Finite Group Databasebeta

← #28299726

Subgroups — #28299726

Order 13568.

468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290643C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289532C212yesmaximal
4C4×#13133206C211yesmaximal, characteristic
5#16811489—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133206C411yescharacteristic
9C2×C4×D212C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810524C2211yescharacteristic
12#16810513—22no—
13#16810488C411yescharacteristic
14C4×#11719921—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C22×#88324—24no—
18C22×D212C2×C411yescharacteristic
19C22×C4×D53—21no—
20#13133206—22no—
21#13133206—22no—
22C4×D212—42no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×#11719921—21no—
25C2×#11719921—22no—
26C2×#11719921—22no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31#13133124—22no—
32#13133124—22no—
33C22×Q8×C53C2×C411yescharacteristic
34C4×Q8×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C42×C53—21no—
38C2×C42×C53C2×C411yescharacteristic
39C23×D53—21no—
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53M1612yes—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—21no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×D212—21no—
53C2×D212—41no—
54C2×D212—41no—
55C2×D212M1612yes—
56C2×C4×D53—44no—
57C2×C4×D53—42no—
58C2×#88324—44no—
59C2×#88324—44no—
60C2×#88324—44no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C42×C53—22no—
64C42×C53—42no—
65#11719921—28no—
66#11719921—42no—
67#11719921—44no—
68C4⋊Dic53—41no—
69C4×Dic53—21no—
70C4×Dic53—21no—
71C4×Dic53—24no—
72C4×Dic53—42no—
73C4×Dic53—41no—
74C23×C53C2×C22⋊C411yescharacteristic
75C22×D53—22no—
76C22×D53—41no—
77C22×D53—41no—
78C22×D53—42no—
79C22×D53—42no—
80Q8×C53—42no—
81Q8×C53—42no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53#10012yes—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53#13514yes—
103D212—42no—
104D212—42no—
105C4×D53—84no—
106#88324—84no—
107#7649—531nomaximal, Sylow
108C22×C53#40711yescharacteristic
109C22×C53#41011yescharacteristic
110C22×C53C2×#10011yescharacteristic
111C22×C53#41612yes—
112C22×C53#42112yes—
113C2×D53—42no—
114C2×D53—44no—
115C2×D53—81no—
116C2×D53—82no—
117C2×D53—81no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—41no—
122C4×C53—42no—
123C4×C53—81no—
124Dic53—24no—
125Dic53—42no—
126Dic53—81no—
127C4×#124—531no—
128#1839—532no—
129#1272—532no—
130#1270—532no—
131C2×C53#146611yescharacteristic
132C2×C53#148812yes—
133C2×C53#152112yes—
134C2×C53#153311yescharacteristic
135C2×C53#154311yescharacteristic
136D53—82no—
137C2×#124—531no—
138C2×C4×Q8#615711yescharacteristic
139C2×C4×D4—531no—
140#391—531no—
141#390—531no—
142#388—1062no—
143#382—531no—
144C4×C22⋊C4—1062no—
145C43—531no—
146#336—534no—
147#336—1062no—
148#336—1062no—
149C53#764911yescharacteristic, complemented, Sylow
150C22×Q8C2×#615711yescharacteristic
151C22×D4—531no—
152C23×C4—1061no—
153C22×C8—1064no—
154#124—1062no—
155#124—1062no—
156C4×Q8—42no—
157C4×D4—2122no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C22⋊C4—1062no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42—21no—
165C2×C42—532no—
166C2×C42—531no—
167C2×C42—532no—
168C2×C42—531no—
169C2×C42C2×#615711yescharacteristic
170#95—1062no—
171#95—1062no—
172C24—1061no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8#1171993612yes—
177C2×D4—532no—
178C2×D4—1061no—
179C2×D4—2121no—
180C2×D4—2121no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—532no—
184C22×C4—532no—
185C22×C4—1061no—
186C22×C4—2124no—
187C22×C4—2122no—
188C22×C4C22⋊#615711yescharacteristic
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C2×C8—2124no—
193C2×C8—2124no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—2121no—
197C22⋊C4—1068no—
198C22⋊C4—2122no—
199C22⋊C4—2124no—
200C42—22no—
201C42—42no—
202C42—1064no—
203C42—1061no—
204C42—1061no—
205C42—2121no—
206C42—2122no—
207C23—1062no—
208C23—2121no—
209C23—2121no—
210C23—2122no—
211C23—2122no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—2122no—
216D4—2122no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—534no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—4244no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4#1313327712yes—
237C2×C4D53⋊M1611yescharacteristic
238C2×C4C2×#1171993611yescharacteristic
239C8—4244no—
240C22—4242no—
241C22—4241no—
242C22—4241no—
243C22—2124no—
244C22—2122no—
245C22#1681159411yescharacteristic
246C22#1681160211yescharacteristic
247C22C2×#1313327711yescharacteristic
248C22#1681161312yes—
249C22#1681162212yes—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—42no—
255C4—81no—
256C4—1064no—
257C4—4241no—
258C4—2122no—
259C2—4242no—
260C2#2028997411yescharacteristic
261C2#2029004912yes—
262C2#2029015912yes—
263C2#2029019611yescharacteristic
264C2#2029021811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7649
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20289535C2semidirect
149. C53#7649semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895352
#202895322
C4×#131332062
#764953