Finite Group Databasebeta

← C2×#20254347 #27087491

Subgroups — C2×#20254347 #27087491

Order 12032.

2186 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic, complemented
2C2×#15689674C211yesmaximal, characteristic
3#20254347C2116yesmaximal, direct product factor
4C2×#15689502C211yesmaximal, characteristic, complemented
5C2×#15689496C211yesmaximal, characteristic, complemented
6C2×#15689471C211yesmaximal, characteristic, complemented
7C2×#15689348C214yesmaximal, complemented
8C2×#15689288C214yesmaximal, complemented
9C2×#15689174C211yesmaximal, characteristic, complemented
10C2×C47×#487C211yesmaximal, characteristic, complemented
11C22×#10625354C2212yes—
12C22×#10625337C2214yes—
13C22×#10625325C2214yes—
14#15689678—24no—
15#15689678C2218yescomplemented
16#15689674—24no—
17#15689674C2218yes—
18C2×#13094024C2211yescharacteristic
19C2×#13094021C2211yescharacteristic
20C2×#13094015—21no—
21C2×#13094013C2212yes—
22C2×#13094004C2212yes—
23C2×#13094000—21no—
24C2×#13093999—21no—
25C2×#13093999—21no—
26C2×Q8×Dic47—21no—
27C2×#13093989C2212yes—
28C2×#13093986—21no—
29C2×#13093986—21no—
30C2×#13093982—21no—
31C2×C42×D47C2211yescharacteristic
32#15689502C2218yescomplemented
33#15689496C2218yescomplemented
34#15689471C2218yescomplemented
35#15689348C22132yescomplemented
36#15689288C22132yescomplemented
37#15689174C2218yescomplemented
38C2×Dic47⋊C8C2212yes—
39C2×#13093944C2214yes—
40C2×(C2×D47)⋊C8C2212yes—
41C2×C4×C47⋊C8C2211yescharacteristic
42C2×C4⋊Q8×C47C2211yescharacteristic
43C2×C4⋊Q8×C47C2211yescharacteristic
44C47×#487C2218yescomplemented
45C2×#103×C47C2214yes—
46C2×C4×C8×C47C2211yescharacteristic
47C2×#10625354—24no—
48C2×#10625354—48no—
49C2×#10625354C2318yes—
50C22×Dic94C2312yes—
51C22×D188—22no—
52C22×D188—21no—
53C22×D188—21no—
54C22×D188C2312yes—
55C22×C4×D47—22no—
56C22×C4×D47D411yescharacteristic
57C22×C4×D47D411yescharacteristic
58C22×C4×D47C2311yescharacteristic
59C2×#10625337—48no—
60C2×#10625337C23116yes—
61C2×#10625325—48no—
62C2×#10625325C23116yes—
63C22×C47⋊C8C2312yes—
64#13094024—22no—
65#13094024C2314yes—
66#13094021—22no—
67#13094021C2314yes—
68#13094015—28no—
69#13094013—24no—
70#13094013C2318yes—
71#13094004—24no—
72#13094004C2318yes—
73#13094000—28no—
74#13093999—28no—
75#13093999—28no—
76Q8×Dic47—28no—
77#13093989—24no—
78#13093989C2318yes—
79#13093986—28no—
80#13093986—28no—
81#13093982—28no—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—21no—
84C2×C4⋊Dic47—22no—
85C2×Dic47⋊C4—22no—
86C2×(C2×D47)⋊C4—22no—
87C2×(C2×D47)⋊C4—22no—
88C2×(C2×D47)⋊C4—22no—
89C2×C4×Dic47—22no—
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93C42×D47D414yes—
94C42×D47C2314yes—
95Dic47⋊C8C2318yes—
96#13093944C23116yes—
97(C2×D47)⋊C8C2318yes—
98C4×C47⋊C8C2314yes—
99C22×Q8×C47—21no—
100C22×Q8×C47C2312yes—
101C22×C8×C47C2312yes—
102C4⋊Q8×C47—22no—
103C4⋊Q8×C47—22no—
104C4⋊Q8×C47C2314yes—
105C4⋊Q8×C47C2314yes—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47C2312yes—
110C2×C42×C47C2311yescharacteristic
111#103×C47C23116yes—
112C4×C8×C47C2314yes—
113C23×D47—22no—
114C23×D47C2×D411yescharacteristic
115C2×Q8×C47—22no—
116C2×Q8×C47—24no—
117C2×Q8×C47—28no—
118C2×Q8×C47—44no—
119C2×Q8×C47C2414yes—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C22×Dic47—22no—
126C22×Dic47C2×D411yescharacteristic
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129#10625354—416no—
130#10625354—416no—
131C2×D188—28no—
132C2×D188—28no—
133C2×D188—24no—
134C2×D188—24no—
135C2×D188—22no—
136C2×D188—48no—
137C2×D188—44no—
138C2×D188—44no—
139C2×D188C2414yes—
140C2×C4×D47—24no—
141C2×C4×D47—24no—
142C2×C4×D47—24no—
143C2×C4×D47—28no—
144C2×C4×D47—48no—
145C2×C4×D47C2×D412yes—
146C2×C4×D47C2×D414yes—
147C2×C4×D47C2×D414yes—
148C2×C4×D47C2412yes—
149C2×Dic94—28no—
150C2×Dic94—22no—
151C2×Dic94C2414yes—
152C2×C8×C47—24no—
153C2×C8×C47C2414yes—
154C4⋊C4×C47—24no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47C2414yes—
159C42×C47C2×D412yes—
160C42×C47C2412yes—
161#10625337—416no—
162(C2×D47)⋊C4—28no—
163(C2×D47)⋊C4—28no—
164(C2×D47)⋊C4—28no—
165C4⋊Dic47—24no—
166C4⋊Dic47—24no—
167C4⋊Dic47—28no—
168Dic47⋊C4—28no—
169C4×Dic47—28no—
170C4×Dic47C2×D412yes—
171C4×Dic47C2×D414yes—
172C4×Dic47C2×D414yes—
173C4×Dic47C2412yes—
174C2×C47⋊C8—24no—
175C2×C47⋊C8C2414yes—
176#10625325—416no—
177C23×C47C22×D411yescharacteristic
178C22×D47—22no—
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—24no—
182C22×D47—44no—
183C22×D47—44no—
184C22×D47—44no—
185C22×D47C22×D412yes—
186Q8×C47—28no—
187Q8×C47—44no—
188Q8×C47—48no—
189C2×C4×C47—24no—
190C2×C4×C47—24no—
191C2×C4×C47—44no—
192C2×C4×C47—44no—
193C2×C4×C47#13514yes—
194C2×C4×C47#13714yes—
195C2×C4×C47C22×D411yescharacteristic
196C2×C4×C47C22×D412yes—
197C2×C4×C47C22×D412yes—
198C2×C4×C47H(5,𝔽2)14yes—
199C2×C4×C47C2511yescharacteristic
200C2×Dic47—22no—
201C2×Dic47—22no—
202C2×Dic47—22no—
203C2×Dic47—24no—
204C2×Dic47—44no—
205C2×Dic47C22×D412yes—
206C2×Dic47C22×D412yes—
207C2×Dic47C22×D412yes—
208D188—28no—
209D188—416no—
210D188—44no—
211D188—48no—
212D188—48no—
213C4×D47—24no—
214C4×D47—28no—
215C4×D47—28no—
216C4×D47—24no—
217C4×D47—416no—
218Dic94—28no—
219Dic94—44no—
220C8×C47—24no—
221C47⋊C8—24no—
222C2×#3210—471nomaximal, Sylow
223C22×C47#53512yes—
224C22×C47C2×#13512yes—
225C22×C47C2×#13712yes—
226C22×C47C23×D411yescharacteristic
227C2×D47—22no—
228C2×D47—22no—
229C2×D47—22no—
230C2×D47—44no—
231C2×D47—44no—
232C2×D47—44no—
233C2×D47—44no—
234C2×D47—88no—
235C4×C47—44no—
236C4×C47—44no—
237C4×C47#53512yes—
238C4×C47C2×#13514yes—
239C4×C47C2×#13714yes—
240C4×C47C2×H(5,𝔽2)12yes—
241Dic47—22no—
242Dic47—22no—
243Dic47—22no—
244Dic47—44no—
245C2×#533—471no—
246C2×#532—471no—
247#3210—4716no—
248C2×#494—471no—
249C2×#492—471no—
250C2×#487D4711yescharacteristic, complemented
251C2×#486—471no—
252C2×#464—474no—
253C2×#452—474no—
254C2×#431—471no—
255C2×C47#321014yescomplemented
256C2×C47C2×#53511yescharacteristic
257C2×C47C22×#13511yescharacteristic
258C2×C47C22×#13711yescharacteristic
259D47—42no—
260D47—84no—
261C22×#41—472no—
262C22×Q16—474no—
263C22×SD16—474no—
264#533—478no—
265#533—944no—
266#532—478no—
267#532—944no—
268C2×C4⋊Q8—471no—
269C2×C4⋊Q8C2×D4711yescharacteristic
270C2×C4⋊Q8C2×D4711yescharacteristic
271C2×#127—471no—
272C2×#125—941no—
273C2×#124—472no—
274C2×C22⋊Q8—472no—
275C2×C22⋊Q8—941no—
276C2×#121—941no—
277C2×#121—941no—
278C2×C4×Q8—472no—
279C2×C4×Q8—941no—
280C2×C4×D4—941no—
281C2×C4×D4—941no—
282C2×#117—941no—
283C22×C42—471no—
284#494—478no—
285#492—478no—
286#487C2×D4718yescomplemented
287#486—478no—
288#464—4732no—
289#452—4732no—
290#431—478no—
291C2×C4⋊C8—472no—
292C2×#103C2×D4714yes—
293C2×#102—474no—
294C2×C22⋊C8—472no—
295C2×C4×C8—471no—
296C2×C4×C8C2×D4711yescharacteristic
297C47C2×#321011yescharacteristic, complemented, Sylow
298C2×#41—478no—
299C2×#41—944no—
300C2×#41—1888no—
301C22×Q8—21no—
302C22×Q8—472no—
303C22×Q8C22×D4712yes—
304C22×D4—472no—
305C22×D4—942no—
306C22×D4—941no—
307C22×D4—941no—
308C23×C4—471no—
309C23×C4—471no—
310C23×C4—471no—
311C23×C4—942no—
312C2×Q16—4716no—
313C2×Q16—1888no—
314C2×SD16—4716no—
315C2×SD16—1888no—
316C22×C8—472no—
317C22×C8C22×D4712yes—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—474no—
321C4⋊Q8—942no—
322C4⋊Q8C22×D4714yes—
323C4⋊Q8C22×D4714yes—
324#127—474no—
325#127—942no—
326#125—948no—
327#124—478no—
328#124—944no—
329C22⋊Q8—478no—
330C22⋊Q8—944no—
331C22⋊Q8—948no—
332#121—948no—
333#121—948no—
334C4×Q8—478no—
335C4×Q8—944no—
336C4×Q8—948no—
337C4×D4—948no—
338C4×D4—948no—
339#117—948no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—942no—
344C2×C4⋊C4—942no—
345C2×C4⋊C4—941no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4C22×D4712yes—
348C2×C22⋊C4—942no—
349C2×C22⋊C4—942no—
350C2×C22⋊C4—942no—
351C2×C42—471no—
352C2×C42—471no—
353C2×C42—471no—
354C2×C42—474no—
355C2×C42—474no—
356C2×C42—942no—
357C2×C42C22×D4711yescharacteristic
358C4⋊C8—478no—
359#103C22×D47116yes—
360#102—4716no—
361C22⋊C8—478no—
362C4×C8—474no—
363C4×C8C22×D4714yes—
364C24—471no—
365C24—942no—
366#41—18816no—
367#41—18816no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—28no—
371C2×Q8—44no—
372C2×Q8—474no—
373C2×Q8—942no—
374C2×Q8—948no—
375C2×Q8C23×D4714yes—
376C2×D4—474no—
377C2×D4—948no—
378C2×D4—944no—
379C2×D4—944no—
380C2×D4—942no—
381C2×D4—948no—
382C2×D4—1888no—
383C2×D4—1884no—
384C2×D4—1884no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—471no—
388C22×C4—471no—
389C22×C4—471no—
390C22×C4—474no—
391C22×C4—474no—
392C22×C4—472no—
393C22×C4—472no—
394C22×C4—948no—
395C22×C4—944no—
396C22×C4—944no—
397C22×C4—944no—
398C22×C4—942no—
399C22×C4—1888no—
400C22×C4D4×D4711yescharacteristic
401C22×C4D4×D4711yescharacteristic
402C22×C4C23×D4711yescharacteristic
403Q16—18816no—
404SD16—18816no—
405C2×C8—24no—
406C2×C8—474no—
407C2×C8—944no—
408C2×C8C23×D4714yes—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—948no—
414C4⋊C4—948no—
415C4⋊C4—944no—
416C4⋊C4—944no—
417C4⋊C4C23×D4714yes—
418C22⋊C4—948no—
419C22⋊C4—948no—
420C22⋊C4—948no—
421C42—474no—
422C42—474no—
423C42—472no—
424C42—472no—
425C42—948no—
426C42D4×D4712yes—
427C42C23×D4712yes—
428C23—472no—
429C23—942no—
430C23—942no—
431C23—942no—
432C23—944no—
433C23—1884no—
434C23—1884no—
435C23—1884no—
436C23C2×D4×D4711yescharacteristic
437Q8—28no—
438Q8—48no—
439Q8—44no—
440Q8—948no—
441Q8—1884no—
442D4—948no—
443D4—18816no—
444D4—1888no—
445D4—1888no—
446D4—1884no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—472no—
452C2×C4—472no—
453C2×C4—472no—
454C2×C4—942no—
455C2×C4—942no—
456C2×C4—942no—
457C2×C4—948no—
458C2×C4—948no—
459C2×C4—944no—
460C2×C4—944no—
461C2×C4—944no—
462C2×C4—18816no—
463C2×C4—1884no—
464C2×C4#1309404514yes—
465C2×C4#1309405614yes—
466C2×C4C2×D4×D4712yes—
467C2×C4C2×D4×D4712yes—
468C2×C4C2×D4×D4711yescharacteristic
469C2×C4#1309407014yes—
470C2×C4C24×D4711yescharacteristic
471C8—24no—
472C8—944no—
473C22—942no—
474C22—942no—
475C22—942no—
476C22—3768no—
477C22—1884no—
478C22—1884no—
479C22—1884no—
480C22—1884no—
481C22#1568969112yes—
482C22C2×#1309404512yes—
483C22C2×#1309405612yes—
484C22C22×D4×D4711yescharacteristic
485C4—44no—
486C4—44no—
487C4—942no—
488C4—942no—
489C4—942no—
490C4—1884no—
491C4#1568969112yes—
492C4C2×#1309404514yes—
493C4C2×#1309405614yes—
494C4C2×#1309407012yes—
495C2—3764no—
496C2—1882no—
497C2#2025434714yesdirect product factor
498C2C2×#1568969111yescharacteristic
499C2C22×#1309404511yescharacteristic
500C2C22×#1309405611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#487C2
socleC23×C47C22×D4
prime radicalC2×#15689674C2
hypercenterC2×#487D47
hypocenterC47C2×#3210
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×#15689678C2semidirect
3. #20254347C2direct (N × H)
4. C2×#15689502C2semidirect
5. C2×#15689496C2semidirect
6. C2×#15689471C2semidirect
7. C2×#15689348C2semidirect
8. C2×#15689288C2semidirect
9. C2×#15689174C2semidirect
10. C2×C47×#487C2semidirect
15. #15689678C22semidirect
32. #15689502C22semidirect
33. #15689496C22semidirect
34. #15689471C22semidirect
35. #15689348C22semidirect
36. #15689288C22semidirect
37. #15689174C22semidirect
44. C47×#487C22semidirect
250. C2×#487D47semidirect
255. C2×C47#3210semidirect
286. #487C2×D47semidirect
297. C47C2×#3210semidirect
497. C2#20254347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896782
C2×#156896742
#202543472
C2×#156895022
C2×#156894962
C2×#156894712
C2×#156893482
C2×#156892882
C2×#156891742
C2×C47×#4872
C2×#321047