Finite Group Databasebeta

← #27281712

Subgroups — #27281712

Order 13568.

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

normal subgroups only

This group has 490 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#20277827C211yesmaximal, characteristic
2#20273339C211yesmaximal, characteristic
3#20273227C212yesmaximal, complemented
4#20272609C212yesmaximal, complemented
5C53×#3140C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810530—21no—
8#16810516—22no—
9#16810503—21no—
10C22⋊C4×Dic53—22no—
11#16810469C2212yes—
12#16810363—22no—
13#16810360—22no—
14#16810360—22no—
15C53×#533C2211yescharacteristic
16C2×#124×C53C2211yescharacteristic
17C53×#489C414yescomplemented
18C53×#488—22no—
19C53×#419—21no—
20C53×#418—21no—
21C4×M16×C53C2211yescharacteristic
22C23×Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—21no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30#13133128—22no—
31#13133124—22no—
32#13133124—22no—
33#13133124D412yes—
34#13133124D412yes—
35C22×Q8×C53D411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C2×M16×C53—21no—
38C4⋊Q8×C53C2×C412yescomplemented
39#125×C53—21no—
40#124×C53—22no—
41#124×C53C2×C412yes—
42#124×C53D412yes—
43C22⋊Q8×C53—22no—
44C4×Q8×C53—22no—
45#117×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53C2311yescharacteristic
48#103×C53—24no—
49#102×C53—24no—
50#97×C53C2×C412yescomplemented
51C4×C8×C53—21no—
52C24×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53C22⋊C412yes—
58C2×D4×C53—21no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53C22⋊C412yes—
62C22×C4×C53—21no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—44no—
68C22×Dic53—81no—
69M16×C53—41no—
70M16×C53—41no—
71C2×C8×C53—22no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—42no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—24no—
77C22⋊C4×C53—41no—
78C42×C53—41no—
79C42×C53C22×C411yescharacteristic
80C42×C53C2×D411yescharacteristic
81C42×C53#4112yes—
82C22⋊Dic53—44no—
83C22⋊Dic53—81no—
84C4⋊Dic53—81no—
85C4×Dic53—44no—
86C4×Dic53—81no—
87C2×#88322—42no—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#12011yescharacteristic
94Q8×C53—42no—
95Q8×C53—81no—
96Q8×C53—82no—
97D4×C53—42no—
98D4×C53—81no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53C4×D412yes—
108C2×C4×C53#12112yes—
109C2×C4×C53#12311yescharacteristic
110C2×Dic53—24no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—84no—
117C8×C53—42no—
118#88322—82no—
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—
134C4×C53—82no—
135Dic53—44no—
136Dic53—82no—
137#3140D5311yescharacteristic
138#2372—531no—
139#1533—531no—
140#1506—532no—
141#1300—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—82no—
145C2×C53#188511yescharacteristic
146C2×C53#190612yes—
147#533C2×D5311yescharacteristic
148C2×#124C2×D5311yescharacteristic
149#489Dic5314yescomplemented
150#488—22no—
151#432—532no—
152#419—21no—
153#418—21no—
154C4×M16C2×D5311yescharacteristic
155#391—1061no—
156#388—1062no—
157#386—1061no—
158C4×C22⋊C4—1062no—
159#376—532no—
160#329—1062no—
161#328—1062no—
162#328—1062no—
163C53#943911yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D5311yescharacteristic
165C22×D4C22⋊D5311yescharacteristic
166C23×C4—1061no—
167C2×M16—21no—
168C4⋊Q8C2×Dic5312yescomplemented
169#125—21no—
170#124—22no—
171#124C2×Dic5312yes—
172#124C22⋊D5312yes—
173C22⋊Q8—22no—
174C4×Q8—22no—
175#117—21no—
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—1062no—
184#103—24no—
185#102—24no—
186C22⋊C8—1062no—
187#97—1062no—
188#97C2×Dic5312yescomplemented
189C4×C8—21no—
190#95—532no—
191#95—532no—
192#95—1062no—
193#95—1062no—
194C24—21no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C22⋊Dic5312yes—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C22⋊Dic5312yes—
204C22×C4—21no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—4241no—
208C22×C4—2124no—
209C22×C4D4×D5312yes—
210C22×C4C2×C22⋊D5311yescharacteristic
211M16—41no—
212M16—41no—
213C2×C8—22no—
214C2×C8—2122no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—4241no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—41no—
222C22⋊C4—4241no—
223C22⋊C4—2124no—
224C42—41no—
225C42—4241no—
226C42—2124no—
227C42#1171994712yes—
228C42C22×Dic5311yescharacteristic
229C42C2×C22⋊D5311yescharacteristic
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#1313318511yescharacteristic
236Q8—42no—
237Q8—81no—
238Q8—82no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—1064no—
249C2×C4—4244no—
250C2×C4—2122no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4C2×C22⋊Dic5311yescharacteristic
256C2×C4D4×Dic5312yes—
257C2×C4#1313318912yes—
258C2×C4#1313319811yescharacteristic
259C8—42no—
260C8—4242no—
261C22—22no—
262C22—44no—
263C22—42no—
264C22—81no—
265C22—82no—
266C22—81no—
267C22#1681050311yescharacteristic
268C22#1681080011yescharacteristic
269C22#1681080011yescharacteristic
270C4—22no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—4242no—
277C4—2124no—
278C2—21no—
279C2—21no—
280C2—82no—
281C2#2027510411yescharacteristic
282C2#2027521812yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3140C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#3140D53
hypocenterC53#9439
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3140

Complements

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

NH ≅ G/Ntype
3. #20273227C2semidirect
4. #20272609C2semidirect
17. C53×#489C4semidirect
38. C4⋊Q8×C53C2×C4semidirect
50. #97×C53C2×C4semidirect
149. #489Dic53semidirect
163. C53#9439semidirect
168. C4⋊Q8C2×Dic53semidirect
188. #97C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778272
#202733392
#202732272
#202726092
C53×#31402
#943953