Finite Group Databasebeta

← #27295628

Subgroups — #27295628

Order 13568.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276527C212yesmaximal, complemented
2#20272756C212yesmaximal, complemented
3#20272754C212yesmaximal, complemented
4C53×#3152C211yesmaximal, characteristic
5#16810444C2214yes—
6#16810405—22no—
7#16810401—22no—
8C2×#137×C53—22no—
9C53×#532C411yescharacteristic
10C2×C4⋊Q8×C53C411yescharacteristic
11C53×#497—22no—
12C53×#494C411yescharacteristic
13C53×#494C2211yescharacteristic
14C53×#492C411yescharacteristic
15C53×#492C2211yescharacteristic
16C4×M16×C53C2211yescharacteristic
17#13133157—24no—
18C2×#41×C53—21no—
19C22×Q8×C53—21no—
20#137×C53—48no—
21C2×Q16×C53—22no—
22C2×Q16×C53—22no—
23C2×SD16×C53—22no—
24C2×SD16×C53—22no—
25C2×M16×C53D412yes—
26C4⋊Q8×C53—21no—
27C4⋊Q8×C53—41no—
28C4⋊Q8×C53C2×C411yescharacteristic
29C4⋊Q8×C53C2×C411yescharacteristic
30C4⋊Q8×C53C2×C411yescharacteristic
31#127×C53C2×C411yescharacteristic
32#124×C53—21no—
33#121×C53—41no—
34C4×D4×C53—41no—
35C2×C4⋊C4×C53—41no—
36C2×C42×C53C2×C411yescharacteristic
37#97×C53D412yes—
38C4×C8×C53C2×C411yescharacteristic
39C4×C8×C53C2311yescharacteristic
40#41×C53—44no—
41C2×Q8×C53—21no—
42C2×Q8×C53—21no—
43C2×Q8×C53—21no—
44C2×Q8×C53—44no—
45C2×Q8×C53—41no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53C22⋊C411yescharacteristic
53Q16×C53—44no—
54Q16×C53—44no—
55SD16×C53—44no—
56SD16×C53—44no—
57M16×C53—24no—
58C2×C8×C53C22⋊C412yes—
59C2×C8×C53C2×D412yes—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—42no—
65C42×C53—21no—
66C42×C53C22⋊C411yescharacteristic
67C42×C53C22×C411yescharacteristic
68#11719908—44no—
69C23×C53—41no—
70C23×C53#10011yescharacteristic
71Q8×C53—42no—
72Q8×C53—42no—
73Q8×C53—42no—
74Q8×C53—82no—
75D4×C53—42no—
76D4×C53—82no—
77D4×C53—82no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—42no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—82no—
86C2×C4×C53—82no—
87C2×C4×C53#10011yescharacteristic
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C8×C53—22no—
90C8×C53AGL(1,ℤ/8)14yes—
91#11993—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—82no—
95C22×C53—81no—
96C22×C53C2×#10011yescharacteristic
97C4×C53—22no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—81no—
101C4×C53—81no—
102C4×C53—81no—
103C4×C53#41812yes—
104#3152D5311yescharacteristic
105#2165—532no—
106#1359—532no—
107#1358—532no—
108C2×C53—21no—
109C2×C53—41no—
110C2×C53—161no—
111C2×C53#155511yescharacteristic
112C2×#137—22no—
113#532Dic5311yescharacteristic
114C2×C4⋊Q8Dic5311yescharacteristic
115#497—22no—
116#494Dic5311yescharacteristic
117#494C2×D5311yescharacteristic
118#492Dic5311yescharacteristic
119#492C2×D5311yescharacteristic
120C4×M16C2×D5311yescharacteristic
121#364—534no—
122#349—1062no—
123#347—1062no—
124C53#1199311yescharacteristic, complemented, Sylow
125C2×#41—21no—
126C22×Q8—21no—
127#137—48no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×SD16—22no—
131C2×SD16—22no—
132C2×M16C22⋊D5312yes—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×Dic5311yescharacteristic
136C4⋊Q8C2×Dic5311yescharacteristic
137C4⋊Q8C2×Dic5311yescharacteristic
138#127C2×Dic5311yescharacteristic
139#124—21no—
140#121—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C42C2×Dic5311yescharacteristic
144M32—1064no—
145#97C22⋊D5312yes—
146C4×C8C2×Dic5311yescharacteristic
147C4×C8C22×D5311yescharacteristic
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—44no—
153C2×Q8—41no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊Dic5311yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊Dic5312yes—
167C2×C8C2×C22⋊D5312yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊Dic5311yescharacteristic
175C42C22×Dic5311yescharacteristic
176C16—2124no—
177C23—41no—
178C23#1313313411yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#1313313411yescharacteristic
196C2×C4C2×C22⋊Dic5311yescharacteristic
197C8—22no—
198C8#1313324314yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#1313313411yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4#1681062812yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2#2027342911yescharacteristic
subgroupG/H
centerC2#20273429
commutatorC42×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#3152C2
socleC2×C53#1555
prime radicalC53×#532C4
cosocleC4×C8×C53C23
hypercenter#3152D53
hypocenterC53#11993
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #20276527C2semidirect
2. #20272756C2semidirect
3. #20272754C2semidirect
124. C53#11993semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202727562
#202727542
C53×#31522
#1199353