Finite Group Databasebeta

← #27293249

Subgroups — #27293249

Order 13568.

462 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277120C211yesmaximal, characteristic
2#20273028C212yesmaximal
3#20272661C211yesmaximal, characteristic, complemented
4#20272658C212yesmaximal, complemented
5C53×#1325C211yesmaximal, characteristic, complemented
6C2×#13133175C411yescharacteristic
7C2×#13133175C2211yescharacteristic
8C2×C42×D53C411yescharacteristic
9#16810673—22no—
10#16810493C411yescharacteristic
11#16810488C411yescharacteristic
12#16810488C2211yescharacteristic
13#16810480—22no—
14C4⋊C4×Dic53—22no—
15#16810349C2212yes—
16#16810346—22no—
17C53×#384C2211yescharacteristic
18C53×#324C2212yes—
19C22×C4×D53—21no—
20C22×C4×D53C2×C411yescharacteristic
21#13133175—41no—
22#13133175—41no—
23#13133175—41no—
24#13133175—41no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719921—21no—
29C2×#11719921D412yes—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C42×D53—22no—
35C42×D53C814yescomplemented
36#13133146—42no—
37C22⋊#88322—22no—
38C8×Dic53—42no—
39#13133124—22no—
40#13133124—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C22⋊C8×C53—22no—
46C23×D53C22⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C22⋊C411yescharacteristic
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×C4×D53—22no—
58C2×C4×D53—24no—
59C2×C4×D53—42no—
60C2×C4×D53—42no—
61C2×C4×D53C2×C812yes—
62C2×C4×D53M1612yes—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—44no—
66C4⋊C4×C53—41no—
67C42×C53—21no—
68C42×C53C2×C812yes—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—41no—
72C4⋊Dic53—44no—
73C4⋊Dic53—41no—
74C4×Dic53—21no—
75C4×Dic53—24no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—41no—
79C4×Dic53—41no—
80C4×Dic53C2×C812yes—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53C22⋊C812yes—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53C22×C811yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53C22⋊C812yes—
107C2×Dic53#9911yescharacteristic
108C2×Dic53#9911yescharacteristic
109C2×Dic53#10112yes—
110C4×D53—22no—
111C4×D53—22no—
112C4×D53—22no—
113C4×D53—22no—
114C4×D53—48no—
115C8×C53—82no—
116#88322—82no—
117#11447—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53C2×C22⋊C811yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10111yescharacteristic
123C2×D53—22no—
124C2×D53—42no—
125C2×D53—41no—
126C2×D53—41no—
127C2×D53—42no—
128C4×C53—24no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—42no—
132C4×C53—81no—
133Dic53—42no—
134Dic53—42no—
135Dic53—81no—
136Dic53#32412yes—
137Dic53#32412yes—
138#2285—531no—
139#1458—532no—
140#1325—531no—
141#1325D5311yescharacteristic, complemented
142#1324—532no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53C2×#32411yescharacteristic
146C2×C53#212611yescharacteristic
147C2×C53#213211yescharacteristic
148D53—42no—
149C2×#117—531no—
150C2×#117—531no—
151C22×C42—531no—
152#431—1062no—
153#384—531no—
154#384C2×D5311yescharacteristic
155#382—531no—
156#382—531no—
157#380—1062no—
158C4×C4⋊C4—1062no—
159#324—532no—
160#324C2×D5312yes—
161#323—1062no—
162C53#1144711yescharacteristic, complemented, Sylow
163C23×C4—531no—
164C23×C4—1061no—
165#117—2121no—
166#117—2121no—
167#117—2121no—
168#117—2121no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4C4×D5311yescharacteristic
174C2×C4⋊C4C22×D5311yescharacteristic
175C2×C22⋊C4—532no—
176C2×C22⋊C4—1061no—
177C2×C42—531no—
178C2×C42—534no—
179C2×C42—532no—
180C2×C42—1062no—
181C2×C42—1061no—
182C2×C42—1061no—
183C2×C42C4×D5311yescharacteristic
184C4⋊C8—2122no—
185C22⋊C8—22no—
186C22⋊C8—1062no—
187C4×C8—2122no—
188#95—1062no—
189#95—1062no—
190C24—531no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—531no—
194C22×C4—532no—
195C22×C4—532no—
196C22×C4—532no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—1062no—
200C22×C4—1064no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—2122no—
204C22×C4—2122no—
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4D4×D5312yes—
207C2×C8—42no—
208C2×C8—2122no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2124no—
215C22⋊C4—2122no—
216C22⋊C4—2122no—
217C42—21no—
218C42—532no—
219C42—1062no—
220C42—1062no—
221C42—1064no—
222C42—1061no—
223C42—2121no—
224C42—2121no—
225C42C8×D5312yes—
226C23—532no—
227C23—1061no—
228C23—1061no—
229C23—2121no—
230C23—2121no—
231C23C22⋊C4×D5311yescharacteristic
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—532no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—532no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—2128no—
252C2×C4—2122no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4C2×C8×D5311yescharacteristic
258C2×C4M16×D5311yescharacteristic
259C8—82no—
260C8—4242no—
261C22—21no—
262C22—21no—
263C22—1062no—
264C22—2122no—
265C22—2122no—
266C22—2121no—
267C22—2121no—
268C22C22⋊C8×D5311yescharacteristic
269C22#99×D5311yescharacteristic
270C22#101×D5311yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—81no—
276C4—532no—
277C4—532no—
278C4—4241no—
279C4—2122no—
280C4—2122no—
281C2—21no—
282C2—21no—
283C2—2122no—
284C2#324×D5311yescharacteristic
285C2#2027637711yescharacteristic
286C2#2027639911yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC23×D53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11447
LCSG ▷ C2×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. #20272658C2semidirect
5. C53×#1325C2semidirect
35. C42×D53C8semidirect
141. #1325D53semidirect
162. C53#11447semidirect

Maximal subgroups

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

subgroupindex
#202771202
#202730282
#202726612
#202726582
C53×#13252
#1144753