Finite Group Databasebeta

← #28299644

Subgroups — #28299644

Order 13568.

874 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 116 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#20290441C212yesmaximal, complemented
2#20289531C214yesmaximal, complemented
3C2×#16810526C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272—22no—
6#16811489C2214yescomplemented
7C22×C4⋊Dic53C411yescharacteristic
8C22×C22⋊Dic53—22no—
9C22×C4×Dic53C2211yescharacteristic
10#16810526—22no—
11#16810526C412yes—
12#16810526C2212yes—
13C23×D4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—44no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53—24no—
24C2×C22⋊Dic53—48no—
25C2×C4×Dic53—42no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29C25×C53—21no—
30C22×D4×C53—24no—
31C22×D4×C53—21no—
32C22×D4×C53—42no—
33C22×D4×C53—42no—
34C22×D4×C53C2×C412yes—
35C23×C4×C53C2×C411yescharacteristic
36C24×C53—24no—
37C24×C53—22no—
38C24×C53—42no—
39C24×C53—44no—
40C24×C53—42no—
41C24×C53—44no—
42C24×C53C22⋊C411yescharacteristic
43C2×D4×C53—24no—
44C2×D4×C53—22no—
45C2×D4×C53—416no—
46C2×D4×C53—42no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—42no—
50C2×D4×C53M1614yes—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—24no—
57C22×Dic53—24no—
58C22×Dic53—22no—
59C22×Dic53—42no—
60C22×Dic53—44no—
61C22×Dic53—42no—
62C22×Dic53C22⋊C412yes—
63C22×Dic53C2×D412yes—
64C2×#88324—28no—
65C2×#88324—44no—
66C2×#88324—44no—
67C22⋊Dic53—48no—
68C22⋊Dic53—48no—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71C4×Dic53—24no—
72C4×Dic53—41no—
73C4×Dic53—41no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—22no—
77C23×C53—24no—
78C23×C53—48no—
79C23×C53—416no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—44no—
83C23×C53—44no—
84C23×C53—84no—
85C23×C53—88no—
86C23×C53—84no—
87C23×C53#10012yes—
88C23×C53#10414yes—
89C23×C53C2×C22⋊C411yescharacteristic
90D4×C53—44no—
91D4×C53—48no—
92D4×C53—44no—
93C2×C4×C53—28no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53#9912yes—
99C2×C4×C53C2×M1612yes—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—48no—
106C2×Dic53—42no—
107C2×Dic53—44no—
108C2×Dic53—84no—
109C2×Dic53AGL(1,ℤ/8)14yes—
110#88324—48no—
111#7626—531nomaximal, Sylow
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—24no—
115C22×C53—24no—
116C22×C53—44no—
117C22×C53—48no—
118C22×C53—816no—
119C22×C53—82no—
120C22×C53—84no—
121C22×C53—82no—
122C22×C53#32914yes—
123C22×C53#40711yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53C2×#10011yescharacteristic
126C22×C53#41812yes—
127C22×C53C2×#10412yes—
128C4×C53—22no—
129C4×C53—22no—
130Dic53—24no—
131Dic53—42no—
132Dic53—82no—
133C2×#390—531no—
134#1749—532no—
135#1270—534no—
136C2×C53—24no—
137C2×C53—84no—
138C2×C53#146211yescharacteristic
139C2×C53#149012yes—
140C2×C53#150712yes—
141C2×C53C2×#32911yescharacteristic
142C2×C53#155511yescharacteristic
143C23×D4#615711yescharacteristic
144C22×C4⋊C4—531no—
145C22×C22⋊C4—1062no—
146C22×C42—531no—
147C2×C22⋊C8—1062no—
148C2×#97—1062no—
149#390—532no—
150#390—532no—
151#390—1062no—
152#336—534no—
153C53#762611yescharacteristic, complemented, Sylow
154C25—21no—
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4C2×#615712yes—
160C23×C4—532no—
161C23×C4—1061no—
162C23×C4C2×#615711yescharacteristic
163C22×C8—1064no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4—2122no—
167C2×C22⋊C4—1064no—
168C2×C22⋊C4—1064no—
169C2×C22⋊C4—2128no—
170C2×C42—532no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—2122no—
174C22⋊C8—1068no—
175#97—2124no—
176C24—24no—
177C24—22no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24—44no—
182C24C22⋊#615711yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—416no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4#1171994014yes—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—532no—
195C22×C4—532no—
196C22×C4—1064no—
197C22×C4—1064no—
198C22×C4—1062no—
199C22×C4—2122no—
200C22×C4—2124no—
201C22×C4—2122no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22×#615711yescharacteristic
204C2×C8—1068no—
205C2×C8—2124no—
206C2×C8—2124no—
207C4⋊C4—2122no—
208C4⋊C4—2122no—
209C22⋊C4—2128no—
210C22⋊C4—2128no—
211C42—1064no—
212C42—2121no—
213C42—2121no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—24no—
218C23—48no—
219C23—416no—
220C23—42no—
221C23—42no—
222C23—44no—
223C23—44no—
224C23—88no—
225C23—84no—
226C23—84no—
227C23#1313327712yes—
228C23#1313328114yes—
229C23C2×C22⋊#615711yescharacteristic
230D4—44no—
231D4—48no—
232D4—44no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—534no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—4244no—
245C2×C4—2128no—
246C2×C4—2122no—
247C2×C4—2124no—
248C2×C4#1313327512yes—
249C2×C4C2×#1171994012yes—
250C8—2128no—
251C22—24no—
252C22—24no—
253C22—22no—
254C22—22no—
255C22—48no—
256C22—44no—
257C22—816no—
258C22—84no—
259C22—82no—
260C22—82no—
261C22#1681147814yes—
262C22#1681159311yescharacteristic
263C22C2×#1313327511yescharacteristic
264C22C2×#1313327711yescharacteristic
265C22#1681161512yes—
266C22C2×#1313328112yes—
267C4—22no—
268C4—22no—
269C4—1064no—
270C4—4242no—
271C4—2122no—
272C2—24no—
273C2—84no—
274C2#2028996011yescharacteristic
275C2#2029005412yes—
276C2#2029011112yes—
277C2C2×#1681147811yescharacteristic
278C2#2029024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC23×D4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC23×D4#6157
hypocenterC53#7626
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. #20289531C2semidirect
6. #16811489C22semidirect
153. C53#7626semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895312
C2×#168105262
#762653