Finite Group Databasebeta

← #27274412

Subgroups — #27274412

Order 13568.

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

normal subgroups only

This group has 994 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#20277093C212yesmaximal
2#20277093C211yesmaximal, characteristic
3#20272619C212yesmaximal
4#20272619C211yesmaximal, characteristic
5C53×#1304C211yesmaximal, characteristic
6C24×Dic53C2211yescharacteristic
7C2×#13133173—22no—
8C2×#13133173C412yescomplemented
9#16810484—22no—
10#16810484C412yescomplemented
11#16810481—22no—
12#16810481—22no—
13#16810481C414yescomplemented
14#16810479C2212yes—
15#16810479C2211yescharacteristic
16C22⋊C4×Dic53—22no—
17C22⋊C4×Dic53—22no—
18C22⋊C4×Dic53C414yescomplemented
19C53×#379C2212yes—
20C53×#379C2211yescharacteristic
21C23×Dic53—44no—
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2×C412yes—
24C23×Dic53C2×C412yes—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53D412yes—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C23×Dic53D412yes—
30C23×Dic53Q811yescharacteristic
31C23×Dic53Q811yescharacteristic
32#13133173—48no—
33#13133173—44no—
34C2×#11719919—24no—
35C2×#11719919—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53C2×C412yes—
40C2×C22⋊Dic53C2×C414yes—
41C2×C4×Dic53—24no—
42C2×C4×Dic53—22no—
43#13133124—24no—
44#13133124—24no—
45#13133124—24no—
46C25×C53C2311yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53C2×C412yes—
51C2×C22⋊C4×C53C2×C414yes—
52C24×C53—44no—
53C24×C53—42no—
54C24×C53C22×C411yescharacteristic
55C24×C53C22×C412yes—
56C24×C53C2×D412yes—
57C24×C53C2×D411yescharacteristic
58C24×C53C2×Q811yescharacteristic
59C22×C4×C53—24no—
60C22×C4×C53—22no—
61C22×Dic53—24no—
62C22×Dic53—28no—
63C22×Dic53—24no—
64C22×Dic53—22no—
65C22×Dic53—44no—
66C22×Dic53—44no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—42no—
72C22×Dic53—44no—
73C22×Dic53C4211yescharacteristic
74C22×Dic53C4211yescharacteristic
75C22×Dic53C4212yes—
76C22×Dic53C22⋊C412yes—
77C22×Dic53C22⋊C412yes—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C412yes—
80C22×Dic53C22⋊C412yes—
81C22×Dic53C22⋊C412yes—
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C4⋊C411yescharacteristic
84C22×Dic53C4⋊C411yescharacteristic
85C22×Dic53C4⋊C412yes—
86C22×Dic53C4⋊C412yes—
87C22×Dic53C4⋊C412yes—
88C22×Dic53C4⋊C412yes—
89C22×Dic53C4⋊C411yescharacteristic
90C22×Dic53C4⋊C411yescharacteristic
91C22⋊C4×C53—44no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—44no—
94C22⋊C4×C53—44no—
95C22⋊C4×C53—42no—
96C22⋊Dic53—44no—
97C22⋊Dic53—44no—
98C22⋊Dic53—44no—
99C22⋊Dic53—44no—
100C22⋊Dic53—42no—
101#11719919—44no—
102#11719919—48no—
103C4×Dic53—48no—
104C4×Dic53—44no—
105C23×C53—22no—
106C23×C53—24no—
107C23×C53—24no—
108C23×C53—24no—
109C23×C53—24no—
110C23×C53—42no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—416no—
118C23×C53C2×C4211yescharacteristic
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53C2×C22⋊C412yes—
121C23×C53C2×C4⋊C411yescharacteristic
122C23×C53C2×C4⋊C412yes—
123C2×C4×C53—44no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—42no—
127C2×C4×C53—44no—
128C2×Dic53—24no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—24no—
133C2×Dic53—24no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—24no—
137C2×Dic53—24no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—44no—
146C2×Dic53—44no—
147C2×Dic53—44no—
148C2×Dic53—42no—
149C2×Dic53—44no—
150C2×Dic53#9511yescharacteristic
151C2×Dic53#9511yescharacteristic
152C2×Dic53#9512yes—
153C2×Dic53#9512yes—
154C2×Dic53#9511yescharacteristic
155C2×Dic53#9511yescharacteristic
156C2×Dic53#9914yes—
157#8100—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—48no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—48no—
172C22×C53—44no—
173C22×C53C2×#9511yescharacteristic
174C22×C53C2×#9912yes—
175C22×C53#41014yes—
176C4×C53—82no—
177C4×C53—84no—
178Dic53—22no—
179Dic53—22no—
180Dic53—21no—
181Dic53—21no—
182Dic53—82no—
183Dic53—84no—
184Dic53#34212yes—
185Dic53#34212yes—
186#2280—531no—
187#2280—532no—
188#1304—532no—
189#1304—531no—
190#1304D5311yescharacteristic
191C2×C53—22no—
192C2×C53—24no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53—42no—
196C2×C53C2×#34211yescharacteristic
197C2×C53#174012yes—
198C24×C4—531no—
199C2×#117—532no—
200C2×#117—1062no—
201#381—532no—
202#381—1062no—
203#380—534no—
204#380—1062no—
205#380—1062no—
206#379—531no—
207#379—532no—
208#379C2×D5312yes—
209#379C2×D5311yescharacteristic
210C4×C22⋊C4—534no—
211C4×C22⋊C4—1062no—
212C4×C22⋊C4—1062no—
213C53#810011yescharacteristic, complemented, Sylow
214C25C22×D5311yescharacteristic
215C23×C4—531no—
216C23×C4—532no—
217C23×C4—532no—
218C23×C4—532no—
219C23×C4—531no—
220C23×C4—531no—
221C23×C4—531no—
222C23×C4—532no—
223C23×C4—531no—
224C23×C4—531no—
225C23×C4—2124no—
226#117—2128no—
227#117—2124no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1064no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—532no—
234C2×C22⋊C4—534no—
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4C4×D5312yes—
239C2×C22⋊C4C4×D5314yes—
240C2×C42—1062no—
241C2×C42—1064no—
242#95—1064no—
243#95—1064no—
244#95—1064no—
245C24—42no—
246C24—44no—
247C24C2×C4×D5311yescharacteristic
248C24C2×C4×D5312yes—
249C24D4×D5312yes—
250C24D4×D5311yescharacteristic
251C24Q8×D5311yescharacteristic
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—532no—
264C22×C4—532no—
265C22×C4—532no—
266C22×C4—532no—
267C22×C4—532no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—1068no—
273C22×C4—1064no—
274C22×C4—1064no—
275C22×C4—1062no—
276C22×C4—2122no—
277C22×C4—2124no—
278C22×C4—2122no—
279C22×C4—2124no—
280C22×C4—2122no—
281C22×C4—2122no—
282C22×C4—2124no—
283C22×C4—2122no—
284C4⋊C4—2128no—
285C4⋊C4—2124no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—2124no—
292C22⋊C4—2124no—
293C22⋊C4—2124no—
294C22⋊C4—2122no—
295C22⋊C4—2124no—
296C42—2124no—
297C42—2128no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—42no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—416no—
311C23C42×D5311yescharacteristic
312C23C22⋊C4×D5311yescharacteristic
313C23C22⋊C4×D5312yes—
314C23C4⋊C4×D5311yescharacteristic
315C23C4⋊C4×D5312yes—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—534no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—532no—
327C2×C4—532no—
328C2×C4—1064no—
329C2×C4—1064no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4—1064no—
333C2×C4—1062no—
334C2×C4—1064no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—1064no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2121no—
342C2×C4—2122no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4—2124no—
346C2×C4—2124no—
347C2×C4—2124no—
348C2×C4—2122no—
349C2×C4—2124no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—48no—
361C22—44no—
362C22—44no—
363C22—48no—
364C22—44no—
365C22#95×D5311yescharacteristic
366C22#99×D5312yes—
367C22#1681059514yes—
368C4—82no—
369C4—84no—
370C4—532no—
371C4—532no—
372C4—1062no—
373C4—1062no—
374C4—1061no—
375C4—1061no—
376C4—4242no—
377C4—4244no—
378C2—22no—
379C2—24no—
380C2—41no—
381C2—41no—
382C2—42no—
383C2#342×D5311yescharacteristic
384C2#2027444612yes—
subgroupG/H
centerC22#95×D53
commutatorC23×C53C2×C42
FrattiniC25C22×D53
FittingC53×#1304C2
socleC22×C53C2×#95
prime radical, cosocleC25×C53C23
hypercenter#1304D53
hypocenterC53#8100
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1304

Complements

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

NH ≅ G/Ntype
8. C2×#13133173C4semidirect
10. #16810484C4semidirect
13. #16810481C4semidirect
18. C22⋊C4×Dic53C4semidirect
213. C53#8100semidirect

Maximal subgroups

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

subgroupindex
#202770932
#202726192
C53×#13042
#810053