Finite Group Databasebeta

← D53×#1306 #27274452

Subgroups — D53×#1306 #27274452

Order 13568.

942 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 159 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×D53C212yesmaximal
2#381×D53C211yesmaximal, characteristic
3#20272624C212yesmaximal, complemented
4#20272623C211yesmaximal, characteristic, complemented
5C53×#1306C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×M16×D53—22no—
8C2×C22⋊C4×D53C412yes—
9#16810587C2212yes—
10#16810586—22no—
11#16810585—22no—
12#16810585C414yescomplemented
13C22⋊C8×D53—22no—
14C22⋊C8×D53C414yescomplemented
15#16810486C2211yescharacteristic
16#16810485—22no—
17#16810484C412yes—
18#95×D53—22no—
19C53×#407C2212yes—
20C53×#381C2211yescharacteristic
21C24×D53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C22×C4×D53—22no—
24C22×C4×D53—44no—
25C22×C4×D53C2×C412yes—
26C22×C4×D53C2×C412yes—
27C22×C4×D53D412yes—
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D412yes—
31C22×C4×D53Q811yescharacteristic
32C22×C4×D53Q811yescharacteristic
33C2×#11719917—22no—
34C2×#11719912—24no—
35M16×D53—48no—
36C2×C8×D53—24no—
37C2×C22⋊Dic53C2×C412yes—
38C2×#11719921—22no—
39C22⋊C4×D53—44no—
40C22⋊#88322—22no—
41C22⋊#88322C2×C414yescomplemented
42#13133129—24no—
43#13133129—24no—
44#13133124—22no—
45#13133124—24no—
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C22⋊C8×C53—22no—
50C22⋊C8×C53C2×C414yescomplemented
51#95×C53—22no—
52C24×C53C22×C411yescharacteristic
53C23×D53—24no—
54C23×D53—44no—
55C23×D53C4211yescharacteristic
56C23×D53C4211yescharacteristic
57C23×D53C22⋊C411yescharacteristic
58C23×D53C22⋊C411yescharacteristic
59C23×D53C4⋊C411yescharacteristic
60C23×D53C4⋊C411yescharacteristic
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53C22×C412yes—
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×Q811yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53C4212yes—
70C22×Dic53C22⋊C412yes—
71C22×Dic53C4⋊C411yescharacteristic
72C22×Dic53C4⋊C411yescharacteristic
73C2×C4×D53—28no—
74C2×C4×D53—44no—
75C2×C4×D53—44no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53—44no—
81C2×C4×D53—48no—
82C2×C4×D53—44no—
83C2×C4×D53C22⋊C412yes—
84C2×C4×D53C22⋊C412yes—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53C22⋊C412yes—
87C2×C4×D53C4⋊C412yes—
88C2×C4×D53C4⋊C412yes—
89C2×C4×D53C4⋊C412yes—
90C2×C4×D53C4⋊C412yes—
91M16×C53—44no—
92C2×C8×C53—24no—
93C22⋊C4×C53—42no—
94C22⋊Dic53—42no—
95#11719921—44no—
96#11719917—44no—
97C2×#88322—24no—
98#11719912—48no—
99C8×D53—48no—
100C23×C53—22no—
101C23×C53—42no—
102C23×C53C2×C4211yescharacteristic
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C2×C4⋊C411yescharacteristic
105C22×D53—24no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—24no—
112C22×D53—24no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—416no—
117C22×D53#9511yescharacteristic
118C22×D53#9511yescharacteristic
119C22×D53#9511yescharacteristic
120C22×D53#9511yescharacteristic
121C22×D53#9914yes—
122C2×C4×C53—24no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—44no—
127C2×C4×C53—42no—
128C2×C4×C53C2×C22⋊C412yes—
129C2×C4×C53C2×C4⋊C412yes—
130C2×Dic53—24no—
131C2×Dic53—42no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—44no—
135C2×Dic53—42no—
136C2×Dic53#9512yes—
137C2×Dic53#9512yes—
138C4×D53—24no—
139C4×D53—24no—
140C4×D53—24no—
141C4×D53—24no—
142C4×D53—42no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—88no—
147C8×C53—44no—
148#88322—44no—
149C2×#1306—531nomaximal, Sylow
150C22×C53—22no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—44no—
156C22×C53C2×#9511yescharacteristic
157C22×C53C2×#9912yes—
158C2×D53—22no—
159C2×D53—28no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—44no—
165C2×D53—44no—
166C2×D53—44no—
167C2×D53—44no—
168C2×D53—44no—
169C2×D53#34212yes—
170C2×D53#34212yes—
171C2×D53#34414yes—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—82no—
175Dic53—22no—
176Dic53—22no—
177Dic53—82no—
178C2×#407—532no—
179C2×#381—531no—
180#1306—532no—
181#1306—531no—
182#1306D5311yescharacteristic, direct product factor
183C2×C53—22no—
184C2×C53—41no—
185C2×C53—41no—
186C2×C53C2×#34211yescharacteristic
187C2×C53C2×#34412yes—
188D53—22no—
189D53—41no—
190D53—41no—
191D53#130614yesdirect product factor
192C24×C4—531no—
193C22×M16—1062no—
194C22×C22⋊C4—532no—
195#407—532no—
196#407—534no—
197#407—1062no—
198#407—1062no—
199#407C2×D5312yes—
200C2×C22⋊C8—534no—
201C2×C22⋊C8—1062no—
202#381—531no—
203#381—532no—
204#381—1062no—
205#381C2×D5311yescharacteristic
206C2×#95—1062no—
207C53C2×#130611yescharacteristic, complemented, Sylow
208C25—531no—
209C23×C4—531no—
210C23×C4—532no—
211C23×C4—532no—
212C23×C4—532no—
213C23×C4—532no—
214C23×C4—531no—
215C23×C4—531no—
216C23×C4—531no—
217C23×C4—531no—
218C23×C4—1062no—
219C23×C4—2124no—
220C23×C4C22×D5311yescharacteristic
221C2×M16—22no—
222C2×M16—1064no—
223C2×M16—1062no—
224C2×M16—2128no—
225C22×C8—1064no—
226C2×C22⋊C4—532no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4—2124no—
229C2×C22⋊C4C4×D5312yes—
230C22⋊C8—22no—
231C22⋊C8—534no—
232C22⋊C8—1064no—
233C22⋊C8—1064no—
234C22⋊C8—1062no—
235C22⋊C8C4×D5314yescomplemented
236#95—22no—
237#95—1064no—
238#95—1062no—
239C24—531no—
240C24—531no—
241C24—531no—
242C24—531no—
243C24—531no—
244C24—531no—
245C24—1064no—
246C24—2124no—
247C24C2×C4×D5311yescharacteristic
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—532no—
251C22×C4—532no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—532no—
261C22×C4—532no—
262C22×C4—1068no—
263C22×C4—1062no—
264C22×C4—2124no—
265C22×C4—2124no—
266C22×C4—2124no—
267C22×C4—2122no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4—2124no—
272C22×C4—2128no—
273C22×C4—2122no—
274C22×C4C2×C4×D5312yes—
275C22×C4D4×D5312yes—
276C22×C4D4×D5311yescharacteristic
277C22×C4Q8×D5311yescharacteristic
278M16—44no—
279M16—2128no—
280M16—2124no—
281C2×C8—24no—
282C2×C8—1064no—
283C2×C8—2128no—
284C22⋊C4—42no—
285C22⋊C4—2124no—
286C22⋊C4—2122no—
287C23—22no—
288C23—42no—
289C23—531no—
290C23—534no—
291C23—531no—
292C23—531no—
293C23—531no—
294C23—1064no—
295C23—1062no—
296C23—1062no—
297C23—1062no—
298C23—1062no—
299C23—1064no—
300C23—1064no—
301C23—1062no—
302C23—2122no—
303C23—2122no—
304C23—2122no—
305C23—21216no—
306C23C42×D5311yescharacteristic
307C23C22⋊C4×D5311yescharacteristic
308C23C4⋊C4×D5311yescharacteristic
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—1064no—
318C2×C4—1064no—
319C2×C4—1064no—
320C2×C4—1064no—
321C2×C4—1064no—
322C2×C4—4248no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2124no—
331C2×C4—2122no—
332C2×C4C22⋊C4×D5312yes—
333C2×C4C4⋊C4×D5312yes—
334C8—44no—
335C8—2124no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—44no—
342C22—532no—
343C22—534no—
344C22—532no—
345C22—1068no—
346C22—1062no—
347C22—1061no—
348C22—1061no—
349C22—1062no—
350C22—1062no—
351C22—2124no—
352C22—2124no—
353C22—2124no—
354C22—2124no—
355C22—2124no—
356C22#95×D5311yescharacteristic
357C22#99×D5312yes—
358C4—22no—
359C4—22no—
360C4—82no—
361C4—1062no—
362C4—1062no—
363C4—4242no—
364C2—22no—
365C2—41no—
366C2—41no—
367C2—534no—
368C2—1062no—
369C2—2121no—
370C2—2121no—
371C2#342×D5311yescharacteristic
372C2#344×D5312yes—
subgroupG/H
centerC22#95×D53
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53C2×#95
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1306D53
hypocenterC53C2×#1306
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. #20272624C2semidirect
4. #20272623C2semidirect
5. C53×#1306C2semidirect
12. #16810585C4semidirect
14. C22⋊C8×D53C4semidirect
41. C22⋊#88322C2×C4semidirect
50. C22⋊C8×C53C2×C4semidirect
182. #1306D53direct (N × H)
191. D53#1306direct (N × H)
207. C53C2×#1306semidirect
235. C22⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#407×D532
#381×D532
#202726242
#202726232
C53×#13062
C2×#130653