Finite Group Databasebeta

← #27287236

Subgroups — #27287236

Order 13568.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#20275696C211yesmaximal, characteristic
3#20273215C212yesmaximal
4#20272617C212yesmaximal
5C53×#1274C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810647C2211yescharacteristic
8#16810565—22no—
9#16810557—22no—
10C4×#11719919—22no—
11C4×C4⋊Dic53C2212yes—
12#16810383—22no—
13#16810361—22no—
14#16810361—22no—
15C53×#431C2212yes—
16C53×#376C2211yescharacteristic
17C2×#11719917—21no—
18#13133217Q812yes—
19#13133209—21no—
20#13133196—22no—
21C4×Dic106—22no—
22#13133176—21no—
23C2×C4⋊Dic53—22no—
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×C4×Dic53—22no—
27C4⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124D412yes—
31#13133124D412yes—
32C2×C42×C53D412yes—
33C2×C42×C53C2311yescharacteristic
34C4⋊C8×C53—22no—
35C22⋊C8×C53—22no—
36C4×C8×C53—22no—
37#95×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53C2×D412yes—
40C22×C4×C53C2×D411yescharacteristic
41C22×Dic53—22no—
42C22×Dic53—22no—
43C2×Dic106—22no—
44C2×C8×C53—42no—
45C42×C53—22no—
46C42×C53—22no—
47C42×C53C2×Q811yescharacteristic
48C42×C53#4112yes—
49C42×C53#4111yescharacteristic
50C22⋊Dic53—41no—
51C4⋊Dic53—22no—
52C4⋊Dic53—22no—
53C4⋊Dic53—44no—
54#11719919—44no—
55#11719919—44no—
56#11719919—42no—
57C4×Dic53—41no—
58C4×Dic53—42no—
59C4×Dic53—42no—
60#11719917—42no—
61C2×#88322—41no—
62C23×C53#12011yescharacteristic
63C2×C4×C53—42no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53C22⋊Q812yes—
68C2×C4×C53C22⋊Q811yescharacteristic
69C2×C4×C53#12412yes—
70C2×C4×C53#12411yescharacteristic
71C2×C4×C53#13512yes—
72C2×C4×C53#13712yes—
73C2×Dic53—24no—
74C2×Dic53—42no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—42no—
79C2×Dic53—42no—
80Dic106—82no—
81C8×C53—82no—
82#88322—81no—
83#10373—531nomaximal, Sylow
84C22×C53—22no—
85C22×C53#39311yescharacteristic
86C22×C53#45212yes—
87C4×C53—22no—
88C4×C53—22no—
89C4×C53—82no—
90C4×C53#47912yes—
91C4×C53#48811yescharacteristic
92C4×C53#49111yescharacteristic
93Dic53—44no—
94Dic53—82no—
95Dic53—82no—
96#2402—531no—
97#1995—531no—
98#1503—532no—
99#1303—532no—
100#1274D5311yescharacteristic
101C2×C53—22no—
102C2×C53#202611yescharacteristic
103C2×C53#203112yes—
104#533—531no—
105#431C2×D5312yes—
106#423—531no—
107#400—1062no—
108#398—1062no—
109C4×C4⋊C4—532no—
110C4×C4⋊C4—1062no—
111#376C2×D5311yescharacteristic
112#339—1062no—
113#328—1062no—
114#328—1062no—
115C53#1037311yescharacteristic, complemented, Sylow
116C2×M16—1061no—
117C4⋊Q8—532no—
118#125—1061no—
119C22⋊Q8—1062no—
120C4×Q8—1062no—
121#117—1061no—
122C2×C4⋊C4—1062no—
123C2×C4⋊C4—1062no—
124C2×C4⋊C4—1062no—
125C2×C42—1062no—
126C2×C42D21212yes—
127C2×C42C22×D5311yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—1062no—
130C22⋊C8—22no—
131C4×C8—22no—
132#95—22no—
133#95—532no—
134#95—532no—
135#95—1062no—
136#95—1062no—
137C2×Q8—1062no—
138C22×C4—22no—
139C22×C4—1062no—
140C22×C4—1062no—
141C22×C4C2×D21211yescharacteristic
142C22×C4D4×D5312yes—
143M16—2122no—
144C2×C8—42no—
145C2×C8—2121no—
146C4⋊C4—1062no—
147C4⋊C4—1062no—
148C4⋊C4—2124no—
149C4⋊C4—2124no—
150C4⋊C4—2124no—
151C4⋊C4—2122no—
152C22⋊C4—2121no—
153C42—22no—
154C42—22no—
155C42—2121no—
156C42—2122no—
157C42—2122no—
158C42#1171994512yes—
159C42#1171994711yescharacteristic
160C42Q8×D5311yescharacteristic
161C23#1313318711yescharacteristic
162Q8—4242no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—1064no—
168C2×C4—2122no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4#1313319512yes—
175C2×C4#1313319711yescharacteristic
176C2×C4#1313320312yes—
177C2×C4#1313320511yescharacteristic
178C2×C4#1313323912yes—
179C2×C4#1313325012yes—
180C8—82no—
181C8—4241no—
182C22—22no—
183C22#1681053611yescharacteristic
184C22#1681078712yes—
185C4—22no—
186C4—22no—
187C4—82no—
188C4—4242no—
189C4—4242no—
190C4—2124no—
191C4#1681093112yes—
192C4#1681097811yescharacteristic
193C4#1681099211yescharacteristic
194C2—22no—
195C2#2027586711yescharacteristic
196C2#2027588612yes—
subgroupG/H
centerC2×C4#13133205
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1274C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1274D53
hypocenterC53#10373
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

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
#202756962
#202732152
#202726172
C53×#12742
#1037353