Finite Group Databasebeta

← #28311342

Subgroups — #28311342

Order 13568.

566 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 80 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
1#20290549C212yesmaximal
2#20289812C214yesmaximal, complemented
3#20277299C211yesmaximal, characteristic
4C2×#13133278—22no—
5C2×#13133278C2214yes—
6#16811593—22no—
7C23×Dic106C2211yescharacteristic
8C2×#13133176C411yescharacteristic
9#16810529C412yes—
10#16810529C2212yes—
11#16810528—22no—
12#16810484—22no—
13C2×#117×C53C411yescharacteristic
14C2×#11719940—24no—
15#13133278—28no—
16#13133278—44no—
17C22⋊#88324—44no—
18C22×Dic106—24no—
19C22×Dic106—42no—
20C22×Dic106—42no—
21C22×Dic106C2×C411yescharacteristic
22C22×Dic106D412yes—
23C22×Dic106C2311yescharacteristic
24C23×Dic53D412yes—
25#13133176—42no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C22⋊Dic53—21no—
28C2×C4×Dic53—21no—
29#13133124—24no—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C22×C411yescharacteristic
43C22×Dic53—24no—
44C22×Dic53—24no—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53C22⋊C412yes—
48C22×Dic53C2×D412yes—
49C2×Dic106—24no—
50C2×Dic106—22no—
51C2×Dic106—24no—
52C2×Dic106—41no—
53C2×Dic106—41no—
54C2×Dic106—42no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106—416no—
58C2×Dic106#4114yes—
59#11719940—44no—
60#11719940—44no—
61C2×#88324—44no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C42×C53—42no—
65C22⋊Dic53—42no—
66C4⋊Dic53—42no—
67C4×Dic53—42no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—42no—
71C23×C53#9912yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53—28no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—44no—
79C2×C4×C53—42no—
80C2×C4×C53#9912yes—
81C2×C4×C53#11712yes—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—48no—
88C2×Dic53—44no—
89C2×Dic53—42no—
90C2×Dic53#12314yes—
91Dic106—44no—
92Dic106—48no—
93Dic106—44no—
94#88324—84no—
95#12862—531nomaximal, Sylow
96C22×C53—22no—
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53#38111yescharacteristic
102C22×C53C2×#9911yescharacteristic
103C22×C53C2×#9911yescharacteristic
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—82no—
107Dic53—24no—
108Dic53—42no—
109Dic53—82no—
110#2310—531no—
111#1784—532no—
112#1394—534no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—42no—
116C2×C53#181911yescharacteristic
117C2×C53#213312yes—
118C23×Q8—531no—
119C2×#117—531no—
120C2×#117#615711yescharacteristic
121C2×#101—534no—
122C2×#101—1062no—
123#407—1062no—
124#391—532no—
125#391—532no—
126#391—1062no—
127#381—1062no—
128C53#1286211yescharacteristic, complemented, Sylow
129C22×Q8—532no—
130C22×Q8—531no—
131C22×Q8—531no—
132C22×Q8—1064no—
133C22×Q8—2122no—
134C22×Q8—2122no—
135C23×C4—532no—
136C23×C4C2×#615711yescharacteristic
137C2×M16—1064no—
138#117—42no—
139#117—2122no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—532no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—1061no—
144C2×C42—1061no—
145C2×C42C2×#615712yes—
146#101—1068no—
147#101—2124no—
148C22⋊C8—2124no—
149#95—1064no—
150#95—1064no—
151C24C22⋊#615711yescharacteristic
152C2×Q8—534no—
153C2×Q8—1064no—
154C2×Q8—1064no—
155C2×Q8—1062no—
156C2×Q8—2121no—
157C2×Q8—2121no—
158C2×Q8—2122no—
159C2×Q8—2121no—
160C2×Q8—2121no—
161C2×Q8—21216no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—1064no—
169C22×C4—1064no—
170C22×C4—1062no—
171C22×C4—2122no—
172C22×C4C22⋊#615711yescharacteristic
173C22×C4C22×#615711yescharacteristic
174M16—2124no—
175M16—2124no—
176C2×C8—2124no—
177C4⋊C4—42no—
178C4⋊C4—2122no—
179C22⋊C4—42no—
180C22⋊C4—2122no—
181C42—42no—
182C42—2122no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1313327612yes—
187C23C2×C22⋊#615711yescharacteristic
188Q8—2124no—
189Q8—2128no—
190Q8—2124no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—534no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—2128no—
205C2×C4—2124no—
206C2×C4—2122no—
207C2×C4#1313327512yes—
208C2×C4#1313329612yes—
209C8—4244no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1681156611yescharacteristic
216C22C2×#1313327511yescharacteristic
217C22C2×#1313327611yescharacteristic
218C4—22no—
219C4—22no—
220C4—82no—
221C4—1064no—
222C4—4242no—
223C4—2122no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#2029061011yescharacteristic
228C2#2029101212yes—
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Dic106C23
hypercenterC2×#117#6157
hypocenterC53#12862
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289812C2semidirect
128. C53#12862semidirect

Maximal subgroups

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

subgroupindex
#202905492
#202898122
#202772992
#1286253