Finite Group Databasebeta

← D53×#2030 #27411490

Subgroups — D53×#2030 #27411490

Order 13568.

1482 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 217 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
1#543×D53C211yesmaximal, characteristic
2#381×D53C213yesmaximal
3#20275883C213yesmaximal, complemented
4#20275882C213yesmaximal, complemented
5#20275881C211yesmaximal, characteristic, complemented
6#342×D53C213yesmaximal
7C53×#2030C211yesmaximal, characteristic, complemented
8C23×C4×D53C2211yescharacteristic
9#16811249C2213yes—
10#16811248—26no—
11C22⋊Q8×D53—26no—
12C2×C22⋊C4×D53C2213yes—
13C2×C22⋊C4×D53C2213yes—
14#16810486C2213yes—
15#16810485—26no—
16#16810484C2213yes—
17#16810484C2213yes—
18#95×D53—26no—
19#16810390C2213yes—
20#16810389C2216yes—
21C53×#543C2211yescharacteristic
22C53×#381C2213yes—
23C53×#342C2213yes—
24C24×D53C2311yescharacteristic
25C2×Q8×D53—22no—
26C23×Dic53C2311yescharacteristic
27C22×C4×D53—23no—
28C22×C4×D53—23no—
29C22×C4×D53—26no—
30C22×C4×D53—23no—
31C22×C4×D53D416yes—
32C22×C4×D53Q814yescomplemented
33#13133197—26no—
34#13133196—26no—
35#13133195—212no—
36#13133194—212no—
37#13133193—26no—
38C4⋊C4×D53—26no—
39C2×C22⋊Dic53C2313yes—
40C2×C22⋊Dic53C2313yes—
41C2×#11719921—23no—
42C2×#11719921D416yes—
43C22⋊C4×D53—26no—
44C22⋊C4×D53—26no—
45C22⋊C4×D53—212no—
46#13133124—26no—
47#13133124—26no—
48#13133124—26no—
49C23×C4×C53C2311yescharacteristic
50C22⋊Q8×C53—26no—
51C2×C22⋊C4×C53C2313yes—
52C2×C22⋊C4×C53C2313yes—
53#95×C53—26no—
54C24×C53C2411yescharacteristic
55C23×D53—44no—
56C23×D53C2×D413yes—
57C23×D53C2×D413yes—
58C23×D53C2×Q812yes—
59C23×D53#4116yes—
60C2×Q8×C53—22no—
61C22×C4×C53—23no—
62C22×C4×C53—23no—
63C22×C4×C53—23no—
64C22×C4×C53C2×D416yes—
65C22×C4×C53C2×Q812yescomplemented
66C22×Dic53—23no—
67C22×Dic53—23no—
68C22×Dic53—23no—
69C22×Dic53C2×D416yes—
70C22×Dic53C2×Q812yescomplemented
71Q8×D53—84no—
72C2×C4×D53—26no—
73C2×C4×D53—212no—
74C2×C4×D53—212no—
75C2×C4×D53—26no—
76C2×C4×D53—43no—
77C2×C4×D53—41no—
78C2×C4×D53—46no—
79C2×C4×D53—46no—
80C2×C4×D53—412no—
81C2×C4×D53—412no—
82C2×C4×D53—46no—
83C2×Dic106—26no—
84C4⋊C4×C53—26no—
85C22⋊C4×C53—26no—
86C22⋊C4×C53—26no—
87C22⋊Dic53—26no—
88C22⋊Dic53—26no—
89#11719921—212no—
90#11719921—46no—
91C4⋊Dic53—26no—
92#11719919—212no—
93C23×C53—42no—
94C23×C53C22×D413yes—
95C23×C53C22×Q811yescharacteristic
96C23×C53C2×#4113yes—
97C22×D53—212no—
98C22×D53—44no—
99C22×D53—412no—
100C22×D53—46no—
101C22×D53#12013yes—
102C22×D53#12011yescharacteristic
103C22×D53C22⋊Q816yes—
104C22×D53C22⋊Q816yes—
105C22×D53#12416yes—
106C22×D53#12416yes—
107Q8×C53—82no—
108C2×C4×C53—26no—
109C2×C4×C53—26no—
110C2×C4×C53—41no—
111C2×C4×C53—43no—
112C2×C4×C53—46no—
113C2×C4×C53—46no—
114C2×C4×C53—43no—
115C2×Dic53—26no—
116C2×Dic53—26no—
117C2×Dic53—41no—
118C2×Dic53—43no—
119C2×Dic53—46no—
120C2×Dic53—46no—
121C2×Dic53—43no—
122C4×D53—412no—
123C4×D53—44no—
124C4×D53—412no—
125C4×D53—86no—
126C4×D53—812no—
127Dic106—86no—
128C2×#2030—531nomaximal, Sylow
129C22×C53—26no—
130C22×C53—41no—
131C22×C53—43no—
132C22×C53C2×#12011yescharacteristic
133C22×C53C2×C22⋊Q813yes—
134C22×C53C2×#12413yes—
135C2×D53—26no—
136C2×D53—212no—
137C2×D53—212no—
138C2×D53—46no—
139C2×D53—46no—
140C2×D53#39316yes—
141C2×D53#39312yes—
142C2×D53#45814yes—
143C4×C53—42no—
144C4×C53—46no—
145C4×C53—83no—
146Dic53—42no—
147Dic53—46no—
148Dic53—83no—
149C2×#543—531no—
150C2×#381—533no—
151#2030—533no—
152#2030—533no—
153#2030—531no—
154#2030D5311yescharacteristic, direct product factor
155C2×#342—533no—
156C2×C53—23no—
157C2×C53—23no—
158C2×C53C2×#39311yescharacteristic
159C2×C53C2×#45812yes—
160D53—26no—
161D53#203014yesdirect product factor
162C24×C4—531no—
163#543—533no—
164#543—1066no—
165#543C2×D5311yescharacteristic
166C2×C22⋊Q8—1066no—
167C22×C22⋊C4—533no—
168C22×C22⋊C4—533no—
169#381—533no—
170#381—533no—
171#381—533no—
172#381—1066no—
173#381C2×D5313yes—
174C2×#95—1066no—
175#342—536no—
176#342—533no—
177#342C2×D5313yes—
178C53C2×#203011yescharacteristic, complemented, Sylow
179C25—531no—
180C22×Q8—1062no—
181C23×C4—531no—
182C23×C4—534no—
183C23×C4—536no—
184C23×C4—1063no—
185C23×C4—1063no—
186C23×C4—1066no—
187C23×C4—1063no—
188C23×C4C22×D5311yescharacteristic
189C22⋊Q8—26no—
190C22⋊Q8—10612no—
191C22⋊Q8—10612no—
192C22⋊Q8—1066no—
193C22⋊Q8—1066no—
194C22⋊Q8—1066no—
195C2×C4⋊C4—1066no—
196C2×C22⋊C4—533no—
197C2×C22⋊C4—536no—
198C2×C22⋊C4—533no—
199C2×C22⋊C4—1066no—
200C2×C22⋊C4—1066no—
201C2×C22⋊C4—10612no—
202C2×C22⋊C4—1063no—
203C2×C22⋊C4C22×D5313yes—
204C2×C22⋊C4C22×D5313yes—
205#95—26no—
206#95—1066no—
207#95—1066no—
208#95—1066no—
209C24—533no—
210C24—533no—
211C24—532no—
212C24—536no—
213C24—2124no—
214C24C23×D5311yescharacteristic
215C2×Q8—22no—
216C2×Q8—1066no—
217C2×Q8—4244no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—532no—
222C22×C4—536no—
223C22×C4—1066no—
224C22×C4—1066no—
225C22×C4—10612no—
226C22×C4—10612no—
227C22×C4—1063no—
228C22×C4—1063no—
229C22×C4—1063no—
230C22×C4—2126no—
231C22×C4—21212no—
232C22×C4—21212no—
233C22×C4—2121no—
234C22×C4—2123no—
235C22×C4—2126no—
236C22×C4—2126no—
237C22×C4D4×D5316yes—
238C22×C4Q8×D5312yescomplemented
239C4⋊C4—26no—
240C4⋊C4—10612no—
241C4⋊C4—1066no—
242C22⋊C4—26no—
243C22⋊C4—26no—
244C22⋊C4—10612no—
245C22⋊C4—1066no—
246C22⋊C4—1066no—
247C22⋊C4—2126no—
248C23—42no—
249C23—531no—
250C23—533no—
251C23—536no—
252C23—536no—
253C23—536no—
254C23—536no—
255C23—10612no—
256C23—2126no—
257C23—2124no—
258C23—21212no—
259C23C2×D4×D5313yes—
260C23C2×Q8×D5311yescharacteristic
261C23#41×D5313yes—
262Q8—82no—
263Q8—4246no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—41no—
267C2×C4—43no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—43no—
271C2×C4—1066no—
272C2×C4—1066no—
273C2×C4—4246no—
274C2×C4—42412no—
275C2×C4—2124no—
276C2×C4—21212no—
277C2×C4—21212no—
278C2×C4—2126no—
279C2×C4—2123no—
280C2×C4—2121no—
281C2×C4—2123no—
282C2×C4—2126no—
283C22—26no—
284C22—41no—
285C22—43no—
286C22—532no—
287C22—534no—
288C22—536no—
289C22—10612no—
290C22—10612no—
291C22—1066no—
292C22—2126no—
293C22—2126no—
294C22#120×D5311yescharacteristic
295C22C22⋊Q8×D5313yes—
296C22#124×D5313yes—
297C4—42no—
298C4—46no—
299C4—83no—
300C4—4243no—
301C4—2122no—
302C4—2126no—
303C2—23no—
304C2—23no—
305C2—534no—
306C2—1066no—
307C2#393×D5311yescharacteristic
308C2#458×D5312yes—
subgroupG/H
centerC22#120×D53
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#2030C2
socleC22×C53C2×#120
prime radicalC24×D53C23
hypercenter#2030D53
hypocenterC53C2×#2030
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #2030

Complements

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

NH ≅ G/Ntype
3. #20275883C2semidirect
4. #20275882C2semidirect
5. #20275881C2semidirect
7. C53×#2030C2semidirect
32. C22×C4×D53Q8semidirect
65. C22×C4×C53C2×Q8semidirect
70. C22×Dic53C2×Q8semidirect
154. #2030D53direct (N × H)
161. D53#2030direct (N × H)
178. C53C2×#2030semidirect
238. C22×C4Q8×D53semidirect

Maximal subgroups

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

subgroupindex
#543×D532
#381×D532
#202758832
#202758822
#202758812
#342×D532
C53×#20302
C2×#203053