Finite Group Databasebeta

← C2×#20273302 #27351975

Subgroups — C2×#20273302 #27351975

Order 13568.

1166 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811158C211yesmaximal, characteristic
2C2×#16810575C211yesmaximal, characteristic, complemented
3C2×#97×D53C211yesmaximal, characteristic, complemented
4#20273302C218yesmaximal, direct product factor
5C2×#16810365C211yesmaximal, characteristic, complemented
6C2×#16810362C211yesmaximal, characteristic, complemented
7C2×#16810362C211yesmaximal, characteristic, complemented
8C2×C53×#330C211yesmaximal, characteristic, complemented
9C22×#11719912—21no—
10C22×C8×D53—21no—
11#16811158—22no—
12#16811158—22no—
13#16811158C414yes—
14#16811158C2214yes—
15C2×#13133212—21no—
16C2×#13133205C2211yescharacteristic
17C2×#13133202—21no—
18C2×#13133197C411yescharacteristic
19C2×#13133192C411yescharacteristic
20C2×#13133182C411yescharacteristic
21C2×#13133179—21no—
22C2×#13133179C411yescharacteristic
23C2×#13133175—21no—
24C2×C42×D53C2211yescharacteristic
25#16810575—22no—
26#16810575C2214yescomplemented
27C2×#13133128C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×#13133127—21no—
30#97×D53—22no—
31#97×D53C2214yescomplemented
32C2×C8×Dic53—21no—
33#16810365C2214yescomplemented
34#16810362C2214yescomplemented
35#16810362C2214yescomplemented
36C2×#125×C53C2211yescharacteristic
37C2×#97×C53C2211yescharacteristic
38C2×#97×C53C2211yescharacteristic
39C53×#330C2214yescomplemented
40C22×Dic106C2×C411yescharacteristic
41C22×D212—21no—
42C22×D212C2×C411yescharacteristic
43C22×C4×D53—21no—
44C22×C4×D53C2×C411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C2×#11719912—28no—
48C2×#11719912—24no—
49C22×#88322—21no—
50C22×#88322—21no—
51C2×C8×D53—28no—
52C2×C8×D53—24no—
53#13133212—24no—
54#13133212—24no—
55#13133205—21no—
56#13133205—21no—
57#13133205C2×C412yes—
58#13133205C2312yes—
59#13133202—24no—
60#13133202—24no—
61#13133197—22no—
62#13133197C2×C414yes—
63#13133192—22no—
64#13133192C2×C414yes—
65#13133182—22no—
66#13133182C2×C414yes—
67#13133179—22no—
68#13133179—24no—
69#13133179—24no—
70#13133179C2×C414yes—
71#13133175—24no—
72#13133175—24no—
73C2×C4⋊Dic53—21no—
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C42×D53C2×C412yes—
85C42×D53D412yes—
86C42×D53D412yes—
87C42×D53C2312yes—
88#13133128—21no—
89#13133128—21no—
90#13133128C2312yes—
91#13133128C2312yes—
92#13133127—24no—
93C8×Dic53—24no—
94C22×C8×C53—21no—
95C22×C8×C53—21no—
96#125×C53—21no—
97#125×C53—21no—
98#125×C53C2×C412yes—
99#125×C53C2312yes—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C42×C53C2311yescharacteristic
105#97×C53—21no—
106#97×C53—21no—
107#97×C53C2312yes—
108#97×C53C2312yes—
109C23×D53—21no—
110C23×D53C22⋊C411yescharacteristic
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×Dic53—21no—
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C22⋊C411yescharacteristic
119C22×Dic53C22⋊C411yescharacteristic
120C2×D212—24no—
121C2×D212—21no—
122C2×D212—42no—
123C2×D212—44no—
124C2×D212C22×C412yes—
125C2×C4×D53—24no—
126C2×C4×D53—24no—
127C2×C4×D53—24no—
128C2×C4×D53—42no—
129C2×C4×D53—44no—
130C2×C4×D53C22⋊C412yes—
131C2×C4×D53C22⋊C412yes—
132C2×C4×D53C22⋊C412yes—
133C2×C4×D53C22×C412yes—
134C2×C4×D53C2×D412yes—
135C2×C4×D53C2×D412yes—
136C2×Dic106—21no—
137C2×Dic106—42no—
138C2×Dic106C22×C412yes—
139C2×C8×C53—24no—
140C2×C8×C53—24no—
141C2×C8×C53—22no—
142C2×C8×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C22×C412yes—
150C4⋊C4×C53C22×C412yes—
151C42×C53C22×C411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53C2411yescharacteristic
155#11719921—24no—
156#11719921—22no—
157#11719921—22no—
158#11719921—22no—
159#11719921—22no—
160#11719921—24no—
161#11719921—24no—
162C4⋊Dic53—24no—
163#11719919—24no—
164C4×Dic53—24no—
165C4×Dic53C22⋊C412yes—
166C4×Dic53C22⋊C412yes—
167C4×Dic53C22⋊C412yes—
168C4×Dic53C22×C412yes—
169C4×Dic53C2×D412yes—
170C4×Dic53C2×D412yes—
171C2×#88322—24no—
172C2×#88322—24no—
173C2×#88322—22no—
174C2×#88322—22no—
175#11719912—216no—
176C8×D53—216no—
177C23×C53C2×C22⋊C411yescharacteristic
178C22×D53—22no—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—42no—
182C22×D53—42no—
183C22×D53—42no—
184C22×D53#10414yes—
185C22×D53C2×C22⋊C412yes—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53C2×C22⋊C411yescharacteristic
194C2×C4×C53C2×C22⋊C411yescharacteristic
195C2×C4×C53C2×C22⋊C411yescharacteristic
196C2×C4×C53C23×C411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×Dic53—22no—
200C2×Dic53—22no—
201C2×Dic53—22no—
202C2×Dic53—42no—
203C2×Dic53#10414yes—
204C2×Dic53C2×C22⋊C412yes—
205C2×Dic53C2×C22⋊C412yes—
206C2×Dic53C2×C22⋊C412yes—
207D212—42no—
208D212—42no—
209D212—48no—
210C4×D53—24no—
211C4×D53—24no—
212C4×D53—24no—
213C4×D53—24no—
214C4×D53—42no—
215C4×D53—42no—
216C4×D53—48no—
217Dic106—42no—
218Dic106—42no—
219C8×C53—24no—
220C8×C53—24no—
221#88322—24no—
222#88322—24no—
223C2×#1524—531nomaximal, Sylow
224C22×C53#41312yes—
225C22×C53C2×#10412yes—
226C22×C53C2×#10412yes—
227C22×C53C22×C22⋊C411yescharacteristic
228C2×D53—22no—
229C2×D53—22no—
230C2×D53—28no—
231C2×D53—42no—
232C2×D53—42no—
233C2×D53—42no—
234C2×D53—84no—
235C2×D53C2×#10414yes—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—42no—
243C4×C53—42no—
244Dic53—22no—
245Dic53—22no—
246Dic53—42no—
247Dic53C2×#10414yes—
248C2×#529—531no—
249C2×C4×M16—531no—
250C22×#97—531no—
251#1524—538no—
252C2×#330—531no—
253C2×#330D5311yescharacteristic, complemented
254C2×#329—531no—
255C2×#329—531no—
256C2×C53#152414yescomplemented
257C2×C53C2×#41311yescharacteristic
258C2×C53C22×#10411yescharacteristic
259C2×C53C22×#10411yescharacteristic
260D53—24no—
261D53—82no—
262C22×M16—1061no—
263C23×C8—1061no—
264#529—534no—
265#529—534no—
266#529—1062no—
267#529—1062no—
268C2×#126—1061no—
269C2×#125C2×D5311yescharacteristic
270C2×#124—531no—
271C2×#123—1061no—
272C2×C22⋊Q8—531no—
273C2×#121—531no—
274C2×C4×Q8—531no—
275C2×C4×D4—531no—
276C2×C4×D4—1061no—
277C2×#117—1061no—
278C22×C42—531no—
279C4×M16—534no—
280C4×M16—1062no—
281C2×#97—531no—
282C2×#97—531no—
283C2×#97—534no—
284C2×#97—1062no—
285C2×#97—1061no—
286C2×#97C2×D5311yescharacteristic
287C2×#97C2×D5311yescharacteristic
288C2×C4×C8—1061no—
289#330—534no—
290#330C2×D5314yescomplemented
291#329—534no—
292#329—534no—
293C53C2×#152411yescharacteristic, complemented, Sylow
294C22×Q8—531no—
295C22×D4—531no—
296C22×D4—1061no—
297C23×C4—531no—
298C23×C4—531no—
299C23×C4—531no—
300C23×C4—1061no—
301C2×M16—1064no—
302C2×M16—1068no—
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—1064no—
306C22×C8—1068no—
307C22×C8—1061no—
308C22×C8—1061no—
309#126—1064no—
310#126—1064no—
311#125—21no—
312#125—21no—
313#125C4×D5312yes—
314#125C22×D5312yes—
315#124—532no—
316#124—532no—
317#124—1061no—
318#124—1061no—
319#123—1064no—
320#123—1064no—
321C22⋊Q8—534no—
322C22⋊Q8—1062no—
323#121—534no—
324#121—1062no—
325C4×Q8—534no—
326C4×Q8—1062no—
327C4×D4—534no—
328C4×D4—1062no—
329C4×D4—1064no—
330C4×D4—1064no—
331#117—1064no—
332#117—1064no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—1061no—
336C2×C4⋊C4—1061no—
337C2×C4⋊C4C4×D5311yescharacteristic
338C2×C4⋊C4C4×D5311yescharacteristic
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—1061no—
341C2×C22⋊C4—1061no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—1061no—
344C2×C42—531no—
345C2×C42—531no—
346C2×C42—531no—
347C2×C42—532no—
348C2×C42—532no—
349C2×C42—532no—
350C2×C42—532no—
351C2×C42—1061no—
352C2×C42C22×D5311yescharacteristic
353#97—21no—
354#97—21no—
355#97—532no—
356#97—532no—
357#97—1064no—
358#97—1061no—
359#97—1061no—
360#97C22×D5312yes—
361#97C22×D5312yes—
362C4×C8—1064no—
363C24—531no—
364C24—1061no—
365C2×Q8—532no—
366C2×Q8—1061no—
367C2×Q8—2122no—
368C2×D4—532no—
369C2×D4—1064no—
370C2×D4—1061no—
371C2×D4—2122no—
372C2×D4—2124no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—531no—
378C22×C4—532no—
379C22×C4—532no—
380C22×C4—532no—
381C22×C4—532no—
382C22×C4—532no—
383C22×C4—532no—
384C22×C4—1064no—
385C22×C4—1064no—
386C22×C4—1064no—
387C22×C4—1061no—
388C22×C4—2122no—
389C22×C4—2124no—
390C22×C4C2×C4×D5311yescharacteristic
391C22×C4D4×D5311yescharacteristic
392C22×C4D4×D5311yescharacteristic
393M16—10616no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—10616no—
399C2×C8—1064no—
400C2×C8—1064no—
401C2×C8—1062no—
402C2×C8—1062no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—1064no—
410C4⋊C4—1064no—
411C4⋊C4C2×C4×D5312yes—
412C4⋊C4C2×C4×D5312yes—
413C22⋊C4—1064no—
414C22⋊C4—1064no—
415C22⋊C4—1064no—
416C22⋊C4—1062no—
417C22⋊C4—1062no—
418C22⋊C4—1062no—
419C22⋊C4—1062no—
420C42—532no—
421C42—532no—
422C42—532no—
423C42—532no—
424C42—532no—
425C42—532no—
426C42—1064no—
427C42C2×C4×D5311yescharacteristic
428C42D4×D5311yescharacteristic
429C42D4×D5311yescharacteristic
430C42C23×D5311yescharacteristic
431C23—532no—
432C23—534no—
433C23—1062no—
434C23—1062no—
435C23—1062no—
436C23—2122no—
437C23—2122no—
438C23—2122no—
439C23C22⋊C4×D5311yescharacteristic
440Q8—2122no—
441Q8—2122no—
442D4—2122no—
443D4—2122no—
444D4—2128no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—532no—
453C2×C4—534no—
454C2×C4—532no—
455C2×C4—532no—
456C2×C4—1064no—
457C2×C4—1064no—
458C2×C4—1064no—
459C2×C4—1064no—
460C2×C4—1062no—
461C2×C4—1062no—
462C2×C4—1062no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4—2128no—
466C2×C4—2122no—
467C2×C4C22⋊C4×D5311yescharacteristic
468C2×C4C22⋊C4×D5311yescharacteristic
469C2×C4C22⋊C4×D5311yescharacteristic
470C2×C4C22×C4×D5311yescharacteristic
471C2×C4C2×D4×D5311yescharacteristic
472C2×C4C2×D4×D5311yescharacteristic
473C8—24no—
474C8—24no—
475C8—1064no—
476C8—1064no—
477C22—534no—
478C22—1062no—
479C22—1062no—
480C22—1068no—
481C22—4244no—
482C22—2122no—
483C22—2122no—
484C22—2122no—
485C22#1681060612yes—
486C22#104×D5312yes—
487C22#104×D5312yes—
488C22C2×C22⋊C4×D5311yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—534no—
498C4—1062no—
499C4—1062no—
500C4—2122no—
501C2—1064no—
502C2—4242no—
503C2#2027330214yesdirect product factor
504C2C2×#1681060611yescharacteristic
505C2C2×#104×D5311yescharacteristic
506C2C2×#104×D5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×#330C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133192C4
cosocleC42×C53C24
hypercenterC2×#330D53
hypocenterC53C2×#1524
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
2. C2×#16810575C2semidirect
3. C2×#97×D53C2semidirect
4. #20273302C2direct (N × H)
5. C2×#16810365C2semidirect
6. C2×#16810362C2semidirect
7. C2×#16810362C2semidirect
8. C2×C53×#330C2semidirect
26. #16810575C22semidirect
31. #97×D53C22semidirect
33. #16810365C22semidirect
34. #16810362C22semidirect
35. #16810362C22semidirect
39. C53×#330C22semidirect
253. C2×#330D53semidirect
256. C2×C53#1524semidirect
290. #330C2×D53semidirect
293. C53C2×#1524semidirect
503. C2#20273302direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111582
C2×#168105752
C2×#97×D532
#202733022
C2×#168103652
C2×#168103622
C2×C53×#3302
C2×#152453