Finite Group Databasebeta

← #28308263

Subgroups — #28308263

Order 13568.

460 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 78 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
2#20289656C212yesmaximal, complemented
3#20289645C212yesmaximal, complemented
4C4×#13133206C211yesmaximal, characteristic
5#16811594—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133206C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810530C2211yescharacteristic
11#16810524C411yescharacteristic
12#16810513—22no—
13#16810488C2211yescharacteristic
14C4×#11719921—22no—
15C42×Dic53C411yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C2×#11719936—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C22×D212C2×C411yescharacteristic
21C22×C4×D53D412yes—
22#13133206—22no—
23#13133206—22no—
24C4×D212—41no—
25C4×D212—41no—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×#11719921—22no—
28C2×#11719921—22no—
29C2×#11719921D412yes—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C23×D53C22⋊C412yes—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53M1612yescomplemented
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53C2×D412yes—
56C2×D212—21no—
57C2×D212—21no—
58C2×D212—41no—
59C2×D212—41no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—44no—
63C2×#88324—44no—
64#11719936—84no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—41no—
70C42×C53—41no—
71#11719921—28no—
72#11719921—42no—
73#11719921—44no—
74C4⋊Dic53—41no—
75C4×Dic53—21no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53—41no—
79C4×Dic53M1612yescomplemented
80C23×C53C2×C22⋊C411yescharacteristic
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—42no—
85C22×D53—42no—
86Q8×C53—42no—
87Q8×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53#9911yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110D212—42no—
111D212—42no—
112C4×D53—44no—
113C4×D53—82no—
114#88324—84no—
115#11531—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53#40711yescharacteristic
119C22×C53C2×#9911yescharacteristic
120C22×C53#41311yescharacteristic
121C2×D53—44no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—82no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53—42no—
131C4×C53—81no—
132C4×C53#35412yes—
133Dic53—42no—
134Dic53—42no—
135Dic53—42no—
136C4×#124—531no—
137#1465—532no—
138#1324—532no—
139#1319—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53#146511yescharacteristic
143C2×C53C2×#35411yescharacteristic
144C2×C53#213211yescharacteristic
145D53—82no—
146C2×#124—531no—
147C2×C4×Q8#615711yescharacteristic
148C2×C4×D4—531no—
149#407—1062no—
150#391—531no—
151#390—531no—
152#388—1062no—
153#382—531no—
154C4×C22⋊C4—1062no—
155C43—531no—
156#324—534no—
157#323—1062no—
158C53#1153111yescharacteristic, complemented, Sylow
159C22×Q8C2×#615711yescharacteristic
160C22×D4—531no—
161C23×C4—532no—
162C2×M16—2122no—
163#124—1062no—
164#124—1062no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—2121no—
168C4×D4—2121no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4C2×#615711yescharacteristic
172C2×C22⋊C4—532no—
173C2×C22⋊C4—1062no—
174C2×C22⋊C4—1062no—
175C2×C42—21no—
176C2×C42—531no—
177C2×C42—531no—
178C2×C42—1061no—
179C2×C42—1061no—
180C2×C42C2×#615711yescharacteristic
181C22⋊C8—1064no—
182C22⋊C8—2122no—
183#95—1062no—
184#95—1062no—
185C24—532no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1171993612yescomplemented
190C2×D4—1061no—
191C2×D4—1061no—
192C2×D4—2121no—
193C2×D4—2121no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—532no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—2124no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22⋊#615711yescharacteristic
204C22×C4C22×#615711yescharacteristic
205M16—4244no—
206C2×C8—2124no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—2121no—
210C22⋊C4—1068no—
211C22⋊C4—2122no—
212C22⋊C4—2124no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—532no—
218C42—1061no—
219C42—2121no—
220C42—2122no—
221C42—2122no—
222C23—1062no—
223C23—1062no—
224C23—1062no—
225C23—2122no—
226C23—2122no—
227C23C2×C22⋊#615711yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—2122no—
231D4—2122no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—4242no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2124no—
251C2×C4—2122no—
252C2×C4#1313327511yescharacteristic
253C2×C4#1313327511yescharacteristic
254C2×C4D53⋊M1611yescharacteristic
255C2×C4C2×#1171993611yescharacteristic
256C8—4244no—
257C22—21no—
258C22—21no—
259C22—4242no—
260C22—2124no—
261C22—2122no—
262C22—2122no—
263C22—2122no—
264C22#1681159411yescharacteristic
265C22C2×#1313327511yescharacteristic
266C22#1681160911yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—41no—
272C4—81no—
273C4—2122no—
274C4—2122no—
275C4—2122no—
276C4#1681152912yes—
277C2—21no—
278C2—21no—
279C2—4242no—
280C2#2028997011yescharacteristic
281C2C2×#1681152911yescharacteristic
282C2#2029100611yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#11531
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #20289656C2semidirect
3. #20289645C2semidirect
47. C2×Q8×C53M16semidirect
79. C4×Dic53M16semidirect
158. C53#11531semidirect
189. C2×Q8#11719936semidirect

Maximal subgroups

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

subgroupindex
#202899722
#202896562
#202896452
C4×#131332062
#1153153