Finite Group Databasebeta

← C53×#9442 #27216954

Subgroups — C53×#9442 #27216954

Order 13568.

456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3142C211yesmaximal, characteristic
2C53×#2371C211yesmaximal, characteristic, complemented
3C53×#1534C211yesmaximal, characteristic, complemented
4C53×#1505C212yesmaximal
5C53×#1300C212yesmaximal, complemented
6C53×#532C2211yescharacteristic
7C2×#125×C53C2211yescharacteristic
8C53×#491C412yescomplemented
9C53×#490C412yescomplemented
10C53×#488—22no—
11C53×#432C2212yes—
12C53×#417—22no—
13C4×M16×C53C2211yescharacteristic
14C53×#387—22no—
15C53×#387—22no—
16C4×C4⋊C4×C53—22no—
17C53×#376C2212yes—
18C53×#330—22no—
19C53×#327—22no—
20C53×#327—22no—
21C2×#41×C53—21no—
22C2×M16×C53—21no—
23C4⋊Q8×C53C2×C411yescharacteristic
24#127×C53C2×C411yescharacteristic
25#125×C53—22no—
26#125×C53C2×C412yes—
27#125×C53D412yes—
28#124×C53—21no—
29#121×C53—22no—
30C4×D4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53D412yes—
34C2×C42×C53—22no—
35C2×C42×C53—22no—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38#103×C53—24no—
39#102×C53—24no—
40C22⋊C8×C53—22no—
41#97×C53—22no—
42#97×C53C2×C412yes—
43C4×C8×C53—21no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53D412yes—
48#95×C53D412yes—
49#41×C53—44no—
50C2×Q8×C53—21no—
51C2×D4×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—41no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60M16×C53—42no—
61C2×C8×C53—22no—
62C2×C8×C53—42no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—24no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—44no—
68C4⋊C4×C53C22⋊C414yes—
69C22⋊C4×C53—42no—
70C42×C53—44no—
71C42×C53—44no—
72C42×C53C22×C411yescharacteristic
73C42×C53C2×D411yescharacteristic
74C42×C53#4112yes—
75C23×C53—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—42no—
78D4×C53—42no—
79D4×C53—82no—
80D4×C53—82no—
81C2×C4×C53—22no—
82C2×C4×C53—24no—
83C2×C4×C53—24no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—42no—
93C2×C4×C53—82no—
94C2×C4×C53C2×C22⋊C411yescharacteristic
95C2×C4×C53C4×D412yes—
96C2×C4×C53#12112yes—
97C2×C4×C53#12311yescharacteristic
98C8×C53—42no—
99C8×C53—82no—
100#9442C5311yesmaximal, characteristic, direct product factor, Sylow
101C22×C53—22no—
102C22×C53—81no—
103C22×C53—82no—
104C22×C53#38611yescharacteristic
105C22×C53#45412yes—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—44no—
111C4×C53—44no—
112C4×C53—82no—
113C4×C53—81no—
114#3142C2×C5311yescharacteristic
115#2371C2×C5311yescharacteristic, complemented
116#1534C2×C5311yescharacteristic, complemented
117#1505C2×C5312yes—
118#1300C2×C5312yescomplemented
119C2×C53—22no—
120C2×C53—161no—
121C2×C53#188811yescharacteristic
122C2×C53#190612yes—
123#532C22×C5311yescharacteristic
124C2×#125C22×C5311yescharacteristic
125#491C4×C5312yescomplemented
126#490C4×C5312yescomplemented
127#488—22no—
128#432C22×C5312yes—
129#417—22no—
130C4×M16C22×C5311yescharacteristic
131#387—22no—
132#387—22no—
133C4×C4⋊C4—22no—
134#376C22×C5312yes—
135#330—22no—
136#327—22no—
137#327—22no—
138C53#944211yescharacteristic, direct product factor, Sylow
139C2×#41—21no—
140C2×M16—21no—
141C4⋊Q8C2×C4×C5311yescharacteristic
142#127C2×C4×C5311yescharacteristic
143#125—22no—
144#125C2×C4×C5312yes—
145#125D4×C5312yes—
146#124—21no—
147#121—22no—
148C4×D4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4D4×C5312yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C23×C5311yescharacteristic
155C4⋊C8—22no—
156#103—24no—
157#102—24no—
158C22⋊C8—22no—
159#97—22no—
160#97C2×C4×C5312yes—
161C4×C8—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95D4×C5312yes—
166#95D4×C5312yes—
167#41—44no—
168C2×Q8—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—41no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4C2×D4×C5311yescharacteristic
177C22×C4C2×D4×C5312yes—
178M16—42no—
179C2×C8—22no—
180C2×C8—42no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—44no—
186C4⋊C4C22⋊C4×C5314yes—
187C22⋊C4—42no—
188C42—44no—
189C42—44no—
190C42C22×C4×C5311yescharacteristic
191C42C2×D4×C5311yescharacteristic
192C42#41×C5312yes—
193C23—41no—
194C23#120×C5311yescharacteristic
195Q8—42no—
196D4—42no—
197D4—82no—
198D4—82no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4C2×C22⋊C4×C5311yescharacteristic
213C2×C4C4×D4×C5312yes—
214C2×C4#121×C5312yes—
215C2×C4#123×C5311yescharacteristic
216C8—42no—
217C8—82no—
218C22—22no—
219C22—81no—
220C22—82no—
221C22C53×#38611yescharacteristic
222C22C53×#45412yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—81no—
231C2—22no—
232C2—161no—
233C2C53×#188811yescharacteristic
234C2C53×#190612yes—
subgroupG/H
center, socleC22×C53#386
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×#2371C2semidirect
3. C53×#1534C2semidirect
5. C53×#1300C2semidirect
8. C53×#491C4semidirect
9. C53×#490C4semidirect
100. #9442C53direct (N × H)
115. #2371C2×C53semidirect
116. #1534C2×C53semidirect
118. #1300C2×C53semidirect
125. #491C4×C53semidirect
126. #490C4×C53semidirect
138. C53#9442direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31422
C53×#23712
C53×#15342
C53×#15052
C53×#13002
#944253