Finite Group Databasebeta

← #27271462

Subgroups — #27271462

Order 13568.

200 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 39 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#20274790C211yesmaximal, characteristic
2#20272645C211yesmaximal, characteristic
3C53×#1317C211yesmaximal, characteristic
4C2×#13133176C411yescharacteristic
5#16810486C411yescharacteristic
6#16810389—21no—
7#16810389—21no—
8#16810347—21no—
9C53×#381C2211yescharacteristic
10C53×#323—21no—
11#13133176—41no—
12#13133176—41no—
13C2×C4⋊Dic53—21no—
14C2×C22⋊Dic53—21no—
15C2×C22⋊Dic53—21no—
16C2×C4×Dic53C812yescomplemented
17C22⋊#88322—41no—
18#13133124—41no—
19C23×C4×C53C2×C411yescharacteristic
20C2×C22⋊C4×C53D411yescharacteristic
21C2×C22⋊C4×C53D411yescharacteristic
22C22⋊C8×C53—41no—
23#95×C53—41no—
24C24×C53C22⋊C411yescharacteristic
25C22×C4×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53C2×C811yescharacteristic
32C22×C4×C53M1611yescharacteristic
33C22×Dic53—22no—
34C22×Dic53—41no—
35C2×C8×C53—81no—
36C22⋊C4×C53—41no—
37C22⋊C4×C53—41no—
38C22⋊Dic53—41no—
39C22⋊Dic53—41no—
40C22⋊Dic53—41no—
41C22⋊Dic53—41no—
42C4⋊Dic53—42no—
43C4×Dic53—42no—
44C2×#88322—81no—
45C23×C53—21no—
46C23×C53—21no—
47C23×C53—81no—
48C23×C53C22⋊C811yescharacteristic
49C23×C53#9911yescharacteristic
50C23×C53#10011yescharacteristic
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—42no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—81no—
61C2×C4×C53—81no—
62C2×Dic53—42no—
63C2×Dic53—82no—
64C2×Dic53—81no—
65C2×Dic53—82no—
66C8×C53—161no—
67#88322—161no—
68#7348—531nomaximal, Sylow
69C22×C53—21no—
70C22×C53—21no—
71C22×C53—41no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—81no—
75C22×C53—81no—
76C22×C53#32311yescharacteristic
77C22×C53#35211yescharacteristic
78C22×C53#35211yescharacteristic
79C4×C53—41no—
80C4×C53—41no—
81C4×C53—81no—
82C4×C53—81no—
83Dic53—82no—
84Dic53—161no—
85#1820—531no—
86#1317—531no—
87#1317D5311yescharacteristic
88C2×C53—21no—
89C2×C53—21no—
90C2×C53—21no—
91C2×C53—41no—
92C2×C53—41no—
93C2×C53#131711yescharacteristic
94C2×#117—531no—
95#381—531no—
96#381C2×D5311yescharacteristic
97#342—1061no—
98#342—1061no—
99#323—21no—
100#323—1061no—
101C53#734811yescharacteristic, complemented, Sylow
102C23×C4C4×D5311yescharacteristic
103#117—2121no—
104#117—2121no—
105C2×C4⋊C4—1061no—
106C2×C22⋊C4—1061no—
107C2×C22⋊C4—1061no—
108C2×C22⋊C4D21211yescharacteristic
109C2×C22⋊C4C22⋊D5311yescharacteristic
110C2×C42—532no—
111C22⋊C8—41no—
112C22⋊C8—2121no—
113#95—41no—
114#95—2121no—
115C24#1171992111yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—1062no—
123C22×C4—2121no—
124C22×C4C8×D5311yescharacteristic
125C22×C4#1171991211yescharacteristic
126C2×C8—81no—
127C2×C8—4241no—
128C4⋊C4—2122no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—2121no—
132C22⋊C4—2121no—
133C22⋊C4—2121no—
134C22⋊C4—2121no—
135C42—2122no—
136C23—21no—
137C23—21no—
138C23—81no—
139C23#1313312911yescharacteristic
140C23#1313313111yescharacteristic
141C23#1313313311yescharacteristic
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—4241no—
154C2×C4—4242no—
155C2×C4—4242no—
156C2×C4—2122no—
157C8—161no—
158C8—8481no—
159C22—21no—
160C22—21no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—81no—
165C22—81no—
166C22#1681034611yescharacteristic
167C22#1681041211yescharacteristic
168C22#1681041411yescharacteristic
169C4—41no—
170C4—41no—
171C4—81no—
172C4—81no—
173C4—8481no—
174C4—4242no—
175C2—21no—
176C2—21no—
177C2—21no—
178C2—41no—
179C2—41no—
180C2#2027264411yescharacteristic
subgroupG/H
centerC2#20272644
commutatorC22×C4×C53C2×C8
Frattini#381C2×D53
FittingC53×#1317C2
socleC2×C53#1317
prime radicalC24×C53C22⋊C4
cosocleC53×#381C22
hypercenter#1317D53
hypocenterC53#7348
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1317

Complements

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

NH ≅ G/Ntype
16. C2×C4×Dic53C8semidirect
101. C53#7348semidirect

Maximal subgroups

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

subgroupindex
#202747902
#202726452
C53×#13172
#734853