Finite Group Databasebeta

← C2×#20280817 #27550634

Subgroups — C2×#20280817 #27550634

Order 13568.

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×#13133191C216yesmaximal, complemented
2#20280817C2116yesmaximal, direct product factor
3C2×#16810560C214yesmaximal, complemented
4C2×#16810533C214yesmaximal, complemented
5C22×#95×C53C211yesmaximal, characteristic, complemented
6C23×D212C2216yes—
7C23×C22⋊D53C2214yes—
8C2×#13133191—248no—
9C2×#13133191C22148yescomplemented
10C22×C4⋊Dic53C2216yes—
11C22×#11719921C22112yes—
12#16810560C22132yescomplemented
13#16810533C22132yescomplemented
14C24×C4×C53C2213yes—
15C2×#95×C53C2214yes—
16C2×#95×C53C2218yescomplemented
17C24×D53C2314yes—
18C22×D212—224no—
19C22×D212—424no—
20C22×D212C23124yes—
21C22×C22⋊D53—248no—
22C22×C22⋊D53C23116yes—
23C23×Dic53C2314yes—
24#13133191—2192no—
25C2×C4⋊Dic53—224no—
26C2×C4⋊Dic53C23124yes—
27C2×#11719921—248no—
28C2×#11719921C23148yes—
29C25×C53C2311yescharacteristic
30C23×C4×C53—412no—
31C23×C4×C53D4124yescomplemented
32C23×C4×C53C2316yes—
33C23×C4×C53C23112yes—
34#95×C53C23116yes—
35C24×C53—26no—
36C24×C53—22no—
37C24×C53C2×D4112yes—
38C24×C53C2412yes—
39C24×C53C2411yescharacteristic
40C23×D53—224no—
41C23×D53—224no—
42C23×D53—28no—
43C23×D53C2418yes—
44C22×C4×C53—224no—
45C22×C4×C53—448no—
46C22×C4×C53—424no—
47C22×C4×C53C2×D4124yes—
48C22×C4×C53C2×D4148yescomplemented
49C22×C4×C53C24112yes—
50C2×C22⋊D53—296no—
51C2×C22⋊D53—464no—
52C22×Dic53—224no—
53C22×Dic53C2418yes—
54C2×D212—248no—
55C2×D212—448no—
56C2×D212—496no—
57#11719921—296no—
58C4⋊Dic53—248no—
59C23×C53—224no—
60C23×C53—224no—
61C23×C53—26no—
62C23×C53—22no—
63C23×C53#127116yes—
64C23×C53C22×D4112yes—
65C23×C53C22×D416yes—
66C23×C53H(5,𝔽2)16yes—
67C23×C532−1+412yes—
68C23×C53C2511yescharacteristic
69C22×D53—224no—
70C22×D53—224no—
71C22×D53—28no—
72C22×D53—416no—
73C22×D53—448no—
74C22×D53—448no—
75C2×C4×C53—248no—
76C2×C4×C53—224no—
77C2×C4×C53—448no—
78C2×C4×C53C22×D4124yes—
79C22⋊D53—464no—
80C2×Dic53—224no—
81C2×Dic53—416no—
82D212—496no—
83C2×#2656—531nomaximal, Sylow
84C22×C53—232no—
85C22×C53—212no—
86C22×C53—212no—
87C22×C53C2×#12718yes—
88C22×C53C2×#12714yes—
89C22×C53#534112yes—
90C22×C53#537112yes—
91C22×C53C23×D413yes—
92C22×C53C2×H(5,𝔽2)13yes—
93C22×C53C2×2−1+411yescharacteristic
94C2×D53—48no—
95C2×D53—424no—
96C2×D53—424no—
97C2×D53—832no—
98C4×C53—224no—
99Dic53—48no—
100C22×#121—536no—
101#2656—5316no—
102C2×#399—534no—
103C2×#392—534no—
104C22×#95D5311yescharacteristic, complemented
105C2×C53—28no—
106C2×C53#265618yescomplemented
107C2×C53C22×#12711yescharacteristic
108C2×C53C2×#53413yes—
109C2×C53C2×#53713yes—
110D53—88no—
111C23×D4—534no—
112C23×D4—536no—
113C24×C4C2×D5313yes—
114C2×#121—5348no—
115C2×#121—10648no—
116C22×C4⋊C4—536no—
117C22×C22⋊C4—5312no—
118#399—5332no—
119#392—5332no—
120C2×#95C2×D5314yes—
121C2×#95C2×D5318yescomplemented
122C53C2×#265611yescharacteristic, complemented, Sylow
123C25—534no—
124C25C22×D5311yescharacteristic
125C22×D4—5316no—
126C22×D4—5324no—
127C22×D4—10648no—
128C22×D4—10624no—
129C22×D4—21224no—
130C23×C4—412no—
131C23×C4—534no—
132C23×C4D212124yescomplemented
133C23×C4C22×D5316yes—
134C23×C4C22×D53112yes—
135#121—106192no—
136C2×C4⋊C4—5324no—
137C2×C4⋊C4—10624no—
138C2×C22⋊C4—5348no—
139C2×C22⋊C4—10648no—
140#95C22×D53116yes—
141C24—26no—
142C24—22no—
143C24—538no—
144C24—10624no—
145C24—10624no—
146C24—1068no—
147C24C2×D212112yes—
148C24C23×D5312yes—
149C24C23×D5311yescharacteristic
150C2×D4—10696no—
151C2×D4—10648no—
152C2×D4—21248no—
153C2×D4—21296no—
154C2×D4—21264no—
155C22×C4—224no—
156C22×C4—448no—
157C22×C4—424no—
158C22×C4—538no—
159C22×C4—10624no—
160C22×C4C2×D212124yes—
161C22×C4C2×D212148yescomplemented
162C22×C4C23×D53112yes—
163C4⋊C4—10648no—
164C22⋊C4—10696no—
165C23—224no—
166C23—26no—
167C23—22no—
168C23—224no—
169C23—10624no—
170C23—10624no—
171C23—1068no—
172C23—21216no—
173C23—21248no—
174C23—21248no—
175C23#13133214116yes—
176C23C22×D212112yes—
177C23C22×D21216yes—
178C23#1313326216yes—
179C23#1313326512yes—
180C23C24×D5311yescharacteristic
181D4—21296no—
182D4—21264no—
183C2×C4—248no—
184C2×C4—224no—
185C2×C4—448no—
186C2×C4—10624no—
187C2×C4—21216no—
188C2×C4C22×D212124yes—
189C22—212no—
190C22—212no—
191C22—232no—
192C22—42432no—
193C22—21224no—
194C22—21224no—
195C22—2128no—
196C22C2×#1313321418yes—
197C22C2×#1313321414yes—
198C22#16811185112yes—
199C22#16811204112yes—
200C22C23×D21213yes—
201C22C2×#1313326213yes—
202C22C2×#1313326511yescharacteristic
203C4—224no—
204C4—2128no—
205C2—28no—
206C2—4248no—
207C2#2028081718yesdirect product factor
208C2C22×#1313321411yescharacteristic
209C2C2×#1681118513yes—
210C2C2×#1681120413yes—
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC22×#95×C53C2
socleC24×C53C24
hypercenterC22×#95D53
hypocenterC53C2×#2656
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C22×#13133191C2semidirect
2. #20280817C2direct (N × H)
3. C2×#16810560C2semidirect
4. C2×#16810533C2semidirect
5. C22×#95×C53C2semidirect
9. C2×#13133191C22semidirect
12. #16810560C22semidirect
13. #16810533C22semidirect
16. C2×#95×C53C22semidirect
31. C23×C4×C53D4semidirect
48. C22×C4×C53C2×D4semidirect
104. C22×#95D53semidirect
106. C2×C53#2656semidirect
121. C2×#95C2×D53semidirect
122. C53C2×#2656semidirect
132. C23×C4D212semidirect
161. C22×C4C2×D212semidirect
207. C2#20280817direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331912
#202808172
C2×#168105602
C2×#168105332
C22×#95×C532
C2×#265653