Finite Group Databasebeta

← #27296504

Subgroups — #27296504

Order 13568.

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

normal subgroups only

This group has 424 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#20275939C211yesmaximal, characteristic
2#20273717C212yesmaximal
3#20272721C212yesmaximal, complemented
4#20272661C211yesmaximal, characteristic, complemented
5C53×#2582C211yesmaximal, characteristic, complemented
6#16811204C2211yescharacteristic
7#16810936—22no—
8C2×#13133145—21no—
9#16810622—21no—
10#16810392C2211yescharacteristic
11#16810356—22no—
12#16810349C2212yes—
13C2×C4×Q8×C53—21no—
14C53×#398—21no—
15C53×#398C2212yes—
16C53×#391—22no—
17C53×#389—22no—
18C53×#384C2211yescharacteristic
19C2×#11719945—21no—
20C2×#11719917—21no—
21#13133197—41no—
22#13133192—41no—
23#13133191—22no—
24#13133150—42no—
25#13133145—42no—
26#13133140—41no—
27#13133137—41no—
28#13133137—42no—
29C22⋊#88322—22no—
30C22×Q8×C53—21no—
31C4×Q8×C53—42no—
32C4×Q8×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C42×C53—22no—
41C2×C42×C53D411yescharacteristic
42#95×C53—22no—
43#95×C53—21no—
44#95×C53—22no—
45#95×C53D412yes—
46C2×Q8×C53—24no—
47C2×Q8×C53—42no—
48C2×Q8×C53—42no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D412yes—
55C2×C22⋊D53—41no—
56#11719945—41no—
57#11719945—41no—
58#11719945—41no—
59#11719945—41no—
60C2×D212—21no—
61C2×D212—22no—
62C2×C4×D53—41no—
63C2×Dic106—21no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—44no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—81no—
72C42×C53—21no—
73C42×C53—44no—
74C42×C53D812yes—
75#11719921—42no—
76C4⋊Dic53—22no—
77#11719917—41no—
78#11719917—41no—
79C2×#88322—42no—
80C2×#88322—41no—
81C23×C53#12011yescharacteristic
82C22×D53—41no—
83Q8×C53—48no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—24no—
90C2×C4×C53—22no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53C2×D811yescharacteristic
100C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
101C22⋊D53—81no—
102C22⋊D53—81no—
103C2×Dic53—41no—
104D212—41no—
105D212—41no—
106D212—82no—
107C4×D53—81no—
108C4×D53—81no—
109Dic106—41no—
110Dic106—41no—
111#88322—81no—
112#88322—82no—
113#12153—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53#44711yescharacteristic
117C22×C53#45411yescharacteristic
118C22×C53B(4,𝔽2)11yescharacteristic
119C2×D53—81no—
120C2×D53—81no—
121C2×D53—81no—
122C4×C53—24no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—44no—
126C4×C53—42no—
127C4×C53—82no—
128C4×C53—81no—
129Dic53—81no—
130#2582D5311yescharacteristic, complemented
131#2039—531no—
132#1610—532no—
133#1346—532no—
134#1325—531no—
135C2×C53—21no—
136C2×C53—21no—
137C2×C53#159811yescharacteristic
138C2×C53#219811yescharacteristic
139C2×C53#220511yescharacteristic
140D53—161no—
141#537—531no—
142C2×C4×Q8—21no—
143#480—1062no—
144C2×#104—1061no—
145#417—1061no—
146#398—21no—
147#398C2×D5312yes—
148#391—22no—
149#389—22no—
150#384C2×D5311yescharacteristic
151#343—531no—
152#327—1062no—
153#324—532no—
154C53#1215311yescharacteristic, complemented, Sylow
155C2×#41—1061no—
156C22×Q8—21no—
157C2×M16—1061no—
158C22⋊Q8—2121no—
159#121—1062no—
160#121—2121no—
161C4×Q8—42no—
162C4×Q8—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C22⋊D5311yescharacteristic
168C2×C4⋊C4C22⋊D5312yes—
169C2×C4⋊C4C22×D5311yescharacteristic
170C2×C42—22no—
171C2×C42C22⋊D5311yescharacteristic
172#107—2122no—
173#104—2122no—
174#103—2121no—
175#102—2121no—
176#102—2122no—
177C22⋊C8—1062no—
178#95—22no—
179#95—21no—
180#95—22no—
181#95C22⋊D5312yes—
182#41—2121no—
183#41—2121no—
184#41—2121no—
185#41—2121no—
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—1061no—
190C2×D4—1062no—
191C2×D4—1061no—
192C2×D4—2121no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—2121no—
198C22×C4C2×C22⋊D5311yescharacteristic
199C22×C4C2×C22⋊D5312yes—
200M16—2121no—
201M16—2121no—
202C2×C8—2122no—
203C2×C8—2121no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—1062no—
213C22⋊C4—2122no—
214C42—21no—
215C42—44no—
216C42#1171992212yes—
217C23—2121no—
218C23#1313318611yescharacteristic
219Q8—48no—
220Q8—2121no—
221Q8—2121no—
222D4—4242no—
223D4—4241no—
224D4—4241no—
225D4—2121no—
226D4—2121no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—4241no—
243C2×C4—4241no—
244C2×C4—2121no—
245C2×C4C2×#1171992211yescharacteristic
246C2×C4#1313324311yescharacteristic
247C8—4241no—
248C8—4242no—
249C22—21no—
250C22—21no—
251C22—4241no—
252C22—4241no—
253C22—4241no—
254C22#1681075711yescharacteristic
255C22#1681079611yescharacteristic
256C22#1681081011yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—4241no—
265C2—21no—
266C2—21no—
267C2—8481no—
268C2#2027365011yescharacteristic
269C2#2027671111yescharacteristic
270C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2582C2
socleC22×C53B(4,𝔽2)
prime radical#16811204C22
hypercenter#2582D53
hypocenterC53#12153
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
4. #20272661C2semidirect
5. C53×#2582C2semidirect
130. #2582D53semidirect
154. C53#12153semidirect

Maximal subgroups

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

subgroupindex
#202759392
#202737172
#202727212
#202726612
C53×#25822
#1215353