Finite Group Databasebeta

← C2×#20284638 #28251870

Subgroups — C2×#20284638 #28251870

Order 13568.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2#20284638C2116yesmaximal, direct product factor
3C2×#16810958C212yesmaximal, complemented
4C2×#16810941C212yesmaximal, complemented
5C2×#16810953C212yesmaximal, complemented
6C2×#16810936C212yesmaximal, complemented
7C2×#16810663C212yesmaximal, complemented
8C2×#16810622C212yesmaximal, complemented
9C2×#16810587C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C53C211yesmaximal, characteristic, complemented
11C22×#11719945C2211yescharacteristic
12C22×#11719917C2212yes—
13#16811134—24no—
14#16811134C2218yes—
15C2×#13133200—21no—
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C2×#13133191C2211yescharacteristic
20C2×#13133186—21no—
21C2×C4×C22⋊D53—21no—
22C2×C4×C22⋊D53—21no—
23C2×#13133176C2211yescharacteristic
24#16810958C22116yescomplemented
25#16810953C22116yescomplemented
26#16810941C22116yescomplemented
27#16810936C22116yescomplemented
28#16810663C22116yescomplemented
29C2×#13133150C2214yescomplemented
30C2×#13133148C2214yescomplemented
31#16810622C22116yescomplemented
32C2×#13133140C2214yescomplemented
33C2×#13133137C2214yescomplemented
34#16810587C2218yescomplemented
35C2×C22⋊#88322C2212yes—
36C2×C22⋊#88322C2212yes—
37C24×C4×C53—21no—
38C24×C4×C53C2211yescharacteristic
39C22×C4⋊C4×C53—24no—
40C22×C4⋊C4×C53—21no—
41C22×C4⋊C4×C53—21no—
42C22×C4⋊C4×C53—21no—
43C22×C4⋊C4×C53C2212yes—
44C22×C4⋊C4×C53C2212yes—
45C22×C4⋊C4×C53C2212yes—
46C22×C4⋊C4×C53C2218yescomplemented
47C2×#11719945—22no—
48C2×#11719945—28no—
49C2×#11719945C2314yes—
50C22×Dic106C2311yescharacteristic
51C22×D212C2311yescharacteristic
52C22×C22⋊D53—21no—
53C22×C22⋊D53—21no—
54C22×C22⋊D53—21no—
55C22×C4×D53—21no—
56C2×#11719917—28no—
57C2×#11719917C2318yes—
58C22×#88322C2314yes—
59#13133200—28no—
60#13133196—22no—
61#13133196—22no—
62#13133196C2314yes—
63#13133196C2314yes—
64#13133191—22no—
65#13133191—22no—
66#13133191C2314yes—
67#13133191C2314yes—
68#13133186—28no—
69C4×C22⋊D53—28no—
70C4×C22⋊D53—28no—
71#13133176—22no—
72#13133176C2314yes—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83#13133150C23116yescomplemented
84#13133148C23116yescomplemented
85#13133140C23116yescomplemented
86#13133137C23116yescomplemented
87C22⋊#88322C2318yes—
88C22⋊#88322C2318yes—
89C25×C53D411yescharacteristic
90C23×C4×C53—28no—
91C23×C4×C53—22no—
92C23×C4×C53—22no—
93C23×C4×C53—22no—
94C23×C4×C53—28no—
95C23×C4×C53—48no—
96C23×C4×C53D411yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99C23×C4×C53D414yes—
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2311yescharacteristic
102C23×C4×C53C2314yes—
103C23×C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53—24no—
107C2×C4⋊C4×C53—24no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53—24no—
110C2×C4⋊C4×C53—28no—
111C2×C4⋊C4×C53—28no—
112C2×C4⋊C4×C53—28no—
113C2×C4⋊C4×C53C2314yes—
114C2×C4⋊C4×C53C2318yes—
115C2×C4⋊C4×C53C2318yes—
116C2×C4⋊C4×C53C2318yes—
117C24×C53—24no—
118C24×C53—24no—
119C24×C53—24no—
120C24×C53C2×D414yes—
121C24×C53C2×D411yescharacteristic
122C24×C53C2×D411yescharacteristic
123C24×C53C2×D411yescharacteristic
124C23×D53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—24no—
127C22×C4×C53—24no—
128C22×C4×C53—216no—
129C22×C4×C53—24no—
130C22×C4×C53—24no—
131C22×C4×C53—216no—
132C22×C4×C53—28no—
133C22×C4×C53—28no—
134C22×C4×C53—28no—
135C22×C4×C53—24no—
136C22×C4×C53—48no—
137C22×C4×C53—48no—
138C22×C4×C53—432no—
139C22×C4×C53—48no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D414yes—
143C22×C4×C53C2×D414yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53#4118yes—
150C22×C4×C53C2412yes—
151C22×C4×C53C2412yes—
152C22×C4×C53C2411yescharacteristic
153C22×C4×C53C2412yes—
154C2×C22⋊D53—24no—
155C2×C22⋊D53—24no—
156C2×C22⋊D53—24no—
157C2×C22⋊D53—44no—
158C2×C22⋊D53—44no—
159C2×C22⋊D53—44no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163#11719945—216no—
164#11719945—48no—
165C2×D212—24no—
166C2×D212—21no—
167C2×D212C2412yes—
168C2×C4×D53—24no—
169C2×C4×D53—44no—
170C2×Dic106—24no—
171C2×Dic106—21no—
172C2×Dic106C2412yes—
173C4⋊C4×C53—28no—
174C4⋊C4×C53—28no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—28no—
177C4⋊C4×C53C2418yes—
178C22⋊Dic53—24no—
179C22⋊Dic53—24no—
180C22⋊Dic53—24no—
181#11719921—24no—
182#11719921—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53C2412yes—
186C4⋊Dic53C2412yes—
187#11719919—24no—
188#11719919—24no—
189C4×Dic53—24no—
190#11719917—216no—
191C2×#88322—28no—
192C2×#88322C2418yes—
193C23×C53—232no—
194C23×C53—28no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—28no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—28no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53#12314yes—
204C23×C53AGL(1,ℤ/8)14yes—
205C23×C53#13714yes—
206C23×C53C22×D412yes—
207C23×C53C22×D412yes—
208C23×C53C22×D412yes—
209C23×C53C22×D411yescharacteristic
210C22×D53—22no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C2×C4×C53—28no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—28no—
220C2×C4×C53—28no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—28no—
224C2×C4×C53—416no—
225C2×C4×C53—416no—
226C2×C4×C53—416no—
227C2×C4×C53—48no—
228C2×C4×C53#12314yes—
229C2×C4×C53#12318yes—
230C2×C4×C53C22×D412yes—
231C2×C4×C53C22×D412yes—
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D412yes—
234C2×C4×C53C2×#4114yes—
235C2×C4×C53C2×#4118yes—
236C2×C4×C53C2511yescharacteristic
237C22⋊D53—48no—
238C22⋊D53—48no—
239C22⋊D53—48no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246D212—24no—
247D212—42no—
248C4×D53—48no—
249Dic106—24no—
250Dic106—42no—
251#88322—28no—
252C2×#3092—531nomaximal, Sylow
253C22×C53—232no—
254C22×C53—28no—
255C22×C53—28no—
256C22×C53—28no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53C2×#12314yes—
264C22×C53C2×#12312yes—
265C22×C53C2×AGL(1,ℤ/8)14yes—
266C22×C53C2×AGL(1,ℤ/8)12yes—
267C22×C53C2×#13714yes—
268C22×C53C2×#13712yes—
269C22×C53C23×D411yescharacteristic
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—84no—
274C4×C53—22no—
275C4×C53—22no—
276C4×C53—48no—
277C4×C53C2×#12312yes—
278C4×C53C2×#12314yes—
279C4×C53C22×#4112yes—
280Dic53—42no—
281Dic53—42no—
282Dic53—42no—
283C2×#525—531no—
284C23×C4⋊C4D5311yescharacteristic, complemented
285#3092—5316no—
286C2×#484—532no—
287C2×#481—532no—
288C2×#483—532no—
289C2×#480—532no—
290C2×#428—532no—
291C2×#417—532no—
292C2×#407—531no—
293C2×C53—24no—
294C2×C53—24no—
295C2×C53—24no—
296C2×C53#309214yescomplemented
297C2×C53C22×#12311yescharacteristic
298C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
299C2×C53C22×#13711yescharacteristic
300D53—82no—
301C22×#41—531no—
302C24×C4—21no—
303C24×C4C2×D5311yescharacteristic
304C22×M16—532no—
305#525—538no—
306#525—1064no—
307C2×#123—1061no—
308C2×C22⋊Q8—531no—
309C2×C22⋊Q8—531no—
310C2×#121—531no—
311C2×#121—531no—
312C2×#120—1061no—
313C2×C4×D4—1061no—
314C2×C4×D4—1061no—
315C2×#117—531no—
316C22×C4⋊C4—24no—
317C22×C4⋊C4—21no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4C2×D5312yes—
321C22×C4⋊C4C2×D5312yes—
322C22×C4⋊C4C2×D5312yes—
323C22×C4⋊C4C2×D5318yescomplemented
324#484—5316no—
325#483—5316no—
326#481—5316no—
327#480—5316no—
328#428—5316no—
329C2×#107—534no—
330C2×#106—534no—
331#417—5316no—
332C2×#103—534no—
333C2×#102—534no—
334#407—538no—
335C2×C22⋊C8—532no—
336C2×C22⋊C8—532no—
337C53C2×#309211yescharacteristic, complemented, Sylow
338C25C22⋊D5311yescharacteristic
339C2×#41—534no—
340C2×#41—1062no—
341C2×#41—1068no—
342C22×Q8—531no—
343C22×D4—531no—
344C22×D4—1061no—
345C22×D4—1061no—
346C22×D4—1061no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—1061no—
354C23×C4C22⋊D5311yescharacteristic
355C23×C4C22⋊D5311yescharacteristic
356C23×C4C22⋊D5311yescharacteristic
357C23×C4C22⋊D5314yes—
358C23×C4C22×D5311yescharacteristic
359C23×C4C22×D5311yescharacteristic
360C23×C4C22×D5314yes—
361C23×C4C22×D5311yescharacteristic
362C2×M16—538no—
363C2×M16—1068no—
364C22×C8—534no—
365#123—1068no—
366C22⋊Q8—534no—
367C22⋊Q8—534no—
368C22⋊Q8—1062no—
369C22⋊Q8—1062no—
370#121—534no—
371#121—534no—
372#121—1062no—
373#121—1062no—
374#120—1068no—
375C4×D4—1068no—
376C4×D4—1068no—
377#117—534no—
378#117—1062no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C22×D5314yes—
393C2×C4⋊C4C22×D5318yes—
394C2×C4⋊C4C22×D5318yes—
395C2×C4⋊C4C22×D5318yes—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—1061no—
402#107—5316no—
403#106—5316no—
404#103—5316no—
405#102—5316no—
406C22⋊C8—538no—
407C22⋊C8—538no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—1061no—
412C24C2×C22⋊D5314yes—
413C24C2×C22⋊D5311yescharacteristic
414C24C2×C22⋊D5311yescharacteristic
415C24C2×C22⋊D5311yescharacteristic
416#41—10616no—
417#41—2128no—
418C2×Q8—532no—
419C2×Q8—1061no—
420C2×Q8—1064no—
421C2×D4—532no—
422C2×D4—1064no—
423C2×D4—1064no—
424C2×D4—1064no—
425C2×D4—1061no—
426C2×D4—1064no—
427C2×D4—2124no—
428C2×D4—2124no—
429C2×D4—2124no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1064no—
449C22×C4—2124no—
450C22×C4#1171994518yes—
451C22×C4C2×C22⋊D5314yes—
452C22×C4C2×C22⋊D5314yes—
453C22×C4C2×C22⋊D5312yes—
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5314yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5312yes—
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C23×D5312yes—
461C22×C4C23×D5312yes—
462C22×C4C23×D5312yes—
463C22×C4C23×D5311yescharacteristic
464M16—10616no—
465C2×C8—538no—
466C2×C8—1068no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—532no—
472C4⋊C4—532no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1064no—
476C4⋊C4—1064no—
477C4⋊C4C23×D5318yes—
478C22⋊C4—1064no—
479C22⋊C4—1064no—
480C22⋊C4—1064no—
481C22⋊C4—1064no—
482C22⋊C4—1064no—
483C42—1064no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—1062no—
495C23—2122no—
496C23—2122no—
497C23—2122no—
498C23#1313320014yes—
499C23#1313324314yes—
500C23#1313324814yes—
501C23C22×C22⋊D5312yes—
502C23C22×C22⋊D5312yes—
503C23C22×C22⋊D5312yes—
504C23C22×C22⋊D5311yescharacteristic
505Q8—1064no—
506Q8—2122no—
507D4—1064no—
508D4—2128no—
509D4—2128no—
510D4—2128no—
511D4—2122no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2128no—
533C2×C4#1313320014yes—
534C2×C4#1313320018yes—
535C2×C4C22×C22⋊D5312yes—
536C2×C4C22×C22⋊D5312yes—
537C2×C4C22×C22⋊D5312yes—
538C2×C4C22×C22⋊D5311yescharacteristic
539C2×C4C2×#1171994514yes—
540C2×C4C2×#1171994518yes—
541C2×C4C24×D5311yescharacteristic
542C8—1068no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—4244no—
554C22—2122no—
555C22—2122no—
556C22—2122no—
557C22C2×#1313320014yes—
558C22C2×#1313320012yes—
559C22C2×#1313324314yes—
560C22C2×#1313324312yes—
561C22C2×#1313324814yes—
562C22C2×#1313324812yes—
563C22C23×C22⋊D5311yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—2122no—
568C4—2122no—
569C4—2122no—
570C4C2×#1313320014yes—
571C4C2×#1313320012yes—
572C4C22×#1171994512yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—4242no—
577C2#2028463814yesdirect product factor
578C2C22×#1313320011yescharacteristic
579C2C22×#1313324311yescharacteristic
580C2C22×#1313324811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC23×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC2×#16811134C2
hypercenterC23×C4⋊C4D53
hypocenterC53C2×#3092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20284638C2direct (N × H)
3. C2×#16810958C2semidirect
4. C2×#16810941C2semidirect
5. C2×#16810953C2semidirect
6. C2×#16810936C2semidirect
7. C2×#16810663C2semidirect
8. C2×#16810622C2semidirect
9. C2×#16810587C2semidirect
10. C23×C4⋊C4×C53C2semidirect
24. #16810958C22semidirect
25. #16810953C22semidirect
26. #16810941C22semidirect
27. #16810936C22semidirect
28. #16810663C22semidirect
29. C2×#13133150C22semidirect
30. C2×#13133148C22semidirect
31. #16810622C22semidirect
32. C2×#13133140C22semidirect
33. C2×#13133137C22semidirect
34. #16810587C22semidirect
46. C22×C4⋊C4×C53C22semidirect
83. #13133150C23semidirect
84. #13133148C23semidirect
85. #13133140C23semidirect
86. #13133137C23semidirect
284. C23×C4⋊C4D53semidirect
296. C2×C53#3092semidirect
323. C22×C4⋊C4C2×D53semidirect
337. C53C2×#3092semidirect
577. C2#20284638direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111342
#202846382
C2×#168109582
C2×#168109412
C2×#168109532
C2×#168109362
C2×#168106632
C2×#168106222
C2×#168105872
C23×C4⋊C4×C532
C2×#309253