Finite Group Databasebeta

← #28299559

Subgroups — #28299559

Order 13568.

492 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290630C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal
4#20277254C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133205C411yescharacteristic
9C2×#13133182C411yescharacteristic
10C2×#13133179C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810524C2211yescharacteristic
13#16810512—22no—
14#16810470—22no—
15C42×Dic53C2211yescharacteristic
16C53×#382C411yescharacteristic
17C22×#88324—24no—
18C4⋊#88324—44no—
19C22×Dic106C2×C411yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133205—22no—
23#13133205C814yescomplemented
24#13133182—42no—
25#13133179—42no—
26C2×#11719919—21no—
27C2×#11719921—22no—
28C2×#11719921—21no—
29C2×#11719921—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35#13133124—22no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#95×C53—22no—
40C23×D53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53C22⋊C412yes—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53—21no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C2×D412yes—
49C2×D212—21no—
50C2×D212—41no—
51C2×D212—41no—
52C2×D212C2×C812yes—
53C2×C4×D53—44no—
54C2×C4×D53—42no—
55C2×Dic106—21no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106C2×C812yes—
59C2×#88324—44no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C42×C53C2×C812yes—
65C42×C53M1612yes—
66#11719921—28no—
67#11719921—42no—
68#11719921—42no—
69#11719921—42no—
70#11719919—42no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—42no—
75C4×Dic53—41no—
76C4×Dic53—42no—
77C23×C53C2×C22⋊C411yescharacteristic
78C22×D53—22no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53—42no—
82C22×D53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53C22⋊C814yes—
92C2×C4×C53C22×C811yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53#10012yes—
103C2×Dic53AGL(1,ℤ/8)12yes—
104C2×Dic53#13712yes—
105D212—42no—
106D212—42no—
107C4×D53—84no—
108Dic106—42no—
109Dic106—42no—
110#88324—84no—
111#7601—531nomaximal, Sylow
112C22×C53C2×C22⋊C811yescharacteristic
113C22×C53#41011yescharacteristic
114C22×C53C2×#10011yescharacteristic
115C22×C53#41712yes—
116C22×C53#42112yes—
117C2×D53—42no—
118C2×D53—44no—
119C2×D53—81no—
120C2×D53—82no—
121C2×D53—81no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—24no—
125C4×C53—81no—
126C4×C53—81no—
127Dic53—22no—
128Dic53—22no—
129Dic53—22no—
130Dic53—41no—
131Dic53—42no—
132C4×#124—531no—
133#1831—532no—
134#1272—532no—
135#1270—532no—
136C2×C53#146011yescharacteristic
137C2×C53#147712yes—
138C2×C53#151212yes—
139C2×C53#153311yescharacteristic
140C2×C53#156011yescharacteristic
141D53—82no—
142C2×#124—531no—
143C2×C4×Q8—531no—
144C2×C4×D4—531no—
145C2×C4⋊C8—1062no—
146#391—531no—
147#390—531no—
148#388—1062no—
149#382#615711yescharacteristic
150C4×C22⋊C4—1062no—
151C43—531no—
152#336—534no—
153#336—1062no—
154C53#760111yescharacteristic, complemented, Sylow
155C22×Q8—531no—
156C22×D4—531no—
157C23×C4—1061no—
158C22×C8—1064no—
159#124—534no—
160#124—1062no—
161C4×Q8—2122no—
162C4×D4—2122no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4C2×#615711yescharacteristic
165C2×C4⋊C4C2×#615711yescharacteristic
166C2×C22⋊C4—1062no—
167C2×C22⋊C4—1061no—
168C2×C22⋊C4—1062no—
169C2×C42—532no—
170C2×C42—531no—
171C2×C42—532no—
172C2×C42—531no—
173C2×C42—1061no—
174C2×C42C2×#615711yescharacteristic
175C4⋊C8—2124no—
176#95—22no—
177#95—1062no—
178C24—1061no—
179C2×Q8—532no—
180C2×Q8—1061no—
181C2×Q8—2121no—
182C2×Q8—2121no—
183C2×D4—532no—
184C2×D4—1061no—
185C2×D4—2121no—
186C2×D4—2121no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—531no—
190C22×C4—532no—
191C22×C4—531no—
192C22×C4—1061no—
193C22×C4—2122no—
194C22×C4—2124no—
195C22×C4C22⋊#615712yes—
196C22×C4C22×#615711yescharacteristic
197C2×C8—2124no—
198C2×C8—2124no—
199C2×C8—2124no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—2122no—
203C22⋊C4—1068no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C22⋊C4—2122no—
207C42—1062no—
208C42—1062no—
209C42—1062no—
210C42—2121no—
211C42—2122no—
212C42—2122no—
213C42D53⋊C812yes—
214C42#1171993612yes—
215C23—1062no—
216C23—2121no—
217C23—2121no—
218C23—2122no—
219C23—2122no—
220C23C2×C22⋊#615711yescharacteristic
221Q8—2122no—
222Q8—2122no—
223D4—2122no—
224D4—2122no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—532no—
234C2×C4—532no—
235C2×C4—532no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—4244no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4#1313327414yes—
246C2×C4C2×D53⋊C811yescharacteristic
247C2×C4C2×#1171993611yescharacteristic
248C8—4244no—
249C22—4242no—
250C22—4241no—
251C22—4241no—
252C22—2122no—
253C22—2124no—
254C22C2×#1313327411yescharacteristic
255C22#1681160411yescharacteristic
256C22D53⋊#10011yescharacteristic
257C22#1681161412yes—
258C22#1681162012yes—
259C4—21no—
260C4—21no—
261C4—24no—
262C4—81no—
263C4—81no—
264C4—1062no—
265C4—1062no—
266C4—1062no—
267C4—2122no—
268C4—2121no—
269C2—4242no—
270C2#2028995611yescharacteristic
271C2#2029001212yes—
272C2#2029012912yes—
273C2#2029019511yescharacteristic
274C2#2029025311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7601
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #20290630C2semidirect
2. #20289536C2semidirect
23. #13133205C8semidirect
154. C53#7601semidirect

Maximal subgroups

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

subgroupindex
#202906302
#202895362
#202895322
#202772542
#760153