Finite Group Databasebeta

← #28308273

Subgroups — #28308273

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#20289655C212yesmaximal, complemented
3#20289644C212yesmaximal, complemented
4#20277440C211yesmaximal, characteristic
5#16811594—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133214C411yescharacteristic
9C2×C4×D212C411yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810525C411yescharacteristic
12#16810525C2211yescharacteristic
13#16810501—22no—
14#16810501—22no—
15#16810489C2211yescharacteristic
16C53×#376C411yescharacteristic
17C2×#11719936—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C22×D212—22no—
21C22×D212C2×C411yescharacteristic
22C22×D212C2×C411yescharacteristic
23C22×C4×D53—21no—
24C22×C4×D53D412yes—
25#13133214—22no—
26#13133214—22no—
27C4×D212—42no—
28C4×D212—42no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719921—22no—
32C2×#11719921—21no—
33C2×#11719921—22no—
34C2×#11719921D412yes—
35#13133124—22no—
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#95×C53—21no—
40#95×C53—21no—
41C23×D53—21no—
42C23×D53C22⋊C412yes—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—21no—
49C22×Dic53C2×D412yes—
50C2×D212—21no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212—28no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212—82no—
59C2×D212—82no—
60C2×D212M1612yescomplemented
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—42no—
64C2×C4×D53—82no—
65C2×C4×D53—82no—
66C2×#88324—44no—
67#11719936—84no—
68C42×C53—22no—
69C42×C53—21no—
70C42×C53—41no—
71C42×C53M1612yescomplemented
72#11719921—42no—
73#11719921—42no—
74#11719921—82no—
75#11719921—82no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C23×C53C2×C22⋊C411yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—42no—
84C22×D53—42no—
85C22×D53—42no—
86C22×D53—42no—
87C22×D53—41no—
88C22×D53—41no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53#9912yes—
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C2×Dic53—22no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—41no—
107D212—42no—
108D212—42no—
109D212—42no—
110D212—42no—
111D212—88no—
112C4×D53—44no—
113C4×D53—82no—
114C4×D53—84no—
115#88324—84no—
116#11533—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53#40711yescharacteristic
120C22×C53C2×#9911yescharacteristic
121C22×C53#41311yescharacteristic
122C2×D53—44no—
123C2×D53—44no—
124C2×D53—42no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—82no—
129C2×D53—81no—
130C2×D53—81no—
131C2×D53—82no—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—42no—
136C4×C53—41no—
137C4×C53—81no—
138Dic53—42no—
139Dic53—81no—
140#2328—531no—
141#1465—532no—
142#1324—532no—
143#1318—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53#146511yescharacteristic
147C2×C53#212711yescharacteristic
148C2×C53#213411yescharacteristic
149D53—82no—
150D53—82no—
151C2×#127—531no—
152C2×C4×D4—531no—
153C2×C4×D4—531no—
154#407—1062no—
155#390—531no—
156#390—531no—
157#386—1062no—
158#386—1062no—
159#382—531no—
160#376#615711yescharacteristic
161#324—534no—
162#323—1062no—
163C53#1153311yescharacteristic, complemented, Sylow
164C22×D4—531no—
165C22×D4—531no—
166C22×D4—1062no—
167C23×C4—532no—
168C23×C4—1061no—
169C2×M16—2122no—
170#127—1062no—
171#127—1062no—
172C4×D4—2122no—
173C4×D4—2122no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—531no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4—1061no—
180C2×C42C2×#615711yescharacteristic
181C2×C42C2×#615711yescharacteristic
182C2×C42C2×#615711yescharacteristic
183C22⋊C8—1064no—
184C22⋊C8—2122no—
185#95—21no—
186#95—21no—
187#95—1062no—
188C24—532no—
189C24—1061no—
190C2×D4—532no—
191C2×D4—1068no—
192C2×D4—1061no—
193C2×D4—1061no—
194C2×D4—1061no—
195C2×D4—4242no—
196C2×D4—4242no—
197C2×D4—2121no—
198C2×D4—2121no—
199C2×D4—2121no—
200C2×D4—2121no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—532no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4—4242no—
208C22×C4—4242no—
209C22×C4—2122no—
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22⋊#615711yescharacteristic
212C22×C4C22×#615711yescharacteristic
213M16—4244no—
214C2×C8—2124no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C22⋊C4—4242no—
218C22⋊C4—4242no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—22no—
222C42—21no—
223C42—41no—
224C42#1171993612yescomplemented
225C23—1062no—
226C23—1062no—
227C23—1062no—
228C23—1062no—
229C23—2122no—
230C23—2122no—
231C23—2122no—
232C23—2122no—
233C23—2121no—
234C23—2121no—
235C23C2×C22⋊#615711yescharacteristic
236D4—4248no—
237D4—2122no—
238D4—2122no—
239D4—2122no—
240D4—2122no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—1062no—
253C2×C4—4242no—
254C2×C4—4244no—
255C2×C4—2124no—
256C2×C4—2122no—
257C2×C4—2121no—
258C2×C4—2122no—
259C2×C4#1313327612yes—
260C2×C4D53⋊M1611yescharacteristic
261C2×C4C2×#1171993611yescharacteristic
262C8—4244no—
263C22—21no—
264C22—21no—
265C22—4242no—
266C22—4242no—
267C22—4241no—
268C22—4241no—
269C22—2124no—
270C22—2124no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22—2122no—
275C22#1681159411yescharacteristic
276C22C2×#1313327611yescharacteristic
277C22#1681160911yescharacteristic
278C4—22no—
279C4—21no—
280C4—21no—
281C4—42no—
282C4—41no—
283C4—81no—
284C4—4241no—
285C4—2122no—
286C2—21no—
287C2—21no—
288C2—4242no—
289C2—4242no—
290C2#2028997211yescharacteristic
291C2#2029099211yescharacteristic
292C2#2029101611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#376C4
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×C4⋊Dic53C23
hypercenter#376#6157
hypocenterC53#11533
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
1. #20289972C2semidirect
2. #20289655C2semidirect
3. #20289644C2semidirect
60. C2×D212M16semidirect
71. C42×C53M16semidirect
163. C53#11533semidirect
224. C42#11719936semidirect

Maximal subgroups

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

subgroupindex
#202899722
#202896552
#202896442
#202774402
#1153353