Finite Group Databasebeta

← C2×#20275082 #27385580

Subgroups — C2×#20275082 #27385580

Order 13568.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133247C211yesmaximal, characteristic
2C2×#16811106C211yesmaximal, characteristic
3C2×#16810633C211yesmaximal, characteristic, complemented
4C2×#16810529C211yesmaximal, characteristic, complemented
5#20275082C218yesmaximal, direct product factor
6C2×#16810391C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C2×#13133247—22no—
11C2×#13133247—24no—
12C2×#13133247C418yescomplemented
13C2×#13133247C2214yes—
14C22×#11719925—21no—
15C22×#11719923—21no—
16C22×#11719917C2211yescharacteristic
17#16811106—22no—
18#16811106—22no—
19#16811106—22no—
20#16811106C2214yes—
21C2×Q8×Dic53—21no—
22C2×D4×Dic53—22no—
23C2×D4×Dic53—21no—
24C2×#13133176—21no—
25C2×#13133176C2211yescharacteristic
26C22×C4⋊Dic53—21no—
27C22×C4⋊Dic53C2211yescharacteristic
28#16810633—22no—
29#16810633C2214yescomplemented
30C2×#13133142—21no—
31C2×#13133142—21no—
32#16810529—22no—
33#16810529C2214yescomplemented
34C2×#13133124—22no—
35#16810391C2214yescomplemented
36#16810383C2214yescomplemented
37C22×#41×C53C2211yescharacteristic
38C22×M16×C53C2211yescharacteristic
39C22×C42×C53C2211yescharacteristic
40C2×#104×C53—24no—
41C2×#104×C53C2214yes—
42C22×Dic106—21no—
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—22no—
46C22×Dic106—21no—
47C22×Dic106—22no—
48C22×Dic106—21no—
49C22×Dic106—22no—
50C22×Dic106C2×C414yescomplemented
51C22×Dic106C2312yes—
52C23×Dic53—21no—
53C23×Dic53—22no—
54C23×Dic53D412yes—
55#13133247—216no—
56#13133247—28no—
57#13133247C2×C4116yescomplemented
58C2×#11719925—28no—
59C2×#11719925—24no—
60C2×#11719923—28no—
61C2×#11719923—24no—
62C2×#11719917—21no—
63C2×#11719917—22no—
64C2×#11719917C2×C414yescomplemented
65C2×#11719917C2312yes—
66C22×#88322—21no—
67Q8×Dic53—24no—
68Q8×Dic53—24no—
69D4×Dic53—24no—
70D4×Dic53—28no—
71D4×Dic53—24no—
72D4×Dic53—28no—
73#13133176—24no—
74#13133176—24no—
75#13133176—21no—
76#13133176—21no—
77#13133176D412yes—
78#13133176C2312yes—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53—24no—
81C2×C4⋊Dic53—24no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53D411yescharacteristic
88C2×C4⋊Dic53D412yes—
89C2×C4⋊Dic53C2312yes—
90C2×C22⋊Dic53—22no—
91C2×C22⋊Dic53—21no—
92C2×C4×Dic53—22no—
93C2×C4×Dic53—21no—
94#13133142—24no—
95#13133142—24no—
96#13133124—28no—
97C2×#41×C53—21no—
98C2×#41×C53—21no—
99C2×#41×C53—22no—
100C2×#41×C53—22no—
101C2×#41×C53C2×C414yes—
102C2×#41×C53D414yes—
103C2×#41×C53D412yes—
104C2×#41×C53C2312yes—
105C22×Q8×C53D411yescharacteristic
106C22×D4×C53—21no—
107C22×D4×C53D411yescharacteristic
108C23×C4×C53—21no—
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C2×M16×C53—21no—
112C2×M16×C53—22no—
113C2×M16×C53—22no—
114C2×M16×C53C2312yes—
115C22×C8×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53—22no—
118C2×C42×C53—22no—
119C2×C42×C53C2312yes—
120#104×C53—28no—
121C24×C53—21no—
122C24×C53C2×D411yescharacteristic
123#41×C53—24no—
124#41×C53—24no—
125#41×C53—24no—
126#41×C53—24no—
127#41×C53—24no—
128#41×C53—22no—
129#41×C53—22no—
130#41×C53C22⋊C418yes—
131#41×C53C22×C414yes—
132#41×C53C2×D414yes—
133C2×Q8×C53—21no—
134C2×Q8×C53—22no—
135C2×Q8×C53C22⋊C414yes—
136C2×Q8×C53C2×D412yes—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—24no—
140C2×D4×C53—22no—
141C2×D4×C53—21no—
142C2×D4×C53—22no—
143C2×D4×C53C22⋊C414yes—
144C2×D4×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—24no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—22no—
150C22×C4×C53—24no—
151C22×C4×C53—22no—
152C22×C4×C53—42no—
153C22×C4×C53—42no—
154C22×C4×C53C22×C412yes—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53#4112yes—
162C22×C4×C53#4112yes—
163C22×C4×C53C2411yescharacteristic
164C22×Dic53—24no—
165C22×Dic53—21no—
166C22×Dic53—22no—
167C22×Dic53—21no—
168C22×Dic53—22no—
169C22×Dic53—28no—
170C22×Dic53—22no—
171C22×Dic53—41no—
172C22×Dic53—41no—
173C22×Dic53—42no—
174C22×Dic53—44no—
175C22×Dic53—84no—
176C22×Dic53C2×D414yes—
177C2×Dic106—24no—
178C2×Dic106—24no—
179C2×Dic106—24no—
180C2×Dic106—24no—
181C2×Dic106—22no—
182C2×Dic106—22no—
183C2×Dic106—22no—
184C2×Dic106—22no—
185C2×Dic106—24no—
186C2×Dic106—22no—
187C2×Dic106—41no—
188C2×Dic106—41no—
189C2×Dic106—42no—
190C2×Dic106—42no—
191C2×Dic106—42no—
192C2×Dic106—42no—
193C2×Dic106—42no—
194C2×Dic106C22×C414yescomplemented
195M16×C53—22no—
196M16×C53—22no—
197M16×C53—42no—
198C2×C8×C53—22no—
199C2×C8×C53—42no—
200C42×C53—22no—
201C42×C53—22no—
202C42×C53—42no—
203C22⋊Dic53—28no—
204C22⋊Dic53—22no—
205C22⋊Dic53—22no—
206#11719925—216no—
207#11719923—216no—
208C4⋊Dic53—28no—
209C4⋊Dic53—21no—
210C4⋊Dic53—22no—
211C4⋊Dic53—22no—
212C4⋊Dic53—21no—
213C4⋊Dic53—22no—
214C4⋊Dic53—22no—
215C4⋊Dic53C2×D412yes—
216C4⋊Dic53C2×D412yes—
217C4×Dic53—28no—
218C4×Dic53—22no—
219C4×Dic53—22no—
220#11719917—24no—
221#11719917—22no—
222#11719917C22×C414yescomplemented
223C2×#88322—24no—
224C2×#88322—22no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—24no—
229C23×C53—42no—
230C23×C53—42no—
231C23×C53C4×D412yes—
232C23×C53#12011yescharacteristic
233C23×C53#12011yescharacteristic
234C23×C53#12112yes—
235C23×C53C22×D411yescharacteristic
236Q8×C53—22no—
237Q8×C53—24no—
238Q8×C53C2×C22⋊C414yes—
239D4×C53—24no—
240D4×C53—24no—
241D4×C53—28no—
242D4×C53—22no—
243D4×C53—24no—
244D4×C53C2×C22⋊C414yes—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—28no—
250C2×C4×C53—24no—
251C2×C4×C53—48no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53C2×C22⋊C412yes—
255C2×C4×C53C2×C22⋊C412yes—
256C2×C4×C53C4×D412yes—
257C2×C4×C53C4×D412yes—
258C2×C4×C53C4×D412yes—
259C2×C4×C53#12011yescharacteristic
260C2×C4×C53#12011yescharacteristic
261C2×C4×C53#12112yes—
262C2×C4×C53#12112yes—
263C2×C4×C53#12112yes—
264C2×C4×C53#12312yes—
265C2×C4×C53#12312yes—
266C2×C4×C53C23×C411yescharacteristic
267C2×C4×C53C22×D411yescharacteristic
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×Dic53—24no—
273C2×Dic53—22no—
274C2×Dic53—24no—
275C2×Dic53—22no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—44no—
279C2×Dic53—44no—
280C2×Dic53—44no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—44no—
284C2×Dic53—82no—
285C2×Dic53—88no—
286Dic106—24no—
287Dic106—24no—
288Dic106—42no—
289Dic106—42no—
290Dic106—44no—
291Dic106—44no—
292C8×C53—42no—
293#88322—24no—
294C2×#1882—531nomaximal, Sylow
295C22×C53—24no—
296C22×C53—24no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—24no—
302C22×C53—48no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53#38612yes—
306C22×C53#38612yes—
307C22×C53#45512yes—
308C22×C53#45612yes—
309C22×C53C2×C4×D411yescharacteristic
310C22×C53C2×#12011yescharacteristic
311C22×C53C2×#12111yescharacteristic
312C4×C53—24no—
313C4×C53—44no—
314C4×C53#38612yes—
315C4×C53#38612yes—
316C4×C53C22×C22⋊C411yescharacteristic
317C4×C53C2×C4×D411yescharacteristic
318C4×C53C2×#12111yescharacteristic
319C4×C53C2×#12311yescharacteristic
320Dic53—44no—
321Dic53—42no—
322Dic53—42no—
323Dic53—44no—
324C22×#137—531no—
325C2×#520—531no—
326C22×#104D5311yescharacteristic
327C2×#419—531no—
328C2×#391—531no—
329#1882—538no—
330C2×#343—531no—
331C2×#339—531no—
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—22no—
335C2×C53—44no—
336C2×C53#188214yescomplemented
337C2×C53C2×#38611yescharacteristic
338C2×C53C2×#45511yescharacteristic
339C2×C53C2×#45611yescharacteristic
340C22×#41C2×D5311yescharacteristic
341C23×Q8—531no—
342C2×#137—538no—
343C2×#137—534no—
344C2×#137—1062no—
345C2×#137—1064no—
346C22×Q16—1061no—
347C22×SD16—1061no—
348C22×M16—531no—
349C22×M16C2×D5311yescharacteristic
350#520—534no—
351#520—1062no—
352#520—1062no—
353#520—1062no—
354C2×C4×Q8—1061no—
355C2×C4×D4—1062no—
356C2×C4×D4—1061no—
357C2×#117—531no—
358C2×#117—1061no—
359C22×C4⋊C4—531no—
360C22×C4⋊C4—1061no—
361C22×C42C2×D5311yescharacteristic
362C2×#104—24no—
363C2×#104C2×D5314yes—
364#419—534no—
365#419—1062no—
366C2×#103—1061no—
367C2×#103—1061no—
368#391—534no—
369#391—1062no—
370C2×#95—1062no—
371#343—534no—
372#339—534no—
373C53C2×#188211yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C2×#41—21no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41C4×D5314yes—
379C2×#41D21212yes—
380C2×#41C22⋊D5314yes—
381C2×#41C22×D5312yes—
382C22×Q8—534no—
383C22×Q8—532no—
384C22×Q8—1061no—
385C22×Q8—1062no—
386C22×Q8—1061no—
387C22×Q8—1061no—
388C22×Q8—1061no—
389C22×Q8—1061no—
390C22×Q8—1062no—
391C22×Q8—1062no—
392C22×Q8D21211yescharacteristic
393C22×D4—21no—
394C22×D4D21211yescharacteristic
395C23×C4—21no—
396C23×C4—21no—
397C23×C4—532no—
398C23×C4—1062no—
399C23×C4—1061no—
400C23×C4C22×D5311yescharacteristic
401#137—5316no—
402#137—1068no—
403#137—10616no—
404C2×Q16—1064no—
405C2×Q16—1068no—
406C2×SD16—1064no—
407C2×SD16—1068no—
408C2×M16—21no—
409C2×M16—22no—
410C2×M16—22no—
411C2×M16—534no—
412C2×M16—532no—
413C2×M16—1061no—
414C2×M16—1062no—
415C2×M16C22×D5312yes—
416C22×C8—21no—
417C22×C8—1061no—
418C4×Q8—1064no—
419C4×Q8—1064no—
420C4×D4—1068no—
421C4×D4—1068no—
422C4×D4—1064no—
423C4×D4—1064no—
424#117—532no—
425#117—532no—
426#117—1064no—
427#117—1064no—
428#117—1061no—
429#117—1061no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—532no—
433C2×C4⋊C4—532no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4—1064no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C22⋊C4—1062no—
442C2×C22⋊C4—1061no—
443C2×C42—22no—
444C2×C42—22no—
445C2×C42—21no—
446C2×C42—1062no—
447C2×C42—1061no—
448C2×C42C22×D5312yes—
449#104—28no—
450#103—1064no—
451#103—1064no—
452#95—1068no—
453C24—21no—
454C24D4×D5311yescharacteristic
455#41—24no—
456#41—24no—
457#41—24no—
458#41—24no—
459#41—24no—
460#41—22no—
461#41—22no—
462#41#1171992118yes—
463#41C2×C4×D5314yes—
464#41C2×C22⋊D5314yes—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—534no—
468C2×Q8—1064no—
469C2×Q8—1062no—
470C2×Q8—1064no—
471C2×Q8—1064no—
472C2×Q8—1064no—
473C2×Q8—1064no—
474C2×Q8—1062no—
475C2×Q8—1062no—
476C2×Q8—1062no—
477C2×Q8—1062no—
478C2×Q8—2121no—
479C2×Q8—2121no—
480C2×Q8—2122no—
481C2×Q8—2122no—
482C2×Q8—2122no—
483C2×Q8—2122no—
484C2×Q8—2122no—
485C2×Q8#1171992114yes—
486C2×Q8C2×D21212yes—
487C2×D4—24no—
488C2×D4—22no—
489C2×D4—24no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4#1171992114yes—
494C2×D4C2×D21212yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—534no—
505C22×C4—1068no—
506C22×C4—1062no—
507C22×C4—1064no—
508C22×C4—1061no—
509C22×C4—1061no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—4244no—
513C22×C4—2124no—
514C22×C4—2121no—
515C22×C4—2121no—
516C22×C4—2122no—
517C22×C4C2×C4×D5312yes—
518C22×C4C2×D21211yescharacteristic
519C22×C4C2×D21211yescharacteristic
520C22×C4D4×D5311yescharacteristic
521C22×C4D4×D5311yescharacteristic
522C22×C4D4×D5311yescharacteristic
523C22×C4#1171994712yes—
524C22×C4#1171994712yes—
525C22×C4C2×C22⋊D5312yes—
526C22×C4C23×D5311yescharacteristic
527Q16—10616no—
528SD16—10616no—
529M16—22no—
530M16—22no—
531M16—42no—
532M16—534no—
533M16—1064no—
534M16—1062no—
535C2×C8—22no—
536C2×C8—42no—
537C2×C8—1064no—
538C2×C8—1062no—
539C4⋊C4—532no—
540C4⋊C4—532no—
541C4⋊C4—1068no—
542C4⋊C4—1061no—
543C4⋊C4—1062no—
544C4⋊C4—1062no—
545C4⋊C4—1061no—
546C4⋊C4—1062no—
547C4⋊C4—1062no—
548C22⋊C4—1068no—
549C22⋊C4—1062no—
550C22⋊C4—1062no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—1068no—
555C42—1062no—
556C42—1062no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23#1313317712yes—
564C23#1313318711yescharacteristic
565C23#1313318711yescharacteristic
566C23#1313318912yes—
567C23C2×D4×D5311yescharacteristic
568Q8—24no—
569Q8—22no—
570Q8—1064no—
571Q8—1064no—
572Q8—2124no—
573Q8—2124no—
574Q8—2122no—
575Q8—2122no—
576Q8C2×#1171992114yes—
577D4—28no—
578D4—24no—
579D4—24no—
580D4—24no—
581D4—22no—
582D4C2×#1171992114yes—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—1064no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1064no—
596C2×C4—4242no—
597C2×C4—4248no—
598C2×C4—2124no—
599C2×C4—2124no—
600C2×C4—2122no—
601C2×C4—2124no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2124no—
606C2×C4C2×#1171992112yes—
607C2×C4C2×#1171992112yes—
608C2×C4#1313317712yes—
609C2×C4#1313317712yes—
610C2×C4#1313317712yes—
611C2×C4#1313318711yescharacteristic
612C2×C4#1313318711yescharacteristic
613C2×C4#1313318912yes—
614C2×C4#1313318912yes—
615C2×C4#1313318912yes—
616C2×C4#1313320112yes—
617C2×C4#1313320112yes—
618C2×C4C22×C4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4C22×C22⋊D5311yescharacteristic
621C2×C4C22×D21211yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C8—42no—
625C8—1064no—
626C22—24no—
627C22—24no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—24no—
633C22—48no—
634C22—42no—
635C22—42no—
636C22#1681050012yes—
637C22#1681050012yes—
638C22#1681080212yes—
639C22#1681080712yes—
640C22C2×#1313317711yescharacteristic
641C22C2×#1313318711yescharacteristic
642C22C2×#1313318911yescharacteristic
643C4—24no—
644C4—44no—
645C4—2124no—
646C4—2122no—
647C4—2124no—
648C4—2122no—
649C4#1681050012yes—
650C4#1681050012yes—
651C4C22×#1171992111yescharacteristic
652C4C2×#1313317711yescharacteristic
653C4C2×#1313318911yescharacteristic
654C4C2×#1313320111yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2—44no—
659C2#2027508214yesdirect product factor
660C2C2×#1681050011yescharacteristic
661C2C2×#1681080211yescharacteristic
662C2C2×#1681080711yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53#120
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC22×#104D53
hypocenterC53C2×#1882
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
3. C2×#16810633C2semidirect
4. C2×#16810529C2semidirect
5. #20275082C2direct (N × H)
6. C2×#16810391C2semidirect
7. C2×#16810383C2semidirect
12. C2×#13133247C4semidirect
29. #16810633C22semidirect
33. #16810529C22semidirect
35. #16810391C22semidirect
36. #16810383C22semidirect
50. C22×Dic106C2×C4semidirect
57. #13133247C2×C4semidirect
64. C2×#11719917C2×C4semidirect
194. C2×Dic106C22×C4semidirect
222. #11719917C22×C4semidirect
336. C2×C53#1882semidirect
373. C53C2×#1882semidirect
659. C2#20275082direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332472
C2×#168111062
C2×#168106332
C2×#168105292
#202750822
C2×#168103912
C2×#168103832
C22×#104×C532
C2×#188253