Finite Group Databasebeta

← #27281711

Subgroups — #27281711

Order 13568.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#20273339C211yesmaximal, characteristic
3#20273226C212yesmaximal, complemented
4#20272608C212yesmaximal, complemented
5C53×#2372C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810982C414yescomplemented
8#16810977—22no—
9#16810680C2212yes—
10#16810632—21no—
11#16810628—21no—
12C4×#11719917C2211yescharacteristic
13#16810363—22no—
14#16810359—22no—
15#16810359—22no—
16C2×#124×C53C2211yescharacteristic
17C53×#391—21no—
18C53×#388—22no—
19C53×#386—21no—
20C4×C22⋊C4×C53—22no—
21C53×#376C2212yes—
22C2×#11719917—21no—
23#13133217C2×C412yescomplemented
24#13133209—21no—
25#13133196—22no—
26C4×Dic106—22no—
27#13133176—21no—
28C4⋊#88322—22no—
29#13133141—24no—
30#13133138—24no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133128C2×C412yescomplemented
34C4×#88322—21no—
35C22×Q8×C53D411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C23×C4×C53—21no—
38#124×C53—22no—
39#124×C53C2×C412yes—
40#124×C53D412yes—
41C2×C4⋊C4×C53—21no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53—21no—
46C2×C42×C53—22no—
47C2×C42×C53C2311yescharacteristic
48#95×C53—22no—
49#95×C53—22no—
50#95×C53D412yes—
51#95×C53D412yes—
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—22no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—44no—
65C22×C4×C53—81no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D412yes—
68C2×Dic106—22no—
69C4⋊C4×C53—81no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53—44no—
73C22⋊C4×C53—81no—
74C42×C53—44no—
75C42×C53—81no—
76C42×C53C22×C411yescharacteristic
77C42×C53C2×D411yescharacteristic
78C42×C53#4112yes—
79C22⋊Dic53—41no—
80C4⋊Dic53—22no—
81C4⋊Dic53—22no—
82#11719919—42no—
83C4×Dic53—41no—
84#11719917—41no—
85#11719917—41no—
86C2×#88322—22no—
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—
96D4×C53—42no—
97D4×C53—81no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—24no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—84no—
111C2×C4×C53C2×C22⋊C411yescharacteristic
112C2×C4×C53C4×D412yes—
113C2×C4×C53#12112yes—
114C2×C4×C53#12311yescharacteristic
115C2×Dic53—42no—
116Dic106—82no—
117#88322—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—44no—
134C4×C53—42no—
135C4×C53—82no—
136Dic53—82no—
137#3140—531no—
138#2372D5311yescharacteristic
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#533—531no—
148C2×#124C2×D5311yescharacteristic
149#489—534no—
150#488—1062no—
151#432—532no—
152#419—1061no—
153#418—1061no—
154C4×M16—531no—
155#391—21no—
156#388—22no—
157#386—21no—
158C4×C22⋊C4—22no—
159#376C2×D5312yes—
160#329—1062no—
161#328—1062no—
162#328—1062no—
163C53#943911yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D5311yescharacteristic
165C22×D4C22⋊D5311yescharacteristic
166C23×C4—21no—
167C2×M16—1061no—
168C4⋊Q8—532no—
169#125—1061no—
170#124—22no—
171#124C4×D5312yes—
172#124D21212yes—
173C22⋊Q8—1062no—
174C4×Q8—1062no—
175#117—1061no—
176C2×C4⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42C22×D5311yescharacteristic
183C4⋊C8—1062no—
184#103—1064no—
185#102—1064no—
186C22⋊C8—1062no—
187#97—532no—
188#97—1062no—
189C4×C8—1061no—
190#95—22no—
191#95—22no—
192#95C22⋊D5312yes—
193#95C22⋊D5312yes—
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—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—44no—
208C22×C4—81no—
209C22×C4C2×C22⋊D5311yescharacteristic
210C22×C4C2×C22⋊D5312yes—
211M16—2121no—
212M16—2121no—
213C2×C8—1062no—
214C2×C8—2122no—
215C4⋊C4—81no—
216C4⋊C4—1062no—
217C4⋊C4—1062no—
218C4⋊C4—2122no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—44no—
222C22⋊C4—81no—
223C22⋊C4—2121no—
224C42—44no—
225C42—81no—
226C42—2121no—
227C42C2×C4×D5311yescharacteristic
228C42C2×D21211yescharacteristic
229C42#1171994512yes—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#1313318611yescharacteristic
236Q8—42no—
237Q8—81no—
238Q8—4242no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—84no—
254C2×C4—2122no—
255C2×C4C2×#1171992111yescharacteristic
256C2×C4C4×C22⋊D5312yes—
257C2×C4#1313319112yes—
258C2×C4#1313320011yescharacteristic
259C8—4242no—
260C8—2122no—
261C22—22no—
262C22—44no—
263C22—42no—
264C22—81no—
265C22—81no—
266C22—82no—
267C22#1681050211yescharacteristic
268C22#1681080111yescharacteristic
269C22#1681080111yescharacteristic
270C4—22no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—42no—
276C4—82no—
277C4—4242no—
278C2—21no—
279C2—21no—
280C2—82no—
281C2#2027510311yescharacteristic
282C2#2027521612yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2372C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2372D53
hypocenterC53#9439
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #20273226C2semidirect
4. #20272608C2semidirect
7. #16810982C4semidirect
23. #13133217C2×C4semidirect
33. #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
#202733392
#202732262
#202726082
C53×#23722
#943953