Finite Group Databasebeta

← C22×#16810533 #27550269

Subgroups — C22×#16810533 #27550269

Order 13568.

13646 conjugacy classes of subgroups; only the 1539 normal subgroups are shown.

normal subgroups only

This group has 13646 conjugacy classes of subgroups — too many to draw. Only its 1539 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
1C24×D212C213yesmaximal, complemented
2C23×#11719921C213yesmaximal, complemented
3C2×#16810533C2124yesmaximal, direct product factor
4C22×#95×C53C211yesmaximal, characteristic, complemented
5C25×D53C2213yes—
6C23×D212—248no—
7C23×D212—224no—
8C23×D212C22136yescomplemented
9C24×Dic53C2211yescharacteristic
10C22×#11719921—224no—
11C22×#11719921C22136yescomplemented
12#16810533C22164yesdirect product factor
13C24×C4×C53C2213yes—
14C2×#95×C53C22112yescomplemented
15C24×D53—212no—
16C24×D53—224no—
17C24×D53—224no—
18C24×D53—212no—
19C24×D53D4124yescomplemented
20C24×D53C23118yes—
21C22×D212—2384no—
22C22×D212—2288no—
23C22×D212—2144no—
24C22×D212C23148yescomplemented
25C23×Dic53—212no—
26C23×Dic53C2316yes—
27C2×#11719921—2144no—
28C2×#11719921C23148yescomplemented
29C25×C53C2311yescharacteristic
30C23×C4×C53—224no—
31C23×C4×C53D4124yescomplemented
32C23×C4×C53C23118yes—
33#95×C53C23116yescomplemented
34C24×C53C2×D4112yes—
35C24×C53C2×D4112yes—
36C24×C53#4114yes—
37C24×C53C2413yes—
38C23×D53—2192no—
39C23×D53—296no—
40C23×D53—236no—
41C23×D53—272no—
42C23×D53—272no—
43C23×D53—236no—
44C23×D53—448no—
45C23×D53—496no—
46C23×D53—448no—
47C23×D53C2×D4172yescomplemented
48C23×D53C24112yes—
49C22×C4×C53—296no—
50C22×C4×C53—272no—
51C22×C4×C53C2×D4172yescomplemented
52C22×C4×C53C24112yes—
53C22×Dic53—236no—
54C22×Dic53—416no—
55C22×Dic53C2414yes—
56C2×D212—21152no—
57C2×D212—2192no—
58C2×D212—296no—
59#11719921—296no—
60C23×C53#120148yes—
61C23×C53#121148yes—
62C23×C53#127116yes—
63C23×C53C22×D4118yes—
64C23×C53C22×D4118yes—
65C23×C53C2×#4116yes—
66C23×C53C2511yescharacteristic
67C22×D53—2288no—
68C22×D53—2144no—
69C22×D53—224no—
70C22×D53—224no—
71C22×D53—212no—
72C22×D53—212no—
73C22×D53—472no—
74C22×D53—4144no—
75C22×D53—472no—
76C22×D53—4384no—
77C22×D53C22×D4124yescomplemented
78C2×C4×C53—2144no—
79C2×C4×C53—224no—
80C2×C4×C53C22×D4124yescomplemented
81C2×Dic53—212no—
82C2×Dic53—424no—
83D212—2384no—
84C22×#392—531nomaximal, Sylow
85C22×C53#392164yescomplemented
86C22×C53C2×#120136yes—
87C22×C53C2×#121136yes—
88C22×C53C2×#127112yes—
89C22×C53C23×D413yes—
90C22×C53C23×D413yes—
91C22×C53C22×#4111yescharacteristic
92C2×D53—248no—
93C2×D53—224no—
94C2×D53—4288no—
95C2×D53—412no—
96C2×D53—424no—
97C2×D53—412no—
98C4×C53—224no—
99Dic53—44no—
100C24×D4—533no—
101C23×C22⋊C4—533no—
102C2×#392—5324no—
103C22×#95D5311yescharacteristic, complemented
104C2×C53C2×#392124yescomplemented
105C2×C53C22×#12013yes—
106C2×C53C22×#12113yes—
107C2×C53C22×#12711yescharacteristic
108D53—424no—
109C26—533no—
110C23×D4—5336no—
111C23×D4—10648no—
112C23×D4—10624no—
113C24×C4—531no—
114C24×C4C2×D5313yes—
115C22×C22⋊C4—5336no—
116C22×C22⋊C4—10624no—
117#392—5364no—
118C2×#95C2×D53112yescomplemented
119C53C22×#39211yescharacteristic, complemented, Sylow
120C25—5324no—
121C25—5318no—
122C25—10612no—
123C25—10624no—
124C25—10624no—
125C25—10612no—
126C25C22×D5311yescharacteristic
127C22×D4—5348no—
128C22×D4—106288no—
129C22×D4—106384no—
130C22×D4—106144no—
131C23×C4—224no—
132C23×C4—536no—
133C23×C4—10612no—
134C23×C4D212124yescomplemented
135C23×C4C22×D53118yes—
136C2×C22⋊C4—5348no—
137C2×C22⋊C4—106144no—
138#95C22×D53116yescomplemented
139C24—5372no—
140C24—5312no—
141C24—106192no—
142C24—10696no—
143C24—10672no—
144C24—10636no—
145C24—10672no—
146C24—10636no—
147C24—21248no—
148C24—21296no—
149C24—21248no—
150C24C2×D212112yes—
151C24D4×D53112yes—
152C24#1171994914yes—
153C24C23×D5313yes—
154C2×D4—106192no—
155C2×D4—1061152no—
156C2×D4—10696no—
157C22×C4—296no—
158C22×C4—272no—
159C22×C4—534no—
160C22×C4—10636no—
161C22×C4—21216no—
162C22×C4C2×D212172yescomplemented
163C22×C4C23×D53112yes—
164C22⋊C4—10696no—
165C23—5324no—
166C23—106288no—
167C23—106144no—
168C23—10624no—
169C23—10612no—
170C23—10624no—
171C23—10612no—
172C23—212384no—
173C23—21272no—
174C23—212144no—
175C23—21272no—
176C23#13133187148yes—
177C23#13133192148yes—
178C23#13133214116yes—
179C23C2×D4×D53118yes—
180C23C22×D212118yes—
181C23C2×#1171994916yes—
182C23C24×D5311yescharacteristic
183D4—106384no—
184C2×C4—2144no—
185C2×C4—224no—
186C2×C4—10612no—
187C2×C4—21224no—
188C2×C4C22×D212124yescomplemented
189C22—10648no—
190C22—10624no—
191C22—212288no—
192C22—21212no—
193C22—21224no—
194C22—21212no—
195C22#16810533164yesdirect product factor
196C22C2×#13133187136yes—
197C22C2×#13133192136yes—
198C22C2×#13133214112yes—
199C22C22×D4×D5313yes—
200C22C23×D21213yes—
201C22C22×#1171994911yescharacteristic
202C4—224no—
203C4—2124no—
204C2—21224no—
205C2C2×#16810533124yesdirect product factor
206C2C22×#1313318713yes—
207C2C22×#1313319213yes—
208C2C22×#1313321411yescharacteristic
subgroupG/H
centerC25C22×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC22×#95×C53C2
socleC25×C53C23
hypercenterC22×#95D53
hypocenterC53C22×#392
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C25 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C24×D212C2semidirect
2. C23×#11719921C2semidirect
3. C2×#16810533C2direct (N × H)
4. C22×#95×C53C2semidirect
8. C23×D212C22semidirect
11. C22×#11719921C22semidirect
12. #16810533C22direct (N × H)
14. C2×#95×C53C22semidirect
19. C24×D53D4semidirect
24. C22×D212C23semidirect
28. C2×#11719921C23semidirect
31. C23×C4×C53D4semidirect
33. #95×C53C23semidirect
47. C23×D53C2×D4semidirect
51. C22×C4×C53C2×D4semidirect
77. C22×D53C22×D4semidirect
80. C2×C4×C53C22×D4semidirect
85. C22×C53#392semidirect
103. C22×#95D53semidirect
104. C2×C53C2×#392semidirect
118. C2×#95C2×D53semidirect
119. C53C22×#392semidirect
134. C23×C4D212semidirect
138. #95C22×D53semidirect
162. C22×C4C2×D212semidirect
188. C2×C4C22×D212semidirect
195. C22#16810533direct (N × H)
205. C2C2×#16810533direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D2122
C23×#117199212
C2×#168105332
C22×#95×C532
C22×#39253