Finite Group Databasebeta

← C2×#20274998 #27385175

Subgroups — C2×#20274998 #27385175

Order 13568.

1994 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811098C211yesmaximal, characteristic
2C2×#16810503C211yesmaximal, characteristic
3C2×#16810498C212yesmaximal, complemented
4#20274998C218yesmaximal, direct product factor
5C2×#16810389C212yesmaximal, complemented
6C22×#99×C53C211yesmaximal, characteristic
7C24×Dic53C2211yescharacteristic
8#16811098—24no—
9#16811098—22no—
10#16811098C2214yes—
11C2×D4×Dic53—22no—
12C2×D4×Dic53—22no—
13C2×#13133176—21no—
14C22×C4⋊Dic53—22no—
15C22×C22⋊Dic53—21no—
16C22×C22⋊Dic53—21no—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2212yes—
19#16810503—22no—
20#16810503C2214yes—
21#16810498—24no—
22#16810498C2218yescomplemented
23C2×#13133124—22no—
24C2×#13133124—21no—
25#16810389C2218yescomplemented
26C23×D4×C53C2211yescharacteristic
27C22×C22⋊C4×C53C2212yes—
28C2×#99×C53C418yescomplemented
29C2×#99×C53C2214yes—
30C23×Dic53—21no—
31C23×Dic53—22no—
32C23×Dic53—21no—
33C23×Dic53—21no—
34C23×Dic53—22no—
35C23×Dic53—21no—
36C23×Dic53—21no—
37C23×Dic53—44no—
38C23×Dic53D412yes—
39C23×Dic53D412yes—
40C23×Dic53C2312yes—
41D4×Dic53—28no—
42D4×Dic53—28no—
43D4×Dic53—28no—
44D4×Dic53—28no—
45#13133176—24no—
46#13133176—24no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—28no—
50C2×C4⋊Dic53—48no—
51C2×C22⋊Dic53—22no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—22no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—24no—
61C2×C22⋊Dic53—44no—
62C2×C22⋊Dic53D414yes—
63C2×C22⋊Dic53C2312yes—
64C2×C22⋊Dic53C2314yes—
65C2×C4×Dic53—22no—
66#13133124—24no—
67#13133124—28no—
68C25×C53D411yescharacteristic
69C25×C53C2311yescharacteristic
70C22×D4×C53—22no—
71C22×D4×C53—22no—
72C22×D4×C53—24no—
73C22×D4×C53C2×C414yes—
74C22×D4×C53D412yes—
75C22×D4×C53Q814yes—
76C22×D4×C53C2312yes—
77C23×C4×C53—22no—
78C23×C4×C53D411yescharacteristic
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53—24no—
81C2×C22⋊C4×C53C2×C418yescomplemented
82C2×C22⋊C4×C53C2314yes—
83#99×C53C2×C4116yescomplemented
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—22no—
87C24×C53—42no—
88C24×C53—42no—
89C24×C53—42no—
90C24×C53—42no—
91C24×C53C4⋊C414yescomplemented
92C24×C53C22×C412yes—
93C24×C53C2×D411yescharacteristic
94C24×C53C2×D412yes—
95C24×C53C2×D412yes—
96C24×C53C2×D411yescharacteristic
97C24×C53C2×D412yes—
98C24×C53C2×Q812yes—
99C24×C53#4114yes—
100C24×C53C2411yescharacteristic
101C2×D4×C53—28no—
102C2×D4×C53—28no—
103C2×D4×C53—24no—
104C2×D4×C53—28no—
105C2×D4×C53—28no—
106C2×D4×C53—24no—
107C2×D4×C53—24no—
108C2×D4×C53—24no—
109C2×D4×C53C4⋊C418yescomplemented
110C2×D4×C53C22×C414yes—
111C2×D4×C53C2×Q814yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—28no—
116C22×C4×C53—24no—
117C22×C4×C53—44no—
118C22×C4×C53—44no—
119C22×C4×C53C4⋊C414yescomplemented
120C22×C4×C53C2×D412yes—
121C22×Dic53—24no—
122C22×Dic53—28no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—24no—
127C22×Dic53—24no—
128C22×Dic53—24no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—44no—
133C22×Dic53—48no—
134C22×Dic53—44no—
135C22×Dic53—44no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53—41no—
139C22×Dic53—41no—
140C22×Dic53—44no—
141C22×Dic53—44no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—82no—
146C22×Dic53—82no—
147C22×Dic53C2×D414yes—
148C22×Dic53C2×D414yes—
149C22⋊C4×C53—28no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53C22×C418yescomplemented
152C22⋊Dic53—28no—
153C22⋊Dic53—24no—
154C22⋊Dic53—24no—
155C22⋊Dic53—24no—
156C22⋊Dic53—24no—
157C22⋊Dic53—24no—
158C22⋊Dic53—48no—
159C22⋊Dic53—42no—
160C4⋊Dic53—24no—
161C4⋊Dic53—24no—
162C4⋊Dic53—416no—
163C4×Dic53—28no—
164C23×C53—24no—
165C23×C53—22no—
166C23×C53—24no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—24no—
170C23×C53—22no—
171C23×C53—24no—
172C23×C53—41no—
173C23×C53—42no—
174C23×C53—41no—
175C23×C53—44no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—44no—
179C23×C53—48no—
180C23×C53—41no—
181C23×C53—48no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—84no—
187C23×C53—81no—
188C23×C53—81no—
189C23×C53C2×C4⋊C412yes—
190C23×C53C2×C4⋊C412yes—
191C23×C53C2×C4⋊C414yescomplemented
192C23×C53C4×D414yes—
193C23×C53C4×D414yes—
194C23×C53#12012yes—
195C23×C53#12011yescharacteristic
196C23×C53#12011yescharacteristic
197C23×C53C22⋊Q814yes—
198C23×C53C22⋊Q814yes—
199C23×C53#12312yes—
200C23×C53#12312yes—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D412yes—
203C23×C53C22×D411yescharacteristic
204C23×C53C22×Q811yescharacteristic
205C23×C53C2×#4112yes—
206D4×C53—28no—
207D4×C53—28no—
208D4×C53—216no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—28no—
214C2×C4×C53—416no—
215C2×C4×C53—44no—
216C2×C4×C53—44no—
217C2×C4×C53C2×C4⋊C414yescomplemented
218C2×Dic53—28no—
219C2×Dic53—24no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222C2×Dic53—44no—
223C2×Dic53—42no—
224C2×Dic53—48no—
225C2×Dic53—44no—
226C2×Dic53—42no—
227C2×Dic53—44no—
228C2×Dic53—44no—
229C2×Dic53—44no—
230C2×Dic53—44no—
231C2×Dic53—44no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—82no—
236C2×Dic53—82no—
237C2×Dic53—84no—
238C2×Dic53—84no—
239C2×Dic53—84no—
240C2×#1867—531nomaximal, Sylow
241C22×C53—24no—
242C22×C53—28no—
243C22×C53—24no—
244C22×C53—28no—
245C22×C53—22no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—21no—
251C22×C53—22no—
252C22×C53—21no—
253C22×C53—24no—
254C22×C53—42no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—42no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—42no—
261C22×C53—44no—
262C22×C53—48no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—82no—
266C22×C53—84no—
267C22×C53#38512yes—
268C22×C53#38514yes—
269C22×C53#38512yes—
270C22×C53B(4,𝔽2)12yes—
271C22×C53#45812yes—
272C22×C53C22×C4⋊C411yescharacteristic
273C22×C53C2×C4×D412yes—
274C22×C53C2×#12011yescharacteristic
275C22×C53C2×C22⋊Q812yes—
276C22×C53C2×#12311yescharacteristic
277C4×C53—24no—
278C4×C53—48no—
279Dic53—44no—
280Dic53—44no—
281Dic53—44no—
282Dic53—82no—
283C2×#518—531no—
284C22×#99D5311yescharacteristic
285C2×#386—531no—
286C2×#385—532no—
287#1867—538no—
288C2×#342—532no—
289C2×C53—21no—
290C2×C53—22no—
291C2×C53—21no—
292C2×C53—22no—
293C2×C53—22no—
294C2×C53—24no—
295C2×C53—44no—
296C2×C53#186714yescomplemented
297C2×C53C2×#38511yescharacteristic
298C2×C53C2×B(4,𝔽2)11yescharacteristic
299C2×C53C2×#45811yescharacteristic
300C23×D4C2×D5311yescharacteristic
301C24×C4—531no—
302#518—534no—
303#518—1064no—
304#518—1062no—
305C2×C4×D4—1062no—
306C2×C4×D4—1062no—
307C2×#117—1061no—
308C22×C4⋊C4—1062no—
309C22×C22⋊C4—532no—
310C22×C22⋊C4—531no—
311C22×C22⋊C4—1061no—
312C22×C22⋊C4—1061no—
313C22×C22⋊C4C2×D5312yes—
314C2×#99Dic5318yescomplemented
315C2×#99C2×D5314yes—
316#386—534no—
317#386—1062no—
318#385—538no—
319#385—1064no—
320C2×#95—1062no—
321C2×#95—1061no—
322#342—538no—
323C53C2×#186711yescharacteristic, complemented, Sylow
324C25D21211yescharacteristic
325C25C22×D5311yescharacteristic
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—24no—
329C22×D4Dic10614yes—
330C22×D4D21212yes—
331C22×D4C2×Dic5314yes—
332C22×D4C22×D5312yes—
333C23×C4—22no—
334C23×C4—532no—
335C23×C4—532no—
336C23×C4—532no—
337C23×C4—1061no—
338C23×C4—1061no—
339C23×C4—1061no—
340C23×C4—1062no—
341C23×C4—1061no—
342C23×C4—1061no—
343C23×C4—1062no—
344C23×C4—2124no—
345C23×C4D21211yescharacteristic
346C4×D4—1068no—
347C4×D4—1068no—
348C4×D4—1068no—
349C4×D4—1068no—
350#117—1064no—
351#117—1064no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1068no—
355C2×C4⋊C4—2128no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—534no—
359C2×C22⋊C4—534no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1064no—
363C2×C22⋊C4—1062no—
364C2×C22⋊C4—1062no—
365C2×C22⋊C4—1064no—
366C2×C22⋊C4—1062no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1064no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1064no—
371C2×C22⋊C4—2124no—
372C2×C22⋊C4C2×Dic5318yescomplemented
373C2×C22⋊C4C22×D5314yes—
374C2×C42—1062no—
375#99C2×Dic53116yescomplemented
376#95—1068no—
377#95—1064no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C4⋊Dic5314yescomplemented
386C24C2×Dic10612yes—
387C24C2×D21211yescharacteristic
388C24C2×D21212yes—
389C24C2×D21211yescharacteristic
390C24D4×D5312yes—
391C24D4×D5312yes—
392C24#1171994714yes—
393C24C22×Dic5312yes—
394C24C23×D5311yescharacteristic
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4C4⋊Dic5318yescomplemented
404C2×D4C2×Dic10614yes—
405C2×D4C22×Dic5314yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—534no—
414C22×C4—534no—
415C22×C4—1064no—
416C22×C4—1064no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1064no—
420C22×C4—1068no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4—1064no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4—4242no—
427C22×C4—4242no—
428C22×C4—2124no—
429C22×C4—2124no—
430C22×C4—2128no—
431C22×C4—2122no—
432C22×C4—2122no—
433C22×C4—2122no—
434C22×C4—2124no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4—2122no—
438C22×C4—2121no—
439C22×C4—2122no—
440C22×C4—2121no—
441C22×C4C4⋊Dic5314yescomplemented
442C22×C4C2×D21212yes—
443C4⋊C4—1064no—
444C4⋊C4—1064no—
445C4⋊C4—21216no—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—1068no—
449C22⋊C4—1064no—
450C22⋊C4—1064no—
451C22⋊C4—1064no—
452C22⋊C4—1064no—
453C22⋊C4—1064no—
454C22⋊C4—2128no—
455C22⋊C4—2122no—
456C22⋊C4C22×Dic5318yescomplemented
457C42—1068no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—44no—
470C23—41no—
471C23—42no—
472C23—48no—
473C23—44no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—84no—
481C23—81no—
482C23—81no—
483C23C2×C4⋊Dic5312yes—
484C23C2×C4⋊Dic5312yes—
485C23C2×C4⋊Dic5314yescomplemented
486C23D4×Dic5314yes—
487C23D4×Dic5314yes—
488C23#1313318711yescharacteristic
489C23#1313318712yes—
490C23#1313318711yescharacteristic
491C23#1313319414yes—
492C23#1313319414yes—
493C23#1313320112yes—
494C23#1313320112yes—
495C23C23×Dic5311yescharacteristic
496C23C2×D4×D5312yes—
497C23C22×D21211yescharacteristic
498C23C22×Dic10611yescharacteristic
499C23C2×#1171994712yes—
500D4—216no—
501D4—28no—
502D4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—44no—
509C2×C4—416no—
510C2×C4—44no—
511C2×C4—1064no—
512C2×C4—1068no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—4244no—
516C2×C4—4242no—
517C2×C4—4242no—
518C2×C4—4244no—
519C2×C4—4244no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2124no—
524C2×C4—2122no—
525C2×C4—2128no—
526C2×C4—2124no—
527C2×C4—2124no—
528C2×C4—2124no—
529C2×C4—2124no—
530C2×C4—2122no—
531C2×C4—2124no—
532C2×C4—2124no—
533C2×C4C2×C4⋊Dic5314yescomplemented
534C22—28no—
535C22—24no—
536C22—24no—
537C22—28no—
538C22—22no—
539C22—21no—
540C22—22no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—24no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—44no—
555C22—48no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—84no—
560C22#1681049812yes—
561C22#1681049814yes—
562C22#1681049812yes—
563C22#1681081112yes—
564C22#1681081412yes—
565C22C22×C4⋊Dic5311yescharacteristic
566C22C2×D4×Dic5312yes—
567C22C2×#1313318711yescharacteristic
568C22C2×#1313319412yes—
569C22C2×#1313320111yescharacteristic
570C4—24no—
571C4—48no—
572C4—4242no—
573C4—2124no—
574C4—2124no—
575C4—2124no—
576C2—22no—
577C2—21no—
578C2—21no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—44no—
583C2#2027499814yesdirect product factor
584C2C2×#1681049811yescharacteristic
585C2C2×#1681081111yescharacteristic
586C2C2×#1681081411yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC22×#99×C53C2
socleC23×C53#120
prime radicalC23×D4×C53C22
cosocleC24×C53C24
hypercenterC22×#99D53
hypocenterC53C2×#1867
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. C2×#16810498C2semidirect
4. #20274998C2direct (N × H)
5. C2×#16810389C2semidirect
22. #16810498C22semidirect
25. #16810389C22semidirect
28. C2×#99×C53C4semidirect
81. C2×C22⋊C4×C53C2×C4semidirect
83. #99×C53C2×C4semidirect
91. C24×C53C4⋊C4semidirect
109. C2×D4×C53C4⋊C4semidirect
119. C22×C4×C53C4⋊C4semidirect
151. C22⋊C4×C53C22×C4semidirect
191. C23×C53C2×C4⋊C4semidirect
217. C2×C4×C53C2×C4⋊C4semidirect
296. C2×C53#1867semidirect
314. C2×#99Dic53semidirect
323. C53C2×#1867semidirect
372. C2×C22⋊C4C2×Dic53semidirect
375. #99C2×Dic53semidirect
385. C24C4⋊Dic53semidirect
403. C2×D4C4⋊Dic53semidirect
441. C22×C4C4⋊Dic53semidirect
456. C22⋊C4C22×Dic53semidirect
485. C23C2×C4⋊Dic53semidirect
533. C2×C4C2×C4⋊Dic53semidirect
583. C2#20274998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110982
C2×#168105032
C2×#168104982
#202749982
C2×#168103892
C22×#99×C532
C2×#186753