Finite Group Databasebeta

← #27296911

Subgroups — #27296911

Order 13568.

460 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275871C211yesmaximal, characteristic
2#20273688C212yesmaximal, complemented
3#20272745C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic
5C53×#2683C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810779—22no—
8#16810628—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×C22⋊Q8×C53—22no—
14C53×#396—22no—
15C53×#395C2212yes—
16C53×#390C2211yescharacteristic
17C53×#389—21no—
18C53×#386—21no—
19C2×#11719925—42no—
20C2×#11719917—21no—
21#13133216—41no—
22#13133208—41no—
23#13133196—22no—
24#13133176—21no—
25#13133141—42no—
26#13133138—41no—
27#13133138—41no—
28C22⋊#88322—22no—
29C22×Q8×C53D412yes—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C22⋊Q8×C53—48no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53C2311yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C42×C53D411yescharacteristic
42#95×C53—22no—
43#95×C53D412yes—
44C24×C53—21no—
45C2×Q8×C53—44no—
46C2×Q8×C53—42no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53Q1612yescomplemented
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—44no—
55C22×C4×C53—81no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×Dic106—22no—
59C4⋊C4×C53—24no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—44no—
62C4⋊C4×C53—44no—
63C4⋊C4×C53—81no—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—44no—
66C22⋊C4×C53—81no—
67C42×C53—41no—
68C22⋊Dic53—41no—
69#11719925—84no—
70C4⋊Dic53—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53—22no—
73#11719919—42no—
74C4×Dic53—41no—
75#11719917—41no—
76#11719917—41no—
77C2×#88322—42no—
78C2×#88322—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85Q8×C53—44no—
86Q8×C53—82no—
87D4×C53—42no—
88D4×C53—81no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—84no—
103C2×C4×C53C2×Q1611yescharacteristic
104C2×C4×C53#13711yescharacteristic
105C2×Dic53—41no—
106C2×Dic53—41no—
107Dic106—82no—
108#88322—81no—
109#88322—82no—
110#12220—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—44no—
114C22×C53—42no—
115C22×C53—82no—
116C22×C53—81no—
117C22×C53—81no—
118C22×C53#45111yescharacteristic
119C22×C53#45511yescharacteristic
120C22×C53B(4,𝔽2)11yescharacteristic
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—44no—
126C4×C53—82no—
127Dic53—81no—
128Dic53—81no—
129#2683D5311yescharacteristic
130#2027—531no—
131#1605—532no—
132#1354—532no—
133#1318—531no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53—82no—
137C2×C53#160211yescharacteristic
138C2×C53#219611yescharacteristic
139C2×C53#220311yescharacteristic
140#544—531no—
141C2×C22⋊Q8—22no—
142#451—1062no—
143#418—1061no—
144#396—22no—
145#395C2×D5312yes—
146#390C2×D5311yescharacteristic
147#389—21no—
148#386—21no—
149#343—531no—
150#339—1061no—
151#328—1062no—
152#324—532no—
153C53#1222011yescharacteristic, complemented, Sylow
154C22×Q8C22⋊D5312yes—
155C22×D4C22⋊D5311yescharacteristic
156C23×C4—21no—
157C2×Q16—2122no—
158C2×M16—1061no—
159C4⋊Q8—2121no—
160#125—2121no—
161C22⋊Q8—48no—
162C22⋊Q8—1062no—
163#117—1061no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×D5311yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42C22⋊D5311yescharacteristic
173#103—2122no—
174#102—2121no—
175#102—2121no—
176C22⋊C8—1062no—
177#95—22no—
178#95C22⋊D5312yes—
179C24—21no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—1062no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1171992512yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×C22⋊D5311yescharacteristic
193C22×C4C2×C22⋊D5312yes—
194Q16—4244no—
195M16—2121no—
196M16—2121no—
197C2×C8—2122no—
198C2×C8—2121no—
199C4⋊C4—24no—
200C4⋊C4—41no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—1061no—
205C4⋊C4—1061no—
206C4⋊C4—1062no—
207C4⋊C4—2122no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—81no—
211C22⋊C4—2121no—
212C42—41no—
213C42—2121no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1313318611yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—4242no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—2121no—
240C2×C4—2121no—
241C2×C4C2×#1171992511yescharacteristic
242C2×C4#1313324811yescharacteristic
243C8—4241no—
244C8—4242no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1681077911yescharacteristic
253C22#1681080111yescharacteristic
254C22#1681081011yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—4241no—
262C4—4241no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2027367211yescharacteristic
267C2#2027669911yescharacteristic
268C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2683C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2683D53
hypocenterC53#12220
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #20273688C2semidirect
3. #20272745C2semidirect
50. C2×D4×C53Q16semidirect
153. C53#12220semidirect
186. C2×D4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758712
#202736882
#202727452
#202726472
C53×#26832
#1222053