Finite Group Databasebeta

← #28299752

Subgroups — #28299752

Order 13568.

370 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290429C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5#20283101C211yesmaximal, characteristic
6C2×C4×#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810522C2211yescharacteristic
13#16810521C411yescharacteristic
14#16810519C2211yescharacteristic
15#16810509—21no—
16#16810508—21no—
17#16810505—21no—
18#16810504—21no—
19#16810488C411yescharacteristic
20#16810488C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4×#88324—24no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48#95×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—21no—
55C22×Dic53—21no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×#88324—24no—
61C2×#88324—42no—
62C2×#88324—42no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C42×C53—22no—
70C4⋊Dic53—22no—
71C4⋊Dic53—42no—
72C4⋊Dic53—41no—
73#11719919—42no—
74#11719919—42no—
75C4×Dic53—22no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C4×Dic53—41no—
80C4×Dic53—41no—
81C23×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53#10412yes—
96C2×C4×C53#13112yes—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53#10112yes—
113C2×Dic53#13512yes—
114C2×Dic53#13512yes—
115#88324—44no—
116#88324—82no—
117#7657—531nomaximal, Sylow
118C22×C53#40811yescharacteristic
119C22×C53#41011yescharacteristic
120C22×C53C2×#10111yescharacteristic
121C22×C53#41611yescharacteristic
122C22×C53#41611yescharacteristic
123C22×C53C2×#10411yescharacteristic
124C22×C53#42111yescharacteristic
125C4×C53—22no—
126C4×C53—22no—
127C4×C53—41no—
128C4×C53—81no—
129C4×C53—81no—
130Dic53—22no—
131Dic53—22no—
132Dic53—22no—
133Dic53—41no—
134Dic53—81no—
135Dic53—81no—
136#2864—531no—
137#1839—531no—
138#1744—531no—
139#1271—532no—
140#1271—532no—
141C2×C53#147011yescharacteristic
142C2×C53#148211yescharacteristic
143C2×C53#149411yescharacteristic
144C2×C53#150011yescharacteristic
145C2×C53#152111yescharacteristic
146C2×C53#153111yescharacteristic
147C2×C53#154411yescharacteristic
148C2×C4×C8—1061no—
149#389—531no—
150#389—531no—
151#389—531no—
152#389#615711yescharacteristic
153#387—1061no—
154#387—1061no—
155#387—1061no—
156#387—1061no—
157#382—531no—
158#382—531no—
159C43—531no—
160#336—532no—
161#336—532no—
162#336—1061no—
163#336—1061no—
164#336—1061no—
165C53#765711yescharacteristic, complemented, Sylow
166C22×C8—1062no—
167C22×C8—1062no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4—1061no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4C2×#615711yescharacteristic
175C2×C4⋊C4C2×#615711yescharacteristic
176C2×C42—531no—
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—531no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42C2×#615711yescharacteristic
183C4×C8—1064no—
184#95—21no—
185#95—1061no—
186#95—1061no—
187#95—1061no—
188#95—1061no—
189#95—1061no—
190#95—1061no—
191#95—1061no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—1061no—
199C22×C4—1061no—
200C22×C4C22⋊#615711yescharacteristic
201C22×C4C22⋊#615711yescharacteristic
202C22×C4C22×#615711yescharacteristic
203C2×C8—1064no—
204C2×C8—2122no—
205C2×C8—2122no—
206C2×C8—2122no—
207C2×C8—2122no—
208C2×C8—2122no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—1062no—
213C4⋊C4—2121no—
214C4⋊C4—2122no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C42—22no—
218C42—1062no—
219C42—1062no—
220C42—1062no—
221C42—1062no—
222C42—2121no—
223C42—2121no—
224C23C2×C22⋊#615711yescharacteristic
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—532no—
239C2×C4—532no—
240C2×C4—532no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4#1313328112yes—
257C2×C4#1313330412yes—
258C8—4242no—
259C8—2124no—
260C22#1681159611yescharacteristic
261C22#1681160211yescharacteristic
262C22D53⋊#10111yescharacteristic
263C22#1681161311yescharacteristic
264C22#1681161311yescharacteristic
265C22C2×#1313328111yescharacteristic
266C22#1681162211yescharacteristic
267C4—22no—
268C4—22no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4—1062no—
273C4—1062no—
274C4—1062no—
275C4—4241no—
276C4—4241no—
277C4—2121no—
278C2#2028998711yescharacteristic
279C2#2029002811yescharacteristic
280C2#2029006611yescharacteristic
281C2#2029008611yescharacteristic
282C2#2029015911yescharacteristic
283C2#2029018811yescharacteristic
284C2#2029022111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7657
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
165. C53#7657semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202904292
#202895342
#202895332
#202831012
#765753