Finite Group Databasebeta

← C2×#20275749 #27410990

Subgroups — C2×#20275749 #27410990

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133247C211yesmaximal, characteristic
2C22×#13133215C211yesmaximal, characteristic, complemented
3C22×#13133144C211yesmaximal, characteristic
4C2×#16810631C211yesmaximal, characteristic, complemented
5C22×#13133136C211yesmaximal, characteristic, complemented
6#20275749C218yesmaximal, direct product factor
7C2×#16810382C211yesmaximal, characteristic, complemented
8C2×C53×#536C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C2×#13133247—24no—
11C2×#13133247—24no—
12C2×#13133247—22no—
13C2×#13133247C2214yes—
14C22×#11719925—21no—
15C22×#11719923—21no—
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133215—24no—
19C2×#13133215—24no—
20C2×#13133215—22no—
21C2×#13133215C2214yescomplemented
22C22×#11719919—22no—
23C22×C4×Dic53C2211yescharacteristic
24C2×#13133144—24no—
25C2×#13133144C2214yes—
26#16810631—22no—
27#16810631C2214yescomplemented
28C2×#13133140—22no—
29C2×#13133136—24no—
30C2×#13133136C2214yescomplemented
31#16810382C2214yescomplemented
32C22×#41×C53C2211yescharacteristic
33C23×Q8×C53C2211yescharacteristic
34C53×#536—22no—
35C53×#536—22no—
36C53×#536—22no—
37C53×#536C2214yes—
38C2×C4⋊Q8×C53—21no—
39C2×#124×C53—21no—
40C2×#123×C53—22no—
41C2×C22⋊Q8×C53—22no—
42C2×#117×C53C2211yescharacteristic
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—21no—
46C22×Dic106—22no—
47C22×Dic106—21no—
48C22×Dic106—22no—
49C22×Dic106—21no—
50C22×Dic106—22no—
51C22×Dic106D414yes—
52C22×Dic106C2312yes—
53C23×Dic53—21no—
54C23×Dic53—21no—
55#13133247—216no—
56#13133247—28no—
57#13133247—28no—
58C2×#11719925—28no—
59C2×#11719925—24no—
60C2×#11719923—28no—
61C2×#11719923—24no—
62C2×#11719917—22no—
63C2×#11719917—21no—
64C2×#11719917—21no—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917D414yes—
68C2×#11719917C2312yes—
69C2×#11719917C2312yes—
70C22×#88322—21no—
71C22×#88322—21no—
72#13133215—216no—
73#13133215—28no—
74#13133215—28no—
75C2×#11719919—216no—
76C2×#11719919—28no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D414yes—
80C2×C4×Dic53C2312yes—
81#13133144—28no—
82#13133140—28no—
83#13133136—28no—
84C2×#41×C53—21no—
85C2×#41×C53—21no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53D412yes—
91C2×#41×C53C2312yes—
92C22×Q8×C53—21no—
93C22×Q8×C53—21no—
94C22×Q8×C53—21no—
95C22×Q8×C53—21no—
96C22×Q8×C53—22no—
97C22×Q8×C53—22no—
98C22×Q8×C53—22no—
99C22×Q8×C53—22no—
100C22×Q8×C53D411yescharacteristic
101C22×Q8×C53D412yes—
102C22×Q8×C53C2312yes—
103C22×D4×C53—21no—
104C22×D4×C53D411yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53D412yes—
107C23×C4×C53C2311yescharacteristic
108C4⋊Q8×C53—24no—
109C4⋊Q8×C53—24no—
110#124×C53—24no—
111#124×C53—24no—
112#123×C53—28no—
113#123×C53—28no—
114C22⋊Q8×C53—28no—
115C22⋊Q8×C53—28no—
116#117×C53—21no—
117#117×C53—21no—
118#117×C53D412yes—
119#117×C53C2312yes—
120C2×C4⋊C4×C53—24no—
121C2×C4⋊C4×C53D412yes—
122C2×C22⋊C4×C53—24no—
123C2×C22⋊C4×C53—21no—
124C2×C42×C53—21no—
125C24×C53—21no—
126C24×C53C2×D411yescharacteristic
127#41×C53—24no—
128#41×C53—24no—
129#41×C53—22no—
130#41×C53—22no—
131#41×C53—22no—
132#41×C53—22no—
133#41×C53—42no—
134#41×C53—42no—
135#41×C53—42no—
136#41×C53—42no—
137C2×Q8×C53—22no—
138C2×Q8×C53—22no—
139C2×Q8×C53—21no—
140C2×Q8×C53—24no—
141C2×Q8×C53—24no—
142C2×Q8×C53—22no—
143C2×Q8×C53—22no—
144C2×Q8×C53—22no—
145C2×Q8×C53—22no—
146C2×Q8×C53—22no—
147C2×Q8×C53—22no—
148C2×Q8×C53—22no—
149C2×Q8×C53—22no—
150C2×Q8×C53—42no—
151C2×Q8×C53—42no—
152C2×Q8×C53—42no—
153C2×Q8×C53—42no—
154C2×Q8×C53—42no—
155C2×Q8×C53—42no—
156C2×Q8×C53—42no—
157C2×Q8×C53—42no—
158C2×Q8×C53C2×D412yes—
159C2×D4×C53—22no—
160C2×D4×C53—22no—
161C2×D4×C53—22no—
162C2×D4×C53—21no—
163C2×D4×C53—22no—
164C2×D4×C53—42no—
165C2×D4×C53—42no—
166C2×D4×C53C2×D412yes—
167C22×C4×C53—22no—
168C22×C4×C53—22no—
169C22×C4×C53—21no—
170C22×C4×C53—22no—
171C22×C4×C53—24no—
172C22×C4×C53—24no—
173C22×C4×C53—22no—
174C22×C4×C53—22no—
175C22×C4×C53—42no—
176C22×C4×C53—42no—
177C22×C4×C53C2×D414yes—
178C22×C4×C53C2×D411yescharacteristic
179C22×C4×C53C2×D412yes—
180C22×C4×C53C2×D411yescharacteristic
181C22×C4×C53C2×D411yescharacteristic
182C22×C4×C53C2×D412yes—
183C22×C4×C53C2×D412yes—
184C22×C4×C53C2×D411yescharacteristic
185C22×C4×C53C2×D411yescharacteristic
186C22×C4×C53#4112yes—
187C22×C4×C53C2411yescharacteristic
188C22×Dic53—28no—
189C22×Dic53—24no—
190C22×Dic53—22no—
191C22×Dic53—24no—
192C22×Dic53—22no—
193C22×Dic53—41no—
194C22×Dic53—41no—
195C22×Dic53—42no—
196C2×Dic106—24no—
197C2×Dic106—24no—
198C2×Dic106—24no—
199C2×Dic106—24no—
200C2×Dic106—22no—
201C2×Dic106—22no—
202C2×Dic106—22no—
203C2×Dic106—22no—
204C2×Dic106—24no—
205C2×Dic106—22no—
206C2×Dic106—41no—
207C2×Dic106—41no—
208C2×Dic106—42no—
209C2×Dic106—42no—
210C2×Dic106—42no—
211C2×Dic106—42no—
212C2×Dic106—42no—
213C2×Dic106C2×D414yes—
214C4⋊C4×C53—216no—
215C4⋊C4×C53—22no—
216C4⋊C4×C53C2×D414yes—
217C22⋊C4×C53—216no—
218C22⋊C4×C53—22no—
219C22⋊C4×C53—22no—
220C42×C53—22no—
221C42×C53—22no—
222#11719925—216no—
223#11719923—216no—
224#11719919—232no—
225C4×Dic53—24no—
226C4×Dic53—22no—
227C4×Dic53C2×D414yes—
228#11719917—24no—
229#11719917—22no—
230#11719917—22no—
231#11719917—22no—
232#11719917—42no—
233#11719917C2×D414yes—
234C2×#88322—24no—
235C2×#88322—22no—
236C2×#88322—22no—
237C2×#88322—42no—
238C23×C53—24no—
239C23×C53—22no—
240C23×C53—42no—
241C23×C53—44no—
242C23×C53#12012yes—
243C23×C53#12011yescharacteristic
244C23×C53#12011yescharacteristic
245C23×C53#12112yes—
246C23×C53C22×D411yescharacteristic
247Q8×C53—22no—
248Q8×C53—22no—
249Q8×C53—42no—
250Q8×C53—42no—
251Q8×C53—42no—
252Q8×C53—42no—
253Q8×C53—44no—
254Q8×C53—44no—
255Q8×C53—42no—
256D4×C53—22no—
257D4×C53—22no—
258D4×C53—42no—
259D4×C53—42no—
260D4×C53—44no—
261D4×C53—42no—
262C2×C4×C53—22no—
263C2×C4×C53—24no—
264C2×C4×C53—24no—
265C2×C4×C53—24no—
266C2×C4×C53—24no—
267C2×C4×C53—44no—
268C2×C4×C53—44no—
269C2×C4×C53—42no—
270C2×C4×C53—44no—
271C2×C4×C53—42no—
272C2×C4×C53—42no—
273C2×C4×C53—44no—
274C2×C4×C53#12012yes—
275C2×C4×C53#12012yes—
276C2×C4×C53#12012yes—
277C2×C4×C53#12011yescharacteristic
278C2×C4×C53#12011yescharacteristic
279C2×C4×C53#12112yes—
280C2×C4×C53#12112yes—
281C2×C4×C53#12112yes—
282C2×C4×C53#12112yes—
283C2×C4×C53#12112yes—
284C2×C4×C53#12712yes—
285C2×C4×C53#12712yes—
286C2×C4×C53C22×D411yescharacteristic
287C2×C4×C53C22×D411yescharacteristic
288C2×C4×C53C22×D411yescharacteristic
289C2×C4×C53C22×D411yescharacteristic
290C2×C4×C53C22×D411yescharacteristic
291C2×C4×C53C2×#4111yescharacteristic
292C2×Dic53—28no—
293C2×Dic53—216no—
294C2×Dic53—24no—
295C2×Dic53—24no—
296C2×Dic53—42no—
297C2×Dic53—42no—
298C2×Dic53—44no—
299C2×Dic53—44no—
300Dic106—24no—
301Dic106—24no—
302Dic106—42no—
303Dic106—42no—
304Dic106—44no—
305Dic106—44no—
306#88322—24no—
307#88322—42no—
308C2×#2005—531nomaximal, Sylow
309C22×C53—22no—
310C22×C53—22no—
311C22×C53—24no—
312C22×C53—24no—
313C22×C53—42no—
314C22×C53—44no—
315C22×C53—84no—
316C22×C53#39212yes—
317C22×C53#39212yes—
318C22×C53#45614yes—
319C22×C53C2×#12011yescharacteristic
320C22×C53C2×#12011yescharacteristic
321C22×C53C2×#12111yescharacteristic
322C4×C53—44no—
323C4×C53—42no—
324C4×C53—44no—
325C4×C53#39212yes—
326C4×C53#39212yes—
327C4×C53C2×#12011yescharacteristic
328C4×C53C2×#12111yescharacteristic
329C4×C53C2×#12111yescharacteristic
330C4×C53C2×#12711yescharacteristic
331Dic53—28no—
332Dic53—44no—
333C22×#137—531no—
334C2×#536D5311yescharacteristic
335C22×C4⋊Q8—531no—
336C22×#104—531no—
337C2×#419—531no—
338C22×#101—531no—
339#2005—538no—
340C2×#339—531no—
341C2×C53—21no—
342C2×C53—21no—
343C2×C53—22no—
344C2×C53—82no—
345C2×C53#200514yescomplemented
346C2×C53C2×#39211yescharacteristic
347C2×C53C2×#45612yes—
348C22×#41C2×D5311yescharacteristic
349C23×Q8—531no—
350C23×Q8C2×D5311yescharacteristic
351C2×#137—534no—
352C2×#137—1064no—
353C2×#137—1064no—
354C2×#137—1062no—
355C22×Q16—1061no—
356C22×SD16—1061no—
357C22×M16—531no—
358C22×M16—531no—
359#536—22no—
360#536—22no—
361#536—22no—
362#536C2×D5314yes—
363C2×C4⋊Q8—21no—
364C2×C4⋊Q8—534no—
365C2×C4⋊Q8—1064no—
366C2×C4⋊Q8—1064no—
367C2×C4⋊Q8—1062no—
368C2×#124—21no—
369C2×#123—22no—
370C2×C22⋊Q8—22no—
371C2×#117C2×D5311yescharacteristic
372C22×C4⋊C4—1062no—
373C22×C42—531no—
374C2×#104—534no—
375C2×#104—1064no—
376#419—534no—
377#419—1062no—
378C2×#103—1062no—
379C2×#101—534no—
380C2×#101—1064no—
381#339—534no—
382C53C2×#200511yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D5312yes—
390C2×#41C22×D5312yes—
391C22×Q8—22no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—534no—
400C22×Q8—532no—
401C22×Q8—1062no—
402C22×Q8—1061no—
403C22×Q8—1062no—
404C22×Q8—1062no—
405C22×Q8—1061no—
406C22×Q8—1061no—
407C22×Q8—1061no—
408C22×Q8—1061no—
409C22×Q8C22⋊D5311yescharacteristic
410C22×Q8C22⋊D5312yes—
411C22×Q8C22×D5312yes—
412C22×D4—21no—
413C22×D4C22⋊D5311yescharacteristic
414C23×C4—21no—
415C23×C4—1061no—
416C23×C4—1061no—
417C23×C4C22⋊D5312yes—
418C23×C4C22×D5311yescharacteristic
419#137—1068no—
420#137—1068no—
421#137—10616no—
422C2×Q16—1068no—
423C2×Q16—1064no—
424C2×SD16—1068no—
425C2×SD16—1064no—
426C2×M16—534no—
427C2×M16—532no—
428C2×M16—532no—
429C2×M16—1062no—
430C2×M16—1061no—
431C2×M16—1061no—
432C2×M16—1062no—
433C2×M16—1062no—
434C22×C8—1061no—
435C22×C8—1061no—
436C4⋊Q8—24no—
437C4⋊Q8—24no—
438C4⋊Q8—1068no—
439C4⋊Q8—1068no—
440C4⋊Q8—10616no—
441#124—24no—
442#124—24no—
443#123—28no—
444#123—28no—
445C22⋊Q8—28no—
446C22⋊Q8—28no—
447#117—21no—
448#117—21no—
449#117C22⋊D5312yes—
450#117C22×D5312yes—
451C2×C4⋊C4—24no—
452C2×C4⋊C4—10616no—
453C2×C4⋊C4—1068no—
454C2×C4⋊C4C22⋊D5312yes—
455C2×C22⋊C4—24no—
456C2×C22⋊C4—21no—
457C2×C42—21no—
458C2×C42—534no—
459C2×C42—532no—
460C2×C42—1062no—
461C2×C42—1061no—
462#104—1068no—
463#103—1068no—
464#101—1068no—
465C24—21no—
466C24C2×C22⋊D5311yescharacteristic
467#41—24no—
468#41—24no—
469#41—22no—
470#41—22no—
471#41—22no—
472#41—22no—
473#41—42no—
474#41—42no—
475#41—42no—
476#41—42no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—534no—
499C2×Q8—1064no—
500C2×Q8—1062no—
501C2×Q8—1064no—
502C2×Q8—1064no—
503C2×Q8—1064no—
504C2×Q8—1064no—
505C2×Q8—1062no—
506C2×Q8—1062no—
507C2×Q8—1062no—
508C2×Q8—1062no—
509C2×Q8—2121no—
510C2×Q8—2121no—
511C2×Q8—2122no—
512C2×Q8—2122no—
513C2×Q8—2122no—
514C2×Q8—2122no—
515C2×Q8—2122no—
516C2×Q8C2×C22⋊D5312yes—
517C2×D4—22no—
518C2×D4—22no—
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—22no—
522C2×D4—42no—
523C2×D4—42no—
524C2×D4C2×C22⋊D5312yes—
525C22×C4—24no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—24no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—42no—
534C22×C4—42no—
535C22×C4—1068no—
536C22×C4—1064no—
537C22×C4—1062no—
538C22×C4—1064no—
539C22×C4—1062no—
540C22×C4—2121no—
541C22×C4—2121no—
542C22×C4—2122no—
543C22×C4D4×D5312yes—
544C22×C4D4×D5312yes—
545C22×C4D4×D5312yes—
546C22×C4#1171994712yes—
547C22×C4C2×C22⋊D5314yes—
548C22×C4C2×C22⋊D5311yescharacteristic
549C22×C4C2×C22⋊D5311yescharacteristic
550C22×C4C2×C22⋊D5311yescharacteristic
551C22×C4C2×C22⋊D5311yescharacteristic
552C22×C4C2×C22⋊D5311yescharacteristic
553C22×C4C23×D5311yescharacteristic
554Q16—10616no—
555SD16—10616no—
556M16—534no—
557M16—1064no—
558M16—1062no—
559M16—1062no—
560M16—1062no—
561M16—2122no—
562C2×C8—1064no—
563C2×C8—1062no—
564C2×C8—1062no—
565C2×C8—2122no—
566C4⋊C4—216no—
567C4⋊C4—22no—
568C4⋊C4—10632no—
569C4⋊C4C2×C22⋊D5314yes—
570C22⋊C4—216no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C42—22no—
574C42—22no—
575C42—534no—
576C42—1064no—
577C42—1062no—
578C23—24no—
579C23—22no—
580C23—42no—
581C23—44no—
582C23#1313318512yes—
583C23#1313318611yescharacteristic
584C23#1313318611yescharacteristic
585C23#1313318912yes—
586C23C22×C22⋊D5311yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—44no—
590Q8—44no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—1064no—
597Q8—1064no—
598Q8—2124no—
599Q8—2124no—
600Q8—2122no—
601Q8—2122no—
602D4—22no—
603D4—22no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—10616no—
621C2×C4—1068no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—2124no—
625C2×C4—2124no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4#1313318512yes—
629C2×C4#1313318512yes—
630C2×C4#1313318512yes—
631C2×C4#1313318611yescharacteristic
632C2×C4#1313318611yescharacteristic
633C2×C4#1313318812yes—
634C2×C4#1313318812yes—
635C2×C4#1313318912yes—
636C2×C4#1313318912yes—
637C2×C4#1313318912yes—
638C2×C4#1313321312yes—
639C2×C4#1313321312yes—
640C2×C4C2×D4×D5311yescharacteristic
641C2×C4C2×D4×D5311yescharacteristic
642C2×C4C2×D4×D5311yescharacteristic
643C2×C4C22×C22⋊D5311yescharacteristic
644C2×C4C22×C22⋊D5311yescharacteristic
645C2×C4C2×#1171994711yescharacteristic
646C8—1064no—
647C8—2122no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—42no—
653C22—44no—
654C22—84no—
655C22#1681053112yes—
656C22#1681053112yes—
657C22#1681080614yes—
658C22C2×#1313318511yescharacteristic
659C22C2×#1313318611yescharacteristic
660C22C2×#1313318911yescharacteristic
661C4—44no—
662C4—42no—
663C4—44no—
664C4—1068no—
665C4—2124no—
666C4#1681053112yes—
667C4#1681053112yes—
668C4C2×#1313318511yescharacteristic
669C4C2×#1313318811yescharacteristic
670C4C2×#1313318911yescharacteristic
671C4C2×#1313321311yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—82no—
676C2#2027574914yesdirect product factor
677C2C2×#1681053111yescharacteristic
678C2C2×#1681080612yes—
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#536C2
socleC23×C53#120
prime radicalC22×#41×C53C22
hypercenterC2×#536D53
hypocenterC53C2×#2005
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
2. C22×#13133215C2semidirect
4. C2×#16810631C2semidirect
5. C22×#13133136C2semidirect
6. #20275749C2direct (N × H)
7. C2×#16810382C2semidirect
21. C2×#13133215C22semidirect
27. #16810631C22semidirect
30. C2×#13133136C22semidirect
31. #16810382C22semidirect
345. C2×C53#2005semidirect
382. C53C2×#2005semidirect
676. C2#20275749direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332472
C22×#131332152
C22×#131331442
C2×#168106312
C22×#131331362
#202757492
C2×#168103822
C2×C53×#5362
C2×#200553