Finite Group Databasebeta

← #27281836

Subgroups — #27281836

Order 13568.

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

normal subgroups only

This group has 610 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#20279182C211yesmaximal, characteristic
2#20274795C211yesmaximal, characteristic, complemented
3#20273243C212yesmaximal
4#20272567C212yesmaximal, complemented
5C53×#2313C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133200—21no—
8C2×#13133191C2211yescharacteristic
9#16810587C2212yes—
10#16810558—22no—
11#16810524C412yescomplemented
12#16810492C412yescomplemented
13C4×C22⋊Dic53C2211yescharacteristic
14#16810392—22no—
15#16810356—24no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#387—24no—
18C53×#381C2212yes—
19C53×#380—21no—
20C4×C22⋊C4×C53—21no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23C2×#11719917—22no—
24#13133200—44no—
25#13133200—44no—
26#13133191—22no—
27#13133191—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×C22⋊Dic53—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C2×C4×Dic53C2×C412yes—
36C22⋊#88322—22no—
37C22⋊#88322—22no—
38#13133124—21no—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—22no—
48C2×C42×C53—22no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53—22no—
52#95×C53D414yes—
53C24×C53C2×D411yescharacteristic
54C23×D53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—24no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53#4112yes—
69C2×C22⋊D53—42no—
70C2×C22⋊D53—42no—
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C2×D212—22no—
76C2×D212—42no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—82no—
80C22⋊C4×C53—44no—
81C42×C53—82no—
82C22⋊Dic53—24no—
83#11719921—44no—
84#11719921—44no—
85#11719921—42no—
86C4⋊Dic53—22no—
87#11719919—44no—
88#11719919—44no—
89C4×Dic53—42no—
90#11719917—44no—
91C2×#88322—42no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—41no—
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53#12011yescharacteristic
97C23×C53#12311yescharacteristic
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—42no—
101C22×D53—41no—
102C22×D53—42no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—44no—
114C2×C4×C53—44no—
115C2×C4×C53—44no—
116C2×C4×C53—81no—
117C2×C4×C53—81no—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12112yes—
120C22⋊D53—82no—
121C22⋊D53—82no—
122C2×Dic53—24no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128D212—82no—
129#88322—82no—
130#9464—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53#38611yescharacteristic
140C22×C53#41012yes—
141C22×C53#45312yes—
142C2×D53—82no—
143C2×D53—82no—
144C2×D53—81no—
145C2×D53—81no—
146C2×D53—81no—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—42no—
150C4×C53—42no—
151C4×C53—84no—
152Dic53—44no—
153Dic53—81no—
154#2501—531no—
155#2313D5311yescharacteristic, complemented
156#1821—531no—
157#1510—532no—
158#1284—532no—
159C2×C53—22no—
160C2×C53—22no—
161C2×C53—41no—
162C2×C53#188711yescharacteristic
163C2×C53#189911yescharacteristic
164C2×C53#191211yescharacteristic
165D53—161no—
166C2×#123—1061no—
167C2×#123—1061no—
168C2×#121—531no—
169C22×C4⋊C4C2×D5311yescharacteristic
170#407—532no—
171#399—1062no—
172#390—532no—
173#387—24no—
174#384—532no—
175#381C2×D5312yes—
176#380—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—531no—
179#343—1062no—
180#327—1064no—
181C53#946411yescharacteristic, complemented, Sylow
182C22×D4—531no—
183C22×D4—1061no—
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C23×C4C22×D5311yescharacteristic
187C2×M16—1062no—
188#123—2124no—
189#123—2124no—
190#121—1062no—
191#121—1062no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4C4×D5312yes—
199C2×C4⋊C4D21212yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1061no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1062no—
205C2×C42—22no—
206C2×C42—532no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—1061no—
213#95C22⋊D5314yes—
214C24—1061no—
215C24C2×C22⋊D5311yescharacteristic
216C2×D4—1062no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2122no—
220C2×D4—2121no—
221C2×D4—2121no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1062no—
231C22×C4—1061no—
232C22×C4#1171992112yes—
233C22×C4#1171992112yes—
234C22×C4C2×C4×D5311yescharacteristic
235C22×C4C2×D21211yescharacteristic
236C22×C4#1171994512yes—
237C22×C4C2×C22⋊D5312yes—
238M16—2124no—
239C2×C8—2122no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—82no—
243C4⋊C4—1062no—
244C4⋊C4—2124no—
245C4⋊C4—2124no—
246C22⋊C4—44no—
247C22⋊C4—1064no—
248C22⋊C4—2124no—
249C22⋊C4—2124no—
250C22⋊C4—2122no—
251C42—82no—
252C42—2122no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—2121no—
257C23—2121no—
258C23—2122no—
259C23—2121no—
260C23—2122no—
261C23C2×#1171992111yescharacteristic
262C23#1313318611yescharacteristic
263C23#1313320011yescharacteristic
264D4—4242no—
265D4—4242no—
266D4—4242no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—1064no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2122no—
287C2×C4—2122no—
288C2×C4C4×C22⋊D5312yes—
289C2×C4#1313319112yes—
290C8—4242no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—4242no—
300C22—4242no—
301C22—4241no—
302C22—4241no—
303C22—4241no—
304C22#1681050211yescharacteristic
305C22#1681059612yes—
306C22#1681079112yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4—4241no—
313C4—2124no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—8481no—
318C2#2027511311yescharacteristic
319C2#2027517311yescharacteristic
320C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2313C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2313D53
hypocenterC53#9464
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #20274795C2semidirect
4. #20272567C2semidirect
5. C53×#2313C2semidirect
11. #16810524C4semidirect
12. #16810492C4semidirect
155. #2313D53semidirect
181. C53#9464semidirect

Maximal subgroups

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

subgroupindex
#202791822
#202747952
#202732432
#202725672
C53×#23132
#946453