Finite Group Databasebeta

← #28301860

Subgroups — #28301860

Order 13568.

528 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290432C212yesmaximal, complemented
2C2×#16811479C212yesmaximal
3#20289591C212yesmaximal, complemented
4C4×#13133206C211yesmaximal, characteristic
5C2×#13133272C2214yes—
6#16811489—22no—
7#16811489—22no—
8#16811479C418yescomplemented
9C2×#13133206C2211yescharacteristic
10C2×C4×D212C411yescharacteristic
11#16810530C411yescharacteristic
12#16810524C411yescharacteristic
13#16810513—22no—
14#16810488C2211yescharacteristic
15C4×#11719921—22no—
16C42×Dic53C2211yescharacteristic
17C2×C4×Q8×C53C411yescharacteristic
18C22×#88324—24no—
19#13133272—24no—
20#13133272C2×C418yescomplemented
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23#13133206—22no—
24#13133206C2×C414yes—
25C4×D212—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719921—21no—
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32C2×C4×Dic53D412yes—
33C2×C4×Dic53C2311yescharacteristic
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C4×Q8×C53—42no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C23×D53—21no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53C4212yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D412yes—
55C2×D212—21no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212C4212yes—
59C2×C4×D53—44no—
60C2×C4×D53—42no—
61C2×#88324—28no—
62C2×#88324—44no—
63C2×#88324—44no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—41no—
66C42×C53—22no—
67C42×C53—42no—
68#11719921—28no—
69#11719921—44no—
70#11719921—42no—
71C4⋊Dic53—41no—
72C4×Dic53—41no—
73C4×Dic53—42no—
74C4×Dic53—42no—
75C4×Dic53C22×C412yes—
76C4×Dic53#4112yes—
77C23×C53C2×C22⋊C411yescharacteristic
78C22×D53—22no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53—42no—
82C22×D53—42no—
83Q8×C53—42no—
84Q8×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53#10012yes—
96C2×C4×C53C2×C4211yescharacteristic
97C2×C4×C53#11711yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53C4×D414yes—
106D212—42no—
107D212—42no—
108C4×D53—84no—
109#88324—48no—
110#8224—531nomaximal, Sylow
111C22×C53C4×C22⋊C411yescharacteristic
112C22×C53C2×#10011yescharacteristic
113C22×C53#41311yescharacteristic
114C22×C53#42112yes—
115C22×C53#42112yes—
116C2×D53—42no—
117C2×D53—44no—
118C2×D53—81no—
119C2×D53—81no—
120C2×D53—82no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—22no—
124C4×C53—41no—
125C4×C53—42no—
126C4×C53—81no—
127Dic53—24no—
128Dic53—42no—
129Dic53—81no—
130C4×#124—531no—
131#1745—532no—
132C2×#329—532no—
133#1295—532no—
134C2×C53#153311yescharacteristic
135C2×C53#153512yes—
136C2×C53#153611yescharacteristic
137C2×C53C4×#10011yescharacteristic
138C2×C53#175912yes—
139D53—82no—
140C2×#124—531no—
141C2×C4×Q8#615711yescharacteristic
142C2×C4×D4—531no—
143C2×#97—534no—
144#391—531no—
145#390—531no—
146#388—1062no—
147#382—531no—
148C4×C22⋊C4—1062no—
149C43—531no—
150#336—1062no—
151#336—1062no—
152#329—538no—
153C53#822411yescharacteristic, complemented, Sylow
154C22×Q8C2×#615711yescharacteristic
155C22×D4—531no—
156C23×C4—1061no—
157C22×C8—1064no—
158#124—534no—
159#124—1062no—
160C4×Q8—42no—
161C4×D4—2122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4C2×#615711yescharacteristic
165C2×C22⋊C4—1062no—
166C2×C22⋊C4—1062no—
167C2×C22⋊C4—1061no—
168C2×C42—21no—
169C2×C42—531no—
170C2×C42—532no—
171C2×C42—532no—
172C2×C42—531no—
173C2×C42C2×#615711yescharacteristic
174#97—538no—
175#97—1064no—
176#95—1062no—
177#95—1062no—
178C24—1061no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C4×#615712yes—
183C2×D4—532no—
184C2×D4—1061no—
185C2×D4—2121no—
186C2×D4—2121no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1061no—
192C22×C4—2124no—
193C22×C4—2122no—
194C22×C4C22⋊#615711yescharacteristic
195C22×C4C22⋊#615711yescharacteristic
196C22×C4C22×#615711yescharacteristic
197C2×C8—1068no—
198C2×C8—2124no—
199C2×C8—2124no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—2121no—
203C22⋊C4—1068no—
204C22⋊C4—2122no—
205C22⋊C4—2124no—
206C42—22no—
207C42—42no—
208C42—532no—
209C42—532no—
210C42—2121no—
211C42—2122no—
212C42—2122no—
213C23—1062no—
214C23—2121no—
215C23—2121no—
216C23—2122no—
217C23—2122no—
218C23C2×C22⋊#615711yescharacteristic
219Q8—42no—
220Q8—42no—
221D4—2122no—
222D4—2122no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—534no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—4244no—
240C2×C4—2121no—
241C2×C4—2122no—
242C2×C4#1313327712yes—
243C2×C4C2×C4×#615711yescharacteristic
244C2×C4#1313329611yescharacteristic
245C8—2128no—
246C22—4241no—
247C22—4242no—
248C22—4241no—
249C22—2122no—
250C22—2124no—
251C22C4×C22⋊#615711yescharacteristic
252C22C2×#1313327711yescharacteristic
253C22#1681160911yescharacteristic
254C22#1681162012yes—
255C22#1681162212yes—
256C4—22no—
257C4—21no—
258C4—21no—
259C4—41no—
260C4—42no—
261C4—81no—
262C4—1064no—
263C4—4241no—
264C4—2122no—
265C2—4242no—
266C2#2029019611yescharacteristic
267C2#2029020112yes—
268C2#2029020311yescharacteristic
269C2C4×#1313327711yescharacteristic
270C2#2029047912yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#8224
LCSG ▷ C2×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. #20290432C2semidirect
3. #20289591C2semidirect
8. #16811479C4semidirect
20. #13133272C2×C4semidirect
153. C53#8224semidirect

Maximal subgroups

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

subgroupindex
#202904322
C2×#168114792
#202895912
C4×#131332062
#822453