Finite Group Databasebeta

← C53×#12122 #27219634

Subgroups — C53×#12122 #27219634

Order 13568.

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

normal subgroups only

This group has 458 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×#1613C212yesmaximal
4C53×#1352C212yesmaximal, complemented
5C53×#1317C211yesmaximal, characteristic, complemented
6C53×#534C2211yescharacteristic
7C2×#123×C53—21no—
8C53×#481—22no—
9C2×#99×C53—21no—
10C53×#399C2212yes—
11C53×#393—21no—
12C53×#391—22no—
13C53×#387—22no—
14C53×#381C2211yescharacteristic
15C4×C22⋊C4×C53—21no—
16C53×#342C2211yescharacteristic
17C53×#327—22no—
18C53×#323C2212yes—
19C22×Q8×C53—21no—
20C22×D4×C53—21no—
21C23×C4×C53D411yescharacteristic
22#123×C53—41no—
23#123×C53—42no—
24#123×C53—42no—
25#121×C53—22no—
26#120×C53—41no—
27C2×C4⋊C4×C53—22no—
28C2×C4⋊C4×C53—41no—
29C2×C4⋊C4×C53D412yes—
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53—21no—
32C2×C22⋊C4×C53—41no—
33C2×C22⋊C4×C53D411yescharacteristic
34C2×C22⋊C4×C53C2311yescharacteristic
35C2×C42×C53—21no—
36C2×C42×C53—21no—
37#106×C53—42no—
38#102×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—81no—
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×C53SD1612yes—
67C22×C4×C53C2×D412yes—
68C2×C8×C53—42no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—44no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—44no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—42no—
77C22⋊C4×C53—44no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C42×C53—81no—
81C42×C53—81no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—42no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53—82no—
89C23×C53#12011yescharacteristic
90C23×C53C2×SD1611yescharacteristic
91C23×C53#13711yescharacteristic
92Q8×C53—84no—
93D4×C53—82no—
94D4×C53—81no—
95D4×C53—81no—
96D4×C53—81no—
97D4×C53—81no—
98C2×C4×C53—24no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—81no—
117C2×C4×C53—82no—
118C8×C53—82no—
119#12122C5311yesmaximal, characteristic, direct product factor, Sylow
120C22×C53—22no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—42no—
126C22×C53—81no—
127C22×C53—82no—
128C22×C53—81no—
129C22×C53—82no—
130C22×C53—161no—
131C22×C53#44811yescharacteristic
132C22×C53#45311yescharacteristic
133C22×C53B(4,𝔽2)11yescharacteristic
134C4×C53—44no—
135C4×C53—42no—
136C4×C53—42no—
137C4×C53—81no—
138C4×C53—82no—
139C4×C53—81no—
140C4×C53—82no—
141#2607C2×C5311yescharacteristic, complemented
142#2022C2×C5311yescharacteristic
143#1613C2×C5312yes—
144#1352C2×C5312yescomplemented
145#1317C2×C5311yescharacteristic, complemented
146C2×C53—21no—
147C2×C53—21no—
148C2×C53—41no—
149C2×C53—41no—
150C2×C53—161no—
151C2×C53#159611yescharacteristic
152C2×C53#219812yes—
153#534C22×C5311yescharacteristic
154C2×#123—21no—
155#481—22no—
156C2×#99—21no—
157#399C22×C5312yes—
158#393—21no—
159#391—22no—
160#387—22no—
161#381C22×C5311yescharacteristic
162C4×C22⋊C4—21no—
163#342C22×C5311yescharacteristic
164#327—22no—
165#323C22×C5312yes—
166C53#1212211yescharacteristic, direct product factor, Sylow
167C22×Q8—21no—
168C22×D4—21no—
169C23×C4D4×C5311yescharacteristic
170#123—42no—
171#123—41no—
172#123—42no—
173#121—22no—
174#120—41no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4D4×C5312yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D4×C5311yescharacteristic
182C2×C22⋊C4C23×C5311yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185#106—42no—
186#102—42no—
187#99—42no—
188C22⋊C8—22no—
189#95—22no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95D4×C5312yes—
194C24—21no—
195C24C2×D4×C5311yescharacteristic
196C2×Q8—44no—
197C2×Q8—81no—
198C2×D4—22no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4SD16×C5312yes—
215C22×C4C2×D4×C5312yes—
216C2×C8—42no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—22no—
222C22⋊C4—41no—
223C22⋊C4—44no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—81no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23#120×C5311yescharacteristic
238C23C2×SD16×C5311yescharacteristic
239C23#137×C5311yescharacteristic
240Q8—84no—
241D4—82no—
242D4—81no—
243D4—81no—
244D4—81no—
245D4—81no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—82no—
266C8—82no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—161no—
278C22C53×#44811yescharacteristic
279C22C53×#45311yescharacteristic
280C22C53×B(4,𝔽2)11yescharacteristic
281C4—44no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—82no—
286C4—82no—
287C4—81no—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2—161no—
293C2C53×#159611yescharacteristic
294C2C53×#219812yes—
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×#1352C2semidirect
5. C53×#1317C2semidirect
119. #12122C53direct (N × H)
141. #2607C2×C53semidirect
144. #1352C2×C53semidirect
145. #1317C2×C53semidirect
166. C53#12122direct (N × H)

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×#16132
C53×#13522
C53×#13172
#1212253