Finite Group Databasebeta

← #28309614

Subgroups — #28309614

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#20289720C212yesmaximal, complemented
3#20289717C212yesmaximal, complemented
4#20285115C211yesmaximal, characteristic
5#16811546—22no—
6#16811519—22no—
7#16811515C2214yes—
8C2×#13133209C411yescharacteristic
9#16810991C411yescharacteristic
10#16810991C2211yescharacteristic
11#16810987C411yescharacteristic
12#16810987C2211yescharacteristic
13#16810977—22no—
14#16810624—22no—
15C4×#11719917C2211yescharacteristic
16C53×#532C411yescharacteristic
17#13133291—24no—
18C2×#11719917D412yes—
19#13133209—21no—
20#13133209—41no—
21#13133209C2×C411yescharacteristic
22#13133209C2×C411yescharacteristic
23C2×C4⋊Dic53—21no—
24C2×#11719919—41no—
25#13133141—22no—
26#13133141—22no—
27#13133138—22no—
28#13133138—22no—
29#13133128C2×C411yescharacteristic
30#13133128C2311yescharacteristic
31C4×#88322D412yes—
32C2×#41×C53—21no—
33C4⋊Q8×C53C2×C411yescharacteristic
34#127×C53C2×C411yescharacteristic
35#124×C53—21no—
36#121×C53—41no—
37C4×D4×C53—41no—
38C2×C42×C53C2×C411yescharacteristic
39#41×C53—44no—
40C2×Q8×C53—21no—
41C2×D4×C53—21no—
42C2×D4×C53—41no—
43C2×D4×C53—41no—
44C22×C4×C53—21no—
45C22×C4×C53—41no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×Dic53—41no—
48C4⋊C4×C53—41no—
49C22⋊C4×C53—42no—
50C42×C53—21no—
51C42×C53C22⋊C411yescharacteristic
52C42×C53C22×C411yescharacteristic
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—41no—
56#11719919—42no—
57#11719919—41no—
58#11719919—41no—
59#11719917—42no—
60C2×#88322C22⋊C412yes—
61C2×#88322C2×D412yes—
62#11719910—44no—
63C23×C53—41no—
64C23×C53#10011yescharacteristic
65Q8×C53—42no—
66D4×C53—42no—
67D4×C53—82no—
68D4×C53—82no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—41no—
73C2×C4×C53—82no—
74C2×C4×C53—81no—
75C2×C4×C53#10011yescharacteristic
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—82no—
80#88322—22no—
81#88322—22no—
82#11917—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—41no—
85C22×C53—82no—
86C22×C53—81no—
87C22×C53C2×#10011yescharacteristic
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—41no—
91C4×C53—81no—
92C4×C53#42112yes—
93Dic53—81no—
94Dic53—81no—
95#3142—531no—
96#2165—532no—
97#1356—532no—
98#1355—532no—
99C2×C53—21no—
100C2×C53—41no—
101C2×C53—161no—
102C2×C53#153311yescharacteristic
103#532#615711yescharacteristic
104C2×#125—531no—
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—21no—
117C2×M16—532no—
118C4⋊Q8C2×#615711yescharacteristic
119#127C2×#615711yescharacteristic
120#125—531no—
121#125—531no—
122#125—1061no—
123#125—2121no—
124#124—21no—
125#121—41no—
126C4×D4—41no—
127C2×C4⋊C4—1061no—
128C2×C4⋊C4—2121no—
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—44no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—41no—
142C2×D4—41no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—2121no—
146C22×C4C22⋊#615711yescharacteristic
147M16—2122no—
148C2×C8—532no—
149C2×C8—532no—
150C4⋊C4—41no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—2121no—
154C4⋊C4—2121no—
155C4⋊C4—2122no—
156C4⋊C4—2121no—
157C22⋊C4—42no—
158C42—21no—
159C42C22⋊#615711yescharacteristic
160C42C22×#615711yescharacteristic
161C16—2124no—
162C23—41no—
163C23#1313327711yescharacteristic
164Q8—42no—
165D4—42no—
166D4—82no—
167D4—82no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—4242no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4#1313327711yescharacteristic
178C2×C4C2×C22⋊#615711yescharacteristic
179C8—1062no—
180C8—1062no—
181C22—21no—
182C22—41no—
183C22—82no—
184C22—81no—
185C22C2×#1313327711yescharacteristic
186C4—21no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4—4241no—
191C4—4241no—
192C4#1681162112yes—
193C2—21no—
194C2—41no—
195C2—161no—
196C2#2029019411yescharacteristic
subgroupG/H
centerC2#20290194
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
Fitting, prime radicalC53×#532C4
socleC2×C53#1533
cosocle#13133128C23
hypercenter#532#6157
hypocenterC53#11917
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289720C2semidirect
3. #20289717C2semidirect
115. C53#11917semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897202
#202897172
#202851152
#1191753