Finite Group Databasebeta

← #28309616

Subgroups — #28309616

Order 13568.

280 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 60 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#20291084C212yesmaximal, complemented
2#20289722C212yesmaximal, complemented
3#20289718C212yesmaximal, complemented
4#20285118C211yesmaximal, characteristic
5#16811546—22no—
6#16811519—22no—
7#16811515C2214yes—
8#16811173C411yescharacteristic
9#16810993C411yescharacteristic
10#16810993C2211yescharacteristic
11#16810989C411yescharacteristic
12#16810989C2211yescharacteristic
13#16810980—22no—
14#16810622—22no—
15C4×#11719917C2211yescharacteristic
16C2×#125×C53C411yescharacteristic
17#13133291—24no—
18C2×#11719945—21no—
19C2×#11719917D412yes—
20#13133217C2×C411yescharacteristic
21#13133214C2×C411yescharacteristic
22#13133205—21no—
23#13133191—41no—
24C4×D212—41no—
25#13133140—22no—
26#13133140—22no—
27#13133137—22no—
28#13133137—22no—
29#13133128C2×C411yescharacteristic
30#13133128C2311yescharacteristic
31C4×#88322D412yes—
32#125×C53—21no—
33#125×C53—41no—
34#125×C53C2×C411yescharacteristic
35#125×C53C2×C411yescharacteristic
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—41no—
38C2×C42×C53C2×C411yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53—41no—
41C22×C4×C53C22⋊C411yescharacteristic
42C2×C22⋊D53—41no—
43#11719945—44no—
44C2×D212—21no—
45C2×D212—41no—
46C2×C4×D53—41no—
47C2×Dic106—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—41no—
54C42×C53—21no—
55C42×C53C22⋊C411yescharacteristic
56C42×C53C22×C411yescharacteristic
57#11719921—42no—
58C4⋊Dic53—41no—
59#11719917—42no—
60C2×#88322C22⋊C412yes—
61C2×#88322C2×D412yes—
62#11719910—44no—
63C23×C53#10011yescharacteristic
64C22×D53—41no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—81no—
71C2×C4×C53—82no—
72C2×C4×C53#10011yescharacteristic
73C2×C4×C53C2×C22⋊C411yescharacteristic
74C22⋊D53—82no—
75C2×Dic53—41no—
76D212—42no—
77D212—82no—
78C4×D53—82no—
79Dic106—42no—
80#88322—22no—
81#88322—22no—
82#11917—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—41no—
85C22×C53C2×#10011yescharacteristic
86C2×D53—82no—
87C2×D53—81no—
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—41no—
91C4×C53—81no—
92C4×C53—81no—
93C4×C53#42112yes—
94Dic53—81no—
95#3142—531no—
96#2165—532no—
97#1356—532no—
98#1355—532no—
99C2×C53—21no—
100C2×C53—41no—
101C2×C53#153311yescharacteristic
102D53—161no—
103#532—531no—
104C2×#125#615711yescharacteristic
105#491—531no—
106#491—531no—
107#490—531no—
108#490—531no—
109#488—1062no—
110#417—1062no—
111C4×M16—531no—
112#364—1062no—
113#349—1062no—
114#347—534no—
115C53#1191711yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C2×M16—532no—
118C4⋊Q8—531no—
119#127—531no—
120#125—21no—
121#125—41no—
122#125C2×#615711yescharacteristic
123#125C2×#615711yescharacteristic
124#124—1061no—
125#121—2121no—
126C4×D4—2121no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C42C2×#615711yescharacteristic
130M32—1064no—
131#103—1062no—
132#103—1062no—
133#102—1062no—
134#102—1062no—
135#97—531no—
136#97—531no—
137C4×C8—532no—
138#41—2124no—
139C2×Q8—1061no—
140C2×D4—1061no—
141C2×D4—2121no—
142C2×D4—2121no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—2121no—
146C22×C4C22⋊#615711yescharacteristic
147M16—2122no—
148C2×C8—532no—
149C2×C8—532no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—2121no—
157C22⋊C4—2122no—
158C42—21no—
159C42C22⋊#615711yescharacteristic
160C42C22×#615711yescharacteristic
161C16—2124no—
162C23—2121no—
163C23#1313327711yescharacteristic
164Q8—2122no—
165D4—4242no—
166D4—4242no—
167D4—2122no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—81no—
174C2×C4—82no—
175C2×C4—4242no—
176C2×C4—2121no—
177C2×C4#1313327711yescharacteristic
178C2×C4C2×C22⋊#615711yescharacteristic
179C8—1062no—
180C8—1062no—
181C22—21no—
182C22—41no—
183C22—4242no—
184C22—4241no—
185C22C2×#1313327711yescharacteristic
186C4—21no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4—81no—
191C4—4241no—
192C4#1681162212yes—
193C2—21no—
194C2—41no—
195C2—8481no—
196C2#2029019611yescharacteristic
subgroupG/H
centerC2#20290196
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#125×C53C4
socleC2×C53#1533
prime radical#16811173C4
cosocle#13133128C23
hypercenterC2×#125#6157
hypocenterC53#11917
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289722C2semidirect
3. #20289718C2semidirect
115. C53#11917semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897222
#202897182
#202851182
#1191753