Finite Group Databasebeta

← C2×#20249490 #26382820

Subgroups — C2×#20249490 #26382820

Order 12032.

4886 conjugacy classes of subgroups; only the 907 normal subgroups are shown.

normal subgroups only

This group has 4886 conjugacy classes of subgroups — too many to draw. Only its 907 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×#13093998C216yesmaximal, complemented
2#20249490C2116yesmaximal, direct product factor
3C2×#15689060C214yesmaximal, complemented
4C2×#15689033C214yesmaximal, complemented
5C22×#95×C47C211yesmaximal, characteristic, complemented
6C23×D188C2216yes—
7C23×C22⋊D47C2214yes—
8C2×#13093998—248no—
9C2×#13093998C22148yescomplemented
10C22×C4⋊Dic47C2216yes—
11C22×(C2×D47)⋊C4C22112yes—
12#15689060C22132yescomplemented
13#15689033C22132yescomplemented
14C24×C4×C47C2213yes—
15C2×#95×C47C2214yes—
16C2×#95×C47C2218yescomplemented
17C24×D47C2314yes—
18C22×D188—224no—
19C22×D188—424no—
20C22×D188C23124yes—
21C22×C22⋊D47—248no—
22C22×C22⋊D47C23116yes—
23C23×Dic47C2314yes—
24#13093998—2192no—
25C2×C4⋊Dic47—224no—
26C2×C4⋊Dic47C23124yes—
27C2×(C2×D47)⋊C4—248no—
28C2×(C2×D47)⋊C4C23148yes—
29C25×C47C2311yescharacteristic
30C23×C4×C47—412no—
31C23×C4×C47D4124yescomplemented
32C23×C4×C47C2316yes—
33C23×C4×C47C23112yes—
34#95×C47C23116yes—
35C24×C47—26no—
36C24×C47—22no—
37C24×C47C2×D4112yes—
38C24×C47C2412yes—
39C24×C47C2411yescharacteristic
40C23×D47—224no—
41C23×D47—224no—
42C23×D47—28no—
43C23×D47C2418yes—
44C22×C4×C47—224no—
45C22×C4×C47—448no—
46C22×C4×C47—424no—
47C22×C4×C47C2×D4124yes—
48C22×C4×C47C2×D4148yescomplemented
49C22×C4×C47C24112yes—
50C2×C22⋊D47—296no—
51C2×C22⋊D47—464no—
52C22×Dic47—224no—
53C22×Dic47C2418yes—
54C2×D188—248no—
55C2×D188—448no—
56C2×D188—496no—
57(C2×D47)⋊C4—296no—
58C4⋊Dic47—248no—
59C23×C47—224no—
60C23×C47—224no—
61C23×C47—26no—
62C23×C47—22no—
63C23×C47#127116yes—
64C23×C47C22×D4112yes—
65C23×C47C22×D416yes—
66C23×C47H(5,𝔽2)16yes—
67C23×C472−1+412yes—
68C23×C47C2511yescharacteristic
69C22×D47—224no—
70C22×D47—224no—
71C22×D47—28no—
72C22×D47—416no—
73C22×D47—448no—
74C22×D47—448no—
75C2×C4×C47—248no—
76C2×C4×C47—224no—
77C2×C4×C47—448no—
78C2×C4×C47C22×D4124yes—
79C22⋊D47—464no—
80C2×Dic47—224no—
81C2×Dic47—416no—
82D188—496no—
83C2×#2656—471nomaximal, Sylow
84C22×C47—232no—
85C22×C47—212no—
86C22×C47—212no—
87C22×C47C2×#12718yes—
88C22×C47C2×#12714yes—
89C22×C47#534112yes—
90C22×C47#537112yes—
91C22×C47C23×D413yes—
92C22×C47C2×H(5,𝔽2)13yes—
93C22×C47C2×2−1+411yescharacteristic
94C2×D47—48no—
95C2×D47—424no—
96C2×D47—424no—
97C2×D47—832no—
98C4×C47—224no—
99Dic47—48no—
100C22×#121—476no—
101#2656—4716no—
102C2×#399—474no—
103C2×#392—474no—
104C22×#95D4711yescharacteristic, complemented
105C2×C47—28no—
106C2×C47#265618yescomplemented
107C2×C47C22×#12711yescharacteristic
108C2×C47C2×#53413yes—
109C2×C47C2×#53713yes—
110D47—88no—
111C23×D4—474no—
112C23×D4—476no—
113C24×C4C2×D4713yes—
114C2×#121—4748no—
115C2×#121—9448no—
116C22×C4⋊C4—476no—
117C22×C22⋊C4—4712no—
118#399—4732no—
119#392—4732no—
120C2×#95C2×D4714yes—
121C2×#95C2×D4718yescomplemented
122C47C2×#265611yescharacteristic, complemented, Sylow
123C25—474no—
124C25C22×D4711yescharacteristic
125C22×D4—4716no—
126C22×D4—4724no—
127C22×D4—9448no—
128C22×D4—9424no—
129C22×D4—18824no—
130C23×C4—412no—
131C23×C4—474no—
132C23×C4D188124yescomplemented
133C23×C4C22×D4716yes—
134C23×C4C22×D47112yes—
135#121—94192no—
136C2×C4⋊C4—4724no—
137C2×C4⋊C4—9424no—
138C2×C22⋊C4—4748no—
139C2×C22⋊C4—9448no—
140#95C22×D47116yes—
141C24—26no—
142C24—22no—
143C24—478no—
144C24—9424no—
145C24—9424no—
146C24—948no—
147C24C2×D188112yes—
148C24C23×D4712yes—
149C24C23×D4711yescharacteristic
150C2×D4—9496no—
151C2×D4—9448no—
152C2×D4—18848no—
153C2×D4—18896no—
154C2×D4—18864no—
155C22×C4—224no—
156C22×C4—448no—
157C22×C4—424no—
158C22×C4—478no—
159C22×C4—9424no—
160C22×C4C2×D188124yes—
161C22×C4C2×D188148yescomplemented
162C22×C4C23×D47112yes—
163C4⋊C4—9448no—
164C22⋊C4—9496no—
165C23—224no—
166C23—26no—
167C23—22no—
168C23—224no—
169C23—9424no—
170C23—9424no—
171C23—948no—
172C23—18816no—
173C23—18848no—
174C23—18848no—
175C23#13094021116yes—
176C23C22×D188112yes—
177C23C22×D18816yes—
178C23#1309406916yes—
179C23#1309407212yes—
180C23C24×D4711yescharacteristic
181D4—18896no—
182D4—18864no—
183C2×C4—248no—
184C2×C4—224no—
185C2×C4—448no—
186C2×C4—9424no—
187C2×C4—18816no—
188C2×C4C22×D188124yes—
189C22—232no—
190C22—212no—
191C22—212no—
192C22—37632no—
193C22—1888no—
194C22—18824no—
195C22—18824no—
196C22C2×#1309402118yes—
197C22C2×#1309402114yes—
198C22#15689685112yes—
199C22#15689704112yes—
200C22C23×D18813yes—
201C22C2×#1309406913yes—
202C22C2×#1309407211yescharacteristic
203C4—224no—
204C4—1888no—
205C2—28no—
206C2—3768no—
207C2#2024949018yesdirect product factor
208C2C22×#1309402111yescharacteristic
209C2C2×#1568968513yes—
210C2C2×#1568970413yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC22×#95×C47C2
socleC24×C47C24
hypercenterC22×#95D47
hypocenterC47C2×#2656
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C22×#13093998C2semidirect
2. #20249490C2direct (N × H)
3. C2×#15689060C2semidirect
4. C2×#15689033C2semidirect
5. C22×#95×C47C2semidirect
9. C2×#13093998C22semidirect
12. #15689060C22semidirect
13. #15689033C22semidirect
16. C2×#95×C47C22semidirect
31. C23×C4×C47D4semidirect
48. C22×C4×C47C2×D4semidirect
104. C22×#95D47semidirect
106. C2×C47#2656semidirect
121. C2×#95C2×D47semidirect
122. C47C2×#2656semidirect
132. C23×C4D188semidirect
161. C22×C4C2×D188semidirect
207. C2#20249490direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939982
#202494902
C2×#156890602
C2×#156890332
C22×#95×C472
C2×#265647