Finite Group Databasebeta

← #27280168

Subgroups — #27280168

Order 13568.

578 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 111 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#20280457C211yesmaximal, characteristic
2#20277318C211yesmaximal, characteristic, complemented
3#20273284C212yesmaximal
4#20272583C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8#16810553C412yescomplemented
9#16810545—22no—
10#16810540C412yescomplemented
11#16810509—24no—
12#16810502—21no—
13#16810486—21no—
14#16810483C2211yescharacteristic
15#16810483C2212yes—
16C4×C22⋊Dic53—21no—
17#16810358—24no—
18C22×C4⋊C4×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#324—22no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23#13133200—44no—
24#13133191—22no—
25#13133191—22no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—22no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—22no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C2×C4×Dic53—22no—
36#13133124—22no—
37#13133124—24no—
38#13133124—22no—
39#13133124C2×C412yes—
40#13133124D414yes—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C22×C8×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C4⋊C4×C53D412yes—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C24×C53C2×D411yescharacteristic
52C23×D53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—81no—
60C22×C4×C53—81no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C2×C22⋊D53—41no—
68C2×C22⋊D53—41no—
69C2×C22⋊D53—42no—
70C2×C22⋊D53—42no—
71C22×Dic53—21no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C2×D212—22no—
75C2×D212—81no—
76C2×C8×C53—42no—
77C2×C8×C53—82no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—24no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83#11719921—44no—
84#11719921—42no—
85#11719921—82no—
86C4⋊Dic53—22no—
87C4⋊Dic53—82no—
88C4⋊Dic53—81no—
89#11719919—44no—
90C4×Dic53—82no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#12011yescharacteristic
95C23×C53#12311yescharacteristic
96C23×C53#13512yes—
97C23×C53AGL(1,ℤ/8)12yes—
98C22×D53—42no—
99C22×D53—42no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—81no—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12112yes—
118C22⋊D53—82no—
119C22⋊D53—82no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—44no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—42no—
128C2×Dic53—44no—
129D212—82no—
130C8×C53—82no—
131#9206—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53#38611yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53#41811yescharacteristic
143C22×C53#44912yes—
144C22×C53#48212yes—
145C2×D53—82no—
146C2×D53—82no—
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—42no—
153C4×C53—42no—
154Dic53—82no—
155Dic53—84no—
156Dic53—81no—
157#2618—531no—
158#2313—531no—
159#1519—532no—
160C2×#324D5311yescharacteristic, complemented
161#1289—532no—
162C2×C53—21no—
163C2×C53—22no—
164C2×C53—21no—
165C2×C53—22no—
166C2×C53#187511yescharacteristic
167C2×C53#191211yescharacteristic
168C2×C53#191311yescharacteristic
169D53—161no—
170C2×#123—1061no—
171C2×#121—531no—
172C22×C4⋊C4C2×D5311yescharacteristic
173C2×C22⋊C8C2×D5312yes—
174#398—532no—
175#396—1062no—
176#394—532no—
177#387—1064no—
178#386—1061no—
179#381—1061no—
180#380—531no—
181#380—532no—
182C4×C22⋊C4—1061no—
183#327—1064no—
184#324—22no—
185C53#920611yescharacteristic, complemented, Sylow
186C22×D4—531no—
187C22×D4—1061no—
188C23×C4D21211yescharacteristic
189C23×C4D21211yescharacteristic
190C23×C4C22×D5311yescharacteristic
191C22×C8—22no—
192#123—2124no—
193#121—1062no—
194#121—1062no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4C4×D5312yes—
202C2×C4⋊C4C22⋊D5312yes—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1062no—
208C2×C42—1062no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—532no—
212#95—534no—
213#95—1062no—
214#95—1064no—
215#95—1062no—
216C24—1061no—
217C24C2×D21211yescharacteristic
218C2×D4—1062no—
219C2×D4—4241no—
220C2×D4—2121no—
221C2×D4—2121no—
222C2×D4—2122no—
223C2×D4—2122no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—1062no—
233C22×C4—1061no—
234C22×C4—1064no—
235C22×C4#1171992112yes—
236C22×C4#1171992112yes—
237C22×C4C2×C4×D5311yescharacteristic
238C22×C4D4×D5312yes—
239C22×C4#1171994712yes—
240C22×C4C2×C22⋊D5311yescharacteristic
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—1062no—
246C4⋊C4—4242no—
247C4⋊C4—4241no—
248C4⋊C4—2124no—
249C22⋊C4—4242no—
250C22⋊C4—2124no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C22⋊C4—2122no—
254C22⋊C4—2122no—
255C42—4242no—
256C23—41no—
257C23—41no—
258C23—2122no—
259C23—2122no—
260C23—2121no—
261C23—2121no—
262C23—2121no—
263C23C2×#1171992111yescharacteristic
264C23#1313318711yescharacteristic
265C23#1313320111yescharacteristic
266C23#1313323912yes—
267C23#1313324512yes—
268D4—4242no—
269D4—4242no—
270D4—4242no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—2124no—
285C2×C4—2122no—
286C2×C4—2124no—
287C2×C4—2122no—
288C2×C4—2121no—
289C2×C4—2124no—
290C2×C4—2121no—
291C2×C4—2122no—
292C2×C4—2121no—
293C2×C4#1313317712yes—
294C2×C4#1313318912yes—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—4242no—
305C22—4242no—
306C22—4241no—
307C22—4241no—
308C22—4241no—
309C22#1681050011yescharacteristic
310C22#1681062011yescharacteristic
311C22#1681062911yescharacteristic
312C22#1681077012yes—
313C22#1681094812yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—4242no—
319C4—4244no—
320C4—4241no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—8481no—
326C2#2027503511yescharacteristic
327C2#2027525011yescharacteristic
328C2#2027525511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#9206
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #20277318C2semidirect
4. #20272583C2semidirect
5. C2×C53×#324C2semidirect
8. #16810553C4semidirect
10. #16810540C4semidirect
160. C2×#324D53semidirect
185. C53#9206semidirect

Maximal subgroups

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

subgroupindex
#202804572
#202773182
#202732842
#202725832
C2×C53×#3242
#920653