Finite Group Databasebeta

← C53×#8322 #27215834

Subgroups — C53×#8322 #27215834

Order 13568.

1146 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C2×C53×#518C211yesmaximal, characteristic
2C53×#1478C214yesmaximal, complemented
3C53×#1458C212yesmaximal, complemented
4C23×D4×C53C411yescharacteristic
5C53×#518—28no—
6C2×C4×D4×C53—22no—
7C2×C4×D4×C53C412yes—
8C2×C4×D4×C53C2212yes—
9C2×#117×C53C411yescharacteristic
10C2×#117×C53C2211yescharacteristic
11C22×C22⋊C4×C53—22no—
12C53×#431C2214yescomplemented
13C53×D4⋊C8—28no—
14C53×#324—22no—
15C53×#323—22no—
16C25×C53—21no—
17C22×D4×C53—24no—
18C22×D4×C53—24no—
19C22×D4×C53—21no—
20C22×D4×C53—22no—
21C22×D4×C53C814yescomplemented
22C22×D4×C53C2×C412yes—
23C23×C4×C53—22no—
24C23×C4×C53—22no—
25C23×C4×C53C2×C411yescharacteristic
26C4×D4×C53—216no—
27C4×D4×C53—24no—
28C4×D4×C53—24no—
29C4×D4×C53—24no—
30C4×D4×C53D418yes—
31#117×C53—24no—
32#117×C53—24no—
33C2×C4⋊C4×C53C2×C412yes—
34C2×C4⋊C4×C53D412yes—
35C2×C22⋊C4×C53—216no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53D412yes—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—24no—
44C22⋊C8×C53—24no—
45C4×C8×C53—24no—
46C24×C53—24no—
47C24×C53—24no—
48C24×C53—22no—
49C24×C53—24no—
50C24×C53—84no—
51C24×C53C22⋊C411yescharacteristic
52C2×D4×C53—24no—
53C2×D4×C53—24no—
54C2×D4×C53—216no—
55C2×D4×C53—216no—
56C2×D4×C53—24no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53C22⋊C414yes—
61C2×D4×C53C2×C814yes—
62C2×D4×C53M1614yes—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—28no—
67C22×C4×C53—28no—
68C22×C4×C53—84no—
69C22×C4×C53—84no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C2×C812yes—
73C22×C4×C53M1612yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D412yes—
76C2×C8×C53—44no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53C22⋊C414yes—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—28no—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—28no—
84C42×C53—24no—
85C42×C53—22no—
86C42×C53C2×D414yes—
87C23×C53—216no—
88C23×C53—24no—
89C23×C53—21no—
90C23×C53—24no—
91C23×C53—21no—
92C23×C53—24no—
93C23×C53—42no—
94C23×C53—44no—
95C23×C53—44no—
96C23×C53—88no—
97C23×C53—88no—
98C23×C53—88no—
99C23×C53C22⋊C812yes—
100C23×C53C2×C22⋊C411yescharacteristic
101D4×C53—216no—
102D4×C53—24no—
103D4×C53—24no—
104D4×C53—24no—
105D4×C53C22⋊C818yes—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—24no—
109C2×C4×C53—24no—
110C2×C4×C53—24no—
111C2×C4×C53—22no—
112C2×C4×C53—42no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—88no—
116C2×C4×C53—88no—
117C2×C4×C53C22⋊C812yes—
118C2×C4×C53C2×C22⋊C412yes—
119C2×C4×C53#12014yes—
120C2×C4×C53C22×C811yescharacteristic
121C2×C4×C53C2×M1612yes—
122C2×C4×C53C2×M1611yescharacteristic
123C8×C53—84no—
124#8322C5311yesmaximal, characteristic, direct product factor, Sylow
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—28no—
128C22×C53—42no—
129C22×C53—48no—
130C22×C53—48no—
131C22×C53—42no—
132C22×C53—44no—
133C22×C53—44no—
134C22×C53—42no—
135C22×C53—816no—
136C22×C53#37911yescharacteristic
137C22×C53C2×C22⋊C811yescharacteristic
138C22×C53#40711yescharacteristic
139C4×C53—24no—
140C4×C53—22no—
141C4×C53—44no—
142C4×C53—42no—
143C4×C53C2×C22⋊C812yes—
144C4×C53#40712yes—
145C2×#518C2×C5311yescharacteristic
146#1478C2×C5314yescomplemented
147#1458C2×C5312yescomplemented
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—48no—
151C2×C53—42no—
152C2×C53#177911yescharacteristic
153C2×C53#179512yes—
154C23×D4C4×C5311yescharacteristic
155#518—28no—
156C2×C4×D4—22no—
157C2×C4×D4C4×C5312yes—
158C2×C4×D4C22×C5312yes—
159C2×#117C4×C5311yescharacteristic
160C2×#117C22×C5311yescharacteristic
161C22×C22⋊C4—22no—
162#431C22×C5314yescomplemented
163D4⋊C8—28no—
164#324—22no—
165#323—22no—
166C53#832211yescharacteristic, direct product factor, Sylow
167C25—21no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—21no—
171C22×D4—22no—
172C22×D4C8×C5314yescomplemented
173C22×D4C2×C4×C5312yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C2×C4×C5311yescharacteristic
177C4×D4—216no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4D4×C5318yes—
182#117—24no—
183#117—24no—
184C2×C4⋊C4C2×C4×C5312yes—
185C2×C4⋊C4D4×C5312yes—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4D4×C5312yes—
191C2×C42—21no—
192C2×C42C2×C4×C5311yescharacteristic
193C2×C42C23×C5311yescharacteristic
194C4⋊C8—24no—
195C22⋊C8—24no—
196C4×C8—24no—
197C24—24no—
198C24—24no—
199C24—22no—
200C24—24no—
201C24—84no—
202C24C22⋊C4×C5311yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—216no—
206C2×D4—216no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C22⋊C4×C5314yes—
212C2×D4C2×C8×C5314yes—
213C2×D4M16×C5314yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4C22⋊C4×C5311yescharacteristic
222C22×C4C22⋊C4×C5312yes—
223C22×C4C2×C8×C5312yes—
224C22×C4M16×C5312yes—
225C22×C4C22×C4×C5311yescharacteristic
226C22×C4C2×D4×C5312yes—
227C2×C8—44no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4C22⋊C4×C5314yes—
231C22⋊C4—24no—
232C22⋊C4—28no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C42—24no—
236C42—22no—
237C42C2×D4×C5314yes—
238C23—216no—
239C23—21no—
240C23—24no—
241C23—24no—
242C23—21no—
243C23—24no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23—88no—
248C23—88no—
249C23—88no—
250C23C22⋊C8×C5312yes—
251C23C2×C22⋊C4×C5311yescharacteristic
252D4—216no—
253D4—24no—
254D4—24no—
255D4—24no—
256D4C22⋊C8×C5318yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4C22⋊C8×C5312yes—
269C2×C4C2×C22⋊C4×C5312yes—
270C2×C4#120×C5314yes—
271C2×C4C22×C8×C5311yescharacteristic
272C2×C4C2×M16×C5312yes—
273C2×C4C2×M16×C5311yescharacteristic
274C8—84no—
275C22—22no—
276C22—22no—
277C22—28no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—48no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—816no—
286C22C53×#37911yescharacteristic
287C22C2×C22⋊C8×C5311yescharacteristic
288C22C53×#40711yescharacteristic
289C4—22no—
290C4—24no—
291C4—44no—
292C4—42no—
293C4C2×C22⋊C8×C5312yes—
294C4C53×#40712yes—
295C2—21no—
296C2—21no—
297C2—48no—
298C2—42no—
299C2C53×#177911yescharacteristic
300C2C53×#179512yes—
subgroupG/H
center, socleC22×C53#379
commutatorC2×C4C22×C8×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC23×D4×C53C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1478C2semidirect
3. C53×#1458C2semidirect
12. C53×#431C22semidirect
21. C22×D4×C53C8semidirect
124. #8322C53direct (N × H)
146. #1478C2×C53semidirect
147. #1458C2×C53semidirect
162. #431C22×C53semidirect
166. C53#8322direct (N × H)
172. C22×D4C8×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5182
C53×#14782
C53×#14582
#832253