Finite Group Databasebeta

← C2×#20251640 #26422685

Subgroups — C2×#20251640 #26422685

Order 12032.

3550 conjugacy classes of subgroups; only the 627 normal subgroups are shown.

normal subgroups only

This group has 3550 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#20251640C2116yesmaximal, direct product factor
2C2×#15689036C216yesmaximal, complemented
3C2×#15689035C213yesmaximal, complemented
4C2×#15689034C212yesmaximal, complemented
5C2×#15688978C213yesmaximal
6C2×C47×#393C211yesmaximal, characteristic, complemented
7C25×D47C2211yescharacteristic
8C23×Dic94C2213yescomplemented
9C22×C22⋊Dic47C2213yes—
10C22×(C2×D47)⋊C4C2216yes—
11C22×(C2×D47)⋊C4C2216yes—
12#15689036C22148yescomplemented
13#15689035C22124yescomplemented
14#15689034C22116yescomplemented
15#15688978C22124yes—
16C2×C47⋊#95C2216yescomplemented
17C2×C47⋊#95C2213yescomplemented
18C23×Q8×C47C2211yescharacteristic, complemented
19C22×C22⋊C4×C47C2213yes—
20C47×#393C2218yescomplemented
21C2×#95×C47C2213yescomplemented
22C24×D47—46no—
23C24×D47—46no—
24C24×D47Q818yescomplemented
25C24×D47C2312yes—
26C24×D47C2314yes—
27C22×Dic94—46no—
28C22×Dic94—412no—
29C22×Dic94C23112yescomplemented
30C23×Dic47C2316yes—
31C2×C22⋊Dic47—46no—
32C2×C22⋊Dic47C23112yes—
33C2×(C2×D47)⋊C4—412no—
34C2×(C2×D47)⋊C4—412no—
35C2×(C2×D47)⋊C4C23124yes—
36C2×(C2×D47)⋊C4C23124yes—
37C47⋊#95C23124yescomplemented
38C47⋊#95C23112yescomplemented
39C25×C47C2311yescharacteristic
40C22×Q8×C47—46no—
41C22×Q8×C47C2314yescomplemented
42C23×C4×C47C2313yes—
43C23×C4×C47C2313yes—
44C2×C22⋊C4×C47—46no—
45C2×C22⋊C4×C47C23112yes—
46#95×C47C23112yescomplemented
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—448no—
58C23×D47—416no—
59C23×D47C2×Q818yes—
60C23×D47C2×Q8116yescomplemented
61C23×D47C2414yes—
62C2×Q8×C47—412no—
63C2×Q8×C47—416no—
64C22×C4×C47—26no—
65C22×C4×C47—26no—
66C22×C4×C47—212no—
67C22×C4×C47—212no—
68C22×C4×C47C2416yes—
69C22×C4×C47C2416yes—
70C22×Dic47—212no—
71C22×Dic47—212no—
72C22×Dic47—212no—
73C22×Dic47C24112yes—
74C2×Dic94—412no—
75C2×Dic94—424no—
76C2×Dic94—448no—
77C22⋊C4×C47—412no—
78C22⋊Dic47—412no—
79(C2×D47)⋊C4—424no—
80(C2×D47)⋊C4—424no—
81C23×C47—224no—
82C23×C47—26no—
83C23×C47—26no—
84C23×C47—412no—
85C23×C47—44no—
86C23×C47C22×Q814yes—
87C23×C47C22×Q812yes—
88C23×C47H(5,𝔽2)16yes—
89C23×C47H(5,𝔽2)16yes—
90C23×C47C2511yescharacteristic
91C22×D47—248no—
92C22×D47—212no—
93C22×D47—212no—
94C22×D47—448no—
95C22×D47—4128no—
96C22×D47—448no—
97C22×D47—416no—
98C22×D47—424no—
99C22×D47—48no—
100C22×D47C22×Q818yes—
101Q8×C47—416no—
102C2×C4×C47—212no—
103C2×C4×C47—212no—
104C2×C4×C47—26no—
105C2×C4×C47—26no—
106C2×C4×C47—412no—
107C2×C4×C47—412no—
108C2×Dic47—212no—
109C2×Dic47—212no—
110C2×Dic47—212no—
111C2×Dic47—424no—
112Dic94—448no—
113C2×#2840—471nomaximal, Sylow
114C22×C47—212no—
115C22×C47—416no—
116C22×C47—46no—
117C22×C47—42no—
118C22×C47#543112yes—
119C22×C47H(3,𝔽4)112yes—
120C22×C47H(3,𝔽4)14yes—
121C22×C47C23×Q811yescharacteristic
122C22×C47C2×H(5,𝔽2)13yes—
123C22×C47C2×H(5,𝔽2)13yes—
124C2×D47—224no—
125C2×D47—464no—
126C2×D47—432no—
127C2×D47—412no—
128C2×D47—44no—
129C4×C47—46no—
130C4×C47—46no—
131Dic47—412no—
132#2840—4716no—
133C2×#393—476no—
134C2×#393—472no—
135C2×#393—473no—
136C2×#393D4711yescharacteristic, complemented
137C2×#379—473no—
138C2×C47—44no—
139C2×C47#284018yescomplemented
140C2×C47C2×#54313yes—
141C2×C47C2×H(3,𝔽4)13yes—
142C2×C47C2×H(3,𝔽4)11yescharacteristic
143D47—48no—
144C26—471no—
145C23×Q8—473no—
146C23×Q8C2×D4711yescharacteristic, complemented
147C22×C22⋊C4—473no—
148C22×C22⋊C4—476no—
149C22×C22⋊C4—476no—
150C22×C22⋊C4C2×D4713yes—
151#393—4716no—
152#393—4724no—
153#393—4748no—
154#393C2×D4718yescomplemented
155#379—4724no—
156C2×#95—473no—
157C2×#95—476no—
158C2×#95C2×D4713yescomplemented
159C47C2×#284011yescharacteristic, complemented, Sylow
160C25—472no—
161C25—478no—
162C25—474no—
163C25—1886no—
164C25—1886no—
165C25C22×D4711yescharacteristic
166C22×Q8—46no—
167C22×Q8—4712no—
168C22×Q8—1886no—
169C22×Q8—18812no—
170C22×Q8C22×D4714yescomplemented
171C23×C4—476no—
172C23×C4C22×D4713yes—
173C23×C4C22×D4713yes—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—4712no—
176C2×C22⋊C4—4724no—
177C2×C22⋊C4—4724no—
178C2×C22⋊C4—18812no—
179C2×C22⋊C4—18812no—
180C2×C22⋊C4—1886no—
181C2×C22⋊C4C22×D47112yes—
182#95—4724no—
183#95—4712no—
184#95C22×D47112yescomplemented
185C24—26no—
186C24—26no—
187C24—478no—
188C24—474no—
189C24—4716no—
190C24—9412no—
191C24—9412no—
192C24—18848no—
193C24—18812no—
194C24—18812no—
195C24—18848no—
196C24—18816no—
197C24Q8×D4714yes—
198C24C23×D4712yes—
199C24C23×D4711yescharacteristic
200C2×Q8—412no—
201C2×Q8—416no—
202C2×Q8—18812no—
203C2×Q8—18824no—
204C2×Q8—18848no—
205C22×C4—26no—
206C22×C4—26no—
207C22×C4—212no—
208C22×C4—212no—
209C22×C4—4712no—
210C22×C4—9412no—
211C22×C4—9412no—
212C22×C4—9412no—
213C22×C4C23×D4716yes—
214C22×C4C23×D4716yes—
215C22⋊C4—412no—
216C22⋊C4—18824no—
217C22⋊C4—18824no—
218C22⋊C4—18812no—
219C23—224no—
220C23—26no—
221C23—26no—
222C23—412no—
223C23—44no—
224C23—478no—
225C23—9448no—
226C23—9412no—
227C23—9412no—
228C23—18848no—
229C23—18848no—
230C23—18816no—
231C23—188128no—
232C23—18824no—
233C23—1888no—
234C23C2×Q8×D4714yes—
235C23C2×Q8×D4712yes—
236C23#1309406916yes—
237C23#1309407016yes—
238C23C24×D4711yescharacteristic
239Q8—416no—
240Q8—18848no—
241C2×C4—26no—
242C2×C4—26no—
243C2×C4—212no—
244C2×C4—212no—
245C2×C4—412no—
246C2×C4—412no—
247C2×C4—9412no—
248C2×C4—9412no—
249C2×C4—9412no—
250C2×C4—18824no—
251C22—212no—
252C22—46no—
253C22—42no—
254C22—416no—
255C22—9424no—
256C22—18864no—
257C22—18812no—
258C22—1884no—
259C22—18832no—
260C22#15689748112yes—
261C22#15689865112yes—
262C22#1568986614yes—
263C22C22×Q8×D4711yescharacteristic
264C22C2×#1309406913yes—
265C22C2×#1309407013yes—
266C4—46no—
267C4—46no—
268C4—18812no—
269C2—44no—
270C2—1888no—
271C2#2025164018yesdirect product factor
272C2C2×#1568974813yes—
273C2C2×#1568986513yes—
274C2C2×#1568986611yescharacteristic
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#393C2
socleC24×C47C24
prime radicalC25×D47C22
hypercenterC2×#393D47
hypocenterC47C2×#2840
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. #20251640C2direct (N × H)
2. C2×#15689036C2semidirect
3. C2×#15689035C2semidirect
4. C2×#15689034C2semidirect
6. C2×C47×#393C2semidirect
8. C23×Dic94C22semidirect
12. #15689036C22semidirect
13. #15689035C22semidirect
14. #15689034C22semidirect
16. C2×C47⋊#95C22semidirect
17. C2×C47⋊#95C22semidirect
18. C23×Q8×C47C22semidirect
20. C47×#393C22semidirect
21. C2×#95×C47C22semidirect
24. C24×D47Q8semidirect
29. C22×Dic94C23semidirect
37. C47⋊#95C23semidirect
38. C47⋊#95C23semidirect
41. C22×Q8×C47C23semidirect
46. #95×C47C23semidirect
60. C23×D47C2×Q8semidirect
136. C2×#393D47semidirect
139. C2×C47#2840semidirect
146. C23×Q8C2×D47semidirect
154. #393C2×D47semidirect
158. C2×#95C2×D47semidirect
159. C47C2×#2840semidirect
170. C22×Q8C22×D47semidirect
184. #95C22×D47semidirect
271. C2#20251640direct (N × H)

Maximal subgroups

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

subgroupindex
#202516402
C2×#156890362
C2×#156890352
C2×#156890342
C2×#156889782
C2×C47×#3932
C2×#284047