Finite Group Databasebeta

← #27281837

Subgroups — #27281837

Order 13568.

610 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277318C211yesmaximal, characteristic, complemented
2#20274795C211yesmaximal, characteristic, complemented
3#20273244C212yesmaximal
4#20272568C212yesmaximal, complemented
5C53×#2501C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810509—24no—
8#16810486C2212yes—
9#16810483—21no—
10C4×C22⋊Dic53—21no—
11#16810392—22no—
12#16810357—24no—
13C2×#123×C53—21no—
14C2×#123×C53—21no—
15C2×#121×C53C2211yescharacteristic
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#399—22no—
18C53×#390C412yescomplemented
19C53×#384C412yescomplemented
20C4×C22⋊C4×C53C2211yescharacteristic
21C2×#11719917—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—22no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124D414yes—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36#123×C53—44no—
37#123×C53—44no—
38#121×C53—22no—
39#121×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53—21no—
51C2×C42×C53C2×C412yes—
52#95×C53—21no—
53C24×C53—21no—
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D412yes—
75C22×C4×C53#4112yes—
76C22×Dic53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—44no—
81C4⋊C4×C53—44no—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—44no—
84C22⋊C4×C53—44no—
85C22⋊C4×C53—42no—
86C42×C53—42no—
87C22⋊Dic53—44no—
88C4⋊Dic53—82no—
89C4×Dic53—82no—
90#11719917—44no—
91C2×#88322—42no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—42no—
98C23×C53—41no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#12011yescharacteristic
102C23×C53#12311yescharacteristic
103D4×C53—82no—
104D4×C53—82no—
105D4×C53—82no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53C4×D412yes—
125C2×C4×C53#12112yes—
126C2×Dic53—44no—
127C2×Dic53—44no—
128C2×Dic53—44no—
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—82no—
140C22×C53—82no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53#38611yescharacteristic
145C22×C53#41012yes—
146C22×C53#45312yes—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—44no—
150C4×C53—42no—
151C4×C53—42no—
152C4×C53—81no—
153Dic53—84no—
154#2501D5311yescharacteristic
155#2313—531no—
156#1821—531no—
157#1510—532no—
158#1284—532no—
159C2×C53—22no—
160C2×C53—22no—
161C2×C53—41no—
162C2×C53—161no—
163C2×C53#188711yescharacteristic
164C2×C53#189911yescharacteristic
165C2×C53#191211yescharacteristic
166C2×#123—21no—
167C2×#123—21no—
168C2×#121C2×D5311yescharacteristic
169C22×C4⋊C4C2×D5311yescharacteristic
170#407—532no—
171#399—22no—
172#390Dic5312yescomplemented
173#387—1064no—
174#384Dic5312yescomplemented
175#381—532no—
176#380—1061no—
177C4×C22⋊C4—1061no—
178C4×C22⋊C4C2×D5311yescharacteristic
179#343—1062no—
180#327—1064no—
181C53#946411yescharacteristic, complemented, Sylow
182C22×D4—21no—
183C22×D4C2×Dic5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C23×C4C22×D5311yescharacteristic
187C2×M16—1062no—
188#123—44no—
189#123—44no—
190#121—22no—
191#121—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5312yes—
199C2×C4⋊C4C22⋊D5312yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1062no—
205C2×C42—1062no—
206C2×C42C2×Dic5312yes—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#95—21no—
210#95—534no—
211#95—1062no—
212#95—1062no—
213#95—1062no—
214C24—21no—
215C24C2×C22⋊D5311yescharacteristic
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—1064no—
232C22×C4C22⋊Dic5312yes—
233C22×C4C22⋊Dic5312yes—
234C22×C4D4×D5312yes—
235C22×C4#1171994712yes—
236C22×C4C22×Dic5311yescharacteristic
237C22×C4C2×C22⋊D5311yescharacteristic
238M16—2124no—
239C2×C8—2122no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—4242no—
246C22⋊C4—24no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C22⋊C4—2124no—
251C42—42no—
252C42—4242no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23C2×C22⋊Dic5311yescharacteristic
262C23#1313318511yescharacteristic
263C23#1313319811yescharacteristic
264D4—82no—
265D4—82no—
266D4—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4D4×Dic5312yes—
289C2×C4#1313318912yes—
290C8—4242no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#1681050311yescharacteristic
305C22#1681059712yes—
306C22#1681079012yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—81no—
313C4—4244no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2#2027511411yescharacteristic
319C2#2027517411yescharacteristic
320C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2501C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2501D53
hypocenterC53#9464
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
1. #20277318C2semidirect
2. #20274795C2semidirect
4. #20272568C2semidirect
18. C53×#390C4semidirect
19. C53×#384C4semidirect
172. #390Dic53semidirect
174. #384Dic53semidirect
181. C53#9464semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202747952
#202732442
#202725682
C53×#25012
#946453