Finite Group Databasebeta

← #27294791

Subgroups — #27294791

Order 13568.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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#20273049C211yesmaximal, characteristic, complemented
3#20272717C214yesmaximal, complemented
4C53×#1822C211yesmaximal, characteristic, complemented
5C2×#13133214C411yescharacteristic
6C2×#13133191—21no—
7#16810680—21no—
8#16810540—21no—
9#16810540C2212yes—
10#16810532—21no—
11#16810532C412yes—
12#16810525—21no—
13#16810489C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C2×#117×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#343—21no—
19C53×#343C2212yes—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C22×C22⋊D53—22no—
24#13133214—41no—
25#13133214—41no—
26#13133191—44no—
27C2×C4⋊Dic53—21no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C4⋊#88322—41no—
32C22⋊#88322—22no—
33#13133128—41no—
34#13133124—21no—
35#13133124C2×C412yes—
36C23×C4×C53D411yescharacteristic
37C2×M16×C53—22no—
38#117×C53—41no—
39#117×C53—41no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C22⋊C8×C53—41no—
47C22⋊C8×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C23×D53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—41no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C2×C22⋊D53—44no—
61C2×C22⋊D53—82no—
62C2×C22⋊D53—82no—
63C22×Dic53—22no—
64C2×D212—42no—
65C2×D212—42no—
66C2×D212—42no—
67C2×D212—42no—
68C2×D212—41no—
69C2×D212—41no—
70C2×D212—82no—
71C2×D212—82no—
72M16×C53—42no—
73M16×C53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83#11719921—44no—
84#11719921—82no—
85#11719921—82no—
86C4⋊Dic53—42no—
87C2×#88322—42no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—42no—
100C22×D53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53AGL(1,ℤ/8)12yes—
118C22⋊D53—88no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122D212—82no—
123D212—82no—
124D212—84no—
125D212—84no—
126C8×C53—82no—
127#88322—82no—
128#11806—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—81no—
136C22×C53C2×#9911yescharacteristic
137C22×C53#41811yescharacteristic
138C22×C53#42111yescharacteristic
139C2×D53—82no—
140C2×D53—82no—
141C2×D53—82no—
142C2×D53—82no—
143C2×D53—82no—
144C2×D53—82no—
145C2×D53—82no—
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153Dic53—82no—
154#2619—531no—
155#1822D5311yescharacteristic, complemented
156#1464—531no—
157#1345—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—81no—
161C2×C53#151011yescharacteristic
162C2×C53#212211yescharacteristic
163C2×C53#213411yescharacteristic
164D53—162no—
165C2×#127—531no—
166C2×#121—1061no—
167C2×#117C2×D5311yescharacteristic
168#432—1061no—
169#407—21no—
170#394—532no—
171#394—1061no—
172#392—532no—
173#392—1061no—
174#390—1061no—
175#382—531no—
176#343—21no—
177#343C2×D5312yes—
178#324—532no—
179#323—1061no—
180C53#1180611yescharacteristic, complemented, Sylow
181C22×D4—532no—
182C22×D4—1062no—
183C22×D4—1061no—
184C22×D4—1061no—
185C23×C4C22⋊D5311yescharacteristic
186C2×M16—22no—
187#127—2121no—
188#127—2121no—
189#121—2124no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1062no—
198C2×C22⋊C4C4×D5311yescharacteristic
199C2×C22⋊C4D21211yescharacteristic
200C2×C42C4×D5311yescharacteristic
201C2×C42D21211yescharacteristic
202C4⋊C8—2121no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—1062no—
206#97—2121no—
207#95—532no—
208#95—1061no—
209C24—1062no—
210C24#1171992111yescharacteristic
211C2×D4—4242no—
212C2×D4—4242no—
213C2×D4—4242no—
214C2×D4—4242no—
215C2×D4—2122no—
216C2×D4—2122no—
217C2×D4—2122no—
218C2×D4—2122no—
219C2×D4—2124no—
220C2×D4—2121no—
221C2×D4—2121no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—1062no—
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4#1171992111yescharacteristic
230C22×C4C2×C4×D5311yescharacteristic
231C22×C4C2×D21211yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—2122no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—2122no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—4242no—
243C22⋊C4—4242no—
244C22⋊C4—2124no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—2122no—
255C23—2122no—
256C23—2122no—
257C23—2122no—
258C23—2122no—
259C23—2122no—
260C23—2122no—
261C23C2×#1171992111yescharacteristic
262D4—4242no—
263D4—4242no—
264D4—4244no—
265D4—4244no—
266D4—4248no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4#1313313111yescharacteristic
285C2×C4#1313313111yescharacteristic
286C2×C4#1313324212yes—
287C8—82no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—4242no—
297C22—4242no—
298C22—4242no—
299C22—4242no—
300C22—4242no—
301C22—4242no—
302C22—4242no—
303C22C2×#1313313111yescharacteristic
304C22#1681062711yescharacteristic
305C22#1681064111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—4242no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—8482no—
318C2#2027324311yescharacteristic
319C2#2027636611yescharacteristic
320C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1822C2
socleC22×C53C2×#99
prime radical#20280465C2
cosocleC2×C4⋊C4×C53C23
hypercenter#1822D53
hypocenterC53#11806
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #20273049C2semidirect
3. #20272717C2semidirect
4. C53×#1822C2semidirect
155. #1822D53semidirect
180. C53#11806semidirect

Maximal subgroups

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

subgroupindex
#202804652
#202730492
#202727172
C53×#18222
#1180653