Finite Group Databasebeta

← C2×#20251653 #26422690

Subgroups — C2×#20251653 #26422690

Order 12032.

3294 conjugacy classes of subgroups; only the 627 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 627 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251653C2116yesmaximal, direct product factor
2C2×#15689059C213yesmaximal, complemented
3C2×#15689047C212yesmaximal, complemented
4C2×#15689045C216yesmaximal, complemented
5C2×#15688978C213yesmaximal
6C2×C47×#399C211yesmaximal, characteristic, complemented
7C25×D47C2211yescharacteristic
8C22×C4⋊Dic47C2213yescomplemented
9C22×Dic47⋊C4C2216yescomplemented
10C22×C22⋊Dic47C2213yes—
11C22×(C2×D47)⋊C4C2216yes—
12C22×(C2×D47)⋊C4C2216yes—
13#15689059C22124yescomplemented
14#15689047C22116yescomplemented
15#15689045C22148yescomplemented
16#15688978C22124yes—
17C2×C47⋊#95C2213yescomplemented
18C22×C4⋊C4×C47C2213yescomplemented
19C22×C22⋊C4×C47C2213yes—
20C47×#399C2218yescomplemented
21C2×#95×C47C2211yescharacteristic, complemented
22C24×D47—46no—
23C24×D47—46no—
24C24×D47Q818yes—
25C24×D47C2312yes—
26C24×D47C2314yes—
27C23×Dic47C2316yes—
28C2×C4⋊Dic47—46no—
29C2×C4⋊Dic47C23112yescomplemented
30C2×Dic47⋊C4—412no—
31C2×Dic47⋊C4C23124yescomplemented
32C2×C22⋊Dic47—46no—
33C2×C22⋊Dic47C23112yes—
34C2×(C2×D47)⋊C4—412no—
35C2×(C2×D47)⋊C4—412no—
36C2×(C2×D47)⋊C4C23124yes—
37C2×(C2×D47)⋊C4C23124yes—
38C47⋊#95C23112yescomplemented
39C25×C47C2311yescharacteristic
40C23×C4×C47C2313yes—
41C23×C4×C47C2313yes—
42C2×C4⋊C4×C47—46no—
43C2×C4⋊C4×C47C23112yescomplemented
44C2×C22⋊C4×C47—46no—
45C2×C22⋊C4×C47C23112yes—
46#95×C47C2314yescomplemented
47C24×C47—26no—
48C24×C47—26no—
49C24×C47C2×Q814yes—
50C24×C47C2412yes—
51C24×C47C2411yescharacteristic
52C23×D47—212no—
53C23×D47—212no—
54C23×D47—448no—
55C23×D47—412no—
56C23×D47—412no—
57C23×D47—416no—
58C23×D47—448no—
59C23×D47C2×Q818yes—
60C23×D47C2×Q8116yes—
61C23×D47C2414yes—
62C22×C4×C47—26no—
63C22×C4×C47—26no—
64C22×C4×C47—212no—
65C22×C4×C47—212no—
66C22×C4×C47C2416yes—
67C22×C4×C47C2416yes—
68C22×Dic47—212no—
69C22×Dic47—212no—
70C22×Dic47—212no—
71C22×Dic47C24112yes—
72C4⋊C4×C47—412no—
73C22⋊C4×C47—412no—
74C22⋊Dic47—412no—
75(C2×D47)⋊C4—424no—
76(C2×D47)⋊C4—424no—
77C4⋊Dic47—412no—
78Dic47⋊C4—424no—
79C23×C47—224no—
80C23×C47—26no—
81C23×C47—26no—
82C23×C47—44no—
83C23×C47—412no—
84C23×C47C22×Q814yes—
85C23×C47C22×Q812yes—
86C23×C47H(5,𝔽2)16yes—
87C23×C47H(5,𝔽2)16yes—
88C23×C47C2511yescharacteristic
89C22×D47—248no—
90C22×D47—212no—
91C22×D47—212no—
92C22×D47—448no—
93C22×D47—4128no—
94C22×D47—416no—
95C22×D47—448no—
96C22×D47—48no—
97C22×D47—424no—
98C22×D47C22×Q818yes—
99C2×C4×C47—212no—
100C2×C4×C47—26no—
101C2×C4×C47—212no—
102C2×C4×C47—26no—
103C2×C4×C47—412no—
104C2×C4×C47—412no—
105C2×Dic47—212no—
106C2×Dic47—212no—
107C2×Dic47—212no—
108C2×Dic47—424no—
109C2×#2842—471nomaximal, Sylow
110C22×C47—212no—
111C22×C47—416no—
112C22×C47—42no—
113C22×C47—46no—
114C22×C47#543112yes—
115C22×C47#56014yes—
116C22×C47#560112yes—
117C22×C47C23×Q811yescharacteristic
118C22×C47C2×H(5,𝔽2)13yes—
119C22×C47C2×H(5,𝔽2)13yes—
120C2×D47—224no—
121C2×D47—464no—
122C2×D47—432no—
123C2×D47—44no—
124C2×D47—412no—
125C4×C47—46no—
126C4×C47—46no—
127Dic47—412no—
128#2842—4716no—
129C2×#399—473no—
130C2×#399D4711yescharacteristic, complemented
131C2×#396—472no—
132C2×#396—476no—
133C2×#379—473no—
134C2×C47—44no—
135C2×C47#284218yescomplemented
136C2×C47C2×#54313yes—
137C2×C47C2×#56011yescharacteristic
138C2×C47C2×#56013yes—
139D47—48no—
140C26—471no—
141C22×C4⋊C4—476no—
142C22×C4⋊C4—473no—
143C22×C4⋊C4C2×D4713yescomplemented
144C22×C22⋊C4—473no—
145C22×C22⋊C4—476no—
146C22×C22⋊C4—476no—
147C22×C22⋊C4C2×D4713yes—
148#399—4724no—
149#399C2×D4718yescomplemented
150#396—4748no—
151#396—4716no—
152#379—4724no—
153C2×#95—473no—
154C2×#95C2×D4711yescharacteristic, complemented
155C47C2×#284211yescharacteristic, complemented, Sylow
156C25—472no—
157C25—478no—
158C25—474no—
159C25—1886no—
160C25—1886no—
161C25C22×D4711yescharacteristic
162C23×C4—476no—
163C23×C4C22×D4713yes—
164C23×C4C22×D4713yes—
165C2×C4⋊C4—46no—
166C2×C4⋊C4—4724no—
167C2×C4⋊C4—4712no—
168C2×C4⋊C4—18812no—
169C2×C4⋊C4—1886no—
170C2×C4⋊C4C22×D47112yescomplemented
171C2×C22⋊C4—46no—
172C2×C22⋊C4—4712no—
173C2×C22⋊C4—4724no—
174C2×C22⋊C4—4724no—
175C2×C22⋊C4—18812no—
176C2×C22⋊C4—18812no—
177C2×C22⋊C4—1886no—
178C2×C22⋊C4C22×D47112yes—
179#95—4712no—
180#95C22×D4714yescomplemented
181C24—26no—
182C24—26no—
183C24—478no—
184C24—474no—
185C24—4716no—
186C24—9412no—
187C24—9412no—
188C24—18848no—
189C24—18812no—
190C24—18812no—
191C24—18816no—
192C24—18848no—
193C24Q8×D4714yes—
194C24C23×D4712yes—
195C24C23×D4711yescharacteristic
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4—4712no—
201C22×C4—9412no—
202C22×C4—9412no—
203C22×C4—9412no—
204C22×C4C23×D4716yes—
205C22×C4C23×D4716yes—
206C4⋊C4—412no—
207C4⋊C4—18824no—
208C4⋊C4—18812no—
209C22⋊C4—412no—
210C22⋊C4—18824no—
211C22⋊C4—18824no—
212C22⋊C4—18812no—
213C23—224no—
214C23—26no—
215C23—26no—
216C23—44no—
217C23—412no—
218C23—478no—
219C23—9448no—
220C23—9412no—
221C23—9412no—
222C23—18848no—
223C23—188128no—
224C23—18816no—
225C23—18848no—
226C23—1888no—
227C23—18824no—
228C23C2×Q8×D4714yes—
229C23C2×Q8×D4712yes—
230C23#1309406916yes—
231C23#1309407016yes—
232C23C24×D4711yescharacteristic
233C2×C4—26no—
234C2×C4—26no—
235C2×C4—212no—
236C2×C4—212no—
237C2×C4—412no—
238C2×C4—412no—
239C2×C4—9412no—
240C2×C4—9412no—
241C2×C4—9412no—
242C2×C4—18824no—
243C22—212no—
244C22—416no—
245C22—42no—
246C22—46no—
247C22—9424no—
248C22—18864no—
249C22—18832no—
250C22—1884no—
251C22—18812no—
252C22#15689748112yes—
253C22#1568986014yes—
254C22#15689863112yes—
255C22C22×Q8×D4711yescharacteristic
256C22C2×#1309406913yes—
257C22C2×#1309407013yes—
258C4—46no—
259C4—46no—
260C4—18812no—
261C2—44no—
262C2—1888no—
263C2#2025165318yesdirect product factor
264C2C2×#1568974813yes—
265C2C2×#1568986011yescharacteristic
266C2C2×#1568986313yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#399C2
socleC24×C47C24
prime radicalC25×D47C22
hypercenterC2×#399D47
hypocenterC47C2×#2842
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #20251653C2direct (N × H)
2. C2×#15689059C2semidirect
3. C2×#15689047C2semidirect
4. C2×#15689045C2semidirect
6. C2×C47×#399C2semidirect
8. C22×C4⋊Dic47C22semidirect
9. C22×Dic47⋊C4C22semidirect
13. #15689059C22semidirect
14. #15689047C22semidirect
15. #15689045C22semidirect
17. C2×C47⋊#95C22semidirect
18. C22×C4⋊C4×C47C22semidirect
20. C47×#399C22semidirect
21. C2×#95×C47C22semidirect
29. C2×C4⋊Dic47C23semidirect
31. C2×Dic47⋊C4C23semidirect
38. C47⋊#95C23semidirect
43. C2×C4⋊C4×C47C23semidirect
46. #95×C47C23semidirect
130. C2×#399D47semidirect
135. C2×C47#2842semidirect
143. C22×C4⋊C4C2×D47semidirect
149. #399C2×D47semidirect
154. C2×#95C2×D47semidirect
155. C47C2×#2842semidirect
170. C2×C4⋊C4C22×D47semidirect
180. #95C22×D47semidirect
263. C2#20251653direct (N × H)

Maximal subgroups

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

subgroupindex
#202516532
C2×#156890592
C2×#156890472
C2×#156890452
C2×#156889782
C2×C47×#3992
C2×#284247