Finite Group Databasebeta

← C2×#20276693 #27538039

Subgroups — C2×#20276693 #27538039

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811063C211yesmaximal, characteristic, complemented
2C22×#13133163C211yesmaximal, characteristic, complemented
3C2×#16810898C211yesmaximal, characteristic, complemented
4#20276693C218yesmaximal, direct product factor
5C2×#16810439C211yesmaximal, characteristic, complemented
6C2×#16810430C211yesmaximal, characteristic, complemented
7C2×#16810407C211yesmaximal, characteristic, complemented
8C2×C53×#578C211yesmaximal, characteristic
9C2×#13133247—21no—
10C22×Dic212C2211yescharacteristic
11#16811063—22no—
12#16811063C2214yescomplemented
13C2×#13133163—22no—
14C2×#13133163—22no—
15C2×#13133163—21no—
16C2×#13133163C2214yescomplemented
17C2×#13133161—21no—
18C2×#13133157C2211yescharacteristic
19C22×#11719908C2211yescharacteristic
20#16810898—22no—
21#16810898C2214yescomplemented
22C2×#13133153C2211yescharacteristic
23C2×#13133135—21no—
24#16810439C2214yescomplemented
25#16810430C2214yescomplemented
26#16810407C2214yescomplemented
27C2×2−1+4×C53—21no—
28C53×#578—22no—
29C53×#578—22no—
30C53×#578—22no—
31C53×#578C2214yes—
32C2×#137×C53—22no—
33C2×#137×C53—21no—
34C2×#135×C53—21no—
35C2×#135×C53C2211yescharacteristic
36C22×Q16×C53—21no—
37C22×Q16×C53C2211yescharacteristic
38C2×#131×C53C2211yescharacteristic
39C22×Dic106—21no—
40#13133247—24no—
41#13133247—42no—
42C2×Dic212—21no—
43C2×Dic212—21no—
44C2×Dic212—21no—
45C2×Dic212—21no—
46C2×Dic212—21no—
47C2×Dic212C2312yes—
48C2×#11719925—41no—
49C2×#11719923—41no—
50C2×#11719917—21no—
51#13133163—28no—
52#13133163—24no—
53#13133161—24no—
54#13133157—21no—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908—21no—
58C2×#11719908—21no—
59C2×#11719908C2312yes—
60#13133153—21no—
61#13133153C2312yes—
62#13133135—24no—
632−1+4×C53—24no—
642−1+4×C53—24no—
652−1+4×C53—42no—
662−1+4×C53—42no—
67C2×#41×C53—22no—
68C2×#41×C53—21no—
69C2×#41×C53—42no—
70C2×#41×C53—41no—
71C2×#41×C53D411yescharacteristic
72C22×Q8×C53—22no—
73C22×Q8×C53—21no—
74C22×Q8×C53—41no—
75#137×C53—28no—
76#137×C53—24no—
77#137×C53—28no—
78#137×C53—24no—
79#135×C53—24no—
80#135×C53—24no—
81#135×C53—21no—
82#135×C53—21no—
83#135×C53D412yes—
84#135×C53C2312yes—
85C2×Q16×C53—22no—
86C2×Q16×C53—24no—
87C2×Q16×C53—24no—
88C2×Q16×C53—21no—
89C2×Q16×C53—21no—
90C2×Q16×C53—21no—
91C2×Q16×C53D411yescharacteristic
92C2×Q16×C53D412yes—
93C2×Q16×C53D412yes—
94C2×Q16×C53C2312yes—
95C2×SD16×C53—22no—
96C2×SD16×C53—21no—
97C2×D8×C53D411yescharacteristic
98#131×C53—21no—
99#131×C53—21no—
100#131×C53D412yes—
101#131×C53C2312yes—
102C2×M16×C53—21no—
103C2×M16×C53D411yescharacteristic
104C22×C8×C53—21no—
105C22×C8×C53C2311yescharacteristic
106#41×C53—21no—
107#41×C53—21no—
108#41×C53—28no—
109#41×C53—22no—
110#41×C53—22no—
111#41×C53—44no—
112#41×C53—48no—
113#41×C53—48no—
114#41×C53—44no—
115#41×C53—44no—
116#41×C53—41no—
117#41×C53—41no—
118#41×C53D812yes—
119#41×C53C2×D412yes—
120C2×Q8×C53—28no—
121C2×Q8×C53—22no—
122C2×Q8×C53—21no—
123C2×Q8×C53—22no—
124C2×Q8×C53—22no—
125C2×Q8×C53—42no—
126C2×Q8×C53—44no—
127C2×Q8×C53—44no—
128C2×Q8×C53—44no—
129C2×Q8×C53—41no—
130C2×Q8×C53—41no—
131C2×Q8×C53—41no—
132C2×Q8×C53D811yescharacteristic
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—41no—
137C2×D4×C53D811yescharacteristic
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53—41no—
142C22×C4×C53—41no—
143C22×C4×C53C2×D411yescharacteristic
144C22×Dic53—41no—
145C2×Dic106—22no—
146C2×Dic106—41no—
147C2×Dic106—41no—
148C2×Dic106—41no—
149C2×Dic106—41no—
150C2×Dic106—41no—
151Q16×C53—28no—
152Q16×C53—21no—
153Q16×C53—22no—
154Q16×C53—22no—
155Q16×C53—22no—
156Q16×C53C2×D412yes—
157Q16×C53C2×D414yes—
158SD16×C53—28no—
159SD16×C53—22no—
160SD16×C53—22no—
161D8×C53—21no—
162D8×C53C2×D412yes—
163M16×C53—21no—
164M16×C53—22no—
165M16×C53—22no—
166M16×C53C2×D412yes—
167C2×C8×C53—21no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C2×C8×C53C2×D411yescharacteristic
175C2×C8×C53C2411yescharacteristic
176#11719925—44no—
177#11719923—44no—
178#11719917—22no—
179#11719917—41no—
180C2×#88322—41no—
181Dic212—22no—
182Dic212—22no—
183Dic212—41no—
184Dic212—41no—
185#11719908—22no—
186#11719908—22no—
187C23×C53—21no—
188C23×C53—41no—
189C23×C53AGL(1,ℤ/8)11yescharacteristic
190Q8×C53—21no—
191Q8×C53—22no—
192Q8×C53—24no—
193Q8×C53—48no—
194Q8×C53—42no—
195Q8×C53—41no—
196Q8×C53—42no—
197Q8×C53—42no—
198Q8×C53C2×D812yes—
199D4×C53—22no—
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—21no—
203D4×C53—48no—
204D4×C53—42no—
205D4×C53—42no—
206D4×C53—41no—
207D4×C53C2×D812yes—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—48no—
212C2×C4×C53—48no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53C2×D811yescharacteristic
222C2×C4×C53C2×D811yescharacteristic
223C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×Dic53—42no—
226C2×Dic53—81no—
227C2×Dic53—81no—
228Dic106—42no—
229Dic106—42no—
230Dic106—81no—
231Dic106—81no—
232C8×C53—22no—
233C8×C53#12011yescharacteristic
234C8×C53#12011yescharacteristic
235C8×C53C22×D411yescharacteristic
236C8×C53C22×D411yescharacteristic
237#88322—42no—
238C2×#2195—531nomaximal, Sylow
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—82no—
244C22×C53#44711yescharacteristic
245C22×C53#44711yescharacteristic
246C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C53—22no—
248C4×C53—44no—
249C4×C53—42no—
250C4×C53#44711yescharacteristic
251C4×C53#44711yescharacteristic
252C4×C53C2×#12011yescharacteristic
253C4×C53C22×D811yescharacteristic
254Dic53—82no—
255C2×#578D5311yescharacteristic
256C2×#510—531no—
257C22×Q32—531no—
258C2×#473—531no—
259#2195—538no—
260C2×#362—531no—
261C2×#359—531no—
262C2×#350—531no—
263C2×C53—21no—
264C2×C53—21no—
265C2×C53—42no—
266C2×C53—82no—
267C2×C53#219512yescomplemented
268C2×C53C2×#44711yescharacteristic
269C2×2−1+4—21no—
270#578—22no—
271#578—22no—
272#578—22no—
273#578C2×D5314yes—
274C2×#137—22no—
275C2×#137—21no—
276C2×#137—1061no—
277C2×#135—21no—
278C2×#135C2×D5311yescharacteristic
279C22×Q16—21no—
280C22×Q16—531no—
281C22×Q16C2×D5311yescharacteristic
282C2×#131C2×D5311yescharacteristic
283#510—534no—
284#510—1062no—
285C2×Q32—534no—
286C2×Q32—1062no—
287C2×Q32—1062no—
288C2×Q32—1061no—
289C2×SD32—1061no—
290C2×M32—531no—
291C22×C16—531no—
292#473—534no—
293#473—1062no—
294C2×#108—531no—
295C2×#101—1061no—
296#362—534no—
297#359—534no—
298#350—534no—
299C53C2×#219511yescharacteristic, complemented, Sylow
3002−1+4—24no—
3012−1+4—24no—
3022−1+4—42no—
3032−1+4—42no—
304C2×#41—22no—
305C2×#41—21no—
306C2×#41—42no—
307C2×#41—41no—
308C2×#41C22⋊D5311yescharacteristic
309C22×Q8—22no—
310C22×Q8—21no—
311C22×Q8—41no—
312C22×Q8—1061no—
313#137—24no—
314#137—28no—
315#137—28no—
316#137—24no—
317#137—1064no—
318#137—2122no—
319#135—24no—
320#135—24no—
321#135—21no—
322#135—21no—
323#135C22⋊D5312yes—
324#135C22×D5312yes—
325C2×Q16—22no—
326C2×Q16—24no—
327C2×Q16—24no—
328C2×Q16—21no—
329C2×Q16—21no—
330C2×Q16—21no—
331C2×Q16—532no—
332C2×Q16—1061no—
333C2×Q16—1061no—
334C2×Q16—1061no—
335C2×Q16—1061no—
336C2×Q16—1061no—
337C2×Q16—2121no—
338C2×Q16C22⋊D5311yescharacteristic
339C2×Q16C22⋊D5312yes—
340C2×Q16C22⋊D5312yes—
341C2×Q16C22×D5312yes—
342C2×SD16—22no—
343C2×SD16—21no—
344C2×SD16—2121no—
345C2×D8C22⋊D5311yescharacteristic
346#131—21no—
347#131—21no—
348#131C22⋊D5312yes—
349#131C22×D5312yes—
350C2×M16—21no—
351C2×M16—1061no—
352C2×M16C22⋊D5311yescharacteristic
353C22×C8—21no—
354C22×C8C22×D5311yescharacteristic
355Q32—1068no—
356Q32—1064no—
357SD32—1064no—
358M32—532no—
359M32—1061no—
360C2×C16—532no—
361C2×C16—1061no—
362C2×C16—1061no—
363C2×C16—1061no—
364#108—532no—
365#108—1061no—
366#101—1064no—
367#41—21no—
368#41—21no—
369#41—28no—
370#41—22no—
371#41—22no—
372#41—44no—
373#41—48no—
374#41—48no—
375#41—44no—
376#41—44no—
377#41—41no—
378#41—41no—
379#41#1171992212yes—
380#41C2×C22⋊D5312yes—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—1062no—
394C2×Q8—2121no—
395C2×Q8—2121no—
396C2×Q8—2121no—
397C2×Q8—2121no—
398C2×Q8—2121no—
399C2×Q8#1171992211yescharacteristic
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4#1171992211yescharacteristic
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—2121no—
411C22×C4C2×C22⋊D5311yescharacteristic
412Q16—28no—
413Q16—21no—
414Q16—22no—
415Q16—22no—
416Q16—22no—
417Q16—1062no—
418Q16—1062no—
419Q16—2124no—
420Q16—2121no—
421Q16—2121no—
422Q16C2×C22⋊D5312yes—
423Q16C2×C22⋊D5314yes—
424SD16—28no—
425SD16—22no—
426SD16—22no—
427SD16—2124no—
428D8—21no—
429D8C2×C22⋊D5312yes—
430M16—21no—
431M16—22no—
432M16—22no—
433M16—1062no—
434M16—2121no—
435M16C2×C22⋊D5312yes—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—2121no—
440C2×C8C2×C22⋊D5311yescharacteristic
441C2×C8C2×C22⋊D5311yescharacteristic
442C2×C8C2×C22⋊D5311yescharacteristic
443C2×C8C2×C22⋊D5311yescharacteristic
444C2×C8C2×C22⋊D5311yescharacteristic
445C2×C8C23×D5311yescharacteristic
446C16—1062no—
447C16—1062no—
448C23—21no—
449C23—41no—
450C23#1313324311yescharacteristic
451Q8—21no—
452Q8—24no—
453Q8—22no—
454Q8—48no—
455Q8—41no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—4241no—
460Q8—4241no—
461Q8—2122no—
462Q8—2122no—
463Q8C2×#1171992212yes—
464D4—22no—
465D4—22no—
466D4—21no—
467D4—22no—
468D4—48no—
469D4—42no—
470D4—42no—
471D4—41no—
472D4C2×#1171992212yes—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—4241no—
485C2×C4—4241no—
486C2×C4—2122no—
487C2×C4#1313318611yescharacteristic
488C2×C4#1313318611yescharacteristic
489C2×C4C2×#1171992211yescharacteristic
490C2×C4C2×#1171992211yescharacteristic
491C2×C4#1313324311yescharacteristic
492C2×C4C22×C22⋊D5311yescharacteristic
493C8—22no—
494C8—2122no—
495C8#1313318611yescharacteristic
496C8#1313318611yescharacteristic
497C8C22×C22⋊D5311yescharacteristic
498C8C22×C22⋊D5311yescharacteristic
499C22—22no—
500C22—22no—
501C22—42no—
502C22—42no—
503C22—82no—
504C22#1681075711yescharacteristic
505C22#1681075711yescharacteristic
506C22C2×#1313324311yescharacteristic
507C4—22no—
508C4—44no—
509C4—42no—
510C4—4242no—
511C4#1681075711yescharacteristic
512C4#1681075711yescharacteristic
513C4C2×#1313318611yescharacteristic
514C4C22×#1171992211yescharacteristic
515C2—21no—
516C2—21no—
517C2—42no—
518C2—82no—
519C2#2027669312yesdirect product factor
520C2C2×#1681075711yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC2×C53×#578C2
socleC22×C53#447
hypercenterC2×#578D53
hypocenterC53C2×#2195
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#16811063C2semidirect
2. C22×#13133163C2semidirect
3. C2×#16810898C2semidirect
4. #20276693C2direct (N × H)
5. C2×#16810439C2semidirect
6. C2×#16810430C2semidirect
7. C2×#16810407C2semidirect
12. #16811063C22semidirect
16. C2×#13133163C22semidirect
21. #16810898C22semidirect
24. #16810439C22semidirect
25. #16810430C22semidirect
26. #16810407C22semidirect
267. C2×C53#2195semidirect
299. C53C2×#2195semidirect
519. C2#20276693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110632
C22×#131331632
C2×#168108982
#202766932
C2×#168104392
C2×#168104302
C2×#168104072
C2×C53×#5782
C2×#219553