Finite Group Databasebeta

← #27297416

Subgroups — #27297416

Order 13568.

376 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 55 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#20285515C211yesmaximal, characteristic
2#20276929C212yesmaximal, complemented
3#20272705C212yesmaximal, complemented
4#20272651C211yesmaximal, characteristic, complemented
5C53×#2091C211yesmaximal, characteristic, complemented
6C2×#13133191—21no—
7#16810942—22no—
8#16810876—21no—
9#16810876C2212yes—
10#16810759—22no—
11#16810447—22no—
12#16810435—22no—
13#16810405C2212yescomplemented
14C53×#430—21no—
15C53×#430C2211yescharacteristic, complemented
16C53×#407—21no—
17C53×#407C2211yescharacteristic
18C53×#338C2211yescharacteristic, complemented
19C22×D212—21no—
20C22×C22⋊D53—41no—
21C2×D424—21no—
22C2×D424D412yes—
23#13133191—22no—
24#13133191—22no—
25#13133191—41no—
26#13133191—41no—
27C2×C4⋊Dic53—21no—
28C2×#11719921—41no—
29#13133157—22no—
30#13133149—21no—
31#13133149D412yes—
32#13133139—22no—
33#13133139—22no—
34C23×C4×C53D411yescharacteristic
35C2×M16×C53—21no—
36C2×M16×C53—21no—
37C2×M16×C53D411yescharacteristic
38C2×M16×C53C2311yescharacteristic
39#108×C53—22no—
40#108×C53—42no—
41C22⋊C8×C53—22no—
42C22⋊C8×C53—42no—
43C24×C53SD1611yescharacteristic
44C23×D53—41no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53SD1611yescharacteristic
50C22×C4×C53C2×D411yescharacteristic
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—84no—
54C22×Dic53—41no—
55C2×D212—22no—
56C2×D212—41no—
57C2×D212—81no—
58C2×D212—81no—
59M16×C53—41no—
60M16×C53—41no—
61M16×C53—41no—
62M16×C53—41no—
63M16×C53—41no—
64M16×C53—41no—
65C2×C8×C53—22no—
66C2×C8×C53—21no—
67C2×C8×C53—41no—
68C2×C8×C53C2×D412yes—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—22no—
72C4⋊Dic53—41no—
73D424—42no—
74D424—81no—
75#11719908—42no—
76C23×C53—21no—
77C23×C53—42no—
78C23×C53—41no—
79C23×C53C2×SD1611yescharacteristic
80C22×D53—42no—
81C22×D53—82no—
82C22×D53—82no—
83C22×D53—82no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—81no—
91C2×C4×C53#12011yescharacteristic
92C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
93C22⋊D53—84no—
94C22⋊D53—84no—
95C2×Dic53—42no—
96C2×Dic53—82no—
97D212—42no—
98D212—42no—
99D212—82no—
100C8×C53—22no—
101C8×C53—42no—
102C8×C53—41no—
103C8×C53—81no—
104#12298—531nomaximal, Sylow
105C22×C53—22no—
106C22×C53—21no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53—81no—
112C22×C53#45011yescharacteristic
113C2×D53—82no—
114C2×D53—82no—
115C2×D53—82no—
116C2×D53—164no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53#45311yescharacteristic
120C4×C53#45811yescharacteristic
121Dic53—82no—
122#3192—531no—
123#2238—532no—
124#2091D5311yescharacteristic, complemented
125#1341—532no—
126#1320—531no—
127C2×C53—21no—
128C2×C53—42no—
129C2×C53—41no—
130C2×C53#161311yescharacteristic
131D53—162no—
132C2×#121—1061no—
133#481—1062no—
134#469—532no—
135#469—1061no—
136#447—1062no—
137#430—21no—
138#430C2×D5311yescharacteristic, complemented
139#407—21no—
140#407C2×D5311yescharacteristic
141#365—1062no—
142#361—1062no—
143#349—532no—
144#338C2×D5311yescharacteristic, complemented
145C53#1229811yescharacteristic, complemented, Sylow
146C22×D4—1061no—
147C22×D4—2121no—
148C23×C4C22⋊D5311yescharacteristic
149C2×D8—532no—
150C2×D8—1061no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16C22⋊D5311yescharacteristic
154C2×M16C22×D5311yescharacteristic
155#121—1062no—
156#121—1062no—
157#121—2121no—
158#121—2121no—
159C2×C4⋊C4—1061no—
160C2×C22⋊C4—2121no—
161M32—1062no—
162#108—22no—
163#108—42no—
164#106—532no—
165#106—1061no—
166#102—1062no—
167#102—1062no—
168C22⋊C8—22no—
169C22⋊C8—42no—
170C24—2121no—
171C24#1171992411yescharacteristic
172C2×D4—1062no—
173C2×D4—4241no—
174C2×D4—4241no—
175C2×D4—4244no—
176C2×D4—2121no—
177C2×D4—2122no—
178C2×D4—2122no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—2121no—
184C22×C4#1171992411yescharacteristic
185C22×C4C2×C22⋊D5311yescharacteristic
186D8—4241no—
187D8—2122no—
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193M16—41no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8D4×D5312yes—
198C4⋊C4—1062no—
199C4⋊C4—2121no—
200C22⋊C4—2122no—
201C22⋊C4—2122no—
202C16—2122no—
203C23—21no—
204C23—41no—
205C23—42no—
206C23—4242no—
207C23—4242no—
208C23—4242no—
209C23—2122no—
210C23C2×#1171992411yescharacteristic
211D4—4242no—
212D4—4244no—
213D4—4244no—
214D4—2122no—
215D4—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—4242no—
224C2×C4—2122no—
225C2×C4#1313318511yescharacteristic
226C2×C4#1313324211yescharacteristic
227C8—22no—
228C8—42no—
229C8—41no—
230C8—81no—
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—81no—
238C22—8484no—
239C22—4242no—
240C22—4242no—
241C22—4242no—
242C22#1681077311yescharacteristic
243C4—21no—
244C4—41no—
245C4—4242no—
246C4#1681079011yescharacteristic
247C4#1681081211yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—8482no—
252C2#2027373411yescharacteristic
subgroupG/H
centerC2#20273734
commutator, cosocleC2×M16×C53C23
FrattiniC2×M16C22×D53
FittingC53×#2091C2
socleC2×C53#1613
prime radical#20285515C2
hypercenter#2091D53
hypocenterC53#12298
LCSG ▷ C2×M16×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2091

Complements

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

NH ≅ G/Ntype
2. #20276929C2semidirect
3. #20272705C2semidirect
4. #20272651C2semidirect
5. C53×#2091C2semidirect
13. #16810405C22semidirect
15. C53×#430C22semidirect
18. C53×#338C22semidirect
124. #2091D53semidirect
138. #430C2×D53semidirect
144. #338C2×D53semidirect
145. C53#12298semidirect

Maximal subgroups

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

subgroupindex
#202855152
#202769292
#202727052
#202726512
C53×#20912
#1229853