Finite Group Databasebeta

← C53×#12121 #27219633

Subgroups — C53×#12121 #27219633

Order 13568.

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

normal subgroups only

This group has 470 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×#2607C211yesmaximal, characteristic, complemented
2C53×#2022C211yesmaximal, characteristic
3C53×#1601C212yesmaximal
4C53×#1351C212yesmaximal, complemented
5C53×#1317C211yesmaximal, characteristic, complemented
6C53×#534C2211yescharacteristic
7C2×#123×C53—21no—
8C53×#448—22no—
9C2×#99×C53—21no—
10C53×#399—21no—
11C53×#393C2212yes—
12C53×#391—22no—
13C53×#387—22no—
14C53×#381C2211yescharacteristic
15C4×C22⋊C4×C53—21no—
16C53×#342C2211yescharacteristic
17C53×#327—22no—
18C53×#323C2212yes—
19C22×Q8×C53D412yes—
20C22×D4×C53—21no—
21C23×C4×C53D411yescharacteristic
22C2×SD16×C53—42no—
23#123×C53—41no—
24#123×C53—42no—
25#123×C53—42no—
26#121×C53—22no—
27#120×C53—41no—
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—41no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—41no—
34C2×C22⋊C4×C53D411yescharacteristic
35C2×C22⋊C4×C53C2311yescharacteristic
36C2×C42×C53—21no—
37C2×C42×C53—21no—
38#103×C53—42no—
39#99×C53—42no—
40C22⋊C8×C53—22no—
41#95×C53—22no—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yes—
46C24×C53—21no—
47C24×C53C2×D411yescharacteristic
48C2×Q8×C53—44no—
49C2×Q8×C53—42no—
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53SD1612yescomplemented
67C22×C4×C53C2×D412yes—
68SD16×C53—84no—
69C2×C8×C53—42no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—44no—
72C4⋊C4×C53—44no—
73C4⋊C4×C53—81no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—44no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C42×C53—81no—
82C42×C53—81no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—82no—
90C23×C53#12011yescharacteristic
91C23×C53C2×SD1611yescharacteristic
92C23×C53#13711yescharacteristic
93Q8×C53—44no—
94Q8×C53—82no—
95D4×C53—82no—
96D4×C53—81no—
97D4×C53—81no—
98D4×C53—81no—
99D4×C53—81no—
100C2×C4×C53—24no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—81no—
119C2×C4×C53—82no—
120C8×C53—82no—
121#12121C5311yesmaximal, characteristic, direct product factor, Sylow
122C22×C53—22no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—81no—
129C22×C53—82no—
130C22×C53—81no—
131C22×C53—82no—
132C22×C53—161no—
133C22×C53#44811yescharacteristic
134C22×C53#45311yescharacteristic
135C22×C53B(4,𝔽2)11yescharacteristic
136C4×C53—42no—
137C4×C53—42no—
138C4×C53—44no—
139C4×C53—82no—
140C4×C53—81no—
141C4×C53—82no—
142C4×C53—81no—
143#2607C2×C5311yescharacteristic, complemented
144#2022C2×C5311yescharacteristic
145#1601C2×C5312yes—
146#1351C2×C5312yescomplemented
147#1317C2×C5311yescharacteristic, complemented
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—41no—
151C2×C53—41no—
152C2×C53—161no—
153C2×C53#159611yescharacteristic
154C2×C53#219712yes—
155#534C22×C5311yescharacteristic
156C2×#123—21no—
157#448—22no—
158C2×#99—21no—
159#399—21no—
160#393C22×C5312yes—
161#391—22no—
162#387—22no—
163#381C22×C5311yescharacteristic
164C4×C22⋊C4—21no—
165#342C22×C5311yescharacteristic
166#327—22no—
167#323C22×C5312yes—
168C53#1212111yescharacteristic, direct product factor, Sylow
169C22×Q8D4×C5312yes—
170C22×D4—21no—
171C23×C4D4×C5311yescharacteristic
172C2×SD16—42no—
173#123—42no—
174#123—41no—
175#123—42no—
176#121—22no—
177#120—41no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4D4×C5311yescharacteristic
185C2×C22⋊C4C23×C5311yescharacteristic
186C2×C42—21no—
187C2×C42—21no—
188#103—42no—
189#99—42no—
190C22⋊C8—22no—
191#95—22no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95D4×C5312yes—
196C24—21no—
197C24C2×D4×C5311yescharacteristic
198C2×Q8—44no—
199C2×Q8—42no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4SD16×C5312yescomplemented
217C22×C4C2×D4×C5312yes—
218SD16—84no—
219C2×C8—42no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—81no—
224C22⋊C4—22no—
225C22⋊C4—41no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42—81no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—82no—
240C23#120×C5311yescharacteristic
241C23C2×SD16×C5311yescharacteristic
242C23#137×C5311yescharacteristic
243Q8—44no—
244Q8—82no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—81no—
249D4—81no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—82no—
270C8—82no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—82no—
280C22—81no—
281C22—161no—
282C22C53×#44811yescharacteristic
283C22C53×#45311yescharacteristic
284C22C53×B(4,𝔽2)11yescharacteristic
285C4—42no—
286C4—44no—
287C4—42no—
288C4—82no—
289C4—81no—
290C4—82no—
291C4—81no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—161no—
297C2C53×#159611yescharacteristic
298C2C53×#219712yes—
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC53×#534C22
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
1. C53×#2607C2semidirect
4. C53×#1351C2semidirect
5. C53×#1317C2semidirect
66. C22×C4×C53SD16semidirect
121. #12121C53direct (N × H)
143. #2607C2×C53semidirect
146. #1351C2×C53semidirect
147. #1317C2×C53semidirect
168. C53#12121direct (N × H)
216. C22×C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26072
C53×#20222
C53×#16012
C53×#13512
C53×#13172
#1212153