Finite Group Databasebeta

← #27304789

Subgroups — #27304789

Order 13568.

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

normal subgroups only

This group has 284 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#20287852C211yesmaximal, characteristic
2#20277033C211yesmaximal, characteristic, complemented
3#20276837C211yesmaximal, characteristic, complemented
4#20272905C211yesmaximal, characteristic, complemented
5#20272900C211yesmaximal, characteristic, complemented
6#20272856C211yesmaximal, characteristic, complemented
7C2×C53×SD64C211yesmaximal, characteristic, complemented
8#16811424—21no—
9#16811059—21no—
10#16811057—22no—
11#16811055C2211yescharacteristic
12#16811052—21no—
13D16×D53—21no—
14#16811039C2211yescharacteristic
15#16810895—21no—
16#16810463—21no—
17#16810460—21no—
18#16810457C2211yescharacteristic, complemented
19#16810453C2211yescharacteristic, complemented
20#16810439—21no—
21C2×Q32×C53C2211yescharacteristic, complemented
22C2×D16×C53C2211yescharacteristic
23C53×SD64—22no—
24C2×C32×C53C2211yescharacteristic, complemented
25#13133262—41no—
26#13133248—41no—
27#13133243—41no—
28#13133241—42no—
29#13133239—21no—
30#13133236—22no—
31D8×D53—22no—
32#13133223D411yescharacteristic
33Dic424D411yescharacteristic
34#13133162—21no—
35#13133161—22no—
36D848D411yescharacteristic
37#13133158—22no—
38#13133157D411yescharacteristic
39#13133156—21no—
40C16×D53—21no—
41#13133153—21no—
42#13133136—41no—
43#13133123—21no—
44C2×Q16×C53—21no—
45C2×D8×C53—21no—
46Q32×C53—21no—
47Q32×C53—21no—
48D16×C53—21no—
49D16×C53D412yes—
50C2×C16×C53C2311yescharacteristic
51C32×C53—21no—
52C2×Q8×C53—41no—
53C2×D4×C53—41no—
54C2×C22⋊D53—82no—
55#11719947—42no—
56#11719947—81no—
57D4×D53—42no—
58D4×D53—81no—
59#11719945—41no—
60#11719945D811yescharacteristic
61Q16×C53—41no—
62Q16×C53—41no—
63D8×C53—22no—
64D8×C53—41no—
65C2×C8×C53C2×D411yescharacteristic
66#11719925—81no—
67#11719924—81no—
68#11719923—42no—
69#11719922—42no—
70#11719917—41no—
71#11719917D811yescharacteristic
72Dic212—21no—
73D424—21no—
74#11719913—41no—
75#11719912—21no—
76C8×D53—21no—
77C16×C53C2×D411yescharacteristic
78C16×C53C2×D411yescharacteristic
79#11719908—21no—
80C23×C53—81no—
81C22×D53—82no—
82Q8×C53—81no—
83Q8×C53—81no—
84D4×C53—42no—
85D4×C53—81no—
86C2×C4×C53—81no—
87C2×C4×C53C2×D811yescharacteristic
88C22⋊D53—21no—
89C22⋊D53—82no—
90C22⋊D53—82no—
91C22⋊D53—81no—
92C2×Dic53—82no—
93D212—41no—
94D212D1611yescharacteristic
95C4×D53—21no—
96C4×D53—81no—
97Dic106—41no—
98Dic106D1611yescharacteristic
99C8×C53#12011yescharacteristic
100C8×C53AGL(1,ℤ/8)11yescharacteristic
101#88322—21no—
102#88322—81no—
103#13671—531nomaximal, Sylow
104C22×C53—82no—
105C22×C53—321no—
106C22×C53#50911yescharacteristic
107C2×D53—21no—
108C2×D53—81no—
109C2×D53—322no—
110C4×C53—161no—
111C4×C53#44711yescharacteristic
112C4×C53C2×D1611yescharacteristic
113Dic53—21no—
114Dic53—81no—
115#3415—531no—
116#2264—531no—
117C2×SD64D5311yescharacteristic, complemented
118#2222—531no—
119#1419—531no—
120#1417—531no—
121#1401—531no—
122C2×C53—21no—
123C2×C53—162no—
124C2×C53#218411yescharacteristic
125D53—41no—
126D53—161no—
127#576—1061no—
128#509—1061no—
129#509—1062no—
130#508—531no—
131#508—1061no—
132C2×Q32C2×D5311yescharacteristic, complemented
133C2×D16—1061no—
134C2×D16C2×D5311yescharacteristic
135#504—531no—
136#473—1061no—
137Q64—1061no—
138SD64—22no—
139SD64—1061no—
140M64—531no—
141C2×C32C2×D5311yescharacteristic, complemented
142#368—531no—
143#362—1061no—
144C53#1367111yescharacteristic, complemented, Sylow
145H(5,𝔽2)—2121no—
146#137—2121no—
147AGL(1,ℤ/8)—2121no—
148AGL(1,ℤ/8)—2122no—
149#135—1062no—
150#135—1061no—
151C2×Q16—21no—
152C2×D8—21no—
153C2×D8—1062no—
154#131—531no—
155Q32—21no—
156Q32—21no—
157Q32—531no—
158SD32—1062no—
159SD32—1061no—
160D16—21no—
161D16—531no—
162D16—1062no—
163D16C22⋊D5312yes—
164M32—531no—
165M32—1061no—
166C2×C16—1061no—
167C2×C16C22×D5311yescharacteristic
168#108—1061no—
169#101—2121no—
170C32—21no—
171C32—1061no—
172#41—531no—
173#41—4241no—
174#41—2122no—
175#41—2121no—
176C2×Q8—41no—
177C2×D4—41no—
178C2×D4—4241no—
179C2×D4—4242no—
180C2×D4—2122no—
181Q16—41no—
182Q16—41no—
183Q16—1061no—
184Q16—4241no—
185SD16—4241no—
186SD16—2122no—
187SD16—2121no—
188D8—22no—
189D8—41no—
190D8—1061no—
191D8—2122no—
192M16—531no—
193M16—1061no—
194M16—2121no—
195C2×C8—1061no—
196C2×C8D4×D5311yescharacteristic
197C16—1061no—
198C16D4×D5311yescharacteristic
199C16C2×C22⋊D5311yescharacteristic
200C23—81no—
201C23—4242no—
202Q8—81no—
203Q8—81no—
204Q8—531no—
205Q8—2121no—
206D4—42no—
207D4—81no—
208D4—531no—
209D4—1061no—
210D4—4242no—
211D4—4242no—
212D4—4241no—
213D4—2121no—
214C2×C4—81no—
215C2×C4—1061no—
216C2×C4—4242no—
217C2×C4—4241no—
218C2×C4D8×D5311yescharacteristic
219C8—1061no—
220C8—4241no—
221C8#1313318511yescharacteristic
222C8#1313324111yescharacteristic
223C22—82no—
224C22—321no—
225C22—1061no—
226C22—16962no—
227C22—4241no—
228C22#1681105711yescharacteristic
229C4—161no—
230C4—1061no—
231C4—4241no—
232C4#1681075511yescharacteristic
233C4D16×D5311yescharacteristic
234C2—21no—
235C2—162no—
236C2—8481no—
237C2—2121no—
238C2#2027662511yescharacteristic
subgroupG/H
centerC2#20276625
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC2×C53×SD64C2
socleC2×C53#2184
prime radical#20287852C2
hypercenterC2×SD64D53
hypocenterC53#13671
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
2. #20277033C2semidirect
3. #20276837C2semidirect
4. #20272905C2semidirect
5. #20272900C2semidirect
6. #20272856C2semidirect
7. C2×C53×SD64C2semidirect
18. #16810457C22semidirect
19. #16810453C22semidirect
21. C2×Q32×C53C22semidirect
24. C2×C32×C53C22semidirect
117. C2×SD64D53semidirect
132. C2×Q32C2×D53semidirect
141. C2×C32C2×D53semidirect
144. C53#13671semidirect

Maximal subgroups

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

subgroupindex
#202878522
#202770332
#202768372
#202729052
#202729002
#202728562
C2×C53×SD642
#1367153