Finite Group Databasebeta

← #27278760

Subgroups — #27278760

Order 13568.

510 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 119 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#20283800C211yesmaximal, characteristic
2C4×#13133215C211yesmaximal, characteristic
3#20273111C212yesmaximal
4#20272547C212yesmaximal
5C53×#1461C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8#16810754—24no—
9#16810680—22no—
10#16810676C2212yes—
11#16810530C411yescharacteristic
12#16810528C411yescharacteristic
13#16810518—22no—
14#16810492C411yescharacteristic
15C4×#11719919—22no—
16C42×Dic53C2211yescharacteristic
17#16810355—24no—
18C2×C4×Q8×C53C2211yescharacteristic
19C53×#431C2212yes—
20C53×#324—22no—
21C22×Dic106C2×C411yescharacteristic
22#13133215—22no—
23#13133215—22no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C4⋊#88322—24no—
36C4⋊#88322D414yes—
37C22⋊#88322—22no—
38#13133128—24no—
39C4×#88322—22no—
40#13133124—22no—
41#13133124—22no—
42C22×Q8×C53C2×C411yescharacteristic
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53D412yes—
47C2×C42×C53D412yes—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C4×C8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—21no—
54C2×Q8×C53—21no—
55C2×Q8×C53—22no—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—21no—
68C2×Dic106M1612yes—
69C2×C8×C53—42no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—24no—
72C42×C53—22no—
73C42×C53—22no—
74C42×C53C2×D412yes—
75C42×C53#4112yes—
76C4⋊Dic53—22no—
77#11719919—28no—
78#11719919—44no—
79#11719919—42no—
80C4×Dic53—24no—
81C4×Dic53—21no—
82C4×Dic53—22no—
83C4×Dic53—42no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C4×Dic53M1612yes—
87C2×#88322—24no—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—42no—
90Q8×C53—42no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53C4×D412yes—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12112yes—
101C2×C4×C53#12311yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×C4×C53#13112yes—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115C2×Dic53—41no—
116Dic106—24no—
117Dic106—42no—
118C8×C53—82no—
119#88322—44no—
120#8962—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#38611yescharacteristic
124C22×C53#40711yescharacteristic
125C22×C53#40811yescharacteristic
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—42no—
129C4×C53—42no—
130C4×C53#43512yes—
131C4×C53#43612yes—
132C4×C53#45612yes—
133Dic53—24no—
134Dic53—42no—
135Dic53—42no—
136Dic53—81no—
137#2983—531no—
138C4×C4⋊Q8—531no—
139#1479—532no—
140#1461D5311yescharacteristic
141#1276—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53#187011yescharacteristic
145C2×C53#188411yescharacteristic
146C2×C53#188611yescharacteristic
147C2×C4⋊Q8—531no—
148C2×C4×Q8—531no—
149C2×C4×Q8C2×D5311yescharacteristic
150#446—1064no—
151#432—1062no—
152#431—532no—
153#431C2×D5312yes—
154#391—531no—
155#391—531no—
156#389—1062no—
157#384—531no—
158C4×C4⋊C4—1062no—
159C43—531no—
160Q8⋊C8—1064no—
161#324—22no—
162C53#896211yescharacteristic, complemented, Sylow
163C22×Q8—531no—
164C22×Q8C4×D5311yescharacteristic
165C4⋊Q8—1062no—
166C4⋊Q8—1062no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—1062no—
170C4×Q8—1062no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C4⋊C4C22⋊D5312yes—
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—1061no—
180C2×C42—1061no—
181C2×C42—1061no—
182C2×C42D21212yes—
183C2×C42C22×D5311yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—534no—
186C4⋊C8—1064no—
187C22⋊C8—22no—
188C22⋊C8—1062no—
189#97—1064no—
190C4×C8—22no—
191C4×C8—1062no—
192#95—1062no—
193#95—1062no—
194C2×Q8—22no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—22no—
198C2×Q8—532no—
199C2×Q8—1061no—
200C2×Q8—1062no—
201C2×Q8—1062no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4#1171992112yes—
207C22×C4#1171992112yes—
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4C2×D21211yescharacteristic
210C22×C4C2×C22⋊D5311yescharacteristic
211C2×C8—42no—
212C2×C8—1064no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—1068no—
216C4⋊C4—1062no—
217C4⋊C4—2124no—
218C4⋊C4—2122no—
219C42—22no—
220C42—22no—
221C42—532no—
222C42—1064no—
223C42—1061no—
224C42—1062no—
225C42—2122no—
226C42—2121no—
227C42—2121no—
228C42D4×D5312yes—
229C42#1171994712yes—
230C23C2×#1171992111yescharacteristic
231Q8—42no—
232Q8—42no—
233Q8—1064no—
234Q8—2122no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—2122no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4#1313317712yes—
254C2×C4#1313318711yescharacteristic
255C2×C4#1313318912yes—
256C2×C4#1313320111yescharacteristic
257C2×C4M16×D5311yescharacteristic
258C2×C4M16×D5311yescharacteristic
259C2×C4#1313322312yes—
260C8—82no—
261C8—2124no—
262C22—21no—
263C22—21no—
264C22#1681050011yescharacteristic
265C22#1681058611yescharacteristic
266C22#1681058911yescharacteristic
267C4—22no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—1064no—
272C4—4241no—
273C4—2122no—
274C4—2122no—
275C4#1681069712yes—
276C4#1681069812yes—
277C4#1681080812yes—
278C2—21no—
279C2—21no—
280C2#2027501011yescharacteristic
281C2#2027509611yescharacteristic
282C2#2027510611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1461C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1461D53
hypocenterC53#8962
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
162. C53#8962semidirect

Maximal subgroups

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

subgroupindex
#202838002
C4×#131332152
#202731112
#202725472
C53×#14612
#896253