Finite Group Databasebeta

← #27277632

Subgroups — #27277632

Order 13568.

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

normal subgroups only

This group has 474 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#20273042C211yesmaximal, characteristic
3#20272622C212yesmaximal, complemented
4#20272620C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810516—22no—
8#16810516—22no—
9#16810483C2211yescharacteristic
10#16810483C2211yescharacteristic
11#16810479C412yes—
12C4×C22⋊Dic53C412yes—
13#16810347—22no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#407C2212yes—
16C53×#323—22no—
17C2×#11719917—22no—
18C2×C4⋊Dic53—21no—
19C2×C4⋊Dic53—21no—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53C2×C412yes—
23C2×C22⋊Dic53C2×C412yes—
24C2×C4×Dic53—22no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—21no—
29#13133124—21no—
30C25×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53C2311yescharacteristic
33C2×M16×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53D412yes—
37C2×C22⋊C4×C53D412yes—
38C22⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40C24×C53—21no—
41C24×C53—21no—
42C24×C53—41no—
43C24×C53—41no—
44C24×C53—81no—
45C24×C53—81no—
46C24×C53C22⋊C412yes—
47C24×C53C22×C411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—81no—
51C22×C4×C53—81no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53#4112yes—
56C22×C4×C53#4112yes—
57C22×Dic53—22no—
58C22×Dic53—22no—
59M16×C53—82no—
60C2×C8×C53—42no—
61C22⋊C4×C53—24no—
62C22⋊C4×C53—24no—
63C22⋊Dic53—42no—
64C22⋊Dic53—42no—
65C22⋊Dic53—82no—
66C22⋊Dic53—82no—
67C4⋊Dic53—81no—
68C4⋊Dic53—81no—
69C4×Dic53—82no—
70#11719917—82no—
71C2×#88322—42no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—41no—
78C23×C53—41no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53—81no—
84C23×C53—81no—
85C23×C53—81no—
86C23×C53—81no—
87C23×C53—84no—
88C23×C53—84no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#11711yescharacteristic
91C23×C53#11711yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—81no—
99C2×C4×C53—81no—
100C2×C4×C53#12312yes—
101C2×C4×C53#12312yes—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C8×C53—82no—
109#88322—82no—
110#8759—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—84no—
127C22×C53—84no—
128C22×C53#38111yescharacteristic
129C22×C53#41311yescharacteristic
130C22×C53#41311yescharacteristic
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134Dic53—82no—
135Dic53—82no—
136#2312—531no—
137#1462—531no—
138#1462D5311yescharacteristic
139#1305—532no—
140#1305—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—41no—
144C2×C53—41no—
145C2×C53—82no—
146C2×C53#182211yescharacteristic
147C2×C53#182811yescharacteristic
148C2×C53#182911yescharacteristic
149C22×C22⋊C4C2×D5311yescharacteristic
150#407—532no—
151#407C2×D5312yes—
152#388—1062no—
153#388—1062no—
154#380—531no—
155#380—531no—
156#379—532no—
157C4×C22⋊C4—532no—
158#323—22no—
159#323—1062no—
160C53#875911yescharacteristic, complemented, Sylow
161C25C4×D5311yescharacteristic
162C23×C4C4×D5311yescharacteristic
163C23×C4C22×D5311yescharacteristic
164C2×M16—22no—
165C2×M16—1062no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4—1061no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—1062no—
173C2×C22⋊C4—1062no—
174C2×C22⋊C4D21212yes—
175C2×C22⋊C4C22⋊D5312yes—
176C2×C42—1062no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—1062no—
180C22⋊C8—1062no—
181#95—1061no—
182#95—1061no—
183#95—1062no—
184C24—21no—
185C24—21no—
186C24—41no—
187C24—41no—
188C24—81no—
189C24—81no—
190C24#1171992112yes—
191C24C2×C4×D5311yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4#1171992112yes—
199C22×C4C2×D21211yescharacteristic
200C22×C4#1171994712yes—
201C22×C4#1171994712yes—
202C22×C4C2×C22⋊D5311yescharacteristic
203M16—82no—
204M16—4242no—
205C2×C8—42no—
206C2×C8—2122no—
207C4⋊C4—4241no—
208C4⋊C4—4241no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—4242no—
212C22⋊C4—4242no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C42—4242no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—84no—
232C23—84no—
233C23C2×#1171992111yescharacteristic
234C23#1313317311yescharacteristic
235C23#1313317311yescharacteristic
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4#1313319812yes—
251C2×C4#1313320112yes—
252C8—82no—
253C8—4242no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—84no—
271C22#1681048411yescharacteristic
272C22#1681060511yescharacteristic
273C22#1681060511yescharacteristic
274C4—22no—
275C4—22no—
276C4—42no—
277C4—4242no—
278C4—4242no—
279C2—21no—
280C2—21no—
281C2—41no—
282C2—41no—
283C2—82no—
284C2#2027479611yescharacteristic
285C2#2027482111yescharacteristic
286C2#2027482411yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1462C2
socleC22×C53#381
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1462D53
hypocenterC53#8759
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20272622C2semidirect
4. #20272620C2semidirect
160. C53#8759semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202730422
#202726222
#202726202
C53×#14622
#875953