Finite Group Databasebeta

← #27299282

Subgroups — #27299282

Order 13568.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16810420C212yesmaximal
2C4×#13133136C212yesmaximal, complemented
3#20272842C212yesmaximal, complemented
4C53×#2326C211yesmaximal, characteristic
5C2×#13133136C2214yes—
6C4×#11719917—22no—
7#16810420C418yescomplemented
8#16810392—22no—
9C2×#124×C53C2211yescharacteristic
10C2×C4×Q8×C53C2211yescharacteristic
11C2×C4×D4×C53C411yescharacteristic
12C53×#391C2211yescharacteristic
13C53×#390C411yescharacteristic
14C53×#388—22no—
15C53×#385—22no—
16C53×#384C411yescharacteristic
17C53×#376C411yescharacteristic
18C2×#11719917—24no—
19#13133136—24no—
20#13133136C2×C418yescomplemented
21#13133128—42no—
22C4×#88322—42no—
23C22×Q8×C53C2311yescharacteristic
24C22×D4×C53C2×C411yescharacteristic
25C23×C4×C53—21no—
26#124×C53—22no—
27#124×C53C2×C414yes—
28C4×Q8×C53—42no—
29C4×D4×C53—42no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53C2×C411yescharacteristic
33C2×C4⋊C4×C53D412yes—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53D412yes—
41#95×C53—21no—
42#95×C53—21no—
43#95×C53—22no—
44C24×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C22×C412yes—
48C2×Q8×C53#4112yes—
49C2×D4×C53—21no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53C4212yes—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—42no—
57C22×C4×C53—82no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D412yes—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—82no—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—82no—
70C42×C53—22no—
71C42×C53—22no—
72C42×C53—22no—
73C42×C53—21no—
74C42×C53—41no—
75C42×C53C4212yes—
76#11719917—28no—
77#11719917—44no—
78C2×#88322—44no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85Q8×C53—42no—
86Q8×C53—42no—
87D4×C53—42no—
88D4×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—84no—
107C2×C4×C53#9912yes—
108C2×C4×C53C2×C4211yescharacteristic
109C2×C4×C53#11711yescharacteristic
110C2×C4×C53C4×D414yes—
111#88322—48no—
112#12632—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—44no—
116C22×C53—42no—
117C22×C53—82no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53C4×C22⋊C411yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53#41011yescharacteristic
123C4×C53—22no—
124C4×C53—24no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—41no—
130C4×C53—81no—
131C4×C53—81no—
132#2326D5311yescharacteristic
133C2×#355—532no—
134C4×#101—532no—
135#1394—532no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53—82no—
139C2×C53C4×#9911yescharacteristic
140C2×C53#213311yescharacteristic
141C2×C53#213411yescharacteristic
142C2×#124C2×D5311yescharacteristic
143C2×C4×Q8C2×D5311yescharacteristic
144C2×C4×D4Dic5311yescharacteristic
145C2×#101—534no—
146C4×M16—1062no—
147#391C2×D5311yescharacteristic
148#390Dic5311yescharacteristic
149#388—22no—
150#385—22no—
151#384Dic5311yescharacteristic
152#376Dic5311yescharacteristic
153#355—538no—
154#343—1062no—
155C53#1263211yescharacteristic, complemented, Sylow
156C22×Q8C22×D5311yescharacteristic
157C22×D4C2×Dic5311yescharacteristic
158C23×C4—21no—
159C2×M16—1064no—
160#124—22no—
161#124C4×D5314yes—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×Dic5311yescharacteristic
166C2×C4⋊C4C2×Dic5311yescharacteristic
167C2×C4⋊C4C22⋊D5312yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C42C2×Dic5311yescharacteristic
172C2×C42C2×Dic5311yescharacteristic
173C2×C42C2×Dic5311yescharacteristic
174C2×C42C22⋊D5312yes—
175#101—538no—
176#101—1064no—
177#97—2122no—
178C4×C8—2122no—
179#95—22no—
180#95—21no—
181#95—21no—
182C24—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×C4×D5312yes—
186C2×Q8#1171994512yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×Dic5312yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C22⋊Dic5311yescharacteristic
197C22×C4C22⋊Dic5312yes—
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22×Dic5311yescharacteristic
200C22×C4C2×C22⋊D5312yes—
201M16—1068no—
202M16—2124no—
203C2×C8—2124no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42C4×Dic5312yes—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊Dic5311yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#1313313212yes—
246C2×C4C2×C4×Dic5311yescharacteristic
247C2×C4#1313317611yescharacteristic
248C2×C4C4×C22⋊D5314yes—
249C8—2128no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊Dic5311yescharacteristic
258C22C2×#1313313211yescharacteristic
259C22#1681059711yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—22no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#1313313211yescharacteristic
273C2#2027640411yescharacteristic
274C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC53×#2326C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Q8×C53C23
hypercenter#2326D53
hypocenterC53#12632
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C4×#13133136C2semidirect
3. #20272842C2semidirect
7. #16810420C4semidirect
20. #13133136C2×C4semidirect
155. C53#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#168104202
C4×#131331362
#202728422
C53×#23262
#1263253