Finite Group Databasebeta

← C53×#10372 #27217884

Subgroups — C53×#10372 #27217884

Order 13568.

402 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2402C211yesmaximal, characteristic, complemented
2C53×#1963C211yesmaximal, characteristic
3C53×#1502C212yesmaximal
4C53×#1303C212yesmaximal, complemented
5C53×#1274C211yesmaximal, characteristic, complemented
6C53×#532C2211yescharacteristic
7C53×#431C2212yes—
8C53×#423C2211yescharacteristic
9C2×#104×C53—22no—
10C53×#400—22no—
11C53×#398—22no—
12C4×C4⋊C4×C53—22no—
13C4×C4⋊C4×C53C2212yes—
14C53×#376C2211yescharacteristic
15C53×#327—22no—
16C53×#327—22no—
17C2×#41×C53—21no—
18C2×M16×C53—21no—
19C4⋊Q8×C53Q811yescharacteristic
20#127×C53Q811yescharacteristic
21#124×C53—21no—
22#121×C53—22no—
23C4×D4×C53—22no—
24C2×C4⋊C4×C53—22no—
25C2×C4⋊C4×C53—22no—
26C2×C4⋊C4×C53—22no—
27C2×C42×C53—22no—
28C2×C42×C53D412yes—
29C2×C42×C53C2311yescharacteristic
30C4⋊C8×C53—22no—
31C4⋊C8×C53—22no—
32#104×C53—44no—
33C22⋊C8×C53—22no—
34C4×C8×C53—22no—
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38#95×C53D412yes—
39#95×C53D412yes—
40#41×C53—42no—
41#41×C53—42no—
42C2×Q8×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—22no—
45C2×D4×C53—41no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—41no—
50C22×C4×C53C2×D412yes—
51C22×C4×C53C2×D411yescharacteristic
52M16×C53—42no—
53C2×C8×C53—42no—
54C2×C8×C53—41no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53—44no—
58C4⋊C4×C53—44no—
59C22⋊C4×C53—42no—
60C42×C53—22no—
61C42×C53—22no—
62C42×C53—42no—
63C42×C53—42no—
64C42×C53C2×Q811yescharacteristic
65C42×C53#4111yescharacteristic
66C42×C53#4112yes—
67C23×C53—41no—
68C23×C53#12011yescharacteristic
69Q8×C53—42no—
70D4×C53—42no—
71D4×C53—82no—
72D4×C53—82no—
73C2×C4×C53—24no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—41no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—81no—
85C2×C4×C53—81no—
86C2×C4×C53C22⋊Q812yes—
87C2×C4×C53C22⋊Q811yescharacteristic
88C2×C4×C53#12412yes—
89C2×C4×C53#12411yescharacteristic
90C2×C4×C53#13512yes—
91C2×C4×C53AGL(1,ℤ/8)12yes—
92C8×C53—82no—
93C8×C53—81no—
94#10372C5311yesmaximal, characteristic, direct product factor, Sylow
95C22×C53—22no—
96C22×C53—81no—
97C22×C53—82no—
98C22×C53#39311yescharacteristic
99C22×C53#44912yes—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—44no—
103C4×C53—82no—
104C4×C53—82no—
105C4×C53—81no—
106C4×C53#47812yes—
107C4×C53#48811yescharacteristic
108C4×C53#49011yescharacteristic
109#2402C2×C5311yescharacteristic, complemented
110#1963C2×C5311yescharacteristic
111#1502C2×C5312yes—
112#1303C2×C5312yescomplemented
113#1274C2×C5311yescharacteristic, complemented
114C2×C53—22no—
115C2×C53—161no—
116C2×C53#202511yescharacteristic
117C2×C53#203112yes—
118#532C22×C5311yescharacteristic
119#431C22×C5312yes—
120#423C22×C5311yescharacteristic
121C2×#104—22no—
122#400—22no—
123#398—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C22×C5312yes—
126#376C22×C5311yescharacteristic
127#327—22no—
128#327—22no—
129C53#1037211yescharacteristic, direct product factor, Sylow
130C2×#41—21no—
131C2×M16—21no—
132C4⋊Q8Q8×C5311yescharacteristic
133#127Q8×C5311yescharacteristic
134#124—21no—
135#121—22no—
136C4×D4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C42—22no—
141C2×C42D4×C5312yes—
142C2×C42C23×C5311yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—22no—
145#104—44no—
146C22⋊C8—22no—
147C4×C8—22no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95D4×C5312yes—
152#95D4×C5312yes—
153#41—42no—
154#41—42no—
155C2×Q8—21no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4—41no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4C2×D4×C5312yes—
164C22×C4C2×D4×C5311yescharacteristic
165M16—42no—
166C2×C8—42no—
167C2×C8—41no—
168C4⋊C4—22no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C22⋊C4—42no—
173C42—22no—
174C42—22no—
175C42—42no—
176C42—42no—
177C42C2×Q8×C5311yescharacteristic
178C42#41×C5312yes—
179C42#41×C5311yescharacteristic
180C23—41no—
181C23#120×C5311yescharacteristic
182Q8—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4C22⋊Q8×C5312yes—
200C2×C4C22⋊Q8×C5311yescharacteristic
201C2×C4#124×C5312yes—
202C2×C4#124×C5311yescharacteristic
203C2×C4#135×C5312yes—
204C2×C4AGL(1,ℤ/8)×C5312yes—
205C8—82no—
206C8—81no—
207C22—22no—
208C22—81no—
209C22—82no—
210C22C53×#39311yescharacteristic
211C22C53×#44912yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—82no—
216C4—82no—
217C4—81no—
218C4C53×#47812yes—
219C4C53×#48811yescharacteristic
220C4C53×#49011yescharacteristic
221C2—22no—
222C2—161no—
223C2C53×#202511yescharacteristic
224C2C53×#203112yes—
subgroupG/H
centerC2×C4×C53#124
commutator, Frattini, cosocleC2×C42C23×C53
socleC22×C53#393
prime radicalC53×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2402C2semidirect
4. C53×#1303C2semidirect
5. C53×#1274C2semidirect
94. #10372C53direct (N × H)
109. #2402C2×C53semidirect
112. #1303C2×C53semidirect
113. #1274C2×C53semidirect
129. C53#10372direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24022
C53×#19632
C53×#15022
C53×#13032
C53×#12742
#1037253