Finite Group Databasebeta

← #28303397

Subgroups — #28303397

Order 13568.

418 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#16811469C212yesmaximal
2#20289558C214yesmaximal
3#20277317C211yesmaximal, characteristic
4C2×C22⋊#88324C2214yes—
5#16811469—22no—
6#16811469—22no—
7C22×#11719919C2211yescharacteristic
8#16810510—22no—
9#16810504—22no—
10#16810484C412yes—
11#16810483C411yescharacteristic
12C22⋊C4×Dic53C2212yes—
13C53×#380C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C23×Dic53C2×C411yescharacteristic
18C23×Dic53C2311yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×#11719919—24no—
21C2×#11719919D412yes—
22C2×#11719919D412yes—
23C2×C22⋊Dic53C2×C412yes—
24C2×C4×Dic53—22no—
25#13133124—21no—
26#13133124—24no—
27#13133124—22no—
28C23×C4×C53C2×C411yescharacteristic
29C2×C4⋊C4×C53—21no—
30C2×C22⋊C4×C53C2×C412yes—
31#95×C53—21no—
32C24×C53C22×C411yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53—22no—
35C22×C4×C53—81no—
36C22×C4×C53C22⋊C411yescharacteristic
37C22×C4×C53C22⋊C411yescharacteristic
38C22×Dic53—22no—
39C22×Dic53—22no—
40C22×Dic53—22no—
41C22×Dic53—22no—
42C22×Dic53—81no—
43C22×Dic53C22⋊C412yes—
44C22×Dic53C2×D412yes—
45C22×Dic53#4114yes—
46C2×#88324—44no—
47C2×#88324—84no—
48C4⋊C4×C53—81no—
49C22⋊C4×C53—24no—
50C22⋊Dic53—24no—
51C4⋊Dic53—81no—
52#11719919—24no—
53#11719919—24no—
54C4×Dic53—82no—
55C23×C53—41no—
56C23×C53—41no—
57C23×C53C2×C22⋊C411yescharacteristic
58C23×C53#11712yes—
59C23×C53#13112yes—
60C23×C53#13112yes—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—82no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—44no—
73C2×Dic53—42no—
74C2×Dic53—44no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53#12314yes—
79#88324—84no—
80#8737—531nomaximal, Sylow
81C22×C53—21no—
82C22×C53—22no—
83C22×C53—21no—
84C22×C53—22no—
85C22×C53—42no—
86C22×C53—42no—
87C22×C53#38111yescharacteristic
88C22×C53#40811yescharacteristic
89C22×C53#40811yescharacteristic
90C22×C53#43314yes—
91C4×C53—41no—
92C4×C53—41no—
93C4×C53—82no—
94Dic53—24no—
95Dic53—42no—
96Dic53—82no—
97#2313—531no—
98C2×#324—532no—
99#1282—534no—
100C2×C53—21no—
101C2×C53—21no—
102C2×C53—24no—
103C2×C53#181811yescharacteristic
104C2×C53#182811yescharacteristic
105C2×C53#183011yescharacteristic
106C22×C4⋊C4—531no—
107C2×C22⋊C8—534no—
108#387—1062no—
109#387—1062no—
110#381—532no—
111#380—531no—
112#380#615711yescharacteristic
113C4×C22⋊C4—532no—
114#324—1062no—
115#324—1062no—
116C53#873711yescharacteristic, complemented, Sylow
117C23×C4—531no—
118C23×C4—531no—
119C23×C4C2×#615711yescharacteristic
120C22×C8—1064no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—532no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—1064no—
125C2×C4⋊C4—1061no—
126C2×C22⋊C4—532no—
127C2×C22⋊C4C2×#615712yes—
128C2×C42—1062no—
129C22⋊C8—1064no—
130C22⋊C8—1064no—
131#95—21no—
132#95—1062no—
133#95—1064no—
134#95—1061no—
135C24C22×#615711yescharacteristic
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—81no—
139C22×C4—532no—
140C22×C4—534no—
141C22×C4—532no—
142C22×C4—1062no—
143C22×C4—1062no—
144C22×C4—1062no—
145C22×C4—1062no—
146C22×C4—4241no—
147C22×C4C22⋊#615711yescharacteristic
148C22×C4C22⋊#615711yescharacteristic
149C2×C8—4244no—
150C2×C8—2124no—
151C4⋊C4—81no—
152C4⋊C4—1064no—
153C4⋊C4—1064no—
154C4⋊C4—4241no—
155C22⋊C4—24no—
156C22⋊C4—1064no—
157C42—4242no—
158C23—41no—
159C23—41no—
160C23C2×C22⋊#615711yescharacteristic
161C23#1313329612yes—
162C23#1313330412yes—
163C23#1313330512yes—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—534no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—2122no—
177C2×C4—2124no—
178C2×C4—2122no—
179C2×C4—2122no—
180C2×C4—2122no—
181C2×C4—2124no—
182C8—4244no—
183C22—21no—
184C22—22no—
185C22—21no—
186C22—22no—
187C22—42no—
188C22—42no—
189C22#1681156611yescharacteristic
190C22#1681159711yescharacteristic
191C22#1681159811yescharacteristic
192C22#1681164814yes—
193C4—41no—
194C4—41no—
195C4—82no—
196C4—1064no—
197C4—4242no—
198C4—2122no—
199C2—21no—
200C2—21no—
201C2—24no—
202C2#2029060911yescharacteristic
203C2#2029062611yescharacteristic
204C2#2029062911yescharacteristic
subgroupG/H
centerC22#16811566
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC53×#380C4
socleC22×C53#381
cosocleC23×Dic53C23
hypercenter#380#6157
hypocenterC53#8737
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #380

Complements

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

NH ≅ G/Ntype
116. C53#8737semidirect

Maximal subgroups

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

subgroupindex
C2×#168114692
#202895582
#202773172
#873753