Finite Group Databasebeta

← C2×#20276068 #27412222

Subgroups — C2×#20276068 #27412222

Order 13568.

1614 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811249C211yesmaximal, characteristic
2C2×#16810496C212yesmaximal
3#20276068C218yesmaximal, direct product factor
4C2×#16810390C211yesmaximal, characteristic
5C2×#16810389C212yesmaximal
6C2×C53×#385C211yesmaximal, characteristic
7#16811249—24no—
8#16811249—22no—
9#16811249C2214yes—
10C2×#13133196—21no—
11C2×#13133196—21no—
12C2×#13133194—22no—
13C2×#13133194—22no—
14C22×#11719919—22no—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2212yes—
17#16810496—24no—
18#16810496C2218yes—
19C2×#13133124—22no—
20#16810390C2214yes—
21#16810389C2218yes—
22C24×C4×C53C2211yescharacteristic
23C22×C4⋊C4×C53—21no—
24C22×C22⋊C4×C53C2211yescharacteristic
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#385—22no—
27C53×#385C2214yes—
28C2×#95×C53—21no—
29C22×Dic106—21no—
30C22×Dic106—21no—
31C23×Dic53—22no—
32C23×Dic53D412yes—
33C23×Dic53D412yes—
34#13133196—24no—
35#13133196—24no—
36#13133196—24no—
37#13133196—24no—
38#13133194—28no—
39#13133194—28no—
40#13133194—28no—
41#13133194—28no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×#11719919—22no—
45C2×#11719919—22no—
46C2×#11719919—28no—
47C2×#11719919—48no—
48C2×C22⋊Dic53—22no—
49C2×C22⋊Dic53—22no—
50C2×C22⋊Dic53—22no—
51C2×C22⋊Dic53—22no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53D414yes—
55C2×C22⋊Dic53Q818yescomplemented
56C2×C22⋊Dic53C2314yes—
57C2×C22⋊Dic53C2314yes—
58#13133124—28no—
59C25×C53C2311yescharacteristic
60C23×C4×C53—21no—
61C23×C4×C53—21no—
62C23×C4×C53—22no—
63C23×C4×C53—22no—
64C23×C4×C53—21no—
65C23×C4×C53—44no—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2312yes—
69C2×C4⋊C4×C53—24no—
70C2×C4⋊C4×C53—44no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53D412yes—
76C2×C22⋊C4×C53Q814yescomplemented
77C2×C22⋊C4×C53C2312yes—
78C2×C22⋊C4×C53C2312yes—
79#95×C53—24no—
80C24×C53—42no—
81C24×C53—42no—
82C24×C53C2×D412yes—
83C24×C53C2×D411yescharacteristic
84C24×C53C2×D412yes—
85C24×C53C2×D411yescharacteristic
86C24×C53C2×Q812yes—
87C24×C53C2×Q814yes—
88C24×C53#4112yes—
89C24×C53C2411yescharacteristic
90C22×C4×C53—24no—
91C22×C4×C53—28no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—24no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—44no—
101C22×C4×C53—48no—
102C22×C4×C53—44no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53—44no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53C2×D412yes—
113C22×C4×C53C2×D412yes—
114C22×Dic53—22no—
115C22×Dic53—22no—
116C22×Dic53—28no—
117C22×Dic53—24no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—44no—
121C22×Dic53—44no—
122C22×Dic53—44no—
123C22×Dic53—44no—
124C22×Dic53C2×D414yes—
125C22×Dic53C2×D414yes—
126C2×Dic106—24no—
127C2×Dic106—24no—
128C2×Dic106—82no—
129C2×Dic106—82no—
130C4⋊C4×C53—48no—
131C22⋊C4×C53—24no—
132C22⋊C4×C53—24no—
133C22⋊C4×C53—24no—
134C22⋊C4×C53—22no—
135C22⋊C4×C53C2×Q814yescomplemented
136C22⋊Dic53—24no—
137C22⋊Dic53—24no—
138C22⋊Dic53—28no—
139C22⋊Dic53—24no—
140C22⋊Dic53—24no—
141C22⋊Dic53—24no—
142C22⋊Dic53C2×Q818yescomplemented
143C4⋊Dic53—24no—
144C4⋊Dic53—24no—
145#11719919—28no—
146#11719919—28no—
147#11719919—416no—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—24no—
151C23×C53—22no—
152C23×C53—42no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—42no—
156C23×C53—41no—
157C23×C53—44no—
158C23×C53—48no—
159C23×C53—44no—
160C23×C53—41no—
161C23×C53#12012yes—
162C23×C53#12011yescharacteristic
163C23×C53#12011yescharacteristic
164C23×C53C22⋊Q814yes—
165C23×C53C22⋊Q814yes—
166C23×C53C22⋊Q812yes—
167C23×C53C22⋊Q812yes—
168C23×C53C4⋊Q812yes—
169C23×C53C4⋊Q812yes—
170C23×C53C4⋊Q814yes—
171C23×C53C4⋊Q814yes—
172C23×C53C22×D412yes—
173C23×C53C22×D411yescharacteristic
174C23×C53C22×Q812yes—
175C23×C53C22×Q811yescharacteristic
176C23×C53C2×#4111yescharacteristic
177C2×C4×C53—28no—
178C2×C4×C53—24no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—24no—
182C2×C4×C53—48no—
183C2×C4×C53—42no—
184C2×C4×C53—44no—
185C2×C4×C53—48no—
186C2×C4×C53—42no—
187C2×C4×C53—44no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—44no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×Dic53—24no—
198C2×Dic53—24no—
199C2×Dic53—28no—
200C2×Dic53—416no—
201C2×Dic53—44no—
202C2×Dic53—44no—
203C2×Dic53—44no—
204C2×Dic53—44no—
205C2×Dic53—44no—
206C2×Dic53—44no—
207C2×Dic53—82no—
208C2×Dic53—82no—
209Dic106—84no—
210Dic106—84no—
211C2×#2062—531nomaximal, Sylow
212C22×C53—28no—
213C22×C53—24no—
214C22×C53—24no—
215C22×C53—28no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—44no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—44no—
228C22×C53—42no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53#39514yes—
232C22×C53#39512yes—
233C22×C53#39512yes—
234C22×C53B(4,𝔽2)12yes—
235C22×C53#45812yes—
236C22×C53C2×#12011yescharacteristic
237C22×C53C2×C22⋊Q812yes—
238C22×C53C2×C22⋊Q811yescharacteristic
239C22×C53C2×C4⋊Q811yescharacteristic
240C22×C53C2×C4⋊Q812yes—
241C4×C53—44no—
242C4×C53—42no—
243C4×C53—44no—
244C4×C53—42no—
245Dic53—48no—
246Dic53—44no—
247Dic53—44no—
248C2×#543—531no—
249C2×#385—532no—
250C2×#385D5311yescharacteristic
251#2062—538no—
252C2×#342—532no—
253C2×#342—531no—
254C2×C53—22no—
255C2×C53—21no—
256C2×C53—21no—
257C2×C53—22no—
258C2×C53—24no—
259C2×C53—22no—
260C2×C53#206214yescomplemented
261C2×C53C2×#39511yescharacteristic
262C2×C53C2×B(4,𝔽2)11yescharacteristic
263C2×C53C2×#45811yescharacteristic
264C24×C4C2×D5311yescharacteristic
265#543—534no—
266#543—1064no—
267#543—1062no—
268C2×C22⋊Q8—1062no—
269C2×C22⋊Q8—1062no—
270C2×C22⋊Q8—1061no—
271C2×C22⋊Q8—1061no—
272C22×C4⋊C4—21no—
273C22×C4⋊C4—1062no—
274C22×C22⋊C4—532no—
275C22×C22⋊C4—532no—
276C22×C22⋊C4C2×D5311yescharacteristic
277C22×C22⋊C4C2×D5311yescharacteristic
278#385—22no—
279#385—538no—
280#385—1064no—
281#385C2×D5314yes—
282C2×#95—21no—
283C2×#95—1062no—
284#342—538no—
285#342—534no—
286C53C2×#206211yescharacteristic, complemented, Sylow
287C25C22×D5311yescharacteristic
288C22×Q8—1061no—
289C22×Q8—1061no—
290C23×C4—22no—
291C23×C4—21no—
292C23×C4—21no—
293C23×C4—22no—
294C23×C4—21no—
295C23×C4—44no—
296C23×C4—532no—
297C23×C4—532no—
298C23×C4—1062no—
299C23×C4C22⋊D5311yescharacteristic
300C23×C4C22⋊D5311yescharacteristic
301C23×C4C22×D5312yes—
302C22⋊Q8—1068no—
303C22⋊Q8—1068no—
304C22⋊Q8—1068no—
305C22⋊Q8—1068no—
306C22⋊Q8—1064no—
307C22⋊Q8—1064no—
308C22⋊Q8—1064no—
309C22⋊Q8—1064no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—1062no—
313C2×C4⋊C4—1062no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—1068no—
317C2×C4⋊C4—2128no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—538no—
323C2×C22⋊C4—534no—
324C2×C22⋊C4—534no—
325C2×C22⋊C4—534no—
326C2×C22⋊C4—1064no—
327C2×C22⋊C4—1062no—
328C2×C22⋊C4—1062no—
329C2×C22⋊C4—1062no—
330C2×C22⋊C4—1062no—
331C2×C22⋊C4—1062no—
332C2×C22⋊C4Dic10614yescomplemented
333C2×C22⋊C4C22⋊D5312yes—
334C2×C22⋊C4C22×D5312yes—
335C2×C22⋊C4C22×D5312yes—
336#95—24no—
337#95—1068no—
338C24—42no—
339C24—42no—
340C24C2×Dic10612yes—
341C24#1171994512yes—
342C24D4×D5312yes—
343C24D4×D5312yes—
344C24Q8×D5314yes—
345C24C2×C22⋊D5311yescharacteristic
346C24C2×C22⋊D5311yescharacteristic
347C24C23×D5311yescharacteristic
348C2×Q8—1064no—
349C2×Q8—1064no—
350C2×Q8—4242no—
351C2×Q8—4242no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—534no—
375C22×C4—534no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1068no—
381C22×C4—1064no—
382C22×C4—2124no—
383C22×C4—2124no—
384C22×C4—2124no—
385C22×C4—2124no—
386C22×C4C2×C22⋊D5312yes—
387C22×C4C2×C22⋊D5312yes—
388C4⋊C4—48no—
389C4⋊C4—1068no—
390C4⋊C4—1068no—
391C4⋊C4—1064no—
392C4⋊C4—1064no—
393C4⋊C4—21216no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—538no—
399C22⋊C4—1064no—
400C22⋊C4—1064no—
401C22⋊C4—1068no—
402C22⋊C4—1064no—
403C22⋊C4—1064no—
404C22⋊C4—1064no—
405C22⋊C4C2×Dic10614yescomplemented
406C23—22no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—48no—
417C23—41no—
418C23—41no—
419C23#1313318512yes—
420C23#1313318511yescharacteristic
421C23#1313318511yescharacteristic
422C23#1313319514yes—
423C23#1313319514yes—
424C23#1313319612yes—
425C23#1313319612yes—
426C23#1313321512yes—
427C23#1313321512yes—
428C23#1313321614yes—
429C23#1313321614yes—
430C23C2×D4×D5312yes—
431C23C22×C22⋊D5311yescharacteristic
432C23C2×Q8×D5312yes—
433C23C22×Dic10611yescharacteristic
434C23C2×#1171994511yescharacteristic
435Q8—4244no—
436Q8—4244no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—1064no—
458C2×C4—1064no—
459C2×C4—1068no—
460C2×C4—4242no—
461C2×C4—4242no—
462C2×C4—2124no—
463C2×C4—21216no—
464C2×C4—2124no—
465C2×C4—2124no—
466C2×C4—2124no—
467C2×C4—2124no—
468C2×C4—2124no—
469C22—24no—
470C22—28no—
471C22—24no—
472C22—28no—
473C22—21no—
474C22—22no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—22no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22#1681054314yes—
489C22#1681054312yes—
490C22#1681054312yes—
491C22#1681080912yes—
492C22#1681081212yes—
493C22C2×#1313318511yescharacteristic
494C22C2×#1313319512yes—
495C22C2×#1313319611yescharacteristic
496C22C2×#1313321511yescharacteristic
497C22C2×#1313321612yes—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—42no—
502C4—2128no—
503C4—2124no—
504C4—2124no—
505C2—21no—
506C2—22no—
507C2—22no—
508C2—24no—
509C2—21no—
510C2—22no—
511C2#2027606814yesdirect product factor
512C2C2×#1681054311yescharacteristic
513C2C2×#1681080911yescharacteristic
514C2C2×#1681081211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC2×C53×#385C2
socleC23×C53#120
prime radicalC25×C53C23
hypercenterC2×#385D53
hypocenterC53C2×#2062
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. #20276068C2direct (N × H)
55. C2×C22⋊Dic53Q8semidirect
76. C2×C22⋊C4×C53Q8semidirect
135. C22⋊C4×C53C2×Q8semidirect
142. C22⋊Dic53C2×Q8semidirect
260. C2×C53#2062semidirect
286. C53C2×#2062semidirect
332. C2×C22⋊C4Dic106semidirect
405. C22⋊C4C2×Dic106semidirect
511. C2#20276068direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112492
C2×#168104962
#202760682
C2×#168103902
C2×#168103892
C2×C53×#3852
C2×#206253