Finite Group Databasebeta

← C53×#19075 #27226587

Subgroups — C53×#19075 #27226587

Order 13568.

1116 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3548C211yesmaximal, characteristic, complemented
2C53×#3151C212yesmaximal, complemented
3C53×#3138C211yesmaximal, characteristic, complemented
4C53×#1656C218yesmaximal, complemented
5C53×#1526C211yesmaximal, characteristic, complemented
6C53×#1525C212yesmaximal, complemented
7C2×AGL(1,ℤ/8)×C53—24no—
8C53×#562C2214yescomplemented
9C53×#554—22no—
10C53×#550—22no—
11C53×#547—22no—
12C53×#540—22no—
13C53×#537—24no—
14C53×#533C2211yescharacteristic
15C53×#532C2212yes—
16C2×#127×C53C2211yescharacteristic
17C53×#496C2218yes—
18C53×#493—22no—
19C53×#492—22no—
20C53×#490C2214yes—
21C53×#486—22no—
22C53×#418—22no—
23C4×M16×C53C2211yescharacteristic
24C4×M16×C53C2212yes—
25C53×#330C2214yescomplemented
26C53×#329C2218yescomplemented
27C2×#41×C53—24no—
28C2×#41×C53D414yes—
29C22×D4×C53—22no—
30C22×D4×C53—21no—
31AGL(1,ℤ/8)×C53—416no—
32C2×SD16×C53—28no—
33C2×D8×C53—28no—
34C2×M16×C53—22no—
35C2×M16×C53—21no—
36C4⋊Q8×C53D412yes—
37C4⋊Q8×C53D412yes—
38#127×C53—22no—
39#127×C53—21no—
40#127×C53—22no—
41#127×C53D412yes—
42#127×C53C2312yes—
43#125×C53—22no—
44#125×C53C2312yes—
45#124×C53C2314yes—
46#123×C53—28no—
47C22⋊Q8×C53—24no—
48C22⋊Q8×C53—28no—
49C22⋊Q8×C53—22no—
50#121×C53—28no—
51#121×C53—24no—
52#121×C53—28no—
53#121×C53—22no—
54#121×C53—22no—
55C4×Q8×C53—22no—
56C4×D4×C53—24no—
57C4×D4×C53—22no—
58C4×D4×C53—22no—
59#117×C53D412yes—
60C2×C4⋊C4×C53—24no—
61C2×C42×C53C2311yescharacteristic
62#102×C53—28no—
63#97×C53C2312yes—
64#97×C53C2314yes—
65C4×C8×C53—21no—
66C4×C8×C53—22no—
67C24×C53—41no—
68#41×C53—48no—
69#41×C53—88no—
70C2×Q8×C53—22no—
71C2×Q8×C53C2×D414yes—
72C2×D4×C53—28no—
73C2×D4×C53—24no—
74C2×D4×C53—22no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—24no—
78C2×D4×C53—48no—
79C2×D4×C53—44no—
80C2×D4×C53—42no—
81C2×D4×C53—41no—
82C2×D4×C53C2×D414yes—
83C22×C4×C53—28no—
84C22×C4×C53—24no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D412yes—
87SD16×C53—416no—
88D8×C53—416no—
89M16×C53—28no—
90M16×C53—42no—
91C2×C8×C53—24no—
92C2×C8×C53—22no—
93C4⋊C4×C53—28no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53C2×D414yes—
99C22⋊C4×C53—28no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C42×C53—22no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D412yes—
106C42×C53C2411yescharacteristic
107C23×C53—24no—
108C23×C53—42no—
109C23×C53—84no—
110C23×C53—82no—
111C23×C53#12011yescharacteristic
112Q8×C53—44no—
113Q8×C53—44no—
114D4×C53—44no—
115D4×C53—44no—
116D4×C53—48no—
117D4×C53—44no—
118D4×C53—44no—
119D4×C53—84no—
120D4×C53—88no—
121D4×C53—84no—
122D4×C53—82no—
123C2×C4×C53—24no—
124C2×C4×C53—22no—
125C2×C4×C53—24no—
126C2×C4×C53—24no—
127C2×C4×C53—48no—
128C2×C4×C53—88no—
129C2×C4×C53#12012yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53C22×D412yes—
132C2×C4×C53C22×D411yescharacteristic
133C8×C53—28no—
134C8×C53—42no—
135#19075C5311yesmaximal, characteristic, direct product factor, Sylow
136C22×C53—22no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53—44no—
140C22×C53—44no—
141C22×C53—84no—
142C22×C53—82no—
143C22×C53—164no—
144C22×C53C2×#12011yescharacteristic
145C4×C53—22no—
146C4×C53—21no—
147C4×C53—44no—
148C4×C53—44no—
149C4×C53#45314yes—
150C4×C53#45412yes—
151#3548C2×C5311yescharacteristic, complemented
152#3151C2×C5312yescomplemented
153#3138C2×C5311yescharacteristic, complemented
154#1656C2×C5318yescomplemented
155#1526C2×C5311yescharacteristic, complemented
156#1525C2×C5312yescomplemented
157C2×C53—41no—
158C2×C53—84no—
159C2×C53—162no—
160C2×C53C2×#45312yes—
161C2×C53C2×#45411yescharacteristic
162C2×AGL(1,ℤ/8)—24no—
163#562C22×C5314yescomplemented
164#554—22no—
165#550—22no—
166#547—22no—
167#540—22no—
168#537—24no—
169#533C22×C5311yescharacteristic
170#532C22×C5312yes—
171C2×#127C22×C5311yescharacteristic
172#496C22×C5318yes—
173#493—22no—
174#492—22no—
175#490C22×C5314yes—
176#486—22no—
177#418—22no—
178C4×M16C22×C5311yescharacteristic
179C4×M16C22×C5312yes—
180#330C22×C5314yescomplemented
181#329C22×C5318yescomplemented
182C53#1907511yescharacteristic, direct product factor, Sylow
183C2×#41—24no—
184C2×#41D4×C5314yes—
185C22×D4—22no—
186C22×D4—21no—
187AGL(1,ℤ/8)—416no—
188C2×SD16—28no—
189C2×D8—28no—
190C2×M16—22no—
191C2×M16—21no—
192C4⋊Q8D4×C5312yes—
193C4⋊Q8D4×C5312yes—
194#127—22no—
195#127—21no—
196#127—22no—
197#127D4×C5312yes—
198#127C23×C5312yes—
199#125—22no—
200#125C23×C5312yes—
201#124C23×C5314yes—
202#123—28no—
203C22⋊Q8—24no—
204C22⋊Q8—28no—
205C22⋊Q8—22no—
206#121—28no—
207#121—24no—
208#121—28no—
209#121—22no—
210#121—22no—
211C4×Q8—22no—
212C4×D4—24no—
213C4×D4—22no—
214C4×D4—22no—
215#117D4×C5312yes—
216C2×C4⋊C4—24no—
217C2×C42C23×C5311yescharacteristic
218#102—28no—
219#97C23×C5312yes—
220#97C23×C5314yes—
221C4×C8—21no—
222C4×C8—22no—
223C24—41no—
224#41—48no—
225#41—88no—
226C2×Q8—22no—
227C2×Q8C2×D4×C5314yes—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—48no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4C2×D4×C5314yes—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4C2×D4×C5311yescharacteristic
242C22×C4C2×D4×C5312yes—
243SD16—416no—
244D8—416no—
245M16—28no—
246M16—42no—
247C2×C8—24no—
248C2×C8—22no—
249C4⋊C4—28no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4C2×D4×C5314yes—
255C22⋊C4—28no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C42—22no—
260C42C2×D4×C5311yescharacteristic
261C42C2×D4×C5312yes—
262C42C24×C5311yescharacteristic
263C23—24no—
264C23—42no—
265C23—84no—
266C23—82no—
267C23#120×C5311yescharacteristic
268Q8—44no—
269Q8—44no—
270D4—44no—
271D4—48no—
272D4—44no—
273D4—44no—
274D4—44no—
275D4—84no—
276D4—88no—
277D4—84no—
278D4—82no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—48no—
284C2×C4—88no—
285C2×C4#120×C5312yes—
286C2×C4#120×C5311yescharacteristic
287C2×C4C22×D4×C5311yescharacteristic
288C2×C4C22×D4×C5312yes—
289C8—28no—
290C8—42no—
291C22—22no—
292C22—21no—
293C22—44no—
294C22—44no—
295C22—44no—
296C22—84no—
297C22—82no—
298C22—164no—
299C22C2×#120×C5311yescharacteristic
300C4—22no—
301C4—21no—
302C4—44no—
303C4—44no—
304C4C53×#45314yes—
305C4C53×#45412yes—
306C2—41no—
307C2—84no—
308C2—162no—
309C2C2×C53×#45312yes—
310C2C2×C53×#45411yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC42C24×C53
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
1. C53×#3548C2semidirect
2. C53×#3151C2semidirect
3. C53×#3138C2semidirect
4. C53×#1656C2semidirect
5. C53×#1526C2semidirect
6. C53×#1525C2semidirect
8. C53×#562C22semidirect
25. C53×#330C22semidirect
26. C53×#329C22semidirect
135. #19075C53direct (N × H)
151. #3548C2×C53semidirect
152. #3151C2×C53semidirect
153. #3138C2×C53semidirect
154. #1656C2×C53semidirect
155. #1526C2×C53semidirect
156. #1525C2×C53semidirect
163. #562C22×C53semidirect
180. #330C22×C53semidirect
181. #329C22×C53semidirect
182. C53#19075direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35482
C53×#31512
C53×#31382
C53×#16562
C53×#15262
C53×#15252
#1907553