Finite Group Databasebeta

← #27293504

Subgroups — #27293504

Order 13568.

418 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 77 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#20277336C211yesmaximal, characteristic
2#20273048C212yesmaximal
3#20272661C211yesmaximal, characteristic, complemented
4#20272660C212yesmaximal, complemented
5C53×#1325C211yesmaximal, characteristic, complemented
6C2×#13133175C2211yescharacteristic
7C2×#13133174C411yescharacteristic
8C2×C4⋊C4×D53C411yescharacteristic
9#16810677—22no—
10#16810518—22no—
11#16810494—22no—
12#16810492C2211yescharacteristic
13#16810489C411yescharacteristic
14#16810488C411yescharacteristic
15#16810349C2212yes—
16#16810346—22no—
17C53×#384C2211yescharacteristic
18C53×#324C2212yes—
19C22×C4×D53—21no—
20C22×C4×D53C2×C411yescharacteristic
21#13133175—42no—
22#13133174—22no—
23#13133174—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—21no—
28C2×#11719919—21no—
29C4⋊C4×D53—42no—
30C2×#11719921—21no—
31C2×#11719921D412yes—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34#13133146—42no—
35C22⋊#88322—22no—
36#13133127—42no—
37#13133124—22no—
38#13133124—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—22no—
44C23×D53C22⋊C411yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22×C411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C412yes—
53C2×C4×D53—22no—
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—22no—
57C2×C4×D53—42no—
58C2×C4×D53—82no—
59C2×C8×C53—42no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—82no—
63C42×C53—21no—
64C42×C53M1612yes—
65#11719921—24no—
66#11719921—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69#11719919—24no—
70#11719919—42no—
71#11719919—82no—
72C4×Dic53—21no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53M1612yes—
76C2×#88322—42no—
77C23×C53C2×C22⋊C411yescharacteristic
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—41no—
81C22×D53—41no—
82C22×D53#10112yes—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53C2×M1611yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—41no—
103C2×Dic53#9912yes—
104C4×D53—24no—
105C4×D53—24no—
106C4×D53—84no—
107C8×C53—82no—
108#88322—82no—
109#11516—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#40711yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53C2×#10111yescharacteristic
115C2×D53—22no—
116C2×D53—42no—
117C2×D53—41no—
118C2×D53—41no—
119C2×D53—42no—
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—42no—
123C4×C53—42no—
124C4×C53—81no—
125Dic53—22no—
126Dic53—42no—
127Dic53—42no—
128Dic53—81no—
129#2317—531no—
130#1464—532no—
131#1325—531no—
132#1325—532no—
133#1325D5311yescharacteristic, complemented
134C2×C53—21no—
135C2×C53—21no—
136C2×C53#146311yescharacteristic
137C2×C53#212811yescharacteristic
138C2×C53#213511yescharacteristic
139D53—42no—
140C2×#117—531no—
141C2×#117—531no—
142C22×C4⋊C4—531no—
143#432—1062no—
144#389—1062no—
145#385—1062no—
146#384—531no—
147#384C2×D5311yescharacteristic
148#382—531no—
149#382—531no—
150#324—532no—
151#324C2×D5312yes—
152#323—1062no—
153C53#1151611yescharacteristic, complemented, Sylow
154C23×C4—531no—
155C23×C4—1061no—
156#117—1062no—
157#117—1062no—
158#117—2122no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1062no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—2122no—
166C2×C4⋊C4C4×D5311yescharacteristic
167C2×C4⋊C4C22×D5311yescharacteristic
168C2×C22⋊C4—532no—
169C2×C22⋊C4—1061no—
170C2×C42—531no—
171C2×C42—532no—
172C2×C42C4×D5311yescharacteristic
173C4⋊C8—2122no—
174C22⋊C8—22no—
175C22⋊C8—1062no—
176#97—2122no—
177#95—1062no—
178#95—1062no—
179C24—531no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—1061no—
190C22×C4—4242no—
191C22×C4—2122no—
192C22×C4C2×C4×D5311yescharacteristic
193C22×C4D4×D5312yes—
194C2×C8—42no—
195C2×C8—2122no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—1064no—
200C4⋊C4—4242no—
201C4⋊C4—2121no—
202C4⋊C4—2121no—
203C4⋊C4—2122no—
204C22⋊C4—1064no—
205C22⋊C4—2122no—
206C42—21no—
207C42—532no—
208C42—1062no—
209C42—1062no—
210C42—1061no—
211C42#1171991212yes—
212C23—532no—
213C23—1061no—
214C23—1061no—
215C23—2121no—
216C23—2121no—
217C23C22⋊C4×D5311yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—532no—
228C2×C4—1064no—
229C2×C4—1064no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—4244no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4M16×D5311yescharacteristic
241C2×C4C2×#1171991211yescharacteristic
242C8—82no—
243C8—4242no—
244C22—21no—
245C22—21no—
246C22—1062no—
247C22—2122no—
248C22—2122no—
249C22—2121no—
250C22—2121no—
251C22#1681058511yescharacteristic
252C22#99×D5311yescharacteristic
253C22#101×D5311yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—1062no—
260C4—4241no—
261C4—2122no—
262C4—2122no—
263C2—21no—
264C2—21no—
265C2—2122no—
266C2#2027304311yescharacteristic
267C2#2027638511yescharacteristic
268C2#2027640811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11516
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #20272661C2semidirect
4. #20272660C2semidirect
5. C53×#1325C2semidirect
133. #1325D53semidirect
153. C53#11516semidirect

Maximal subgroups

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

subgroupindex
#202773362
#202730482
#202726612
#202726602
C53×#13252
#1151653