Finite Group Databasebeta

← #28299563

Subgroups — #28299563

Order 13568.

608 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 106 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#20289532C212yesmaximal, complemented
3#20289531C212yesmaximal, complemented
4C4×#13133213C211yesmaximal, characteristic
5#16811489—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133213C411yescharacteristic
9C2×C4×D212C411yescharacteristic
10C2×C4×C22⋊D53—22no—
11#16810526C2211yescharacteristic
12#16810524C2211yescharacteristic
13#16810523—22no—
14#16810492C411yescharacteristic
15C42×Dic53C2211yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17C22×#88324—24no—
18C22×D212C2×C411yescharacteristic
19C22×C22⋊D53—22no—
20C22×C4×D53—21no—
21#13133213—22no—
22#13133213C814yescomplemented
23C4×D212—42no—
24C4×C22⋊D53—48no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×C22⋊Dic53—22no—
28C2×#11719921—21no—
29C2×#11719921—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32C2×C4×Dic53D412yes—
33C2×C4×Dic53C2311yescharacteristic
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53—21no—
36C4×D4×C53—42no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C24×C53—21no—
41C23×D53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53C2×C812yes—
46C22×C4×C53—21no—
47C22×C4×C53—44no—
48C22×C4×C53—42no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C2×C22⋊D53—28no—
53C2×C22⋊D53—44no—
54C2×C22⋊D53—44no—
55C22×Dic53—21no—
56C22×Dic53C22⋊C412yes—
57C22×Dic53C2×D412yes—
58C2×D212—21no—
59C2×D212—41no—
60C2×D212—41no—
61C2×D212C2×C812yes—
62C2×C4×D53—44no—
63C2×C4×D53—42no—
64C2×#88324—44no—
65C2×#88324—44no—
66C2×#88324—44no—
67C4⋊C4×C53—41no—
68C22⋊C4×C53—42no—
69C42×C53—22no—
70C22⋊Dic53—44no—
71#11719921—42no—
72#11719921—44no—
73C4⋊Dic53—41no—
74#11719919—44no—
75C4×Dic53—24no—
76C4×Dic53—42no—
77C4×Dic53—41no—
78C4×Dic53C2×C812yes—
79C4×Dic53M1612yes—
80C23×C53—22no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53C2×C22⋊C411yescharacteristic
86C22×D53—22no—
87C22×D53—41no—
88C22×D53—41no—
89C22×D53—42no—
90C22×D53—42no—
91D4×C53—42no—
92D4×C53—42no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—84no—
101C2×C4×C53#10012yes—
102C2×C4×C53C22×C811yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C22⋊D53—416no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53C22⋊C814yes—
112C2×Dic53AGL(1,ℤ/8)14yes—
113D212—42no—
114D212—42no—
115C4×D53—84no—
116#88324—84no—
117#7603—531nomaximal, Sylow
118C22×C53—42no—
119C22×C53—44no—
120C22×C53—81no—
121C22×C53—81no—
122C22×C53—82no—
123C22×C53C2×C22⋊C811yescharacteristic
124C22×C53#41011yescharacteristic
125C22×C53C2×#10011yescharacteristic
126C22×C53#41812yes—
127C22×C53#42112yes—
128C2×D53—42no—
129C2×D53—44no—
130C2×D53—81no—
131C2×D53—81no—
132C2×D53—82no—
133C4×C53—22no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—81no—
138Dic53—24no—
139Dic53—24no—
140Dic53—81no—
141C4×#127—531no—
142#1839—532no—
143#1270—532no—
144#1270—532no—
145C2×C53—82no—
146C2×C53#146011yescharacteristic
147C2×C53#147812yes—
148C2×C53#151412yes—
149C2×C53#153611yescharacteristic
150C2×C53#155511yescharacteristic
151D53—82no—
152C2×#127—531no—
153C2×C4×D4—531no—
154C2×C4×D4—1062no—
155C2×C4×D4#615711yescharacteristic
156#390—531no—
157#390—531no—
158#390—1062no—
159#384—531no—
160C43—531no—
161#336—534no—
162#336—1062no—
163#336—1062no—
164C53#760311yescharacteristic, complemented, Sylow
165C22×D4—531no—
166C22×D4—1062no—
167C22×D4C2×#615711yescharacteristic
168C23×C4—21no—
169C23×C4—1061no—
170C22×C8—1064no—
171#127—534no—
172#127—1062no—
173C4×D4—42no—
174C4×D4—2122no—
175C4×D4—2128no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1062no—
178C2×C4⋊C4C2×#615711yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—1062no—
181C2×C22⋊C4—1062no—
182C2×C22⋊C4—1061no—
183C2×C42—531no—
184C2×C42—532no—
185C2×C42—532no—
186C2×C42—531no—
187C2×C42C2×#615711yescharacteristic
188C24—21no—
189C24—1061no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—532no—
194C2×D4—1068no—
195C2×D4—1061no—
196C2×D4—2121no—
197C2×D4—2121no—
198C2×D4—2124no—
199C2×D4—2124no—
200C2×D4D53⋊C812yes—
201C22×C4—21no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—532no—
205C22×C4—532no—
206C22×C4—1061no—
207C22×C4—2124no—
208C22×C4—2122no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212C2×C8—2124no—
213C2×C8—2124no—
214C2×C8—2124no—
215C4⋊C4—41no—
216C4⋊C4—2121no—
217C4⋊C4—2124no—
218C22⋊C4—42no—
219C22⋊C4—2122no—
220C22⋊C4—2124no—
221C22⋊C4—2124no—
222C42—22no—
223C42—532no—
224C42—532no—
225C42—1064no—
226C42—2121no—
227C42—2122no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—1062no—
234C23—2121no—
235C23—2121no—
236C23—2122no—
237C23—2122no—
238C23C2×C22⋊#615711yescharacteristic
239D4—42no—
240D4—42no—
241D4—2122no—
242D4—2122no—
243D4—21216no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—534no—
253C2×C4—534no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—4244no—
259C2×C4—2121no—
260C2×C4—2122no—
261C2×C4#1313327712yes—
262C2×C4C2×D53⋊C811yescharacteristic
263C2×C4D53⋊M1611yescharacteristic
264C8—4244no—
265C22—42no—
266C22—44no—
267C22—82no—
268C22—81no—
269C22—81no—
270C22—4242no—
271C22—4241no—
272C22—4241no—
273C22—2122no—
274C22—2124no—
275C22D53⋊(C22⋊C8)11yescharacteristic
276C22#1681160211yescharacteristic
277C22C2×#1313327711yescharacteristic
278C22#1681161512yes—
279C22#1681162012yes—
280C4—21no—
281C4—21no—
282C4—22no—
283C4—41no—
284C4—81no—
285C4—1064no—
286C4—1064no—
287C4—4241no—
288C2—82no—
289C2—4242no—
290C2#2028995411yescharacteristic
291C2#2029001512yes—
292C2#2029013412yes—
293C2#2029020311yescharacteristic
294C2#2029024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7603
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20289532C2semidirect
3. #20289531C2semidirect
22. #13133213C8semidirect
164. C53#7603semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895322
#202895312
C4×#131332132
#760353