Finite Group Databasebeta

← #27281774

Subgroups — #27281774

Order 13568.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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
2#20279700C211yesmaximal, characteristic
3#20273440C212yesmaximal
4#20272570C212yesmaximal
5C53×#1467C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133215C412yes—
8#16810754—24no—
9#16810680—21no—
10#16810680C2212yes—
11#16810676—21no—
12#16810553—21no—
13#16810553—21no—
14#16810528C412yes—
15#16810521—21no—
16#16810521—21no—
17C4×C4⋊Dic53C2211yescharacteristic
18#16810371—24no—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432C2212yes—
21C53×#324—21no—
22C53×#324—21no—
23C22×Dic106C2×C412yes—
24#13133217—22no—
25#13133217—22no—
26#13133215—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×C4×Dic53C2×C412yes—
33C4⋊#88322—22no—
34C4⋊#88322—22no—
35C4⋊#88322D414yes—
36C22⋊#88322—22no—
37#13133128—22no—
38#13133128—22no—
39C4×#88322—22no—
40#13133124—22no—
41#13133124—22no—
42#13133124—21no—
43C22×Q8×C53C2×C411yescharacteristic
44C4×Q8×C53—22no—
45C4×Q8×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54#97×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—22no—
57C2×Q8×C53—41no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C2×Dic106—24no—
68C2×Dic106—42no—
69C2×Dic106—42no—
70C2×C8×C53—42no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53C2×D412yes—
79C42×C53#4112yes—
80C4⋊Dic53—24no—
81C4⋊Dic53—41no—
82C4⋊Dic53—42no—
83C4⋊Dic53—41no—
84#11719919—44no—
85#11719919—44no—
86C4×Dic53—42no—
87C2×#88322—24no—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—44no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53#10012yes—
104C2×C4×C53C4×D412yes—
105C2×C4×C53#12011yescharacteristic
106C2×C4×C53#12112yes—
107C2×C4×C53#12311yescharacteristic
108C2×Dic53—24no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114Dic106—84no—
115C8×C53—82no—
116#88322—44no—
117#9452—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#38611yescharacteristic
121C22×C53C2×#10011yescharacteristic
122C22×C53#41311yescharacteristic
123C4×C53—21no—
124C4×C53—22no—
125C4×C53—22no—
126C4×C53—22no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53#45612yes—
131Dic53—44no—
132Dic53—82no—
133#2983—531no—
134#2546—531no—
135#1558—532no—
136#1467D5311yescharacteristic
137#1285—532no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53#188611yescharacteristic
141C2×C53#190011yescharacteristic
142C2×C53#190911yescharacteristic
143C2×C4⋊Q8—532no—
144C2×C4⋊Q8—531no—
145C2×C4×Q8C2×D5311yescharacteristic
146#446—1064no—
147#432—532no—
148#432—1061no—
149#432C2×D5312yes—
150#431—1061no—
151#398—1061no—
152#398—1061no—
153#391—532no—
154#389—1061no—
155#389—1061no—
156C4×C4⋊C4—531no—
157#333—1064no—
158#324—21no—
159#324—21no—
160C53#945211yescharacteristic, complemented, Sylow
161C22×Q8—532no—
162C22×Q8C4×D5311yescharacteristic
163C4⋊Q8—1062no—
164C4⋊Q8—1062no—
165C4⋊Q8—2124no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4C4×D5311yescharacteristic
174C2×C4⋊C4C22⋊D5311yescharacteristic
175C2×C4⋊C4C22⋊D5311yescharacteristic
176C2×C42—532no—
177C2×C42D21211yescharacteristic
178C2×C42D21211yescharacteristic
179C2×C42C22×D5311yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—534no—
182C4⋊C8—1062no—
183C4⋊C8—1062no—
184C22⋊C8—22no—
185C22⋊C8—1062no—
186#97—22no—
187#97—1062no—
188#97—1062no—
189C4×C8—1062no—
190#95—1061no—
191#95—1062no—
192#95—1062no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—1064no—
197C2×Q8—2122no—
198C2×Q8—2122no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4#1171992111yescharacteristic
202C22×C4#1171992111yescharacteristic
203C22×C4#1171992111yescharacteristic
204C22×C4#1171992111yescharacteristic
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4C2×D21211yescharacteristic
207C22×C4C2×C22⋊D5311yescharacteristic
208C2×C8—42no—
209C2×C8—1064no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—1064no—
214C4⋊C4—2124no—
215C4⋊C4—2124no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C4⋊C4—2121no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—2122no—
224C42D4×D5312yes—
225C42#1171994712yes—
226C23C2×#1171992111yescharacteristic
227Q8—44no—
228Q8—4244no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—1064no—
243C2×C4—2122no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4#1313313312yes—
249C2×C4#1313317712yes—
250C2×C4#1313318711yescharacteristic
251C2×C4#1313318912yes—
252C2×C4#1313320111yescharacteristic
253C8—82no—
254C8—2124no—
255C22—21no—
256C22—21no—
257C22#1681050011yescharacteristic
258C22C2×#1313313311yescharacteristic
259C22#1681060511yescharacteristic
260C4—22no—
261C4—21no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—4242no—
268C4—2124no—
269C4#1681080812yes—
270C2—21no—
271C2—21no—
272C2#2027510611yescharacteristic
273C2#2027517611yescharacteristic
274C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1467C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1467D53
hypocenterC53#9452
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
160. C53#9452semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202797002
#202734402
#202725702
C53×#14672
#945253