Finite Group Databasebeta

← #27276451

Subgroups — #27276451

Order 13568.

692 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 131 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#20275548C211yesmaximal, characteristic
2#20273445C212yesmaximal, complemented
3#20273350C211yesmaximal, characteristic
4#20273227C212yesmaximal, complemented
5C53×#3550C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810647C2211yescharacteristic
8C2×#13133144—22no—
9#16810469C2211yescharacteristic
10#16810372—22no—
11#16810369—22no—
12#16810363—22no—
13#16810360—24no—
14C53×#563—22no—
15C53×#559C412yescomplemented
16Q82×C53C412yescomplemented
17C53×#556—22no—
18D4×Q8×C53—22no—
19C53×#542—24no—
20C53×#536—22no—
21C53×#533C2212yes—
22C53×#532C2211yescharacteristic
23C2×#11719917—21no—
24C2×C4×Dic53—21no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133144—44no—
28C22⋊#88322—22no—
29#13133128—22no—
30#13133124—22no—
31C2×#41×C53D412yes—
32C22×Q8×C53—22no—
33C4⋊Q8×C53—24no—
34C4⋊Q8×C53—24no—
35C4⋊Q8×C53C2×C411yescharacteristic
36C4⋊Q8×C53D412yes—
37C4⋊Q8×C53D412yes—
38#127×C53C2×C411yescharacteristic
39#126×C53—28no—
40#125×C53—22no—
41#124×C53—24no—
42#124×C53—24no—
43#124×C53D412yes—
44#123×C53—24no—
45#123×C53—24no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53—28no—
48C22⋊Q8×C53—22no—
49C22⋊Q8×C53D414yes—
50#121×C53—22no—
51C4×Q8×C53—22no—
52C4×Q8×C53—24no—
53C4×Q8×C53C2×C414yes—
54C4×D4×C53—24no—
55C4×D4×C53—22no—
56#117×C53—22no—
57#117×C53—22no—
58C2×C42×C53C2311yescharacteristic
59#41×C53—42no—
60#41×C53—42no—
61C2×Q8×C53—24no—
62C2×Q8×C53—22no—
63C2×Q8×C53—82no—
64C2×Q8×C53—82no—
65C2×Q8×C53C22⋊C414yes—
66C2×Q8×C53C22⋊C412yes—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53C22⋊C412yes—
70C22×C4×C53—24no—
71C22×C4×C53—22no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D412yes—
74C22×Dic53—41no—
75C4⋊C4×C53—28no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53C22⋊C414yes—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—28no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C42×C53—24no—
88C42×C53—24no—
89C42×C53C22×C411yescharacteristic
90C42×C53C2×D412yes—
91C42×C53C2×D411yescharacteristic
92C4×Dic53—22no—
93C4×Dic53—41no—
94#11719917—42no—
95C2×#88322—42no—
96C2×#88322—41no—
97C23×C53—22no—
98C23×C53#12011yescharacteristic
99Q8×C53—44no—
100Q8×C53—42no—
101Q8×C53—42no—
102Q8×C53—42no—
103Q8×C53—84no—
104D4×C53—42no—
105D4×C53—44no—
106D4×C53—82no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—84no—
116C2×C4×C53C2×C22⋊C412yes—
117C2×C4×C53C2×C22⋊C411yescharacteristic
118C2×C4×C53#12012yes—
119C2×C4×C53#12011yescharacteristic
120C2×Dic53—42no—
121C2×Dic53—82no—
122#88322—82no—
123#88322—81no—
124#8492—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—22no—
127C22×C53—42no—
128C22×C53—44no—
129C22×C53#37911yescharacteristic
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134C4×C53—44no—
135C4×C53—44no—
136C4×C53#45311yescharacteristic
137C4×C53#45511yescharacteristic
138Dic53—82no—
139#3550D5311yescharacteristic
140#1963—531no—
141#1559—532no—
142#1536—531no—
143#1506—532no—
144C2×C53—41no—
145C2×C53—82no—
146C2×C53#179211yescharacteristic
147C2×C53#179612yes—
148#563—22no—
149#559Dic5312yescomplemented
150Q82Dic5312yescomplemented
151#556—22no—
152D4×Q8—22no—
153#542—24no—
154#536—22no—
155#533C2×D5312yes—
156#532C2×D5311yescharacteristic
157#432—532no—
158#423—531no—
159C2×#104—1062no—
160#376—531no—
161#333—1062no—
162#332—1062no—
163#329—1062no—
164#328—1064no—
165C53#849211yescharacteristic, complemented, Sylow
166C2×#41C22⋊D5312yes—
167C22×Q8—22no—
168C2×M16—1061no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8C2×Dic5311yescharacteristic
172C4⋊Q8C22⋊D5312yes—
173C4⋊Q8C22⋊D5312yes—
174#127C2×Dic5311yescharacteristic
175#126—28no—
176#125—22no—
177#124—24no—
178#124—24no—
179#124C22⋊D5312yes—
180#123—24no—
181#123—24no—
182C22⋊Q8—24no—
183C22⋊Q8—28no—
184C22⋊Q8—22no—
185C22⋊Q8C22⋊D5314yes—
186#121—22no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8C2×Dic5314yes—
190C4×D4—24no—
191C4×D4—22no—
192#117—22no—
193#117—22no—
194C2×C42—1061no—
195C2×C42C22×D5311yescharacteristic
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198#104—2124no—
199C22⋊C8—1062no—
200#97—1062no—
201#95—1062no—
202#41—42no—
203#41—42no—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8C22⋊Dic5314yes—
209C2×Q8C22⋊Dic5312yes—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4C22⋊Dic5312yes—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—2121no—
216C22×C4C2×C22⋊D5311yescharacteristic
217C22×C4C2×C22⋊D5312yes—
218M16—2122no—
219C2×C8—2122no—
220C2×C8—2121no—
221C4⋊C4—28no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic5314yes—
229C22⋊C4—24no—
230C22⋊C4—28no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—24no—
235C42—1062no—
236C42—2121no—
237C42C22×Dic5311yescharacteristic
238C42C2×C22⋊D5311yescharacteristic
239C42C2×C22⋊D5312yes—
240C23—22no—
241C23#1313318611yescharacteristic
242Q8—44no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8—84no—
247D4—42no—
248D4—44no—
249D4—82no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—4242no—
260C2×C4—2122no—
261C2×C4C2×C22⋊Dic5312yes—
262C2×C4C2×C22⋊Dic5311yescharacteristic
263C2×C4#1313318612yes—
264C2×C4#1313318611yescharacteristic
265C8—4242no—
266C8—4241no—
267C22—22no—
268C22—21no—
269C22—44no—
270C22—42no—
271C22#1681047911yescharacteristic
272C4—22no—
273C4—22no—
274C4—21no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4—4242no—
279C4#1681079111yescharacteristic
280C4#1681080111yescharacteristic
281C2—41no—
282C2—82no—
283C2#2027466911yescharacteristic
284C2#2027469012yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3550C2
socleC22×C53#379
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3550D53
hypocenterC53#8492
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #20273445C2semidirect
4. #20273227C2semidirect
15. C53×#559C4semidirect
16. Q82×C53C4semidirect
149. #559Dic53semidirect
150. Q82Dic53semidirect
165. C53#8492semidirect

Maximal subgroups

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

subgroupindex
#202755482
#202734452
#202733502
#202732272
C53×#35502
#849253