Finite Group Databasebeta

← #28311204

Subgroups — #28311204

Order 13568.

480 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 84 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#20290582C212yesmaximal, complemented
3#20289812C212yesmaximal, complemented
4#20277425C211yesmaximal, characteristic
5#16811626—22no—
6C2×#13133278C2214yes—
7#16811531—24no—
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#13133284—42no—
20C4⋊#88324—42no—
21#13133278—24no—
22#13133278—24no—
23C22×Dic106C2311yescharacteristic
24C23×Dic53—21no—
25#13133204—22no—
26#13133204D412yes—
27#13133204Q812yes—
28#13133182—42no—
29D4×Dic53—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—22no—
32C2×#11719919D412yes—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D412yes—
38#13133124—21no—
39#13133124—21no—
40#13133124—22no—
41C22×D4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C42×C53C2×C411yescharacteristic
45C24×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53C4⋊C412yes—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53—42no—
56C22×Dic53—82no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D412yes—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106C2×D411yescharacteristic
63C2×Dic106C2×Q811yescharacteristic
64C2×Dic106#4112yes—
65#11719940—44no—
66#11719940—44no—
67C2×#88324—44no—
68C4⋊C4×C53—41no—
69C22⋊C4×C53—82no—
70C42×C53—41no—
71C22⋊Dic53—24no—
72C22⋊Dic53—24no—
73C22⋊Dic53—42no—
74C4⋊Dic53—41no—
75#11719919—42no—
76#11719919—82no—
77C4×Dic53—21no—
78C4×Dic53—22no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53C4⋊C412yes—
82C23×C53—22no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88D4×C53—42no—
89D4×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×C4⋊C411yescharacteristic
98C2×C4×C53#11711yescharacteristic
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—42no—
108C2×Dic53—41no—
109C2×Dic53—84no—
110C2×Dic53#9912yes—
111C2×Dic53#12112yes—
112C2×Dic53C22⋊Q812yes—
113Dic106—41no—
114Dic106—41no—
115Dic106—42no—
116#88324—84no—
117#12803—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—44no—
121C22×C53—42no—
122C22×C53—82no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53#38011yescharacteristic
126C22×C53C2×#9911yescharacteristic
127C22×C53#41011yescharacteristic
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—42no—
131C4×C53—81no—
132Dic53—22no—
133Dic53—22no—
134Dic53—22no—
135Dic53—42no—
136Dic53—41no—
137Dic53—81no—
138#2326—531no—
139C2×#355—532no—
140#1801—532no—
141#1394—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—82no—
145C2×C53#179911yescharacteristic
146C2×C53#213311yescharacteristic
147C2×C53#213411yescharacteristic
148C2×#124—531no—
149C2×C4×Q8—531no—
150C2×C4×D4—531no—
151#423—1062no—
152C2×#101—534no—
153#391—531no—
154#390#615711yescharacteristic
155#388—1062no—
156#385—1062no—
157#384—531no—
158#376—531no—
159#355—1064no—
160#343—1062no—
161C53#1280311yescharacteristic, complemented, Sylow
162C22×Q8—531no—
163C22×D4C2×#615711yescharacteristic
164C23×C4—1061no—
165C2×M16—1064no—
166#124—532no—
167#124—532no—
168#124—1062no—
169C4×Q8—2122no—
170C4×D4—2122no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4C2×#615711yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—1062no—
177C2×C22⋊C4—1061no—
178C2×C42—531no—
179C2×C42—532no—
180C2×C42—531no—
181C2×C42C2×#615711yescharacteristic
182C4⋊C8—2122no—
183C4⋊C8—2122no—
184#101—1064no—
185#101—1064no—
186#95—1061no—
187#95—1061no—
188#95—1062no—
189C24—21no—
190C2×Q8—531no—
191C2×Q8—531no—
192C2×Q8—532no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4C4⋊#615712yes—
199C22×C4—21no—
200C22×C4—532no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—4242no—
206C22×C4—2122no—
207C22×C4C22⋊#615712yes—
208C22×C4C22×#615711yescharacteristic
209M16—2124no—
210M16—2124no—
211C2×C8—2124no—
212C4⋊C4—41no—
213C4⋊C4—4242no—
214C4⋊C4—2122no—
215C4⋊C4—2121no—
216C22⋊C4—82no—
217C22⋊C4—1064no—
218C22⋊C4—1064no—
219C22⋊C4—2122no—
220C42—41no—
221C42—532no—
222C42—1062no—
223C42—1062no—
224C42—1062no—
225C42—1061no—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23C2×C22⋊#615711yescharacteristic
232Q8—2121no—
233Q8—2121no—
234Q8—2122no—
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—532no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—4244no—
252C2×C4—2121no—
253C2×C4—2122no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4C2×C4⋊#615711yescharacteristic
259C2×C4#1313329611yescharacteristic
260C8—4244no—
261C22—21no—
262C22—21no—
263C22—44no—
264C22—42no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22#1681156511yescharacteristic
269C22D53⋊#9911yescharacteristic
270C22#1681160311yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—81no—
275C4—1062no—
276C4—1062no—
277C4—1062no—
278C4—4241no—
279C4—2122no—
280C4—2121no—
281C2—21no—
282C2—21no—
283C2—82no—
284C2#2029057711yescharacteristic
285C2#2029101111yescharacteristic
286C2#2029101511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Dic106C23
hypercenter#390#6157
hypocenterC53#12803
LCSG ▷ C22×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. #20290582C2semidirect
3. #20289812C2semidirect
161. C53#12803semidirect

Maximal subgroups

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

subgroupindex
C2×#168115312
#202905822
#202898122
#202774252
#1280353