Finite Group Databasebeta

← #28299727

Subgroups — #28299727

Order 13568.

468 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290643C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal
4#20277254C211yesmaximal, characteristic
5#16811489—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133205C411yescharacteristic
9C2×#13133182C411yescharacteristic
10C2×#13133179C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810524C2211yescharacteristic
13#16810512—22no—
14#16810470—22no—
15C42×Dic53C2211yescharacteristic
16C53×#382C411yescharacteristic
17C22×#88324—24no—
18C22×Dic106C2×C411yescharacteristic
19C22×D212C2×C411yescharacteristic
20C22×C4×D53—21no—
21#13133205—22no—
22#13133205—22no—
23#13133182—42no—
24#13133179—42no—
25C2×#11719919—21no—
26C2×#11719921—22no—
27C2×#11719921—21no—
28C2×#11719921—22no—
29C2×C4×Dic53—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32C2×C4×Dic53D412yes—
33C2×C4×Dic53C2311yescharacteristic
34#13133124—22no—
35C2×C4⋊C4×C53C2×C411yescharacteristic
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38#95×C53—22no—
39C23×D53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53C22⋊C412yes—
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—21no—
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C2×D412yes—
48C2×D212—21no—
49C2×D212—41no—
50C2×D212—41no—
51C2×D212M1612yes—
52C2×C4×D53—44no—
53C2×C4×D53—42no—
54C2×Dic106—21no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106M1612yes—
58C2×#88324—44no—
59C2×#88324—44no—
60C2×#88324—44no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C42×C53—21no—
64C42×C53—21no—
65#11719921—28no—
66#11719921—42no—
67#11719921—44no—
68#11719919—42no—
69C4×Dic53—22no—
70C4×Dic53—24no—
71C4×Dic53—42no—
72C4×Dic53—41no—
73C4×Dic53—42no—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—22no—
76C22×D53—41no—
77C22×D53—41no—
78C22×D53—42no—
79C22×D53—42no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53C2×M1611yescharacteristic
90C2×C4×C53C2×M1611yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53#10012yes—
100C2×Dic53#13514yes—
101D212—42no—
102D212—42no—
103C4×D53—84no—
104Dic106—42no—
105Dic106—42no—
106#88324—84no—
107#7649—531nomaximal, Sylow
108C22×C53#40711yescharacteristic
109C22×C53#41011yescharacteristic
110C22×C53C2×#10011yescharacteristic
111C22×C53#41612yes—
112C22×C53#42112yes—
113C2×D53—42no—
114C2×D53—44no—
115C2×D53—81no—
116C2×D53—82no—
117C2×D53—81no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—81no—
122C4×C53—81no—
123Dic53—24no—
124Dic53—22no—
125Dic53—41no—
126Dic53—42no—
127C4×#124—531no—
128#1839—532no—
129#1272—532no—
130#1270—532no—
131C2×C53#146611yescharacteristic
132C2×C53#148812yes—
133C2×C53#152112yes—
134C2×C53#153311yescharacteristic
135C2×C53#154311yescharacteristic
136D53—82no—
137C2×#124—531no—
138C2×C4×Q8—531no—
139C2×C4×D4—531no—
140#391—531no—
141#390—531no—
142#388—1062no—
143#382#615711yescharacteristic
144C4×C22⋊C4—1062no—
145C43—531no—
146#336—534no—
147#336—1062no—
148#336—1062no—
149C53#764911yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×D4—531no—
152C23×C4—1061no—
153C22×C8—1064no—
154#124—1062no—
155#124—1062no—
156C4×Q8—2122no—
157C4×D4—2122no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4C2×#615711yescharacteristic
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C22⋊C4—1062no—
162C2×C22⋊C4—1061no—
163C2×C22⋊C4—1062no—
164C2×C42—532no—
165C2×C42—531no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—1061no—
169C2×C42C2×#615711yescharacteristic
170#95—22no—
171#95—1062no—
172C24—1061no—
173C2×Q8—532no—
174C2×Q8—1061no—
175C2×Q8—2121no—
176C2×Q8—2121no—
177C2×D4—532no—
178C2×D4—1061no—
179C2×D4—2121no—
180C2×D4—2121no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—531no—
184C22×C4—532no—
185C22×C4—531no—
186C22×C4—1061no—
187C22×C4—2122no—
188C22×C4—2124no—
189C22×C4C22⋊#615712yes—
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C2×C8—2124no—
193C2×C8—2124no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—2122no—
197C22⋊C4—1068no—
198C22⋊C4—2124no—
199C22⋊C4—2122no—
200C42—21no—
201C42—21no—
202C42—1062no—
203C42—1064no—
204C42—2121no—
205C42—2122no—
206C42—2122no—
207C23—1062no—
208C23—2121no—
209C23—2121no—
210C23—2122no—
211C23—2122no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—2122no—
214Q8—2122no—
215D4—2122no—
216D4—2122no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—532no—
227C2×C4—534no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—4244no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4D53⋊M1611yescharacteristic
238C2×C4D53⋊M1611yescharacteristic
239C8—4244no—
240C22—4242no—
241C22—4241no—
242C22—4241no—
243C22—2124no—
244C22—2122no—
245C22#1681159511yescharacteristic
246C22#1681160411yescharacteristic
247C22D53⋊#10011yescharacteristic
248C22#1681161312yes—
249C22#1681162012yes—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—81no—
254C4—81no—
255C4—1064no—
256C4—1062no—
257C4—2121no—
258C4—2122no—
259C2—4242no—
260C2#2028997611yescharacteristic
261C2#2029004812yes—
262C2#2029015812yes—
263C2#2029019511yescharacteristic
264C2#2029021911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7649
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20289536C2semidirect
149. C53#7649semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895362
#202895322
#202772542
#764953