Finite Group Databasebeta

← #27277601

Subgroups — #27277601

Order 13568.

538 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 91 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#20277313C211yesmaximal, characteristic
2#20274789C211yesmaximal, characteristic
3#20272619C212yesmaximal
4#20272619C212yesmaximal
5C53×#1819C211yesmaximal, characteristic
6#16810516—22no—
7#16810516—22no—
8#16810483C412yescomplemented
9#16810479C2212yes—
10#16810479C2211yescharacteristic
11#16810479C2211yescharacteristic
12C4×C22⋊Dic53C412yescomplemented
13#16810390—22no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#379C2212yes—
16C53×#342—22no—
17C2×C4⋊Dic53—22no—
18C2×C22⋊Dic53—21no—
19C2×C22⋊Dic53—21no—
20C2×C22⋊Dic53—21no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53C2×C412yes—
26C2×C22⋊Dic53C2×C412yes—
27C2×C4×Dic53—22no—
28#13133124—22no—
29#13133124—22no—
30C25×C53C2311yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53D412yes—
39C2×C22⋊C4×C53D412yes—
40C24×C53—42no—
41C24×C53—81no—
42C24×C53—81no—
43C24×C53C22×C411yescharacteristic
44C24×C53C2×D411yescharacteristic
45C24×C53C2×D411yescharacteristic
46C24×C53#4112yes—
47C24×C53#4112yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—42no—
53C22×C4×C53—81no—
54C22×C4×C53—81no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22⋊C412yes—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53—42no—
60C22⋊C4×C53—24no—
61C22⋊C4×C53—24no—
62C22⋊C4×C53—44no—
63C22⋊C4×C53—82no—
64C22⋊C4×C53—82no—
65C22⋊Dic53—42no—
66C22⋊Dic53—42no—
67C22⋊Dic53—44no—
68C22⋊Dic53—81no—
69C22⋊Dic53—81no—
70C22⋊Dic53—81no—
71C22⋊Dic53—81no—
72C22⋊Dic53—82no—
73C22⋊Dic53—82no—
74C4⋊Dic53—82no—
75C4×Dic53—82no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—41no—
80C23×C53—41no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—88no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53#11711yescharacteristic
94C23×C53#11711yescharacteristic
95C23×C53#12312yes—
96C23×C53#12312yes—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—81no—
102C2×C4×C53—81no—
103C2×C4×C53—81no—
104C2×C4×C53—81no—
105C2×C4×C53—82no—
106C2×C4×C53—82no—
107C2×C4×C53—82no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—82no—
115C2×Dic53—82no—
116C2×Dic53—82no—
117#8750—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—22no—
121C22×C53—22no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—44no—
133C22×C53—82no—
134C22×C53—82no—
135C22×C53—84no—
136C22×C53—84no—
137C22×C53#38111yescharacteristic
138C22×C53#41012yes—
139C4×C53—42no—
140C4×C53—42no—
141C4×C53—162no—
142Dic53—82no—
143Dic53—82no—
144Dic53—162no—
145#2312—531no—
146#1819—531no—
147#1819D5311yescharacteristic
148#1304—532no—
149#1304—532no—
150C2×C53—22no—
151C2×C53—42no—
152C2×C53—42no—
153C2×C53—41no—
154C2×C53—41no—
155C2×C53#182011yescharacteristic
156C2×C53#182811yescharacteristic
157C2×C53#182811yescharacteristic
158C22×C22⋊C4C2×D5311yescharacteristic
159#388—1062no—
160#388—1062no—
161#380—532no—
162#379—531no—
163#379—531no—
164#379—532no—
165#379C2×D5312yes—
166C4×C22⋊C4—532no—
167#342—22no—
168#342—1062no—
169C53#875011yescharacteristic, complemented, Sylow
170C25C22×D5311yescharacteristic
171C23×C4C4×D5311yescharacteristic
172C23×C4C4×D5311yescharacteristic
173C2×C4⋊C4—1062no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—1061no—
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1061no—
184C2×C22⋊C4—1061no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1062no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4D21212yes—
189C2×C22⋊C4C22⋊D5312yes—
190C2×C42—1062no—
191#95—1062no—
192#95—1062no—
193C24—42no—
194C24—81no—
195C24—81no—
196C24C2×C4×D5311yescharacteristic
197C24C2×D21211yescharacteristic
198C24#1171994712yes—
199C24#1171994712yes—
200C24C2×C22⋊D5311yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—1062no—
209C22×C4—1062no—
210C22×C4—2122no—
211C22×C4#1171992112yes—
212C22×C4#1171992112yes—
213C4⋊C4—4242no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—4242no—
220C22⋊C4—4242no—
221C22⋊C4—4241no—
222C22⋊C4—4241no—
223C22⋊C4—4241no—
224C22⋊C4—4241no—
225C22⋊C4—2124no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C42—4242no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—88no—
245C23C2×#1171992111yescharacteristic
246C23#1313317311yescharacteristic
247C23#1313317311yescharacteristic
248C23#1313319812yes—
249C23#1313320112yes—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—4242no—
262C2×C4—4242no—
263C2×C4—4242no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4—2122no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—44no—
285C22—82no—
286C22—82no—
287C22—84no—
288C22—84no—
289C22#1681048411yescharacteristic
290C22#1681059612yes—
291C4—42no—
292C4—42no—
293C4—162no—
294C4—8482no—
295C4—4242no—
296C4—4242no—
297C2—22no—
298C2—42no—
299C2—42no—
300C2—41no—
301C2—41no—
302C2#2027479011yescharacteristic
303C2#2027482111yescharacteristic
304C2#2027482111yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#1819C2
socleC22×C53#381
prime radical, cosocleC25×C53C23
hypercenter#1819D53
hypocenterC53#8750
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
8. #16810483C4semidirect
12. C4×C22⋊Dic53C4semidirect
169. C53#8750semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202747892
#202726192
C53×#18192
#875053