Finite Group Databasebeta

← #27279835

Subgroups — #27279835

Order 13568.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#20283046C211yesmaximal, characteristic
2C2×#16810469C211yesmaximal, characteristic, complemented
3C2×#16810358C212yesmaximal
4#20272595C212yesmaximal, complemented
5C53×#1817C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8C22×C4×Dic53—21no—
9#16810525C412yescomplemented
10#16810521C412yescomplemented
11#16810514—22no—
12#16810502—21no—
13#16810483C2211yescharacteristic
14#16810469—24no—
15C2×#13133124—21no—
16C2×#13133124C2212yes—
17#16810358—24no—
18C22×C42×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#336—22no—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—21no—
23C23×Dic53—22no—
24#13133200—44no—
25#13133191—22no—
26#13133191—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×C22⋊Dic53—21no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—21no—
34C2×C4×Dic53—28no—
35C2×C4×Dic53—24no—
36#13133124—24no—
37#13133124—22no—
38#13133124—21no—
39#13133124—22no—
40#13133124D414yes—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C22×C8×C53—22no—
45C2×C42×C53—22no—
46C2×C42×C53—22no—
47C2×C42×C53C2×C412yes—
48C2×C42×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—24no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53#4112yes—
67C2×C22⋊D53—41no—
68C2×C22⋊D53—41no—
69C2×C22⋊D53—42no—
70C2×C22⋊D53—42no—
71C22×Dic53—28no—
72C22×Dic53—24no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C22×Dic53—44no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C2×D212—22no—
79C2×D212—42no—
80C2×C8×C53—44no—
81C2×C8×C53—42no—
82C42×C53—24no—
83C42×C53—24no—
84C22⋊Dic53—42no—
85#11719921—44no—
86#11719921—44no—
87#11719921—42no—
88C4⋊Dic53—22no—
89C4⋊Dic53—42no—
90#11719919—44no—
91C4×Dic53—216no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53#12011yescharacteristic
96C23×C53#12311yescharacteristic
97C23×C53#13512yes—
98C23×C53AGL(1,ℤ/8)12yes—
99C22×D53—41no—
100C22×D53—42no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—42no—
104C2×C4×C53—24no—
105C2×C4×C53—24no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53#10414yes—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12112yes—
120C22⋊D53—82no—
121C22⋊D53—82no—
122C2×Dic53—28no—
123C2×Dic53—44no—
124C2×Dic53—48no—
125C2×Dic53—48no—
126C2×Dic53—44no—
127C2×Dic53—41no—
128C2×Dic53—42no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133D212—82no—
134C8×C53—82no—
135#9149—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—24no—
141C22×C53—24no—
142C22×C53#38611yescharacteristic
143C22×C53#41012yes—
144C22×C53#41611yescharacteristic
145C22×C53#41811yescharacteristic
146C22×C53C2×#10412yes—
147C22×C53#44912yes—
148C22×C53#48212yes—
149C2×D53—81no—
150C2×D53—81no—
151C2×D53—81no—
152C2×D53—82no—
153C2×D53—82no—
154C4×C53—24no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—42no—
158Dic53—48no—
159Dic53—81no—
160Dic53—82no—
161#2854—531no—
162C2×#376—531no—
163#1817D5311yescharacteristic, complemented
164C2×#327—532no—
165#1294—532no—
166C2×C53—22no—
167C2×C53—22no—
168C2×C53#150012yes—
169C2×C53#150512yes—
170C2×C53#187511yescharacteristic
171C2×C53#189611yescharacteristic
172C2×C53#189911yescharacteristic
173D53—161no—
174C2×#123—1061no—
175C2×#121—531no—
176C22×C42—1061no—
177C22×C42C2×D5311yescharacteristic
178C2×C22⋊C8C2×D5312yes—
179#390—532no—
180#389—532no—
181#388—1062no—
182#386—1061no—
183#380—531no—
184#376—1064no—
185C2×#95—532no—
186C2×#95—1061no—
187#336—22no—
188#327—1064no—
189C53#914911yescharacteristic, complemented, Sylow
190C22×D4—531no—
191C22×D4—1061no—
192C23×C4—1062no—
193C23×C4D21211yescharacteristic
194C23×C4D21211yescharacteristic
195C23×C4C22×D5311yescharacteristic
196C22×C8—22no—
197#123—2124no—
198#121—1062no—
199#121—1062no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—532no—
202C2×C4⋊C4—1062no—
203C2×C22⋊C4—1061no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1062no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—1064no—
210C2×C42—1068no—
211C2×C42C4×D5312yes—
212C2×C42C22⋊D5312yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—534no—
216#95—1064no—
217#95—1062no—
218#95—1061no—
219#95—1062no—
220C24—1061no—
221C24C2×D21211yescharacteristic
222C2×D4—1062no—
223C2×D4—2122no—
224C2×D4—2121no—
225C2×D4—2121no—
226C2×D4—2122no—
227C2×D4—2122no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—1068no—
237C22×C4—1064no—
238C22×C4—1062no—
239C22×C4—1061no—
240C22×C4—2124no—
241C22×C4—2122no—
242C22×C4—2122no—
243C22×C4#1171992112yes—
244C22×C4#1171992112yes—
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4D4×D5312yes—
247C22×C4#1171994712yes—
248C22×C4C2×C22⋊D5311yescharacteristic
249C2×C8—44no—
250C2×C8—42no—
251C4⋊C4—1062no—
252C4⋊C4—2122no—
253C4⋊C4—2124no—
254C22⋊C4—2124no—
255C22⋊C4—2122no—
256C22⋊C4—2124no—
257C22⋊C4—2122no—
258C42—24no—
259C42—24no—
260C42—10616no—
261C23—22no—
262C23—22no—
263C23—2121no—
264C23—2122no—
265C23—2121no—
266C23—2121no—
267C23—2122no—
268C23C2×#1171992111yescharacteristic
269C23#1313318711yescharacteristic
270C23#1313320111yescharacteristic
271C23#1313323912yes—
272C23#1313324512yes—
273D4—4242no—
274D4—4242no—
275D4—4242no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—1068no—
290C2×C4—2121no—
291C2×C4—2122no—
292C2×C4—2121no—
293C2×C4—2122no—
294C2×C4—2121no—
295C2×C4—2122no—
296C2×C4—2128no—
297C2×C4—2128no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4#1313314314yes—
301C2×C4#1313317712yes—
302C2×C4#1313318912yes—
303C8—82no—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—4242no—
311C22—4242no—
312C22—4241no—
313C22—4241no—
314C22—4241no—
315C22#1681050011yescharacteristic
316C22#1681059612yes—
317C22#1681062011yescharacteristic
318C22#1681062911yescharacteristic
319C22C2×#1313314312yes—
320C22#1681077012yes—
321C22#1681094812yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—4242no—
327C4—4241no—
328C4—2128no—
329C2—22no—
330C2—22no—
331C2—8481no—
332C2#2027320212yes—
333C2#2027322412yes—
334C2#2027503511yescharacteristic
335C2#2027515611yescharacteristic
336C2#2027517111yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1817C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1817D53
hypocenterC53#9149
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#16810469C2semidirect
4. #20272595C2semidirect
5. C53×#1817C2semidirect
9. #16810525C4semidirect
10. #16810521C4semidirect
163. #1817D53semidirect
189. C53#9149semidirect

Maximal subgroups

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

subgroupindex
#202830462
C2×#168104692
C2×#168103582
#202725952
C53×#18172
#914953