Finite Group Databasebeta

← #27276463

Subgroups — #27276463

Order 13568.

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

normal subgroups only

This group has 732 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#20275697C211yesmaximal, characteristic, complemented
2#20273437C212yesmaximal, complemented
3#20273353C211yesmaximal, characteristic, complemented
4#20273223C212yesmaximal, complemented
5C53×#3548C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810647C2211yescharacteristic
8#16810469C2211yescharacteristic
9#16810382—22no—
10#16810369—22no—
11#16810367—22no—
12#16810365—22no—
13#16810357—24no—
14C53×#562—22no—
15C53×#554—22no—
16C53×#550—22no—
17C53×#547C414yescomplemented
18C53×#540—22no—
19C53×#537—24no—
20C53×#533C2211yescharacteristic
21C53×#532C2212yes—
22C2×#11719917—21no—
23C2×C4×Dic53—21no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C22⋊#88322—22no—
27#13133128—22no—
28#13133124—22no—
29C2×#41×C53—24no—
30C2×#41×C53—22no—
31C4⋊Q8×C53C2×C412yes—
32C4⋊Q8×C53D412yes—
33#127×C53—22no—
34#127×C53D412yes—
35#125×C53—22no—
36#125×C53D412yes—
37#124×C53—22no—
38#123×C53—28no—
39C22⋊Q8×C53—28no—
40C22⋊Q8×C53—24no—
41C22⋊Q8×C53—22no—
42#121×C53—28no—
43#121×C53—28no—
44#121×C53—24no—
45#121×C53—22no—
46#121×C53D414yes—
47C4×Q8×C53—22no—
48C4×D4×C53—24no—
49C4×D4×C53—22no—
50C4×D4×C53C2×C414yes—
51#117×C53D412yes—
52C2×C4⋊C4×C53—24no—
53C2×C42×C53C2311yescharacteristic
54#41×C53—84no—
55#41×C53—88no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×D4×C53—28no—
59C2×D4×C53—24no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53C22⋊C414yes—
63C22×C4×C53—28no—
64C22×C4×C53—24no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×Dic53—41no—
68C4⋊C4×C53—28no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53C22⋊C414yes—
73C4⋊C4×C53C22⋊C414yes—
74C22⋊C4×C53—28no—
75C22⋊C4×C53—22no—
76C22⋊C4×C53—24no—
77C22⋊C4×C53—24no—
78C42×C53—22no—
79C42×C53C22×C411yescharacteristic
80C42×C53C2×D412yes—
81C42×C53C2×D411yescharacteristic
82C4×Dic53—22no—
83C4×Dic53—41no—
84#11719917—42no—
85C2×#88322—42no—
86C2×#88322—41no—
87C23×C53—24no—
88C23×C53#12011yescharacteristic
89Q8×C53—44no—
90Q8×C53—82no—
91D4×C53—44no—
92D4×C53—42no—
93D4×C53—42no—
94D4×C53—82no—
95D4×C53—84no—
96D4×C53—88no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—24no—
100C2×C4×C53—24no—
101C2×C4×C53—42no—
102C2×C4×C53—48no—
103C2×C4×C53—88no—
104C2×C4×C53C2×C22⋊C412yes—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53#12012yes—
107C2×C4×C53#12011yescharacteristic
108C2×Dic53—42no—
109C2×Dic53—82no—
110#88322—82no—
111#88322—81no—
112#8494—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—22no—
115C22×C53—44no—
116C22×C53—44no—
117C22×C53—44no—
118C22×C53#37911yescharacteristic
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—44no—
123C4×C53—44no—
124C4×C53#45411yescharacteristic
125C4×C53#45611yescharacteristic
126Dic53—82no—
127#3548D5311yescharacteristic
128#1995—531no—
129#1557—532no—
130#1537—531no—
131#1505—532no—
132C2×C53—41no—
133C2×C53—84no—
134C2×C53#179311yescharacteristic
135C2×C53#179512yes—
136#562—22no—
137#554—22no—
138#550—22no—
139#547Dic5314yescomplemented
140#540—22no—
141#537—24no—
142#533C2×D5311yescharacteristic
143#532C2×D5312yes—
144#432—532no—
145#423—531no—
146#376—531no—
147#339—1062no—
148#332—1062no—
149#331—1062no—
150#330—1062no—
151#327—1064no—
152C53#849411yescharacteristic, complemented, Sylow
153C2×#41—24no—
154C2×#41—22no—
155C2×M16—1061no—
156C4⋊Q8C2×Dic5312yes—
157C4⋊Q8C22⋊D5312yes—
158#127—22no—
159#127C22⋊D5312yes—
160#125—22no—
161#125C22⋊D5312yes—
162#124—22no—
163#123—28no—
164C22⋊Q8—28no—
165C22⋊Q8—22no—
166C22⋊Q8—24no—
167#121—28no—
168#121—28no—
169#121—24no—
170#121—22no—
171#121C22⋊D5314yes—
172C4×Q8—22no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4C2×Dic5314yes—
176#117C22⋊D5312yes—
177C2×C4⋊C4—24no—
178C2×C42—1061no—
179C2×C42C22×D5311yescharacteristic
180C4⋊C8—1062no—
181C4⋊C8—1062no—
182C22⋊C8—1062no—
183#97—1062no—
184#95—1062no—
185#41—84no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C22⋊Dic5314yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—2121no—
197C22×C4C2×C22⋊D5311yescharacteristic
198C22×C4C2×C22⋊D5312yes—
199M16—2122no—
200C2×C8—2122no—
201C2×C8—2121no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4C22⋊Dic5314yes—
207C4⋊C4C22⋊Dic5314yes—
208C22⋊C4—28no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C42—22no—
213C42—1062no—
214C42—2121no—
215C42C22×Dic5311yescharacteristic
216C42C2×C22⋊D5311yescharacteristic
217C42C2×C22⋊D5312yes—
218C23—24no—
219C23#1313318611yescharacteristic
220Q8—44no—
221Q8—82no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—82no—
226D4—84no—
227D4—88no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—88no—
235C2×C4—4242no—
236C2×C4—2122no—
237C2×C4C2×C22⋊Dic5311yescharacteristic
238C2×C4C2×C22⋊Dic5312yes—
239C2×C4#1313318612yes—
240C2×C4#1313318611yescharacteristic
241C8—4242no—
242C8—4241no—
243C22—21no—
244C22—22no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22#1681047911yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—4242no—
255C4#1681079611yescharacteristic
256C4#1681080611yescharacteristic
257C2—41no—
258C2—84no—
259C2#2027467411yescharacteristic
260C2#2027468512yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3548C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3548D53
hypocenterC53#8494
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #20275697C2semidirect
2. #20273437C2semidirect
3. #20273353C2semidirect
4. #20273223C2semidirect
17. C53×#547C4semidirect
139. #547Dic53semidirect
152. C53#8494semidirect

Maximal subgroups

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

subgroupindex
#202756972
#202734372
#202733532
#202732232
C53×#35482
#849453