Finite Group Databasebeta

← #27293387

Subgroups — #27293387

Order 13568.

392 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277433C211yesmaximal, characteristic
2#20273037C212yesmaximal
3#20272658C212yesmaximal
4#20272649C211yesmaximal, characteristic
5C53×#1319C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133182C2211yescharacteristic
9#16810673—22no—
10#16810529C411yescharacteristic
11#16810510—22no—
12#16810505—22no—
13#16810488C2211yescharacteristic
14#16810469C411yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#391C2211yescharacteristic
18C53×#324C2212yes—
19C22×Dic106C2×C411yescharacteristic
20#13133215—22no—
21#13133215C814yescomplemented
22Q8×Dic53—42no—
23#13133182—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919D412yes—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D412yes—
32#13133146—42no—
33C22⋊#88322—22no—
34C8×Dic53—42no—
35#13133124—22no—
36#13133124—21no—
37#13133124—22no—
38#13133124—21no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C22⋊C8×C53—22no—
43#95×C53—22no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C2×C812yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106C2×C812yes—
60C2×Dic106M1612yes—
61C2×C8×C53—42no—
62C4⋊C4×C53—41no—
63C42×C53—41no—
64C4⋊Dic53—42no—
65C4⋊Dic53—41no—
66#11719919—28no—
67#11719919—42no—
68C4×Dic53—21no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—42no—
73C4×Dic53C2×C812yes—
74C2×#88322—42no—
75C23×C53C2×C22⋊C411yescharacteristic
76Q8×C53—42no—
77Q8×C53—42no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53C22×C811yescharacteristic
89C2×C4×C53C2×M1611yescharacteristic
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—21no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—42no—
99C2×Dic53—41no—
100C2×Dic53C22⋊C814yes—
101C2×Dic53#9912yes—
102Dic106—42no—
103Dic106—42no—
104C8×C53—82no—
105#88322—82no—
106#11482—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53C2×C22⋊C811yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53#41311yescharacteristic
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53—81no—
117Dic53—22no—
118Dic53—24no—
119Dic53—42no—
120Dic53—41no—
121Dic53—81no—
122#2327—531no—
123#1461—532no—
124#1324—532no—
125#1319—531no—
126#1319D5311yescharacteristic
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#145911yescharacteristic
130C2×C53#212911yescharacteristic
131C2×C53#213311yescharacteristic
132C2×C4⋊Q8—531no—
133C2×C4×Q8—531no—
134C2×C4×Q8—531no—
135#431—1062no—
136#391—531no—
137#391C2×D5311yescharacteristic
138#387—1062no—
139#387—1062no—
140#382—531no—
141#376—531no—
142#324—532no—
143#324—1062no—
144#324C2×D5312yes—
145C53#1148211yescharacteristic, complemented, Sylow
146C22×Q8—531no—
147C22×Q8C4×D5311yescharacteristic
148C4⋊Q8—534no—
149C4⋊Q8—1062no—
150C4×Q8—2122no—
151C4×Q8—2122no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4C22×D5311yescharacteristic
157C2×C42—531no—
158C2×C42—531no—
159C2×C42—532no—
160C2×C42—1061no—
161C2×C42C4×D5311yescharacteristic
162C4⋊C8—2122no—
163C22⋊C8—22no—
164C22⋊C8—1062no—
165C4×C8—2122no—
166#95—22no—
167#95—1061no—
168#95—1061no—
169#95—1062no—
170#95—1062no—
171C2×Q8—21no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—532no—
175C2×Q8—532no—
176C2×Q8—2121no—
177C2×Q8—2121no—
178C2×Q8C8×D5312yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—532no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4C2×C4×D5311yescharacteristic
187C22×C4D4×D5312yes—
188C2×C8—42no—
189C2×C8—2122no—
190C4⋊C4—41no—
191C4⋊C4—1068no—
192C4⋊C4—2122no—
193C4⋊C4—2121no—
194C4⋊C4—2122no—
195C42—41no—
196C42—532no—
197C42—1062no—
198C42—1062no—
199C42—1062no—
200C42—1061no—
201C42—2122no—
202C23C22⋊C4×D5311yescharacteristic
203Q8—42no—
204Q8—42no—
205Q8—2122no—
206Q8—2122no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—534no—
218C2×C4—532no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1061no—
223C2×C4—2121no—
224C2×C4—2122no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4C2×C8×D5311yescharacteristic
230C2×C4M16×D5311yescharacteristic
231C8—82no—
232C8—4242no—
233C22—21no—
234C22—21no—
235C22C22⋊C8×D5311yescharacteristic
236C22#99×D5311yescharacteristic
237C22#1681060411yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—81no—
243C4—1064no—
244C4—1062no—
245C4—4241no—
246C4—2122no—
247C4—2121no—
248C2—21no—
249C2—21no—
250C2#2027303011yescharacteristic
251C2#2027638811yescharacteristic
252C2#2027640311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1319C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1319D53
hypocenterC53#11482
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
21. #13133215C8semidirect
145. C53#11482semidirect

Maximal subgroups

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

subgroupindex
#202774332
#202730372
#202726582
#202726492
C53×#13192
#1148253