Finite Group Databasebeta

← C2×#20276679 #27538031

Subgroups — C2×#20276679 #27538031

Order 13568.

1098 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic, complemented
2C22×#13133160C211yesmaximal, characteristic, complemented
3C2×#16810893C211yesmaximal, characteristic, complemented
4#20276679C218yesmaximal, direct product factor
5C2×#16810439C211yesmaximal, characteristic, complemented
6C2×#16810430C211yesmaximal, characteristic, complemented
7C2×#16810407C211yesmaximal, characteristic, complemented
8C2×C53×#578C211yesmaximal, characteristic, complemented
9C2×#13133242—21no—
10C22×D424C2211yescharacteristic
11#16811058—22no—
12#16811058C2214yescomplemented
13C2×#13133160—22no—
14C2×#13133160—22no—
15C2×#13133160—21no—
16C2×#13133160C2214yescomplemented
17C2×#13133158—21no—
18C2×#13133157C2211yescharacteristic
19C22×#11719908C2211yescharacteristic
20#16810893—22no—
21#16810893C2214yescomplemented
22C2×#13133153C2211yescharacteristic
23C2×#13133133—21no—
24#16810439C2214yescomplemented
25#16810430C2214yescomplemented
26#16810407C2214yescomplemented
27C2×2−1+4×C53—21no—
28C53×#578—22no—
29C53×#578—22no—
30C53×#578—22no—
31C53×#578C2214yescomplemented
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×D212—21no—
40#13133242—24no—
41#13133242—42no—
42C2×#11719924—41no—
43C2×D424—21no—
44C2×D424—21no—
45C2×D424—21no—
46C2×D424—21no—
47C2×D424—21no—
48C2×D424C2312yes—
49C2×#11719922—41no—
50C2×#11719917—21no—
51#13133160—28no—
52#13133160—24no—
53#13133158—24no—
54#13133157—21no—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908—21no—
58C2×#11719908—21no—
59C2×#11719908C2312yes—
60#13133153—21no—
61#13133153C2312yes—
62#13133133—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
106C23×D53—41no—
107#41×C53—21no—
108#41×C53—21no—
109#41×C53—28no—
110#41×C53—22no—
111#41×C53—22no—
112#41×C53—44no—
113#41×C53—48no—
114#41×C53—48no—
115#41×C53—44no—
116#41×C53—44no—
117#41×C53—41no—
118#41×C53—41no—
119#41×C53D812yes—
120#41×C53C2×D412yes—
121C2×Q8×C53—28no—
122C2×Q8×C53—22no—
123C2×Q8×C53—21no—
124C2×Q8×C53—22no—
125C2×Q8×C53—22no—
126C2×Q8×C53—42no—
127C2×Q8×C53—44no—
128C2×Q8×C53—44no—
129C2×Q8×C53—44no—
130C2×Q8×C53—41no—
131C2×Q8×C53—41no—
132C2×Q8×C53—41no—
133C2×Q8×C53D811yescharacteristic
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—42no—
137C2×D4×C53—41no—
138C2×D4×C53D811yescharacteristic
139C22×C4×C53—21no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—41no—
143C22×C4×C53—41no—
144C22×C4×C53C2×D411yescharacteristic
145C2×D212—22no—
146C2×D212—41no—
147C2×D212—41no—
148C2×D212—41no—
149C2×D212—41no—
150C2×D212—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#11719924—44no—
177#11719922—44no—
178#11719917—22no—
179#11719917—41no—
180C2×#88322—41no—
181D424—22no—
182D424—22no—
183D424—41no—
184D424—41no—
185#11719908—22no—
186#11719908—22no—
187C23×C53—21no—
188C23×C53—41no—
189C23×C53AGL(1,ℤ/8)11yescharacteristic
190C22×D53—42no—
191C22×D53—81no—
192C22×D53—81no—
193C22×D53—162no—
194Q8×C53—21no—
195Q8×C53—22no—
196Q8×C53—24no—
197Q8×C53—48no—
198Q8×C53—42no—
199Q8×C53—41no—
200Q8×C53—42no—
201Q8×C53—42no—
202Q8×C53C2×D812yes—
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—22no—
206D4×C53—21no—
207D4×C53—48no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—41no—
211D4×C53C2×D812yes—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—48no—
216C2×C4×C53—48no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53#12011yescharacteristic
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53C2×D811yescharacteristic
226C2×C4×C53C2×D811yescharacteristic
227C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229D212—42no—
230D212—42no—
231D212—81no—
232D212—81no—
233C8×C53—22no—
234C8×C53#12011yescharacteristic
235C8×C53#12011yescharacteristic
236C8×C53C22×D411yescharacteristic
237C8×C53C22×D411yescharacteristic
238#88322—42no—
239C2×#2193—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—82no—
245C22×C53#44711yescharacteristic
246C22×C53#44711yescharacteristic
247C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
248C2×D53—82no—
249C2×D53—162no—
250C2×D53—162no—
251C2×D53—162no—
252C4×C53—22no—
253C4×C53—44no—
254C4×C53—42no—
255C4×C53#44711yescharacteristic
256C4×C53#44711yescharacteristic
257C4×C53C2×#12011yescharacteristic
258C4×C53C22×D811yescharacteristic
259C2×#578D5311yescharacteristic, complemented
260C2×#509—531no—
261C22×SD32—531no—
262C2×#472—531no—
263#2193—538no—
264C2×#362—531no—
265C2×#359—531no—
266C2×#350—531no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53—42no—
270C2×C53—82no—
271C2×C53#219312yescomplemented
272C2×C53C2×#44711yescharacteristic
273D53—162no—
274C2×2−1+4—21no—
275#578—22no—
276#578—22no—
277#578—22no—
278#578C2×D5314yescomplemented
279C2×#137—22no—
280C2×#137—21no—
281C2×AGL(1,ℤ/8)—1061no—
282C2×#135—21no—
283C2×#135C2×D5311yescharacteristic
284C22×Q16—21no—
285C22×Q16C2×D5311yescharacteristic
286C22×D8—531no—
287C2×#131C2×D5311yescharacteristic
288#509—534no—
289#509—1062no—
290C2×SD32—534no—
291C2×SD32—1062no—
292C2×SD32—1062no—
293C2×SD32—1061no—
294C2×D16—1061no—
295C2×M32—531no—
296C22×C16—531no—
297#472—534no—
298#472—1062no—
299C2×#108—531no—
300C2×#100—1061no—
301#362—534no—
302#359—534no—
303#350—534no—
304C53C2×#219311yescharacteristic, complemented, Sylow
3052−1+4—24no—
3062−1+4—24no—
3072−1+4—42no—
3082−1+4—42no—
309C2×#41—22no—
310C2×#41—21no—
311C2×#41—42no—
312C2×#41—41no—
313C2×#41C22⋊D5311yescharacteristic
314C22×Q8—22no—
315C22×Q8—21no—
316C22×Q8—41no—
317C22×D4—1061no—
318#137—24no—
319#137—28no—
320#137—28no—
321#137—24no—
322AGL(1,ℤ/8)—1064no—
323AGL(1,ℤ/8)—2122no—
324#135—24no—
325#135—24no—
326#135—21no—
327#135—21no—
328#135C22⋊D5312yes—
329#135C22×D5312yes—
330C2×Q16—22no—
331C2×Q16—24no—
332C2×Q16—24no—
333C2×Q16—21no—
334C2×Q16—21no—
335C2×Q16—21no—
336C2×Q16C22⋊D5311yescharacteristic
337C2×Q16C22⋊D5312yes—
338C2×Q16C22⋊D5312yes—
339C2×Q16C22×D5312yes—
340C2×SD16—22no—
341C2×SD16—21no—
342C2×SD16—2121no—
343C2×D8—532no—
344C2×D8—1061no—
345C2×D8—1061no—
346C2×D8—1061no—
347C2×D8—1061no—
348C2×D8—1061no—
349C2×D8—2121no—
350C2×D8C22⋊D5311yescharacteristic
351#131—21no—
352#131—21no—
353#131C22⋊D5312yes—
354#131C22×D5312yes—
355C2×M16—21no—
356C2×M16—1061no—
357C2×M16C22⋊D5311yescharacteristic
358C22×C8—21no—
359C22×C8C22×D5311yescharacteristic
360SD32—1068no—
361SD32—1064no—
362D16—1064no—
363M32—532no—
364M32—1061no—
365C2×C16—532no—
366C2×C16—1061no—
367C2×C16—1061no—
368C2×C16—1061no—
369#108—532no—
370#108—1061no—
371#100—1064no—
372C24—2121no—
373#41—21no—
374#41—21no—
375#41—28no—
376#41—22no—
377#41—22no—
378#41—44no—
379#41—48no—
380#41—48no—
381#41—44no—
382#41—44no—
383#41—41no—
384#41—41no—
385#41#1171992212yes—
386#41C2×C22⋊D5312yes—
387C2×Q8—28no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8#1171992211yescharacteristic
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—1062no—
405C2×D4—2121no—
406C2×D4—2121no—
407C2×D4—2121no—
408C2×D4—2121no—
409C2×D4—2121no—
410C2×D4#1171992211yescharacteristic
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C2×C22⋊D5311yescharacteristic
417Q16—28no—
418Q16—21no—
419Q16—22no—
420Q16—22no—
421Q16—22no—
422Q16C2×C22⋊D5312yes—
423Q16C2×C22⋊D5314yes—
424SD16—28no—
425SD16—22no—
426SD16—22no—
427SD16—2124no—
428D8—21no—
429D8—1062no—
430D8—1062no—
431D8—2124no—
432D8—2121no—
433D8—2121no—
434D8C2×C22⋊D5312yes—
435M16—21no—
436M16—22no—
437M16—22no—
438M16—1062no—
439M16—2121no—
440M16C2×C22⋊D5312yes—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—2121no—
445C2×C8C2×C22⋊D5311yescharacteristic
446C2×C8C2×C22⋊D5311yescharacteristic
447C2×C8C2×C22⋊D5311yescharacteristic
448C2×C8C2×C22⋊D5311yescharacteristic
449C2×C8C2×C22⋊D5311yescharacteristic
450C2×C8C23×D5311yescharacteristic
451C16—1062no—
452C16—1062no—
453C23—21no—
454C23—41no—
455C23—8482no—
456C23—4241no—
457C23—4241no—
458C23—2122no—
459C23#1313324311yescharacteristic
460Q8—21no—
461Q8—24no—
462Q8—22no—
463Q8—48no—
464Q8—41no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8C2×#1171992212yes—
469D4—22no—
470D4—22no—
471D4—21no—
472D4—22no—
473D4—48no—
474D4—42no—
475D4—42no—
476D4—41no—
477D4—4241no—
478D4—4241no—
479D4—2122no—
480D4—2122no—
481D4C2×#1171992212yes—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4#1313318611yescharacteristic
494C2×C4#1313318611yescharacteristic
495C2×C4C2×#1171992211yescharacteristic
496C2×C4C2×#1171992211yescharacteristic
497C2×C4#1313324311yescharacteristic
498C2×C4C22×C22⋊D5311yescharacteristic
499C8—22no—
500C8—2122no—
501C8#1313318611yescharacteristic
502C8#1313318611yescharacteristic
503C8C22×C22⋊D5311yescharacteristic
504C8C22×C22⋊D5311yescharacteristic
505C22—22no—
506C22—22no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—8482no—
511C22—8482no—
512C22—8482no—
513C22—4242no—
514C22#1681075711yescharacteristic
515C22#1681075711yescharacteristic
516C22C2×#1313324311yescharacteristic
517C4—22no—
518C4—44no—
519C4—42no—
520C4#1681075711yescharacteristic
521C4#1681075711yescharacteristic
522C4C2×#1313318611yescharacteristic
523C4C22×#1171992211yescharacteristic
524C2—21no—
525C2—21no—
526C2—42no—
527C2—82no—
528C2—8482no—
529C2#2027667912yesdirect product factor
530C2C2×#1681075711yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#578C2
socleC22×C53#447
hypercenterC2×#578D53
hypocenterC53C2×#2193
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×#16811058C2semidirect
2. C22×#13133160C2semidirect
3. C2×#16810893C2semidirect
4. #20276679C2direct (N × H)
5. C2×#16810439C2semidirect
6. C2×#16810430C2semidirect
7. C2×#16810407C2semidirect
8. C2×C53×#578C2semidirect
12. #16811058C22semidirect
16. C2×#13133160C22semidirect
21. #16810893C22semidirect
24. #16810439C22semidirect
25. #16810430C22semidirect
26. #16810407C22semidirect
31. C53×#578C22semidirect
259. C2×#578D53semidirect
271. C2×C53#2193semidirect
278. #578C2×D53semidirect
304. C53C2×#2193semidirect
529. C2#20276679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110582
C22×#131331602
C2×#168108932
#202766792
C2×#168104392
C2×#168104302
C2×#168104072
C2×C53×#5782
C2×#219353