Finite Group Databasebeta

← #27299592

Subgroups — #27299592

Order 13568.

1182 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16810416C214yesmaximal, complemented
2#20274633C212yesmaximal, complemented
3C2×C53×#535C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yescomplemented
6#16810486—22no—
7#16810416—28no—
8C22×#41×C53C2211yescharacteristic
9C23×D4×C53C411yescharacteristic, complemented
10C53×#535—28no—
11C2×#127×C53C2212yes—
12C2×#124×C53C412yescomplemented
13C2×#121×C53—22no—
14C2×#120×C53—22no—
15C2×#117×C53C411yescharacteristic, complemented
16C2×C22⋊Dic53—24no—
17#13133132—28no—
18#13133132—44no—
19#13133124—44no—
20C25×C53—21no—
21C2×#41×C53—24no—
22C2×#41×C53—42no—
23C2×#41×C53—42no—
24C22×Q8×C53C2×C411yescharacteristic
25C22×D4×C53—24no—
26C22×D4×C53—24no—
27C22×D4×C53—21no—
28C22×D4×C53—42no—
29C22×D4×C53—42no—
30C22×D4×C53C2×C412yes—
31C22×D4×C53D412yes—
32C22×D4×C53C2311yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D412yes—
35#127×C53—24no—
36#127×C53D418yes—
37#124×C53—24no—
38#124×C53—24no—
39#121×C53—216no—
40#120×C53—216no—
41#117×C53—24no—
42C2×C4⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—24no—
45C2×C42×C53C2×C412yes—
46C24×C53—24no—
47C24×C53—22no—
48C24×C53—42no—
49C24×C53—42no—
50C24×C53—84no—
51C24×C53C22⋊C411yescharacteristic
52C24×C53C2×D412yes—
53#41×C53—416no—
54C2×Q8×C53—22no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×D4×C53—24no—
58C2×D4×C53—24no—
59C2×D4×C53—24no—
60C2×D4×C53—216no—
61C2×D4×C53—22no—
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—416no—
66C2×D4×C53—42no—
67C2×D4×C53—42no—
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C2×D4×C53—88no—
71C2×D4×C53C22⋊C414yes—
72C2×D4×C53C2×D414yes—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—24no—
78C22×C4×C53—24no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×Dic53—44no—
84C4⋊C4×C53—24no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—216no—
87C42×C53—22no—
88C42×C53C22⋊C414yes—
89C22⋊Dic53—44no—
90C22⋊Dic53—44no—
91C23×C53—21no—
92C23×C53—22no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53—24no—
96C23×C53—42no—
97C23×C53—44no—
98C23×C53—48no—
99C23×C53—48no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—88no—
103C23×C53—88no—
104C23×C53—88no—
105C23×C53#9912yes—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53#12014yes—
108Q8×C53—44no—
109D4×C53—44no—
110D4×C53—44no—
111D4×C53—48no—
112D4×C53—44no—
113D4×C53—48no—
114D4×C53—816no—
115C2×C4×C53—28no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—48no—
125C2×C4×C53—44no—
126C2×C4×C53#9912yes—
127C2×C4×C53C2×C22⋊C412yes—
128C2×Dic53—84no—
129C2×Dic53—88no—
130#12697—531nomaximal, Sylow
131C22×C53—22no—
132C22×C53—24no—
133C22×C53—22no—
134C22×C53—44no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—48no—
138C22×C53—44no—
139C22×C53—42no—
140C22×C53—44no—
141C22×C53—82no—
142C22×C53—84no—
143C22×C53—816no—
144C22×C53—84no—
145C22×C53#37911yescharacteristic
146C22×C53C2×#9911yescharacteristic
147C22×C53C2×#9911yescharacteristic
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—44no—
151C4×C53—42no—
152Dic53—164no—
153C2×#535D5311yescharacteristic
154C2×#353—534no—
155#1782—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—44no—
159C2×C53—42no—
160C2×C53—84no—
161C2×C53#178111yescharacteristic
162C2×C53#212812yes—
163C22×#41C2×D5311yescharacteristic
164C23×D4Dic5311yescharacteristic, complemented
165#535—28no—
166C2×#127C2×D5312yes—
167C2×#124Dic5312yescomplemented
168C2×#121—22no—
169C2×#120—22no—
170C2×#117Dic5311yescharacteristic, complemented
171C2×#99—534no—
172C2×#99—1062no—
173#381—1062no—
174#353—1068no—
175C53#1269711yescharacteristic, complemented, Sylow
176C25—21no—
177C2×#41—24no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×Dic5311yescharacteristic
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×Dic5312yes—
187C22×D4C22⋊D5312yes—
188C22×D4C22×D5311yescharacteristic
189C23×C4C2×Dic5311yescharacteristic
190C23×C4C22⋊D5312yes—
191#127—24no—
192#127C22⋊D5318yes—
193#124—24no—
194#124—24no—
195#121—216no—
196#120—216no—
197#117—24no—
198C2×C4⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—1064no—
202C2×C42C2×Dic5312yes—
203#99—1068no—
204#99—2124no—
205#95—2124no—
206C24—24no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—84no—
211C24C22⋊Dic5311yescharacteristic
212C24C2×C22⋊D5312yes—
213#41—416no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—216no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—416no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—88no—
231C2×D4C22⋊Dic5314yes—
232C2×D4C2×C22⋊D5314yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—2124no—
241C22×C4C22⋊Dic5312yes—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22×Dic5311yescharacteristic
244C4⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—216no—
247C22⋊C4—2124no—
248C22⋊C4—2124no—
249C42—22no—
250C42C22⋊Dic5314yes—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—24no—
256C23—42no—
257C23—44no—
258C23—48no—
259C23—48no—
260C23—42no—
261C23—42no—
262C23—88no—
263C23—88no—
264C23—88no—
265C23#1313313212yes—
266C23C2×C22⋊Dic5311yescharacteristic
267C23#1313318614yes—
268Q8—44no—
269D4—44no—
270D4—44no—
271D4—48no—
272D4—44no—
273D4—48no—
274D4—816no—
275C2×C4—28no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—4244no—
287C2×C4—4248no—
288C2×C4#1313313212yes—
289C2×C4C2×C22⋊Dic5312yes—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—48no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—82no—
301C22—84no—
302C22—816no—
303C22—84no—
304C22#1681047911yescharacteristic
305C22C2×#1313313211yescharacteristic
306C22C2×#1313313211yescharacteristic
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—8484no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—42no—
316C2—84no—
317C2#2027463011yescharacteristic
318C2#2027638712yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC2×C53×#535C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenterC2×#535D53
hypocenterC53#12697
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×#16810416C2semidirect
2. #20274633C2semidirect
5. C2×#13133132C22semidirect
9. C23×D4×C53C4semidirect
12. C2×#124×C53C4semidirect
15. C2×#117×C53C4semidirect
164. C23×D4Dic53semidirect
167. C2×#124Dic53semidirect
170. C2×#117Dic53semidirect
175. C53#12697semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
#202746332
C2×C53×#5352
#1269753