Finite Group Databasebeta

← #27277634

Subgroups — #27277634

Order 13568.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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#20273042C212yesmaximal
2#20272621C214yesmaximal, complemented
3C53×#2312C211yesmaximal, characteristic
4#16810587C2214yes—
5#16810347—24no—
6C22×C22⋊C4×C53C2211yescharacteristic
7C53×#388—24no—
8C53×#380C2212yes—
9C53×#379C412yes—
10C4×C22⋊C4×C53C412yes—
11C2×#11719917—24no—
12C22⋊#88322—24no—
13C22⋊#88322—24no—
14C25×C53C2×C411yescharacteristic
15C23×C4×C53C2×C411yescharacteristic
16C23×C4×C53C2311yescharacteristic
17C2×C4⋊C4×C53—22no—
18C2×C22⋊C4×C53—22no—
19C2×C22⋊C4×C53—22no—
20C2×C22⋊C4×C53—24no—
21C2×C22⋊C4×C53C2×C414yes—
22C2×C22⋊C4×C53D414yes—
23C2×C42×C53—22no—
24#95×C53—22no—
25#95×C53—22no—
26C24×C53—22no—
27C24×C53—42no—
28C24×C53—81no—
29C24×C53—81no—
30C24×C53C22⋊C412yes—
31C24×C53C22×C411yescharacteristic
32C22×C4×C53—22no—
33C22×C4×C53—24no—
34C22×C4×C53—81no—
35C22×C4×C53—81no—
36C22×C4×C53C22⋊C412yes—
37C22×C4×C53C2×D412yes—
38C22×C4×C53#4114yes—
39C4⋊C4×C53—82no—
40C22⋊C4×C53—28no—
41C22⋊C4×C53—44no—
42C22⋊C4×C53—82no—
43C22⋊C4×C53—82no—
44C42×C53—82no—
45#11719917—84no—
46C2×#88322—44no—
47C23×C53—22no—
48C23×C53—22no—
49C23×C53—24no—
50C23×C53—41no—
51C23×C53—41no—
52C23×C53—44no—
53C23×C53—42no—
54C23×C53—42no—
55C23×C53—82no—
56C23×C53—82no—
57C23×C53—88no—
58C23×C53C2×C22⋊C411yescharacteristic
59C23×C53#11712yes—
60C2×C4×C53—22no—
61C2×C4×C53—42no—
62C2×C4×C53—44no—
63C2×C4×C53—42no—
64C2×C4×C53—44no—
65C2×C4×C53—44no—
66C2×C4×C53—44no—
67C2×C4×C53—82no—
68C2×C4×C53#12314yes—
69#88322—84no—
70#8759—531nomaximal, Sylow
71C22×C53—21no—
72C22×C53—21no—
73C22×C53—22no—
74C22×C53—22no—
75C22×C53—22no—
76C22×C53—44no—
77C22×C53—42no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—84no—
81C22×C53—88no—
82C22×C53#38111yescharacteristic
83C22×C53#41312yes—
84C4×C53—24no—
85C4×C53—42no—
86C4×C53—84no—
87#2312D5311yescharacteristic
88#1462—532no—
89#1305—534no—
90C2×C53—22no—
91C2×C53—42no—
92C2×C53—82no—
93C2×C53#182211yescharacteristic
94C2×C53#182811yescharacteristic
95C2×C53#182911yescharacteristic
96C22×C22⋊C4C2×D5311yescharacteristic
97#407—534no—
98#388—24no—
99#380C2×D5312yes—
100#379Dic5312yes—
101C4×C22⋊C4Dic5312yes—
102#323—1064no—
103C53#875911yescharacteristic, complemented, Sylow
104C25C2×Dic5311yescharacteristic
105C23×C4C2×Dic5311yescharacteristic
106C23×C4C22×D5311yescharacteristic
107C2×M16—1064no—
108C2×C4⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4C2×Dic5314yes—
113C2×C22⋊C4C22⋊D5314yes—
114C2×C42—22no—
115C22⋊C8—1064no—
116C22⋊C8—1064no—
117#95—22no—
118#95—22no—
119C24—22no—
120C24—42no—
121C24—81no—
122C24—81no—
123C24C22⋊Dic5312yes—
124C24C22×Dic5311yescharacteristic
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—81no—
128C22×C4—81no—
129C22×C4C22⋊Dic5312yes—
130C22×C4#1171994514yes—
131C22×C4C2×C22⋊D5312yes—
132M16—4244no—
133C2×C8—2124no—
134C4⋊C4—82no—
135C22⋊C4—28no—
136C22⋊C4—44no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C42—82no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23—41no—
144C23—41no—
145C23—44no—
146C23—42no—
147C23—42no—
148C23—82no—
149C23—82no—
150C23—88no—
151C23C2×C22⋊Dic5311yescharacteristic
152C23#1313317612yes—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—82no—
161C2×C4#1313320014yes—
162C8—4244no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—84no—
173C22—88no—
174C22#1681048611yescharacteristic
175C22#1681060712yes—
176C4—24no—
177C4—42no—
178C4—84no—
179C2—22no—
180C2—42no—
181C2—82no—
182C2#2027479811yescharacteristic
183C2#2027482311yescharacteristic
184C2#2027482611yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2312C2
socleC22×C53#381
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2312D53
hypocenterC53#8759
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20272621C2semidirect
103. C53#8759semidirect

Maximal subgroups

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

subgroupindex
#202730422
#202726212
C53×#23122
#875953