Finite Group Databasebeta

← #28308201

Subgroups — #28308201

Order 13568.

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

normal subgroups only

This group has 434 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#20289963C212yesmaximal
2#20289657C212yesmaximal, complemented
3#20289655C212yesmaximal, complemented
4#20277338C211yesmaximal, characteristic
5#16811593—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133176C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810522—22no—
11#16810498—22no—
12#16810493C411yescharacteristic
13#16810493C2211yescharacteristic
14#16810489C411yescharacteristic
15#16810489C2211yescharacteristic
16C2×#117×C53C411yescharacteristic
17C2×#11719940—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C23×Dic53D412yes—
21#13133176—42no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×C4⋊Dic53C2311yescharacteristic
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53—21no—
32#13133124—22no—
33#13133124—22no—
34C23×C4×C53C2×C411yescharacteristic
35#117×C53—22no—
36#117×C53—22no—
37C2×C4⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53C2×C411yescharacteristic
41C24×C53C22⋊C411yescharacteristic
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—82no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C2×D412yes—
57#11719940—84no—
58C2×#88324—44no—
59C4⋊C4×C53—24no—
60C22⋊C4×C53—24no—
61C42×C53—21no—
62C42×C53—21no—
63C42×C53M1612yescomplemented
64C42×C53M1612yescomplemented
65C22⋊Dic53—42no—
66C4⋊Dic53—41no—
67C4⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4⋊Dic53—82no—
72C4⋊Dic53—82no—
73C4×Dic53—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—41no—
77C23×C53—41no—
78C23×C53#9912yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53#10112yes—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—44no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—82no—
105#88324—84no—
106#11514—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53#40711yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122Dic53—42no—
123Dic53—42no—
124Dic53—81no—
125Dic53—81no—
126#2317—531no—
127#1463—532no—
128#1325—532no—
129#1324—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53—42no—
133C2×C53#146311yescharacteristic
134C2×C53#212711yescharacteristic
135C2×C53#213311yescharacteristic
136C2×#117—531no—
137C2×#117#615711yescharacteristic
138C22×C4⋊C4—531no—
139#407—1062no—
140#389—1062no—
141#385—1062no—
142#384—531no—
143#384—531no—
144#382—531no—
145#382—531no—
146#324—534no—
147#324—1062no—
148C53#1151411yescharacteristic, complemented, Sylow
149C23×C4—532no—
150C23×C4C2×#615711yescharacteristic
151C2×M16—2122no—
152#117—22no—
153#117—22no—
154#117—2122no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—2122no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—1061no—
166C2×C42—1061no—
167C2×C42C2×#615711yescharacteristic
168C2×C42C2×#615711yescharacteristic
169C22⋊C8—1064no—
170C22⋊C8—2122no—
171#95—1062no—
172#95—1062no—
173C24C22⋊#615711yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—532no—
181C22×C4—532no—
182C22×C4—1062no—
183C22×C4—1062no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4—4242no—
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189M16—4244no—
190C2×C8—2124no—
191C4⋊C4—24no—
192C4⋊C4—4242no—
193C4⋊C4—4242no—
194C4⋊C4—2122no—
195C4⋊C4—2121no—
196C4⋊C4—2121no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C22⋊C4—24no—
200C22⋊C4—2122no—
201C42—21no—
202C42—21no—
203C42—2122no—
204C42#1171994012yescomplemented
205C42#1171994012yescomplemented
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#1313327612yes—
211C23C2×C22⋊#615711yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—4242no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4—2124no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4#1313327812yes—
235C2×C4C2×#1171994011yescharacteristic
236C2×C4C2×#1171994011yescharacteristic
237C8—4244no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#1681159311yescharacteristic
246C22C2×#1313327611yescharacteristic
247C22C2×#1313327811yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—4241no—
254C4—4241no—
255C4—2122no—
256C4—2122no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2028996311yescharacteristic
261C2#2029099211yescharacteristic
262C2#2029101211yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11514
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. #20289657C2semidirect
3. #20289655C2semidirect
63. C42×C53M16semidirect
64. C42×C53M16semidirect
148. C53#11514semidirect
204. C42#11719940semidirect
205. C42#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896572
#202896552
#202773382
#1151453