Finite Group Databasebeta

← #28307925

Subgroups — #28307925

Order 13568.

686 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289946C212yesmaximal, complemented
2#20289644C214yesmaximal, complemented
3#20277295C211yesmaximal, characteristic
4#16811639—22no—
5#16811468C2214yescomplemented
6#16811466—22no—
7C23×D212C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810525C412yes—
10#16810525C2212yes—
11#16810524—22no—
12#16810478—22no—
13C2×#117×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17C24×D53—21no—
18C22×D212—21no—
19C22×D212—22no—
20C22×D212—42no—
21C22×D212—42no—
22C22×D212C814yescomplemented
23C22×D212C2×C412yes—
24#13133176—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53D412yes—
28C2×#11719921—24no—
29C2×#11719921—24no—
30C2×C4×Dic53D412yes—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C23×D53—22no—
38C23×D53—24no—
39C23×D53—42no—
40C23×D53—42no—
41C23×D53—82no—
42C23×D53—82no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C2×C812yes—
48C22×C4×C53M1612yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×D212—24no—
53C2×D212—22no—
54C2×D212—48no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—42no—
58C2×D212—42no—
59C2×D212—48no—
60C2×D212C2×C814yes—
61C2×#88324—44no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C42×C53—42no—
65C22⋊Dic53—42no—
66#11719921—84no—
67#11719921—84no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53C22⋊C812yes—
77C23×C53C2×C22⋊C411yescharacteristic
78C22×D53—24no—
79C22×D53—48no—
80C22×D53—44no—
81C22×D53—44no—
82C22×D53—42no—
83C22×D53—42no—
84C22×D53—88no—
85C22×D53—88no—
86C22×D53—84no—
87C22×D53—84no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—24no—
91C2×C4×C53—24no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53C22⋊C812yes—
96C2×C4×C53#9912yes—
97C2×C4×C53#10012yes—
98C2×C4×C53C22×C811yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104D212—44no—
105D212—44no—
106D212—48no—
107#88324—84no—
108#11433—531nomaximal, Sylow
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—22no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53C2×C22⋊C811yescharacteristic
117C22×C53C2×#9911yescharacteristic
118C22×C53C2×#10011yescharacteristic
119C2×D53—48no—
120C2×D53—44no—
121C2×D53—84no—
122C2×D53—816no—
123C2×D53—82no—
124C2×D53—82no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—82no—
128C4×C53#32314yes—
129Dic53—42no—
130Dic53—42no—
131#2309—531no—
132#1458—532no—
133#1318—534no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53—42no—
137C2×C53C2×#32311yescharacteristic
138C2×C53#212811yescharacteristic
139C2×C53#213411yescharacteristic
140D53—84no—
141C23×D4—531no—
142C2×#117—531no—
143C2×#117#615711yescharacteristic
144#431—1062no—
145#390—532no—
146#390—532no—
147#390—1062no—
148#379—1062no—
149#324—534no—
150#323—1062no—
151C53#1143311yescharacteristic, complemented, Sylow
152C25—1061no—
153C22×D4—532no—
154C22×D4—534no—
155C22×D4—1062no—
156C22×D4—1061no—
157C22×D4—2122no—
158C22×D4—2122no—
159C23×C4C2×#615711yescharacteristic
160#117—42no—
161#117—2122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—1064no—
168C2×C22⋊C4—1064no—
169C2×C42—532no—
170C2×C42C2×#615712yes—
171C4⋊C8—2122no—
172C22⋊C8—1064no—
173C4×C8—2122no—
174C24—1064no—
175C24—1062no—
176C24—4242no—
177C24—4242no—
178C24—2122no—
179C24—2122no—
180C24C22⋊#615711yescharacteristic
181C2×D4—534no—
182C2×D4—1064no—
183C2×D4—1062no—
184C2×D4—2122no—
185C2×D4—2122no—
186C2×D4—2128no—
187C2×D4—2128no—
188C2×D4—2122no—
189C2×D4—2122no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—532no—
194C22×C4—532no—
195C22×C4D53⋊C812yes—
196C22×C4#1171993612yes—
197C22×C4C22⋊#615711yescharacteristic
198C22×C4C22×#615711yescharacteristic
199C2×C8—2124no—
200C4⋊C4—42no—
201C4⋊C4—2121no—
202C4⋊C4—2121no—
203C22⋊C4—42no—
204C22⋊C4—4244no—
205C22⋊C4—4244no—
206C22⋊C4—2122no—
207C42—42no—
208C42—1062no—
209C42—1062no—
210C23—21no—
211C23—21no—
212C23—41no—
213C23—41no—
214C23—1064no—
215C23—4248no—
216C23—4244no—
217C23—4244no—
218C23—4248no—
219C23—2124no—
220C23—2124no—
221C23—2128no—
222C23—2122no—
223C23—2122no—
224C23#1313327412yes—
225C23C2×C22⋊#615711yescharacteristic
226D4—2124no—
227D4—2124no—
228D4—2128no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4#1313327412yes—
241C2×C4#1313327612yes—
242C2×C4#1313327712yes—
243C2×C4C2×D53⋊C811yescharacteristic
244C2×C4C2×#1171993611yescharacteristic
245C8—4244no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—41no—
252C22—41no—
253C22—4244no—
254C22—42416no—
255C22—4242no—
256C22—4242no—
257C22—2128no—
258C22—2124no—
259C22C2×#1313327411yescharacteristic
260C22C2×#1313327611yescharacteristic
261C22C2×#1313327711yescharacteristic
262C4—21no—
263C4—21no—
264C4—82no—
265C4—2122no—
266C4—2122no—
267C4#1681146714yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—4244no—
272C2C2×#1681146711yescharacteristic
273C2#2029099611yescharacteristic
274C2#2029101611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC23×D212C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11433
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20289946C2semidirect
2. #20289644C2semidirect
5. #16811468C22semidirect
22. C22×D212C8semidirect
151. C53#11433semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896442
#202772952
#1143353