Finite Group Databasebeta

← #27293577

Subgroups — #27293577

Order 13568.

424 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 75 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#20277425C211yesmaximal, characteristic
2#20273057C212yesmaximal, complemented
3#20272660C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic
5C53×#1318C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×#13133182C2211yescharacteristic
8C2×D4×Dic53C411yescharacteristic
9#16810677—22no—
10#16810529C411yescharacteristic
11#16810517—22no—
12#16810495—22no—
13#16810492C2211yescharacteristic
14#16810469C411yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#390C2211yescharacteristic
18C53×#324C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C23×Dic53—21no—
21#13133204—22no—
22#13133204—22no—
23#13133182—42no—
24D4×Dic53—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919D412yes—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33#13133146—42no—
34C22⋊#88322—22no—
35#13133127—42no—
36#13133124—21no—
37#13133124—22no—
38#13133124—21no—
39C22×D4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—22no—
44C24×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53M1612yes—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—42no—
55C22×Dic53—82no—
56C22×Dic53C22⋊C412yes—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C2×Dic106—21no—
60C2×Dic106—21no—
61C2×Dic106—41no—
62C2×Dic106—41no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C22⋊C4×C53—82no—
66C42×C53—41no—
67C22⋊Dic53—28no—
68C22⋊Dic53—42no—
69C4⋊Dic53—41no—
70#11719919—42no—
71#11719919—82no—
72C4×Dic53—21no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53M1612yes—
77C2×#88322—42no—
78C23×C53—22no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53C2×C22⋊C411yescharacteristic
84D4×C53—42no—
85D4×C53—42no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53—84no—
107C2×Dic53#9912yes—
108Dic106—42no—
109Dic106—42no—
110C8×C53—82no—
111#88322—82no—
112#11534—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—44no—
116C22×C53—42no—
117C22×C53—82no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53#40711yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53#41311yescharacteristic
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—81no—
127Dic53—22no—
128Dic53—42no—
129Dic53—41no—
130Dic53—42no—
131Dic53—81no—
132#2326—531no—
133#1467—532no—
134#1325—532no—
135#1318—531no—
136#1318D5311yescharacteristic
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—82no—
140C2×C53#146511yescharacteristic
141C2×C53#212811yescharacteristic
142C2×C53#213311yescharacteristic
143C2×#124—531no—
144C2×C4×Q8—531no—
145C2×C4×D4—531no—
146#432—1062no—
147#391—531no—
148#390C2×D5311yescharacteristic
149#388—1062no—
150#385—1062no—
151#384—531no—
152#376—531no—
153#324—532no—
154#324—1062no—
155#324C2×D5312yes—
156C53#1153411yescharacteristic, complemented, Sylow
157C22×Q8—531no—
158C22×D4C4×D5311yescharacteristic
159C23×C4—1061no—
160#124—1062no—
161#124—1062no—
162C4×Q8—2122no—
163C4×D4—2122no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4C22×D5311yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—1062no—
170C2×C22⋊C4—1061no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—532no—
174C2×C42C4×D5311yescharacteristic
175C4⋊C8—2122no—
176C22⋊C8—22no—
177C22⋊C8—1062no—
178#97—2122no—
179#95—1061no—
180#95—1061no—
181#95—1062no—
182C24—21no—
183C2×Q8—1061no—
184C2×Q8—1061no—
185C2×Q8—2121no—
186C2×Q8—2121no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4#1171991212yes—
191C22×C4—21no—
192C22×C4—531no—
193C22×C4—532no—
194C22×C4—531no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4—4242no—
198C22×C4—2122no—
199C22×C4C2×C4×D5311yescharacteristic
200C22×C4D4×D5312yes—
201C2×C8—42no—
202C2×C8—2122no—
203C4⋊C4—41no—
204C4⋊C4—4242no—
205C4⋊C4—2122no—
206C4⋊C4—2121no—
207C22⋊C4—82no—
208C22⋊C4—1068no—
209C22⋊C4—2122no—
210C42—41no—
211C42—532no—
212C42—1062no—
213C42—1062no—
214C42—1062no—
215C42—1061no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23C22⋊C4×D5311yescharacteristic
222Q8—2122no—
223Q8—2122no—
224D4—42no—
225D4—42no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—532no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1061no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—4244no—
240C2×C4—2121no—
241C2×C4—2122no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4M16×D5311yescharacteristic
247C2×C4C2×#1171991211yescharacteristic
248C8—82no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22#1681058511yescharacteristic
258C22#99×D5311yescharacteristic
259C22#1681060411yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—81no—
264C4—1062no—
265C4—4241no—
266C4—2122no—
267C4—2122no—
268C4—2121no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2#2027305011yescharacteristic
273C2#2027638511yescharacteristic
274C2#2027640111yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1318C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1318D53
hypocenterC53#11534
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #20273057C2semidirect
3. #20272660C2semidirect
156. C53#11534semidirect

Maximal subgroups

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

subgroupindex
#202774252
#202730572
#202726602
#202726472
C53×#13182
#1153453