Finite Group Databasebeta

← #28301886

Subgroups — #28301886

Order 13568.

684 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 132 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#20290500C212yesmaximal, complemented
2C2×#16811481C212yesmaximal
3#20289593C212yesmaximal, complemented
4#20277279C211yesmaximal, characteristic
5C2×#13133284C2214yes—
6C2×#13133272—22no—
7#16811489—22no—
8#16811481C418yescomplemented
9C2×#13133213C2211yescharacteristic
10C2×D4×Dic53C411yescharacteristic
11C2×#13133179C411yescharacteristic
12C2×#13133177—22no—
13#16810525C411yescharacteristic
14#16810523—22no—
15#16810492C2211yescharacteristic
16C42×Dic53C2211yescharacteristic
17C53×#390C411yescharacteristic
18C22×#88324—24no—
19#13133284—24no—
20#13133284C2×C418yescomplemented
21#13133272—44no—
22C22×D212C2×C411yescharacteristic
23C22×C22⋊D53—22no—
24C23×Dic53—21no—
25C22×C4×D53—21no—
26#13133213—22no—
27#13133213C2×C414yes—
28D4×Dic53—42no—
29#13133179—42no—
30#13133177—48no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—22no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—22no—
35C2×#11719921—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39C2×C4×Dic53C2311yescharacteristic
40C22×D4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C24×C53—21no—
45C23×D53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53C4212yes—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C2×C22⋊D53—28no—
54C2×C22⋊D53—44no—
55C2×C22⋊D53—42no—
56C2×C22⋊D53—42no—
57C22×Dic53—21no—
58C22×Dic53—44no—
59C22×Dic53—42no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D412yes—
63C2×D212—21no—
64C2×D212—41no—
65C2×D212—41no—
66C2×D212C4212yes—
67C2×C4×D53—44no—
68C2×C4×D53—42no—
69C2×#88324—28no—
70C2×#88324—44no—
71C2×#88324—44no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—44no—
74C42×C53—41no—
75C22⋊Dic53—42no—
76#11719921—42no—
77#11719921—44no—
78C4⋊Dic53—41no—
79#11719919—44no—
80C4×Dic53—22no—
81C4×Dic53—24no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53C22×C412yes—
85C4×Dic53#4112yes—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—22no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—42no—
96C22×D53—42no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53C2×C4211yescharacteristic
107C2×C4×C53#11711yescharacteristic
108C22⋊D53—48no—
109C22⋊D53—48no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115C2×Dic53—84no—
116C2×Dic53#10012yes—
117C2×Dic53C4×D414yes—
118C2×Dic53AGL(1,ℤ/8)14yes—
119C2×Dic53AGL(1,ℤ/8)12yes—
120C2×Dic53#13712yes—
121D212—42no—
122D212—42no—
123C4×D53—84no—
124#88324—48no—
125#8231—531nomaximal, Sylow
126C22×C53—42no—
127C22×C53—44no—
128C22×C53—81no—
129C22×C53—82no—
130C22×C53—81no—
131C22×C53C4×C22⋊C411yescharacteristic
132C22×C53C2×#10011yescharacteristic
133C22×C53#41311yescharacteristic
134C22×C53#41712yes—
135C22×C53#41812yes—
136C2×D53—42no—
137C2×D53—44no—
138C2×D53—81no—
139C2×D53—82no—
140C2×D53—81no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—42no—
144C4×C53—81no—
145Dic53—22no—
146Dic53—41no—
147Dic53—81no—
148Dic53#44014yes—
149Dic53#44214yes—
150C4×#127—531no—
151#1767—532no—
152C2×#331—532no—
153#1296—532no—
154C2×C53—82no—
155C2×C53#155511yescharacteristic
156C2×C53#155712yes—
157C2×C53#156011yescharacteristic
158C2×C53C4×#10011yescharacteristic
159C2×C53#176212yes—
160D53—82no—
161C2×#127—531no—
162C2×C4×D4—531no—
163C2×C4×D4—531no—
164C2×C4×D4—1062no—
165C2×C4⋊C8—534no—
166C2×#97—1062no—
167#390—531no—
168#390—1062no—
169#390#615711yescharacteristic
170#384—531no—
171C43—531no—
172#336—1062no—
173#331—538no—
174C53#823111yescharacteristic, complemented, Sylow
175C22×D4—531no—
176C22×D4—1062no—
177C22×D4C2×#615711yescharacteristic
178C23×C4—1061no—
179C23×C4—1061no—
180C22×C8—1064no—
181#127—534no—
182#127—1062no—
183C4×D4—2128no—
184C4×D4—2122no—
185C4×D4—2122no—
186C2×C4⋊C4—531no—
187C2×C4⋊C4—1062no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—1061no—
191C2×C22⋊C4—1061no—
192C2×C22⋊C4—1062no—
193C2×C42—531no—
194C2×C42—532no—
195C2×C42—532no—
196C2×C42—531no—
197C2×C42C2×#615711yescharacteristic
198C4⋊C8—538no—
199C4⋊C8—1064no—
200#97—2124no—
201C24—21no—
202C24—1061no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—532no—
207C2×D4—1068no—
208C2×D4—1061no—
209C2×D4—2121no—
210C2×D4—2121no—
211C2×D4—2124no—
212C2×D4—2122no—
213C2×D4—2122no—
214C2×D4C4×#615712yes—
215C22×C4—21no—
216C22×C4—532no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—2122no—
221C22×C4—2124no—
222C22×C4—2122no—
223C22×C4—2124no—
224C22×C4C22⋊#615712yes—
225C22×C4C22×#615711yescharacteristic
226C2×C8—1068no—
227C2×C8—2124no—
228C2×C8—2124no—
229C4⋊C4—41no—
230C4⋊C4—2121no—
231C4⋊C4—2124no—
232C22⋊C4—44no—
233C22⋊C4—2124no—
234C22⋊C4—2122no—
235C22⋊C4—2122no—
236C42—41no—
237C42—532no—
238C42—532no—
239C42—1062no—
240C42—1062no—
241C42—1064no—
242C42—1062no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—1062no—
249C23—2121no—
250C23—2121no—
251C23—2122no—
252C23—2122no—
253C23C2×C22⋊#615711yescharacteristic
254D4—42no—
255D4—42no—
256D4—2122no—
257D4—2122no—
258D4—2128no—
259D4—2128no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—534no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—534no—
271C2×C4—532no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—1062no—
275C2×C4—4244no—
276C2×C4—4244no—
277C2×C4—2121no—
278C2×C4—2122no—
279C2×C4C2×C4×#615711yescharacteristic
280C2×C4#1313329611yescharacteristic
281C8—2128no—
282C22—42no—
283C22—44no—
284C22—81no—
285C22—82no—
286C22—81no—
287C22—4241no—
288C22—4242no—
289C22—4241no—
290C22—2122no—
291C22—2124no—
292C22C4×C22⋊#615711yescharacteristic
293C22D53⋊#10011yescharacteristic
294C22#1681161011yescharacteristic
295C22#1681161412yes—
296C22#1681161512yes—
297C4—21no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—534no—
302C4—534no—
303C4—1062no—
304C4—4241no—
305C4—2121no—
306C2—82no—
307C2—4242no—
308C2#2029024311yescharacteristic
309C2#2029024712yes—
310C2#2029025311yescharacteristic
311C2#2029046511yescharacteristic
312C2#2029048512yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C22
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#8231
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290500C2semidirect
3. #20289593C2semidirect
8. #16811481C4semidirect
20. #13133284C2×C4semidirect
174. C53#8231semidirect

Maximal subgroups

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

subgroupindex
#202905002
C2×#168114812
#202895932
#202772792
#823153