Finite Group Databasebeta

← D53×#1297 #27274210

Subgroups — D53×#1297 #27274210

Order 13568.

1226 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 179 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#407×D53C211yesmaximal, characteristic
2C4×C22⋊C4×D53C212yesmaximal
3#20272601C212yesmaximal, complemented
4#20272600C211yesmaximal, characteristic, complemented
5C53×#1297C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×M16×D53C412yescomplemented
8C2×C22⋊C4×D53—22no—
9C2×C42×D53—22no—
10#16810587C2211yescharacteristic
11#16810586C412yescomplemented
12#16810585—22no—
13C22⋊C8×D53—22no—
14C22⋊C4×Dic53—22no—
15C4×#11719921—22no—
16C4×C22⋊Dic53C2212yes—
17#16810470C414yescomplemented
18#95×D53C414yescomplemented
19C53×#407C2211yescharacteristic
20C4×C22⋊C4×C53C2212yes—
21C24×D53D411yescharacteristic
22C23×Dic53D411yescharacteristic
23C22×C4×D53—24no—
24C22×C4×D53—44no—
25C22×C4×D53C2×C411yescharacteristic
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53C2×C412yes—
28C22×C4×D53C2×C412yes—
29C22×C4×D53D412yes—
30C22×C4×D53D412yes—
31C22×C4×D53Q811yescharacteristic
32C22×C4×D53Q811yescharacteristic
33C2×#11719917C2×C412yescomplemented
34C2×#11719912—22no—
35M16×D53—44no—
36C2×C8×D53—22no—
37C2×C22⋊Dic53—22no—
38C2×#11719921—24no—
39C22⋊C4×D53—216no—
40C2×C4×Dic53—22no—
41C2×C4×Dic53—24no—
42C42×D53—48no—
43C22⋊#88322—22no—
44#13133129—24no—
45#13133124—24no—
46#13133124C2×C414yescomplemented
47C23×C4×C53C2311yescharacteristic
48C2×M16×C53C2×C412yescomplemented
49C2×C22⋊C4×C53—22no—
50C2×C42×C53—22no—
51C22⋊C8×C53—22no—
52#95×C53C2×C414yescomplemented
53C24×C53C2×D411yescharacteristic
54C23×D53—28no—
55C23×D53—42no—
56C23×D53C22⋊C412yes—
57C23×D53C22⋊C412yes—
58C23×D53C4⋊C411yescharacteristic
59C23×D53C4⋊C411yescharacteristic
60C22×C4×C53—24no—
61C22×C4×C53—42no—
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C22×C412yes—
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×Q811yescharacteristic
66C22×Dic53—24no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C4⋊C411yescharacteristic
71C22×Dic53C4⋊C411yescharacteristic
72C2×C4×D53—28no—
73C2×C4×D53—24no—
74C2×C4×D53—216no—
75C2×C4×D53—44no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—44no—
79C2×C4×D53—44no—
80C2×C4×D53—42no—
81C2×C4×D53—48no—
82C2×C4×D53—48no—
83C2×C4×D53C4211yescharacteristic
84C2×C4×D53C4211yescharacteristic
85C2×C4×D53C4212yes—
86C2×C4×D53C22⋊C412yes—
87C2×C4×D53C22⋊C411yescharacteristic
88C2×C4×D53C22⋊C411yescharacteristic
89C2×C4×D53C4⋊C412yes—
90C2×C4×D53C4⋊C412yes—
91C2×C4×D53C4⋊C412yes—
92C2×C4×D53C4⋊C412yes—
93M16×C53—42no—
94C2×C8×C53—22no—
95C22⋊C4×C53—28no—
96C42×C53—44no—
97C22⋊Dic53—28no—
98#11719921—216no—
99C4×Dic53—48no—
100C4×Dic53—44no—
101#11719917—42no—
102C2×#88322—22no—
103#11719912—44no—
104C8×D53—44no—
105C23×C53—24no—
106C23×C53—41no—
107C23×C53C2×C22⋊C412yes—
108C23×C53C2×C4⋊C411yescharacteristic
109C22×D53—24no—
110C22×D53—24no—
111C22×D53—24no—
112C22×D53—24no—
113C22×D53—24no—
114C22×D53—24no—
115C22×D53—22no—
116C22×D53—42no—
117C22×D53—41no—
118C22×D53—416no—
119C22×D53#9511yescharacteristic
120C22×D53#9512yes—
121C22×D53#9511yescharacteristic
122C22×D53#10418yes—
123C2×C4×C53—22no—
124C2×C4×C53—24no—
125C2×C4×C53—28no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—44no—
129C2×C4×C53—44no—
130C2×C4×C53C2×C4211yescharacteristic
131C2×C4×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53C2×C4⋊C412yes—
133C2×Dic53—24no—
134C2×Dic53—22no—
135C2×Dic53—28no—
136C2×Dic53—42no—
137C2×Dic53—41no—
138C2×Dic53—44no—
139C2×Dic53—44no—
140C2×Dic53#9511yescharacteristic
141C2×Dic53#9512yes—
142C2×Dic53#9511yescharacteristic
143C4×D53—22no—
144C4×D53—24no—
145C4×D53—24no—
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—24no—
149C4×D53—42no—
150C4×D53—42no—
151C4×D53—42no—
152C4×D53—432no—
153C4×D53#9914yes—
154C8×C53—42no—
155#88322—42no—
156C2×#1297—531nomaximal, Sylow
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—21no—
161C22×C53—44no—
162C22×C53C2×#9511yescharacteristic
163C22×C53C2×#10414yes—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—216no—
167C2×D53—22no—
168C2×D53—21no—
169C2×D53—44no—
170C2×D53—48no—
171C2×D53—44no—
172C2×D53#33914yes—
173C2×D53#33914yes—
174C2×D53#34212yes—
175C4×C53—21no—
176C4×C53—22no—
177C4×C53—48no—
178C4×C53C2×#9912yes—
179Dic53—21no—
180Dic53—22no—
181Dic53—48no—
182Dic53#34212yes—
183C2×#407—531no—
184C2×C4×C22⋊C4—532no—
185#1297—531no—
186#1297—532no—
187#1297D5311yescharacteristic, direct product factor
188C2×C53—24no—
189C2×C53—41no—
190C2×C53C2×#33912yes—
191C2×C53C2×#34211yescharacteristic
192D53—24no—
193D53—41no—
194D53#129714yesdirect product factor
195C24×C4—531no—
196C22×M16—532no—
197C22×C22⋊C4—1062no—
198C22×C42—1062no—
199#407—531no—
200#407—532no—
201#407—1062no—
202#407C2×D5311yescharacteristic
203C2×C22⋊C8—1062no—
204C4×C22⋊C4—532no—
205C4×C22⋊C4—534no—
206C4×C22⋊C4—1062no—
207C4×C22⋊C4—1062no—
208C4×C22⋊C4C2×D5312yes—
209C2×#95—534no—
210C53C2×#129711yescharacteristic, complemented, Sylow
211C25—531no—
212C23×C4—531no—
213C23×C4—531no—
214C23×C4—532no—
215C23×C4—532no—
216C23×C4—532no—
217C23×C4—531no—
218C23×C4—532no—
219C23×C4—531no—
220C23×C4—531no—
221C23×C4—1064no—
222C23×C4—2124no—
223C23×C4C22×D5311yescharacteristic
224C2×M16—532no—
225C2×M16—1062no—
226C2×M16—2124no—
227C2×M16C4×D5312yescomplemented
228C22×C8—1062no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—10616no—
231C2×C22⋊C4—1064no—
232C2×C22⋊C4—1062no—
233C2×C42—22no—
234C2×C42—1064no—
235C2×C42—1062no—
236C2×C42—2128no—
237C22⋊C8—22no—
238C22⋊C8—1064no—
239C22⋊C8—1062no—
240#95—534no—
241#95—1064no—
242#95C4×D5314yescomplemented
243C24—532no—
244C24—531no—
245C24—532no—
246C24—531no—
247C24—1068no—
248C24—2122no—
249C24D4×D5311yescharacteristic
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—531no—
253C22×C4—532no—
254C22×C4—531no—
255C22×C4—532no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—532no—
260C22×C4—531no—
261C22×C4—532no—
262C22×C4—532no—
263C22×C4—532no—
264C22×C4—532no—
265C22×C4—532no—
266C22×C4—1068no—
267C22×C4—1064no—
268C22×C4—10616no—
269C22×C4—1064no—
270C22×C4—2124no—
271C22×C4—2122no—
272C22×C4—2124no—
273C22×C4—2122no—
274C22×C4—2124no—
275C22×C4—2122no—
276C22×C4—2128no—
277C22×C4—2128no—
278C22×C4—2122no—
279C22×C4C2×C4×D5311yescharacteristic
280C22×C4C2×C4×D5312yes—
281C22×C4D4×D5312yes—
282C22×C4Q8×D5311yescharacteristic
283M16—42no—
284M16—2124no—
285M16—2122no—
286C2×C8—22no—
287C2×C8—1062no—
288C2×C8—2124no—
289C22⋊C4—28no—
290C22⋊C4—10616no—
291C22⋊C4—1068no—
292C42—44no—
293C42—2124no—
294C42—2128no—
295C23—24no—
296C23—41no—
297C23—531no—
298C23—538no—
299C23—532no—
300C23—531no—
301C23—1064no—
302C23—1064no—
303C23—1064no—
304C23—1062no—
305C23—1064no—
306C23—1064no—
307C23—1064no—
308C23—2122no—
309C23—2121no—
310C23—21216no—
311C23C22⋊C4×D5312yes—
312C23C4⋊C4×D5311yescharacteristic
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—531no—
321C2×C4—534no—
322C2×C4—532no—
323C2×C4—531no—
324C2×C4—1062no—
325C2×C4—1064no—
326C2×C4—1062no—
327C2×C4—1064no—
328C2×C4—1064no—
329C2×C4—1062no—
330C2×C4—1062no—
331C2×C4—1064no—
332C2×C4—1068no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—21232no—
337C2×C4—2122no—
338C2×C4—2121no—
339C2×C4—2124no—
340C2×C4—2124no—
341C2×C4C42×D5311yescharacteristic
342C2×C4C22⋊C4×D5311yescharacteristic
343C2×C4C4⋊C4×D5312yes—
344C8—42no—
345C8—2122no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—22no—
350C22—44no—
351C22—532no—
352C22—534no—
353C22—534no—
354C22—10616no—
355C22—1062no—
356C22—1062no—
357C22—1061no—
358C22—1062no—
359C22—2128no—
360C22—2124no—
361C22—2124no—
362C22#95×D5311yescharacteristic
363C22#104×D5314yes—
364C4—21no—
365C4—22no—
366C4—48no—
367C4—532no—
368C4—1062no—
369C4—1061no—
370C4—2128no—
371C4#99×D5312yes—
372C2—24no—
373C2—41no—
374C2—534no—
375C2—1064no—
376C2—2121no—
377C2#339×D5312yes—
378C2#342×D5311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1297C2
socleC22×C53C2×#95
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1297D53
hypocenterC53C2×#1297
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
3. #20272601C2semidirect
4. #20272600C2semidirect
5. C53×#1297C2semidirect
7. C2×M16×D53C4semidirect
11. #16810586C4semidirect
17. #16810470C4semidirect
18. #95×D53C4semidirect
33. C2×#11719917C2×C4semidirect
46. #13133124C2×C4semidirect
48. C2×M16×C53C2×C4semidirect
52. #95×C53C2×C4semidirect
187. #1297D53direct (N × H)
194. D53#1297direct (N × H)
210. C53C2×#1297semidirect
227. C2×M16C4×D53semidirect
242. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
#407×D532
C4×C22⋊C4×D532
#202726012
#202726002
C53×#12972
C2×#129753