Finite Group Databasebeta

← #27281709

Subgroups — #27281709

Order 13568.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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#20285101C211yesmaximal, characteristic
2#20277827C211yesmaximal, characteristic
3#20273228C212yesmaximal, complemented
4#20272607C212yesmaximal, complemented
5C53×#1533C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810982C414yescomplemented
8#16810977—22no—
9#16810632—21no—
10#16810628—21no—
11C4×#11719917C2211yescharacteristic
12#16810530—21no—
13#16810516—22no—
14#16810503—21no—
15C22⋊C4×Dic53—22no—
16#16810469C2212yes—
17#16810361—22no—
18#16810361—22no—
19C2×#124×C53C2211yescharacteristic
20C53×#432C2212yes—
21C53×#329—22no—
22C23×Dic53—21no—
23C2×#11719917—21no—
24#13133217C2×C412yescomplemented
25#13133209—21no—
26#13133196—22no—
27C4×Dic106—22no—
28#13133176—21no—
29C2×C4⋊Dic53—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—22no—
34#13133141—24no—
35#13133138—24no—
36#13133128C2×C412yescomplemented
37C4×#88322—21no—
38#13133124—22no—
39#13133124—22no—
40#13133124D412yes—
41#13133124D412yes—
42C22×Q8×C53D411yescharacteristic
43C22×D4×C53D411yescharacteristic
44#124×C53—22no—
45#124×C53C2×C412yes—
46#124×C53D412yes—
47C2×C22⋊C4×C53—22no—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51#97×C53—22no—
52C24×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53C22⋊C412yes—
57C2×D4×C53—21no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53C22⋊C412yes—
61C22×C4×C53—21no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D412yes—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—44no—
67C22×Dic53—81no—
68C2×Dic106—22no—
69C2×C8×C53—42no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C42×C53C22×C411yescharacteristic
73C42×C53C2×D411yescharacteristic
74C42×C53#4112yes—
75C22⋊Dic53—44no—
76C22⋊Dic53—41no—
77C22⋊Dic53—81no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80C4⋊Dic53—81no—
81#11719919—42no—
82C4×Dic53—44no—
83C4×Dic53—41no—
84C4×Dic53—81no—
85#11719917—41no—
86#11719917—41no—
87C2×#88322—22no—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#12011yescharacteristic
94Q8×C53—42no—
95Q8×C53—81no—
96D4×C53—42no—
97D4×C53—81no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53C4×D412yes—
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—
114C2×Dic53—42no—
115C2×Dic53—84no—
116Dic106—82no—
117C8×C53—82no—
118#88322—42no—
119#9439—531nomaximal, Sylow
120C22×C53—22no—
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53—82no—
126C22×C53#38611yescharacteristic
127C22×C53#45511yescharacteristic
128C22×C53#45511yescharacteristic
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134Dic53—44no—
135Dic53—82no—
136Dic53—82no—
137#3140—531no—
138#2372—531no—
139#1533D5311yescharacteristic
140#1506—532no—
141#1300—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—82no—
145C2×C53#188511yescharacteristic
146C2×C53#190612yes—
147#533—531no—
148C2×#124C2×D5311yescharacteristic
149#489—534no—
150#488—1062no—
151#432C2×D5312yes—
152#419—1061no—
153#418—1061no—
154C4×M16—531no—
155#391—1061no—
156#388—1062no—
157#386—1061no—
158C4×C22⋊C4—1062no—
159#376—532no—
160#329—22no—
161#328—1062no—
162#328—1062no—
163C53#943911yescharacteristic, complemented, Sylow
164C22×Q8D21211yescharacteristic
165C22×D4D21211yescharacteristic
166C23×C4—1061no—
167C2×M16—1061no—
168C4⋊Q8—532no—
169#125—1061no—
170#124—22no—
171#124C4×D5312yes—
172#124C22⋊D5312yes—
173C22⋊Q8—1062no—
174C4×Q8—1062no—
175#117—1061no—
176C2×C4⋊C4—1061no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4—1062no—
180C2×C42—1061no—
181C2×C42—1062no—
182C2×C42C22×D5311yescharacteristic
183C4⋊C8—22no—
184#103—1064no—
185#102—1064no—
186C22⋊C8—22no—
187#97—22no—
188#97—532no—
189C4×C8—1061no—
190#95—532no—
191#95—532no—
192#95—1062no—
193#95—1062no—
194C24—21no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—1062no—
199C2×Q8#1171992112yes—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4#1171992112yes—
204C22×C4—21no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—4241no—
208C22×C4—2124no—
209C22×C4C2×D21211yescharacteristic
210C22×C4D4×D5312yes—
211M16—2121no—
212M16—2121no—
213C2×C8—42no—
214C2×C8—1062no—
215C4⋊C4—1062no—
216C4⋊C4—1062no—
217C4⋊C4—4241no—
218C4⋊C4—2122no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—4241no—
222C22⋊C4—2124no—
223C22⋊C4—2121no—
224C42—4241no—
225C42—2124no—
226C42—2121no—
227C42C2×C4×D5311yescharacteristic
228C42#1171994712yes—
229C42C2×C22⋊D5311yescharacteristic
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#1313318711yescharacteristic
236Q8—42no—
237Q8—81no—
238Q8—4242no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—1064no—
248C2×C4—4244no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4C2×#1171992111yescharacteristic
256C2×C4#1313317712yes—
257C2×C4#1313318912yes—
258C2×C4#1313320111yescharacteristic
259C8—82no—
260C8—2122no—
261C22—22no—
262C22—44no—
263C22—42no—
264C22—81no—
265C22—82no—
266C22—81no—
267C22#1681050011yescharacteristic
268C22#1681080311yescharacteristic
269C22#1681080311yescharacteristic
270C4—22no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—4242no—
276C4—4242no—
277C4—2124no—
278C2—21no—
279C2—21no—
280C2—82no—
281C2#2027510111yescharacteristic
282C2#2027521412yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1533C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1533D53
hypocenterC53#9439
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
3. #20273228C2semidirect
4. #20272607C2semidirect
7. #16810982C4semidirect
24. #13133217C2×C4semidirect
36. #13133128C2×C4semidirect
163. C53#9439semidirect

Maximal subgroups

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

subgroupindex
#202851012
#202778272
#202732282
#202726072
C53×#15332
#943953