Finite Group Databasebeta

← #27304715

Subgroups — #27304715

Order 13568.

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

normal subgroups only

This group has 194 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#20277028C211yesmaximal, characteristic
2#20276602C211yesmaximal, characteristic, complemented
3#20272920C211yesmaximal, characteristic, complemented
4#20272905C211yesmaximal, characteristic, complemented
5#20272901C211yesmaximal, characteristic, complemented
6#20272851C211yesmaximal, characteristic, complemented
7C53×#2177C211yesmaximal, characteristic, complemented
8#16811055C2211yescharacteristic
9#16810741—21no—
10#16810461C412yescomplemented
11#16810458C412yescomplemented
12#16810457C2211yescharacteristic
13#16810457C2211yescharacteristic
14#16810453C2211yescharacteristic
15#16810431—21no—
16C2×D16×C53C2211yescharacteristic
17C4×D8×C53—21no—
18C53×#367C2211yescharacteristic
19C53×#357—21no—
20C53×#346C2211yescharacteristic
21#13133239—21no—
22#13133223—41no—
23Dic424C2×C411yescharacteristic, complemented
24#13133162—21no—
25D848C2×C411yescharacteristic, complemented
26#13133153—21no—
27#13133145—41no—
28#13133123—21no—
29#13133123C2×C412yescomplemented
30C2×D8×C53—21no—
31C4×D4×C53—41no—
32D16×C53—21no—
33D16×C53C2×C412yes—
34C2×C16×C53D411yescharacteristic
35C2×C16×C53C2311yescharacteristic
36#107×C53—21no—
37#102×C53—41no—
38C4×C8×C53D411yescharacteristic
39C2×D4×C53—41no—
40C22×C4×C53—81no—
41#11719945—41no—
42D8×C53—22no—
43D8×C53—41no—
44C2×C8×C53D811yescharacteristic
45C2×C8×C53C2×D411yescharacteristic
46C4⋊C4×C53—41no—
47C22⋊C4×C53—81no—
48C42×C53D811yescharacteristic
49#11719917—41no—
50Dic212—21no—
51D424—21no—
52#11719913—41no—
53#11719912—81no—
54C8×D53—81no—
55C16×C53—21no—
56C16×C53C22×C411yescharacteristic
57C16×C53#4111yescharacteristic
58C23×C53—81no—
59D4×C53—42no—
60D4×C53—81no—
61C2×C4×C53—21no—
62C2×C4×C53—81no—
63C2×C4×C53—162no—
64C2×C4×C53C2×D811yescharacteristic
65C22⋊D53—81no—
66D212—41no—
67C4×D53—81no—
68Dic106—41no—
69C8×C53—21no—
70C8×C53C4×D411yescharacteristic
71C8×C53#13511yescharacteristic
72#88322—81no—
73#13658—531nomaximal, Sylow
74C22×C53—82no—
75C22×C53—161no—
76C22×C53—161no—
77C22×C53#50911yescharacteristic
78C2×D53—81no—
79C4×C53—41no—
80C4×C53—161no—
81C4×C53C4×D811yescharacteristic
82C4×C53#50911yescharacteristic
83Dic53—81no—
84#2263—531no—
85#2179—531no—
86#2177D5311yescharacteristic, complemented
87#1426—531no—
88#1419—531no—
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—21no—
99SD64—532no—
100D32—532no—
101M64—531no—
102M64—531no—
103#368—531no—
104#367C2×D5311yescharacteristic
105#359—1061no—
106#357—21no—
107#346C2×D5311yescharacteristic
108C53#1365811yescharacteristic, complemented, Sylow
109#135—1061no—
110C2×D8—21no—
111#131—2121no—
112C4×D4—41no—
113Q32—531no—
114SD32—1061no—
115D16—21no—
116D16—531no—
117D16C4×D5312yes—
118C2×C16C22⋊D5311yescharacteristic
119C2×C16C22×D5311yescharacteristic
120#108—1061no—
121#107—21no—
122#104—2121no—
123#102—41no—
124C4×C8C22⋊D5311yescharacteristic
125C32—532no—
126C32—1061no—
127#41—2121no—
128C2×D4—41no—
129C22×C4—81no—
130Q16—1061no—
131SD16—2121no—
132D8—22no—
133D8—41no—
134D8—1061no—
135M16—4241no—
136M16—2121no—
137C2×C8—4241no—
138C2×C8#1171992211yescharacteristic
139C2×C8C2×C22⋊D5311yescharacteristic
140C4⋊C4—41no—
141C22⋊C4—81no—
142C42#1171992211yescharacteristic
143C16—21no—
144C16C2×C4×D5311yescharacteristic
145C16#1171994511yescharacteristic
146C23—81no—
147Q8—2121no—
148D4—42no—
149D4—81no—
150D4—4241no—
151D4—2121no—
152C2×C4—21no—
153C2×C4—81no—
154C2×C4—162no—
155C2×C4—4241no—
156C2×C4C2×#1171992211yescharacteristic
157C8—21no—
158C8—4241no—
159C8C4×C22⋊D5311yescharacteristic
160C8#1313323711yescharacteristic
161C22—82no—
162C22—161no—
163C22—161no—
164C22—4241no—
165C22#1681105911yescharacteristic
166C4—41no—
167C4—161no—
168C4—4241no—
169C4C4×#1171992211yescharacteristic
170C4#1681105811yescharacteristic
171C2—21no—
172C2—162no—
173C2—8481no—
174C2#2027658911yescharacteristic
subgroupG/H
centerC2#20276589
commutatorC16×C53C22×C4
FrattiniC2×C16C22×D53
FittingC53×#2177C2
socleC2×C53#2177
prime radical#20277028C2
cosocleC2×C16×C53C23
hypercenter#2177D53
hypocenterC53#13658
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #20276602C2semidirect
3. #20272920C2semidirect
4. #20272905C2semidirect
5. #20272901C2semidirect
6. #20272851C2semidirect
7. C53×#2177C2semidirect
10. #16810461C4semidirect
11. #16810458C4semidirect
23. Dic424C2×C4semidirect
25. D848C2×C4semidirect
29. #13133123C2×C4semidirect
86. #2177D53semidirect
108. C53#13658semidirect

Maximal subgroups

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

subgroupindex
#202770282
#202766022
#202729202
#202729052
#202729012
#202728512
C53×#21772
#1365853