Finite Group Databasebeta

← C53×#12203 #27219715

Subgroups — C53×#12203 #27219715

Order 13568.

532 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2483C211yesmaximal, characteristic
2C53×#2029C211yesmaximal, characteristic
3C53×#1608C212yesmaximal, complemented
4C53×#1344C212yesmaximal, complemented
5C53×#1316C211yesmaximal, characteristic
6C53×#543C2211yescharacteristic
7C22×C22⋊C4×C53—21no—
8C53×#483—22no—
9C53×#397—21no—
10C53×#397C2212yes—
11C53×#385—22no—
12C53×#381—22no—
13C53×#379—21no—
14C53×#379C2211yescharacteristic
15C53×#342—21no—
16C53×#342C2211yescharacteristic
17C53×#328—22no—
18C53×#323C2212yes—
19C25×C53D411yescharacteristic
20C23×C4×C53—21no—
21C23×C4×C53—21no—
22C22⋊Q8×C53—22no—
23C22⋊Q8×C53—42no—
24C2×C4⋊C4×C53—21no—
25C2×C4⋊C4×C53D412yes—
26C2×C22⋊C4×C53—22no—
27C2×C22⋊C4×C53—22no—
28C2×C22⋊C4×C53—21no—
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—42no—
32C2×C22⋊C4×C53—42no—
33C2×C22⋊C4×C53—42no—
34C2×C22⋊C4×C53D411yescharacteristic
35C2×C22⋊C4×C53C2311yescharacteristic
36#106×C53—42no—
37#103×C53—42no—
38C22⋊C8×C53—22no—
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C24×C53—21no—
44C24×C53—21no—
45C24×C53—41no—
46C24×C53—44no—
47C24×C53—41no—
48C24×C53SD1612yes—
49C24×C53C2×D411yescharacteristic
50C2×Q8×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—44no—
58C22×C4×C53—41no—
59C22×C4×C53—81no—
60C22×C4×C53—81no—
61C22×C4×C53C2×D412yes—
62C2×C8×C53—42no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—81no—
67C22⋊C4×C53—41no—
68C22⋊C4×C53—44no—
69C22⋊C4×C53—44no—
70C22⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—84no—
78C23×C53—24no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—42no—
83C23×C53—44no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—44no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—44no—
90C23×C53—81no—
91C23×C53—84no—
92C23×C53—82no—
93C23×C53#12011yescharacteristic
94C23×C53C2×SD1611yescharacteristic
95C23×C53AGL(1,ℤ/8)11yescharacteristic
96Q8×C53—82no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53—44no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—84no—
108C2×C4×C53—84no—
109C2×C4×C53—82no—
110C2×C4×C53—81no—
111C2×C4×C53—82no—
112C8×C53—82no—
113#12203C5311yesmaximal, characteristic, direct product factor, Sylow
114C22×C53—22no—
115C22×C53—24no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—44no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—44no—
126C22×C53—81no—
127C22×C53—84no—
128C22×C53—82no—
129C22×C53#45011yescharacteristic
130C22×C53#45511yescharacteristic
131C22×C53B(4,𝔽2)11yescharacteristic
132C4×C53—42no—
133C4×C53—81no—
134C4×C53—82no—
135C4×C53—82no—
136C4×C53—82no—
137C4×C53—161no—
138#2483C2×C5311yescharacteristic
139#2029C2×C5311yescharacteristic
140#1608C2×C5312yescomplemented
141#1344C2×C5312yescomplemented
142#1316C2×C5311yescharacteristic
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—41no—
146C2×C53—44no—
147C2×C53—41no—
148C2×C53#160111yescharacteristic
149C2×C53#219712yes—
150#543C22×C5311yescharacteristic
151C22×C22⋊C4—21no—
152#483—22no—
153#397—21no—
154#397C22×C5312yes—
155#385—22no—
156#381—22no—
157#379—21no—
158#379C22×C5311yescharacteristic
159#342—21no—
160#342C22×C5311yescharacteristic
161#328—22no—
162#323C22×C5312yes—
163C53#1220311yescharacteristic, direct product factor, Sylow
164C25D4×C5311yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C22⋊Q8—22no—
168C22⋊Q8—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4D4×C5312yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4D4×C5311yescharacteristic
180C2×C22⋊C4C23×C5311yescharacteristic
181#106—42no—
182#103—42no—
183C22⋊C8—22no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95D4×C5312yes—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—44no—
192C24—41no—
193C24SD16×C5312yes—
194C24C2×D4×C5311yescharacteristic
195C2×Q8—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4C2×D4×C5312yes—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C22⋊C4—41no—
213C22⋊C4—44no—
214C22⋊C4—41no—
215C22⋊C4—44no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—84no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—42no—
228C23—42no—
229C23—44no—
230C23—42no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—44no—
235C23—81no—
236C23—84no—
237C23—82no—
238C23#120×C5311yescharacteristic
239C23C2×SD16×C5311yescharacteristic
240C23AGL(1,ℤ/8)×C5311yescharacteristic
241Q8—82no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—82no—
257C8—82no—
258C22—22no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—81no—
271C22—84no—
272C22—82no—
273C22C53×#45011yescharacteristic
274C22C53×#45511yescharacteristic
275C22C53×B(4,𝔽2)11yescharacteristic
276C4—42no—
277C4—82no—
278C4—81no—
279C4—82no—
280C4—82no—
281C4—161no—
282C2—21no—
283C2—21no—
284C2—41no—
285C2—44no—
286C2—41no—
287C2C53×#160111yescharacteristic
288C2C53×#219712yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC25×C53D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1608C2semidirect
4. C53×#1344C2semidirect
113. #12203C53direct (N × H)
140. #1608C2×C53semidirect
141. #1344C2×C53semidirect
163. C53#12203direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24832
C53×#20292
C53×#16082
C53×#13442
C53×#13162
#1220353