Finite Group Databasebeta

← #28299098

Subgroups — #28299098

Order 13568.

182 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 32 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#20289648C211yesmaximal, characteristic
2#20289597C211yesmaximal, characteristic
3#20276230C211yesmaximal, characteristic
4D53⋊(C22⋊C8)—21no—
5#16811519—21no—
6#16810671—21no—
7#16810586—21no—
8#16810586C2211yescharacteristic
9#16810381C411yescharacteristic
10C53×#430C411yescharacteristic
11C2×D53⋊C8—41no—
12#13133291—41no—
13#13133274—22no—
14C22⋊#88324—41no—
15C22×C4×D53D411yescharacteristic
16C2×#11719917—21no—
17C2×#11719917D411yescharacteristic
18#13133152—42no—
19#13133129—41no—
20#13133129—42no—
21C2×M16×C53—21no—
22C2×M16×C53C2×C411yescharacteristic
23#108×C53—41no—
24C23×D53Q1611yescharacteristic
25C22×C4×C53C22⋊C411yescharacteristic
26C22×Dic53SD1611yescharacteristic
27C2×C4×D53—21no—
28C2×C4×D53—21no—
29C2×C4×D53—41no—
30C2×C4×D53—41no—
31C2×#88324—42no—
32D53⋊C8—81no—
33D53⋊C8—81no—
34M16×C53—41no—
35M16×C53—41no—
36M16×C53—41no—
37C2×C8×C53—21no—
38C2×C8×C53—41no—
39#11719917—41no—
40#11719917—41no—
41#11719917—41no—
42C2×#88322—22no—
43C2×#88322—21no—
44#11719910—81no—
45C23×C53#10311yescharacteristic
46C22×D53—21no—
47C22×D53—42no—
48C22×D53—41no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53#9911yescharacteristic
52C2×C4×C53#10411yescharacteristic
53C2×Dic53—21no—
54C2×Dic53—41no—
55C4×D53—41no—
56C4×D53—41no—
57C4×D53—81no—
58#88324—82no—
59C8×C53—41no—
60C8×C53—81no—
61#88322—42no—
62#88322—41no—
63#7433—531nomaximal, Sylow
64C22×C53—21no—
65C22×C53—41no—
66C22×C53#32811yescharacteristic
67C2×D53—22no—
68C2×D53—41no—
69C2×D53—41no—
70C2×D53—42no—
71C2×D53—81no—
72C4×C53—21no—
73C4×C53#35211yescharacteristic
74C4×C53#35311yescharacteristic
75Dic53—41no—
76#2091—531no—
77#1320—531no—
78#1298—531no—
79C2×C53—21no—
80C2×C53—41no—
81C2×C53#135211yescharacteristic
82D53—42no—
83#430—1061no—
84#430#615711yescharacteristic
85#407—531no—
86#407—1061no—
87C2×C22⋊C8—1061no—
88#349—1061no—
89#338—531no—
90C53#743311yescharacteristic, complemented, Sylow
91C23×C4—531no—
92C2×M16—21no—
93C2×M16—531no—
94C2×M16—1061no—
95C2×M16C2×#615711yescharacteristic
96C22×C8—2121no—
97M32—2121no—
98#108—41no—
99#108—2122no—
100C22⋊C8—1062no—
101C22⋊C8—2121no—
102C22⋊C8—2122no—
103C22⋊C8—2121no—
104C24—531no—
105C22×C4—531no—
106C22×C4—1061no—
107C22×C4—1061no—
108C22×C4—2121no—
109C22×C4—2121no—
110C22×C4C22⋊#615711yescharacteristic
111M16—41no—
112M16—41no—
113M16—41no—
114M16—2121no—
115M16—2121no—
116M16—2121no—
117C2×C8—21no—
118C2×C8—41no—
119C2×C8—1062no—
120C2×C8—1061no—
121C2×C8—4241no—
122C2×C8—4241no—
123C2×C8—2122no—
124C16—4241no—
125C23—1061no—
126C23—2121no—
127C23—2122no—
128C23#1313328011yescharacteristic
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—1061no—
132C2×C4—4241no—
133C2×C4—2121no—
134C2×C4—2121no—
135C2×C4—2121no—
136C2×C4#1313327511yescharacteristic
137C2×C4#1313328211yescharacteristic
138C8—41no—
139C8—81no—
140C8—4242no—
141C8—2122no—
142C8—2121no—
143C22—21no—
144C22—41no—
145C22—1062no—
146C22—4241no—
147C22—2121no—
148C22—2122no—
149C22—2121no—
150C22#1681147711yescharacteristic
151C4—21no—
152C4—2121no—
153C4#1681152511yescharacteristic
154C4#1681152711yescharacteristic
155C2—21no—
156C2—41no—
157C2—2122no—
158C2#2028971211yescharacteristic
subgroupG/H
centerC4#16811527
commutatorC2×M16×C53C2×C4
FrattiniC2×M16C2×#6157
FittingC53×#430C4
socleC2×C53#1352
prime radicalC23×D53Q16
cosocle#16810586C22
hypercenter#430#6157
hypocenterC53#7433
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16 ◁ #430

Complements

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

NH ≅ G/Ntype
90. C53#7433semidirect

Maximal subgroups

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

subgroupindex
#202896482
#202895972
#202762302
#743353