Finite Group Databasebeta

← #27287235

Subgroups — #27287235

Order 13568.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#20278020C211yesmaximal, characteristic
2#20273214C212yesmaximal
3#20272616C212yesmaximal
4#20272544C211yesmaximal, characteristic
5C53×#1995C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810565—22no—
8#16810557—22no—
9C4×#11719919—22no—
10C4×C4⋊Dic53C2212yes—
11#16810360—22no—
12#16810360—22no—
13C53×#533C2211yescharacteristic
14C53×#423C2211yescharacteristic
15C53×#376C2211yescharacteristic
16C53×#339—22no—
17C2×C4⋊Dic53—22no—
18C2×#11719919—22no—
19C2×#11719919—22no—
20C2×C4×Dic53—22no—
21C4⋊#88322—22no—
22C22⋊#88322—22no—
23C4×#88322—22no—
24#13133124—22no—
25#13133124—22no—
26#13133124D412yes—
27#13133124D412yes—
28C2×M16×C53—21no—
29C4⋊Q8×C53Q812yes—
30#125×C53—21no—
31C22⋊Q8×C53—22no—
32C4×Q8×C53—22no—
33#117×C53—21no—
34C2×C42×C53D412yes—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37#95×C53—22no—
38C2×Q8×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C2×D412yes—
41C22×C4×C53C2×D411yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—22no—
44M16×C53—42no—
45C2×C8×C53—41no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—42no—
49C22⋊C4×C53—41no—
50C42×C53—22no—
51C42×C53—22no—
52C42×C53—41no—
53C42×C53C2×Q811yescharacteristic
54C42×C53#4111yescharacteristic
55C42×C53#4112yes—
56C4⋊Dic53—44no—
57#11719919—44no—
58#11719919—44no—
59C4×Dic53—42no—
60C4×Dic53—42no—
61C2×#88322—42no—
62C23×C53#12011yescharacteristic
63Q8×C53—82no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—42no—
69C2×C4×C53C22⋊Q812yes—
70C2×C4×C53C22⋊Q811yescharacteristic
71C2×C4×C53#12412yes—
72C2×C4×C53#12411yescharacteristic
73C2×C4×C53#13512yes—
74C2×C4×C53#13712yes—
75C2×Dic53—24no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81C8×C53—81no—
82#88322—82no—
83#10373—531nomaximal, Sylow
84C22×C53—22no—
85C22×C53#39311yescharacteristic
86C22×C53#45212yes—
87C4×C53—22no—
88C4×C53—22no—
89C4×C53—82no—
90C4×C53—82no—
91C4×C53#47912yes—
92C4×C53#48811yescharacteristic
93C4×C53#49111yescharacteristic
94Dic53—44no—
95Dic53—82no—
96#2402—531no—
97#1995D5311yescharacteristic
98#1503—532no—
99#1303—532no—
100#1274—531no—
101C2×C53—22no—
102C2×C53#202611yescharacteristic
103C2×C53#203112yes—
104#533C2×D5311yescharacteristic
105#431—532no—
106#423C2×D5311yescharacteristic
107#400—1062no—
108#398—1062no—
109C4×C4⋊C4—532no—
110C4×C4⋊C4—1062no—
111#376C2×D5311yescharacteristic
112#339—22no—
113#328—1062no—
114#328—1062no—
115C53#1037311yescharacteristic, complemented, Sylow
116C2×M16—21no—
117C4⋊Q8Dic10612yes—
118#125—21no—
119C22⋊Q8—22no—
120C4×Q8—22no—
121#117—21no—
122C2×C4⋊C4—1062no—
123C2×C4⋊C4—1062no—
124C2×C4⋊C4—1062no—
125C2×C42—1062no—
126C2×C42C22⋊D5312yes—
127C2×C42C22×D5311yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—1062no—
130C22⋊C8—1062no—
131C4×C8—1062no—
132#95—22no—
133#95—532no—
134#95—532no—
135#95—1062no—
136#95—1062no—
137C2×Q8—22no—
138C22×C4—22no—
139C22×C4—1062no—
140C22×C4—1062no—
141C22×C4D4×D5312yes—
142C22×C4C2×C22⋊D5311yescharacteristic
143M16—42no—
144C2×C8—41no—
145C2×C8—2122no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—2124no—
150C4⋊C4—2124no—
151C4⋊C4—2124no—
152C22⋊C4—41no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—2122no—
157C42—2122no—
158C42C2×Dic10611yescharacteristic
159C42#1171994511yescharacteristic
160C42#1171994712yes—
161C23#1313318511yescharacteristic
162Q8—82no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—1064no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4#1313319412yes—
175C2×C4#1313319611yescharacteristic
176C2×C4#1313320312yes—
177C2×C4#1313320411yescharacteristic
178C2×C4#1313323712yes—
179C2×C4#1313324812yes—
180C8—81no—
181C8—4242no—
182C22—22no—
183C22#1681053511yescharacteristic
184C22#1681078312yes—
185C4—22no—
186C4—22no—
187C4—82no—
188C4—82no—
189C4—4242no—
190C4—2124no—
191C4#1681093012yes—
192C4#1681097711yescharacteristic
193C4#1681099111yescharacteristic
194C2—22no—
195C2#2027586611yescharacteristic
196C2#2027588512yes—
subgroupG/H
centerC2×C4#13133204
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1995C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1995D53
hypocenterC53#10373
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
115. C53#10373semidirect

Maximal subgroups

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

subgroupindex
#202780202
#202732142
#202726162
#202725442
C53×#19952
#1037353