Finite Group Databasebeta

← #28310623

Subgroups — #28310623

Order 13568.

628 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 102 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×#16811527C212yesmaximal
2C4×#13133276C212yesmaximal, complemented
3#20289810C212yesmaximal, complemented
4#20277437C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6C4×C22⋊#6157—22no—
7#16811527C418yescomplemented
8#16811501—22no—
9C2×#13133213C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810526C411yescharacteristic, complemented
12#16810524C2211yescharacteristic
13#16810502—22no—
14#16810499—22no—
15#16810488C411yescharacteristic, complemented
16#16810469C411yescharacteristic, complemented
17C2×C4×D4×C53C411yescharacteristic, complemented
18C2×C22⋊#6157—24no—
19C2×C4×#6157—42no—
20#13133276—24no—
21#13133276C2×C418yescomplemented
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C22⋊D53—22no—
25C22×C4×D53D412yes—
26#13133213—22no—
27#13133213C2×C414yes—
28C4×D212—42no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C22⋊Dic53—22no—
31C2×#11719921—22no—
32C2×#11719921D412yes—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35#13133124—21no—
36#13133124—21no—
37#13133124—22no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53—21no—
40C4×D4×C53—42no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C23×D53C2×D412yes—
46C22×#6157—44no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53C4212yescomplemented
51C22×C4×C53—21no—
52C22×C4×C53—42no—
53C22×C4×C53—82no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C2×C22⋊D53—28no—
58C2×C22⋊D53—82no—
59C2×C22⋊D53—82no—
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C412yes—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212C22×C412yes—
66C2×D212#4112yes—
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—82no—
70C22⋊#6157—28no—
71C22⋊#6157—44no—
72C4×#6157—84no—
73C4⋊C4×C53—41no—
74C22⋊C4×C53—42no—
75C42×C53—22no—
76C22⋊Dic53—82no—
77#11719921—42no—
78#11719921—82no—
79C4⋊Dic53—41no—
80C4×Dic53—21no—
81C4×Dic53—41no—
82C4×Dic53C4212yescomplemented
83C23×C53—22no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—42no—
92C22×D53—42no—
93C22×D53C4×D414yes—
94C2×#6157—48no—
95C2×#6157—84no—
96C2×#6157—84no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—84no—
107C2×C4×C53#9912yes—
108C2×C4×C53C2×C4211yescharacteristic
109C2×C4×C53#11711yescharacteristic
110C22⋊D53—88no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119D212—42no—
120D212—42no—
121C4×D53—44no—
122C4×D53—82no—
123#12628—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53C4×C22⋊C411yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C22×C53#41011yescharacteristic
134C2×D53—24no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—44no—
138C2×D53—82no—
139#6157—88no—
140C4×C53—22no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—81no—
145Dic53—42no—
146Dic53—42no—
147Dic53—81no—
148#2328—531no—
149C2×#353—532no—
150C4×#99—532no—
151#1393—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—82no—
155C2×C53C4×#9911yescharacteristic
156C2×C53#212711yescharacteristic
157C2×C53#212811yescharacteristic
158D53—44no—
159C2×#127—531no—
160C2×C4×D4—531no—
161C2×C4×D4#615711yescharacteristic, complemented
162C2×#99—534no—
163#390—531no—
164#390—531no—
165#386—1062no—
166#386—1062no—
167#382—531no—
168C4×C22⋊C4—1062no—
169#376—531no—
170#353—538no—
171#342—1062no—
172C53#1262811yescharacteristic, complemented, Sylow
173C22×D4—531no—
174C22×D4—1062no—
175C22×D4C2×#615711yescharacteristic
176C23×C4—21no—
177C23×C4—532no—
178#127—534no—
179#127—1062no—
180C4×D4—42no—
181C4×D4—2122no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4C2×#615711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—1064no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—2122no—
192C2×C42C2×#615711yescharacteristic
193#99—538no—
194#99—1064no—
195#95—1062no—
196#95—1061no—
197#95—1061no—
198#95—2122no—
199C24—21no—
200C24—532no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—532no—
205C2×D4—532no—
206C2×D4—1068no—
207C2×D4—4242no—
208C2×D4—4242no—
209C2×D4—2121no—
210C2×D4—2121no—
211C2×D4C4×#615712yescomplemented
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—82no—
215C22×C4—532no—
216C22×C4—1061no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—1061no—
220C22×C4—4242no—
221C22×C4—2124no—
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22×#615711yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—2121no—
227C22⋊C4—42no—
228C22⋊C4—1068no—
229C22⋊C4—4242no—
230C22⋊C4—4242no—
231C22⋊C4—2122no—
232C22⋊C4—2124no—
233C42—22no—
234C42—532no—
235C42—1061no—
236C42—4244no—
237C42—2121no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—534no—
244C23—1062no—
245C23—1062no—
246C23—2122no—
247C23—2122no—
248C23C2×C22⋊#615711yescharacteristic
249D4—42no—
250D4—42no—
251D4—4248no—
252D4—2122no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—4242no—
265C2×C4—4244no—
266C2×C4—4244no—
267C2×C4—2124no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2128no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2122no—
275C2×C4#1313327512yes—
276C2×C4C2×C4×#615711yescharacteristic
277C2×C4#1313329611yescharacteristic
278C22—21no—
279C22—21no—
280C22—44no—
281C22—42no—
282C22—82no—
283C22—81no—
284C22—81no—
285C22—1064no—
286C22—4242no—
287C22—2124no—
288C22—2122no—
289C22—2122no—
290C22C4×C22⋊#615711yescharacteristic
291C22C2×#1313327511yescharacteristic
292C22#1681160211yescharacteristic
293C4—21no—
294C4—21no—
295C4—22no—
296C4—41no—
297C4—81no—
298C4—4241no—
299C4—4248no—
300C4—2122no—
301C4—2122no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2—2124no—
306C2C4×#1313327511yescharacteristic
307C2#2029099011yescharacteristic
308C2#2029099411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D212C23
hypercenterC2×C4×D4#6157
hypocenterC53#12628
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×#13133276C2semidirect
3. #20289810C2semidirect
7. #16811527C4semidirect
11. #16810526C4semidirect
15. #16810488C4semidirect
16. #16810469C4semidirect
17. C2×C4×D4×C53C4semidirect
21. #13133276C2×C4semidirect
50. C2×D4×C53C42semidirect
82. C4×Dic53C42semidirect
161. C2×C4×D4#6157semidirect
172. C53#12628semidirect
211. C2×D4C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115272
C4×#131332762
#202898102
#202774372
#1262853