Finite Group Databasebeta

← #27287342

Subgroups — #27287342

Order 13568.

762 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 75 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#20282968C211yesmaximal, characteristic, complemented
2#20274630C212yesmaximal
3#20272619C212yesmaximal, complemented
4#20272619C211yesmaximal, characteristic, complemented
5C53×#1781C211yesmaximal, characteristic
6C2×#13133132—24no—
7#16810535—24no—
8#16810535—22no—
9#16810479C2212yes—
10#16810479C2212yes—
11C23×D4×C53C2211yescharacteristic
12C2×#99×C53—22no—
13C53×#379C2211yescharacteristic
14C53×#379C2211yescharacteristic
15C22×Dic106—22no—
16C2×C22⋊Dic53—22no—
17C2×C22⋊Dic53—22no—
18C2×C22⋊Dic53—22no—
19C2×C22⋊Dic53—22no—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53D414yes—
22#13133132—48no—
23#13133124—26no—
24C25×C53Q811yescharacteristic, complemented
25C25×C53C2311yescharacteristic
26C22×D4×C53—22no—
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53—22no—
30C22×D4×C53—44no—
31C23×C4×C53Q811yescharacteristic, complemented
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—21no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53D412yes—
36#99×C53—44no—
37C24×C53—22no—
38C24×C53—21no—
39C24×C53—42no—
40C24×C53—42no—
41C24×C53—82no—
42C24×C53—82no—
43C24×C53C2×D412yes—
44C24×C53C2×D411yescharacteristic
45C24×C53C2×Q811yescharacteristic
46C24×C53#4111yescharacteristic
47C24×C53#4112yes—
48C2×D4×C53—24no—
49C2×D4×C53—28no—
50C2×D4×C53—42no—
51C2×D4×C53—44no—
52C2×D4×C53—42no—
53C2×D4×C53—44no—
54C2×D4×C53—44no—
55C2×D4×C53—42no—
56C2×D4×C53—41no—
57C2×D4×C53—42no—
58C2×D4×C53—41no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—41no—
64C22×Dic53—24no—
65C22×Dic53—42no—
66C2×Dic106—84no—
67C2×Dic106—82no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—82no—
71C22⋊C4×C53—82no—
72C22⋊Dic53—44no—
73C22⋊Dic53—44no—
74C22⋊Dic53—82no—
75C22⋊Dic53—82no—
76C22⋊Dic53—82no—
77C22⋊Dic53—82no—
78C23×C53—24no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—24no—
82C23×C53—41no—
83C23×C53—42no—
84C23×C53—41no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—82no—
96C23×C53—82no—
97C23×C53—82no—
98C23×C53—88no—
99C23×C53—88no—
100C23×C53#12011yescharacteristic
101C23×C53C22⋊Q812yes—
102C23×C53C22⋊Q811yescharacteristic
103C23×C53#12412yes—
104C23×C53#12411yescharacteristic
105D4×C53—48no—
106D4×C53—44no—
107D4×C53—44no—
108C2×C4×C53—22no—
109C2×C4×C53—24no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53—82no—
117C2×Dic53—44no—
118C2×Dic53—44no—
119C2×Dic53—44no—
120C2×Dic53—82no—
121C2×Dic53—82no—
122C2×Dic53—82no—
123Dic106—88no—
124#10390—531nomaximal, Sylow
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—84no—
139C22×C53—82no—
140C22×C53—82no—
141C22×C53—84no—
142C22×C53—82no—
143C22×C53—84no—
144C22×C53—84no—
145C22×C53—82no—
146C22×C53—82no—
147C22×C53#39311yescharacteristic
148C22×C53B(4,𝔽2)12yes—
149C4×C53—42no—
150C4×C53—82no—
151C4×C53—161no—
152Dic53—84no—
153Dic53—162no—
154#2840—531no—
155#1781—532no—
156#1781D5311yescharacteristic
157#1304—532no—
158#1304—531no—
159C2×C53—22no—
160C2×C53—42no—
161C2×C53—41no—
162C2×C53—42no—
163C2×C53—41no—
164C2×C53—82no—
165C2×C53#202911yescharacteristic
166C2×C53#203212yes—
167C23×D4C2×D5311yescharacteristic
168C2×#99—22no—
169C2×#99—1064no—
170#393—1064no—
171#393—1062no—
172#379—532no—
173#379—532no—
174#379C2×D5311yescharacteristic
175#379C2×D5311yescharacteristic
176C53#1039011yescharacteristic, complemented, Sylow
177C25Dic10611yescharacteristic, complemented
178C25C22×D5311yescharacteristic
179C22×Q8—1062no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—44no—
185C23×C4Dic10611yescharacteristic, complemented
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—534no—
190C2×C22⋊C4—1062no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4C22⋊D5312yes—
196#99—44no—
197#99—2128no—
198#95—1066no—
199C24—22no—
200C24—21no—
201C24—42no—
202C24—42no—
203C24—82no—
204C24—82no—
205C24C2×Dic10611yescharacteristic
206C24#1171994511yescharacteristic
207C24D4×D5312yes—
208C24#1171994712yes—
209C24C2×C22⋊D5311yescharacteristic
210C2×Q8—4244no—
211C2×Q8—4242no—
212C2×D4—24no—
213C2×D4—28no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—41no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—1064no—
229C22×C4—2122no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—4242no—
235C22⋊C4—4242no—
236C22⋊C4—4242no—
237C22⋊C4—4242no—
238C22⋊C4—2124no—
239C22⋊C4—2124no—
240C23—22no—
241C23—24no—
242C23—24no—
243C23—22no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—82no—
258C23—82no—
259C23—82no—
260C23—88no—
261C23—88no—
262C23#1313318511yescharacteristic
263C23#1313319412yes—
264C23#1313319611yescharacteristic
265C23#1313320312yes—
266C23#1313320411yescharacteristic
267Q8—4248no—
268D4—44no—
269D4—48no—
270D4—44no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—4242no—
281C2×C4—4242no—
282C2×C4—4242no—
283C2×C4—2124no—
284C2×C4—2124no—
285C2×C4—2124no—
286C22—22no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—44no—
298C22—42no—
299C22—82no—
300C22—84no—
301C22—84no—
302C22—82no—
303C22—84no—
304C22—82no—
305C22—84no—
306C22—82no—
307C22—82no—
308C22#1681053511yescharacteristic
309C22#1681080912yes—
310C4—42no—
311C4—82no—
312C4—161no—
313C4—8482no—
314C4—4244no—
315C2—22no—
316C2—41no—
317C2—42no—
318C2—42no—
319C2—41no—
320C2—82no—
321C2#2027587911yescharacteristic
322C2#2027589212yes—
subgroupG/H
centerC22#16810535
commutator, cosocleC25×C53C23
FrattiniC25C22×D53
FittingC53×#1781C2
socleC22×C53#393
prime radicalC23×D4×C53C22
hypercenter#1781D53
hypocenterC53#10390
LCSG ▷ C25×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
1. #20282968C2semidirect
3. #20272619C2semidirect
4. #20272619C2semidirect
24. C25×C53Q8semidirect
31. C23×C4×C53Q8semidirect
176. C53#10390semidirect
177. C25Dic106semidirect
185. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202829682
#202746302
#202726192
C53×#17812
#1039053