Finite Group Databasebeta

← #27304720

Subgroups — #27304720

Order 13568.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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#20277028C211yesmaximal, characteristic
2#20276600C211yesmaximal, characteristic, complemented
3#20276591C211yesmaximal, characteristic, complemented
4#20272922C211yesmaximal, characteristic, complemented
5#20272903C211yesmaximal, characteristic, complemented
6#20272850C211yesmaximal, characteristic, complemented
7C53×#1419C211yesmaximal, characteristic, complemented
8#16811055C2211yescharacteristic
9#16810739—21no—
10D8×Dic53—21no—
11#16810461C412yescomplemented
12#16810458C412yescomplemented
13#16810457C2211yescharacteristic
14#16810451C2211yescharacteristic
15#16810429—21no—
16#16810424—21no—
17#16810398C2211yescharacteristic
18C2×D16×C53C2211yescharacteristic
19C53×M64C2211yescharacteristic
20C53×#368C2211yescharacteristic
21#13133239—21no—
22#13133225—41no—
23D4×Dic53—41no—
24Dic424C2×C411yescharacteristic, complemented
25#13133162—21no—
26D848C2×C411yescharacteristic, complemented
27C2×#11719908D411yescharacteristic
28#13133151—21no—
29#13133143—41no—
30#13133138—41no—
31C8×Dic53D411yescharacteristic
32#13133123C2×C412yescomplemented
33C2×D8×C53—21no—
34D16×C53—21no—
35D16×C53C2×C412yes—
36C2×C16×C53C2311yescharacteristic
37#108×C53—21no—
38C32×C53—21no—
39C2×D4×C53—41no—
40C22×Dic53—81no—
41#11719945—41no—
42D8×C53—22no—
43D8×C53—41no—
44M16×C53—41no—
45C2×C8×C53C2×D411yescharacteristic
46C22⋊Dic53—81no—
47C4⋊Dic53—41no—
48C4×Dic53D811yescharacteristic
49C2×#88322D811yescharacteristic
50Dic212—21no—
51D424—21no—
52#11719913—41no—
53#11719912—81no—
54C8×D53—81no—
55C16×C53C22×C411yescharacteristic
56C16×C53#4111yescharacteristic
57#11719908—21no—
58C23×C53—81no—
59D4×C53—42no—
60D4×C53—81no—
61C2×C4×C53C2×D811yescharacteristic
62C22⋊D53—81no—
63C2×Dic53—21no—
64C2×Dic53—81no—
65C2×Dic53—162no—
66D212—41no—
67C4×D53—81no—
68Dic106—41no—
69C8×C53—81no—
70C8×C53C4×D411yescharacteristic
71C8×C53#13511yescharacteristic
72#88322—21no—
73#13658—531nomaximal, Sylow
74C22×C53—82no—
75C22×C53—161no—
76C22×C53—161no—
77C22×C53#50911yescharacteristic
78C2×D53—81no—
79C4×C53C4×D811yescharacteristic
80C4×C53#50911yescharacteristic
81Dic53—41no—
82Dic53—81no—
83Dic53—161no—
84#2263—531no—
85#2179—531no—
86#2177—531no—
87#1426—531no—
88#1419D5311yescharacteristic, complemented
89#1418—531no—
90#1398—531no—
91C2×C53—21no—
92C2×C53—162no—
93C2×C53#217711yescharacteristic
94D53—161no—
95#508—531no—
96C2×D16C2×D5311yescharacteristic
97#443—1061no—
98C4×D8—1061no—
99SD64—532no—
100D32—532no—
101M64—531no—
102M64C2×D5311yescharacteristic
103#368C2×D5311yescharacteristic
104#367—531no—
105#359—1061no—
106#357—1061no—
107#346—531no—
108C53#1365811yescharacteristic, complemented, Sylow
109#135—1061no—
110C2×D8—21no—
111#131—2121no—
112C4×D4—2121no—
113Q32—531no—
114SD32—1061no—
115D16—21no—
116D16—531no—
117D16C4×D5312yes—
118C2×C16—531no—
119C2×C16C22×D5311yescharacteristic
120#108—21no—
121#107—1061no—
122#104—2121no—
123#102—2121no—
124C4×C8—531no—
125C32—21no—
126C32—532no—
127#41—2121no—
128C2×D4—41no—
129C22×C4—4241no—
130Q16—1061no—
131SD16—2121no—
132D8—22no—
133D8—41no—
134D8—1061no—
135M16—41no—
136M16—4241no—
137C2×C8—531no—
138C2×C8—4241no—
139C2×C8D4×D5311yescharacteristic
140C4⋊C4—2121no—
141C22⋊C4—4241no—
142C42—531no—
143C16—1061no—
144C16C2×C4×D5311yescharacteristic
145C16#1171994711yescharacteristic
146C23—81no—
147Q8—2121no—
148D4—42no—
149D4—81no—
150D4—4241no—
151D4—2121no—
152C2×C4—8482no—
153C2×C4—1061no—
154C2×C4—4241no—
155C2×C4—4241no—
156C2×C4D8×D5311yescharacteristic
157C8—81no—
158C8—1061no—
159C8#1313317711yescharacteristic
160C8#1313323811yescharacteristic
161C22—82no—
162C22—161no—
163C22—161no—
164C22—4241no—
165C22#1681106011yescharacteristic
166C4—8481no—
167C4—4241no—
168C4—2121no—
169C4#1681070311yescharacteristic
170C4#1681105711yescharacteristic
171C2—21no—
172C2—162no—
173C2—8481no—
174C2#2027658811yescharacteristic
subgroupG/H
centerC2#20276588
commutatorC16×C53C22×C4
FrattiniC2×C16C22×D53
FittingC53×#1419C2
socleC2×C53#2177
prime radical#20277028C2
cosocleC2×C16×C53C23
hypercenter#1419D53
hypocenterC53#13658
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
2. #20276600C2semidirect
3. #20276591C2semidirect
4. #20272922C2semidirect
5. #20272903C2semidirect
6. #20272850C2semidirect
7. C53×#1419C2semidirect
11. #16810461C4semidirect
12. #16810458C4semidirect
24. Dic424C2×C4semidirect
26. D848C2×C4semidirect
32. #13133123C2×C4semidirect
88. #1419D53semidirect
108. C53#13658semidirect

Maximal subgroups

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

subgroupindex
#202770282
#202766002
#202765912
#202729222
#202729032
#202728502
C53×#14192
#1365853