Finite Group Databasebeta

← #28299646

Subgroups — #28299646

Order 13568.

874 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 112 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#20289532C214yesmaximal, complemented
3C2×#16810524C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272—22no—
6#16811489C2214yescomplemented
7C23×D212C411yescharacteristic
8C22×#11719921—22no—
9C22×C4×Dic53C2211yescharacteristic
10#16810524—22no—
11#16810524C412yes—
12#16810524C2212yes—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—44no—
17C24×D53—21no—
18C22×D212—21no—
19C22×D212—24no—
20C22×D212—42no—
21C22×D212—42no—
22C22×D212C2×C412yes—
23C23×Dic53D412yes—
24C2×#11719921—24no—
25C2×#11719921—24no—
26C2×#11719921—48no—
27C2×C4×Dic53—42no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—42no—
35C2×C4⋊C4×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C23×D53—24no—
38C23×D53—22no—
39C23×D53—44no—
40C23×D53—42no—
41C23×D53—42no—
42C23×D53—44no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—44no—
48C22×C4×C53—42no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—24no—
52C22×Dic53—24no—
53C22×Dic53—42no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×D212—24no—
57C2×D212—22no—
58C2×D212—416no—
59C2×D212—42no—
60C2×D212—42no—
61C2×D212—42no—
62C2×D212—42no—
63C2×D212M1614yes—
64C2×#88324—28no—
65C2×#88324—44no—
66C2×#88324—44no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69#11719921—48no—
70#11719921—48no—
71C4×Dic53—24no—
72C4×Dic53—41no—
73C4×Dic53—41no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—22no—
77C23×C53#10012yes—
78C23×C53#10414yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C22×D53—24no—
81C22×D53—48no—
82C22×D53—416no—
83C22×D53—42no—
84C22×D53—42no—
85C22×D53—44no—
86C22×D53—44no—
87C22×D53—84no—
88C22×D53—88no—
89C22×D53—84no—
90C2×C4×C53—28no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53—44no—
97C2×C4×C53—84no—
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×Dic53AGL(1,ℤ/8)14yes—
107D212—44no—
108D212—48no—
109D212—44no—
110#88324—48no—
111#7626—531nomaximal, Sylow
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—24no—
115C22×C53—24no—
116C22×C53#32914yes—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53C2×#10011yescharacteristic
120C22×C53#41812yes—
121C22×C53C2×#10412yes—
122C2×D53—44no—
123C2×D53—48no—
124C2×D53—816no—
125C2×D53—82no—
126C2×D53—84no—
127C2×D53—82no—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—82no—
131Dic53—24no—
132Dic53—42no—
133C2×#390—531no—
134#1749—532no—
135#1270—534no—
136C2×C53—24no—
137C2×C53#146211yescharacteristic
138C2×C53#149012yes—
139C2×C53#150712yes—
140C2×C53C2×#32911yescharacteristic
141C2×C53#155511yescharacteristic
142D53—84no—
143C23×D4—531no—
144C22×C4⋊C4#615711yescharacteristic
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—1061no—
155C22×D4—532no—
156C22×D4—1064no—
157C22×D4—1061no—
158C22×D4—2122no—
159C22×D4—2122no—
160C23×C4—21no—
161C23×C4—532no—
162C23×C4C2×#615711yescharacteristic
163C22×C8—1064no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×#615712yes—
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—1064no—
177C24—1062no—
178C24—2124no—
179C24—2122no—
180C24—2122no—
181C24—2124no—
182C24C22⋊#615711yescharacteristic
183C2×D4—534no—
184C2×D4—1064no—
185C2×D4—1062no—
186C2×D4—2122no—
187C2×D4—2122no—
188C2×D4—21216no—
189C2×D4—2122no—
190C2×D4—2122no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—532no—
198C22×C4—532no—
199C22×C4—1064no—
200C22×C4—1064no—
201C22×C4—2122no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22×#615711yescharacteristic
204C2×C8—1068no—
205C2×C8—2124no—
206C2×C8—2124no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C22⋊C4—2128no—
210C22⋊C4—2128no—
211C42—1064no—
212C42—2121no—
213C42—2121no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—1064no—
218C23—4248no—
219C23—4244no—
220C23—4244no—
221C23—2128no—
222C23—21216no—
223C23—2122no—
224C23—2122no—
225C23—2124no—
226C23—2124no—
227C23#1313327712yes—
228C23#1313328214yes—
229C23C2×C22⋊#615711yescharacteristic
230D4—2124no—
231D4—2128no—
232D4—2124no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—84no—
241C2×C4—534no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—2128no—
248C2×C4#1313327612yes—
249C2×C4D53⋊M1612yes—
250C8—2128no—
251C22—24no—
252C22—24no—
253C22—22no—
254C22—22no—
255C22—42416no—
256C22—4244no—
257C22—4242no—
258C22—4242no—
259C22—2128no—
260C22—2124no—
261C22#1681147914yes—
262C22#1681159511yescharacteristic
263C22C2×#1313327611yescharacteristic
264C22C2×#1313327711yescharacteristic
265C22#1681161512yes—
266C22C2×#1313328212yes—
267C4—22no—
268C4—22no—
269C4—82no—
270C4—1064no—
271C4—2122no—
272C2—24no—
273C2—4244no—
274C2#2028996211yescharacteristic
275C2#2029005612yes—
276C2#2029011212yes—
277C2C2×#1681147911yescharacteristic
278C2#2029024211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D212C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7626
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20290441C2semidirect
2. #20289532C2semidirect
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
#202895322
C2×#168105242
#762653