Finite Group Databasebeta

← #27293317

Subgroups — #27293317

Order 13568.

438 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16810686C211yesmaximal, characteristic
2#20276324C212yesmaximal, complemented
3#20272654C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic, complemented
5C53×#1314C211yesmaximal, characteristic, complemented
6C23×#88322C411yescharacteristic
7C22×C8×D53C2211yescharacteristic
8C2×C4×C22⋊D53C411yescharacteristic
9C2×#13133156—22no—
10#16810686—22no—
11#16810686—22no—
12C2×#13133146C2211yescharacteristic
13C2×#13133129C411yescharacteristic
14C2×C8×Dic53C411yescharacteristic
15C22⋊#11719908C2212yes—
16#16810402—22no—
17C2×C22⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×C22⋊D53C811yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21C22×#88322—21no—
22C22×#88322—21no—
23C22×#88322—42no—
24C22×#88322C2×C411yescharacteristic
25C22×#88322C2×C411yescharacteristic
26C2×C8×D53—42no—
27C2×C8×D53D412yes—
28C2×C8×D53D412yes—
29C4×C22⋊D53—22no—
30C4×C22⋊D53—22no—
31C2×#11719919C811yescharacteristic
32C2×C22⋊Dic53C811yescharacteristic
33C2×#11719921C811yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35#13133156—44no—
36C2×#11719908—22no—
37#13133146—22no—
38#13133129—22no—
39C8×Dic53—21no—
40C8×Dic53—21no—
41C23×C4×C53C2×C411yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43C22×C8×C53C2311yescharacteristic
44C2×C16×C53—22no—
45C22⋊C8×C53—22no—
46C24×C53C2×C811yescharacteristic
47C23×D53C2×C811yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53C2×C811yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—22no—
57C22×Dic53C2×C811yescharacteristic
58C22×Dic53C2×C811yescharacteristic
59C22×Dic53C2×C811yescharacteristic
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53C22⋊C412yes—
63C2×C4×D53M1612yes—
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C2×C8×C53C2×D412yes—
68C22⋊Dic53—22no—
69#11719921—22no—
70#11719919—22no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C2×#88322—22no—
74C2×#88322—21no—
75C2×#88322—42no—
76C2×#88322—42no—
77C2×#88322—41no—
78C2×#88322C22⋊C411yescharacteristic
79C2×#88322C22⋊C411yescharacteristic
80C8×D53—24no—
81C8×D53—42no—
82C16×C53—42no—
83#11719908—42no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53M3212yescomplemented
89C23×C53C22×C811yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53C22⋊C812yes—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C22⋊D53—28no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53C22⋊C812yes—
111C2×Dic53M3212yescomplemented
112C4×D53—22no—
113C4×D53—22no—
114C4×D53—42no—
115C8×C53—22no—
116C8×C53—41no—
117#88322—21no—
118#88322—41no—
119#88322#9912yes—
120#11466—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53C2×C22⋊C811yescharacteristic
129C22×C53C2×M3211yescharacteristic
130C22×C53C2×M3211yescharacteristic
131C2×D53—22no—
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—42no—
135C2×D53—42no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53C2×#9911yescharacteristic
139C4×C53#41311yescharacteristic
140Dic53—22no—
141Dic53—22no—
142Dic53—41no—
143C2×C8×D4—531no—
144#2112—532no—
145#1322—532no—
146#1314—531no—
147#1314D5311yescharacteristic, complemented
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—42no—
151C2×C53#145911yescharacteristic
152C2×C53#211211yescharacteristic
153C2×C53#211511yescharacteristic
154D53—42no—
155C23×C8—531no—
156C23×C8—531no—
157C2×C4×D4—531no—
158C2×M32—1062no—
159C8×D4—1062no—
160C8×D4—1062no—
161C2×C4⋊C8—531no—
162C2×C22⋊C8—531no—
163C2×C22⋊C8C2×D5311yescharacteristic
164C2×C4×C8—531no—
165C22⋊C16—532no—
166C22⋊C16—1062no—
167C22⋊C16C2×D5312yes—
168C53#1146611yescharacteristic, complemented, Sylow
169C22×D4—531no—
170C23×C4—531no—
171C23×C4C4×D5311yescharacteristic
172C22×C8—531no—
173C22×C8—532no—
174C22×C8—532no—
175C22×C8—531no—
176C22×C8—1061no—
177C22×C8—1061no—
178C22×C8—2122no—
179C22×C8—2122no—
180C22×C8C4×D5311yescharacteristic
181C22×C8C22×D5311yescharacteristic
182C4×D4—1062no—
183C4×D4—1062no—
184C2×C4⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—531no—
187C2×C42—531no—
188M32—2124no—
189C2×C16—22no—
190C2×C16—1062no—
191C4⋊C8—1062no—
192C22⋊C8—22no—
193C22⋊C8—1062no—
194C4×C8—1061no—
195C4×C8—1061no—
196C24—531no—
197C24C8×D5311yescharacteristic
198C2×D4—1062no—
199C2×D4—1062no—
200C2×D4—1062no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—532no—
208C22×C4—532no—
209C22×C4—531no—
210C22×C4—2121no—
211C22×C4—2121no—
212C22×C4C8×D5311yescharacteristic
213C22×C4C2×C4×D5311yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—531no—
218C2×C8—531no—
219C2×C8—1064no—
220C2×C8—1062no—
221C2×C8—1061no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2121no—
226C2×C8D4×D5312yes—
227C4⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C42—1061no—
231C42—1061no—
232C16—42no—
233C16—2122no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—532no—
239C23—1061no—
240C23—1061no—
241C23—2121no—
242C23—2121no—
243C23#1313315612yescomplemented
244C23C2×C8×D5311yescharacteristic
245D4—1068no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—532no—
254C2×C4—532no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—2122no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4C22⋊C4×D5311yescharacteristic
264C2×C4M16×D5311yescharacteristic
265C8—22no—
266C8—41no—
267C8—532no—
268C8—1061no—
269C8—2121no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—1062no—
278C22—1062no—
279C22—1062no—
280C22—2122no—
281C22—2122no—
282C22C22⋊C8×D5311yescharacteristic
283C22M32×D5311yescharacteristic
284C22C2×#1313315611yescharacteristic
285C4—21no—
286C4—41no—
287C4—1062no—
288C4—1062no—
289C4—2121no—
290C4#99×D5311yescharacteristic
291C4#1681060411yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—2122no—
296C2#2027303011yescharacteristic
297C2#2027632411yescharacteristic
298C2#2027633411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1314C2
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×C8×C53C23
hypercenter#1314D53
hypocenterC53#11466
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20276324C2semidirect
3. #20272654C2semidirect
4. #20272639C2semidirect
5. C53×#1314C2semidirect
88. C23×C53M32semidirect
111. C2×Dic53M32semidirect
147. #1314D53semidirect
168. C53#11466semidirect
243. C23#13133156semidirect

Maximal subgroups

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

subgroupindex
C2×#168106862
#202763242
#202726542
#202726392
C53×#13142
#1146653