Finite Group Databasebeta

← #27294778

Subgroups — #27294778

Order 13568.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#20274795C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272721C214yesmaximal, complemented
4C2×C53×#398C211yesmaximal, characteristic
5#16810587—21no—
6#16810587—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C22×C4⋊C4×C53—21no—
12C22×C4⋊C4×C53C411yescharacteristic
13C22×C4⋊C4×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398—21no—
16C53×#398C412yes—
17C53×#398C2212yes—
18C2×#95×C53—21no—
19C2×#95×C53C411yescharacteristic
20C2×#11719917—22no—
21C2×#11719917—41no—
22C22⋊#88322—22no—
23C22⋊#88322—41no—
24C22⋊#88322—41no—
25C22⋊#88322—41no—
26C23×C4×C53—21no—
27C23×C4×C53—21no—
28C23×C4×C53C2×C411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—41no—
35C2×C4⋊C4×C53—41no—
36C2×C4⋊C4×C53—44no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44#95×C53—22no—
45#95×C53—21no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—24no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—41no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—44no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—44no—
82#11719917—42no—
83#11719917—42no—
84#11719917—82no—
85C2×#88322—42no—
86C2×#88322—42no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53#9911yescharacteristic
92C23×C53#9911yescharacteristic
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—24no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—48no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—81no—
120C2×C4×C53—82no—
121C2×C4×C53—82no—
122C2×C4×C53AGL(1,ℤ/8)12yes—
123#88322—82no—
124#88322—82no—
125#11803—531nomaximal, Sylow
126C22×C53—24no—
127C22×C53—24no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53#35212yes—
135C22×C53#35612yes—
136C22×C53C2×#9911yescharacteristic
137C22×C53#41811yescharacteristic
138C22×C53#42111yescharacteristic
139C4×C53—22no—
140C4×C53—44no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—82no—
147C2×#398D5311yescharacteristic
148#1821—531no—
149#1463—531no—
150#1346—534no—
151C2×C53—21no—
152C2×C53—22no—
153C2×C53—22no—
154C2×C53—21no—
155C2×C53#151011yescharacteristic
156C2×C53C2×#35211yescharacteristic
157C2×C53C2×#35611yescharacteristic
158C22×C4⋊C4—21no—
159C22×C4⋊C4Dic5311yescharacteristic
160C22×C4⋊C4C2×D5311yescharacteristic
161#407—1061no—
162#407—1061no—
163#398—21no—
164#398—21no—
165#398Dic5312yes—
166#398C2×D5312yes—
167C2×#95—21no—
168C2×#95Dic5311yescharacteristic
169#343—532no—
170#343—1061no—
171#324—532no—
172#324—1061no—
173C53#1180311yescharacteristic, complemented, Sylow
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C2×Dic5311yescharacteristic
177C23×C4C22⋊D5311yescharacteristic
178C23×C4C22⋊D5311yescharacteristic
179C2×M16—1062no—
180C2×M16—2121no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×Dic5312yes—
190C2×C4⋊C4C2×Dic5311yescharacteristic
191C2×C4⋊C4C22⋊D5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22×D5311yescharacteristic
194C22⋊C8—1062no—
195C22⋊C8—2121no—
196C22⋊C8—2121no—
197C22⋊C8—2121no—
198#95—22no—
199#95—22no—
200#95—21no—
201#95C2×Dic5312yes—
202C24C22⋊Dic5311yescharacteristic
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22×Dic5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—4242no—
229M16—2122no—
230M16—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23#1313313211yescharacteristic
246C23#1313313211yescharacteristic
247C23C2×C22⋊Dic5311yescharacteristic
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#1313324312yes—
277C8—4242no—
278C8—4242no—
279C22—24no—
280C22—24no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22#1681041312yes—
288C22#1681042212yes—
289C22C2×#1313313211yescharacteristic
290C22#1681062811yescharacteristic
291C22#1681064011yescharacteristic
292C4—22no—
293C4—44no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—22no—
304C2#2027324411yescharacteristic
305C2C2×#1681041311yescharacteristic
306C2C2×#1681042211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#398C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#398D53
hypocenterC53#11803
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
173. C53#11803semidirect

Maximal subgroups

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

subgroupindex
#202747952
#202730452
#202727212
C2×C53×#3982
#1180353