Finite Group Databasebeta

← C53×#9462 #27216974

Subgroups — C53×#9462 #27216974

Order 13568.

700 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2434C211yesmaximal, characteristic
2C53×#2310C211yesmaximal, characteristic
3C53×#1784C211yesmaximal, characteristic
4C53×#1511C212yesmaximal
5C53×#1284C212yesmaximal
6C23×Q8×C53C2211yescharacteristic
7C2×C4⋊Q8×C53C412yescomplemented
8C2×C22⋊Q8×C53C2211yescharacteristic
9C2×#117×C53—21no—
10C2×#101×C53—22no—
11C53×#407C2212yes—
12C53×#395—22no—
13C53×#391—24no—
14C53×#391C412yescomplemented
15C53×#385—21no—
16C53×#385—21no—
17C53×#381—21no—
18C53×#381C2212yes—
19C4×C22⋊C4×C53C2211yescharacteristic
20C53×#328—24no—
21C22×Q8×C53—22no—
22C22×Q8×C53—22no—
23C22×Q8×C53—44no—
24C22×Q8×C53C2×C411yescharacteristic
25C22×Q8×C53C2×C412yes—
26C22×Q8×C53D412yes—
27C23×C4×C53D411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C23×C4×C53C2311yescharacteristic
30C2×M16×C53—22no—
31C4⋊Q8×C53—44no—
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—22no—
34#117×C53—44no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—21no—
43C2×C42×C53—22no—
44C2×C42×C53C2×C412yes—
45#101×C53—44no—
46C22⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48#95×C53—24no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53—21no—
52#95×C53—22no—
53#95×C53D414yes—
54C24×C53C2×D411yescharacteristic
55C2×Q8×C53—24no—
56C2×Q8×C53—24no—
57C2×Q8×C53—24no—
58C2×Q8×C53—24no—
59C2×Q8×C53—22no—
60C2×Q8×C53—42no—
61C2×Q8×C53—42no—
62C2×Q8×C53—48no—
63C2×Q8×C53—44no—
64C2×Q8×C53—44no—
65C2×Q8×C53—42no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—24no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—24no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22⋊C412yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×C4×C53#4112yes—
83M16×C53—44no—
84C2×C8×C53—42no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—44no—
87C4⋊C4×C53—44no—
88C4⋊C4×C53—44no—
89C4⋊C4×C53—42no—
90C22⋊C4×C53—24no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—44no—
95C42×C53—42no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#12011yescharacteristic
101C23×C53#12311yescharacteristic
102Q8×C53—48no—
103Q8×C53—48no—
104Q8×C53—82no—
105C2×C4×C53—24no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—44no—
123C2×C4×C53—42no—
124C2×C4×C53—44no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—44no—
129C2×C4×C53#9914yes—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12112yes—
132C8×C53—82no—
133#9462C5311yesmaximal, characteristic, direct product factor, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—44no—
142C22×C53#38611yescharacteristic
143C22×C53C2×#9912yes—
144C22×C53#45612yes—
145C4×C53—22no—
146C4×C53—24no—
147C4×C53—22no—
148C4×C53—44no—
149C4×C53—42no—
150C4×C53—84no—
151C4×C53—81no—
152C4×C53—81no—
153#2434C2×C5311yescharacteristic
154#2310C2×C5311yescharacteristic
155#1784C2×C5311yescharacteristic
156#1511C2×C5312yes—
157#1284C2×C5312yes—
158C2×C53—22no—
159C2×C53—22no—
160C2×C53—41no—
161C2×C53#188611yescharacteristic
162C2×C53#189811yescharacteristic
163C2×C53#191211yescharacteristic
164C23×Q8C22×C5311yescharacteristic
165C2×C4⋊Q8C4×C5312yescomplemented
166C2×C22⋊Q8C22×C5311yescharacteristic
167C2×#117—21no—
168C2×#101—22no—
169#407C22×C5312yes—
170#395—22no—
171#391—24no—
172#391C4×C5312yescomplemented
173#385—21no—
174#385—21no—
175#381—21no—
176#381C22×C5312yes—
177C4×C22⋊C4C22×C5311yescharacteristic
178#328—24no—
179C53#946211yescharacteristic, direct product factor, Sylow
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×C4×C5311yescharacteristic
184C22×Q8C2×C4×C5312yes—
185C22×Q8D4×C5312yes—
186C23×C4D4×C5311yescharacteristic
187C23×C4D4×C5311yescharacteristic
188C23×C4C23×C5311yescharacteristic
189C2×M16—22no—
190C4⋊Q8—44no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C2×C4×C5311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C42—22no—
203C2×C42C2×C4×C5312yes—
204#101—44no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207#95—24no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95D4×C5314yes—
213C24C2×D4×C5311yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊C4×C5312yes—
237C22×C4C22⋊C4×C5312yes—
238C22×C4C22×C4×C5311yescharacteristic
239C22×C4C2×D4×C5311yescharacteristic
240C22×C4C2×D4×C5312yes—
241C22×C4#41×C5312yes—
242M16—44no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C22⋊C4—24no—
250C22⋊C4—44no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C42—42no—
254C42—44no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×C22⋊C4×C5311yescharacteristic
259C23#120×C5311yescharacteristic
260C23#123×C5311yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4#99×C5314yes—
289C2×C4C4×D4×C5312yes—
290C2×C4#121×C5312yes—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22C53×#38611yescharacteristic
301C22C2×#99×C5312yes—
302C22C53×#45612yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—84no—
309C4—81no—
310C4—81no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2C53×#188611yescharacteristic
315C2C53×#189811yescharacteristic
316C2C53×#191211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C4⋊Q8×C53C4semidirect
14. C53×#391C4semidirect
133. #9462C53direct (N × H)
165. C2×C4⋊Q8C4×C53semidirect
172. #391C4×C53semidirect
179. C53#9462direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#24342
C53×#23102
C53×#17842
C53×#15112
C53×#12842
#946253