Finite Group Databasebeta

← #28309612

Subgroups — #28309612

Order 13568.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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#20291086C212yesmaximal
2#20289722C211yesmaximal, characteristic, complemented
3#20289721C211yesmaximal, characteristic, complemented
4#20289720C212yesmaximal, complemented
5#20285094C211yesmaximal, characteristic
6#16811545—22no—
7#16811520—22no—
8#16811515C2212yes—
9#16811515C2212yes—
10#16811179C411yescharacteristic
11C4⋊Q8×D53C411yescharacteristic
12#16810993C2211yescharacteristic
13#16810992C411yescharacteristic
14#16810991C2211yescharacteristic
15#16810972—22no—
16#16810630—22no—
17#16810575C2211yescharacteristic
18C53×#491C411yescharacteristic
19#13133291—22no—
20#13133291—22no—
21C2×Q8×D53—21no—
22C2×#11719912D412yes—
23#13133217C2×C411yescharacteristic
24#13133216—21no—
25#13133216—41no—
26#13133215—21no—
27#13133209C2×C411yescharacteristic
28#13133195—41no—
29#13133182—41no—
30#13133175—21no—
31C4⋊C4×D53—41no—
32C42×D53C2×C411yescharacteristic
33#13133142—22no—
34#13133141—22no—
35#13133140—22no—
36#13133128C2311yescharacteristic
37C8×Dic53D412yes—
38C4⋊Q8×C53C2×C411yescharacteristic
39#125×C53C2×C411yescharacteristic
40#103×C53—22no—
41#97×C53C2×C411yescharacteristic
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×C53C22⋊C412yes—
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×#88322C2×D412yes—
65#11719912—42no—
66#11719910—42no—
67#11719910—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—532no—
101#1356—531no—
102#1356—531no—
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#491#615711yescharacteristic
112#487—1062no—
113#419—1062no—
114C4×M16—531no—
115#364—1062no—
116#349—1062no—
117#347—532no—
118#347—532no—
119C53#1191611yescharacteristic, complemented, Sylow
120C22×Q8—1061no—
121C2×M16—532no—
122C4⋊Q8—531no—
123C4⋊Q8—1061no—
124C4⋊Q8—1061no—
125C4⋊Q8—2121no—
126C4⋊Q8C2×#615711yescharacteristic
127#125—531no—
128#125C2×#615711yescharacteristic
129C22⋊Q8—2121no—
130C4×Q8—2121no—
131#117—1061no—
132C2×C4⋊C4—2121no—
133C2×C42—531no—
134M32—1062no—
135M32—1062no—
136#103—22no—
137#103—1062no—
138#103—1062no—
139#103—1062no—
140#97—531no—
141#97C2×#615711yescharacteristic
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×C8C22⋊#615712yes—
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—
166C42C22×#615711yescharacteristic
167C16—2122no—
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×C4C2×C22⋊#615711yescharacteristic
185C8—22no—
186C8—1062no—
187C22—1061no—
188C22—2121no—
189C22D53⋊#10011yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—1062no—
194C4—1061no—
195C4—4241no—
196C4—4241no—
197C4#1681162012yes—
198C2—21no—
199C2—2121no—
200C2#2029019511yescharacteristic
subgroupG/H
centerC2#20290195
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#491C4
socleC2×C53#1533
prime radicalC22×D53#100
cosocle#13133128C23
hypercenter#491#6157
hypocenterC53#11916
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #20289722C2semidirect
3. #20289721C2semidirect
4. #20289720C2semidirect
119. C53#11916semidirect

Maximal subgroups

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

subgroupindex
#202910862
#202897222
#202897212
#202897202
#202850942
#1191653