Finite Group Databasebeta

← C2×#20251428 #26422835

Subgroups — C2×#20251428 #26422835

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
2#20251428C2116yesmaximal, direct product factor
3C2×#15689040C216yesmaximal, complemented
4C2×#15689032C216yesmaximal, complemented
5C2×#15688991C211yesmaximal, characteristic, complemented
6C22×#126×C47C211yesmaximal, characteristic, complemented
7C23×D188C2216yes—
8C23×C22⋊D47C2214yes—
9C2×#13094021—28no—
10C2×#13094021C2218yescomplemented
11C22×(C2×D47)⋊C4C22112yes—
12#15689040C22148yescomplemented
13#15689032C22148yescomplemented
14#15688991C2218yescomplemented
15C2×C47⋊#95C2216yes—
16C2×#126×C47—28no—
17C2×#126×C47C2218yescomplemented
18C22×C4⋊C4×C47C2213yes—
19C22×C22⋊C4×C47C2213yes—
20C22×C42×C47C2211yescharacteristic
21C24×D47C2314yes—
22C22×D188—224no—
23C22×D188—424no—
24C22×D188C23124yes—
25C22×C22⋊D47—424no—
26C22×C22⋊D47C23116yes—
27C23×Dic47C2314yes—
28#13094021—232no—
29C2×(C2×D47)⋊C4—424no—
30C2×(C2×D47)⋊C4C23148yes—
31C47⋊#95C23124yes—
32C25×C47C2311yescharacteristic
33C23×C4×C47C2313yes—
34C23×C4×C47C2313yes—
35#126×C47—232no—
36C2×C4⋊C4×C47—212no—
37C2×C4⋊C4×C47C23112yes—
38C2×C22⋊C4×C47—212no—
39C2×C22⋊C4×C47C23112yes—
40C2×C42×C47D418yescomplemented
41C2×C42×C47C2314yes—
42C24×C47—22no—
43C24×C47—212no—
44C24×C47C2412yes—
45C24×C47C2411yescharacteristic
46C23×D47—28no—
47C23×D47—224no—
48C23×D47—224no—
49C23×D47C2418yes—
50C22×C4×C47—212no—
51C22×C4×C47—26no—
52C22×C4×C47—212no—
53C22×C4×C47C2×D4112yes—
54C22×C4×C47C2416yes—
55C22×C4×C47C2416yes—
56C2×C22⋊D47—448no—
57C2×C22⋊D47—464no—
58C22×Dic47—224no—
59C22×Dic47C2418yes—
60C2×D188—248no—
61C2×D188—448no—
62C2×D188—496no—
63C4⋊C4×C47—224no—
64C22⋊C4×C47—224no—
65C42×C47C2×D4116yescomplemented
66(C2×D47)⋊C4—448no—
67C23×C47—22no—
68C23×C47—212no—
69C23×C47—412no—
70C23×C47—416no—
71C23×C47C22×D412yes—
72C23×C47H(5,𝔽2)112yes—
73C23×C47C2511yescharacteristic
74C22×D47—28no—
75C22×D47—224no—
76C22×D47—224no—
77C22×D47—448no—
78C22×D47—416no—
79C22×D47—448no—
80C2×C4×C47—224no—
81C2×C4×C47—212no—
82C2×C4×C47—26no—
83C2×C4×C47—212no—
84C2×C4×C47—412no—
85C2×C4×C47C22×D4112yes—
86C22⋊D47—464no—
87C2×Dic47—224no—
88C2×Dic47—416no—
89D188—496no—
90C2×#2818—471nomaximal, Sylow
91C22×C47—46no—
92C22×C47—48no—
93C22×C47—88no—
94C22×C47#535112yes—
95C22×C47#560116yes—
96C22×C47C23×D411yescharacteristic
97C22×C47C2×H(5,𝔽2)16yes—
98C2×D47—424no—
99C2×D47—48no—
100C2×D47—424no—
101C2×D47—832no—
102C4×C47—212no—
103C4×C47—46no—
104Dic47—48no—
105C22×#127—471no—
106C22×#126D4711yescharacteristic, complemented
107#2818—4716no—
108C2×#394—476no—
109C2×#392—476no—
110C2×#383—471no—
111C2×C47—82no—
112C2×C47#281818yescomplemented
113C2×C47C2×#53513yes—
114C2×C47C2×#56014yes—
115D47—88no—
116C23×D4—474no—
117C23×D4—476no—
118C2×#127—478no—
119C2×#127—948no—
120C2×#126—28no—
121C2×#126C2×D4718yescomplemented
122C22×C4⋊C4C2×D4713yes—
123C22×C22⋊C4—4712no—
124C22×C22⋊C4C2×D4713yes—
125C22×C42C2×D4711yescharacteristic
126#394—4748no—
127#392—4748no—
128#383—478no—
129C2×#95—476no—
130C47C2×#281811yescharacteristic, complemented, Sylow
131C25—474no—
132C25C22×D4711yescharacteristic
133C22×D4—4716no—
134C22×D4—4724no—
135C22×D4—9424no—
136C22×D4—18824no—
137C22×D4—18824no—
138C23×C4—474no—
139C23×C4C22×D4713yes—
140C23×C4C22×D4713yes—
141#127—9432no—
142#126—232no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4C22×D47112yes—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—4748no—
147C2×C22⋊C4—18824no—
148C2×C22⋊C4C22×D47112yes—
149C2×C42C22⋊D4718yescomplemented
150C2×C42C22×D4714yes—
151#95—4724no—
152C24—22no—
153C24—212no—
154C24—478no—
155C24—948no—
156C24—9424no—
157C24—9424no—
158C24C23×D4712yes—
159C24C23×D4711yescharacteristic
160C2×D4—9448no—
161C2×D4—18848no—
162C2×D4—18896no—
163C2×D4—18848no—
164C2×D4—18864no—
165C22×C4—212no—
166C22×C4—26no—
167C22×C4—212no—
168C22×C4—478no—
169C22×C4—9424no—
170C22×C4C2×C22⋊D47112yes—
171C22×C4C23×D4716yes—
172C22×C4C23×D4716yes—
173C4⋊C4—224no—
174C22⋊C4—224no—
175C22⋊C4—18848no—
176C42C2×C22⋊D47116yescomplemented
177C23—22no—
178C23—212no—
179C23—416no—
180C23—412no—
181C23—948no—
182C23—9424no—
183C23—9424no—
184C23—18816no—
185C23—18848no—
186C23—18848no—
187C23C22×C22⋊D4712yes—
188C23#13094070112yes—
189C23C24×D4711yescharacteristic
190D4—18896no—
191D4—18864no—
192C2×C4—224no—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—412no—
197C2×C4—9424no—
198C2×C4—18816no—
199C2×C4C22×C22⋊D47112yes—
200C22—46no—
201C22—48no—
202C22—88no—
203C22—37632no—
204C22—18824no—
205C22—1888no—
206C22—18824no—
207C22#15689687112yes—
208C22#15689864116yes—
209C22C23×C22⋊D4711yescharacteristic
210C22C2×#1309407016yes—
211C4—212no—
212C4—46no—
213C4—1888no—
214C2—82no—
215C2—3768no—
216C2#2025142818yesdirect product factor
217C2C2×#1568968713yes—
218C2C2×#1568986414yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#126×C47C2
socleC24×C47C24
hypercenterC22×#126D47
hypocenterC47C2×#2818
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×#13094021C2semidirect
2. #20251428C2direct (N × H)
3. C2×#15689040C2semidirect
4. C2×#15689032C2semidirect
5. C2×#15688991C2semidirect
6. C22×#126×C47C2semidirect
10. C2×#13094021C22semidirect
12. #15689040C22semidirect
13. #15689032C22semidirect
14. #15688991C22semidirect
17. C2×#126×C47C22semidirect
40. C2×C42×C47D4semidirect
65. C42×C47C2×D4semidirect
106. C22×#126D47semidirect
112. C2×C47#2818semidirect
121. C2×#126C2×D47semidirect
130. C47C2×#2818semidirect
149. C2×C42C22⋊D47semidirect
176. C42C2×C22⋊D47semidirect
216. C2#20251428direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940212
#202514282
C2×#156890402
C2×#156890322
C2×#156889912
C22×#126×C472
C2×#281847