Finite Group Databasebeta

← C53×#9464 #27216976

Subgroups — C53×#9464 #27216976

Order 13568.

610 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2501C211yesmaximal, characteristic
2C53×#2313C211yesmaximal, characteristic, complemented
3C53×#1821C211yesmaximal, characteristic, complemented
4C53×#1510C212yesmaximal
5C53×#1284C212yesmaximal, complemented
6C2×#123×C53—21no—
7C2×#123×C53—21no—
8C2×#121×C53C2211yescharacteristic
9C22×C4⋊C4×C53C2211yescharacteristic
10C53×#407C2212yes—
11C53×#399—22no—
12C53×#390C412yescomplemented
13C53×#387—24no—
14C53×#384C412yescomplemented
15C53×#381C2212yes—
16C53×#380—21no—
17C4×C22⋊C4×C53—21no—
18C4×C22⋊C4×C53C2211yescharacteristic
19C53×#343—22no—
20C53×#327—24no—
21C22×D4×C53—21no—
22C22×D4×C53C2×C411yescharacteristic
23C23×C4×C53D411yescharacteristic
24C23×C4×C53D411yescharacteristic
25C23×C4×C53C2311yescharacteristic
26C2×M16×C53—22no—
27#123×C53—44no—
28#123×C53—44no—
29#121×C53—22no—
30#121×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53D412yes—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C42×C53—22no—
45C2×C42×C53C2×C412yes—
46C22⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—21no—
51#95×C53—22no—
52#95×C53D414yes—
53C24×C53—21no—
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—22no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—24no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D412yes—
76C22×C4×C53#4112yes—
77M16×C53—44no—
78C2×C8×C53—42no—
79C4⋊C4×C53—24no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—44no—
83C4⋊C4×C53—44no—
84C4⋊C4×C53—82no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—44no—
88C22⋊C4×C53—44no—
89C22⋊C4×C53—42no—
90C42×C53—42no—
91C42×C53—82no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—42no—
98C23×C53—41no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#12011yescharacteristic
102C23×C53#12311yescharacteristic
103D4×C53—82no—
104D4×C53—82no—
105D4×C53—82no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—44no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—44no—
123C2×C4×C53—41no—
124C2×C4×C53—44no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53C4×D412yes—
128C2×C4×C53#12112yes—
129C8×C53—82no—
130#9464C5311yesmaximal, characteristic, direct product factor, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53#38611yescharacteristic
145C22×C53#41012yes—
146C22×C53#45312yes—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—44no—
150C4×C53—42no—
151C4×C53—42no—
152C4×C53—84no—
153C4×C53—81no—
154#2501C2×C5311yescharacteristic
155#2313C2×C5311yescharacteristic, complemented
156#1821C2×C5311yescharacteristic, complemented
157#1510C2×C5312yes—
158#1284C2×C5312yescomplemented
159C2×C53—22no—
160C2×C53—22no—
161C2×C53—41no—
162C2×C53—161no—
163C2×C53#188711yescharacteristic
164C2×C53#189911yescharacteristic
165C2×C53#191211yescharacteristic
166C2×#123—21no—
167C2×#123—21no—
168C2×#121C22×C5311yescharacteristic
169C22×C4⋊C4C22×C5311yescharacteristic
170#407C22×C5312yes—
171#399—22no—
172#390C4×C5312yescomplemented
173#387—24no—
174#384C4×C5312yescomplemented
175#381C22×C5312yes—
176#380—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4C22×C5311yescharacteristic
179#343—22no—
180#327—24no—
181C53#946411yescharacteristic, direct product factor, Sylow
182C22×D4—21no—
183C22×D4C2×C4×C5311yescharacteristic
184C23×C4D4×C5311yescharacteristic
185C23×C4D4×C5311yescharacteristic
186C23×C4C23×C5311yescharacteristic
187C2×M16—22no—
188#123—44no—
189#123—44no—
190#121—22no—
191#121—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×C4×C5311yescharacteristic
198C2×C4⋊C4C2×C4×C5312yes—
199C2×C4⋊C4D4×C5312yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42C2×C4×C5312yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95D4×C5314yes—
214C24—21no—
215C24C2×D4×C5311yescharacteristic
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊C4×C5312yes—
233C22×C4C22⋊C4×C5312yes—
234C22×C4C22×C4×C5311yescharacteristic
235C22×C4C2×D4×C5311yescharacteristic
236C22×C4C2×D4×C5312yes—
237C22×C4#41×C5312yes—
238M16—44no—
239C2×C8—42no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—82no—
246C22⋊C4—24no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C42—42no—
252C42—82no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23C2×C22⋊C4×C5311yescharacteristic
262C23#120×C5311yescharacteristic
263C23#123×C5311yescharacteristic
264D4—82no—
265D4—82no—
266D4—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4C4×D4×C5312yes—
289C2×C4#121×C5312yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22C53×#38611yescharacteristic
305C22C53×#41012yes—
306C22C53×#45312yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—84no—
313C4—81no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2C53×#188711yescharacteristic
319C2C53×#189911yescharacteristic
320C2C53×#191211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
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
2. C53×#2313C2semidirect
3. C53×#1821C2semidirect
5. C53×#1284C2semidirect
12. C53×#390C4semidirect
14. C53×#384C4semidirect
130. #9464C53direct (N × H)
155. #2313C2×C53semidirect
156. #1821C2×C53semidirect
158. #1284C2×C53semidirect
172. #390C4×C53semidirect
174. #384C4×C53semidirect
181. C53#9464direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#25012
C53×#23132
C53×#18212
C53×#15102
C53×#12842
#946453