Finite Group Databasebeta

← C53×#9439 #27216951

Subgroups — C53×#9439 #27216951

Order 13568.

490 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3140C211yesmaximal, characteristic
2C53×#2372C211yesmaximal, characteristic
3C53×#1533C211yesmaximal, characteristic
4C53×#1506C212yesmaximal, complemented
5C53×#1300C212yesmaximal, complemented
6C53×#533C2211yescharacteristic
7C2×#124×C53C2211yescharacteristic
8C53×#489C414yescomplemented
9C53×#488—22no—
10C53×#432C2212yes—
11C53×#419—21no—
12C53×#418—21no—
13C4×M16×C53C2211yescharacteristic
14C53×#391—21no—
15C53×#388—22no—
16C53×#386—21no—
17C4×C22⋊C4×C53—22no—
18C53×#376C2212yes—
19C53×#329—22no—
20C53×#328—22no—
21C53×#328—22no—
22C22×Q8×C53D411yescharacteristic
23C22×D4×C53D411yescharacteristic
24C23×C4×C53—21no—
25C2×M16×C53—21no—
26C4⋊Q8×C53C2×C412yescomplemented
27#125×C53—21no—
28#124×C53—22no—
29#124×C53C2×C412yes—
30#124×C53D412yes—
31C22⋊Q8×C53—22no—
32C4×Q8×C53—22no—
33#117×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53—21no—
39C2×C42×C53—22no—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—22no—
42#103×C53—24no—
43#102×C53—24no—
44C22⋊C8×C53—22no—
45#97×C53—22no—
46#97×C53C2×C412yescomplemented
47C4×C8×C53—21no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53D412yes—
51#95×C53D412yes—
52C24×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53C22⋊C412yes—
58C2×D4×C53—21no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53C22⋊C412yes—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—44no—
66C22×C4×C53—81no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D412yes—
69M16×C53—41no—
70M16×C53—41no—
71C2×C8×C53—22no—
72C2×C8×C53—42no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—81no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—81no—
82C42×C53—44no—
83C42×C53—41no—
84C42×C53—81no—
85C42×C53C22×C411yescharacteristic
86C42×C53C2×D411yescharacteristic
87C42×C53#4112yes—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#12011yescharacteristic
94Q8×C53—42no—
95Q8×C53—81no—
96Q8×C53—82no—
97D4×C53—42no—
98D4×C53—81no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—24no—
102C2×C4×C53—22no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—84no—
113C2×C4×C53C2×C22⋊C411yescharacteristic
114C2×C4×C53C4×D412yes—
115C2×C4×C53#12112yes—
116C2×C4×C53#12311yescharacteristic
117C8×C53—42no—
118C8×C53—82no—
119#9439C5311yesmaximal, characteristic, direct product factor, Sylow
120C22×C53—22no—
121C22×C53—42no—
122C22×C53—44no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53—82no—
126C22×C53#38611yescharacteristic
127C22×C53#45511yescharacteristic
128C22×C53#45511yescharacteristic
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—44no—
134C4×C53—42no—
135C4×C53—82no—
136C4×C53—82no—
137#3140C2×C5311yescharacteristic
138#2372C2×C5311yescharacteristic
139#1533C2×C5311yescharacteristic
140#1506C2×C5312yescomplemented
141#1300C2×C5312yescomplemented
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—82no—
145C2×C53#188511yescharacteristic
146C2×C53#190612yes—
147#533C22×C5311yescharacteristic
148C2×#124C22×C5311yescharacteristic
149#489C4×C5314yescomplemented
150#488—22no—
151#432C22×C5312yes—
152#419—21no—
153#418—21no—
154C4×M16C22×C5311yescharacteristic
155#391—21no—
156#388—22no—
157#386—21no—
158C4×C22⋊C4—22no—
159#376C22×C5312yes—
160#329—22no—
161#328—22no—
162#328—22no—
163C53#943911yescharacteristic, direct product factor, Sylow
164C22×Q8D4×C5311yescharacteristic
165C22×D4D4×C5311yescharacteristic
166C23×C4—21no—
167C2×M16—21no—
168C4⋊Q8C2×C4×C5312yescomplemented
169#125—21no—
170#124—22no—
171#124C2×C4×C5312yes—
172#124D4×C5312yes—
173C22⋊Q8—22no—
174C4×Q8—22no—
175#117—21no—
176C2×C4⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42C23×C5311yescharacteristic
183C4⋊C8—22no—
184#103—24no—
185#102—24no—
186C22⋊C8—22no—
187#97—22no—
188#97C2×C4×C5312yescomplemented
189C4×C8—21no—
190#95—22no—
191#95—22no—
192#95D4×C5312yes—
193#95D4×C5312yes—
194C24—21no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C22⋊C4×C5312yes—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C22⋊C4×C5312yes—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—44no—
208C22×C4—81no—
209C22×C4C2×D4×C5311yescharacteristic
210C22×C4C2×D4×C5312yes—
211M16—41no—
212M16—41no—
213C2×C8—22no—
214C2×C8—42no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—81no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—44no—
222C22⋊C4—41no—
223C22⋊C4—81no—
224C42—44no—
225C42—41no—
226C42—81no—
227C42C22×C4×C5311yescharacteristic
228C42C2×D4×C5311yescharacteristic
229C42#41×C5312yes—
230C23—22no—
231C23—42no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23#120×C5311yescharacteristic
236Q8—42no—
237Q8—81no—
238Q8—82no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4C2×C22⋊C4×C5311yescharacteristic
256C2×C4C4×D4×C5312yes—
257C2×C4#121×C5312yes—
258C2×C4#123×C5311yescharacteristic
259C8—42no—
260C8—82no—
261C22—22no—
262C22—42no—
263C22—44no—
264C22—81no—
265C22—81no—
266C22—82no—
267C22C53×#38611yescharacteristic
268C22C53×#45511yescharacteristic
269C22C53×#45511yescharacteristic
270C4—21no—
271C4—21no—
272C4—22no—
273C4—22no—
274C4—44no—
275C4—42no—
276C4—82no—
277C4—82no—
278C2—21no—
279C2—21no—
280C2—82no—
281C2C53×#188511yescharacteristic
282C2C53×#190612yes—
subgroupG/H
center, socleC22×C53#386
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53D4
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
4. C53×#1506C2semidirect
5. C53×#1300C2semidirect
8. C53×#489C4semidirect
26. C4⋊Q8×C53C2×C4semidirect
46. #97×C53C2×C4semidirect
119. #9439C53direct (N × H)
140. #1506C2×C53semidirect
141. #1300C2×C53semidirect
149. #489C4×C53semidirect
163. C53#9439direct (N × H)
168. C4⋊Q8C2×C4×C53semidirect
188. #97C2×C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#31402
C53×#23722
C53×#15332
C53×#15062
C53×#13002
#943953