Finite Group Databasebeta

← C53×#11993 #27219505

Subgroups — C53×#11993 #27219505

Order 13568.

360 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C53×#3152C211yesmaximal, characteristic
2C53×#2165C212yesmaximal, complemented
3C53×#1359C212yesmaximal, complemented
4C53×#1358C212yesmaximal, complemented
5C2×#137×C53—22no—
6C53×#532C411yescharacteristic
7C2×C4⋊Q8×C53C411yescharacteristic
8C53×#497—22no—
9C53×#494C411yescharacteristic
10C53×#494C2211yescharacteristic
11C53×#492C411yescharacteristic
12C53×#492C2211yescharacteristic
13C4×M16×C53C2211yescharacteristic
14C53×#364C2214yes—
15C53×#349—22no—
16C53×#347—22no—
17C2×#41×C53—21no—
18C22×Q8×C53—21no—
19#137×C53—48no—
20C2×Q16×C53—22no—
21C2×Q16×C53—22no—
22C2×SD16×C53—22no—
23C2×SD16×C53—22no—
24C2×M16×C53D412yes—
25C4⋊Q8×C53—21no—
26C4⋊Q8×C53—41no—
27C4⋊Q8×C53C2×C411yescharacteristic
28C4⋊Q8×C53C2×C411yescharacteristic
29C4⋊Q8×C53C2×C411yescharacteristic
30#127×C53C2×C411yescharacteristic
31#124×C53—21no—
32#121×C53—41no—
33C4×D4×C53—41no—
34C2×C4⋊C4×C53—41no—
35C2×C42×C53C2×C411yescharacteristic
36M32×C53—24no—
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
68C16×C53—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#11993C5311yesmaximal, characteristic, direct product factor, 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#3152C2×C5311yescharacteristic
105#2165C2×C5312yescomplemented
106#1359C2×C5312yescomplemented
107#1358C2×C5312yescomplemented
108C2×C53—21no—
109C2×C53—41no—
110C2×C53—161no—
111C2×C53#155511yescharacteristic
112C2×#137—22no—
113#532C4×C5311yescharacteristic
114C2×C4⋊Q8C4×C5311yescharacteristic
115#497—22no—
116#494C4×C5311yescharacteristic
117#494C22×C5311yescharacteristic
118#492C4×C5311yescharacteristic
119#492C22×C5311yescharacteristic
120C4×M16C22×C5311yescharacteristic
121#364C22×C5314yes—
122#349—22no—
123#347—22no—
124C53#1199311yescharacteristic, direct product factor, Sylow
125C2×#41—21no—
126C22×Q8—21no—
127#137—48no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×SD16—22no—
131C2×SD16—22no—
132C2×M16D4×C5312yes—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8C2×C4×C5311yescharacteristic
136C4⋊Q8C2×C4×C5311yescharacteristic
137C4⋊Q8C2×C4×C5311yescharacteristic
138#127C2×C4×C5311yescharacteristic
139#124—21no—
140#121—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C42C2×C4×C5311yescharacteristic
144M32—24no—
145#97D4×C5312yes—
146C4×C8C2×C4×C5311yescharacteristic
147C4×C8C23×C5311yescharacteristic
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⋊C4×C5311yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊C4×C5312yes—
167C2×C8C2×D4×C5312yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊C4×C5311yescharacteristic
175C42C22×C4×C5311yescharacteristic
176C16—44no—
177C23—41no—
178C23#100×C5311yescharacteristic
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#100×C5311yescharacteristic
196C2×C4C2×C22⋊C4×C5311yescharacteristic
197C8—22no—
198C8AGL(1,ℤ/8)×C5314yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×#100×C5311yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4C53×#41812yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2C53×#155511yescharacteristic
subgroupG/H
center, socleC2×C53#1555
commutatorC42C22×C4×C53
Frattini, cosocleC4×C8C23×C53
prime radicalC53×#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C4×M16×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2165C2semidirect
3. C53×#1359C2semidirect
4. C53×#1358C2semidirect
91. #11993C53direct (N × H)
105. #2165C2×C53semidirect
106. #1359C2×C53semidirect
107. #1358C2×C53semidirect
124. C53#11993direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31522
C53×#21652
C53×#13592
C53×#13582
#1199353