Finite Group Databasebeta

← #27299581

Subgroups — #27299581

Order 13568.

1206 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16810418C214yesmaximal, complemented
2#20274630C212yesmaximal
3C22×#124×C53C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810479—22no—
7#16810418—28no—
8C23×Q8×C53C411yescharacteristic, complemented
9C23×D4×C53C2211yescharacteristic
10C2×#124×C53—22no—
11C2×#124×C53—28no—
12C2×#124×C53C412yescomplemented
13C2×#124×C53C2212yes—
14C22×C22⋊C4×C53—22no—
15C22×C42×C53C411yescharacteristic, complemented
16C2×C22⋊Dic53—24no—
17C2×C22⋊Dic53—44no—
18#13133132—28no—
19#13133132—44no—
20C25×C53D412yes—
21C22×Q8×C53—24no—
22C22×Q8×C53—21no—
23C22×Q8×C53—42no—
24C22×Q8×C53—42no—
25C22×Q8×C53C2×C412yes—
26C22×D4×C53—24no—
27C22×D4×C53—42no—
28C22×D4×C53—42no—
29C22×D4×C53C2×C411yescharacteristic
30C22×D4×C53D412yes—
31C22×D4×C53C2311yescharacteristic
32C23×C4×C53—21no—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35#124×C53—216no—
36#124×C53—24no—
37#124×C53—24no—
38#124×C53—24no—
39#124×C53D418yes—
40C2×C22⋊C4×C53—216no—
41C2×C22⋊C4×C53—24no—
42C2×C22⋊C4×C53—24no—
43C2×C42×C53—24no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C412yes—
46C24×C53—24no—
47C24×C53—24no—
48C24×C53—42no—
49C24×C53—48no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C412yes—
52C24×C53C2×D412yes—
53C2×Q8×C53—24no—
54C2×Q8×C53—22no—
55C2×Q8×C53—416no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C2×Q8×C53C22⋊C414yescomplemented
61C2×D4×C53—24no—
62C2×D4×C53—24no—
63C2×D4×C53—22no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—42no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53—416no—
70C2×D4×C53C2×D414yes—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—24no—
75C22×C4×C53—24no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—44no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×Dic53—44no—
84C22⋊C4×C53—216no—
85C22⋊C4×C53—216no—
86C42×C53—24no—
87C42×C53—22no—
88C42×C53C22⋊C414yescomplemented
89C22⋊Dic53—44no—
90C22⋊Dic53—44no—
91C22⋊Dic53—88no—
92C23×C53—24no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—48no—
101C23×C53—44no—
102C23×C53—44no—
103C23×C53—416no—
104C23×C53—48no—
105C23×C53—84no—
106C23×C53—88no—
107C23×C53#9912yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12014yes—
110Q8×C53—48no—
111Q8×C53—44no—
112Q8×C53—44no—
113D4×C53—44no—
114D4×C53—48no—
115D4×C53—44no—
116C2×C4×C53—28no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—24no—
122C2×C4×C53—24no—
123C2×C4×C53—48no—
124C2×C4×C53—44no—
125C2×C4×C53—48no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53#9912yes—
129C2×C4×C53C2×C22⋊C412yes—
130C2×Dic53—84no—
131C2×Dic53—88no—
132#12694—531nomaximal, Sylow
133C22×C53—24no—
134C22×C53—24no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—24no—
139C22×C53—44no—
140C22×C53—44no—
141C22×C53—44no—
142C22×C53—44no—
143C22×C53—42no—
144C22×C53—82no—
145C22×C53—816no—
146C22×C53—82no—
147C22×C53#35414yescomplemented
148C22×C53#37911yescharacteristic
149C22×C53C2×#9911yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—44no—
154C4×C53—44no—
155Dic53—164no—
156C22×#124D5311yescharacteristic
157C2×#354—534no—
158#1781—532no—
159C2×C53—21no—
160C2×C53—24no—
161C2×C53—21no—
162C2×C53—44no—
163C2×C53—82no—
164C2×C53#178111yescharacteristic
165C2×C53C2×#35412yes—
166C23×Q8Dic5311yescharacteristic, complemented
167C23×D4C2×D5311yescharacteristic
168C2×#124—22no—
169C2×#124—28no—
170C2×#124Dic5312yescomplemented
171C2×#124C2×D5312yes—
172C22×C22⋊C4—22no—
173C22×C42Dic5311yescharacteristic, complemented
174C2×#99—534no—
175C2×#99—1062no—
176#379—1062no—
177#354—1068no—
178C53#1269411yescharacteristic, complemented, Sylow
179C25C22⋊D5312yes—
180C22×Q8—24no—
181C22×Q8—21no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8C2×Dic5312yes—
185C22×D4—24no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4C2×Dic5311yescharacteristic
189C22×D4C22⋊D5312yes—
190C22×D4C22×D5311yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C2×Dic5311yescharacteristic
194#124—216no—
195#124—24no—
196#124—24no—
197#124—24no—
198#124C22⋊D5318yes—
199C2×C22⋊C4—216no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—1064no—
203C2×C22⋊C4—2124no—
204C2×C42—21no—
205C2×C42—24no—
206C2×C42C2×Dic5312yes—
207#99—1068no—
208#99—2124no—
209C24—24no—
210C24—24no—
211C24—42no—
212C24—48no—
213C24C22⋊Dic5312yes—
214C24C22⋊Dic5311yescharacteristic
215C24C2×C22⋊D5312yes—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—416no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8C22⋊Dic5314yescomplemented
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—416no—
233C2×D4C2×C22⋊D5314yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—2124no—
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C22×Dic5311yescharacteristic
247C22⋊C4—216no—
248C22⋊C4—216no—
249C22⋊C4—4248no—
250C22⋊C4—2124no—
251C22⋊C4—2124no—
252C42—24no—
253C42—22no—
254C42C22⋊Dic5314yescomplemented
255C23—21no—
256C23—24no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—416no—
267C23—48no—
268C23—84no—
269C23—88no—
270C23#1313313212yes—
271C23C2×C22⋊Dic5311yescharacteristic
272C23#1313318614yes—
273Q8—44no—
274Q8—48no—
275Q8—44no—
276D4—44no—
277D4—48no—
278D4—44no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—4244no—
292C2×C4—4248no—
293C2×C4#1313313212yes—
294C2×C4C2×C22⋊Dic5312yes—
295C22—24no—
296C22—24no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—24no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—82no—
307C22—816no—
308C22—82no—
309C22#1681041814yescomplemented
310C22#1681047911yescharacteristic
311C22C2×#1313313211yescharacteristic
312C22C2×#1313313211yescharacteristic
313C4—22no—
314C4—22no—
315C4—44no—
316C4—44no—
317C4—8484no—
318C2—21no—
319C2—21no—
320C2—24no—
321C2—44no—
322C2—82no—
323C2#2027463011yescharacteristic
324C2C2×#1681041812yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC22×#124×C53C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenterC22×#124D53
hypocenterC53#12694
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16810418C2semidirect
8. C23×Q8×C53C4semidirect
12. C2×#124×C53C4semidirect
15. C22×C42×C53C4semidirect
60. C2×Q8×C53C22⋊C4semidirect
88. C42×C53C22⋊C4semidirect
147. C22×C53#354semidirect
166. C23×Q8Dic53semidirect
170. C2×#124Dic53semidirect
173. C22×C42Dic53semidirect
178. C53#12694semidirect
223. C2×Q8C22⋊Dic53semidirect
254. C42C22⋊Dic53semidirect
309. C22#16810418semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202746302
C22×#124×C532
#1269453