Finite Group Databasebeta

← C2×#20251431 #26422833

Subgroups — C2×#20251431 #26422833

Order 12032.

3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 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
1C22×#13094021C211yesmaximal, characteristic, complemented
2C22×#13094018C211yesmaximal, characteristic, complemented
3#20251431C2116yesmaximal, direct product factor
4C2×#15689041C216yesmaximal, complemented
5C2×#15689033C216yesmaximal, complemented
6C2×C47×#383C211yesmaximal, characteristic, complemented
7C23×D188C2216yes—
8C23×D188C2214yes—
9C2×#13094021—28no—
10C2×#13094021C2218yescomplemented
11C2×#13094018—28no—
12C2×#13094018C2218yescomplemented
13C22×Dic47⋊C4C2213yes—
14C22×(C2×D47)⋊C4C2213yes—
15C22×(C2×D47)⋊C4C22112yes—
16#15689041C22148yescomplemented
17#15689033C22148yescomplemented
18C22×C42×C47C2211yescharacteristic
19C47×#383C2218yescomplemented
20C2×#95×C47C2216yes—
21C24×D47C2311yescharacteristic
22C24×D47C2314yes—
23C22×D188—224no—
24C22×D188—424no—
25C22×D188—424no—
26C22×D188C23124yes—
27C22×D188C23116yes—
28C23×Dic47C2313yes—
29#13094021—232no—
30#13094018—232no—
31C2×Dic47⋊C4—212no—
32C2×Dic47⋊C4C23112yes—
33C2×(C2×D47)⋊C4—212no—
34C2×(C2×D47)⋊C4—424no—
35C2×(C2×D47)⋊C4C23112yes—
36C2×(C2×D47)⋊C4C23148yes—
37C23×C4×C47C2314yes—
38C23×C4×C47C2313yes—
39C2×C42×C47D418yescomplemented
40C2×C42×C47C2314yes—
41#95×C47C23124yes—
42C24×C47C2411yescharacteristic
43C23×D47—22no—
44C23×D47—28no—
45C23×D47—224no—
46C23×D47—224no—
47C23×D47—212no—
48C23×D47C2412yes—
49C23×D47C2418yes—
50C22×C4×C47—212no—
51C22×C4×C47—224no—
52C22×C4×C47C2×D4112yes—
53C22×C4×C47C2418yes—
54C22×C4×C47C2416yes—
55C22×Dic47—26no—
56C22×Dic47—212no—
57C22×Dic47C2416yes—
58C2×D188—248no—
59C2×D188—448no—
60C2×D188—448no—
61C2×D188—496no—
62C2×D188—464no—
63C42×C47C2×D4116yescomplemented
64(C2×D47)⋊C4—224no—
65(C2×D47)⋊C4—448no—
66Dic47⋊C4—224no—
67C23×C47C22×D412yes—
68C23×C47H(5,𝔽2)112yes—
69C23×C47C2511yescharacteristic
70C22×D47—22no—
71C22×D47—28no—
72C22×D47—224no—
73C22×D47—224no—
74C22×D47—212no—
75C22×D47—448no—
76C22×D47—412no—
77C22×D47—416no—
78C22×D47—416no—
79C22×D47—448no—
80C2×C4×C47—224no—
81C2×C4×C47—212no—
82C2×C4×C47—224no—
83C2×C4×C47—416no—
84C2×C4×C47C22×D4112yes—
85C2×Dic47—26no—
86C2×Dic47—212no—
87C2×Dic47—412no—
88D188—496no—
89D188—464no—
90C2×#2818—471nomaximal, Sylow
91C22×C47#535112yes—
92C22×C47#560116yes—
93C22×C47C23×D411yescharacteristic
94C22×C47C2×H(5,𝔽2)16yes—
95C2×D47—424no—
96C2×D47—46no—
97C2×D47—48no—
98C2×D47—48no—
99C2×D47—424no—
100C2×D47—832no—
101C2×D47—88no—
102C4×C47—212no—
103C4×C47—48no—
104Dic47—46no—
105C22×#127—471no—
106C22×#126—471no—
107#2818—4716no—
108C2×#394—476no—
109C2×#392—476no—
110C2×#383D4711yescharacteristic, complemented
111C2×C47#281818yescomplemented
112C2×C47C2×#53513yes—
113C2×C47C2×#56014yes—
114D47—88no—
115D47—82no—
116C23×D4—476no—
117C23×D4—474no—
118C2×#127—478no—
119C2×#127—948no—
120C2×#126—478no—
121C2×#126—948no—
122C22×C4⋊C4—473no—
123C22×C22⋊C4—473no—
124C22×C22⋊C4—4712no—
125C22×C42C2×D4711yescharacteristic
126#394—4748no—
127#392—4748no—
128#383C2×D4718yescomplemented
129C2×#95C2×D4716yes—
130C47C2×#281811yescharacteristic, complemented, Sylow
131C25—471no—
132C25—474no—
133C22×D4—4724no—
134C22×D4—4716no—
135C22×D4—9424no—
136C22×D4—18824no—
137C22×D4—18824no—
138C23×C4—473no—
139C23×C4C22×D4714yes—
140C23×C4C22×D4713yes—
141#127—9432no—
142#126—9432no—
143C2×C4⋊C4—4712no—
144C2×C4⋊C4—9412no—
145C2×C22⋊C4—4712no—
146C2×C22⋊C4—4748no—
147C2×C22⋊C4—9412no—
148C2×C22⋊C4—18824no—
149C2×C42D18818yescomplemented
150C2×C42C22×D4714yes—
151#95C22×D47124yes—
152C24—472no—
153C24—478no—
154C24—942no—
155C24—948no—
156C24—9424no—
157C24—9412no—
158C24—9424no—
159C24C23×D4711yescharacteristic
160C2×D4—9448no—
161C2×D4—18848no—
162C2×D4—18848no—
163C2×D4—18896no—
164C2×D4—18864no—
165C22×C4—212no—
166C22×C4—224no—
167C22×C4—476no—
168C22×C4—946no—
169C22×C4—9412no—
170C22×C4C2×D188112yes—
171C22×C4C23×D4718yes—
172C22×C4C23×D4716yes—
173C4⋊C4—9424no—
174C22⋊C4—9424no—
175C22⋊C4—18848no—
176C42C2×D188116yescomplemented
177C23—942no—
178C23—948no—
179C23—9424no—
180C23—9424no—
181C23—9412no—
182C23—18816no—
183C23—18816no—
184C23—18848no—
185C23—18848no—
186C23—18812no—
187C23C22×D18812yes—
188C23#13094070112yes—
189C23C24×D4711yescharacteristic
190D4—18896no—
191D4—18864no—
192C2×C4—224no—
193C2×C4—212no—
194C2×C4—224no—
195C2×C4—416no—
196C2×C4—946no—
197C2×C4—9412no—
198C2×C4—18812no—
199C2×C4C22×D188112yes—
200C22—3768no—
201C22—37632no—
202C22—18824no—
203C22—1886no—
204C22—1888no—
205C22—1888no—
206C22—18824no—
207C22#15689692112yes—
208C22#15689864116yes—
209C22C23×D18811yescharacteristic
210C22C2×#1309407016yes—
211C4—212no—
212C4—48no—
213C4—1886no—
214C2—3768no—
215C2—3762no—
216C2#2025143118yesdirect product factor
217C2C2×#1568969213yes—
218C2C2×#1568986414yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC2×C47×#383C2
socleC24×C47C24
hypercenterC2×#383D47
hypocenterC47C2×#2818
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
1. C22×#13094021C2semidirect
2. C22×#13094018C2semidirect
3. #20251431C2direct (N × H)
4. C2×#15689041C2semidirect
5. C2×#15689033C2semidirect
6. C2×C47×#383C2semidirect
10. C2×#13094021C22semidirect
12. C2×#13094018C22semidirect
16. #15689041C22semidirect
17. #15689033C22semidirect
19. C47×#383C22semidirect
39. C2×C42×C47D4semidirect
63. C42×C47C2×D4semidirect
110. C2×#383D47semidirect
111. C2×C47#2818semidirect
128. #383C2×D47semidirect
130. C47C2×#2818semidirect
149. C2×C42D188semidirect
176. C42C2×D188semidirect
216. C2#20251431direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940212
C22×#130940182
#202514312
C2×#156890412
C2×#156890332
C2×C47×#3832
C2×#281847