Finite Group Databasebeta

← C2×#20275173 #27385879

Subgroups — C2×#20275173 #27385879

Order 13568.

2066 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133200C211yesmaximal, characteristic
2C22×#13133131C212yesmaximal
3C2×#16810486C211yesmaximal, characteristic, complemented
4#20275173C218yesmaximal, direct product factor
5C2×#16810389C212yesmaximal, complemented
6C22×#117×C53C211yesmaximal, characteristic, complemented
7C23×C22⋊D53C2211yescharacteristic
8C2×#13133200—22no—
9C2×#13133200—22no—
10C2×#13133200—21no—
11C2×#13133200—21no—
12C2×#13133200—22no—
13C2×#13133200C414yescomplemented
14C2×#13133200C2214yes—
15C22×#11719919C412yescomplemented
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×#11719921C412yescomplemented
19C2×#13133131—28no—
20C2×#13133131C2218yes—
21#16810486—22no—
22#16810486C2214yescomplemented
23C2×#13133124—22no—
24#16810389C2218yescomplemented
25C24×C4×C53C2211yescharacteristic
26C2×#117×C53—22no—
27C2×#117×C53—28no—
28C2×#117×C53C2214yescomplemented
29C22×C4⋊C4×C53—21no—
30C22×C22⋊C4×C53C2212yes—
31C22×C42×C53—21no—
32C24×D53C2×C411yescharacteristic
33C22×C22⋊D53—24no—
34C22×C22⋊D53—21no—
35C22×C22⋊D53—21no—
36C22×C22⋊D53—21no—
37C22×C22⋊D53—21no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53C2×C412yes—
40C22×C22⋊D53C2312yes—
41C23×Dic53—22no—
42C23×Dic53C2×C411yescharacteristic
43C23×Dic53C2×C412yes—
44#13133200—28no—
45#13133200—24no—
46#13133200—24no—
47#13133200—28no—
48#13133200—24no—
49#13133200—24no—
50C2×#11719919—24no—
51C2×#11719919—22no—
52C2×#11719919—22no—
53C2×#11719919C2×C418yescomplemented
54C2×C22⋊Dic53—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—24no—
61C2×C22⋊Dic53C2×C412yes—
62C2×C22⋊Dic53C2314yes—
63C2×C22⋊Dic53C2312yes—
64C2×#11719921—24no—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921C2×C418yescomplemented
68#13133131—216no—
69#13133124—28no—
70C25×C53C2311yescharacteristic
71C23×C4×C53—24no—
72C23×C4×C53—24no—
73C23×C4×C53—22no—
74C23×C4×C53C2×C412yes—
75C23×C4×C53C2×C412yes—
76C23×C4×C53D412yes—
77C23×C4×C53D412yes—
78C23×C4×C53D411yescharacteristic
79C23×C4×C53D411yescharacteristic
80C23×C4×C53D412yes—
81C23×C4×C53C2312yes—
82#117×C53—216no—
83#117×C53—216no—
84C2×C4⋊C4×C53—28no—
85C2×C4⋊C4×C53—24no—
86C2×C22⋊C4×C53—24no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53D418yes—
89C2×C22⋊C4×C53C2314yes—
90C2×C42×C53—28no—
91C2×C42×C53—24no—
92C24×C53—44no—
93C24×C53C22×C411yescharacteristic
94C24×C53C22×C411yescharacteristic
95C24×C53C2×D411yescharacteristic
96C24×C53C2×D411yescharacteristic
97C24×C53C2×D411yescharacteristic
98C24×C53C2×D411yescharacteristic
99C24×C53C2×D414yes—
100C24×C53#4114yes—
101C24×C53C2411yescharacteristic
102C23×D53—21no—
103C23×D53—21no—
104C23×D53—24no—
105C23×D53—44no—
106C23×D53C22×C412yes—
107C22×C4×C53—28no—
108C22×C4×C53—28no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—24no—
112C22×C4×C53—28no—
113C22×C4×C53—28no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53—24no—
119C22×C4×C53—28no—
120C22×C4×C53—24no—
121C22×C4×C53—48no—
122C22×C4×C53C22⋊C414yes—
123C22×C4×C53C22⋊C412yes—
124C22×C4×C53C22⋊C412yes—
125C22×C4×C53C22⋊C414yes—
126C22×C4×C53C22⋊C412yes—
127C22×C4×C53C22⋊C412yes—
128C22×C4×C53C22×C414yes—
129C22×C4×C53C2×D414yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C2×C22⋊D53—22no—
133C2×C22⋊D53—22no—
134C2×C22⋊D53—28no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—22no—
137C2×C22⋊D53—22no—
138C2×C22⋊D53—24no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—44no—
142C2×C22⋊D53—44no—
143C2×C22⋊D53—42no—
144C2×C22⋊D53—42no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—24no—
150C22×Dic53—24no—
151C22×Dic53—44no—
152C22×Dic53—48no—
153C22×Dic53—44no—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C414yes—
156C4⋊C4×C53—216no—
157C22⋊C4×C53—28no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53C2×D418yes—
160C42×C53—216no—
161C22⋊Dic53—22no—
162C22⋊Dic53—22no—
163C22⋊Dic53—22no—
164C22⋊Dic53—22no—
165C22⋊Dic53—24no—
166C22⋊Dic53—24no—
167C22⋊Dic53—44no—
168#11719921—28no—
169#11719921—28no—
170#11719919—28no—
171#11719919—28no—
172C23×C53—24no—
173C23×C53—24no—
174C23×C53—24no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—48no—
179C23×C53—48no—
180C23×C53—42no—
181C23×C53C2×C22⋊C411yescharacteristic
182C23×C53C2×C22⋊C411yescharacteristic
183C23×C53C2×C22⋊C411yescharacteristic
184C23×C53C2×C22⋊C411yescharacteristic
185C23×C53C4×D414yes—
186C23×C53C4×D414yes—
187C23×C53#12012yes—
188C23×C53#12012yes—
189C23×C53#12114yes—
190C23×C53#12114yes—
191C23×C53#12312yes—
192C23×C53#12312yes—
193C23×C53C23×C411yescharacteristic
194C23×C53C22×D411yescharacteristic
195C23×C53C22×D412yes—
196C23×C53C22×D411yescharacteristic
197C23×C53C2×#4112yes—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—44no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—44no—
204C22×D53—44no—
205C22×D53—88no—
206C2×C4×C53—28no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—28no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—28no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—28no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—28no—
219C2×C4×C53—48no—
220C2×C4×C53—416no—
221C2×C4×C53C2×C22⋊C414yes—
222C2×C4×C53C2×C22⋊C414yes—
223C22⋊D53—42no—
224C22⋊D53—44no—
225C22⋊D53—42no—
226C22⋊D53—44no—
227C22⋊D53—44no—
228C2×Dic53—22no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—44no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—44no—
235C2×Dic53—44no—
236C2×Dic53—48no—
237C2×Dic53—44no—
238C2×Dic53—88no—
239C2×#1899—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—28no—
247C22×C53—24no—
248C22×C53—28no—
249C22×C53—24no—
250C22×C53—44no—
251C22×C53—42no—
252C22×C53—44no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53#38612yes—
257C22×C53#38612yes—
258C22×C53#38612yes—
259C22×C53#38612yes—
260C22×C53#41014yes—
261C22×C53C22×C22⋊C411yescharacteristic
262C22×C53C2×C4×D412yes—
263C22×C53C2×#12011yescharacteristic
264C22×C53C2×#12112yes—
265C22×C53C2×#12311yescharacteristic
266C2×D53—44no—
267C2×D53—42no—
268C2×D53—84no—
269C2×D53—84no—
270C2×D53—84no—
271C4×C53—24no—
272C4×C53—24no—
273C4×C53—48no—
274Dic53—44no—
275Dic53—42no—
276Dic53—84no—
277C22×#123—531no—
278C22×#117D5311yescharacteristic, complemented
279C22×#99—532no—
280C2×#381—531no—
281#1899—538no—
282C2×#342—532no—
283C2×C53—22no—
284C2×C53—21no—
285C2×C53—22no—
286C2×C53—22no—
287C2×C53—21no—
288C2×C53—22no—
289C2×C53—22no—
290C2×C53#189914yescomplemented
291C2×C53C2×#38611yescharacteristic
292C2×C53C2×#41012yes—
293D53—82no—
294C23×D4—531no—
295C24×C4C2×D5311yescharacteristic
296C2×#123—534no—
297C2×#123—534no—
298C2×#123—1062no—
299C2×#123—1062no—
300C2×#123—1061no—
301C2×#123—1062no—
302C2×#123—1061no—
303C2×#117—22no—
304C2×#117—28no—
305C2×#117C2×D5314yescomplemented
306C22×C4⋊C4—21no—
307C22×C4⋊C4—532no—
308C22×C22⋊C4—531no—
309C22×C22⋊C4—532no—
310C22×C22⋊C4—532no—
311C22×C22⋊C4C2×D5312yes—
312C22×C42—21no—
313C2×#99—538no—
314C2×#99—1068no—
315#381—534no—
316#381—1062no—
317C2×#95—1062no—
318#342—538no—
319C53C2×#189911yescharacteristic, complemented, Sylow
320C25—531no—
321C25C22×D5311yescharacteristic
322C22×D4—532no—
323C22×D4—532no—
324C22×D4—1061no—
325C22×D4—1061no—
326C22×D4—1064no—
327C22×D4—1061no—
328C22×D4—1061no—
329C22×D4—1064no—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4—531no—
334C23×C4—532no—
335C23×C4—1062no—
336C23×C4C4×D5312yes—
337C23×C4C4×D5312yes—
338C23×C4D21212yes—
339C23×C4D21212yes—
340C23×C4C22⋊D5311yescharacteristic
341C23×C4C22⋊D5312yes—
342C23×C4C22⋊D5311yescharacteristic
343C23×C4C22×D5312yes—
344#123—1068no—
345#123—1064no—
346#123—1064no—
347#123—1068no—
348#123—1064no—
349#123—1064no—
350#117—216no—
351#117—216no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—538no—
355C2×C4⋊C4—1064no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4—1062no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—538no—
363C2×C22⋊C4—534no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C22⋊C4—1061no—
367C2×C22⋊C4—1061no—
368C2×C22⋊C4—1064no—
369C2×C22⋊C4—1062no—
370C2×C22⋊C4—1062no—
371C2×C22⋊C4—1064no—
372C2×C22⋊C4—1064no—
373C2×C22⋊C4—1062no—
374C2×C22⋊C4C22⋊D5318yes—
375C2×C22⋊C4C22×D5314yes—
376C2×C42—28no—
377C2×C42—24no—
378#99—10616no—
379#95—1068no—
380C24—44no—
381C24—532no—
382C24—1064no—
383C24—1061no—
384C24—1061no—
385C24—2124no—
386C24C2×C4×D5311yescharacteristic
387C24C2×C4×D5311yescharacteristic
388C24C2×D21211yescharacteristic
389C24C2×D21211yescharacteristic
390C24#1171994514yes—
391C24C2×C22⋊D5311yescharacteristic
392C24C2×C22⋊D5311yescharacteristic
393C24C2×C22⋊D5314yes—
394C24C23×D5311yescharacteristic
395C2×D4—1062no—
396C2×D4—1062no—
397C2×D4—1068no—
398C2×D4—1064no—
399C2×D4—1062no—
400C2×D4—1062no—
401C2×D4—1064no—
402C2×D4—2122no—
403C2×D4—2122no—
404C2×D4—2124no—
405C2×D4—2124no—
406C2×D4—2122no—
407C2×D4—2122no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—532no—
424C22×C4—534no—
425C22×C4—1062no—
426C22×C4—1061no—
427C22×C4—1062no—
428C22×C4—1061no—
429C22×C4—1064no—
430C22×C4—1064no—
431C22×C4—2128no—
432C22×C4—2124no—
433C22×C4—2124no—
434C22×C4#1171992114yes—
435C22×C4#1171992114yes—
436C22×C4#1171992112yes—
437C22×C4#1171992112yes—
438C22×C4#1171992112yes—
439C22×C4#1171992112yes—
440C22×C4C2×C4×D5314yes—
441C22×C4C2×D21214yes—
442C22×C4C2×C22⋊D5312yes—
443C22×C4C2×C22⋊D5312yes—
444C4⋊C4—216no—
445C4⋊C4—1068no—
446C4⋊C4—1068no—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—1062no—
450C22⋊C4—1062no—
451C22⋊C4—1062no—
452C22⋊C4—1062no—
453C22⋊C4—1068no—
454C22⋊C4—1068no—
455C22⋊C4—1064no—
456C22⋊C4—1064no—
457C22⋊C4—2124no—
458C22⋊C4C2×C22⋊D5318yes—
459C42—216no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—48no—
466C23—42no—
467C23—48no—
468C23—42no—
469C23—1064no—
470C23—1062no—
471C23—4248no—
472C23—2124no—
473C23—2124no—
474C23—2122no—
475C23—2122no—
476C23—2124no—
477C23C2×#1171992111yescharacteristic
478C23C2×#1171992111yescharacteristic
479C23C2×#1171992111yescharacteristic
480C23C2×#1171992111yescharacteristic
481C23C4×C22⋊D5314yes—
482C23C4×C22⋊D5314yes—
483C23#1313318612yes—
484C23#1313318612yes—
485C23#1313319114yes—
486C23#1313319114yes—
487C23#1313320012yes—
488C23#1313320012yes—
489C23C22×C4×D5311yescharacteristic
490C23C22×C22⋊D5311yescharacteristic
491C23C22×C22⋊D5312yes—
492C23C22×D21211yescharacteristic
493C23C2×#1171994512yes—
494D4—2124no—
495D4—2124no—
496D4—2122no—
497D4—2124no—
498D4—2122no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—48no—
513C2×C4—416no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—1064no—
517C2×C4—4248no—
518C2×C4—2128no—
519C2×C4—2124no—
520C2×C4—2124no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2124no—
524C2×C4—2124no—
525C2×C4C2×#1171992114yes—
526C2×C4C2×#1171992114yes—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—28no—
535C22—28no—
536C22—24no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—4244no—
544C22—4244no—
545C22—4244no—
546C22—2124no—
547C22—2122no—
548C22#1681050212yes—
549C22#1681050212yes—
550C22#1681050212yes—
551C22#1681050212yes—
552C22#1681059614yes—
553C22C22×#1171992111yescharacteristic
554C22C2×C4×C22⋊D5312yes—
555C22C2×#1313318611yescharacteristic
556C22C2×#1313319112yes—
557C22C2×#1313320011yescharacteristic
558C4—24no—
559C4—24no—
560C4—48no—
561C4—4244no—
562C4—2124no—
563C4—2122no—
564C2—21no—
565C2—22no—
566C2—22no—
567C2—22no—
568C2—22no—
569C2—22no—
570C2—21no—
571C2—4242no—
572C2#2027517314yesdirect product factor
573C2C2×#1681050211yescharacteristic
574C2C2×#1681059612yes—
subgroupG/H
centerC23C2×#11719921
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC22×#117×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC24×C53C24
hypercenterC22×#117D53
hypocenterC53C2×#1899
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#16810486C2semidirect
4. #20275173C2direct (N × H)
5. C2×#16810389C2semidirect
6. C22×#117×C53C2semidirect
13. C2×#13133200C4semidirect
15. C22×#11719919C4semidirect
18. C22×#11719921C4semidirect
22. #16810486C22semidirect
24. #16810389C22semidirect
28. C2×#117×C53C22semidirect
53. C2×#11719919C2×C4semidirect
67. C2×#11719921C2×C4semidirect
278. C22×#117D53semidirect
290. C2×C53#1899semidirect
305. C2×#117C2×D53semidirect
319. C53C2×#1899semidirect
572. C2#20275173direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332002
C22×#131331312
C2×#168104862
#202751732
C2×#168103892
C22×#117×C532
C2×#189953