Finite Group Databasebeta

← C2×#20273273 #27351429

Subgroups — C2×#20273273 #27351429

Order 13568.

1366 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811168C211yesmaximal, characteristic
2C2×#16810673C211yesmaximal, characteristic, complemented
3C2×#16810487C211yesmaximal, characteristic, complemented
4#20273273C218yesmaximal, direct product factor
5C2×#16810358C211yesmaximal, characteristic, complemented
6C2×#16810357C211yesmaximal, characteristic, complemented
7C2×#16810356C211yesmaximal, characteristic, complemented
8C2×C53×#327C211yesmaximal, characteristic, complemented
9C22×#11719947—21no—
10#16811168—22no—
11#16811168—22no—
12#16811168C414yescomplemented
13#16811168C2214yes—
14C2×#13133216C411yescharacteristic, complemented
15C2×#13133213C411yescharacteristic, complemented
16C2×#13133203—21no—
17C2×#13133191C2211yescharacteristic
18C2×#13133189—21no—
19C2×D4×Dic53C411yescharacteristic, complemented
20C2×#13133179C411yescharacteristic, complemented
21C2×#13133177—21no—
22C22×#11719919—21no—
23C22×C4×Dic53C2211yescharacteristic
24#16810673—22no—
25#16810673C2214yescomplemented
26C2×#13133146—21no—
27C2×C22⋊#88322C2211yescharacteristic
28C2×C8×Dic53—21no—
29#16810487—22no—
30#16810487C2214yescomplemented
31C2×#13133124—21no—
32C2×#13133124C2211yescharacteristic
33#16810358C2214yescomplemented
34#16810357C2214yescomplemented
35#16810356C2214yescomplemented
36C2×#121×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C53×#327C2214yescomplemented
40C2×#11719947—24no—
41C2×#11719947—24no—
42C2×#11719947—44no—
43C2×#11719947—44no—
44C22×Dic106—21no—
45C22×D212C2×C411yescharacteristic
46C22×C22⋊D53—21no—
47C22×C22⋊D53—21no—
48C22×C22⋊D53—21no—
49C23×Dic53—21no—
50C23×Dic53—21no—
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C22×C4×D53—21no—
54C22×#88322—21no—
55#13133216—22no—
56#13133216C2×C414yescomplemented
57#13133213—22no—
58#13133213C2×C414yescomplemented
59#13133203—24no—
60#13133203—24no—
61#13133191—21no—
62#13133191—21no—
63#13133191C2×C412yes—
64#13133191C2312yes—
65#13133189—24no—
66#13133189—24no—
67D4×Dic53—22no—
68D4×Dic53C2×C414yescomplemented
69#13133179—22no—
70#13133179C2×C414yescomplemented
71#13133177—24no—
72#13133177—24no—
73C2×C4⋊Dic53C2×C411yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—24no—
76C2×#11719919—44no—
77C2×C22⋊Dic53—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C412yes—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D412yes—
86C2×C4×Dic53D412yes—
87C2×C4×Dic53C2312yes—
88#13133146—24no—
89C22⋊#88322—21no—
90C22⋊#88322C2312yes—
91C8×Dic53—24no—
92#13133124—21no—
93#13133124—24no—
94#13133124C2312yes—
95C22×D4×C53—21no—
96C22×D4×C53C2×C411yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—21no—
100#121×C53—21no—
101#121×C53—21no—
102#121×C53C2×C412yes—
103#121×C53C2312yes—
104C2×C4⋊C4×C53C2×C411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C22⋊C8×C53—21no—
107C22⋊C8×C53C2312yes—
108#95×C53—21no—
109#95×C53C2312yes—
110C24×C53—21no—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—42no—
119C2×D4×C53C22×C412yes—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C2×C22⋊D53—22no—
133C2×C22⋊D53—22no—
134C2×C22⋊D53—24no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—44no—
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—44no—
140C22×Dic53—21no—
141C22×Dic53—24no—
142C22×Dic53—22no—
143C22×Dic53—42no—
144C22×Dic53—44no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53C22⋊C411yescharacteristic
149C22×Dic53C22⋊C411yescharacteristic
150C22×Dic53C22⋊C412yes—
151C22×Dic53C22⋊C411yescharacteristic
152C22×Dic53C22⋊C411yescharacteristic
153C22×Dic53C22⋊C412yes—
154C22×Dic53D814yes—
155C22×Dic53SD1614yes—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158#11719947—416no—
159#11719947—48no—
160#11719947—48no—
161C2×D212—21no—
162C2×D212—42no—
163C2×D212C22×C412yes—
164C2×C4×D53—24no—
165C2×C4×D53—44no—
166C2×Dic106—24no—
167C2×Dic106—44no—
168C2×C8×C53—22no—
169C2×C8×C53—42no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C22×C412yes—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊Dic53—24no—
175#11719921—22no—
176#11719921—22no—
177#11719921—24no—
178C4⋊Dic53—21no—
179C4⋊Dic53C22×C412yes—
180#11719919—24no—
181#11719919—48no—
182C4×Dic53C22⋊C412yes—
183C4×Dic53C22⋊C412yes—
184C4×Dic53C22⋊C412yes—
185C4×Dic53C22⋊C412yes—
186C4×Dic53C22×C412yes—
187C4×Dic53C22×C412yes—
188C4×Dic53C2×D412yes—
189C4×Dic53C2×D412yes—
190C2×#88322—22no—
191C2×#88322—42no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53C2×C22⋊C411yescharacteristic
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C2×D812yes—
201C23×C53C2×SD1612yes—
202C23×C53C22×D411yescharacteristic
203C22×D53—22no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—42no—
211D4×C53—44no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—84no—
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C23×C411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C22⋊D53—48no—
229C22⋊D53—42no—
230C22⋊D53—42no—
231C22⋊D53—44no—
232C22⋊D53—48no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—48no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—84no—
244C2×Dic53#10214yes—
245C2×Dic53#10214yes—
246C2×Dic53#10218yes—
247C2×Dic53C2×C22⋊C412yes—
248C2×Dic53C2×C22⋊C412yes—
249C2×Dic53C2×C22⋊C412yes—
250C2×Dic53C2×C22⋊C412yes—
251C2×Dic53C2×D814yes—
252C2×Dic53C2×SD1614yes—
253D212—42no—
254D212—42no—
255C4×D53—48no—
256Dic106—48no—
257C8×C53—42no—
258#88322—42no—
259C2×#1517—531nomaximal, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—24no—
265C22×C53—24no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—84no—
270C22×C53#41012yes—
271C22×C53C2×#10212yes—
272C22×C53C2×#10212yes—
273C22×C53#42112yes—
274C22×C53C22×C22⋊C411yescharacteristic
275C22×C53C22×D811yescharacteristic
276C22×C53C22×SD1611yescharacteristic
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—84no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—21no—
285C4×C53—42no—
286C4×C53—82no—
287Dic53—22no—
288Dic53—22no—
289Dic53—42no—
290Dic53—82no—
291Dic53C2×#10214yes—
292Dic53C2×#10214yes—
293C2×#532—531no—
294C2×#431—531no—
295C2×#382—531no—
296#1517—538no—
297C2×#327—531no—
298C2×#327—531no—
299C2×#327—531no—
300C2×#327D5311yescharacteristic, complemented
301C2×C53—21no—
302C2×C53—21no—
303C2×C53—22no—
304C2×C53—82no—
305C2×C53#151714yescomplemented
306C2×C53C2×#41011yescharacteristic
307C2×C53C22×#10211yescharacteristic
308C2×C53C2×#42111yescharacteristic
309D53—82no—
310C22×#41—1061no—
311#532—534no—
312#532—534no—
313#532—1062no—
314#532—1062no—
315C2×C4⋊Q8—531no—
316C2×#127—531no—
317C2×#124—1061no—
318C2×#121—531no—
319C2×#121—1061no—
320C2×#121C2×D5311yescharacteristic
321C2×C4×D4—531no—
322C2×C4×D4—531no—
323C2×C4×D4—1061no—
324C22×C4⋊C4—1061no—
325C22×C42—531no—
326#431—534no—
327#431—1062no—
328C2×C4⋊C8—1061no—
329C2×C22⋊C8—531no—
330C2×C22⋊C8C2×D5311yescharacteristic
331C2×C4×C8—1061no—
332#382—534no—
333#382—1062no—
334C2×#95—531no—
335C2×#95—1061no—
336C2×#95C2×D5311yescharacteristic
337#327—534no—
338#327—534no—
339#327—534no—
340#327C2×D5314yescomplemented
341C53C2×#151711yescharacteristic, complemented, Sylow
342C2×#41—1064no—
343C2×#41—1064no—
344C2×#41—2124no—
345C2×#41—2124no—
346C22×Q8—1061no—
347C22×D4—21no—
348C22×D4—531no—
349C22×D4—1061no—
350C22×D4—1061no—
351C22×D4—1061no—
352C22×D4C4×D5311yescharacteristic
353C23×C4—21no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—1061no—
357C23×C4—1061no—
358C23×C4—1061no—
359C23×C4C22×D5311yescharacteristic
360C22×C8—21no—
361C22×C8—1061no—
362C4⋊Q8—534no—
363C4⋊Q8—1062no—
364#127—534no—
365#127—1062no—
366#124—1064no—
367#124—1064no—
368#121—21no—
369#121—21no—
370#121—532no—
371#121—532no—
372#121—1064no—
373#121—1064no—
374#121—1061no—
375#121—1061no—
376#121C4×D5312yes—
377#121C22×D5312yes—
378C4×D4—534no—
379C4×D4—534no—
380C4×D4—1064no—
381C4×D4—1064no—
382C4×D4—1062no—
383C4×D4—1062no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—2124no—
388C2×C4⋊C4C4×D5311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—532no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—532no—
399C2×C42—532no—
400C2×C42—532no—
401C4⋊C8—1064no—
402C22⋊C8—21no—
403C22⋊C8—532no—
404C22⋊C8—1061no—
405C22⋊C8C22×D5312yes—
406C4×C8—1064no—
407#95—21no—
408#95—532no—
409#95—1064no—
410#95—1061no—
411#95C22×D5312yes—
412C24—21no—
413C24—1061no—
414C24D4×D5311yescharacteristic
415#41—21216no—
416#41—2128no—
417#41—2128no—
418C2×Q8—1064no—
419C2×Q8—2124no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—532no—
427C2×D4—1064no—
428C2×D4—1064no—
429C2×D4—1062no—
430C2×D4—1062no—
431C2×D4—1061no—
432C2×D4—2122no—
433C2×D4—2124no—
434C2×D4—2124no—
435C2×D4—2122no—
436C2×D4—2122no—
437C2×D4C2×C4×D5312yes—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—532no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—532no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—1064no—
455C22×C4—1064no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—2124no—
459C22×C4—2124no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4C2×C4×D5311yescharacteristic
465C22×C4C2×C4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—22no—
471C2×C8—42no—
472C2×C8—1062no—
473C2×C8—2122no—
474C4⋊C4—21no—
475C4⋊C4—532no—
476C4⋊C4—1064no—
477C4⋊C4—1061no—
478C4⋊C4—2128no—
479C4⋊C4C2×C4×D5312yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—1064no—
483C22⋊C4—1064no—
484C22⋊C4—1062no—
485C22⋊C4—1062no—
486C42—532no—
487C42—532no—
488C42—532no—
489C42—532no—
490C42—532no—
491C42—532no—
492C42—532no—
493C42—532no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—1062no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23C22⋊C4×D5311yescharacteristic
505C23C22⋊C4×D5311yescharacteristic
506C23D8×D5312yes—
507C23SD16×D5312yes—
508C23C2×D4×D5311yescharacteristic
509Q8—2128no—
510D4—42no—
511D4—42no—
512D4—44no—
513D4—42no—
514D4—42no—
515D4—2122no—
516D4—2122no—
517D4—2128no—
518D4—2128no—
519D4—2124no—
520D4—2122no—
521D4—2122no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—534no—
537C2×C4—534no—
538C2×C4—538no—
539C2×C4—534no—
540C2×C4—534no—
541C2×C4—532no—
542C2×C4—532no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—4244no—
547C2×C4—2128no—
548C2×C4—2128no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4C22⋊C4×D5311yescharacteristic
556C2×C4C22⋊C4×D5311yescharacteristic
557C2×C4C22×C4×D5311yescharacteristic
558C2×C4C2×D4×D5311yescharacteristic
559C8—42no—
560C8—2122no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—84no—
571C22—4244no—
572C22—2122no—
573C22—2122no—
574C22—2122no—
575C22#1681059512yes—
576C22#102×D5312yes—
577C22#102×D5312yes—
578C22#1681063812yes—
579C22C2×C22⋊C4×D5311yescharacteristic
580C22C2×D8×D5311yescharacteristic
581C22C2×SD16×D5311yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—82no—
588C4—534no—
589C4—534no—
590C4—1062no—
591C4—1062no—
592C4—4242no—
593C4—2122no—
594C2—21no—
595C2—22no—
596C2—21no—
597C2—82no—
598C2—4242no—
599C2#2027327314yesdirect product factor
600C2C2×#1681059511yescharacteristic
601C2C2×#102×D5311yescharacteristic
602C2C2×#1681063811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#327C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811168C2
cosocleC22×C4×C53C24
hypercenterC2×#327D53
hypocenterC53C2×#1517
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#16810673C2semidirect
3. C2×#16810487C2semidirect
4. #20273273C2direct (N × H)
5. C2×#16810358C2semidirect
6. C2×#16810357C2semidirect
7. C2×#16810356C2semidirect
8. C2×C53×#327C2semidirect
12. #16811168C4semidirect
14. C2×#13133216C4semidirect
15. C2×#13133213C4semidirect
19. C2×D4×Dic53C4semidirect
20. C2×#13133179C4semidirect
25. #16810673C22semidirect
30. #16810487C22semidirect
33. #16810358C22semidirect
34. #16810357C22semidirect
35. #16810356C22semidirect
39. C53×#327C22semidirect
56. #13133216C2×C4semidirect
58. #13133213C2×C4semidirect
68. D4×Dic53C2×C4semidirect
70. #13133179C2×C4semidirect
300. C2×#327D53semidirect
305. C2×C53#1517semidirect
340. #327C2×D53semidirect
341. C53C2×#1517semidirect
599. C2#20273273direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111682
C2×#168106732
C2×#168104872
#202732732
C2×#168103582
C2×#168103572
C2×#168103562
C2×C53×#3272
C2×#151753