Finite Group Databasebeta

← #27293330

Subgroups — #27293330

Order 13568.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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#20283412C211yesmaximal, characteristic
2#20276324C212yesmaximal, complemented
3#20272656C212yesmaximal, complemented
4#20272641C211yesmaximal, characteristic, complemented
5C53×#1315C211yesmaximal, characteristic, complemented
6C2×#13133224C411yescharacteristic
7C22×C8×D53C2211yescharacteristic
8#16811130C411yescharacteristic
9C2×#13133156—22no—
10#16810697—22no—
11#16810686—22no—
12#16810649C2211yescharacteristic
13#16810586C411yescharacteristic
14#16810581C411yescharacteristic
15C22⋊#11719908C2212yes—
16#16810402—22no—
17C53×#408C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×C4×D53C2×C411yescharacteristic
20#13133224—42no—
21C2×#11719917—21no—
22C2×#11719917C2×C411yescharacteristic
23C22×#88322—21no—
24C22×#88322C2×C411yescharacteristic
25C2×C8×D53—42no—
26C2×C8×D53D412yes—
27C2×C8×D53D412yes—
28#13133198—21no—
29#13133193C812yes—
30#13133188C812yes—
31#13133185—21no—
32C4×C22⋊D53—22no—
33#13133176C2×C411yescharacteristic
34#13133156—44no—
35C2×#11719908—22no—
36#13133146—22no—
37#13133129—22no—
38#13133127—21no—
39C8×Dic53—21no—
40C2×#41×C53C2×C411yescharacteristic
41C2×M16×C53C2×C411yescharacteristic
42C22×C8×C53C2311yescharacteristic
43C2×C16×C53—22no—
44C22⋊C8×C53—22no—
45C23×D53M1611yescharacteristic
46#41×C53—41no—
47#41×C53—41no—
48C2×Q8×C53C2×C811yescharacteristic
49C2×D4×C53—21no—
50C2×D4×C53C2×C811yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—22no—
54C22×Dic53M1611yescharacteristic
55C2×C4×D53—41no—
56C2×C4×D53—41no—
57C2×C4×D53C22⋊C412yes—
58C2×C4×D53C2×C812yes—
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53—41no—
62C2×C8×C53C2×D412yes—
63C22⋊Dic53—21no—
64C22⋊Dic53—22no—
65#11719921—22no—
66C4⋊Dic53C2×C812yes—
67#11719919—22no—
68C4×Dic53—21no—
69#11719917—42no—
70#11719917—41no—
71C2×#88322—22no—
72C2×#88322—21no—
73C2×#88322—42no—
74C2×#88322C22⋊C411yescharacteristic
75C2×#88322C22⋊C411yescharacteristic
76C8×D53—24no—
77C8×D53—42no—
78C16×C53—42no—
79#11719908—42no—
80C23×C53—21no—
81C23×C53C2×M1611yescharacteristic
82C22×D53—41no—
83C22×D53—42no—
84C22×D53C22⋊C812yes—
85Q8×C53—41no—
86D4×C53—41no—
87D4×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53C22×C811yescharacteristic
95C22⋊D53—44no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C2×Dic53C22⋊C812yes—
99C4×D53—22no—
100C4×D53—22no—
101C4×D53—42no—
102C8×C53—22no—
103C8×C53—41no—
104#88322—21no—
105#88322—41no—
106#88322#9912yes—
107#11469—531nomaximal, Sylow
108C22×C53—22no—
109C22×C53—21no—
110C22×C53—42no—
111C22×C53—41no—
112C22×C53C2×C22⋊C811yescharacteristic
113C2×D53—22no—
114C2×D53—22no—
115C2×D53—22no—
116C2×D53—44no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53C2×#9911yescharacteristic
120C4×C53#41311yescharacteristic
121Dic53—22no—
122Dic53—42no—
123#2929—531no—
124#2112—532no—
125#1323—532no—
126#1315—531no—
127#1315D5311yescharacteristic, complemented
128C2×C53—41no—
129C2×C53—81no—
130C2×C53#145911yescharacteristic
131C2×C53#211712yes—
132D53—42no—
133C2×#131—531no—
134C23×C8—531no—
135#525—531no—
136C2×M32—1062no—
137#435—1062no—
138C8×D4—1062no—
139#424—531no—
140#408C2×D5311yescharacteristic
141#407—531no—
142#405—531no—
143C22⋊C16—532no—
144C22⋊C16—1062no—
145C22⋊C16C2×D5312yes—
146C53#1146911yescharacteristic, complemented, Sylow
147C2×#41C4×D5311yescharacteristic
148C23×C4—531no—
149#131—2122no—
150C2×M16—531no—
151C2×M16—1061no—
152C2×M16C4×D5311yescharacteristic
153C22×C8—531no—
154C22×C8—532no—
155C22×C8—532no—
156C22×C8—1061no—
157C22×C8—2122no—
158C22×C8C22×D5311yescharacteristic
159#123—1061no—
160C22⋊Q8—532no—
161#121—532no—
162#120—1061no—
163C4×D4—1062no—
164#117—531no—
165M32—2124no—
166C2×C16—22no—
167C2×C16—1062no—
168C4⋊C8—1062no—
169C22⋊C8—22no—
170C22⋊C8—1062no—
171#97—1061no—
172C4×C8—1061no—
173C24—531no—
174#41—41no—
175#41—41no—
176C2×Q8C8×D5311yescharacteristic
177C2×D4—21no—
178C2×D4—1062no—
179C2×D4C8×D5311yescharacteristic
180C22×C4—21no—
181C22×C4—531no—
182C22×C4—532no—
183C22×C4—532no—
184C22×C4—2121no—
185C22×C4—2121no—
186C22×C4C2×C4×D5311yescharacteristic
187M16—41no—
188M16—2122no—
189M16—2121no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—531no—
193C2×C8—531no—
194C2×C8—1064no—
195C2×C8—1062no—
196C2×C8—1061no—
197C2×C8—2122no—
198C2×C8—2122no—
199C2×C8D4×D5312yes—
200C4⋊C4—532no—
201C4⋊C4—1062no—
202C22⋊C4—1062no—
203C22⋊C4—1062no—
204C22⋊C4—1061no—
205C42—1061no—
206C16—42no—
207C16—2122no—
208C23—21no—
209C23—532no—
210C23—2121no—
211C23—2122no—
212C23M16×D5311yescharacteristic
213Q8—41no—
214D4—41no—
215D4—42no—
216D4—2124no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—532no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—2122no—
227C2×C4—2121no—
228C2×C4C22⋊C4×D5311yescharacteristic
229C2×C4C2×C8×D5311yescharacteristic
230C8—22no—
231C8—41no—
232C8—532no—
233C8—1061no—
234C8—2121no—
235C22—22no—
236C22—21no—
237C22—42no—
238C22—41no—
239C22—1062no—
240C22—1062no—
241C22—1062no—
242C22—2124no—
243C22C22⋊C8×D5311yescharacteristic
244C4—21no—
245C4—41no—
246C4—1062no—
247C4—2122no—
248C4#99×D5311yescharacteristic
249C4#1681060411yescharacteristic
250C2—41no—
251C2—81no—
252C2—2122no—
253C2#2027303011yescharacteristic
254C2#2027634112yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1315C2
socleC22×C53C2×C22⋊C8
prime radical#16811130C4
cosocleC22×C8×C53C23
hypercenter#1315D53
hypocenterC53#11469
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #20276324C2semidirect
3. #20272656C2semidirect
4. #20272641C2semidirect
5. C53×#1315C2semidirect
127. #1315D53semidirect
146. C53#11469semidirect

Maximal subgroups

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

subgroupindex
#202834122
#202763242
#202726562
#202726412
C53×#13152
#1146953