Finite Group Databasebeta

← #13120553

Subgroups — #13120553

Order 1664.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×D52C211yesmaximal, characteristic, complemented
2C2×#88183C211yesmaximal, characteristic, complemented
3C2×#88136C212yesmaximal, complemented
4C13×#381C211yesmaximal, characteristic, complemented
5#11718616C212yesmaximal, complemented
6#11718333C214yesmaximal, complemented
7#11718332C214yesmaximal, complemented
8C24×D13C2212yes—
9C23×C4×C13C2211yescharacteristic
10C22×C22⋊D13C2211yescharacteristic
11C22×D52—28no—
12C22×D52—24no—
13C22×D52C2214yes—
14C22×D52C2214yescomplemented
15C2×C13×C22⋊C4C2212yes—
16C2×C22⋊Dic13C2211yescharacteristic
17#88183—24no—
18C2×#6103C2214yescomplemented
19C2×#6103C2212yescomplemented
20C2×#6103C2218yescomplemented
21C2×#6101C2212yescomplemented
22#88136—28no—
23C13×#95C2214yescomplemented
24C24×C13C2311yescharacteristic
25C23×D13—24no—
26C23×D13—24no—
27C23×D13—48no—
28C23×D13D418yescomplemented
29C23×D13C2311yescharacteristic
30C23×D13C2314yes—
31C22×C4×C13—24no—
32C22×C4×C13D414yescomplemented
33C22×C4×C13C2314yes—
34C22×C4×C13C2312yes—
35C2×C22⋊D13—24no—
36C2×C22⋊D13—22no—
37C22×Dic13C2311yescharacteristic
38C22×Dic13C2312yes—
39C2×D52—232no—
40C2×D52—216no—
41C2×D52—28no—
42C2×D52—28no—
43C2×D52—48no—
44C13×C22⋊C4—24no—
45C22⋊Dic13—22no—
46#6103—28no—
47#6103—24no—
48#6103—48no—
49#6101—24no—
50#2613—131nomaximal, Sylow
51C23×C13—24no—
52C23×C13C2×D412yes—
53C23×C13C2×D414yes—
54C23×C13C2411yescharacteristic
55C22×D13—216no—
56C22×D13—24no—
57C22×D13—24no—
58C22×D13—28no—
59C22×D13—28no—
60C22×D13—21no—
61C22×D13—22no—
62C22×D13—24no—
63C22×D13—416no—
64C22×D13—416no—
65C22×D13—416no—
66C22×D13C2×D418yes—
67C2×C4×C13—28no—
68C2×C4×C13—24no—
69C2×C4×C13—24no—
70C2×C4×C13—28no—
71C2×C4×C13C2×D414yes—
72C22⋊D13—44no—
73C2×Dic13—21no—
74C2×Dic13—22no—
75C2×Dic13—22no—
76C2×Dic13—24no—
77D52—232no—
78D52—416no—
79C23×D4—131no—
80C2×#123—131no—
81C2×#120—132no—
82#394—134no—
83#392—134no—
84#381D1311yescharacteristic, complemented
85#379—132no—
86C22×C13—28no—
87C22×C13—24no—
88C22×C13#12014yes—
89C22×C13C22×D411yescharacteristic
90C22×C13C22×D412yes—
91C22×C13H(5,𝔽2)14yes—
92C2×D13—28no—
93C2×D13—28no—
94C2×D13—28no—
95C2×D13—432no—
96C2×D13—48no—
97C2×D13—48no—
98C2×D13—41no—
99C2×D13—44no—
100C2×D13—42no—
101C4×C13—24no—
102C4×C13—44no—
103Dic13—41no—
104Dic13—42no—
105C25—132no—
106C22×D4—131no—
107C22×D4—134no—
108C22×D4—134no—
109C22×D4—268no—
110C22×D4—264no—
111C23×C4C2×D1311yescharacteristic
112#123—264no—
113#120—268no—
114C2×C4⋊C4—132no—
115C2×C22⋊C4—131no—
116C2×C22⋊C4—134no—
117C2×C22⋊C4—132no—
118C2×C22⋊C4—138no—
119C2×C22⋊C4C2×D1312yes—
120#95C2×D1314yescomplemented
121C2×C13—24no—
122C2×C13C2×#12011yescharacteristic
123C2×C13#53414yes—
124C2×C13#53512yes—
125D13—48no—
126D13—81no—
127C24—134no—
128C24—138no—
129C24—131no—
130C24—264no—
131C24—264no—
132C24—528no—
133C24C22×D1311yescharacteristic
134C2×D4—2616no—
135C2×D4—2632no—
136C2×D4—268no—
137C2×D4—262no—
138C2×D4—264no—
139C2×D4—268no—
140C2×D4—528no—
141C22×C4—24no—
142C22×C4—132no—
143C22×C4—131no—
144C22×C4D5214yescomplemented
145C22×C4C22×D1314yes—
146C22×C4C22×D1312yes—
147C4⋊C4—264no—
148C22⋊C4—24no—
149C22⋊C4—268no—
150C22⋊C4—262no—
151C22⋊C4—264no—
152C22⋊C4—528no—
153C13#261311yescharacteristic, complemented, Sylow
154C23—24no—
155C23—138no—
156C23—2616no—
157C23—264no—
158C23—264no—
159C23—268no—
160C23—268no—
161C23—261no—
162C23—262no—
163C23—264no—
164C23—5216no—
165C23—5216no—
166C23—5216no—
167C23C2×D5212yes—
168C23D4×D1314yes—
169C23C23×D1311yescharacteristic
170D4—2632no—
171D4—5216no—
172D4—524no—
173C2×C4—28no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—28no—
177C2×C4—262no—
178C2×C4—261no—
179C2×C4—262no—
180C2×C4—264no—
181C2×C4C2×D5214yes—
182C22—24no—
183C22—28no—
184C22—268no—
185C22—268no—
186C22—268no—
187C22—5232no—
188C22—528no—
189C22—528no—
190C22—521no—
191C22—522no—
192C22—524no—
193C22#8813614yes—
194C22C22×D5211yescharacteristic
195C22C2×D4×D1312yes—
196C22#8825614yes—
197C4—24no—
198C4—44no—
199C4—521no—
200C4—522no—
201C2—24no—
202C2—528no—
203C2—1041no—
204C2C2×#8813611yescharacteristic
205C2#1171923012yes—
206C2#1171935814yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#381C2
hypercenter#381D13
hypocenterC13#2613
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×D52C2semidirect
2. C2×#88183C2semidirect
3. C2×#88136C2semidirect
4. C13×#381C2semidirect
5. #11718616C2semidirect
6. #11718333C2semidirect
7. #11718332C2semidirect
14. C22×D52C22semidirect
18. C2×#6103C22semidirect
19. C2×#6103C22semidirect
20. C2×#6103C22semidirect
21. C2×#6101C22semidirect
23. C13×#95C22semidirect
28. C23×D13D4semidirect
32. C22×C4×C13D4semidirect
84. #381D13semidirect
120. #95C2×D13semidirect
144. C22×C4D52semidirect
153. C13#2613semidirect

Maximal subgroups

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

subgroupindex
C23×D522
C2×#881832
C2×#881362
C13×#3812
#117186162
#117183332
#117183322
#261313