Finite Group Databasebeta

← #27299595

Subgroups — #27299595

Order 13568.

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

normal subgroups only

This group has 990 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×#16810418C214yesmaximal, complemented
2#20274633C212yesmaximal
3C2×C53×#536C211yesmaximal, characteristic
4C2×#13133132—22no—
5C2×#13133132C2214yes—
6#16810486—22no—
7#16810418—28no—
8C22×#41×C53C2211yescharacteristic
9C23×Q8×C53C411yescharacteristic, complemented
10C53×#536—28no—
11C2×C4⋊Q8×C53C412yescomplemented
12C2×#124×C53C2212yes—
13C2×#123×C53—22no—
14C2×C22⋊Q8×C53—22no—
15C2×#117×C53C411yescharacteristic, complemented
16C2×C22⋊Dic53—24no—
17#13133132—28no—
18#13133132—44no—
19#13133124—44no—
20C2×#41×C53—24no—
21C2×#41×C53—42no—
22C2×#41×C53—42no—
23C22×Q8×C53—24no—
24C22×Q8×C53—21no—
25C22×Q8×C53—42no—
26C22×Q8×C53—42no—
27C22×Q8×C53C2×C412yes—
28C22×Q8×C53C2×C411yescharacteristic
29C22×D4×C53D412yes—
30C22×D4×C53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53D412yes—
34C4⋊Q8×C53—24no—
35C4⋊Q8×C53—24no—
36#124×C53—24no—
37#124×C53D418yes—
38#123×C53—216no—
39C22⋊Q8×C53—216no—
40#117×C53—24no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—24no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—24no—
45C2×C42×C53C2×C412yes—
46C24×C53C22⋊C411yescharacteristic
47C24×C53C2×D412yes—
48#41×C53—416no—
49C2×Q8×C53—24no—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—416no—
55C2×Q8×C53—42no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53—42no—
59C2×Q8×C53C22⋊C414yes—
60C2×D4×C53—24no—
61C2×D4×C53—24no—
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53C2×D414yes—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—24no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—24no—
73C22×C4×C53—24no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×Dic53—44no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—216no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—216no—
85C42×C53—22no—
86C42×C53C22⋊C414yes—
87C22⋊Dic53—44no—
88C22⋊Dic53—44no—
89C23×C53—21no—
90C23×C53—22no—
91C23×C53—21no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—44no—
95C23×C53#9912yes—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53#12014yes—
98Q8×C53—48no—
99Q8×C53—44no—
100Q8×C53—44no—
101Q8×C53—44no—
102D4×C53—44no—
103D4×C53—48no—
104C2×C4×C53—28no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—48no—
115C2×C4×C53—44no—
116C2×C4×C53—48no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53#9912yes—
120C2×C4×C53C2×C22⋊C412yes—
121C2×Dic53—84no—
122C2×Dic53—88no—
123#12698—531nomaximal, Sylow
124C22×C53—22no—
125C22×C53—24no—
126C22×C53—22no—
127C22×C53—42no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—44no—
131C22×C53—42no—
132C22×C53—82no—
133C22×C53#37911yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#9911yescharacteristic
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—44no—
139C4×C53—42no—
140C4×C53—44no—
141Dic53—164no—
142C2×#536D5311yescharacteristic
143C2×#354—534no—
144#1782—532no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—44no—
148C2×C53—42no—
149C2×C53#178111yescharacteristic
150C2×C53#212912yes—
151C22×#41C2×D5311yescharacteristic
152C23×Q8Dic5311yescharacteristic, complemented
153#536—28no—
154C2×C4⋊Q8Dic5312yescomplemented
155C2×#124C2×D5312yes—
156C2×#123—22no—
157C2×C22⋊Q8—22no—
158C2×#117Dic5311yescharacteristic, complemented
159C2×#99—534no—
160C2×#99—1062no—
161#381—1062no—
162#354—1068no—
163C53#1269811yescharacteristic, complemented, Sylow
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8—24no—
168C22×Q8—21no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8C2×Dic5312yes—
172C22×Q8C2×Dic5311yescharacteristic
173C22×D4C22⋊D5312yes—
174C22×D4C22×D5311yescharacteristic
175C23×C4—21no—
176C23×C4C2×Dic5311yescharacteristic
177C23×C4C22⋊D5312yes—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180#124—24no—
181#124C22⋊D5318yes—
182#123—216no—
183C22⋊Q8—216no—
184#117—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—1064no—
190C2×C42C2×Dic5312yes—
191#99—1068no—
192#99—2124no—
193#95—2124no—
194C24C22⋊Dic5311yescharacteristic
195C24C2×C22⋊D5312yes—
196#41—416no—
197C2×Q8—24no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊Dic5314yes—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4C2×C22⋊D5314yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—2124no—
226C22×C4C22⋊Dic5312yes—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22×Dic5311yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—216no—
231C22⋊C4—24no—
232C22⋊C4—216no—
233C22⋊C4—2124no—
234C22⋊C4—2124no—
235C42—22no—
236C42C22⋊Dic5314yes—
237C23—21no—
238C23—22no—
239C23—21no—
240C23—22no—
241C23—42no—
242C23—44no—
243C23#1313313212yes—
244C23C2×C22⋊Dic5311yescharacteristic
245C23#1313318614yes—
246Q8—44no—
247Q8—48no—
248Q8—44no—
249Q8—44no—
250D4—44no—
251D4—48no—
252C2×C4—28no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—48no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—4244no—
268C2×C4—4248no—
269C2×C4#1313313212yes—
270C2×C4C2×C22⋊Dic5312yes—
271C22—22no—
272C22—22no—
273C22—24no—
274C22—42no—
275C22—44no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—82no—
280C22#1681047911yescharacteristic
281C22C2×#1313313211yescharacteristic
282C22C2×#1313313211yescharacteristic
283C4—22no—
284C4—22no—
285C4—44no—
286C4—42no—
287C4—44no—
288C4—8484no—
289C2—21no—
290C2—21no—
291C2—44no—
292C2—42no—
293C2#2027463011yescharacteristic
294C2#2027639012yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#536C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#536D53
hypocenterC53#12698
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#16810418C2semidirect
9. C23×Q8×C53C4semidirect
11. C2×C4⋊Q8×C53C4semidirect
15. C2×#117×C53C4semidirect
152. C23×Q8Dic53semidirect
154. C2×C4⋊Q8Dic53semidirect
158. C2×#117Dic53semidirect
163. C53#12698semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202746332
C2×C53×#5362
#1269853