Finite Group Databasebeta

← #28308091

Subgroups — #28308091

Order 13568.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#20289952C212yesmaximal
2#20289658C212yesmaximal, complemented
3#20289645C212yesmaximal, complemented
4#20277422C211yesmaximal, characteristic
5D53⋊(C22⋊C8)—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133206C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810530C2211yescharacteristic
11#16810524C411yescharacteristic
12#16810514—22no—
13#16810494—22no—
14#16810492C2211yescharacteristic
15#16810469C411yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C2×D53⋊C8—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C22×D212C2×C411yescharacteristic
21C22×C4×D53D412yes—
22#13133206—22no—
23#13133206C814yescomplemented
24C4×D212—42no—
25C2×C4⋊Dic53C2311yescharacteristic
26C2×#11719919—22no—
27C2×#11719921—22no—
28C2×#11719921—22no—
29C2×#11719921D412yes—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32#13133124—21no—
33#13133124—21no—
34#13133124—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C4×Q8×C53—42no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C23×D53C22⋊C412yes—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53C2×C812yescomplemented
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C2×D412yes—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212C2×C812yes—
57C2×D212M1612yes—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—82no—
61C2×#88324—44no—
62D53⋊C8—84no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C42×C53—22no—
66C42×C53—42no—
67#11719921—28no—
68#11719921—42no—
69#11719921—82no—
70C4⋊Dic53—41no—
71#11719919—82no—
72C4×Dic53—21no—
73C4×Dic53—41no—
74C4×Dic53C2×C812yescomplemented
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—42no—
79C22×D53—42no—
80C22×D53C22⋊C814yes—
81Q8×C53—42no—
82Q8×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53#9912yes—
94C2×C4×C53C22×C811yescharacteristic
95C2×C4×C53C2×M1611yescharacteristic
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104D212—42no—
105D212—42no—
106C4×D53—44no—
107C4×D53—82no—
108#88324—84no—
109#11481—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53C2×C22⋊C811yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53#41311yescharacteristic
115C2×D53—24no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—44no—
119C2×D53—82no—
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—41no—
124C4×C53—42no—
125C4×C53—81no—
126Dic53—42no—
127Dic53—42no—
128Dic53—81no—
129#2326—531no—
130#1459—532no—
131#1325—532no—
132#1319—532no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#145911yescharacteristic
136C2×C53#212711yescharacteristic
137C2×C53#213511yescharacteristic
138D53—44no—
139C2×#124—531no—
140C2×C4×Q8#615711yescharacteristic
141C2×C4×D4—531no—
142C2×C22⋊C8—1062no—
143#391—531no—
144#390—531no—
145#388—1062no—
146#385—1062no—
147#384—531no—
148#376—531no—
149#324—534no—
150#323—1062no—
151C53#1148111yescharacteristic, complemented, Sylow
152C22×Q8C2×#615711yescharacteristic
153C22×D4—531no—
154C23×C4—532no—
155C22×C8—2122no—
156#124—534no—
157#124—1062no—
158C4×Q8—42no—
159C4×D4—2122no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4C2×#615711yescharacteristic
164C2×C22⋊C4—532no—
165C2×C22⋊C4—1062no—
166C2×C22⋊C4—1062no—
167C2×C42—21no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42C2×#615711yescharacteristic
171C22⋊C8—1064no—
172C22⋊C8—2122no—
173#95—1062no—
174#95—1061no—
175#95—1061no—
176C24—532no—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8D53⋊C812yescomplemented
181C2×D4—532no—
182C2×D4—532no—
183C2×D4—2121no—
184C2×D4—2121no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—532no—
188C22×C4—1061no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4—1061no—
192C22×C4—4242no—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22⋊#615711yescharacteristic
195C22×C4C22×#615711yescharacteristic
196C2×C8—4244no—
197C2×C8—2124no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—4242no—
201C4⋊C4—2121no—
202C22⋊C4—1068no—
203C22⋊C4—4242no—
204C22⋊C4—2122no—
205C42—22no—
206C42—42no—
207C42—532no—
208C42—1061no—
209C42—2121no—
210C23—534no—
211C23—1062no—
212C23—1062no—
213C23—2122no—
214C23—2122no—
215C23C2×C22⋊#615711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—2122no—
219D4—2122no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—4242no—
233C2×C4—2124no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4#1313327512yes—
241C2×C4C2×D53⋊C811yescharacteristic
242C2×C4D53⋊M1611yescharacteristic
243C8—4244no—
244C22—21no—
245C22—21no—
246C22—1064no—
247C22—4242no—
248C22—2124no—
249C22—2122no—
250C22—2122no—
251C22D53⋊(C22⋊C8)11yescharacteristic
252C22C2×#1313327511yescharacteristic
253C22#1681160911yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—4241no—
261C4—2122no—
262C4—2122no—
263C2—21no—
264C2—21no—
265C2—2124no—
266C2#2028995011yescharacteristic
267C2#2029099011yescharacteristic
268C2#2029101711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#11481
LCSG ▷ C2×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. #20289658C2semidirect
3. #20289645C2semidirect
23. #13133206C8semidirect
45. C2×Q8×C53C2×C8semidirect
74. C4×Dic53C2×C8semidirect
151. C53#11481semidirect
180. C2×Q8D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202899522
#202896582
#202896452
#202774222
#1148153