22no—
140C2×Q8×C53—22no—
141C2×Q8×C53—42no—
142C2×Q8×C53—42no—
143C2×Q8×C53—42no—
144C2×Q8×C53—42no—
145C2×Q8×C53—42no—
146C2×Q8×C53—42no—
147C2×Q8×C53—42no—
148C2×Q8×C53—42no—
149C2×Q8×C53C2×D412yes—
150C2×D4×C53—22no—
151C2×D4×C53—22no—
152C2×D4×C53—21no—
153C2×D4×C53—22no—
154C2×D4×C53—22no—
155C2×D4×C53—42no—
156C2×D4×C53—42no—
157C2×D4×C53C2×D412yes—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—22no—
161C22×C4×C53—21no—
162C22×C4×C53—22no—
163C22×C4×C53—24no—
164C22×C4×C53—24no—
165C22×C4×C53—22no—
166C22×C4×C53—22no—
167C22×C4×C53—42no—
168C22×C4×C53—42no—
169C22×C4×C53C2×D414yes—
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2×D411yescharacteristic
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2×D412yes—
176C22×C4×C53#4112yes—
177C22×C4×C53#4112yes—
178C22×C4×C53#4112yes—
179C22×C4×C53C2411yescharacteristic
180C22×Dic53—24no—
181C22×Dic53—22no—
182C22×Dic53—22no—
183C22×Dic53—42no—
184C22×Dic53—42no—
185C22×Dic53—44no—
186C22×Dic53—41no—
187C22×Dic53—41no—
188C4⋊C4×C53—28no—
189C4⋊C4×C53—28no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53C2×D412yes—
193C4⋊C4×C53C2×D412yes—
194C22⋊C4×C53—28no—
195C22⋊C4×C53—28no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—22no—
198C42×C53—22no—
199C42×C53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—41no—
203C4⋊Dic53—41no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—42no—
207#11719917—24no—
208#11719917—22no—
209#11719917—22no—
210#11719917—22no—
211#11719917—42no—
212#11719917C2×D414yes—
213C2×#88322—24no—
214C2×#88322—22no—
215C2×#88322—22no—
216C2×#88322—42no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53—42no—
223C23×C53#12011yescharacteristic
224C23×C53#12011yescharacteristic
225C23×C53#12112yes—
226C23×C53#12312yes—
227C23×C53C22×D411yescharacteristic
228Q8×C53—22no—
229Q8×C53—22no—
230Q8×C53—42no—
231Q8×C53—42no—
232Q8×C53—42no—
233Q8×C53—42no—
234Q8×C53—44no—
235Q8×C53—44no—
236Q8×C53—42no—
237D4×C53—22no—
238D4×C53—22no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—44no—
242D4×C53—42no—
243C2×C4×C53—22no—
244C2×C4×C53—24no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—44no—
250C2×C4×C53—44no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—44no—
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53#12011yescharacteristic
259C2×C4×C53#12112yes—
260C2×C4×C53#12112yes—
261C2×C4×C53#12112yes—
262C2×C4×C53#12112yes—
263C2×C4×C53#12112yes—
264C2×C4×C53#12312yes—
265C2×C4×C53#12312yes—
266C2×C4×C53#12312yes—
267C2×C4×C53#12412yes—
268C2×C4×C53#12412yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C2×Dic53—24no—
276C2×Dic53—42no—
277C2×Dic53—48no—
278C2×Dic53—42no—
279C2×Dic53—44no—
280C2×Dic53—42no—
281C2×Dic53—82no—
282C2×Dic53—82no—
283#88322—24no—
284#88322—42no—
285C2×#2038—531nomaximal, Sylow
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—24no—
291C22×C53—24no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—84no—
296C22×C53#39412yes—
297C22×C53#39412yes—
298C22×C53#45412yes—
299C22×C53#45612yes—
300C22×C53C2×#12011yescharacteristic
301C22×C53C2×#12111yescharacteristic
302C22×C53C2×#12311yescharacteristic
303C4×C53—44no—
304C4×C53—42no—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53#39412yes—
308C4×C53#39412yes—
309C4×C53C2×#12111yescharacteristic
310C4×C53C2×#12111yescharacteristic
311C4×C53C2×#12311yescharacteristic
312C4×C53C2×#12411yescharacteristic
313Dic53—44no—
314Dic53—82no—
315C2×#536D5311yescharacteristic
316C2×#428—531no—
317C22×#104—531no—
318C2×#417—531no—
319C22×#101—531no—
320C2×#391—531no—
321#2038—538no—
322C2×#339—531no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—22no—
326C2×C53—82no—
327C2×C53#203814yescomplemented
328C2×C53C2×#39411yescharacteristic
329C2×C53C2×#45411yescharacteristic
330C2×C53C2×#45611yescharacteristic
331C22×#41C2×D5311yescharacteristic
332C23×Q8C2×D5311yescharacteristic
333C22×M16—531no—
334C22×M16—531no—
335#536—22no—
336#536—22no—
337#536—22no—
338#536C2×D5314yes—
339C2×C4⋊Q8—21no—
340C2×#124—21no—
341C2×#123—22no—
342C2×C22⋊Q8—21no—
343C2×C22⋊Q8—21no—
344C2×#117C2×D5311yescharacteristic
345C22×C4⋊C4—531no—
346C22×C42—531no—
347#428—534no—
348#428—1062no—
349C2×#107—1061no—
350C2×#106—1061no—
351C2×#104—534no—
352C2×#104—1064no—
353#417—534no—
354#417—1062no—
355C2×#103—1061no—
356C2×#102—1061no—
357C2×#101—534no—
358C2×#101—1064no—
359#391—534no—
360#391—1062no—
361C2×#95—1062no—
362#339—534no—
363C53C2×#203811yescharacteristic, complemented, Sylow
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41C22⋊D5312yes—
371C2×#41C22×D5312yes—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8—21no—
377C22×Q8—21no—
378C22×Q8—21no—
379C22×Q8—21no—
380C22×Q8C22⋊D5311yescharacteristic
381C22×Q8C22⋊D5312yes—
382C22×Q8C22×D5312yes—
383C22×D4—21no—
384C22×D4C22⋊D5311yescharacteristic
385C23×C4—21no—
386C23×C4—1061no—
387C23×C4—1061no—
388C23×C4C22⋊D5312yes—
389C23×C4C22×D5311yescharacteristic
390C2×M16—534no—
391C2×M16—532no—
392C2×M16—532no—
393C2×M16—1062no—
394C2×M16—1061no—
395C2×M16—1061no—
396C2×M16—1062no—
397C2×M16—1062no—
398C22×C8—1061no—
399C22×C8—1061no—
400C4⋊Q8—24no—
401C4⋊Q8—24no—
402#124—24no—
403#124—24no—
404#123—28no—
405#123—28no—
406C22⋊Q8—24no—
407C22⋊Q8—24no—
408C22⋊Q8—24no—
409C22⋊Q8—24no—
410#117—21no—
411#117—21no—
412#117C22⋊D5312yes—
413#117C22×D5312yes—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—532no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4C22⋊D5311yescharacteristic
423C2×C4⋊C4C22⋊D5311yescharacteristic
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—21no—
427C2×C42—21no—
428C2×C42—532no—
429C2×C42—1062no—
430C2×C42—1062no—
431C2×C42—1061no—
432#107—1064no—
433#106—1064no—
434#104—1068no—
435#103—1064no—
436#102—1064no—
437#101—1068no—
438#95—1068no—
439C24—21no—
440C24C2×C22⋊D5311yescharacteristic
441#41—24no—
442#41—24no—
443#41—22no—
444#41—22no—
445#41—22no—
446#41—22no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—42no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8C2×C22⋊D5312yes—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4C2×C22⋊D5312yes—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—1064no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—2124no—
496C22×C4—2121no—
497C22×C4—2121no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4D4×D5312yes—
501C22×C4#1171994712yes—
502C22×C4#1171994712yes—
503C22×C4#1171994712yes—
504C22×C4C2×C22⋊D5314yes—
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C23×D5311yescharacteristic
511M16—534no—
512M16—1064no—
513M16—1062no—
514M16—1062no—
515M16—1062no—
516M16—2122no—
517C2×C8—1064no—
518C2×C8—1062no—
519C2×C8—1062no—
520C2×C8—2122no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C4⋊C4—2121no—
528C4⋊C4—2121no—
529C4⋊C4C2×C22⋊D5312yes—
530C4⋊C4C2×C22⋊D5312yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C42—22no—
536C42—22no—
537C42—1062no—
538C42—1062no—
539C42—2122no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#1313318611yescharacteristic
547C23#1313318611yescharacteristic
548C23#1313318912yes—
549C23#1313319812yes—
550C23C22×C22⋊D5311yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—44no—
554Q8—44no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560D4—22no—
561D4—22no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—1064no—
581C2×C4—4242no—
582C2×C4—4242no—
583C2×C4—2124no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2128no—
587C2×C4—2122no—
588C2×C4#1313318611yescharacteristic
589C2×C4#1313318611yescharacteristic
590C2×C4#1313318812yes—
591C2×C4#1313318812yes—
592C2×C4#1313318912yes—
593C2×C4#1313318912yes—
594C2×C4#1313318912yes—
595C2×C4#1313319812yes—
596C2×C4#1313319812yes—
597C2×C4#1313319812yes—
598C2×C4#1313320412yes—
599C2×C4#1313320412yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4C22×C22⋊D5311yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994711yescharacteristic
606C8—1064no—
607C8—2122no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22#1681053712yes—
619C22#1681053712yes—
620C22#1681079612yes—
621C22#1681080612yes—
622C22C2×#1313318611yescharacteristic
623C22C2×#1313318911yescharacteristic
624C22C2×#1313319811yescharacteristic
625C4—42no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—4242no—
630C4—2124no—
631C4#1681053712yes—
632C4#1681053712yes—
633C4C2×#1313318811yescharacteristic
634C4C2×#1313318911yescharacteristic
635C4C2×#1313319811yescharacteristic
636C4C2×#1313320411yescharacteristic
637C2—21no—
638C2—22no—
639C2—21no—
640C2—82no—
641C2#2027592914yesdirect product factor
642C2C2×#1681053711yescharacteristic
643C2C2×#1681079611yescharacteristic
644C2C2×#1681080611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53#120
prime radicalC22×#41×C53C22
hypercenterC2×#536D53
hypocenterC53C2×#2038
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
4. C22×#13133136C2semidirect
5. C2×#16810530C2semidirect
6. #20275929C2direct (N × H)
7. C2×#16810382C2semidirect
14. #16810666C22semidirect
24. C2×#13133136C22semidirect
26. #16810530C22semidirect
28. #16810382C22semidirect
327. C2×C53#2038semidirect
363. C53C2×#2038semidirect
641. C2#20275929direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106662
C22×#131331442
C2×#168106242
C22×#131331362
C2×#168105302
#202759292
C2×#168103822
C2×C53×#5362
C2×#203853