Finite Group Databasebeta

← #28299683

Subgroups — #28299683

Order 13568.

636 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290432C212yesmaximal, complemented
2#20289532C212yesmaximal, complemented
3#20289531C212yesmaximal, complemented
4C4×#13133213C211yesmaximal, characteristic
5C2×#13133272—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—
18#13133272—28no—
19C22×D212C2×C411yescharacteristic
20C22×C22⋊D53—22no—
21C22×C4×D53—21no—
22#13133213—22no—
23#13133213—22no—
24C4×D212—42no—
25C4×C22⋊D53—48no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×C22⋊Dic53—22no—
29C2×#11719921—21no—
30C2×#11719921—22no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53—21no—
37C4×D4×C53—42no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—21no—
42C23×D53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53M1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—44no—
49C22×C4×C53—42no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—28no—
54C2×C22⋊D53—44no—
55C2×C22⋊D53—44no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C2×D412yes—
59C2×D212—21no—
60C2×D212—41no—
61C2×D212—41no—
62C2×D212M1612yes—
63C2×C4×D53—44no—
64C2×C4×D53—42no—
65C2×#88324—28no—
66C2×#88324—44no—
67C2×#88324—44no—
68C4⋊C4×C53—41no—
69C22⋊C4×C53—42no—
70C42×C53—22no—
71C22⋊Dic53—44no—
72#11719921—42no—
73#11719921—44no—
74C4⋊Dic53—41no—
75#11719919—44no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—24no—
79C4×Dic53—24no—
80C4×Dic53—41no—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53C2×C22⋊C411yescharacteristic
87C22×D53—22no—
88C22×D53—41no—
89C22×D53—41no—
90C22×D53—42no—
91C22×D53—42no—
92D4×C53—42no—
93D4×C53—42no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—84no—
102C2×C4×C53#10012yes—
103C2×C4×C53C2×M1611yescharacteristic
104C2×C4×C53C2×M1611yescharacteristic
105C22⋊D53—416no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—42no—
112C2×Dic53#10414yes—
113C2×Dic53AGL(1,ℤ/8)14yes—
114D212—42no—
115D212—42no—
116C4×D53—84no—
117#88324—48no—
118#7637—531nomaximal, Sylow
119C22×C53—42no—
120C22×C53—44no—
121C22×C53—81no—
122C22×C53—82no—
123C22×C53—81no—
124C22×C53#40711yescharacteristic
125C22×C53#41011yescharacteristic
126C22×C53C2×#10011yescharacteristic
127C22×C53#41812yes—
128C22×C53C2×#10412yes—
129C2×D53—42no—
130C2×D53—44no—
131C2×D53—81no—
132C2×D53—82no—
133C2×D53—81no—
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53—81no—
139Dic53—24no—
140Dic53—24no—
141Dic53—81no—
142C4×#127—531no—
143#1745—532no—
144#1270—532no—
145#1270—532no—
146C2×C53—82no—
147C2×C53#146611yescharacteristic
148C2×C53#149012yes—
149C2×C53#150512yes—
150C2×C53#152511yescharacteristic
151C2×C53#155511yescharacteristic
152D53—82no—
153C2×#127—531no—
154C2×C4×D4—531no—
155C2×C4×D4—1062no—
156C2×C4×D4#615711yescharacteristic
157C2×#97—1062no—
158#390—531no—
159#390—531no—
160#390—1062no—
161#384—531no—
162C43—531no—
163#336—534no—
164#336—1062no—
165C53#763711yescharacteristic, complemented, Sylow
166C22×D4—531no—
167C22×D4—1062no—
168C22×D4C2×#615711yescharacteristic
169C23×C4—21no—
170C23×C4—1061no—
171C22×C8—1064no—
172#127—1062no—
173#127—1062no—
174C4×D4—42no—
175C4×D4—2122no—
176C4×D4—2128no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—1062no—
179C2×C4⋊C4C2×#615711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—1062no—
182C2×C22⋊C4—1062no—
183C2×C22⋊C4—1061no—
184C2×C42—532no—
185C2×C42—532no—
186C2×C42—531no—
187C2×C42—531no—
188C2×C42C2×#615711yescharacteristic
189#97—1068no—
190C24—21no—
191C24—1061no—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—532no—
196C2×D4—1068no—
197C2×D4—1061no—
198C2×D4—2121no—
199C2×D4—2121no—
200C2×D4—2124no—
201C2×D4—2124no—
202C2×D4#1171993612yes—
203C22×C4—21no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—1061no—
209C22×C4—2124no—
210C22×C4—2122no—
211C22×C4C22⋊#615711yescharacteristic
212C22×C4C22⋊#615711yescharacteristic
213C22×C4C22×#615711yescharacteristic
214C2×C8—1068no—
215C2×C8—2124no—
216C2×C8—2124no—
217C4⋊C4—41no—
218C4⋊C4—2121no—
219C4⋊C4—2124no—
220C22⋊C4—42no—
221C22⋊C4—2122no—
222C22⋊C4—2124no—
223C22⋊C4—2124no—
224C42—22no—
225C42—1064no—
226C42—1064no—
227C42—1061no—
228C42—1061no—
229C42—2121no—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23—1062no—
236C23—2121no—
237C23—2121no—
238C23—2122no—
239C23—2122no—
240C23C2×C22⋊#615711yescharacteristic
241D4—42no—
242D4—42no—
243D4—2122no—
244D4—2122no—
245D4—21216no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—84no—
254C2×C4—534no—
255C2×C4—534no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—4244no—
261C2×C4—2121no—
262C2×C4—2122no—
263C2×C4#1313327712yes—
264C2×C4D53⋊M1611yescharacteristic
265C2×C4C2×#1171993611yescharacteristic
266C8—2128no—
267C22—44no—
268C22—42no—
269C22—82no—
270C22—81no—
271C22—81no—
272C22—4242no—
273C22—4241no—
274C22—4241no—
275C22—2124no—
276C22—2122no—
277C22#1681159411yescharacteristic
278C22#1681160211yescharacteristic
279C22C2×#1313327711yescharacteristic
280C22#1681161512yes—
281C22D53⋊#10412yes—
282C4—21no—
283C4—21no—
284C4—22no—
285C4—41no—
286C4—81no—
287C4—1064no—
288C4—1064no—
289C4—4241no—
290C2—82no—
291C2—4242no—
292C2#2028997411yescharacteristic
293C2#2029005512yes—
294C2#2029010512yes—
295C2#2029016911yescharacteristic
296C2#2029024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7637
LCSG ▷ C22×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. #20290432C2semidirect
2. #20289532C2semidirect
3. #20289531C2semidirect
165. C53#7637semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202895322
#202895312
C4×#131332132
#763753