Finite Group Databasebeta

← C53×#9486 #27216998

Subgroups — C53×#9486 #27216998

Order 13568.

542 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2982C211yesmaximal, characteristic, complemented
2C53×#2536C211yesmaximal, characteristic
3C53×#1801C211yesmaximal, characteristic, complemented
4C53×#1486C212yesmaximal
5C53×#1286C212yesmaximal, complemented
6C2×C22⋊Q8×C53—21no—
7C2×C4×Q8×C53C2211yescharacteristic
8C2×C4×D4×C53C2211yescharacteristic
9C8×Q8×C53—24no—
10C53×#431—22no—
11C53×#431C2212yes—
12C53×#423C2212yes—
13C2×#101×C53—21no—
14C53×#398C412yes—
15C53×#394C412yes—
16C53×#388—22no—
17C53×#382C2211yescharacteristic
18C53×#380—21no—
19C53×#343—21no—
20C53×D4⋊C8—24no—
21C22×Q8×C53D411yescharacteristic
22C22×D4×C53C2×C411yescharacteristic
23C23×C4×C53—21no—
24C2×M16×C53—22no—
25C22⋊Q8×C53—42no—
26C22⋊Q8×C53—42no—
27C4×Q8×C53—22no—
28C4×Q8×C53—22no—
29C4×D4×C53—22no—
30C4×D4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53C2×C411yescharacteristic
34C2×C4⋊C4×C53C2×C412yes—
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—21no—
39C2×C42×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—24no—
42C4⋊C8×C53—22no—
43C4⋊C8×C53—22no—
44C4⋊C8×C53—22no—
45#101×C53—42no—
46C22⋊C8×C53—22no—
47C4×C8×C53—24no—
48C4×C8×C53D414yes—
49#95×C53—22no—
50#95×C53—21no—
51#95×C53C2×C412yes—
52C24×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—22no—
55C2×Q8×C53D812yes—
56C2×Q8×C53SD1612yes—
57C2×D4×C53—22no—
58C2×D4×C53—42no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70M16×C53—42no—
71M16×C53—42no—
72C2×C8×C53—24no—
73C2×C8×C53—42no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—44no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—44no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C42×C53—22no—
85C42×C53—22no—
86C42×C53C2×D412yes—
87C42×C53#4112yes—
88C23×C53—22no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—24no—
97Q8×C53—24no—
98D4×C53—24no—
99D4×C53—42no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—82no—
113C2×C4×C53—82no—
114C2×C4×C53#10214yes—
115C2×C4×C53C4×D412yes—
116C2×C4×C53#12011yescharacteristic
117C2×C4×C53#12112yes—
118C2×C4×C53#12311yescharacteristic
119C2×C4×C53C2×D811yescharacteristic
120C2×C4×C53C2×SD1611yescharacteristic
121C2×C4×C53#13512yes—
122C8×C53—44no—
123C8×C53—82no—
124#9486C5311yesmaximal, characteristic, direct product factor, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—81no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53—81no—
134C22×C53#38611yescharacteristic
135C22×C53C2×#10211yescharacteristic
136C22×C53#41611yescharacteristic
137C4×C53—22no—
138C4×C53—24no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—42no—
142C4×C53—82no—
143C4×C53—81no—
144C4×C53#46512yes—
145C4×C53#46612yes—
146#2982C2×C5311yescharacteristic, complemented
147#2536C2×C5311yescharacteristic
148#1801C2×C5311yescharacteristic, complemented
149#1486C2×C5312yes—
150#1286C2×C5312yescomplemented
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#189111yescharacteristic
155C2×C53#189311yescharacteristic
156C2×C53#191211yescharacteristic
157C2×C22⋊Q8—21no—
158C2×C4×Q8C22×C5311yescharacteristic
159C2×C4×D4C22×C5311yescharacteristic
160C8×Q8—24no—
161#431—22no—
162#431C22×C5312yes—
163#423C22×C5312yes—
164C2×#101—21no—
165#398C4×C5312yes—
166#394C4×C5312yes—
167#388—22no—
168#382C22×C5311yescharacteristic
169#380—21no—
170#343—21no—
171D4⋊C8—24no—
172C53#948611yescharacteristic, direct product factor, Sylow
173C22×Q8D4×C5311yescharacteristic
174C22×D4C2×C4×C5311yescharacteristic
175C23×C4—21no—
176C2×M16—22no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C2×C4×C5311yescharacteristic
186C2×C4⋊C4C2×C4×C5312yes—
187C2×C4⋊C4D4×C5311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42D4×C5312yes—
192C2×C42C23×C5311yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197#101—42no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8D4×C5314yes—
201#95—21no—
202#95—22no—
203#95C2×C4×C5312yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8D8×C5312yes—
208C2×Q8SD16×C5312yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊C4×C5312yes—
218C22×C4C22⋊C4×C5312yes—
219C22×C4C22×C4×C5311yescharacteristic
220C22×C4C2×D4×C5311yescharacteristic
221C22×C4C2×D4×C5311yescharacteristic
222M16—42no—
223M16—42no—
224C2×C8—24no—
225C2×C8—42no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—22no—
237C42—22no—
238C42C2×D4×C5312yes—
239C42#41×C5312yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊C4×C5311yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#102×C5314yes—
267C2×C4C4×D4×C5312yes—
268C2×C4#120×C5311yescharacteristic
269C2×C4#121×C5312yes—
270C2×C4#123×C5311yescharacteristic
271C2×C4C2×D8×C5311yescharacteristic
272C2×C4C2×SD16×C5311yescharacteristic
273C2×C4#135×C5312yes—
274C8—44no—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C53×#38611yescharacteristic
286C22C2×#102×C5311yescharacteristic
287C22C53×#41611yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—24no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4C53×#46512yes—
296C4C53×#46612yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2C53×#189111yescharacteristic
301C2C53×#189311yescharacteristic
302C2C53×#191211yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C22×C4 ▷ 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×#2982C2semidirect
3. C53×#1801C2semidirect
5. C53×#1286C2semidirect
124. #9486C53direct (N × H)
146. #2982C2×C53semidirect
148. #1801C2×C53semidirect
150. #1286C2×C53semidirect
172. C53#9486direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#29822
C53×#25362
C53×#18012
C53×#14862
C53×#12862
#948653