Finite Group Databasebeta

← #28299728

Subgroups — #28299728

Order 13568.

384 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290643C211yesmaximal, characteristic
2#20290643C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5C4×#13133208C211yesmaximal, characteristic
6#16811489—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133208C411yescharacteristic
13#16810522C2211yescharacteristic
14#16810519C2211yescharacteristic
15#16810510—21no—
16#16810506—21no—
17#16810488C411yescharacteristic
18C4×#11719919—21no—
19C4×C4⋊Dic53—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133208—22no—
26#13133208—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C2×#88324—42no—
62C2×#88324—42no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53M1612yes—
70C42×C53—22no—
71C42×C53—42no—
72C4⋊Dic53—24no—
73C4⋊Dic53—21no—
74C4⋊Dic53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53M1612yes—
77#11719919—24no—
78#11719919—42no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—41no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C4×Dic53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53#10112yes—
102C2×C4×C53C2×M1611yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53#13512yes—
121C2×Dic53#13512yes—
122#88324—82no—
123#88324—82no—
124#7650—531nomaximal, Sylow
125C22×C53#40711yescharacteristic
126C22×C53#41011yescharacteristic
127C22×C53C2×#10111yescharacteristic
128C22×C53#41611yescharacteristic
129C22×C53#41611yescharacteristic
130C22×C53#42111yescharacteristic
131C22×C53#42111yescharacteristic
132C4×C53—21no—
133C4×C53—21no—
134C4×C53—22no—
135C4×C53—41no—
136C4×C53—42no—
137C4×C53—81no—
138Dic53—22no—
139Dic53—22no—
140Dic53—41no—
141Dic53—41no—
142Dic53—42no—
143Dic53—81no—
144C4×#125—531no—
145#1839—531no—
146#1839—531no—
147#1271—532no—
148#1271—532no—
149C2×C53#146711yescharacteristic
150C2×C53#148811yescharacteristic
151C2×C53#148811yescharacteristic
152C2×C53#152111yescharacteristic
153C2×C53#152111yescharacteristic
154C2×C53#153411yescharacteristic
155C2×C53#154411yescharacteristic
156C2×#125—531no—
157#389—531no—
158#389—531no—
159#387—1061no—
160#387—1061no—
161#382—531no—
162C4×C4⋊C4—531no—
163C4×C4⋊C4—1061no—
164C4×C4⋊C4—1061no—
165C4×C4⋊C4#615711yescharacteristic
166C43—531no—
167#336—532no—
168#336—532no—
169#336—1061no—
170#336—1061no—
171#336—1061no—
172#336—1061no—
173C53#765011yescharacteristic, complemented, Sylow
174C22×C8—1062no—
175C22×C8—1062no—
176#125—1062no—
177#125—1062no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C2×#615711yescharacteristic
185C2×C4⋊C4C2×#615711yescharacteristic
186C2×C42—21no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—1061no—
194C2×C42C2×#615711yescharacteristic
195#95—21no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212C2×C8—2122no—
213C2×C8—2122no—
214C2×C8—2122no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4—532no—
221C4⋊C4—1064no—
222C4⋊C4—1064no—
223C4⋊C4—1061no—
224C4⋊C4—2121no—
225C4⋊C4—2122no—
226C4⋊C4—2122no—
227C4⋊C4#1171993612yes—
228C42—22no—
229C42—42no—
230C42—1062no—
231C42—1062no—
232C42—1061no—
233C42—1061no—
234C42—2121no—
235C42—2121no—
236C42—2121no—
237C42—2122no—
238C23C2×C22⋊#615711yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—532no—
253C2×C4—532no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4#1313327812yes—
271C2×C4D53⋊M1611yescharacteristic
272C2×C4C2×#1171993611yescharacteristic
273C8—4242no—
274C8—4242no—
275C22#1681159411yescharacteristic
276C22#1681160211yescharacteristic
277C22C2×#1313327811yescharacteristic
278C22#1681161311yescharacteristic
279C22#1681161311yescharacteristic
280C22#1681162111yescharacteristic
281C22#1681162211yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—41no—
286C4—42no—
287C4—81no—
288C4—1062no—
289C4—1062no—
290C4—4241no—
291C4—2121no—
292C4—2122no—
293C4—2121no—
294C2#2028997711yescharacteristic
295C2#2029004711yescharacteristic
296C2#2029004911yescharacteristic
297C2#2029015711yescharacteristic
298C2#2029015911yescharacteristic
299C2#2029019711yescharacteristic
300C2#2029022011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7650
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
173. C53#7650semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895342
#202895332
C4×#131332082
#765053