Finite Group Databasebeta

← #27287201

Subgroups — #27287201

Order 13568.

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

normal subgroups only

This group has 402 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#20283102C211yesmaximal, characteristic, complemented
2#20275543C211yesmaximal, characteristic
3#20273276C212yesmaximal
4#20272574C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810647C2211yescharacteristic
8C2×#13133145—22no—
9#16810521—22no—
10#16810507—24no—
11#16810489C2212yes—
12#16810358—24no—
13C53×#431C2212yes—
14C43×C53C2211yescharacteristic
15C2×#11719945—21no—
16C2×#11719917—21no—
17#13133217Q811yescharacteristic
18#13133214Q811yescharacteristic
19#13133205—21no—
20#13133191—22no—
21C4×D212—22no—
22C2×C4⋊Dic53—22no—
23C2×#11719919—24no—
24C4⋊#88322—22no—
25#13133145—44no—
26#13133124—24no—
27#13133124—22no—
28#13133124D414yes—
29C2×C42×C53—22no—
30C2×C42×C53D412yes—
31C2×C42×C53C2311yescharacteristic
32C4⋊C8×C53—22no—
33C22⋊C8×C53—22no—
34C4×C8×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53C2×D412yes—
37C22×C4×C53C2×D411yescharacteristic
38C2×C22⋊D53—41no—
39C22×Dic53—24no—
40#11719945—42no—
41#11719945—42no—
42C2×D212—21no—
43C2×D212—22no—
44C2×C4×D53—41no—
45C2×Dic106—21no—
46C2×C8×C53—42no—
47C42×C53—22no—
48C42×C53—22no—
49C42×C53—44no—
50C42×C53C2×Q811yescharacteristic
51C42×C53#4112yes—
52C42×C53#4111yescharacteristic
53#11719921—42no—
54C4⋊Dic53—22no—
55C4⋊Dic53—42no—
56C4⋊Dic53—42no—
57#11719919—48no—
58#11719917—42no—
59C2×#88322—41no—
60C23×C53#12011yescharacteristic
61C22×D53—41no—
62C2×C4×C53—24no—
63C2×C4×C53—42no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53C22⋊Q812yes—
67C2×C4×C53C22⋊Q811yescharacteristic
68C2×C4×C53#12412yes—
69C2×C4×C53#12411yescharacteristic
70C2×C4×C53#13512yes—
71C2×C4×C53AGL(1,ℤ/8)12yes—
72C22⋊D53—82no—
73C2×Dic53—48no—
74C2×Dic53—41no—
75C2×Dic53—44no—
76D212—42no—
77D212—82no—
78C4×D53—81no—
79C4×D53—81no—
80Dic106—42no—
81C8×C53—82no—
82#88322—81no—
83#10366—531nomaximal, Sylow
84C22×C53—22no—
85C22×C53#39311yescharacteristic
86C22×C53#44912yes—
87C2×D53—81no—
88C2×D53—82no—
89C4×C53—22no—
90C4×C53—22no—
91C4×C53—44no—
92C4×C53#47812yes—
93C4×C53#48811yescharacteristic
94C4×C53#49011yescharacteristic
95Dic53—84no—
96Dic53—81no—
97#2864—531no—
98#1962—531no—
99#1517—532no—
100#1286—532no—
101#1276D5311yescharacteristic, complemented
102C2×C53—22no—
103C2×C53#202511yescharacteristic
104C2×C53#203112yes—
105D53—161no—
106#532—531no—
107#431C2×D5312yes—
108#423—531no—
109C2×#104—1062no—
110#389—1062no—
111#387—1064no—
112#382—532no—
113C43C2×D5311yescharacteristic
114#327—1064no—
115C53#1036611yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C2×M16—1061no—
118C4⋊Q8—531no—
119#127—531no—
120#124—1061no—
121#121—1062no—
122C4×D4—1062no—
123C2×C4⋊C4—1064no—
124C2×C4⋊C4—1062no—
125C2×C42—22no—
126C2×C42D21212yes—
127C2×C42C22×D5311yescharacteristic
128C4⋊C8—22no—
129C4⋊C8—1062no—
130#104—2124no—
131C22⋊C8—22no—
132C4×C8—22no—
133#95—534no—
134#95—1064no—
135#95—1062no—
136#41—2122no—
137#41—2122no—
138C2×Q8—1061no—
139C2×D4—1062no—
140C2×D4—1061no—
141C2×D4—2121no—
142C22×C4—22no—
143C22×C4—1064no—
144C22×C4—2121no—
145C22×C4C2×D21211yescharacteristic
146C22×C4D4×D5312yes—
147M16—2122no—
148C2×C8—42no—
149C2×C8—2121no—
150C4⋊C4—1062no—
151C4⋊C4—2128no—
152C4⋊C4—2122no—
153C4⋊C4—2122no—
154C22⋊C4—2122no—
155C42—22no—
156C42—22no—
157C42—44no—
158C42#1171994512yes—
159C42#1171994711yescharacteristic
160C42Q8×D5311yescharacteristic
161C23—2121no—
162C23#1313318711yescharacteristic
163Q8—2122no—
164D4—4242no—
165D4—4242no—
166D4—2122no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—4241no—
172C2×C4—4241no—
173C2×C4—2128no—
174C2×C4—2124no—
175C2×C4—2121no—
176C2×C4#1313319512yes—
177C2×C4#1313319711yescharacteristic
178C2×C4#1313320312yes—
179C2×C4#1313320511yescharacteristic
180C2×C4#1313323912yes—
181C2×C4#1313324512yes—
182C8—82no—
183C8—4241no—
184C22—22no—
185C22—4241no—
186C22—4242no—
187C22#1681053611yescharacteristic
188C22#1681077012yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—4244no—
193C4—4241no—
194C4#1681092412yes—
195C4#1681097811yescharacteristic
196C4#1681098811yescharacteristic
197C2—22no—
198C2—8481no—
199C2#2027586211yescharacteristic
200C2#2027588812yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#393
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10366
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #20283102C2semidirect
4. #20272574C2semidirect
5. C53×#1276C2semidirect
101. #1276D53semidirect
115. C53#10366semidirect

Maximal subgroups

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

subgroupindex
#202831022
#202755432
#202732762
#202725742
C53×#12762
#1036653