Finite Group Databasebeta

← C53×#12212 #27219724

Subgroups — C53×#12212 #27219724

Order 13568.

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

normal subgroups only

This group has 460 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×#2733C211yesmaximal, characteristic
2C53×#2028C211yesmaximal, characteristic, complemented
3C53×#1599C212yesmaximal, complemented
4C53×#1345C212yesmaximal, complemented
5C53×#1319C211yesmaximal, characteristic, complemented
6C53×#544C2211yescharacteristic
7C2×#124×C53—21no—
8C53×#450—22no—
9C53×#419—21no—
10C53×#394C2212yes—
11C53×#392—22no—
12C53×#391C2211yescharacteristic
13C53×#390—21no—
14C53×#388—22no—
15C53×#343C2211yescharacteristic
16C53×#339—21no—
17C53×#328—22no—
18C53×#324C2212yes—
19C22×Q8×C53D411yescharacteristic
20C22×D4×C53—21no—
21C22×D4×C53—21no—
22C22×D4×C53D412yes—
23C2×SD16×C53—42no—
24C2×M16×C53—21no—
25C4⋊Q8×C53—41no—
26#125×C53—41no—
27#124×C53—42no—
28#124×C53—42no—
29C22⋊Q8×C53—22no—
30#117×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53C2311yescharacteristic
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C42×C53—21no—
38C2×C42×C53D411yescharacteristic
39#103×C53—41no—
40#103×C53—41no—
41#102×C53—42no—
42C22⋊C8×C53—22no—
43#95×C53—22no—
44#95×C53D412yes—
45C24×C53—22no—
46C2×Q8×C53—21no—
47C2×Q8×C53—22no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53SD1612yescomplemented
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—82no—
55C2×D4×C53—81no—
56C2×D4×C53—81no—
57C2×D4×C53—81no—
58C2×D4×C53—81no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D412yes—
64SD16×C53—84no—
65M16×C53—41no—
66M16×C53—41no—
67C2×C8×C53—42no—
68C2×C8×C53—41no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53—21no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—81no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—82no—
81C42×C53—42no—
82C42×C53—41no—
83C42×C53—41no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92Q8×C53—42no—
93Q8×C53—81no—
94Q8×C53—82no—
95D4×C53—44no—
96D4×C53—82no—
97D4×C53—84no—
98D4×C53—84no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53C2×SD1611yescharacteristic
115C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
116C8×C53—81no—
117C8×C53—82no—
118#12212C5311yesmaximal, characteristic, direct product factor, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53#45011yescharacteristic
129C22×C53#45511yescharacteristic
130C22×C53B(4,𝔽2)11yescharacteristic
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53—42no—
135C4×C53—42no—
136C4×C53—82no—
137C4×C53—81no—
138C4×C53—81no—
139#2733C2×C5311yescharacteristic
140#2028C2×C5311yescharacteristic, complemented
141#1599C2×C5312yescomplemented
142#1345C2×C5312yescomplemented
143#1319C2×C5311yescharacteristic, complemented
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—162no—
147C2×C53#160111yescharacteristic
148C2×C53#219911yescharacteristic
149C2×C53#220411yescharacteristic
150#544C22×C5311yescharacteristic
151C2×#124—21no—
152#450—22no—
153#419—21no—
154#394C22×C5312yes—
155#392—22no—
156#391C22×C5311yescharacteristic
157#390—21no—
158#388—22no—
159#343C22×C5311yescharacteristic
160#339—21no—
161#328—22no—
162#324C22×C5312yes—
163C53#1221211yescharacteristic, direct product factor, Sylow
164C22×Q8D4×C5311yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4D4×C5312yes—
168C2×SD16—42no—
169C2×M16—21no—
170C4⋊Q8—41no—
171#125—41no—
172#124—42no—
173#124—42no—
174C22⋊Q8—22no—
175#117—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C23×C5311yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—21no—
183C2×C42D4×C5311yescharacteristic
184#103—41no—
185#103—41no—
186#102—42no—
187C22⋊C8—22no—
188#95—22no—
189#95D4×C5312yes—
190C24—22no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8SD16×C5312yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×D4×C5311yescharacteristic
208C22×C4C2×D4×C5312yes—
209SD16—84no—
210M16—41no—
211M16—41no—
212C2×C8—42no—
213C2×C8—41no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C42—41no—
227C42—42no—
228C42—41no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23#120×C5311yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—82no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4C2×SD16×C5311yescharacteristic
260C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
261C8—81no—
262C8—82no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22C53×#45011yescharacteristic
273C22C53×#45511yescharacteristic
274C22C53×B(4,𝔽2)11yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2C53×#160111yescharacteristic
287C2C53×#219911yescharacteristic
288C2C53×#220411yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC53×#2733C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2028C2semidirect
3. C53×#1599C2semidirect
4. C53×#1345C2semidirect
5. C53×#1319C2semidirect
50. C2×Q8×C53SD16semidirect
118. #12212C53direct (N × H)
140. #2028C2×C53semidirect
141. #1599C2×C53semidirect
142. #1345C2×C53semidirect
143. #1319C2×C53semidirect
163. C53#12212direct (N × H)
195. C2×Q8SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#27332
C53×#20282
C53×#15992
C53×#13452
C53×#13192
#1221253