Finite Group Databasebeta

← #27280130

Subgroups — #27280130

Order 13568.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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#20280465C211yesmaximal, characteristic
2#20273384C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic, complemented
4#20272575C212yesmaximal, complemented
5C53×#2921C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133191—21no—
8#16810676C2212yes—
9#16810540—21no—
10#16810540C412yes—
11#16810532—21no—
12#16810532C412yes—
13#16810525—21no—
14#16810489C2211yescharacteristic
15#16810366—24no—
16#16810349—21no—
17#16810347—21no—
18C2×#117×C53C2211yescharacteristic
19C53×#433—24no—
20C53×#423C2212yes—
21C53×#407—21no—
22C4×M16×C53—21no—
23C22×D212—21no—
24C22×D212—21no—
25C22×D212C2×C412yes—
26C22×C22⋊D53—22no—
27#13133214—22no—
28#13133214—22no—
29#13133191—44no—
30C2×C4⋊Dic53—21no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C4⋊#88322—22no—
35C22⋊#88322—22no—
36C4×#88322—22no—
37#13133124—21no—
38#13133124C2×C412yes—
39C23×C4×C53D411yescharacteristic
40C2×M16×C53—22no—
41#117×C53—22no—
42#117×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C4⋊C8×C53D414yescomplemented
51C22⋊C8×C53—24no—
52#97×C53—22no—
53C4×C8×C53—22no—
54C24×C53C22⋊C411yescharacteristic
55C23×D53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C2×C22⋊D53—44no—
67C2×C22⋊D53—82no—
68C2×C22⋊D53—82no—
69C22×Dic53—22no—
70C2×D212—24no—
71C2×D212—41no—
72C2×D212—42no—
73C2×D212—41no—
74C2×D212—42no—
75C2×D212—42no—
76C2×D212—82no—
77C2×D212—82no—
78M16×C53—42no—
79M16×C53—42no—
80C2×C8×C53—24no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53C2×D412yes—
88C42×C53#4112yes—
89#11719921—44no—
90#11719921—82no—
91#11719921—82no—
92C4⋊Dic53—42no—
93C2×#88322—42no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C22×D53—42no—
101C22×D53—42no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—42no—
105C22×D53—42no—
106C22×D53—42no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12011yescharacteristic
120C2×C4×C53#12112yes—
121C2×C4×C53#12311yescharacteristic
122C2×C4×C53#13512yes—
123C2×C4×C53AGL(1,ℤ/8)12yes—
124C22⋊D53—88no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128D212—84no—
129D212—84no—
130D212—82no—
131D212—82no—
132C8×C53—44no—
133#88322—82no—
134#9198—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53#38611yescharacteristic
143C22×C53#41611yescharacteristic
144C22×C53#41811yescharacteristic
145C2×D53—82no—
146C2×D53—82no—
147C2×D53—82no—
148C2×D53—82no—
149C2×D53—82no—
150C2×D53—82no—
151C2×D53—82no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53#44912yes—
158C4×C53#48212yes—
159Dic53—82no—
160#2921D5311yescharacteristic, complemented
161#2619—531no—
162#1545—532no—
163#1458—531no—
164#1286—532no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—81no—
168C2×C53#187511yescharacteristic
169C2×C53#191011yescharacteristic
170C2×C53#191411yescharacteristic
171D53—162no—
172C2×#127—531no—
173C2×#121—1061no—
174C2×#117C2×D5311yescharacteristic
175#433—24no—
176#431—532no—
177#423C2×D5312yes—
178#407—21no—
179C4×M16—21no—
180#394—532no—
181#394—1061no—
182#392—532no—
183#392—1061no—
184#390—1061no—
185#382—531no—
186#331—1064no—
187#324—1061no—
188#323—1061no—
189C53#919811yescharacteristic, complemented, Sylow
190C22×D4—532no—
191C22×D4—1062no—
192C22×D4—1061no—
193C22×D4—1061no—
194C23×C4C22⋊D5311yescharacteristic
195C2×M16—22no—
196#127—1062no—
197#127—1062no—
198#121—2124no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4C4×D5311yescharacteristic
203C2×C4⋊C4D21211yescharacteristic
204C2×C22⋊C4—1062no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4C4×D5311yescharacteristic
208C2×C22⋊C4D21211yescharacteristic
209C2×C42C22⋊D5311yescharacteristic
210C2×C42C22×D5311yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—1062no—
213C4⋊C8C22⋊D5314yescomplemented
214C22⋊C8—24no—
215C22⋊C8—1062no—
216#97—22no—
217C4×C8—22no—
218C4×C8—1062no—
219#95—532no—
220#95—1061no—
221C24—1062no—
222C24#1171992111yescharacteristic
223C2×D4—1064no—
224C2×D4—4242no—
225C2×D4—4242no—
226C2×D4—4242no—
227C2×D4—4242no—
228C2×D4—2124no—
229C2×D4—2121no—
230C2×D4—2122no—
231C2×D4—2121no—
232C2×D4—2122no—
233C2×D4—2122no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—1062no—
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4C2×D21211yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—24no—
248C2×C8—2122no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—2122no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—4242no—
255C22⋊C4—4242no—
256C22⋊C4—2124no—
257C42—21no—
258C42—21no—
259C42#1171994512yes—
260C42C2×C22⋊D5312yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—2122no—
267C23—2122no—
268C23—2122no—
269C23—2122no—
270C23—2122no—
271C23—2122no—
272C23—2122no—
273C23C2×#1171992111yescharacteristic
274D4—4244no—
275D4—4244no—
276D4—4248no—
277D4—4242no—
278D4—4242no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4C4×C22⋊D5312yes—
294C2×C4#1313318611yescharacteristic
295C2×C4#1313319112yes—
296C2×C4#1313320011yescharacteristic
297C2×C4#1313323712yes—
298C2×C4#1313324212yes—
299C8—44no—
300C8—4242no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—4242no—
309C22—4242no—
310C22—4242no—
311C22—4242no—
312C22—4242no—
313C22—4242no—
314C22—4242no—
315C22#1681050211yescharacteristic
316C22#1681061711yescharacteristic
317C22#1681062711yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—4242no—
324C4#1681076812yes—
325C4#1681094712yes—
326C2—21no—
327C2—21no—
328C2—81no—
329C2—8482no—
330C2#2027503711yescharacteristic
331C2#2027524211yescharacteristic
332C2#2027526211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2921C2
socleC22×C53#386
prime radical#20280465C2
cosocleC2×C42×C53C23
hypercenter#2921D53
hypocenterC53#9198
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20273384C2semidirect
3. #20273029C2semidirect
4. #20272575C2semidirect
5. C53×#2921C2semidirect
50. C4⋊C8×C53D4semidirect
160. #2921D53semidirect
189. C53#9198semidirect
213. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202804652
#202733842
#202730292
#202725752
C53×#29212
#919853