Finite Group Databasebeta

← #27295287

Subgroups — #27295287

Order 13568.

274 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 61 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#20285094C211yesmaximal, characteristic
2#20276525C212yesmaximal
3#20272750C211yesmaximal, characteristic, complemented
4#20272749C212yesmaximal, complemented
5C53×#1356C211yesmaximal, characteristic, complemented
6#16811179C411yescharacteristic
7C4⋊Q8×D53C411yescharacteristic
8#16810993C411yescharacteristic
9#16810992C2211yescharacteristic
10#16810991C411yescharacteristic
11#16810972—22no—
12#16810630—22no—
13#16810575C2211yescharacteristic
14#16810443—22no—
15#16810404—22no—
16#16810401C2212yes—
17C53×#491C2211yescharacteristic
18C53×#347C2212yes—
19C2×Q8×D53—21no—
20C2×#11719912D412yes—
21#13133217C2×C411yescharacteristic
22#13133216—21no—
23#13133216—41no—
24#13133215—21no—
25#13133209C2×C411yescharacteristic
26#13133195—41no—
27#13133182—41no—
28#13133175—21no—
29C4⋊C4×D53—41no—
30C42×D53C2×C411yescharacteristic
31#13133157—22no—
32#13133142—22no—
33#13133141—22no—
34#13133140—22no—
35#13133128C2×C411yescharacteristic
36C8×Dic53D412yes—
37C4⋊Q8×C53C2×C411yescharacteristic
38#125×C53C2×C411yescharacteristic
39M32×C53—22no—
40#103×C53—22no—
41#97×C53C2311yescharacteristic
42C2×Q8×C53—21no—
43Q8×D53—44no—
44C2×C4×D53—21no—
45C2×C4×D53—41no—
46C2×C4×D53C22⋊C411yescharacteristic
47C2×Dic106—21no—
48C2×Dic106—41no—
49C2×Dic106—41no—
50C2×C8×C53C2×D412yes—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—41no—
54C42×C53C22×C411yescharacteristic
55#11719921—41no—
56C4⋊Dic53—21no—
57C4⋊Dic53—41no—
58#11719919—42no—
59#11719919—41no—
60#11719919—41no—
61C4×Dic53—21no—
62C4×Dic53—41no—
63C4×Dic53C22⋊C411yescharacteristic
64C2×#88322C22⋊C412yes—
65#11719912—42no—
66C16×C53—42no—
67#11719908—42no—
68C22×D53#10011yescharacteristic
69Q8×C53—42no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C2×C22⋊C411yescharacteristic
74C2×Dic53—21no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53#10011yescharacteristic
78C4×D53—22no—
79C4×D53—42no—
80C4×D53—82no—
81Dic106—42no—
82Dic106—82no—
83Dic106—82no—
84C8×C53—22no—
85#88322—22no—
86#11916—531nomaximal, Sylow
87C22×C53C2×#10011yescharacteristic
88C2×D53—21no—
89C2×D53—41no—
90C4×C53—41no—
91C4×C53—81no—
92C4×C53—81no—
93C4×C53#42112yes—
94Dic53—22no—
95Dic53—21no—
96Dic53—81no—
97Dic53—81no—
98#3139—531no—
99#2165—532no—
100#1356—531no—
101#1356—532no—
102#1356D5311yescharacteristic, complemented
103C2×C53—21no—
104C2×C53#153311yescharacteristic
105D53—41no—
106#533—531no—
107C2×C4⋊Q8—531no—
108#491—531no—
109#491—531no—
110#491—531no—
111#491C2×D5311yescharacteristic
112#487—1062no—
113#419—1062no—
114C4×M16—531no—
115#364—1062no—
116#349—1062no—
117#347—532no—
118#347C2×D5312yes—
119C53#1191611yescharacteristic, complemented, Sylow
120C22×Q8—1061no—
121C2×M16—532no—
122C4⋊Q8—531no—
123C4⋊Q8—1061no—
124C4⋊Q8—1061no—
125C4⋊Q8—2121no—
126C4⋊Q8C4×D5311yescharacteristic
127#125—531no—
128#125C4×D5311yescharacteristic
129C22⋊Q8—2121no—
130C4×Q8—2121no—
131#117—1061no—
132C2×C4⋊C4—2121no—
133C2×C42—531no—
134M32—22no—
135M32—1062no—
136#103—22no—
137#103—1062no—
138#103—1062no—
139#103—1062no—
140#97—531no—
141#97C22×D5311yescharacteristic
142C4×C8—532no—
143C2×Q8—21no—
144C2×Q8—1061no—
145C2×Q8—2121no—
146C2×Q8—2121no—
147C2×Q8—2124no—
148C22×C4—531no—
149C22×C4—1061no—
150C22×C4—2121no—
151M16—2122no—
152C2×C8—532no—
153C2×C8D4×D5312yes—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—1061no—
158C4⋊C4—2121no—
159C4⋊C4—2121no—
160C4⋊C4—2122no—
161C4⋊C4—2121no—
162C22⋊C4—2121no—
163C42—531no—
164C42—1061no—
165C42—2121no—
166C42C2×C4×D5311yescharacteristic
167C16—42no—
168C16—2122no—
169C23—531no—
170Q8—42no—
171Q8—4242no—
172Q8—4242no—
173Q8—2122no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—531no—
178C2×C4—1061no—
179C2×C4—1062no—
180C2×C4—4242no—
181C2×C4—2122no—
182C2×C4—2121no—
183C2×C4—2121no—
184C2×C4C22⋊C4×D5311yescharacteristic
185C8—22no—
186C8—1062no—
187C22—1061no—
188C22—2121no—
189C22#100×D5311yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—1062no—
194C4—1061no—
195C4—4241no—
196C4—4241no—
197C4#1681063812yes—
198C2—21no—
199C2—2121no—
200C2#2027333711yescharacteristic
subgroupG/H
centerC2#20273337
commutatorC42×C53C22×C4
Frattini#97C22×D53
FittingC53×#1356C2
socleC2×C53#1533
prime radicalC22×D53#100
cosocle#97×C53C23
hypercenter#1356D53
hypocenterC53#11916
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
3. #20272750C2semidirect
4. #20272749C2semidirect
5. C53×#1356C2semidirect
102. #1356D53semidirect
119. C53#11916semidirect

Maximal subgroups

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

subgroupindex
#202850942
#202765252
#202727502
#202727492
C53×#13562
#1191653