Finite Group Databasebeta

← C2×#20256539 #27122802

Subgroups — C2×#20256539 #27122802

Order 12032.

2022 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689940C211yesmaximal, characteristic, complemented
2#20256539C2116yesmaximal, direct product factor
3C2×#15689564C213yesmaximal, complemented
4C2×#15689563C213yesmaximal, complemented
5C2×#15689553C213yesmaximal, complemented
6C22×#13093970C213yesmaximal, complemented
7C2×#15689540C211yesmaximal, characteristic, complemented
8C2×C47×#578C211yesmaximal, characteristic, complemented
9C2×#13094071—21no—
10#15689940—24no—
11#15689940C2218yescomplemented
12C2×#13094057—23no—
13C2×#13094046C2213yes—
14C22×Dic188C2213yes—
15#15689564C22124yescomplemented
16#15689563C22124yescomplemented
17#15689553C22124yescomplemented
18C2×#13093970C2216yes—
19C2×#13093970C2213yes—
20C2×#13093970C22124yescomplemented
21C2×#13093968C2213yes—
22C2×#13093967C2213yes—
23C2×#13093965C2211yescharacteristic
24#15689540C2218yescomplemented
25C2×#13093964C2213yes—
26C22×C47⋊C16C2213yes—
27C2×2−1+4×C47—21no—
28C47×#578—24no—
29C47×#578C2218yescomplemented
30C2×#137×C47—23no—
31C2×#135×C47C2213yes—
32C22×Q16×C47C2213yes—
33C2×#131×C47C2211yescharacteristic
34#13094071—28no—
35#13094071—44no—
36C2×#10625350—23no—
37C2×#10625352—43no—
38C22×Dic94—23no—
39C2×Q8×D47—41no—
40#13094057—224no—
41#13094046—26no—
42#13094046C23112yes—
43C2×Dic188—23no—
44C2×Dic188—26no—
45C2×Dic188C2313yes—
46C2×Dic188C23112yes—
47C2×#10625325—23no—
48C2×D376C2311yescharacteristic
49#13093970C23112yes—
50#13093970C23124yes—
51#13093968C23112yes—
52#13093967C23112yes—
53#13093965C2314yes—
54#13093964C23112yes—
55C2×C47⋊C16C2311yescharacteristic
56C2×C47⋊C16C2313yes—
57C2×C47⋊C16C23112yes—
582−1+4×C47—28no—
592−1+4×C47—44no—
60C2×#41×C47—23no—
61C2×#41×C47—43no—
62C2×#41×C47D411yescharacteristic
63C22×Q8×C47—23no—
64C22×Q8×C47—41no—
65#137×C47—224no—
66#135×C47—26no—
67#135×C47C23112yes—
68C2×Q16×C47—23no—
69C2×Q16×C47—26no—
70C2×Q16×C47C2313yes—
71C2×Q16×C47C23112yes—
72C2×SD16×C47—23no—
73C2×D8×C47C2311yescharacteristic
74#131×C47D414yes—
75#131×C47C2314yes—
76C2×M16×C47D413yes—
77C22×C8×C47C2313yes—
78#41×C47—212no—
79#41×C47—424no—
80#41×C47—46no—
81#41×C47D814yes—
82#41×C47C2×D414yes—
83C2×Q8×C47—23no—
84C2×Q8×C47—212no—
85C2×Q8×C47—48no—
86C2×Q8×C47—43no—
87C2×Q8×C47—46no—
88C2×Q8×C47D811yescharacteristic
89C2×D4×C47—21no—
90C2×D4×C47—43no—
91C2×D4×C47D813yes—
92C22×C4×C47—43no—
93C22×C4×C47—43no—
94C22×C4×C47C2×D413yes—
95C2×C22⋊D47—43no—
96C22×Dic47—43no—
97Q8×D47—48no—
98#10625352—424no—
99#10625350—212no—
100#10625350—46no—
101C2×D188—21no—
102C2×C4×D47—43no—
103C2×Dic94—23no—
104C2×Dic94—212no—
105C2×Dic94—43no—
106C2×Dic94—46no—
107Q16×C47—23no—
108Q16×C47—212no—
109Q16×C47C2416yes—
110SD16×C47—212no—
111D8×C47—21no—
112D8×C47C2412yes—
113M16×C47C2×D4112yes—
114C2×C8×C47C2×D413yes—
115C2×C8×C47C2×D416yes—
116C2×C8×C47C2411yescharacteristic
117C2×C8×C47C2416yes—
118Dic188—23no—
119Dic188—212no—
120Dic188C2416yes—
121D376—21no—
122D376C2412yes—
123#10625325—212no—
124C47⋊C16C2412yes—
125C47⋊C16C2416yes—
126C23×C47—41no—
127C23×C47C2×D813yes—
128C22×D47—41no—
129Q8×C47—26no—
130Q8×C47—43no—
131Q8×C47—412no—
132Q8×C47C2×D814yes—
133D4×C47—22no—
134D4×C47—412no—
135D4×C47—41no—
136D4×C47C2×D8112yes—
137C2×C4×C47—43no—
138C2×C4×C47—412no—
139C2×C4×C47—412no—
140C2×C4×C47C2×D813yes—
141C2×C4×C47C2×D816yes—
142C2×C4×C47C22×D411yescharacteristic
143C2×C4×C47C22×D416yes—
144C22⋊D47—412no—
145C2×Dic47—43no—
146C2×Dic47—412no—
147D188—22no—
148D188—41no—
149C4×D47—412no—
150Dic94—26no—
151Dic94—43no—
152Dic94—412no—
153C8×C47C22×D416yes—
154C8×C47C22×D411yescharacteristic
155C8×C47C2511yescharacteristic
156C2×#3417—471nomaximal, Sylow
157C22×C47—26no—
158C22×C47—42no—
159C22×C47—82no—
160C22×C47C22×D816yes—
161C22×C47C22×D811yescharacteristic
162C2×D47—42no—
163C2×D47—82no—
164C4×C47—46no—
165C4×C47C22×D811yescharacteristic
166C4×C47C22×D816yes—
167C4×C47C23×D411yescharacteristic
168Dic47—46no—
169C2×#578—471no—
170C2×#578D4711yescharacteristic, complemented
171#3417—4716no—
172C2×#510—473no—
173C2×#510—473no—
174C2×#508—473no—
175C22×Q32—473no—
176C2×#504—471no—
177C2×C47—26no—
178C2×C47—82no—
179C2×C47#341712yescomplemented
180C2×C47C23×D811yescharacteristic
181D47—82no—
182C2×2−1+4—21no—
183C2×2−1+4—941no—
184#578—24no—
185#578—478no—
186#578—944no—
187#578C2×D4718yescomplemented
188C2×#137—23no—
189C2×#137—943no—
190C2×#135—473no—
191C2×#135C2×D4713yes—
192C22×Q16—473no—
193C22×Q16C2×D4713yes—
194C2×#131C2×D4711yescharacteristic
195#510—4724no—
196#510—4724no—
197#508—4724no—
198C2×Q32—473no—
199C2×Q32—476no—
200C2×Q32—4724no—
201C2×SD32—473no—
202C2×SD32—473no—
203C2×D16—471no—
204#504—478no—
205C2×M32—473no—
206C22×C16—473no—
207C47C2×#341711yescharacteristic, complemented, Sylow
2082−1+4—28no—
2092−1+4—44no—
2102−1+4—948no—
2112−1+4—1884no—
212C2×#41—23no—
213C2×#41—43no—
214C2×#41—943no—
215C2×#41—1883no—
216C2×#41C22⋊D4711yescharacteristic
217C22×Q8—23no—
218C22×Q8—41no—
219C22×Q8—943no—
220C22×Q8—1881no—
221#137—224no—
222#137—9424no—
223#135—26no—
224#135—4712no—
225#135—946no—
226#135C22×D47112yes—
227C2×Q16—23no—
228C2×Q16—26no—
229C2×Q16—473no—
230C2×Q16—4712no—
231C2×Q16—943no—
232C2×Q16—946no—
233C2×Q16C22×D4713yes—
234C2×Q16C22×D47112yes—
235C2×SD16—23no—
236C2×SD16—943no—
237C2×D8—471no—
238C2×D8C22×D4711yescharacteristic
239#131C22⋊D4714yes—
240#131C22×D4714yes—
241C2×M16C22⋊D4713yes—
242C22×C8C22×D4713yes—
243Q32—4712no—
244Q32—4724no—
245SD32—4712no—
246SD32—4712no—
247D16—474no—
248M32—4712no—
249C2×C16—471no—
250C2×C16—473no—
251C2×C16—4712no—
252#41—212no—
253#41—424no—
254#41—46no—
255#41—9412no—
256#41—18824no—
257#41—1886no—
258#41#1062533414yes—
259#41C2×C22⋊D4714yes—
260C2×Q8—23no—
261C2×Q8—212no—
262C2×Q8—48no—
263C2×Q8—43no—
264C2×Q8—46no—
265C2×Q8—943no—
266C2×Q8—9412no—
267C2×Q8—1888no—
268C2×Q8—1883no—
269C2×Q8—1886no—
270C2×Q8#1062533411yescharacteristic
271C2×D4—21no—
272C2×D4—43no—
273C2×D4—941no—
274C2×D4—1883no—
275C2×D4#1062533413yes—
276C22×C4—43no—
277C22×C4—43no—
278C22×C4—1883no—
279C22×C4—1883no—
280C22×C4C2×C22⋊D4713yes—
281Q16—23no—
282Q16—212no—
283Q16—476no—
284Q16—943no—
285Q16—9412no—
286Q16C23×D4716yes—
287SD16—212no—
288SD16—9412no—
289D8—21no—
290D8—472no—
291D8—941no—
292D8C23×D4712yes—
293M16C2×C22⋊D47112yes—
294C2×C8C2×C22⋊D4713yes—
295C2×C8C2×C22⋊D4716yes—
296C2×C8C23×D4711yescharacteristic
297C2×C8C23×D4716yes—
298C16—472no—
299C16—476no—
300C23—41no—
301C23—1881no—
302C23C2×#1062533413yes—
303Q8—26no—
304Q8—412no—
305Q8—43no—
306Q8—946no—
307Q8—18812no—
308Q8—1883no—
309Q8C2×#1062533414yes—
310D4—22no—
311D4—412no—
312D4—41no—
313D4—942no—
314D4—18812no—
315D4—1881no—
316D4C2×#10625334112yes—
317C2×C4—412no—
318C2×C4—43no—
319C2×C4—412no—
320C2×C4—18812no—
321C2×C4—1883no—
322C2×C4—18812no—
323C2×C4C2×#1062533416yes—
324C2×C4C2×#1062533413yes—
325C2×C4C22×C22⋊D4716yes—
326C2×C4C22×C22⋊D4711yescharacteristic
327C8C22×C22⋊D4711yescharacteristic
328C8C22×C22⋊D4716yes—
329C8C24×D4711yescharacteristic
330C22—26no—
331C22—42no—
332C22—82no—
333C22—3762no—
334C22—1882no—
335C22C22×#1062533416yes—
336C22C22×#1062533411yescharacteristic
337C4—46no—
338C4—1886no—
339C4C22×#1062533416yes—
340C4C22×#1062533411yescharacteristic
341C4C23×C22⋊D4711yescharacteristic
342C2—26no—
343C2—82no—
344C2—3762no—
345C2#2025653912yesdirect product factor
346C2C23×#1062533411yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC8×C47C25
FrattiniC8C24×D47
FittingC2×C47×#578C2
socleC22×C47C22×D8
hypercenterC2×#578D47
hypocenterC47C2×#3417
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#15689940C2semidirect
2. #20256539C2direct (N × H)
3. C2×#15689564C2semidirect
4. C2×#15689563C2semidirect
5. C2×#15689553C2semidirect
6. C22×#13093970C2semidirect
7. C2×#15689540C2semidirect
8. C2×C47×#578C2semidirect
11. #15689940C22semidirect
15. #15689564C22semidirect
16. #15689563C22semidirect
17. #15689553C22semidirect
20. C2×#13093970C22semidirect
24. #15689540C22semidirect
29. C47×#578C22semidirect
170. C2×#578D47semidirect
179. C2×C47#3417semidirect
187. #578C2×D47semidirect
207. C47C2×#3417semidirect
345. C2#20256539direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156899402
#202565392
C2×#156895642
C2×#156895632
C2×#156895532
C22×#130939702
C2×#156895402
C2×C47×#5782
C2×#341747