Finite Group Databasebeta

← #27281766

Subgroups — #27281766

Order 13568.

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

normal subgroups only

This group has 424 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#20279902C211yesmaximal, characteristic
2#20273341C212yesmaximal
3#20273059C211yesmaximal, characteristic
4#20272570C212yesmaximal
5C53×#2983C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7#16810680C2212yes—
8#16810556—21no—
9#16810556—21no—
10#16810530C411yescharacteristic
11#16810530C411yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810492C411yescharacteristic
15#16810488C411yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17#16810364—24no—
18#16810349—21no—
19#16810349—21no—
20C2×C4×Q8×C53C2211yescharacteristic
21C53×#446—24no—
22C53×#432—21no—
23C53×#432C2212yes—
24C53×#431—21no—
25#13133209—22no—
26#13133209—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C4⋊#88322—22no—
37C22⋊#88322—22no—
38#13133128—22no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C4×Q8×C53—22no—
48C4×Q8×C53—22no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55C4⋊C8×C53—22no—
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58C22⋊C8×C53—22no—
59#97×C53—22no—
60#97×C53D414yes—
61C4×C8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×C8×C53—24no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53C2×D412yes—
85C42×C53#4112yes—
86C4⋊Dic53—22no—
87C4⋊Dic53—22no—
88C4⋊Dic53—42no—
89#11719919—42no—
90#11719919—42no—
91#11719919—41no—
92#11719919—42no—
93#11719919—41no—
94C4×Dic53—41no—
95C4×Dic53—41no—
96C2×#88322—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98Q8×C53—44no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53#10112yes—
113C2×C4×C53C4×D412yes—
114C2×C4×C53#12011yescharacteristic
115C2×C4×C53#12112yes—
116C2×C4×C53#12311yescharacteristic
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—42no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C8×C53—44no—
129#88322—82no—
130#9450—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#38611yescharacteristic
134C22×C53C2×#10111yescharacteristic
135C22×C53#41311yescharacteristic
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144Dic53—42no—
145Dic53—81no—
146Dic53—81no—
147Dic53—81no—
148#2983D5311yescharacteristic
149#2565—531no—
150#1534—532no—
151#1467—531no—
152#1285—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53#188411yescharacteristic
156C2×C53#190111yescharacteristic
157C2×C53#191211yescharacteristic
158C2×#125—531no—
159C2×C4×Q8C2×D5311yescharacteristic
160#446—24no—
161#432—21no—
162#432—532no—
163#432C2×D5312yes—
164#431—21no—
165#398—1061no—
166#398—1061no—
167#391—531no—
168#391—531no—
169#387—1061no—
170#387—1061no—
171#384—531no—
172#382—531no—
173C4×C4⋊C4—531no—
174#330—1064no—
175#324—1061no—
176#324—1061no—
177C53#945011yescharacteristic, complemented, Sylow
178C22×Q8C4×D5311yescharacteristic
179#125—1062no—
180#125—1062no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C4×D5311yescharacteristic
191C2×C4⋊C4D21211yescharacteristic
192C2×C4⋊C4D21211yescharacteristic
193C2×C42—531no—
194C2×C42—531no—
195C2×C42C22⋊D5311yescharacteristic
196C2×C42C22⋊D5311yescharacteristic
197C2×C42C22×D5311yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—1062no—
202C22⋊C8—22no—
203C22⋊C8—1062no—
204#97—22no—
205#97—1062no—
206#97C22⋊D5314yes—
207C4×C8—22no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4#1171992111yescharacteristic
223C22×C4#1171992111yescharacteristic
224C22×C4#1171992111yescharacteristic
225C22×C4#1171992111yescharacteristic
226C22×C4C2×C4×D5311yescharacteristic
227C22×C4C2×D21211yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229C2×C8—24no—
230C2×C8—2122no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—1062no—
235C4⋊C4—1062no—
236C4⋊C4—2122no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C4⋊C4—2121no—
240C4⋊C4—2122no—
241C4⋊C4—2121no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—2121no—
247C42—2121no—
248C42#1171994512yes—
249C42C2×C22⋊D5312yes—
250C23C2×#1171992111yescharacteristic
251Q8—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—1062no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2122no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4#1313313512yes—
277C2×C4C4×C22⋊D5312yes—
278C2×C4#1313318611yescharacteristic
279C2×C4#1313319112yes—
280C2×C4#1313320011yescharacteristic
281C8—44no—
282C8—4242no—
283C22—21no—
284C22—21no—
285C22#1681050211yescharacteristic
286C22C2×#1313313511yescharacteristic
287C22#1681060511yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—4241no—
297C4—4241no—
298C4—4241no—
299C4—2122no—
300C2—21no—
301C2—21no—
302C2#2027509811yescharacteristic
303C2#2027518311yescharacteristic
304C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9450
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
177. C53#9450semidirect

Maximal subgroups

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

subgroupindex
#202799022
#202733412
#202730592
#202725702
C53×#29832
#945053