Finite Group Databasebeta

← #28310646

Subgroups — #28310646

Order 13568.

512 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 100 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×#16811531C212yesmaximal
2#20290468C212yesmaximal, complemented
3#20289812C212yesmaximal, complemented
4#20277425C211yesmaximal, characteristic
5C2×#13133278C2214yes—
6C4×#11719940—22no—
7#16811531C418yescomplemented
8#16811503—22no—
9C2×#13133204C2211yescharacteristic
10C2×#13133182C2211yescharacteristic
11C2×D4×Dic53C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810517—22no—
14#16810495—22no—
15#16810492C411yescharacteristic
16#16810469C411yescharacteristic
17C53×#390C411yescharacteristic
18C2×#11719940—24no—
19#13133278—24no—
20#13133278C2×C418yescomplemented
21#13133272—42no—
22C4×#88324—42no—
23C22×Dic106C2311yescharacteristic
24C23×Dic53—21no—
25#13133204—22no—
26#13133204C2×C414yes—
27#13133182—42no—
28D4×Dic53—42no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—22no—
31C2×#11719919D412yes—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37#13133124—21no—
38#13133124—21no—
39#13133124—22no—
40C22×D4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53C4212yes—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—42no—
55C22×Dic53—82no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D412yes—
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106C22×C412yes—
62C2×Dic106#4112yes—
63#11719940—28no—
64#11719940—44no—
65C2×#88324—44no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—82no—
68C42×C53—41no—
69C22⋊Dic53—28no—
70C22⋊Dic53—42no—
71C4⋊Dic53—41no—
72#11719919—42no—
73#11719919—82no—
74C4×Dic53—21no—
75C4×Dic53—22no—
76C4×Dic53—22no—
77C4×Dic53—22no—
78C4×Dic53C4212yes—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85D4×C53—42no—
86D4×C53—42no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53C2×C4211yescharacteristic
95C2×C4×C53#11711yescharacteristic
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53—84no—
107C2×Dic53#9912yes—
108C2×Dic53C4×D414yes—
109Dic106—42no—
110Dic106—42no—
111#88324—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—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—81no—
127Dic53—22no—
128Dic53—24no—
129Dic53—42no—
130Dic53—41no—
131Dic53—81no—
132#2326—531no—
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×#124—531no—
143C2×C4×Q8—531no—
144C2×C4×D4—531no—
145C2×#101—534no—
146C4×M16—1062no—
147#391—531no—
148#390#615711yescharacteristic
149#388—1062no—
150#385—1062no—
151#384—531no—
152#376—531no—
153#355—538no—
154#343—1062no—
155C53#1263211yescharacteristic, complemented, Sylow
156C22×Q8—531no—
157C22×D4C2×#615711yescharacteristic
158C23×C4—1061no—
159C2×M16—1064no—
160#124—534no—
161#124—1062no—
162C4×Q8—2122no—
163C4×D4—2122no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4C2×#615711yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—1062no—
170C2×C22⋊C4—1061no—
171C2×C42—531no—
172C2×C42—532no—
173C2×C42—531no—
174C2×C42C2×#615711yescharacteristic
175#101—538no—
176#101—1064no—
177#97—2122no—
178C4×C8—2122no—
179#95—1061no—
180#95—1061no—
181#95—1062no—
182C24—21no—
183C2×Q8—532no—
184C2×Q8—532no—
185C2×Q8—2121no—
186C2×Q8—2121no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×#615712yes—
191C22×C4—21no—
192C22×C4—532no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4—4242no—
198C22×C4—2122no—
199C22×C4C22⋊#615712yes—
200C22×C4C22×#615711yescharacteristic
201M16—1068no—
202M16—2124no—
203C2×C8—2124no—
204C4⋊C4—41no—
205C4⋊C4—4242no—
206C4⋊C4—2122no—
207C4⋊C4—2121no—
208C22⋊C4—82no—
209C22⋊C4—1068no—
210C22⋊C4—2122no—
211C42—41no—
212C42—532no—
213C42—1062no—
214C42—1062no—
215C42—1062no—
216C42—1061no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊#615711yescharacteristic
223Q8—2122no—
224Q8—2122no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—534no—
235C2×C4—532no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1061no—
240C2×C4—4244no—
241C2×C4—2121no—
242C2×C4—2122no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2121no—
247C2×C4C2×C4×#615711yescharacteristic
248C2×C4#1313329611yescharacteristic
249C8—2128no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊#615711yescharacteristic
258C22D53⋊#9911yescharacteristic
259C22#1681160311yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—81no—
264C4—1064no—
265C4—1062no—
266C4—4241no—
267C4—2122no—
268C4—2121no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2#2029046211yescharacteristic
273C2#2029101111yescharacteristic
274C2#2029101511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Dic106C23
hypercenter#390#6157
hypocenterC53#12632
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20290468C2semidirect
3. #20289812C2semidirect
7. #16811531C4semidirect
20. #13133278C2×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×#168115312
#202904682
#202898122
#202774252
#1263253