Finite Group Databasebeta

← C53×#7396 #27214908

Subgroups — C53×#7396 #27214908

Order 13568.

296 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2010C211yesmaximal, characteristic
2C53×#1320C211yesmaximal, characteristic, complemented
3C53×#1297C211yesmaximal, characteristic, complemented
4C53×#575C411yescharacteristic, complemented
5C53×#525—21no—
6C53×#421—21no—
7C53×#407C2211yescharacteristic
8C4×C22⋊C4×C53—21no—
9C53×#349—21no—
10C53×H(3,ℤ/4)C411yescharacteristic, complemented
11C2×#41×C53—21no—
12C23×C4×C53D411yescharacteristic
13#137×C53—41no—
14AGL(1,ℤ/8)×C53—41no—
15#135×C53—41no—
16C2×M16×C53C2×C411yescharacteristic
17C2×M16×C53D411yescharacteristic
18#123×C53—41no—
19C22⋊Q8×C53—41no—
20#121×C53—41no—
21#120×C53—41no—
22C4×D4×C53—42no—
23#117×C53—21no—
24C2×C22⋊C4×C53—41no—
25C2×C42×C53—41no—
26M32×C53—41no—
27#104×C53—42no—
28C22⋊C8×C53—41no—
29#95×C53—22no—
30C24×C53D811yescharacteristic
31#41×C53—41no—
32#41×C53—41no—
33#41×C53—41no—
34#41×C53—41no—
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—42no—
38C2×D4×C53—41no—
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—41no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53—42no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53SD1611yescharacteristic
47Q16×C53—41no—
48SD16×C53—42no—
49D8×C53—41no—
50M16×C53—41no—
51M16×C53—41no—
52C2×C8×C53—21no—
53C2×C8×C53—21no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—42no—
56C22⋊C4×C53—44no—
57C22⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C22⋊C4×C53—41no—
60C42×C53—22no—
61C42×C53—81no—
62C42×C53—81no—
63C16×C53—81no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—42no—
67C23×C53—41no—
68C23×C53#10211yescharacteristic
69Q8×C53—41no—
70Q8×C53—41no—
71D4×C53—41no—
72D4×C53—41no—
73D4×C53—81no—
74D4×C53—84no—
75D4×C53—81no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—44no—
85C2×C4×C53—81no—
86C2×C4×C53—81no—
87C2×C4×C53—82no—
88C2×C4×C53—82no—
89C2×C4×C53—81no—
90C2×C4×C53#9911yescharacteristic
91C2×C4×C53#10411yescharacteristic
92C8×C53—41no—
93C8×C53—41no—
94#7396C5311yesmaximal, characteristic, direct product factor, Sylow
95C22×C53—22no—
96C22×C53—21no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53—82no—
103C22×C53—81no—
104C22×C53#32711yescharacteristic
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—84no—
108C4×C53—82no—
109C4×C53—81no—
110C4×C53AΣL(1,𝔽16)11yescharacteristic
111C4×C53#35311yescharacteristic
112#2010C2×C5311yescharacteristic
113#1320C2×C5311yescharacteristic, complemented
114#1297C2×C5311yescharacteristic, complemented
115C2×C53—21no—
116C2×C53—42no—
117C2×C53—41no—
118C2×C53—161no—
119C2×C53#134311yescharacteristic
120#575C4×C5311yescharacteristic, complemented
121#525—21no—
122#421—21no—
123#407C22×C5311yescharacteristic
124C4×C22⋊C4—21no—
125#349—21no—
126H(3,ℤ/4)C4×C5311yescharacteristic, complemented
127C53#739611yescharacteristic, direct product factor, Sylow
128C2×#41—21no—
129C23×C4D4×C5311yescharacteristic
130#137—41no—
131AGL(1,ℤ/8)—41no—
132#135—41no—
133C2×M16C2×C4×C5311yescharacteristic
134C2×M16D4×C5311yescharacteristic
135#123—41no—
136C22⋊Q8—41no—
137#121—41no—
138#120—41no—
139C4×D4—42no—
140#117—21no—
141C2×C22⋊C4—41no—
142C2×C42—41no—
143M32—41no—
144#104—42no—
145C22⋊C8—41no—
146#95—22no—
147C24D8×C5311yescharacteristic
148#41—41no—
149#41—41no—
150#41—41no—
151#41—41no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—42no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4C22⋊C4×C5311yescharacteristic
163C22×C4SD16×C5311yescharacteristic
164Q16—41no—
165SD16—42no—
166D8—41no—
167M16—41no—
168M16—41no—
169C2×C8—21no—
170C2×C8—21no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C22⋊C4—44no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—41no—
177C42—22no—
178C42—81no—
179C42—81no—
180C16—81no—
181C23—21no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23#102×C5311yescharacteristic
186Q8—41no—
187Q8—41no—
188D4—41no—
189D4—41no—
190D4—81no—
191D4—84no—
192D4—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—81no—
207C2×C4#99×C5311yescharacteristic
208C2×C4#104×C5311yescharacteristic
209C8—41no—
210C8—41no—
211C22—22no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22—81no—
218C22—82no—
219C22—81no—
220C22C53×#32711yescharacteristic
221C4—21no—
222C4—41no—
223C4—84no—
224C4—82no—
225C4—81no—
226C4C53×AΣL(1,𝔽16)11yescharacteristic
227C4C53×#35311yescharacteristic
228C2—21no—
229C2—41no—
230C2—42no—
231C2—161no—
232C2C53×#134311yescharacteristic
subgroupG/H
centerC4×C53#353
commutatorC2×M16C2×C4×C53
Frattini, cosocle#407C22×C53
socleC2×C53#1343
prime radicalC53×#2010C2
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C53×#407 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1320C2semidirect
3. C53×#1297C2semidirect
4. C53×#575C4semidirect
10. C53×H(3,ℤ/4)C4semidirect
94. #7396C53direct (N × H)
113. #1320C2×C53semidirect
114. #1297C2×C53semidirect
120. #575C4×C53semidirect
126. H(3,ℤ/4)C4×C53semidirect
127. C53#7396direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#20102
C53×#13202
C53×#12972
#739653