Finite Group Databasebeta

← #27293749

Subgroups — #27293749

Order 13568.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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#20279902C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272660C212yesmaximal
4#20272658C212yesmaximal
5C53×#1467C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic
7#16810676—21no—
8#16810556—21no—
9#16810556—21no—
10#16810530C411yescharacteristic
11#16810530C411yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810492C2211yescharacteristic
15#16810488C2211yescharacteristic
16C4×C4⋊Dic53C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C22×Q8×C53C2×C411yescharacteristic
45C4×Q8×C53—41no—
46C4×Q8×C53—41no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53C2311yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55#97×C53—41no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—41no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C2×C8×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C4⋊Dic53—42no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83#11719919—42no—
84#11719919—42no—
85#11719919—42no—
86#11719919—41no—
87#11719919—41no—
88C4×Dic53—22no—
89C4×Dic53—22no—
90C2×#88322—42no—
91C23×C53C2×C22⋊C411yescharacteristic
92Q8×C53—44no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#10112yes—
110C2×C4×C53#13112yes—
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C8×C53—82no—
123#88322—82no—
124#11571—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#40811yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10111yescharacteristic
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—41no—
138Dic53—42no—
139Dic53—81no—
140Dic53—81no—
141Dic53—81no—
142#2565—531no—
143#1467D5311yescharacteristic
144#1461—531no—
145#1325—532no—
146#1324—532no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53#147211yescharacteristic
150C2×C53#212611yescharacteristic
151C2×C53#213311yescharacteristic
152C2×#125—531no—
153C2×C4×Q8C2×D5311yescharacteristic
154#432—21no—
155#431—1061no—
156#398—1061no—
157#398—1061no—
158#391—531no—
159#391—531no—
160#387—1061no—
161#387—1061no—
162#384—531no—
163#382—531no—
164C4×C4⋊C4—531no—
165#324—21no—
166#324—532no—
167#324—1061no—
168#324C2×D5312yes—
169C53#1157111yescharacteristic, complemented, Sylow
170C22×Q8C4×D5311yescharacteristic
171#125—2121no—
172#125—2121no—
173C4×Q8—41no—
174C4×Q8—41no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4D21211yescharacteristic
183C2×C4⋊C4C22⋊D5311yescharacteristic
184C2×C4⋊C4C22×D5311yescharacteristic
185C2×C42—531no—
186C2×C42—531no—
187C2×C42C4×D5311yescharacteristic
188C2×C42D21211yescharacteristic
189C2×C42C22⋊D5311yescharacteristic
190C4⋊C8—41no—
191C4⋊C8—2121no—
192C22⋊C8—22no—
193C22⋊C8—1062no—
194#97—41no—
195C4×C8—2121no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4#1171992111yescharacteristic
211C22×C4#1171992111yescharacteristic
212C22×C4#1171992111yescharacteristic
213C22×C4#1171992111yescharacteristic
214C22×C4C2×C4×D5311yescharacteristic
215C22×C4C2×D21211yescharacteristic
216C22×C4C2×C22⋊D5311yescharacteristic
217C2×C8—42no—
218C2×C8—2122no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C4⋊C4—2122no—
227C4⋊C4—2122no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1062no—
237C42—1062no—
238C23C2×#1171992111yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1062no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2122no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4#1313313111yescharacteristic
266C2×C4#1313313111yescharacteristic
267C2×C4#1313313512yes—
268C2×C4#1313322512yes—
269C8—82no—
270C8—4242no—
271C22—21no—
272C22—21no—
273C22#1681059111yescharacteristic
274C22C2×#1313313111yescharacteristic
275C22C2×#1313313511yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—4241no—
285C4—4241no—
286C4—4241no—
287C4—2122no—
288C2—21no—
289C2—21no—
290C2#2027308011yescharacteristic
291C2#2027637811yescharacteristic
292C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11571
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
169. C53#11571semidirect

Maximal subgroups

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

subgroupindex
#202799022
#202730392
#202726602
#202726582
C53×#14672
#1157153