Finite Group Databasebeta

← #28299075

Subgroups — #28299075

Order 13568.

146 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 34 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#20289831C211yesmaximal, characteristic, complemented
2#20289785C211yesmaximal, characteristic, complemented
3#20276815C211yesmaximal, characteristic
4D53⋊#108—21no—
5#16811555—21no—
6C2×#13133156C2211yescharacteristic
7#16810856—21no—
8#16810451C411yescharacteristic
9#16810424—21no—
10C2×D16×C53C411yescharacteristic
11D53⋊M16—41no—
12#13133288—41no—
13#13133287—22no—
14#13133267—41no—
15C2×C8×D53D411yescharacteristic
16#13133188—41no—
17#13133156—41no—
18C2×#11719908D411yescharacteristic
19#13133151—21no—
20#13133138—41no—
21C2×D8×C53—21no—
22D16×C53—41no—
23C2×C16×C53C2×C411yescharacteristic
24C2×D4×C53—41no—
25C2×C22⋊D53—81no—
26C2×C4×D53D811yescharacteristic
27#11719940—81no—
28#11719936—42no—
29D53⋊C8—81no—
30D8×C53—41no—
31D8×C53—41no—
32C2×C8×C53C22⋊C411yescharacteristic
33C22⋊Dic53—81no—
34C4⋊Dic53—41no—
35C2×#88322SD1611yescharacteristic
36C8×D53—21no—
37C8×D53—21no—
38C16×C53—21no—
39#11719908—21no—
40C23×C53—81no—
41C22×D53SD3211yescharacteristic
42D4×C53—81no—
43D4×C53—81no—
44C2×C4×C53#10211yescharacteristic
45C22⋊D53—162no—
46C2×Dic53—81no—
47C2×Dic53D1611yescharacteristic
48C4×D53—21no—
49C4×D53—21no—
50#88324—82no—
51C8×C53#9911yescharacteristic
52C8×C53#10411yescharacteristic
53#88322—21no—
54#7424—531nomaximal, Sylow
55C22×C53—161no—
56C22×C53—161no—
57C22×C53—161no—
58C22×C53#35711yescharacteristic
59C2×D53—21no—
60C2×D53—41no—
61C4×C53#32711yescharacteristic
62C4×C53#36011yescharacteristic
63Dic53—21no—
64Dic53—161no—
65#2218—531no—
66#1400—531no—
67#1385—531no—
68C2×C53—21no—
69C2×C53—321no—
70C2×C53#134711yescharacteristic
71D53—41no—
72C2×D16#615711yescharacteristic
73C2×M32—531no—
74#466—1061no—
75C2×#108—1061no—
76M64—1061no—
77#367—531no—
78#357—1061no—
79C53#742411yescharacteristic, complemented, Sylow
80C2×D8—21no—
81C2×M16—2121no—
82C22×C8—531no—
83#121—2121no—
84D16—41no—
85M32—2121no—
86C2×C16—531no—
87C2×C16C2×#615711yescharacteristic
88#108—1062no—
89#108—2121no—
90#107—1061no—
91#102—2121no—
92C32—2121no—
93C2×D4—41no—
94C2×D4—4241no—
95C22×C4—531no—
96D8—41no—
97D8—41no—
98M16—4241no—
99M16—2122no—
100C2×C8—531no—
101C2×C8—1061no—
102C2×C8—1061no—
103C2×C8—4241no—
104C2×C8C22⋊#615711yescharacteristic
105C4⋊C4—2121no—
106C22⋊C4—4241no—
107C16—21no—
108C16—1061no—
109C23—81no—
110C23—531no—
111D4—81no—
112D4—81no—
113D4—8482no—
114C2×C4—531no—
115C2×C4—1061no—
116C2×C4—1061no—
117C2×C4—4241no—
118C2×C4#1313327911yescharacteristic
119C8—1061no—
120C8—4242no—
121C8#1313327511yescharacteristic
122C8#1313328111yescharacteristic
123C22—161no—
124C22—161no—
125C22—161no—
126C22—1061no—
127C22—2121no—
128C22#1681153511yescharacteristic
129C4—8481no—
130C4—1061no—
131C4#1681147411yescharacteristic
132C4#1681153911yescharacteristic
133C2—21no—
134C2—321no—
135C2—2121no—
136C2#2028970111yescharacteristic
subgroupG/H
centerC2#20289701
commutatorC2×C16×C53C2×C4
FrattiniC2×C16C2×#6157
FittingC2×D16×C53C4
socleC2×C53#1347
prime radical#20276815C2
cosocleC2×#13133156C22
hypercenterC2×D16#6157
hypocenterC53#7424
LCSG ▷ C2×C16×C53 ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #20289831C2semidirect
2. #20289785C2semidirect
79. C53#7424semidirect

Maximal subgroups

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

subgroupindex
#202898312
#202897852
#202768152
#742453