Finite Group Databasebeta

← #27296354

Subgroups — #27296354

Order 13568.

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

normal subgroups only

This group has 398 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#20275761C211yesmaximal, characteristic
2#20273751C212yesmaximal
3#20272742C212yesmaximal, complemented
4#20272649C211yesmaximal, characteristic, complemented
5C53×#2809C211yesmaximal, characteristic, complemented
6C2×#13133248—21no—
7#16811204C2211yescharacteristic
8#16810941—22no—
9C2×#13133145—21no—
10#16810392C2211yescharacteristic
11#16810356—22no—
12#16810349C2212yes—
13C53×#400C2212yes—
14C53×#398—22no—
15C53×#391—21no—
16C53×#391C2211yescharacteristic
17C53×#389—21no—
18C53×#387—22no—
19C2×#11719945—21no—
20#13133248—42no—
21#13133248—42no—
22C2×#11719925—41no—
23C2×#11719924—41no—
24C2×#11719917—21no—
25#13133197—41no—
26#13133192—41no—
27#13133191—22no—
28#13133148—42no—
29#13133145—42no—
30#13133137—42no—
31C22⋊#88322—22no—
32C22×Q8×C53D411yescharacteristic
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53—21no—
40C2×C42×C53D411yescharacteristic
41#95×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yes—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53SD1612yes—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D412yes—
56C2×C22⋊D53—41no—
57#11719945—41no—
58#11719945—41no—
59#11719945—41no—
60#11719945—41no—
61C2×D212—21no—
62C2×D212—22no—
63C2×C4×D53—41no—
64C2×Dic106—21no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—44no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—81no—
70C4⋊C4×C53—81no—
71C42×C53—41no—
72C42×C53—81no—
73#11719925—44no—
74#11719924—44no—
75#11719921—42no—
76C4⋊Dic53—22no—
77#11719917—22no—
78#11719917—41no—
79C2×#88322—41no—
80C2×#88322—42no—
81C23×C53#12011yescharacteristic
82C22×D53—41no—
83Q8×C53—42no—
84Q8×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53C2×SD1611yescharacteristic
102C2×C4×C53#13711yescharacteristic
103C22⋊D53—81no—
104C22⋊D53—81no—
105C2×Dic53—41no—
106D212—41no—
107D212—41no—
108D212—82no—
109C4×D53—81no—
110C4×D53—81no—
111Dic106—41no—
112Dic106—41no—
113#88322—42no—
114#88322—82no—
115#12131—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53#44811yescharacteristic
119C22×C53#45311yescharacteristic
120C22×C53B(4,𝔽2)11yescharacteristic
121C2×D53—81no—
122C2×D53—81no—
123C2×D53—81no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—42no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—82no—
130C4×C53—82no—
131Dic53—81no—
132#2809D5311yescharacteristic, complemented
133#2007—531no—
134#1616—532no—
135#1353—532no—
136#1319—531no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53#159611yescharacteristic
140C2×C53#220111yescharacteristic
141C2×C53#220411yescharacteristic
142D53—161no—
143C2×#137—1061no—
144#537—531no—
145#481—1062no—
146C2×#104—1061no—
147#400C2×D5312yes—
148#398—22no—
149#391—21no—
150#391C2×D5311yescharacteristic
151#389—21no—
152#387—22no—
153#343—531no—
154#327—1062no—
155#324—532no—
156C53#1213111yescharacteristic, complemented, Sylow
157C2×#41—1061no—
158C22×Q8C22⋊D5311yescharacteristic
159#137—2122no—
160#137—2122no—
161C2×Q16—2121no—
162C2×SD16—2121no—
163C2×M16—1061no—
164C22⋊Q8—2121no—
165#121—1062no—
166#121—2121no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C22⋊D5312yes—
172C2×C4⋊C4C22×D5311yescharacteristic
173C2×C42—21no—
174C2×C42C22⋊D5311yescharacteristic
175#106—2122no—
176#104—2122no—
177#102—2122no—
178C22⋊C8—1062no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95C22⋊D5312yes—
184#41—2121no—
185#41—2121no—
186#41—2121no—
187#41—2121no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1061no—
192C2×Q8#1171992412yes—
193C2×D4—1062no—
194C2×D4—1061no—
195C2×D4—2121no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—2121no—
201C22×C4C2×C22⋊D5311yescharacteristic
202C22×C4C2×C22⋊D5312yes—
203Q16—2124no—
204SD16—2124no—
205M16—1062no—
206M16—2121no—
207C2×C8—2122no—
208C2×C8—2121no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—1062no—
216C22⋊C4—2122no—
217C42—41no—
218C42—81no—
219C23—2121no—
220C23#1313318611yescharacteristic
221Q8—42no—
222Q8—81no—
223Q8—2121no—
224Q8—2121no—
225D4—4242no—
226D4—4241no—
227D4—4241no—
228D4—2121no—
229D4—2121no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—4241no—
247C2×C4—4241no—
248C2×C4—2121no—
249C2×C4C2×#1171992411yescharacteristic
250C2×C4#1313324811yescharacteristic
251C8—4242no—
252C8—2122no—
253C22—21no—
254C22—21no—
255C22—4241no—
256C22—4241no—
257C22—4241no—
258C22#1681076211yescharacteristic
259C22#1681079111yescharacteristic
260C22#1681081011yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4—82no—
268C4—4241no—
269C2—21no—
270C2—21no—
271C2—8481no—
272C2#2027363811yescharacteristic
273C2#2027672811yescharacteristic
274C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2809C2
socleC22×C53B(4,𝔽2)
prime radical#16811204C22
hypercenter#2809D53
hypocenterC53#12131
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
3. #20272742C2semidirect
4. #20272649C2semidirect
5. C53×#2809C2semidirect
132. #2809D53semidirect
156. C53#12131semidirect

Maximal subgroups

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

subgroupindex
#202757612
#202737512
#202727422
#202726492
C53×#28092
#1213153