Finite Group Databasebeta

← C53×#9757 #27217269

Subgroups — C53×#9757 #27217269

Order 13568.

400 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 134 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×#3131C211yesmaximal, characteristic
2C53×#1835C211yesmaximal, characteristic, complemented
3C53×#1518C212yesmaximal, complemented
4C53×#1517C212yesmaximal
5C53×#1450C211yesmaximal, characteristic, complemented
6C53×#533C2211yescharacteristic
7C53×#532C2211yescharacteristic
8C53×#488—21no—
9C53×#488—21no—
10C53×#487C412yescomplemented
11C53×#486C412yescomplemented
12C53×#431C2212yes—
13C53×#416—22no—
14C2×C4×C8×C53C2211yescharacteristic
15C53×#382C2212yes—
16C53×#340—22no—
17C53×#328—24no—
18C53×#327—24no—
19C53×#322—21no—
20C82×C53—21no—
21C2×#41×C53—21no—
22C22×C8×C53D412yes—
23C4⋊Q8×C53C2×C412yes—
24C4⋊Q8×C53C2×C411yescharacteristic
25#127×C53C2×C411yescharacteristic
26#125×C53—21no—
27#124×C53—21no—
28C22⋊Q8×C53—22no—
29#121×C53—22no—
30C4×Q8×C53—22no—
31C4×D4×C53—22no—
32#117×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C42×C53C2311yescharacteristic
35C4⋊C8×C53—22no—
36#103×C53—22no—
37#103×C53—22no—
38#102×C53—22no—
39#102×C53—22no—
40C22⋊C8×C53D414yes—
41C4×C8×C53—22no—
42C4×C8×C53C2×C412yes—
43C4×C8×C53D411yescharacteristic
44C4×C8×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#41×C53—44no—
48C2×Q8×C53—21no—
49C2×Q8×C53—22no—
50C2×D4×C53—21no—
51C2×D4×C53—22no—
52C2×D4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53—42no—
55C22×C4×C53C2×D412yes—
56C22×C4×C53C2×D411yescharacteristic
57C2×C8×C53—24no—
58C2×C8×C53—24no—
59C2×C8×C53C22⋊C412yes—
60C2×C8×C53C22⋊C412yes—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—44no—
66C22⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C42×C53—41no—
69C42×C53C22×C411yescharacteristic
70C42×C53C2×D411yescharacteristic
71C42×C53#4112yes—
72C23×C53—41no—
73C23×C53#12711yescharacteristic
74Q8×C53—42no—
75Q8×C53—82no—
76D4×C53—42no—
77D4×C53—82no—
78D4×C53—82no—
79C2×C4×C53—42no—
80C2×C4×C53—41no—
81C2×C4×C53—42no—
82C2×C4×C53—82no—
83C2×C4×C53—84no—
84C2×C4×C53—82no—
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53C4×D412yes—
87C2×C4×C53#12112yes—
88C2×C4×C53#12411yescharacteristic
89C2×C4×C53#13514yes—
90C8×C53—22no—
91C8×C53—22no—
92C8×C53—44no—
93#9757C5311yesmaximal, characteristic, direct product factor, Sylow
94C22×C53—22no—
95C22×C53—81no—
96C22×C53—82no—
97C22×C53#39011yescharacteristic
98C22×C53#49512yes—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—81no—
102C4×C53—82no—
103C4×C53—162no—
104C4×C53#41612yes—
105C4×C53#48812yes—
106#3131C2×C5311yescharacteristic
107#1835C2×C5311yescharacteristic, complemented
108#1518C2×C5312yescomplemented
109#1517C2×C5312yes—
110#1450C2×C5311yescharacteristic, complemented
111C2×C53—22no—
112C2×C53—161no—
113C2×C53#195811yescharacteristic
114C2×C53#197612yes—
115#533C22×C5311yescharacteristic
116#532C22×C5311yescharacteristic
117#488—21no—
118#488—21no—
119#487C4×C5312yescomplemented
120#486C4×C5312yescomplemented
121#431C22×C5312yes—
122#416—22no—
123C2×C4×C8C22×C5311yescharacteristic
124#382C22×C5312yes—
125#340—22no—
126#328—24no—
127#327—24no—
128#322—21no—
129C82—21no—
130C53#975711yescharacteristic, direct product factor, Sylow
131C2×#41—21no—
132C22×C8D4×C5312yes—
133C4⋊Q8C2×C4×C5311yescharacteristic
134C4⋊Q8C2×C4×C5312yes—
135#127C2×C4×C5311yescharacteristic
136#125—21no—
137#124—21no—
138C22⋊Q8—22no—
139#121—22no—
140C4×Q8—22no—
141C4×D4—22no—
142#117—21no—
143C2×C4⋊C4—22no—
144C2×C42C23×C5311yescharacteristic
145C4⋊C8—22no—
146#103—22no—
147#103—22no—
148#102—22no—
149#102—22no—
150C22⋊C8D4×C5314yes—
151C4×C8—22no—
152C4×C8C2×C4×C5312yes—
153C4×C8D4×C5311yescharacteristic
154C4×C8D4×C5311yescharacteristic
155#95—22no—
156#95—22no—
157#41—44no—
158C2×Q8—21no—
159C2×Q8—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4C2×D4×C5312yes—
166C22×C4C2×D4×C5311yescharacteristic
167C2×C8—24no—
168C2×C8—24no—
169C2×C8C22⋊C4×C5312yes—
170C2×C8C22⋊C4×C5312yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—41no—
179C42C22×C4×C5311yescharacteristic
180C42C2×D4×C5311yescharacteristic
181C42#41×C5312yes—
182C23—41no—
183C23#127×C5311yescharacteristic
184Q8—42no—
185Q8—82no—
186D4—42no—
187D4—82no—
188D4—82no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4C2×C22⋊C4×C5311yescharacteristic
196C2×C4C4×D4×C5312yes—
197C2×C4#121×C5312yes—
198C2×C4#124×C5311yescharacteristic
199C2×C4#135×C5314yes—
200C8—22no—
201C8—22no—
202C8—44no—
203C22—22no—
204C22—81no—
205C22—82no—
206C22C53×#39011yescharacteristic
207C22C53×#49512yes—
208C4—22no—
209C4—22no—
210C4—81no—
211C4—82no—
212C4—162no—
213C4C53×#41612yes—
214C4C53×#48812yes—
215C2—22no—
216C2—161no—
217C2C53×#195811yescharacteristic
218C2C53×#197612yes—
subgroupG/H
center, socleC22×C53#390
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1835C2semidirect
3. C53×#1518C2semidirect
5. C53×#1450C2semidirect
10. C53×#487C4semidirect
11. C53×#486C4semidirect
93. #9757C53direct (N × H)
107. #1835C2×C53semidirect
108. #1518C2×C53semidirect
110. #1450C2×C53semidirect
119. #487C4×C53semidirect
120. #486C4×C53semidirect
130. C53#9757direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31312
C53×#18352
C53×#15182
C53×#15172
C53×#14502
#975753