Finite Group Databasebeta

← C53×#13674 #27221186

Subgroups — C53×#13674 #27221186

Order 13568.

454 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3291C211yesmaximal, characteristic, complemented
2C53×#2196C212yesmaximal, complemented
3C53×#1629C211yesmaximal, characteristic, complemented
4C53×#1409C211yesmaximal, characteristic, complemented
5C53×#1402C212yesmaximal, complemented
6C53×#577—22no—
7D42×C53C2211yescharacteristic
8C53×#492C2211yescharacteristic
9C53×#476—21no—
10C53×#471—22no—
11C53×B(4,𝔽2)—22no—
12C53×#456—21no—
13C53×#453C2212yes—
14C53×#443—21no—
15C53×#353C2212yes—
16C53×AΣL(1,𝔽16)—22no—
17C53×#331C2211yescharacteristic
18C53×D4⋊C8—21no—
192−1+4×C53—21no—
20H(5,𝔽2)×C53D412yescomplemented
21C22×D4×C53—41no—
22#137×C53—22no—
23#137×C53—42no—
24AGL(1,ℤ/8)×C53—22no—
25AGL(1,ℤ/8)×C53—42no—
26#135×C53—21no—
27#135×C53—42no—
28C2×SD16×C53—22no—
29C2×SD16×C53—42no—
30#131×C53—22no—
31C4⋊Q8×C53D411yescharacteristic, complemented
32#127×C53C2311yescharacteristic
33#121×C53—41no—
34#120×C53—22no—
35#120×C53—42no—
36C4×D4×C53—21no—
37#108×C53—21no—
38#106×C53—41no—
39C4⋊C8×C53—21no—
40#104×C53—22no—
41#102×C53—41no—
42#101×C53—21no—
43#100×C53D412yescomplemented
44#99×C53—22no—
45#99×C53—42no—
46C4×C8×C53D411yescharacteristic, complemented
47C24×C53—22no—
48#41×C53—22no—
49#41×C53—42no—
50#41×C53—42no—
51#41×C53—81no—
52C2×Q8×C53—21no—
53C2×Q8×C53—42no—
54C2×D4×C53—21no—
55C2×D4×C53—41no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—82no—
60C2×D4×C53—82no—
61C2×D4×C53—82no—
62C2×D4×C53C2×D412yes—
63C22×C4×C53—41no—
64Q16×C53—21no—
65Q16×C53—42no—
66SD16×C53—22no—
67SD16×C53—42no—
68SD16×C53—42no—
69SD16×C53—42no—
70SD16×C53—42no—
71D8×C53—21no—
72D8×C53—42no—
73M16×C53—22no—
74M16×C53—42no—
75C2×C8×C53—21no—
76C2×C8×C53—42no—
77C2×C8×C53—41no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—82no—
84C42×C53C2×D411yescharacteristic
85C23×C53—22no—
86C23×C53—41no—
87C23×C53—42no—
88C23×C53—84no—
89C23×C53—82no—
90Q8×C53—22no—
91Q8×C53—42no—
92Q8×C53—81no—
93D4×C53—22no—
94D4×C53—22no—
95D4×C53—42no—
96D4×C53—41no—
97D4×C53—42no—
98D4×C53—81no—
99D4×C53—81no—
100D4×C53—82no—
101D4×C53—81no—
102D4×C53—84no—
103D4×C53—164no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—81no—
110C2×C4×C53—81no—
111C2×C4×C53—82no—
112C2×C4×C53—82no—
113C2×C4×C53#12011yescharacteristic
114C8×C53—21no—
115C8×C53—21no—
116C8×C53—42no—
117C8×C53—81no—
118#13674C5311yesmaximal, characteristic, direct product factor, Sylow
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—82no—
123C22×C53—81no—
124C22×C53—84no—
125C22×C53—81no—
126C22×C53—162no—
127C22×C53—162no—
128C22×C53B(4,𝔽2)11yescharacteristic
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—81no—
134C4×C53—81no—
135C4×C53—162no—
136#3291C2×C5311yescharacteristic, complemented
137#2196C2×C5312yescomplemented
138#1629C2×C5311yescharacteristic, complemented
139#1409C2×C5311yescharacteristic, complemented
140#1402C2×C5312yescomplemented
141C2×C53—21no—
142C2×C53—82no—
143C2×C53—82no—
144C2×C53—82no—
145C2×C53#219611yescharacteristic
146#577—22no—
147D42C22×C5311yescharacteristic
148#492C22×C5311yescharacteristic
149#476—21no—
150#471—22no—
151B(4,𝔽2)—22no—
152#456—21no—
153#453C22×C5312yes—
154#443—21no—
155#353C22×C5312yes—
156AΣL(1,𝔽16)—22no—
157#331C22×C5311yescharacteristic
158D4⋊C8—21no—
159C53#1367411yescharacteristic, direct product factor, Sylow
1602−1+4—21no—
161H(5,𝔽2)D4×C5312yescomplemented
162C22×D4—41no—
163#137—22no—
164#137—42no—
165AGL(1,ℤ/8)—22no—
166AGL(1,ℤ/8)—42no—
167#135—21no—
168#135—42no—
169C2×SD16—22no—
170C2×SD16—42no—
171#131—22no—
172C4⋊Q8D4×C5311yescharacteristic, complemented
173#127C23×C5311yescharacteristic
174#121—41no—
175#120—22no—
176#120—42no—
177C4×D4—21no—
178#108—21no—
179#106—41no—
180C4⋊C8—21no—
181#104—22no—
182#102—41no—
183#101—21no—
184#100D4×C5312yescomplemented
185#99—22no—
186#99—42no—
187C4×C8D4×C5311yescharacteristic, complemented
188C24—22no—
189#41—22no—
190#41—42no—
191#41—42no—
192#41—81no—
193C2×Q8—21no—
194C2×Q8—42no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—82no—
203C2×D4C2×D4×C5312yes—
204C22×C4—41no—
205Q16—21no—
206Q16—42no—
207SD16—22no—
208SD16—42no—
209SD16—42no—
210SD16—42no—
211SD16—42no—
212D8—21no—
213D8—42no—
214M16—22no—
215M16—42no—
216C2×C8—21no—
217C2×C8—42no—
218C2×C8—41no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C22⋊C4—22no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42C2×D4×C5311yescharacteristic
226C23—22no—
227C23—41no—
228C23—42no—
229C23—82no—
230C23—84no—
231Q8—22no—
232Q8—42no—
233Q8—81no—
234D4—22no—
235D4—22no—
236D4—42no—
237D4—41no—
238D4—42no—
239D4—81no—
240D4—81no—
241D4—82no—
242D4—81no—
243D4—84no—
244D4—164no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#120×C5311yescharacteristic
255C8—21no—
256C8—21no—
257C8—42no—
258C8—81no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—82no—
263C22—81no—
264C22—84no—
265C22—81no—
266C22—162no—
267C22—162no—
268C22C53×B(4,𝔽2)11yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—81no—
274C4—81no—
275C4—162no—
276C2—21no—
277C2—82no—
278C2—82no—
279C2—82no—
280C2C53×#219611yescharacteristic
subgroupG/H
center, socleC2×C53#2196
commutator, Frattini, cosocle#127C23×C53
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C22×C53 ◁ C2×C4×C53 ◁ #127×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3291C2semidirect
2. C53×#2196C2semidirect
3. C53×#1629C2semidirect
4. C53×#1409C2semidirect
5. C53×#1402C2semidirect
20. H(5,𝔽2)×C53D4semidirect
31. C4⋊Q8×C53D4semidirect
43. #100×C53D4semidirect
46. C4×C8×C53D4semidirect
118. #13674C53direct (N × H)
136. #3291C2×C53semidirect
137. #2196C2×C53semidirect
138. #1629C2×C53semidirect
139. #1409C2×C53semidirect
140. #1402C2×C53semidirect
159. C53#13674direct (N × H)
161. H(5,𝔽2)D4×C53semidirect
172. C4⋊Q8D4×C53semidirect
184. #100D4×C53semidirect
187. C4×C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32912
C53×#21962
C53×#16292
C53×#14092
C53×#14022
#1367453