Finite Group Databasebeta

← #28308271

Subgroups — #28308271

Order 13568.

504 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289972C212yesmaximal, complemented
2#20289656C212yesmaximal, complemented
3#20289643C212yesmaximal, complemented
4#20277437C211yesmaximal, characteristic
5#16811594—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133213C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810526C2211yescharacteristic
11#16810524C411yescharacteristic
12#16810502—22no—
13#16810499—22no—
14#16810488C2211yescharacteristic
15#16810469C411yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17C2×#11719936—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C22×D212C2×C411yescharacteristic
21C22×C22⋊D53—22no—
22C22×C4×D53D412yes—
23#13133213—22no—
24#13133213—22no—
25C4×D212—42no—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53—22no—
28C2×#11719921—22no—
29C2×#11719921D412yes—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32#13133124—21no—
33#13133124—21no—
34#13133124—22no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53—21no—
37C4×D4×C53—42no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53—21no—
42C23×D53C22⋊C412yes—
43C2×D4×C53—21no—
44C2×D4×C53—41no—
45C2×D4×C53—41no—
46C2×D4×C53M1612yescomplemented
47C22×C4×C53—21no—
48C22×C4×C53—42no—
49C22×C4×C53—82no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—28no—
54C2×C22⋊D53—82no—
55C2×C22⋊D53—82no—
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C2×D412yes—
59C2×D212—21no—
60C2×D212—21no—
61C2×D212—41no—
62C2×D212—41no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—82no—
66C2×#88324—44no—
67#11719936—84no—
68C4⋊C4×C53—41no—
69C22⋊C4×C53—42no—
70C42×C53—22no—
71C22⋊Dic53—82no—
72#11719921—42no—
73#11719921—82no—
74C4⋊Dic53—41no—
75C4×Dic53—21no—
76C4×Dic53—41no—
77C4×Dic53M1612yescomplemented
78C23×C53—22no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—42no—
88C22×D53—42no—
89D4×C53—42no—
90D4×C53—42no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—84no—
99C2×C4×C53#9912yes—
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C22⋊D53—88no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—42no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111D212—42no—
112D212—42no—
113C4×D53—44no—
114C4×D53—82no—
115#88324—84no—
116#11533—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—44no—
120C22×C53—42no—
121C22×C53—82no—
122C22×C53—81no—
123C22×C53—81no—
124C22×C53#40711yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53#41311yescharacteristic
127C2×D53—44no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—82no—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—41no—
136C4×C53—81no—
137Dic53—42no—
138Dic53—42no—
139Dic53—81no—
140#2328—531no—
141#1465—532no—
142#1324—532no—
143#1318—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—82no—
147C2×C53#146511yescharacteristic
148C2×C53#212711yescharacteristic
149C2×C53#213411yescharacteristic
150D53—82no—
151C2×#127—531no—
152C2×C4×D4—531no—
153C2×C4×D4#615711yescharacteristic
154#407—1062no—
155#390—531no—
156#390—531no—
157#386—1062no—
158#386—1062no—
159#382—531no—
160#376—531no—
161#324—534no—
162#323—1062no—
163C53#1153311yescharacteristic, complemented, Sylow
164C22×D4—531no—
165C22×D4—1062no—
166C22×D4C2×#615711yescharacteristic
167C23×C4—21no—
168C23×C4—532no—
169C2×M16—2122no—
170#127—1062no—
171#127—1062no—
172C4×D4—42no—
173C4×D4—2122no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4C2×#615711yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4—1062no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42C2×#615711yescharacteristic
183C22⋊C8—1064no—
184C22⋊C8—2122no—
185#95—1062no—
186#95—1061no—
187#95—1061no—
188C24—21no—
189C24—532no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—1068no—
194C2×D4—1061no—
195C2×D4—1061no—
196C2×D4—4242no—
197C2×D4—4242no—
198C2×D4—2121no—
199C2×D4—2121no—
200C2×D4#1171993612yescomplemented
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—532no—
205C22×C4—1061no—
206C22×C4—1062no—
207C22×C4—1062no—
208C22×C4—1061no—
209C22×C4—4242no—
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22⋊#615711yescharacteristic
212C22×C4C22×#615711yescharacteristic
213M16—4244no—
214C2×C8—2124no—
215C4⋊C4—41no—
216C4⋊C4—2121no—
217C22⋊C4—42no—
218C22⋊C4—4242no—
219C22⋊C4—4242no—
220C22⋊C4—2122no—
221C42—22no—
222C42—532no—
223C42—1061no—
224C42—2121no—
225C23—22no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—2122no—
234C23—2122no—
235C23C2×C22⋊#615711yescharacteristic
236D4—42no—
237D4—42no—
238D4—4248no—
239D4—2122no—
240D4—2122no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—84no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—4242no—
252C2×C4—2124no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2122no—
259C2×C4#1313327512yes—
260C2×C4D53⋊M1611yescharacteristic
261C2×C4C2×#1171993611yescharacteristic
262C8—4244no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22—42no—
267C22—82no—
268C22—81no—
269C22—81no—
270C22—4242no—
271C22—2124no—
272C22—2122no—
273C22—2122no—
274C22—2122no—
275C22#1681159411yescharacteristic
276C22C2×#1313327511yescharacteristic
277C22#1681160911yescharacteristic
278C4—22no—
279C4—21no—
280C4—21no—
281C4—41no—
282C4—81no—
283C4—4241no—
284C4—2122no—
285C4—2122no—
286C2—21no—
287C2—21no—
288C2—82no—
289C2—4242no—
290C2#2028997011yescharacteristic
291C2#2029099011yescharacteristic
292C2#2029101411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133213C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#11533
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20289972C2semidirect
2. #20289656C2semidirect
3. #20289643C2semidirect
46. C2×D4×C53M16semidirect
77. C4×Dic53M16semidirect
163. C53#11533semidirect
200. C2×D4#11719936semidirect

Maximal subgroups

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

subgroupindex
#202899722
#202896562
#202896432
#202774372
#1153353