Finite Group Databasebeta

← #27271688

Subgroups — #27271688

Order 13568.

210 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 37 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#20276231C211yesmaximal, characteristic
2#20272651C211yesmaximal, characteristic
3C53×#1297C211yesmaximal, characteristic
4#16810672C411yescharacteristic, complemented
5#16810671—21no—
6#16810587—21no—
7#16810405—21no—
8#16810381C411yescharacteristic, complemented
9C53×#407C2211yescharacteristic
10C4×C22⋊C4×C53—21no—
11C2×#11719917—21no—
12C2×#11719917—21no—
13#13133157—41no—
14#13133153—41no—
15#13133152—42no—
16C22⋊#88322—42no—
17C23×C4×C53D411yescharacteristic
18C2×M16×C53C2×C411yescharacteristic
19C2×M16×C53D411yescharacteristic
20C2×C22⋊C4×C53—41no—
21C2×C42×C53—41no—
22C22⋊C8×C53—41no—
23#95×C53—22no—
24C24×C53SD1611yescharacteristic
25C22×C4×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53—41no—
28C22×C4×C53—41no—
29C22×C4×C53—42no—
30C22×C4×C53C22⋊C411yescharacteristic
31C22×C4×C53Q1611yescharacteristic
32M16×C53—41no—
33M16×C53—41no—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C22⋊C4×C53—44no—
37C42×C53—81no—
38C42×C53—81no—
39#11719917—41no—
40#11719917—41no—
41#11719917—41no—
42#11719917—41no—
43C2×#88322—22no—
44C2×#88322—41no—
45#11719908—81no—
46C23×C53—21no—
47C23×C53—41no—
48C23×C53—42no—
49C23×C53#10311yescharacteristic
50C2×C4×C53—21no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—44no—
57C2×C4×C53—81no—
58C2×C4×C53—82no—
59C2×C4×C53—82no—
60C2×C4×C53#9911yescharacteristic
61C2×C4×C53#10411yescharacteristic
62C8×C53—41no—
63C8×C53—41no—
64#88322—42no—
65#88322—81no—
66#7432—531nomaximal, Sylow
67C22×C53—22no—
68C22×C53—21no—
69C22×C53—41no—
70C22×C53—41no—
71C22×C53—42no—
72C22×C53—41no—
73C22×C53—81no—
74C22×C53#32811yescharacteristic
75C4×C53—21no—
76C4×C53—41no—
77C4×C53—84no—
78C4×C53#35211yescharacteristic
79C4×C53#35311yescharacteristic
80#2091—531no—
81#1320—531no—
82#1297D5311yescharacteristic
83C2×C53—21no—
84C2×C53—42no—
85C2×C53—41no—
86C2×C53#135211yescharacteristic
87#430—531no—
88#430—1061no—
89#407—1061no—
90#407C2×D5311yescharacteristic
91C4×C22⋊C4—21no—
92#349—1061no—
93#338—531no—
94C53#743211yescharacteristic, complemented, Sylow
95C23×C4C22⋊D5311yescharacteristic
96C2×M16—1061no—
97C2×M16—1061no—
98C2×M16C4×D5311yescharacteristic
99C2×M16D21211yescharacteristic
100C2×C22⋊C4—41no—
101C2×C42—41no—
102M32—2121no—
103#108—2121no—
104#108—2122no—
105C22⋊C8—41no—
106C22⋊C8—2122no—
107#95—22no—
108C24#1171992311yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4#1171992111yescharacteristic
115C22×C4#1171992511yescharacteristic
116M16—41no—
117M16—41no—
118M16—2121no—
119M16—2121no—
120M16—2121no—
121M16—2121no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—1062no—
125C2×C8—2121no—
126C22⋊C4—44no—
127C42—81no—
128C42—81no—
129C16—4241no—
130C23—21no—
131C23—42no—
132C23—41no—
133C23#1313314011yescharacteristic
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—44no—
141C2×C4—81no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#1313313111yescharacteristic
145C2×C4#1313314511yescharacteristic
146C8—41no—
147C8—41no—
148C8—4241no—
149C8—2122no—
150C22—22no—
151C22—21no—
152C22—41no—
153C22—41no—
154C22—42no—
155C22—41no—
156C22—81no—
157C22#1681035911yescharacteristic
158C4—21no—
159C4—41no—
160C4—84no—
161C4#1681041211yescharacteristic
162C4#1681041511yescharacteristic
163C2—21no—
164C2—42no—
165C2—41no—
166C2#2027273811yescharacteristic
subgroupG/H
centerC4#16810415
commutatorC2×M16×C53C2×C4
Frattini#407C2×D53
FittingC53×#1297C2
socleC2×C53#1352
prime radicalC24×C53SD16
cosocleC53×#407C22
hypercenter#1297D53
hypocenterC53#7432
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1297

Complements

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

NH ≅ G/Ntype
4. #16810672C4semidirect
8. #16810381C4semidirect
94. C53#7432semidirect

Maximal subgroups

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

subgroupindex
#202762312
#202726512
C53×#12972
#743253